**Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach**

#### **Francisco J. Peña 1,\*, Oscar Negrete 1,2, Gabriel Alvarado Barrios 2,3, David Zambrano 1, Alejandro González 1, Alvaro S. Nunez 3,4, Pedro A. Orellana <sup>1</sup> and Patricio Vargas 1,3**


Received: 7 January 2019; Accepted: 1 March 2019; Published: 20 May 2019

**Abstract:** We studied the performance of classical and quantum magnetic Otto cycle with a working substance composed of a single quantum dot using the Fock–Darwin model with the inclusion of the Zeeman interaction. Modulating an external/perpendicular magnetic field, in the classical approach, we found an oscillating behavior in the total work extracted that was not present in the quantum formulation.We found that, in the classical approach, the engine yielded a greater performance in terms of total work extracted and efficiency than when compared with the quantum approach. This is because, in the classical case, the working substance can be in thermal equilibrium at each point of the cycle, which maximizes the energy extracted in the adiabatic strokes.

**Keywords:** magnetic cycle; quantum otto cycle; quantum thermodynamics

#### **1. Introduction**

The study of quantum heat engines (QHEs) [1] is focused on the search and design of efficient nanoscale devices operating with a quantum working substance. These devices are characterized by their working substance, the thermodynamic cycle of operation, and the dynamics that govern the cycle [2–26]. Among the cycles in which the engine may operate, the Carnot and Otto cycles have received increasing attention. In particular, the quantum Otto cycle has been considered for various working substances such as spin-1/2 systems [27,28] and harmonic oscillators [29], among others. Recently, an increasing number of experimental realizations for the quantum Otto cycle has been proposed in the literature [30–33]. Furthermore, it has been shown that thermal machines can be reduced to the limits of single atoms [34].

Previous studies of the quantum Otto cycle embedding working substances with magnetic properties have highlighted the role of degeneracy in the energy spectrum on the performance of the engine [35–41]. In this same framework, we highlight the work of Mehta and Johal [38], who studied a quantum Otto engine in the presence of level degeneracy, finding an enhancement of work and efficiency for two-level particles with a degeneracy in the excited state. In addition, Azimi et al. presented the study of a quantum Otto engine operating with a working substance of a single phase multiferroic *LiCu*2*O*<sup>2</sup> tunable by external electromagnetic fields [39], which was extended by Chotorlishvili et al. [40] under the implementation of shortcuts to adiabaticity, finding an optimal output power for the proposed machine.

On the other hand, the classical description of the Otto cycle is characterized by state variables that are well-defined at each point of the cycle. In this sense, the main difference between the classical and quantum approach is that in the classical cycle the working substance can be at thermal equilibrium after each stroke. Classically, the adiabatic strokes are determined by the isentropic condition, which allows determining the state variables. For many systems, such as diamagnetic systems, which were considered in this study, the relation between the thermodynamics variables involved in the adiabatic stroke is not trivial in general and must be solved numerically [41].

In particular, it is interesting to compare the classical and quantum approaches for the same working substance and establish the conditions for each case appropriately. In this framework, several recent studies have focused on employing quantum coherence in the working fluid for enhancing the performance of the engine [42–44]. Recently, an interesting regime called "sudden cycles" [45] has been explored in an incoherent formulation avoiding off-diagonal elements of the density matrix, characterized by finite cooling power [46].

In this work, we study the classical and quantum performance of a multi-level Otto cycle in a diagonal formulation of the density matrix operator, where the working substance comprises a nanosized quantum dot under a controllable external magnetic field. This system is described by the Fock–Darwin model [47,48] that represents an accurate model for a semiconductor quantum dot. For this diamagnetic system, we find the point at which the quantum total work extracted becomes smaller than the classical one and we report, in the classical approach, an oscillating behaviour in the total work extracted that is not perceptible under the quantum formulation.

#### **2. Model**

Let us consider a system given by an electron in the presence of a parabolic potential and external magnetic field **B**. The Hamiltonian that describes the system is given by

$$
\hat{\mathcal{H}} = \frac{1}{2m^\*} \left(\mathbf{p} + e\mathbf{A}\right)^2 + l l\_D(x, y),
\tag{1}
$$

where *m*<sup>∗</sup> is the effective electron mass, **A** is the total vector potential, and the term *UD*(*x*, *y*) is given by

$$M\_D(\mathbf{x}, y) = \frac{1}{2} m^\* \omega\_0^2 \left(\mathbf{x}^2 + y^2\right),\tag{2}$$

which corresponds to an attractive potential describing the effect of the dot on the electron. The quantity *ω*<sup>0</sup> is the parabolic trap frequency and can be controlled geometrically. If we consider a constant perpendicular magnetic field in the form

$$\mathbf{B} = B\mathbf{\hat{z}}\,.\tag{3}$$

and the use of the vector potential **A** in the symmetric gauge (i.e., **A** = *<sup>B</sup>* <sup>2</sup> (−*y*, *x*, 0)), the solution of the eigenvalues of the Schrödinger equation are given by

$$E\_{nm} = \hbar\Omega\left(2n+\mid m\mid +1\right) + \frac{1}{2}\hbar\omega\_{\rm cf}m.\tag{4}$$

where *ω<sup>c</sup>* = *eB <sup>m</sup>*<sup>∗</sup> is the cyclotron frequency, and *n* and *m* are the radial and magnetic quantum numbers (*n* = 0, 1, 2, ... and *m* = −∞, ..., +∞), respectively. Ω is known as the effective frequency of the system corresponding to

$$
\Omega = \omega\_0 \left( 1 + \left(\frac{\omega\_c}{2\omega\_0}\right)^2 \right)^{\frac{1}{2}}.\tag{5}
$$

Notice that, when the parameter *ω*<sup>0</sup> → 0, the energy levels of Equation (4) take the usual form of the Landau energy levels in cylindrical coordinates.

To obtain a more precise expression, especially when we consider the case of strong magnetic fields for the electron trapped in a quantum dot, we also take into account the electron spin of value *h*¯ *σ*ˆ <sup>2</sup> and magnetic moment *<sup>μ</sup>B*, where *<sup>σ</sup>*<sup>ˆ</sup> is the Pauli spin operator and *<sup>μ</sup><sup>B</sup>* <sup>=</sup> *eh*¯ <sup>2</sup>*m*<sup>∗</sup> . Here, the spin can be in two possible states, either ↑ or ↓, with respect to the applied external magnetic field *B* in the z-axis. Therefore, we include the Zeeman term in the Fock–Darwin energy levels in Equation (4). Consequently, the energy spectrum is given by

$$E\_{n,m,\sigma} = \hbar\Omega(2n+|m|+1) + m\frac{\hbar\omega\_c}{2} - \mu\_B\sigma B. \tag{6}$$

The energy spectrum of Equation (6) is presented in Figure 1 for *σ* = −1 and *σ* = 1. It is interesting to note that, for high magnetic fields (*ωc*/2*ω*<sup>0</sup> >> 1), things simplify in Equation (6) and we get the following expression:

$$E\_{n,m,\sigma} = \frac{\hbar\omega\_{\varepsilon}}{2}(n+1/2+|m|+m) - \mu\_{B}\sigma B \,, \tag{7}$$

where we observe that |*m*| + *m* = 0 for *m* < 0, therefore each Landau level labeled by *n* has infinite degeneracy.

**Figure 1.** (**a**) Fock–Darwin energy spectrum with *σ* = −1 for the first six radial number *n* = 0, 1, ..., 6 and for each of them the azimuthal quantum number taking the values between *m* = −6, −5, ..., 5, 6. (**b**) Fock–Darwin energy spectrum with *σ* = +1 for the first six radial number *n* = 0, 1, ..., 6 and for each of them the azimuthal quantum number taking the values between *m* = −6, −5, ..., 5, 6. We clearly observe the confinement of the energy levels at high magnetic fields (*ωc*/2*ω*<sup>0</sup> >> 1).

In this paper, we consider a low-frequency coupling for the parabolic trap given by *ω*<sup>0</sup> ∼ 2.637 THz which in terms of energy units corresponds to a coupling of approximately 1.7 meV. The selection of this particular value is to compare the intensity of the trap with the typical energy of intra-band optical transitions of the quantum dots [47]. The order of this transition is approximately around ∼1 meV for cylindrical GaAs quantum dots with effective mass given by *m*<sup>∗</sup> ∼ 0.067 *me* [47–49].

For the classical approach, we employ the framework of Refs. [50–53], and, in particular, classical thermodynamic quantities for the Fock–Darwin model with spin can be obtained analytically using the treatment of Kumar et al. [54]. For a working substance in thermal equilibrium at inverse temperature *β* = 1/*kBT*, the partition function can be written as:

$$Z\_{dS} = \frac{1}{2} \text{csch}\left(\frac{\hbar \beta \omega\_{+}}{2}\right) \text{csch}\left(\frac{\hbar \beta \omega\_{-}}{2}\right) \cosh\left(\frac{\hbar \beta \omega\_{B}}{2}\right),\tag{8}$$

where the frequencies *ω*<sup>±</sup> are:

$$
\omega\_{\pm} = \Omega \pm \frac{\omega\_{\varepsilon}}{2}.\tag{9}
$$

Therefore, entropy (*S*(*T*, *B*)), internal energy (*U*(*T*, *B*)) and magnetization *M*(*T*, *B*) are simply given by

$$S(T, B) = k\_B \ln \mathcal{Z}\_{dS} + k\_B T \left(\frac{\partial \ln \mathcal{Z}\_{dS}}{\partial T}\right)\_{B} \tag{10}$$

$$
\Delta U(T, B) = k\_B T^2 \left( \frac{\partial \ln Z\_{dS}}{\partial T} \right)\_{B} \tag{11}
$$

$$M(T, B) = k\_B T \left(\frac{\partial \ln Z\_{dS}}{\partial B}\right). \tag{12}$$

Equations (10)–(12) are presented in Figure 2 for a parabolic trap corresponding to an energy of 1.7 meV together with the scheme of the Otto cycle that we consider. A very interesting behavior is observed for the entropy as a function of the magnetic field in Figure 2a. For external magnetic fields ≤1 T, the entropy decreases as the external field increases, but for values higher than 1 T we see the opposite behavior. This can be explained by the energy levels becoming closer to each other as the magnetic field increases, moving towards degeneracy. This behavior in the energy levels causes the entropy growth as the magnetic field increases. In addition, the change in the behavior of the entropy is affected by temperature, finding that the change of slope as a function of external magnetic field moves away from the 1 T value to higher values as we move to higher temperature of the working substance. This can be appreciated in Figure 2a. At the same time, the magnetization shows a crossing in its behavior as a function of magnetic field, as we can see in Figure 2b, where previous to this crossing at lower temperatures higher values of magnetization are obtained. This fact becomes essential for the total work extracted. In the cycle that we propose, the work is directly related to the change in the magnetization of the system as a function of magnetic field and temperature. On the other hand, we can see that the internal energy monotonically decreases in terms of the magnetic field for all temperatures considered. The reason for this is that the internal energy only depends on the derivative of ln Z*dS* (see Equation (11)) with respect to temperature while the entropy has an additional term proportional to ln Z*dS* (see Equation (10)) and the magnetization on its derivative with respect to the external field (see Equation (12)).

**Figure 2.** Classical thermodynamic quantities entropy (*S*), internal energy (*U*) and magnetization (*M*) as a function of: external magnetic field (*B*) (**a**–**c**); and temperature (*T*) (**d**–**f**). In (**a**–**c**), the colors blue to red represent temperatures from 0.1 K to 10 K, respectively. For (**d**–**f**), the colors blue to red represent lower to higher external magnetic field, from 0.1 T to 5 T. The value of the parabolic trap is approximately to 1.7 meV. Additionally, we show how the Otto cycle appears in terms of the thermodynamic quantities considered.

#### **3. First Law of Thermodynamics and the Quantum and Classical Otto Cycle**

The first law of thermodynamics in a quantum context has been discussed extensively in the literature. We follow the treatment in Refs. [50–52], which identifies the heat transferred and work performed during a thermodynamic process by means of the variation of the internal energy of the system.

First, consider a system described by a Hamiltonian, <sup>H</sup><sup>ˆ</sup> , with discrete energy levels, *En*,*m*,*σ*. The internal energy of the system is simply the expectation value of the Hamiltonian *<sup>E</sup>* <sup>=</sup> H<sup>ˆ</sup> <sup>=</sup> ∑*<sup>n</sup>* ∑*<sup>m</sup>* ∑*<sup>σ</sup> pn*,*m*,*σEn*,*m*,*σ*, where *pn*,*m*,*<sup>σ</sup>* are the corresponding occupation probabilities. The infinitesimal change of the internal energy can be written as

$$dE = \sum\_{n} \sum\_{m} \sum\_{\sigma} \left( E\_{n,m,\sigma} dP\_{n,m,\sigma} + P\_{n,m,\sigma} dE\_{n,m,\sigma} \right) \,, \tag{13}$$

where we can identify the infinitesimal work and heat as

$$dQ := \sum\_{n} \sum\_{m} \sum\_{\sigma} E\_{n,m,\sigma} dp\_{n,m,\sigma}, \qquad d\mathcal{W} := \sum\_{n} \sum\_{m} \sum\_{\sigma} p\_{n,m,\sigma} dE\_{n,m,\sigma}. \tag{14}$$

Equation (13) is a formulation of the first law of thermodynamics for quantum working substances. Therefore, work is then related to a change in the eigenenergies *En*,*m*,*σ*, which is in agreement with the fact that work can only be carried out through a change in generalized coordinates. It is important to note that the expressions of Equation (14) is only a particular case of the definition of work and heat for a case of a density matrix operator that is diagonal on the energy eigenbasis [52]. A more complete definition of Equation (14) can be found in Refs. [29–33,46].

The quantum Otto cycle is composed of four strokes: two quantum isochoric processes and two quantum adiabatic processes. This cycle can be seen in Figure 3, replacing the value of *Sl* and *Sh* for *Pn*,*m*,*σ*(*Tl*, *Bh*) and *Pn*,*m*,*σ*(*Th*, *Bl*) in the vertical axis, respectively. For the cases that we consider, the quantum Otto cycle proceeds as follows.

**Figure 3.** The magnetic Otto engine represented as an entropy (*S*) versus a magnetic field (*B*) diagram. The way to perform the cycle is in the form B → A → D → C → B.

1. Step B → A: Quantum adiabatic compression process. The systems, which is initialized in thermal equilibrium at temperature *Tl*, is isolated from the cold reservoir and the magnetic field is changed from *Bh* to *Bl*, with *Bh* > *Bl*. During this stage the populations remain constant, so *Pn*,*m*,*σ*(*Tl*, *Bh*) = *P*<sup>A</sup> *<sup>n</sup>*,*m*,*σ*. We remark that *P*<sup>A</sup> *<sup>n</sup>*,*m*,*<sup>σ</sup>* does not yield a thermal state. No heat is exchanged during this process.

2. Step A → D: The system, at constant magnetic field *Bl*, is brought into contact with a hot thermal reservoir at temperature *Th* until it reaches thermal equilibrium. The corresponding thermal *Entropy* **2019**, *21*, 512

populations *Pn*,*m*,*σ*(*Th*, *Bl*) are given by the Boltzmann distribution with temperature *Th*. No work is done during this stage.

The heat absorbed for the working substance is given by

$$Q\_{\rm in} = \sum\_{n} \sum\_{m} \sum\_{\sigma} \int\_{A}^{D} E\_{n,m,\sigma} dP\_{n,m,\sigma} = \sum\_{n} \sum\_{m} \sum\_{\sigma} E\_{n,m,\sigma}^{l} \left[ P\_{n,m,\sigma} \left( T\_{h'}, B\_{l} \right) - P\_{n,m,\sigma}^{A} \right],\tag{15}$$

where *E<sup>l</sup> <sup>n</sup>*,*m*,*<sup>σ</sup>* is the *n*-th eigenenergy of the system in the quantum isochoric heating process to an external magnetic field of value *Bl* .

3. Step D → C: Quantum adiabatic expansion process. The system is isolated from the hot reservoir, and the magnetic field is changed back from *Bl* to *Bh*. During this stage the populations remains constant, thus *Pn*,*m*,*σ*(*Th*, *Bl*) = *P*<sup>C</sup> *<sup>n</sup>*,*m*,*σ*. Again, we remark that *P*<sup>C</sup> *<sup>n</sup>*,*m*,*<sup>σ</sup>* is not a thermal state. No heat is exchanged during this process.

4. Step C → B : Quantum isochoric cooling process. The working substance at *Bh* is brought into contact with a cold thermal reservoir at temperature *Tl*. Therefore, the heat released is given by

$$Q\_{\rm out} = \sum\_{n} \sum\_{m} \sum\_{\sigma} \int\_{\mathbf{C}}^{\mathbf{B}} E\_{n,m,\sigma} dP\_{n,m,\sigma} = \sum\_{n} \sum\_{m} \sum\_{\sigma} E\_{n,m,\sigma}^{\mathbf{h}} \left[ P\_{n,m,\sigma} (T\_{\mathbf{l}}, B\_{\mathbf{h}}) - P\_{n,m,\sigma}^{\mathbf{C}} \right], \tag{16}$$

where *E<sup>h</sup> <sup>n</sup>*,*m*,*<sup>σ</sup>* is the *n*-th eigenenergy of the system for an external magnetic field *Bh*.

The net work done in a single cycle can be obtained from *W* = *Qin* + *Qout*,

$$\mathcal{W} = \sum\_{n} \sum\_{m} \sum\_{\sigma} \left( E\_{n,m,\sigma}^{l} - E\_{n,m,\sigma}^{h} \right) \left( P\_{n,m,\sigma} (T\_{\mathbf{h}\prime} B\_{\mathbf{l}}) - P\_{n,m,\sigma} (T\_{\mathbf{l}\prime} B\_{\mathbf{h}}) \right), \tag{17}$$

where we use the condition of constant populations along the quantum adiabatic strokes. Furthermore, the efficiency is given by

$$
\eta = \frac{W}{Q\_{\rm in}}\,\,\,\,\,\,\tag{18}
$$

The main difference between the classical and quantum Otto cycle is related to Points *A* and *C* in the cycle. In the classical case, the working substance can be at thermal equilibrium with a well-defined temperature at each point. On the other hand, for the quantum case, the working substance only reaches thermal equilibrium in the isochoric stages at Points *B* and *D*. After the adiabatic stages, the quantum system is in a diagonal state which is not a thermal state.

For the classical engine, the total work extracted by Equation (16) can be calculated by replacing *P<sup>A</sup> <sup>n</sup>*,*m*,*<sup>σ</sup>* with *P*(*TA*, *Bl*) and *P<sup>C</sup> <sup>n</sup>*,*m*,*<sup>σ</sup>* with *P*(*TC*, *Bh*), that is, it is obtained as a difference between the internal energy at adjacent points which can be calculated from the partition function

$$Q\_{in} = \mathcal{U}(T\_{\hbar}, B\_{l}) - \mathcal{U}(T\_{A}, B\_{l}); \qquad Q\_{out} = \mathcal{U}(T\_{l}, B\_{\hbar}) - \mathcal{U}(T\_{\mathbb{C}}, B\_{\hbar}), \tag{19}$$

where *T*<sup>A</sup> and *T*<sup>C</sup> are determined by the condition imposed by the classical isentropic strokes. If we have the classical entropy, the intermediate temperatures *T*<sup>A</sup> and *T*<sup>C</sup> can be determined in two different forms:

• Finding the relation between the magnetic field and the temperature along an isentropic trajectory by solving the differential equation of first order given by

$$dS(B, T) = \left(\frac{\partial S}{\partial B}\right)\_T dB + \left(\frac{\partial S}{\partial T}\right)\_B dT = 0,\tag{20}$$

which can be written as

$$\frac{dB}{dT} = -\frac{\mathcal{C}\_B}{T \left(\frac{\partial \mathcal{S}}{\partial B}\right)\_T},\tag{21}$$

where *CB* is the specific heat at constant magnetic field.

• By matching two points within an isentropic trajectory

$$\begin{aligned} \mathcal{S}(T\_{l\prime}, B\_{h}) &= \mathcal{S}(T\_{\mathcal{A}\prime}, B\_{l}) \\ \mathcal{S}(T\_{h\prime}B\_{l}) &= \mathcal{S}(T\_{\mathcal{C}\prime}B\_{h}) \end{aligned} \tag{22}$$

finding the magnetic field in terms of the temperature, throughout numerical calculation.

Therefore, from Equation (19) and W = *Qin* + *Qout*, the classical work is given by the difference of four internal energy in the form

$$\mathcal{W} = \mathcal{U}\_{\rm D} \left( T\_{\rm h}, B\_{\rm l} \right) - \mathcal{U}\_{\rm A} \left( T\_{\rm A}, B\_{\rm l} \right) + \mathcal{U}\_{\rm B} \left( T\_{\rm l}, B\_{\rm h} \right) - \mathcal{U}\_{\rm C} \left( T\_{\rm C}, B\_{\rm h} \right), \tag{23}$$

It is important to mention that the cycle operation in the counter-clockwise form starting at Point A described in Figure 2 gives negative work extracted, thus, to define a thermal machine correctly, we start the cycle at Point B, and we go through it in a clockwise direction. This is due to the particular behavior of the entropy as a function of magnetic field and temperature in the chosen zone marked with A, B, C and D. Therefore, the cycle described in the next subsection is the form of B → A → D → C → B and is presented in Figure 3.

The maximum values considered in our calculations for the temperatures and external magnetic field were 10 K and 5 T. Therefore, for the quantum cycle calculation (i.e., Equation (17)), we used the quantum numbers *n* = 0 to *n* = 10 and *m* = −33 to *m* = 33 for Equation (6). The selection of this particular energy levels in this model is justified for the values of the thermal populations for the hot and cold temperatures of the reservoirs that we selected. Our numerical calculations indicated that the contributions of the other levels of energy can be neglected.

Finally, it is useful to express our results of total work extracted and efficiency in terms of the relation between the highest value (*Bh*) and the lowest value (*Bl*) of the external magnetic field over the sample. To do that, we used the definition of "magnetic length", which is given by

$$l\_B = \sqrt{\frac{\hbar}{\varepsilon B'}}\tag{24}$$

allowing us to define the parameter

$$r = \frac{I\_{B\_l}}{I\_{B\_h}} = \sqrt{\frac{B\_h}{B\_l}},\tag{25}$$

which represents the analogy of the compression ratio for the classical case. It is important to remember that the Landau radius is inversely proportional to the magnitude of the magnetic field. Therefore, for a major (minor) magnitude of the field, the Landau radius is smaller (bigger), and the *r* is well defined.

#### **4. Results and Discussions**

#### *4.1. Classical Magnetic Otto Cycle*

The condition given by Equation (21) (or Equation (22)) for the classical cycle give us information about the behavior of the external magnetic field and the temperature in the adiabatic stroke. In Figure 4a, we can appreciate the level curves of the entropy function *S*(*T*, *B*) and, Figure 4b shows some examples of isentropic strokes in a plot of *S*(*B*) vs. *B* for different temperatures. That example shows three cases of constant low (red-black curve, *S* = 0.05), medium (yellow-black curve, *S* = 0.10) and high (white-black curve, *S* = 0.13) entropy. We observe in Figure 4a that there is a zone where the external field grows with the temperature of the sample and a zone where the opposite happens to maintain the entropy constant. At low working temperatures, the behavior changes near *B* = 1 T, while as the temperature increases, the slope change occurs at higher values of the magnetic field, approaching *B* = 2 T. Secondly, if we observe Figure 4b showing the case for *S* = 0.13 (white-black

line), we have a restricted area for field values lower to 3 T if we work with a maximum temperature of 10 K. Therefore, the movement of the magnetic field is not arbitrary if we think in a thermodynamic magnetic Otto cycle with two temperature reservoirs fixed at some specific values, more specifically, the reservoir associated to the hot temperature in the cycle. In addition, Figure 4 is the solution of *S*(*T*, *B*) = constant, obtained from the differential Equation (21) with different conditions (i.e., distinct values of the constant value of *S*). Therefore, Figure 4a depicts the entire family of solutions for the isentropic stroke of the engine of this particular system.

In our first example displayed in Figure 5, Point B has the value of the external field given by *B*<sup>h</sup> = 4 T and a temperature of *T*<sup>B</sup> = 6.19 K. The value of the temperature for Point D is fixed to *T*<sup>D</sup> = 10 K. Therefore, the Carnot efficiency of the proposal cycle is given by

$$
\eta\_{Carnot} = 1 - \frac{T\_{\rm B}}{T\_{\rm D}} = 1 - \frac{6.19}{10} = 0.381 \tag{26}
$$

Figure 5e shows different values of total work extracted (W) varying the value of *B*<sup>D</sup> from 4 T to 1.99 T. This variation in the external field is reflected in the movement of *r* in the form of *r* = <sup>4</sup> *B*l . Therefore, *r* is in the range of 1 ≤ *r* ≤ 1.41. The parabolic trap is fixed to the value of 1.7 meV and the effective mass in the value of *m*<sup>∗</sup> = 0.067*me*. In particular, Figure 5a–c shows the exact paths for the magnetic cycle for the maximum point obtained when multiplying W (Figure 5e) and the efficiency (*η*, Figure 5f). That point corresponds to *r* ∼ 1.22 (black point in Figure 5d–f) and constitutes the best configuration of the systems to obtain the best W with the better *η* through the cycle. In addition, W and *η* are presented in Figure 5e,f for the optimal value of *r* parameter mentioned before. We observe that W obtained for that point is in the order of ∼0.038 meV with an efficiency of *η* ∼ 0.28. We have corroborated the numeric result of total work extracted using the area enclosed by the cycle in Figure 5b of *<sup>M</sup>* versus *<sup>B</sup>*, as the work is <sup>W</sup> <sup>=</sup> <sup>−</sup> *MdB* [50–52] when the parameter changed during the operation of the engine in the external field. On the other hand, to obtain the solid lines presented in Figure 5d–f, we needed to make different cycles configuration keeping the values of the isothermal fixed as can be appreciated in the *Supplementary Materials* (see the link after Section 5), made with the Mathematica software [55], where we show each shape that the cycle must have to generate a specific point of work. It is important to recall that we never reach the optimal value of *η* = 0.381, i.e., the Carnot efficiency.

Due to the change of behavior in the entropy as a function of the external field, we obtained very interesting results for W when we explored the zone close to *B* = 1 T. Before that point, the entropy decreases as function of the external field (*B*) and after that point entropy begins to increase. This fact can be used to explore the magnetic cycle in that zone finding an oscillatory behavior for W. In Figure 6, we show the cycle with operating temperatures *T*<sup>B</sup> = 2.69 K and *T*<sup>D</sup> = 5.40 K and external magnetic field moving between 2.995 T and 0.250 T and, consequently, the *r* parameter moving from 1 to 3.46. First, we observe a decreasing efficiency for *r* > 1.75 in Figure 6f with a maximum value of *η* ∼ 0.43 for *r* ∼ 1.75. Therefore, also for this configuration, the Carnot efficiency (*ηCarnot* ∼ 0.5 for this case) cannot be reached. Comparing these results with those previously discussed (when we avoid this particular region), we can see in Figure 5f that the efficiency asymptotically approaches to the efficiency of Carnot if we increase the intensity of the external magnetic field of the starting point of the cycle (Point B).

In Figure 6b, we can understand the oscillations in W interpreting these results using the expression <sup>W</sup> <sup>=</sup> <sup>−</sup> *MdB*. In Figure 6a–c, Points A–D correspond to the black point displayed in Figure 6d–f where we see that the work is still greater than zero but close to a vanishing situation. The reason there is still positive work at this point under study is that the total area enclosed to the right of the crossing point is larger than the other to the left. The magnetization presented in Figure 6b in the zone around the range of external magnetic field explored for this case (from 2.995 to 0.250*T*) clearly reverses his behavior and presents a crossing point close to *B* ∼ 1.2 T for different temperatures. The area to the right of that point can be interpreted as a positive contribution to W while the left area contributes to negative work.

**Figure 4.** Solution of classical isentropic path. (**a**) The entropy as a function of magnetic field (horizontal axis) and temperature (vertical axis). The level curves (constant entropy values) highlight three different cases for *S*: first, red-black curve corresponding to *S* = 0.05; secondly, yellow-black curve, corresponding to *S* = 0.10 and finally, white-black curve for the case of *S* = 0.13. (**b**) The three constant values for the entropy (*S* = 0.05, *S* = 0.10, *S* = 0.13) in a graphic of entropy as a function of *B* for temperatures from 1 K (blue) up to 10 K (red). Due to the form of the entropy obtained for this system, the solution for *S* = 0.13 needs to work with temperatures higher than 10 K for an external magnetic field lower than 3 T (white dots in (**a**,**b**)). The value of the parabolic trap corresponds to 1.7 meV.

**Figure 5.** Proposed magnetic Otto cycle showing three different thermodynamic quantities, Entropy (*S*), Magnetization (*M*) and Internal Energy (*U*) ((**a**–**c**), respectively) as a function of the external magnetic field and different temperatures from 0.1*K* (blue) to 10*K* (red). (**d**) The total work extracted multiplied by efficiency (W*η*); (**e**) the total work extracted (W); and (**f**) the efficiency (*η*) for the classical cycle. The black points in (**d**–**f**) represent exactly the cycle B → A → D → C → B, presented in (**a**–**c**). The value of the parabolic trap corresponds to 1.7 meV. The fixed temperatures are *T*<sup>B</sup> = 6.19 K and *T*<sup>D</sup> = 10 K.

To explore if these oscillations in W are still obtained for higher temperature ranges, we plot in Figure 7 the work W for different values of *T*<sup>D</sup> with *T*<sup>B</sup> = 2.69 K fixed. We note that for higher temperatures than 7 K the oscillations found before disappear. It is only a reinforcement that the quantum effects of the working substance are only significant at low temperatures. On the other hand, as we expect, W grows as the difference between the temperature reservoir is larger, as shown in Figure 7a. However, for this case, the efficiency obtained is increasingly lower for increasingly larger temperature differences, as we can appreciate in Figure 7b.

**Figure 6.** Proposed magnetic Otto cycle in three different thermodynamics quantities, Entropy, Magnetization and internal energy ((**a**–**c**), respectively) as a function of the external magnetic field and different temperatures from 0.1*K* (blue) to 10 *K* (red). Total work extracted multiplied by efficiency (*Wη*) (**d**) total work extracted (*W*) (**e**) and efficiency (*η*) (**d**) for the cycle. The black point in (**d**–**f**) represents the value of 0.02 meV of total work extracted and corresponds exactly to the cycle B → A → D → C → B, shown in (**a**–**c**). The value of the parabolic trap correspond to 1.7 meV. The fixed temperatures are *T*<sup>B</sup> = 2.69 K and *T*<sup>D</sup> = 5.40 K.

**Figure 7.** Work, efficiency and work multiply by efficiency (**a**–**c**) for different values of *T*<sup>D</sup> for *T*<sup>B</sup> = 2.69 fixed. The value of the parabolic trap corresponds to 1.7 meV.

#### *4.2. Magnetic Quantum Otto Cycle*

Next, we show the results of the evaluation of the quantum version of this magnetic Otto cycle for the same cases shown in Figures 5 and 6. In Figure 8a, we plot the classical work (blue line) and the quantum work (red line) for the same sets of parameters in Figure 5. First, we note that the classical and quantum work are equal up to the value of *r* ∼ 1.07. This means, for values close to the starting external magnetic field to Point B, we do not notice a difference between the classic and quantum formulation of the Otto cycle. As shown in Figure 8a, we found a transition from positive work to negative work not reflected in the classic scenario close to *r* ∼ 1.26.

**Figure 8.** (**a**) Total work extracted for classical (blue line) and quantum version of Otto cycle (red line). The parameters for this case displayed are : *T*<sup>D</sup> = 10 K, *T*<sup>B</sup> = 6.19 K and *B*<sup>B</sup> = 4 T as starting value of the external magnetic field. The value of *B*<sup>D</sup> moves from 4 T to 1.99 T and this variation is reflected in the movement of *r* in the form of *r* = <sup>4</sup> *<sup>B</sup>*<sup>D</sup> , same parameter as the results shown in Figure 5. (**b**) Total work extracted (W) presented in Figure 6e versus the values obtaining in the quantum version of the Otto cycle. The parameters for this figure are *T*<sup>D</sup> = 5.40 *K*, *T*<sup>B</sup> = 2.69 *K* and *B*<sup>B</sup> = 2.995 T and *B*<sup>D</sup> moves from 2.995 to 0.250 T. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of *m*∗ = 0.067*me*.

Additionally, we observe that the maximum positive value of the total work extracted for the quantum version of Otto cycle is reduced by approximately 0.01 meV compared to the classical counterpart. In particular, for the quantum version of this cycle, we did not found the oscillations in *W* presented in Figure 6e. Moreover, we found a transition from positive to negative work at some value of the *r* parameter. This is dramatically reflected in Figure 8b, where the absolute value of *W* is highly increased as compared with the classical approach.

In Figure 9, we present the work *W* per energy level and spin value for the most important values of our numerical calculations. We used the same parameter as in Figure 8b. We observed that the contribution given by *σ* = 1 are positive up to *r* close to *r* ∼ 1.6 being the energy levels *E*0,−1, *E*0,−<sup>2</sup> and *E*0,−<sup>3</sup> those that contribute with the most positive values. Contrarily, for the case of *σ* = −1, we found that all contributions per energy level are negative. Therefore, the small region of positive work found in Figure 8b can only be associated to the spin up (*σ* = 1) contributions.

**Figure 9.** Total quantum work extracted (*W*) per energy level for the case of *σ* = 1 (**a**) and for the case of *σ* = −1 (**b**). The lines marked with circles correspond to the sum of all contributions of the energy level for each spin. The parameters used for this figure are the same as the one used in Figure 8b.

To explore other operation regions for the magnetic Otto cycle, we calculated the total work extracted and efficiency for the same Δ*T* = *Th* − *Tl* in a broad range of temperatures and the same Δ*Bmax* = 1.5 T in different regions of the external magnetic field for the classical cycle and its quantum version. This is displayed in Figures 10 and 11 where the dotted lines represent the classical results and the solid lines the quantum results. The three regions of temperature selected for these two figures are 1–4 K (blue lines), 4–7 K (black lines) and 7–10 K (red lines). First, we treat the case of *B*<sup>B</sup> = 3.5 T and *B*<sup>D</sup> moving from 3.5 T to 2.0 T in Figure 10, where we note large differences between the classical and quantum results for *W*, as can be seen in Figure 10b. On the other hand, if we observed the region of 3.5 ≤ *B*<sup>D</sup> ≤ 5.0 T for a *B*<sup>B</sup> fixed, as shown in Figure 11b. The work and efficiency for the region of 1–4 K and 4–7 K present similar behavior for the classical and quantum versions. Only the case of 7–10 K shows a larger difference between this two approaches. For the case of the efficiency, we note in Figure 10c a major difference between the classical results and quantum results compare with the presented in Figure 11c and this is consistent with the reported results for the work *W*.

**Figure 10.** *η* × *W* (**a**); and total work extracted (**b**,**c**) efficiency for the case of Δ*T* = *Th* − *Tl* = 3 *K* for different regions of temperature parameter for classical approach (solid line) and quantum version of the magnetic Otto cycle (dotted line). For all graphics, we use the initial external magnetic field in the value of *B*<sup>B</sup> = 3.5 T and the minimum value of the field, *BD* moves between 3.5 T and 2.0 T. Therefore, the *r* parameter moves between 1 ≤ *r* ≤ 1.32. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of *m*∗ = 0.067*me*.

Summarizing, our results show that it is the classical engine case with larger total work extracted and efficiency compared to the quantum formulation. This can be explained as follows.

The main difference between the classical and quantum version of Otto cycle lies in the fact that, in the classical formulation, the working substance can be in thermal equilibrium at each point in the cycle. In the quantum approach, the working substance is a single system that can only be in a thermal state after thermalizing with the reservoirs, which happens only in the isochoric strokes. After the adiabatic strokes, the working substance is in a diagonal state which is not a thermal state. In our case, the non-thermal points for the quantum case are Points C and A in Figure 3. The quantum work given by Equation (17) can be rewritten in the convenient form

$$\mathcal{W} = \mathcal{U}\_{\rm D} \left( T\_{\rm h}, B\_{\rm l} \right) - \sum\_{n, \mu, \sigma} E\_{n, \mu, \sigma}^{\rm l} P\_{\rm l, \mu, \sigma} \left( T\_{\rm l}, B\_{\rm h} \right) + \mathcal{U}\_{\rm D} \left( T\_{\rm l}, B\_{\rm h} \right) - \sum\_{n, \mu, \sigma} E\_{n, \mu, \sigma}^{\rm h} P\_{\rm n, \mu, \sigma} \left( T\_{\rm h}, B\_{\rm l} \right), \tag{27}$$

where, due to the thermal equilibrium of the two points (Points D and B in Figure 3), we can define the internal energy from equilibrium partition function. If we subtract the classical work given by Equation (23) from the quantum work written in the form of Equation (27), we obtain the following equation

$$\mathcal{W} - \mathcal{W} = \left(\sum\_{n,m,\sigma} E\_{n,m,\sigma}^{l} P\_{n,m,\sigma}(T\_{l\cdot}, B\_{\mathbf{l}}) - \mathcal{U}\_{\mathbf{A}}(T\_{\mathbf{A}\prime}, B\_{\mathbf{l}})\right) \\ + \left(\sum\_{n,m,\sigma} E\_{n,m,\sigma}^{l} P\_{n,m,\sigma}(T\_{l\cdot}, B\_{\mathbf{l}}) - \mathcal{U}\_{\mathbf{C}}(T\_{\mathbf{C}\prime}, B\_{\mathbf{l}})\right) \tag{28}$$

*Entropy* **2019**, *21*, 512

The first summation of Equation (28) is the average of the energy at low magnetic field with thermal probabilities that satisfies the adiabatic condition

$$S = -k\_B \sum\_{n,m,\sigma} P\_{n,m,\sigma}(T\_{l'}B\_h) \ln\left(P\_{n,m,\sigma}(T\_{l'}B\_h)\right),\tag{29}$$

i.e., the entropy at Point A. On the other hand, *U*A(*T*A, *Bl*) is the average value of the energy at low external field and in thermal equilibrium, with the same value of entropy presented in Equation (29). Therefore, *U*A(*T*A, *Bl*) is the absolute minimum according to thermodynamic [53]. The same argument can be made at Point C, thus the difference of classical work minus quantum work always satisfies the following condition

$$\mathcal{W} - \mathcal{W} \ge 0 \tag{30}$$

This result applies to any system in which the occupation probabilities of the energy levels at any magnetic field are replaced with any form, provided that they satisfy the adiabatic condition. This is because the value at equilibrium of any internal parameter (without constrains) of the system, makes the internal energy to be a minimum for a given value of the total Entropy [53].

**Figure 11.** *η* × *W* (**a**); and total work extracted (**b**,**c**) efficiency for the case of Δ*T* = *Th* − *Tl* = 3 *K* for different regions of temperature parameter. For all cases, we use the initial external magnetic field at the value of *B*<sup>B</sup> = 5.0 T and the minimum value of the field, *BD* moves between 5.0 T and 3.5 T. Therefore, the *r* parameter moves between 1 ≤ *r* ≤ 1.19. The parabolic trap is fixed to the value of 1.7 meV and the effective mass value of *m*∗ = 0.067*me*.

#### **5. Conclusions**

In this work, we explored the classical and quantum approach for a magnetic Otto cycle for an ensemble of non-interacting electrons with intrinsic spin where each one is trapped inside a semiconductor quantum dot modeled by a parabolic potential. We analyzed all relevant thermodynamics quantities, and found that the entropy changes it behavior in terms of the external magnetic field at the point where the energy spectrum tends towards degeneracy; this behavior was present at all temperatures considered. This behavior determined the range of parameters such as temperature and external magnetic field that would lead to the operation of the Otto cycle extracting positive total work. In the classical approach, we found oscillations in the total work extracted that are not present in the quantum approach. This happened near the zone of slope change in the behavior of the entropy in terms of the magnetic field. Interestingly, we found that, in the classical approach, the engine yielded a much higher performance in terms of total work extracted and efficiency than in the quantum approach. This is because, in the classical approach, the working substance can be in thermal equilibrium at each point of the cycle, whereas, in the quantum approach, the working substance can only thermalize in the isochoric strokes. Because of the principle of minimum energy, the system is allowed to extract more energy when the adiabatic strokes can lead to states that are in thermal equilibrium, which is only possible in the classical case.

These results are reasonable, since, in our quantum approach, the working substance remains in a diagonal state and does not use quantum resources such as quantum coherence, which in some cases can lead to enhanced performance.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/1099-4300/21/5/512/s1, Video S1: "Work evolution for high field/temperature zones", Video S2: "Oscillatory behaviour of classical work extracted I", Video S3: "Oscillatory behaviour of classical work extracted II". Video S1 shows the behaviour of work and efficiency in the high field/temperature zones for the proposed machine. Video S2 and S3 shows the oscillatory nature of the extracted work for the classical version of the Otto cycle due to the entropy behavior.

**Author Contributions:** F.J.P. , G.A. and P.V. conceived the idea and formulated the theory. O.N., D.Z and A.G. built the computer program and edited figures. A.S.N. and P.A.O. contributed to discussions during the entire work and editing the manuscript. F.J.P. wrote the paper. All authors have read and approved the final manuscript.

**Funding:** This research was funding by Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia, under Project No. FB 0807 (Chile).

**Acknowledgments:** F.J.P. acknowledges the financial support of FONDECYT-postdoctoral 3170010, and D.Z. acknowledges USM-DGIIP. P.V. and G.A.B. acknowledges support from Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia, under Project No. FB 0807 (Chile). The authors acknowledge DTI-USM for the use of "Mathematica Online Unlimited Site" at the Universidad Técnica Federico Santa María.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Non-Thermal Quantum Engine in Transmon Qubits**

#### **Cleverson Cherubim 1,\*, Frederico Brito <sup>1</sup> and Sebastian Deffner <sup>2</sup>**


Received: 8 May 2019; Accepted: 27 May 2019; Published: 29 May 2019

**Abstract:** The design and implementation of quantum technologies necessitates the understanding of thermodynamic processes in the quantum domain. In stark contrast to macroscopic thermodynamics, at the quantum scale processes generically operate far from equilibrium and are governed by fluctuations. Thus, experimental insight and empirical findings are indispensable in developing a comprehensive framework. To this end, we theoretically propose an experimentally realistic quantum engine that uses transmon qubits as working substance. We solve the dynamics analytically and calculate its efficiency.

**Keywords:** quantum heat engines; quantum thermodynamics; nonequilibrium systems

#### **1. Introduction**

Recent advances in nano and quantum technology will necessitate the development of a comprehensive framework for *quantum thermodynamics* [1]. In particular, it will be crucial to investigate whether and how the laws of thermodynamics apply to small systems, whose dynamics are governed by fluctuations and which generically operate far from thermal equilibrium. In addition, it has already been recognized that at the nanoscale many standard assumptions of classical statistical mechanics and thermodynamics are no longer justified and even in equilibrium quantum subsystems are generically not well-described by a Maxwell-Boltzmann distribution, or rather a Gibbs state [2]. Thus, the formulation of the statements of quantum thermodynamics have to be carefully re-formulated to account for potential quantum effects in, for instance, the efficiency of heat engines [3–6].

In good old thermodynamic tradition, however, this conceptual work needs to be guided by experimental insight and empirical findings. To this end, a cornerstone of quantum thermodynamics has been the description of the working principles of quantum heat engines [7–17].

However, to date it is not unambiguously clear whether quantum features can always be exploited to outperform classical engines, since to describe the thermodynamics of non-thermal states one needs to consider different perspectives—different than the one established for equilibrium thermodynamics. For instance, it has been shown that the Carnot efficiency cannot be beaten [4,18] if one accounts for the energy necessary to maintain the non-thermal stationary state [19–22]. However, it has also been argued that Carnot's limit can be overcome, if one carefully separates the "heat" absorbed from the environment into two different types of energy exchange [23,24]: one is associated with a variation in *passive energy* [25,26] which would be the part responsible for changes in entropy, and the other type is a variation in *ergotropy*, a work-like energy that could be extracted by means of a suitable unitary transformation. On the other hand, it has been shown [27] that a complete thermodynamic description in terms of *ergotropy* is also not always well suited. Having several perspectives to explain the same phenomenon is a clear indication of the subtleties and challenges faced by quantum thermodynamics,

and which can only be settled by the execution of purposefully designed experiments. Therefore, theoretical proposals for feasible and relevant experiments appear instrumental.

In this work we propose an experiment to implement a thermodynamic engine with a transmon qubit as the working substance (WS), which interacts with a non-thermal environment composed by two subsystems, an externally excited cavity (a superconducting transmission line) and a classical heat bath [28] with temperature *T*. The WS undergoes a non-conventional cycle (different from Otto, Carnot, etc.) [29] through a succession of non-thermal stationary states obtained by slowly varying its bare energy gap (frequency) and the amplitude of the pumping field applied to the cavity. We calculate the efficiency of this engine for a range of experimentally accessible parameters [28,30–32], obtaining a maximum value of 47%, which is comparable with values from the current literature.

#### **2. System Description**

We consider a multipartite system, comprised of a transmon qubit of tunable frequency *ω*T, which interacts with a transmission line (cavity) of natural frequency *ω*CPW with coupling strength *g*. The cavity is pumped by an external field of amplitude *Ed* and single frequency *ω* (see Figure 1). Both systems are in contact with a classical heat bath at temperature *T*. Such a set-up is experimentally realistic and several implementations have already been reported in different contexts [28,33]. Here and in the following, the transmon is used as a working substance (WS) and the (non-standard) "bath" is represented by the net effect of the other two systems: the cavity and the cryogenic environment (classical bath). There are two subtleties that must be noted here: (i) the bath "seen" by the qubit does not only consist of a classical reservoir at some fixed temperature, but it has an additional component, namely the pumped cavity. By changing the pumping, several cavity states can be realized. Such a feature gives the possibility of making this composed bath *non-thermal* on demand. In addition, (ii), the proposed engine is devised as containing only one bath (cavity + environment), which does not pose any problems considering that it is an out-of-equilibrium bath.

**Figure 1.** Sketch of the quantum engine with a transmon qubit as working substance interacting with an externally pumped (E(t)) transmission line (cavity). Both systems are embedded in the same cryogenic environment, which plays the role of a standard thermal bath of temperature *T*. Such a setup gives a dynamics of a working substance in the presence of a controllable *non-thermal* environment.

We start our analysis from the Hamiltonian describing a tunable qubit interacting with a single mode pumped cavity through a Jaynes-Cummings interaction

$$\begin{split}H(t) &= \frac{\hbar\omega\_{\rm T}}{2}\sigma\_{z} + \hbar\omega\_{\rm CPW}a^{\dagger}a + g\sigma\_{\rm X}(a+a^{\dagger}) \\ &+ E\_{d}\left(a\varepsilon^{i\omega t} + a^{\dagger}\varepsilon^{-i\omega t}\right),\end{split} \tag{1}$$

where *σ<sup>x</sup>* and *σ<sup>z</sup>* are the Pauli matrices, *a*† and *a* are the canonical bosonic creation and annihilation operators associated with the cavity excitations, *g* is the qubit-cavity coupling strength. The last term represents a monochromatic pumping of amplitude *Ed* and frequency *ω* applied to the cavity. The experimental characterization of the qubit-cavity dissipative dynamics emerging from their interaction with the same thermal bath shows that the system's steady state is determined by the master equation [28]

$$\begin{split} \rho(t) &= -\frac{i}{\hbar} [H\_{\text{RWA}}, \rho] + K\_{\text{CPW}}^{-} \mathcal{D}[a] \rho \\ &+ K\_{\text{CPW}}^{+} \mathcal{D}[a^{\dagger}] \rho + \Gamma^{-} \mathcal{D}[\sigma^{-}] \rho + \Gamma^{+} \mathcal{D}[\sigma^{+}] \rho\_{\text{\textquotedblleft}} \end{split} \tag{2}$$

with *K*− CPW(*K*<sup>+</sup> CPW) being the cavity decay (excitation) rate, <sup>Γ</sup>−(Γ+) the qubit relaxation (excitation) rate and D[*A*]*<sup>ρ</sup>* = *<sup>A</sup>ρA*† − 1/2(*A*†*A<sup>ρ</sup>* + *<sup>ρ</sup>A*†*A*). Please note that these rates satisfy detailed balance for the same bath of temperature *T*, *K*<sup>+</sup> CPW/*K*<sup>−</sup> CPW <sup>=</sup> exp (−*h*¯ *<sup>ω</sup>*CPW/*k*B*T*) and <sup>Γ</sup>+/Γ<sup>−</sup> <sup>=</sup> exp (−*h*¯ *<sup>ω</sup>*T/*k*B*T*). The Hamiltonian part

$$\begin{split} H\_{\text{RWA}} &= \frac{\hbar}{2} \left( \omega\_{\text{T}} - \omega \right) \sigma\_{\text{z}} + \hbar \left( \omega\_{\text{CPW}} - \omega \right) a^{\dagger} a \\ &+ g \left( \sigma\_{+} a + \sigma\_{-} a^{\dagger} \right) + E\_{d} (a + a^{\dagger}), \end{split} \tag{3}$$

is the system Hamiltonian in the rotating wave approximation (RWA) [34], with *σ*+(*σ*−) being the spin ladder operators.

Since we are interested in the observed dynamics of the WS, it is necessary to find the qubit's reduced density matrix *ρ*T(*t*) ≡ tra {*ρ*(*t*)}, where tra {·} represents the partial trace on the cavity's degrees of freedom. The system state is in a qubit-cavity product state, i.e., *ρ*(*t*) ≈ *ρ*T(*t*) ⊗ *ρ*C(*t*), which emerges in the effective qubit-cavity weak coupling regime due to decoherence into the global environment. In addition, the cavity's stationary state *ρ*C(*t*) is assumed to be mainly determined by the external pumping, which can be easily found for situations of strong pumping and/or weak coupling strength *g*. This closely resembles a situation, in which the cavity acts as a work source of effectively infinite inertia [35]. Thus, changing the state of the qubit does not affect the state of the cavity, but it is still susceptible to the applied field and the cryogenic bath, and we have

$$\left = \left^{\*} = \frac{E\_d}{\hbar \left[i\kappa\_{\rm CPW}/2 - \left(\omega\_{\rm CPW} - \omega\right)\right]},\tag{4}$$

where we defined *K*− CPW = *κ*CPW. Hence, the reduced master equation (2) can be written as

$$\dot{\rho}\_{\rm T}(t) = -\frac{\dot{t}}{\hbar}[\tilde{H}\_{\rm T,RWA}\rho\_{\rm T}] + \Gamma^{-}\mathcal{D}[\sigma^{-}]\rho\_{\rm T} + \Gamma^{+}\mathcal{D}[\sigma^{+}]\rho\_{\rm T} \tag{5}$$

with

$$H\_{\rm T,RWA} = \frac{\hbar}{2}(\omega\_{\rm T} - \omega)\sigma\_z + \mathcal{g}\left[\langle a \rangle \sigma\_+ + \langle a^\dagger \rangle \sigma\_-\right]. \tag{6}$$

Please note that the effective qubit Hamiltonian carries information about the interaction with the cavity through *a* and *a*†, which are dependent on the cavity state.

#### **3. Non-Equilibrium Thermodynamics**

#### *3.1. Non-Thermal Equilibrium States*

The only processes that are fully describable by means of conventional thermodynamics are infinitely slow successions of equilibrium states. For the operating principles of heat engines, the second law states that the maximum attainable efficiency of a thermal engine operating between two heat baths is limited by Carnot's efficiency.

An extension of this standard description is considering infinitely slow successions along *non-Gibbsian*, but stationary states [4,18–20,36]. In the present case, namely a heat engine with transmon qubit as working substance, non-Gibbsianity is induced by the external excitation applied as a driving field to the cavity. We will see in the following, however, that identifying the thermodynamic work is subtle – and that the energy exchange can exhibit heat-like character, which is crucial when computing the entropy variation during the engine operation.

The stationary state can be found by solving the master equation Equation (5), and is written as

$$
\rho\_{\mathbb{T}}^{ss} = \begin{pmatrix}
\rho\_{\mathbb{T}}^{\varepsilon\varepsilon} & \rho\_{\mathbb{T}}^{\varepsilon\xi} \\
\rho\_{\mathbb{T}}^{\xi\varepsilon} & \rho\_{\mathbb{T}}^{\xi\xi}
\end{pmatrix}
\tag{7}
$$

where the matrix elements can be computed explicitly and are summarized in Appendix A.

We observe that for the case of effective qubit-cavity ultra-weak coupling, i.e., *h*¯ *ω*<sup>T</sup> *gEd*/ |*i h*¯ *κ*CPW/2 − *h*¯(*ω*CPW − *ω*)|, as expected, the obtained non-thermal state asymptotically approaches thermal equilibrium, namely |*ρ eg* <sup>T</sup> | = |*ρ ge* <sup>T</sup> | ≈ 0 and *<sup>ρ</sup>ee* <sup>T</sup> /*ρ gg* <sup>T</sup> ≈ exp (−*βh*¯ *ω*T). In addition, as also expected, in the high temperature limit *h*¯ *ω*T/*kT* 1 the qubit stationary state becomes the thermal, maximally mixed state, given that the cavity is not strongly pumped.

#### *3.2. The Cycle*

In equilibrium thermodynamics cycles are constructed by following a closed path on a surface obtained by the equation of state [29], which characterizes possible equilibrium states for a given set of macroscopic variables. This procedure can be generalized in the context of steady state thermodynamics, where an equation of state is also constructed.

For the present purposes, we use the steady state (7) to devise a cycle for our heat engine. The equation of state in our case is represented by the stationary state's von Neumann entropy *<sup>S</sup>*(*ω*T, *Ed*) = <sup>−</sup>tr *ρss* <sup>T</sup> ln *<sup>ρ</sup>ss* T , which is fully determined by the pair of controllable variables *ω*T, the transmon's frequency, and *Ed*, amplitude field of the pumping applied to the cavity. In order to implement the cycle, the stationary state is slowly varied (quasi-static) (The timescale for which the changes made can be considered slow is such that the conditions imposed to the system state are satisfied, namely the state is a product state and the cavity steady state is a coherent state with Equation (4)) by changing the "knobs" (*ω*T,*Ed*). It is composed of four strokes where we keep one of the two controllable variables constant and vary the other one, for example, at the first stroke we keep *Ed* = *E*<sup>0</sup> and vary *ω*<sup>T</sup> from *ω*<sup>0</sup> to *ω*1. The complete cycle is sketched in Figure 2.

**Figure 2.** Sketch of the thermodynamic cycle obtained by varying the tunable parameters *ω*<sup>T</sup> and *Ed*. Each one of the strokes are obtained by keeping one of the variables constant while quasi-statically varying the other one.

Since we are interested in analyzing the engine as a function of its parameters of operation, we simulated several cycles with boundary values (*ω*1, *E*1), which will range from the minimum value (*ω*0, *E*0) to the maximum one (*ω*1,max, *E*1,max). The corresponding cycles lie on the von Neumann entropy surface depicted in Figure 3. In Appendix A plots of the stationary state's population and quantum coherence *ρee* <sup>T</sup> and |*ρ eg* <sup>T</sup> | as a function of (*ω*T, *Ed*) are shown. There we can observed clearly that the WS exhibits quantum coherence and population changes during its operation.

**Figure 3.** Stationary state's von Neumann entropy in the regime of operation of the thermal engine. Any thermodynamic cycle must be contained on this surface.

Finally, it is worth emphasizing that in the present analysis all parameters were chosen from an *experimentally accessible* regime [28,30–32], under the validity of the approximation of weak-coupling interaction between transmon and cavity. The parameters are collected in Table 1.



#### **4. Work, Heat and Efficiency**

The first law of thermodynamics, Δ*E*(*t*) = *W*(*t*) + *Q*(*t*), states that a variation of the internal energy along a thermodynamic process can be divided into two different parts, work *W*(*t*) and heat *Q*(*t*), where for Lindblad dynamics we have [4,37],

$$\begin{aligned} \mathcal{W}(t) &= \int\_0^t \text{tr}\left\{\rho(t')\dot{H}(t')dt'\right\}, \\ \mathcal{Q}(t) &= \int\_0^t \text{tr}\left\{\dot{\rho}(t')H(t')dt'\right\}. \end{aligned} \tag{8}$$

Typically, work is understood as a controllable energy exchange, which can be used for something useful, while heat cannot be controlled, emerging from the unavoidable interaction of the engine with its environment. As stated before, there are certain situations in which it can be shown that part of *Q*(*t*) does not cause any entropic variation [24]. This has led to proposals for the differentiation of two distinct forms of energy contributions to *Q*: the *passive energy* Q(*t*), which is responsible for the variation in entropy, and the variation in *ergotropy* ΔW(*t*) which is a "work-like energy" that can be extracted by means of a unitary transformation and consequently would not cause any entropic change. Both terms are defined as,

$$\begin{aligned} \mathcal{Q}(t) &= \int\_0^t \text{tr}\left\{\pi(t')H(t')dt'\right\}, \\ \Delta \mathcal{W}(t) &= \int\_0^t \text{tr}\left\{ [\dot{\rho}(t') - \dot{\pi}(t')]H(t')dt' \right\}, \end{aligned} \tag{9}$$

with *π*(*t*) being the passive state [25] associated with the state *ρ*(*t*) at time *t*. To calculate the upper bound on the efficiency for systems that exhibit these different "flavors" of energy one should replace *Q* by Q in statements of the second law, since the *ergotropy* is essentially a mechanical type of energy, and consequently not limited by the second law, resulting in a different upper bound, see also Ref. [24].

Distinguishing these types of energy exchanged with the environment is crucial when one is interested in determining the fundamental upper bounds on the efficiency. However, in the present context, we are more interested in experimentally relevant statements, i.e., computing the efficiency in terms of what can be measured directly. Thus, we consider the ratio of the extracted work to the total energy acquired from the bath, independent of its type [24].

The cycle designed here is such that in each stroke one of the knobs (*ω*T, *Ed*) is kept fixed, while the other one is changed. Recall that the cavity is assumed to be a subpart of the bath seen by the WS, and that its state is modified by *Ed*. Since the WS is always in contact with the environment, one has that heat and work are exchanged in each stroke. Here, such a calculation is done by using Equation (8), considering the stationary state Equation (7) and the effective WS Hamiltonian Equation (6). Then, for the *i*th stroke, the corresponding *Wi* and *Qi* integrals, representing the work and heat delivered (extracted) to (from) the WS, can be parametrized in terms of the respective knob variation as we can see in Appendix B. These quantities are obtained using the WS effective Hamiltonian *H*˜ T,RWA, which already takes into account the interaction with the external bath and pumped cavity.

Once these quantities are determined, we can calculate the efficiency *η* of this engine, defined by

$$\eta = -\frac{\sum\_{i=1}^{4} \mathcal{W}\_i}{Q\_+} \,\tag{10}$$

with the delivered heat to the WS in a complete cycle being given by *Q*<sup>+</sup> = ∑<sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *Q<sup>i</sup>* <sup>+</sup>, with *Q<sup>i</sup>* <sup>+</sup> the given heat (only positive contributions inside the stroke) during the *i*th stroke (see Appendix B). Therefore, this efficiency represents the amount of work extracted from the engine through the use of the delivered heat to the WS.

Figure 4 shows the engine efficiency *η* attained in the execution of the strokes as a function of the boundary values (*ω*1, *E*1), as depicted in Figure 2. Please note that (*ω*1, *E*1) sweeps the entire spectrum of the tunable parameters (*ω*T, *Ed*), going from (*ω*0, *E*0) to (*ω*1,max, *E*1,max) where we find the maximal efficiency. It is worth mentioning here that the highest value of the efficiency is dependent on the chosen regime of parameters, which in our case is based on experimentally attainable values [28,30–32]. As usual, in order to extract the predicted work, one has to couple our engine to another system. We envision using the experimental setup of Ref. [28], where a mechanical nanoresonator is present and weakly driven by the transmon. Thus, under such a configuration, by following the nanoresonator's state (recall that we have assumed infinite inertia, i.e., the transmon is not capable of changing the cavity's state. In situations where such an assumption does not hold, one has to take into account the possibility of having the transmon doing work on the cavity), one can determine the amount of energy transferred in the form of work. In addition, by observing the transmon's state, one can obtain the amount of heat given by the non-standard bath, providing a full characterization of our engine.

**Figure 4.** Efficiency *η* as a function of the upper values (*ω*1, *E*1) for the cycle depicted in Figure 2. The observed highest efficiency of about 47% was attained when (*ω*1, *E*1)=(*ω*1,max, *E*1,max), with *ω*1,max/2*π* = 1000 MHz and *E*1,max/2*πh*¯ = 2 MHz.

#### **5. Conclusions and Final Remarks**

Theoretical research of small heat engines in the quantum domain is common place in quantum thermodynamics [37–46]. In the present work, we have devised a transmon-based heat engine using an experimentally realistic regime of parameters reaching a maximal efficiency of 47%, which turns out to be a reasonable value when compared with the state of the art in quantum heat engines. One of the most recent experiments in quantum heat engine was implemented by Peterson et al. [47] using a spin −1/2 system and nuclear resonance techniques, performing an Otto cycle with efficiency in excess of 42% at maximum power. It is important to stress that implementing small heat engines constitutes a hard task, even when dealing with classical systems. Indeed, a representative example is the single ion confined in a linear Paul trap with a tapered geometry, which was used to implement a Stirling engine [48] with efficiency of only 0.28%. Additional research is being carried out concerning the behavior of this engine influenced by the presence of coherence and the dimension of the WS. By devising this theoretical protocol for the implementation of a quantum engine, we hope to help the community, and in particular experimentalists, in the formidable task to design and implement quantum thermodynamic systems and to consolidate the concepts of this new exiting field of research.

**Author Contributions:** C.C., F.B. and S.D. equally contributed to conceptualization, investigation and writing of the paper.

**Funding:** C.C. and F.B. acknowledge financial support in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. During his stay at UMBC, C.C. was supported by the CAPES scholarship PDSE/process No. 88881.132982/2016-01. F.B. is also supported by the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ) under Grant No. 465469/2014-0/CNPq. S.D. acknowledges support from the U.S. National Science Foundation under Grant No. CHE-1648973.

**Acknowledgments:** We thank F. Rouxinol and V. F. Teizen for valuable discussions. C.C. would like to thank the hospitality of UMBC, where most of this research was conducted.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Non-Thermal Equilibrium States**

Here, we summarize the explicit expressions of the density matrix elements of *ρss* <sup>T</sup> (7), which are plotted as a function of (*ω*T, *E*d) in Figure A1 .

$$\rho\_{\Gamma}^{\rm eff} = \frac{\frac{\text{g}^2 E\_d^2}{\hbar^4 \left[\frac{1}{4} \kappa\_{\text{CPW}}^2 + (\omega \text{CPW} - \omega)^2\right]} + \frac{1}{1 + e^{\beta \hbar \omega\_{\Gamma}}} \left[\frac{1}{4} \frac{\Gamma^2}{\tanh^2 \left(\beta \hbar \omega\_{\Gamma}/2\right)} + \left(\omega\_{\Gamma} - \omega\right)^2\right]}{2g^2 E\_d^2} + \left[\frac{1}{4} \frac{\Gamma^2}{\tanh^2 \left(\beta \hbar \omega\_{\Gamma}/2\right)} + \left(\omega\_{\Gamma} - \omega\right)^2\right]},\tag{A1}$$

$$\rho\_{\Gamma}^{\rm SV} = \frac{\frac{g^2 E\_d^2}{\hbar^4 \left[\frac{1}{4} \kappa\_{\rm CPW}^2 + (\omega \epsilon\_{\rm CPW} - \omega)^2\right]}}{\frac{2g^2 E\_d^2}{\hbar^4 \left[\frac{1}{4} \kappa\_{\rm CPW}^2 + (\omega \epsilon\_{\rm CPW} - \omega)^2\right]} + \left[\frac{1}{4} \frac{\Gamma^2}{\tanh^2\left(\beta \hbar \omega \tau/2\right)} + (\omega \tau - \omega)^2\right]},\tag{A2}$$

$$\rho\_{\Gamma}^{\mathrm{cg}} = \frac{\frac{1}{2\hbar} \left[ \frac{\Gamma}{\tanh\left(\beta \hbar \omega \tau/2\right)} i + 2(\omega\_{\Gamma} - \omega) \right] \frac{\mathrm{g}E\_{d}}{\hbar \left[ \frac{\mathrm{v} \mathrm{C\_{\mathrm{CPW}}}}{2} - (\omega\_{\mathrm{CPW}} - \omega) \right]}}{\frac{2g^{2}E\_{d}^{2}}{\hbar^{4} \left[ \frac{1}{4} \omega\_{\mathrm{CPW}}^{2} + (\omega\_{\mathrm{CPW}} - \omega)^{2} \right]} + \left[ \frac{1}{4} \frac{\Gamma^{2}}{\tanh^{2}\left(\beta \hbar \omega\_{\Gamma}/2\right)} + (\omega\_{\Gamma} - \omega)^{2} \right]} \tanh\left(\beta \hbar \omega\_{\Gamma}/2\right). \tag{A3}$$

**Figure A1.** Stationary state's elements *ρee* <sup>T</sup> and |*ρ eg* <sup>T</sup> | for different values of (*ω*T, *Ed*). Important amounts of population and quantum coherence changes can be reached during the engine operation.

#### **Appendix B. Thermodynamic Quantities along Each Stroke**

In this appendix we summarize the explicit expressions of the thermodynamic quantities *Wi* and *Qi* for *i* = 1, 2, 3, 4 and the heat *Q*<sup>+</sup> given to the WS. These quantities are obtained by changing quasi-statically the parameters *ω*<sup>T</sup> and *Ed* producing a succession of steady states *ρ*ˆ*ss* <sup>T</sup> (*ω*T, *Ed*):

$$\begin{split} \mathcal{W}\_{1} &= \int\_{\omega\_{0}}^{\omega\_{1}} \text{tr}\left\{ \hat{\rho}\_{\text{T}}^{\text{ss}} (\omega\_{\text{T}}, E\_{0}) \left( \frac{\partial \hat{\Omega}\_{\text{LRWA}}}{\partial \omega\_{\text{T}}} \right)\_{E\_{0}} \right\} d\omega\_{\text{T}}, \\ \mathcal{W}\_{2} &= \int\_{E\_{0}}^{E\_{1}} \text{tr}\left\{ \hat{\rho}\_{\text{T}}^{\text{ss}} (\omega\_{\text{T}}, E\_{d}) \left( \frac{\partial \hat{\Omega}\_{\text{LRWA}}}{\partial \hat{\varepsilon}\_{d}} \right)\_{\omega\_{1}} \right\} d\hat{E}\_{d}, \\ \mathcal{W}\_{3} &= \int\_{\omega\_{1}}^{\omega\_{0}} \text{tr}\left\{ \hat{\rho}\_{\text{T}}^{\text{ss}} (\omega\_{\text{T}}, E\_{1}) \left( \frac{\partial \hat{\Omega}\_{\text{LRWA}}}{\partial \omega\_{\text{T}}} \right)\_{E\_{1}} \right\} d\omega\_{\text{T}}, \\ \mathcal{W}\_{4} &= \int\_{E\_{1}}^{E\_{0}} \text{tr}\left\{ \hat{\rho}\_{\text{T}}^{\text{ss}} (\omega\_{0}, E\_{d}) \left( \frac{\partial \hat{\Omega}\_{\text{LRWA}}}{\partial \hat{\varepsilon}\_{d}} \right)\_{\omega\_{0}} \right\} d\hat{E}\_{d}. \end{split} \tag{A4}$$

$$Q\_{1} = \int\_{\omega\_{0}}^{\omega\_{1}} \text{tr}\left\{ \left(\frac{\partial \tilde{p}\_{\text{T}}^{\omega}}{\partial \omega\_{\text{T}}}\right)\_{E\_{\text{T}}} \tilde{H}\_{\text{T},\text{RWA}}(\omega\_{\text{T}}, E\_{0}) \right\} d\omega\_{\text{T}},$$

$$Q\_{2} = \int\_{E\_{0}}^{E\_{1}} \text{tr}\left\{ \left(\frac{\partial \tilde{p}\_{\text{T}}^{\omega}}{\partial \tilde{E}\_{d}}\right)\_{\omega\_{1}} \tilde{H}\_{\text{T},\text{RWA}}(\omega\_{\text{T}}, E\_{d}) \right\} d\tilde{E}\_{d},$$

$$Q\_{3} = \int\_{\omega\_{1}}^{\omega\_{0}} \text{tr}\left\{ \left(\frac{\partial \tilde{p}\_{\text{T}}^{\omega}}{\partial \omega\_{\text{T}}}\right)\_{E\_{1}} \tilde{H}\_{\text{T},\text{RWA}}(\omega\_{\text{T}}, E\_{1}) \right\} d\omega\_{\text{T}},$$

$$Q\_{4} = \int\_{E\_{1}}^{E\_{0}} \text{tr}\left\{ \left(\frac{\partial \tilde{p}\_{\text{T}}^{\omega}}{\partial \tilde{E}\_{d}}\right)\_{\omega\_{0}} \tilde{H}\_{\text{T},\text{RWA}}(\omega\_{0}, E\_{d}) \right\} d\tilde{E}\_{d}.\tag{A5}$$

$$Q\_{+} = \sum\_{i=1}^{4} Q\_{+}^{i} \tag{A6}$$

with *Q<sup>i</sup>* <sup>+</sup> for *i* = 1, 2, 3, 4 given by

*Q*1 <sup>+</sup> <sup>=</sup> *<sup>ω</sup>*<sup>1</sup> *<sup>ω</sup>*<sup>0</sup> tr *∂ρ*ˆ*ss* T *∂ω*<sup>T</sup> *E*0 *H*˜ T,RWA(*ω*T, *E*0) Θ tr *∂ρ*ˆ*ss* T *∂ω*<sup>T</sup> *E*0 *H*˜ T,RWA(*ω*T, *E*0) *dω*<sup>T</sup> , *Q*2 <sup>+</sup> <sup>=</sup> *<sup>E</sup>*<sup>1</sup> *<sup>E</sup>*<sup>0</sup> tr *∂ρ*ˆ*ss* T *∂Ed ω*1 *H*˜ T,RWA(*ω*1, *Ed*) Θ tr *∂ρ*ˆ*ss* T *∂Ed ω*1 *H*˜ T,RWA(*ω*1, *Ed*) *dEd* , *Q*3 <sup>+</sup> <sup>=</sup> *<sup>ω</sup>*<sup>0</sup> *<sup>ω</sup>*<sup>1</sup> tr *∂ρ*ˆ*ss* T *∂ω*<sup>T</sup> *E*1 *H*˜ T,RWA(*ω*T, *E*1) Θ tr *∂ρ*ˆ*ss* T *∂ω*<sup>T</sup> *E*1 *H*˜ T,RWA(*ω*T, *E*1) *dω*<sup>T</sup> , *Q*4 <sup>+</sup> <sup>=</sup> *<sup>E</sup>*<sup>0</sup> *<sup>E</sup>*<sup>1</sup> tr *∂ρ*ˆ*ss* T *∂Ed ω*<sup>0</sup> *H*˜ T,RWA(*ω*0, *Ed*) Θ tr *∂ρ*ˆ*ss* T *∂Ed ω*<sup>0</sup> *H*˜ T,RWA(*ω*0, *Ed*) *dEd* . (A7)

where the Heaviside function Θ[·] is inside the integral, selecting only the positive contributions (heat given to the WS) along the stroke.

#### **References**


c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **A Quantum Heat Exchanger for Nanotechnology**

**Amjad Aljaloud 1,2,\*, Sally A. Peyman 1,3 and Almut Beige <sup>1</sup>**


Received: 6 February 2020; Accepted: 21 March 2020; Published: 26 March 2020

**Abstract:** In this paper, we design a quantum heat exchanger which converts heat into light on relatively short quantum optical time scales. Our scheme takes advantage of heat transfer as well as collective cavity-mediated laser cooling of an atomic gas inside a cavitating bubble. Laser cooling routinely transfers individually trapped ions to nano-Kelvin temperatures for applications in quantum technology. The quantum heat exchanger which we propose here might be able to provide cooling rates of the order of Kelvin temperatures per millisecond and is expected to find applications in micro- and nanotechnology.

**Keywords:** quantum thermodynamics; laser cooling; cavitation; sonoluminescence

#### **1. Introduction**

Since its discovery in 1975 [1,2], laser cooling of individually trapped atomic particles has become a standard technique in quantum optics laboratories worldwide [3,4]. Rapidly oscillating electric fields can be used to strongly confine charged particles, such as single ions, for relatively large amounts of time [5]. Moreover, laser trapping provides unique means to control the dynamics of neutral particles, such as neutral atoms [6,7]. To cool single atomic particles, laser fields are applied which remove vibrational energy at high enough rates to transfer them down to near absolute-zero temperatures [5]. Nowadays, ion traps are used to perform a wide range of high-precision quantum optics experiments. For example, individually trapped ions are at the heart of devices with applications in quantum technology, such as atomic and optical clocks [8,9], quantum computers [10–13], quantum simulators [14,15] and electric and magnetic field sensors [16].

For laser cooling to be at its most efficient, the confinement of individually trapped particles should be so strong that the quantum characteristics of their motion is no longer negligible. This means that their vibrational energy is made up of energy quanta, which have been named phonons. When this applies, an externally applied laser field not only affects the electronic states of a trapped ion, but it also changes its vibrational state. Ideally, laser frequencies should be chosen such that the excitation of the ion should be most likely accompanied by the loss of a phonon. If the ion returns subsequently into its ground state via the spontaneous emission of a photon, its phonon state remains the same. Overall one phonon is permanently lost from the system which implies cooling. On average, every emitted photon lowers the vibrational energy of the trapped ion by the energy of one phonon. Eventually, the cooling process stops when the ion no longer possesses any vibrational energy.

Currently, there are many different ways of designing and fabricating ion traps [17,18]. However, the main requirements for the efficient conversion of vibrational energy into light on relatively short quantum optical time scales are always the same [19,20]:


$$
\Delta\_{\perp} \sim \nu\_{\prime} \tag{1}
$$

the excitation of an ion is more likely accompanied by the annihilation of a phonon than by the creation of a phonon. Transitions which result in the simultaneous excitation of an ion and the creation of a phonon are possible but are less likely to occur as long as their detuning is larger.

(3) When excited, the confined atomic particle needs to be able to emit a photon. In the following, we denote its spontaneous decay rate by Γ. This rate should not be much larger than *ν*,

$$
\nu\_{\perp} \ge \quad \Gamma\_{\prime}.\tag{2}
$$

so that the cooling laser couples efficiently to atomic transitions. At the same time, Γ should not be too small so that de-excitation of the excited atomic state happens often via the spontaneous emission of a photon.

Given these three conditions, the applied laser field results in the conversion of the vibrational energy of individually trapped ions into photons. As mentioned already above, laser cooling can prepare individually trapped atomic particles at low enough temperatures for applications in high-precision quantum optics experiments and in quantum technology.

In this paper, we ask the question whether laser cooling could also have applications in microand nanoscale physics experiments. For example, nanotechnology deals with objects which have dimensions between 1 and 1000 nm and is well known for its applications in information and communication technology, as well as sensing and imaging. Increasing the speed at which information can be processed and the sensitivity of sensors is usually achieved by reducing system dimensions. However, smaller devices are usually more prone to heating as thermal resistances increase [21]. Sometimes, large surface to volume ratios can help to off-set this problem. Another problem for nanoscale sensors is thermal noise. As sensors are reduced in size, their signal to noise ratio usually decreases and thus the thermal energy of the system can limit device sensitivity [22]. Therefore, thermal considerations have to be taken into account and large vacuums or compact heat exchangers have already become an integral part of nanotechnology devices.

Usually, heat exchangers in micro- and nanotechnology rely on fluid flow [23]. In this paper, we propose an alternative approach. More concretely, we propose to use heat transfer as well as a variation of laser cooling, namely cavity-mediated collective laser cooling [24–28], to lower the temperature of a small device. As illustrated in Figure 1, the proposed quantum heat exchanger mainly consist of a liquid which contains a large number of cavitating bubbles filled with noble gas atoms. Transducers constantly change the radius of these bubbles which should resemble optical cavities when they reach their minimum radius during bubble collapse phases. At this point, a continuously applied external laser field rapidly transfers vibrational energy of the atoms into light. If the surrounding liquid contains many cavitating bubbles, their surface area becomes relatively large and there can be a very efficient exchange of heat between the inside and the outside of cavitating bubbles. Any removal of thermal energy from the trapped atomic gas inside bubbles should eventually result in the cooling of the surrounding liquid and of the surface area of the device on which it is placed.

In this paper, we emphasize that cavitating bubbles can provide all of the above listed requirements for laser cooling, especially a very strong confinement of atomic particles, such as nitrogen [29,30]. For example, calculations based on a variation of the Rayleigh–Plesset equations show that the pressure at the location of a cavitating bubble can be significantly larger than the externally applied driving pressure [31]. However, the strongest indication for the presence of phonon modes with sufficiently

large frequencies for laser cooling to work comes from the fact that sonoluminescence experiments are well-known for converting sound into relatively large amounts of thermal energy, while producing light in the optical regime [32–34]. During this process, the atomic gas inside a cavitating bubbles can reach very high temperatures [35,36], which hints at very strong couplings between electronic and vibrational degrees of freedom. In addition, the surfaces of cavitating bubbles can become opaque during the bubble collapse phase [37], thereby creating a spherical optical cavity [38,39] which is an essential requirement for cavity-mediated collective laser cooling.

To initiate the cooling process, an appropriately detuned laser field needs to be applied in addition to the transducers which confine the bubbles with sound waves. Although sonoluminescence has been studied in great detail and the idea of applying laser fields to cavitating bubbles is not new [40], not enough is known about the relevant quantum properties, such as phonon frequencies. Hence, we cannot predict realistic cooling rates for the experimental setup shown in Figure 1. A crude estimate which borrows data from different, already available experiments suggests that it might be possible to achieve cooling rates of the order of Kelvin temperatures per millisecond for volumes of liquid on a cubic micrometer scale. Cavitating bubbles already have applications in sonochemistry, where they are used to provide energy for chemical reactions [41]. Here, we propose to exploit the atom–phonon interactions in sonoluminescence experiments for laser cooling. In the presence of an appropriately detuned laser field, we expect other, highly-detuned heating processes to become secondary.

**Figure 1.** Schematic view of the proposed quantum heat exchanger. It consists of a liquid in close contact with the area which we want to cool. The liquid should contain cavitating bubbles which are filled with atomic particles, such as nitrogen, and should be driven by sounds waves and laser light. The purpose of the sound waves is to constantly change bubble sizes. The purpose of the laser is to convert thermal energy during bubble collapse phases into light.

There are five sections in this paper. The purpose of Section 2 is to provide an introduction to cavity-mediated collective laser cooling of an atomic gas. As we show below, this technique is a variation of standard laser cooling techniques for individually trapped atomic particles. We provide an overview of the experimental requirements and estimate achievable cooling rates. Section 3 studies the effect of thermalization for a large collection of atoms with elastic collisions. Section 4 reviews the main design principles of a quantum heat exchanger for nanotechnology. Finally, we summarize our findings in Section 5.

#### **2. Cavity-Mediated Collective Laser Cooling**

In this section, we first have a closer look at a standard laser cooling technique for an individually trapped atomic particle [19,20]. Afterwards, we review cavity-mediated laser cooling of a single atom [42–46] and of an atomic gas [26–28].

#### *2.1. Laser Cooling of Individually Trapped Particles*

Figure 2a shows a single two-level atom (or ion) with external laser driving inside an approximately harmonic trapping potential. Most importantly, the atom should be so strongly confined that its phonon states are no longer negligible. In the following, *ν* denotes the frequency of the energy quanta in the vibrational energy of the atomic particle and |*m* is a vibrational state with exactly *m* phonons. Moreover, |g and |e denote the ground and the excited electronic state of the trapped particle with energy separation *h*¯ *ω*0. Figure 2b shows the energy level of the combined atom–phonon system with the energy eigenstates |x, *m*.

**Figure 2.** (**a**) Schematic view of the experimental setup for laser cooling of a single trapped ion. Here, |g and |e denote the ground and the excited state of the ion, respectively, with transition frequency *ω*<sup>0</sup> and spontaneous decay rate Γ. The motion of the particle is strongly confined by an external harmonic trapping potential such that it quantum nature can no longer be neglected. Here, *ν* denotes the frequency of the corresponding phonon mode and *ω*<sup>L</sup> is the frequency of the applied cooling laser. (**b**) The purpose of the laser is to excite the ion, while annihilating a phonon, thereby causing transitions between the basis states |x, *m* with x = g, e and *m* = 0, 1, ... of the atom–phonon system. If the excitation of the ion is followed by the spontaneous emission of a photon, a phonon is permanently lost, which implies cooling.

To lower the temperature of the atom, the frequency *ω*<sup>L</sup> of the cooling laser needs to be below its transition frequency *ω*0. Ideally, the laser detuning Δ = *ω*<sup>0</sup> − *ω*<sup>L</sup> equals the phonon frequency *ν* (cf. Equation (1)). In addition, the spontaneous decay rate Γ of the excited atomic state should not exceed *ν* (cf. Equation (2)). When both conditions apply, the cooling laser couples most strongly, i.e., resonantly and efficiently, to transitions for which the excitation of the atom is accompanied by the simultaneously annihilation of a phonon. All other transitions are strongly detuned. Moreover, the spontaneous emission of a photon only affects the electronic but not the vibrational states of the atom. Hence, the spontaneous emission of a photon usually indicates the loss of one phonon. Suppose the atom was initially prepared in a state |g, *m*. Then, its final state equals |g, *m* − 1. One phonon has been permanently removed from the system which implies cooling. As illustrated in Figure 2b, the trapped particle eventually reaches its ground state |g, 0 where it no longer experiences the cooling due to off-resonant driving [19,20].

To a very good approximation, the Hamiltonian of the atom–phonon system equals [20]

$$\begin{array}{rcl} \, \_\text{H} &=& \, \_\text{hg} \left( \sigma \, ^-b \, ^+ + \sigma \, ^+b \right) \end{array} \tag{3}$$

in the interaction picture with respect to its free energy. Here, *g* denotes the (real) atom–phonon coupling constant, while *<sup>σ</sup>*<sup>+</sup> <sup>=</sup> <sup>|</sup>eg<sup>|</sup> and *<sup>σ</sup>*<sup>−</sup> <sup>=</sup> <sup>|</sup>ge<sup>|</sup> are atomic rising and lowering operators. Moreover, *b* and *b*† are phonon annihilation and creation operators with [*b*, *b*†] = 1. To take into

account the spontaneous emission of photons from the excited state of the atom with decay rate Γ, we describe the atom–phonon system in the following by its density matrix *ρ*I(*t*) with

$$\left[\rho\_{\rm I}\right] = -\frac{i}{\hbar}[H\_{\rm I}\rho\_{\rm I}] + \Gamma\left(\sigma^-\rho\_{\rm I}\sigma^+ - \frac{1}{2}\sigma^+\sigma^-\rho\_{\rm I} - \frac{1}{2}\rho\_{\rm I}\sigma^+\sigma^-\right). \tag{4}$$

This equation can be used to analyze the dynamics of the expectation value *A*I = Tr(*A*I*ρ*I) of observables *A*I, since it implies

$$
\langle \dot{A}\_{\rm I} \rangle \quad = -\frac{\mathrm{i}}{\hbar} [A\_{\rm I} H\_{\rm I}] + \Gamma \left\langle \sigma^{+} A\_{\rm I} \sigma^{-} - \frac{1}{2} A\_{\rm I} \sigma^{+} \sigma^{-} - \frac{1}{2} \sigma^{+} \sigma^{-} A\_{\rm I} \right\rangle. \tag{5}
$$

Here, we are especially interested in the dynamics of the mean phonon number *<sup>m</sup>* = *b*†*b*. To obtain a closed set of rate equations, we also need to study the dynamics of the population of the excited atomic state *<sup>s</sup>* <sup>=</sup> *σ*+*σ*− and the dynamics of the atom–phonon coherence *<sup>k</sup>*<sup>1</sup> <sup>=</sup> <sup>i</sup>*σ*−*b*† <sup>−</sup> *<sup>σ</sup>*+*b*. Using Equation (5), one can show that

$$\begin{array}{rcl} \dot{m}\_1 &=& -\lg k\_1 \\ \dot{s}\_1 &=& \lg k\_1 - \Gamma s\_1 \\ \dot{k}\_1 &=& 2\lg(m-s) - 4\lg m s - \frac{1}{2}\Gamma k\_1 \end{array} \tag{6}$$

when assuming that *σ*+*σ*−*b*†*b* <sup>=</sup> *σ*+*σ*−*b*†*b* <sup>=</sup> *ms* to a very good approximation. Having a closer look at the above equations, we see that the system rapidly reaches its stationary state with *m* = *s* = *k*<sup>1</sup> = 0. Eventually, the atom reaches a very low temperature. More detailed calculations reveal that the final phonon *m* of the trapped atom depends on its system parameters but remains small as long as the ratio Γ/*ν* is sufficiently small [20]. The above cooling equations (Equation (6)) also show that the corresponding cooling rate equals

$$
\gamma\_{1\text{ atom}}^{\text{standard}} = \text{ g}^2/\Gamma \tag{7}
$$

to a very good approximation and that the cooling process takes place not on mechanical but on relatively short quantum optical time scales.

#### *2.2. Cavity-Mediated Laser Cooling of a Single Atom*

Suppose we want to cool a single atom whose transition frequency *ω*<sup>0</sup> is well above the optical regime, i.e., much larger than typical laser frequencies *ω*L. In this case, it is impossible to realize the condition Δ ∼ *ν* in Equation (1). Hence, it might seem impossible to lower the temperature of the atom via laser cooling. To overcome this problem, we confine the particle in the following inside an optical resonator (cf. Figure 3) and denote the cavity state with exactly *n* photons by |*n*. Using this notation, the energy eigenstates of the atom–phonon–photon systems can be written as |x, *m*, *n*. Moreover, *ν* is again the phonon frequency, *κ* denotes the spontaneous cavity decay rate and *ω*<sup>L</sup> and *ω*cav denote the laser and the cavity frequency, respectively.

In the experimental setup in Figure 3, all transitions which result in the excitation of the atom are naturally strongly detuned and can be neglected. However, the same does not have to apply to indirect couplings which result in the direct conversion of phonons into cavity photons [27,44]. Suppose the cavity detuning Δcav = *ω*cav − *ω*<sup>L</sup> and the phonon frequency *ν* are approximately the same and the cavity decay rate *κ* does not exceed *ν*,

$$
\Delta\_{\text{cav}} \sim \nu \quad \quad \text{and} \quad \nu \ge \kappa \,\,\,\tag{8}
$$

in analogy to Equations (1) and (2). Then, two-step transitions which excite the atom while annihilating a phonon immediately followed by the de-excitation of the atom while creating a cavity photon become resonant and dominate the dynamics of the atom–phonon–photon system. The overall effect of these two-step transitions is the direct conversion of a phonon into a cavity photon, while the atom remains essentially in its ground state (cf. Figure 3b). When a cavity photon subsequently leaks into the environment, the phonon is permanently lost.

**Figure 3.** (**a**) Schematic view of the experimental setup for cavity-mediated laser cooling of a single atom. The main difference between this setup and the setup shown in Figure 2 is that the atom now couples in addition to an optical cavity with frequency *ω*cav and the spontaneous decay rate *κ*. Here, both the cavity field and the laser are highly detuned from the atomic transition and the direct excitation of the atom remains negligible. However, the cavity detuning Δcav = *ω*cav − *ω*<sup>L</sup> should equal the phonon frequency of the trapped particle. (**b**) As a result, only the annihilation of a phonon accompanied by the simultaneous creation of a cavity photon are in resonance. In cavity-mediated laser cooling, the purpose of the laser is to convert phonons into cavity photons. The subsequent loss of this photon via spontaneous emission results in the permanent loss of a phonon and therefore in the cooling of the trapped particle.

To model the above described dynamics, we describe the experimental setup in Figure 3 in the following by the interaction Hamiltonian [44,45]

$$H\_{\rm I} = \hbar \mathcal{g}\_{\rm eff} \left( b c^{\dagger} + b^{\dagger} c \right) , \tag{9}$$

where *g*eff denotes the effective atom–cavity coupling constant and where *c* with [*c*, *c*†] = 1 is the cavity photon annihilation operator. Since the atom remains essentially in its ground state, its spontaneous photon emission remains negligible. To model the possible leakage photons through the cavity mirrors, we employ again a master equation. Doing so, the time derivative of the density matrix *ρ*I(*t*) of the phonon–photon system equals

$$\dot{\rho}\_{\rm l} = -\frac{\dot{\mathbf{i}}}{\hbar}[H\_{\rm l}\rho\_{\rm l}] + \kappa \left(c\rho\_{\rm l}\mathbf{c}^{\dagger} - \frac{1}{2}\mathbf{c}^{\dagger}c\rho\_{\rm l} - \frac{1}{2}\rho\_{\rm l}\mathbf{c}^{\dagger}\mathbf{c}\right) \tag{10}$$

in the interaction picture. Hence, expectation values *A*I = Tr(*A*I*ρ*I) of phonon–photon observables *A*<sup>I</sup> evolve such that

$$
\langle \dot{A}\_{\rm I} \rangle \quad = \quad -\frac{\mathrm{i}}{\hbar}[A\_{\rm I}, H\_{\rm I}] + \kappa \left\langle \mathbf{c}^{\dagger} A\_{\rm I} \mathbf{c} - \frac{1}{2} A\_{\rm I} \mathbf{c}^{\dagger} \mathbf{c} - \frac{1}{2} \mathbf{c}^{\dagger} \mathbf{c} \, A\_{\rm I} \right\rangle,\tag{11}
$$

in analogy to Equation (5). In the following, we use this equation to study the dynamics of the phonon number *<sup>m</sup>* = *b*†*b*, the photon number *<sup>n</sup>* = *c*†*c*, and the phonon–photon coherence *<sup>k</sup>*<sup>1</sup> = <sup>i</sup>*bc*† − *<sup>b</sup>*†*c*. Proceeding as described in the previous subsection, we now obtain the rate equations

$$\begin{array}{rcl} \dot{m} &=& \mathbb{g}\_{\text{eff}} k\_1 \\\\ \dot{m} &=& -\mathbb{g}\_{\text{eff}} k\_1 - \kappa n\_1 \\\\ \dot{k}\_1 &=& 2\mathbb{g}\_{\text{eff}} (n - m) - \frac{1}{2} \kappa k\_1 \end{array} \tag{12}$$

These describe the continuous conversion of phonons into cavity photons which subsequently escape the system. Hence, it is not surprising to find that the stationary state of the atom–phonon–photon system corresponds to *m* = *n* = *k*<sup>1</sup> = 0. Independent of its initial state, the atom again reaches a very low temperature. In analogy to Equation (7), the effective cooling rate for cavity-mediated laser cooling is now given by [44,45]

$$
\gamma\_{1\text{ atom}} = \ g\_{\text{eff}}^2 / \kappa \,. \tag{13}
$$

Due to the resonant coupling being indirect, *g*eff is in general a few orders of magnitude smaller than *g* in Equation (7), if the spontaneous decay rates *κ* and Γ are of similar size. Cooling a single atom inside an optical resonator might therefore take significantly longer. However, as we show below, this reduction in cooling rate can be compensated for by the collective enhancement of the atom–cavity interaction constant *g*eff [26].

#### *2.3. Cavity-Mediated Collective Laser Cooling of an Atomic Gas*

Finally, we have a closer look at cavity-mediated collective laser cooling of an atomic gas inside an optical resonator [26,27]. To do so, we replace the single atom in the experimental setup in Figure 3 by a collection of *N* atoms. In analogy to Equation (9), the interaction Hamiltonian *H*<sup>I</sup> between phonons and cavity photons now equals

$$H\_{\rm li} = \sum\_{i=1}^{N} \hbar \mathbf{g}\_{\rm eff}^{(i)} \left( b\_i \mathbf{c}^\dagger + b\_i^\dagger \mathbf{c} \right) \,, \tag{14}$$

where *g* (*i*) eff denotes the effective atom–cavity coupling constant of atom *i*. This coupling constant is essentially the same as *g*eff in Equation (13) and depends in general on the position of atom *i*. Moreover, *bi* denotes the phonon annihilation operator of atom *i* with [*bi*, *b*† *<sup>j</sup>* ] = *δij*. To simplify the above Hamiltonian, we introduce a collective phonon annihilation operator *B*,

$$B = \frac{\sum\_{i=1}^{N} \mathcal{g}\_{\text{eff}}^{(i)} b\_i}{\mathcal{\tilde{g}}\_{\text{eff}}} \quad \text{with} \quad \mathcal{\tilde{g}}\_{\text{eff}} = \left( \sum\_{i=1}^{N} |\mathcal{g}\_{\text{eff}}^{(i)}|^2 \right)^{1/2},\tag{15}$$

with [*B*, *B*†] = 1. Using this notation, *H*<sup>I</sup> in Equation (14) simplifies to

$$\begin{array}{rcl} \mathsf{H}\_{\mathrm{I}} &=& \mathsf{H}\tilde{\mathsf{g}}\_{\mathrm{eff}} \left( \mathrm{B}\mathfrak{c}^{\dagger} + \mathrm{B}^{\dagger}\mathfrak{c} \right) \,. \end{array} \tag{16}$$

Notice that the effective coupling constant *g*˜eff scales as the square root of the number of atoms *N* inside the cavity. For example, if all atomic particles couple equally to the cavity field with a coupling constant *g*eff ≡ *g* (*i*) eff, then *<sup>g</sup>*˜eff <sup>=</sup> <sup>√</sup>*N g*eff. This means, in the case of many atoms, the effective phonon–photon coupling is collectively enhanced [26].

When comparing *H*<sup>I</sup> in Equation (9) with *H*<sup>I</sup> in Equation (14), we see that both Hamiltonians are essentially the same. Moreover, the density matrix *ρ*<sup>I</sup> obeys the master equation in Equation (10) in both cases. Hence, we expect the same cooling dynamics in the one atom and in the many atom case. Suppose all atoms experience the same atom–cavity coupling constant *g*eff, the effective cooling rate of the common vibrational mode *B* becomes

$$\gamma\_{\text{N atoms}} = \, \text{Ng}^2\_{\text{eff}} / \kappa \,\, \tag{17}$$

in analogy to Equation (13). This cooling rate is *N* times larger than the cooling rate which we predicted in the previous subsection for cavity-mediated laser cooling of a single atom. Using sufficiently large number of atoms *N*, it is therefore possible to realize cooling rates *γ<sup>N</sup>* atoms with

$$
\gamma\_{\text{N atoms}} \quad \gg \quad \gamma\_{\text{l atom}}^{\text{standard}}.\tag{18}
$$

This suggests that the cooling rate of cavity-mediated laser cooling, i.e., the rate of change of the mean number *n* of *B* phonons in the system, is comparable and might even exceed the cooling rates of standard laser cooling of single trapped ions.

However, the above discussion also shows that cavity-mediated collective laser cooling only removes phonons from a single common vibrational mode *B*, while all other vibrational modes of the atomic gas do not experience the cooling laser. Once the *B* mode reaches its stationary state, the conversion of thermal energy into light stops. To nevertheless take advantage of the relatively high cooling rates of cavity-mediated collective laser cooling, an additional mechanism is needed [27,28]. As we shall see in the next section, one way of transferring energy between different vibrational modes is to intersperse cooling stages with thermalization stages (cf. Figure 4). The purpose of the cooling stages is to rapidly remove energy from the system. The purpose of subsequent thermalization stages is to transfer energy from the surroundings of the bubble and from the different vibrational modes of the atoms into the *B* mode. Repeating thermalization and cooling stages is expected to result in the cooling of the whole setup in Figure 1.

**Figure 4.** Schematic view of the expected dynamics of the temperature of the atomic gas during cavity-mediated collective laser cooling which involves a sequence of cooling stages (blue) and thermalization stages (pink). During thermalization stages, heat is transferred from the different vibrational degrees of freedoms of the atoms into a certain collective vibrational mode *B*, while the mean temperature of the atoms remains the same. During cooling stages, energy from the *B* mode into light. Eventually, the atomic gas becomes very cold.

#### **3. Thermalization of an Atomic Gas with Elastic Collisions**

Thermalization stages occur naturally in cavitating bubbles between collapse stages due to elastic collisions. As we show below, these transfer an atomic gas into its thermal state, thereby re-distributing energy between all if its vibrational degrees of freedom. During bubble expansions, the phonon frequencies of the atoms become very small. It is therefore safe to assume that the atoms do not see the cooling laser during thermalization stages.

#### *3.1. The Thermal State of a Single Harmonic Oscillator*

As in the previous section, we first consider a single trapped atom inside a harmonic trapping potential. Its thermal state equals [47]

$$\rho = \frac{1}{Z} \mathbf{e}^{-\beta H} \quad \text{with} \quad Z = \text{Tr}(\mathbf{e}^{-\beta H}) \,, \tag{19}$$

where *H* is the relevant harmonic oscillator Hamiltonian, *β* = 1/*k*B*T* is the thermal Lagrange parameter for a given temperature *T*, *k*<sup>B</sup> is Boltzmann's constant and *Z* denotes the partition function which normalizes the density matrix *ρ* of the atom. For sufficiently large atomic transition frequencies *ω*0, the thermal state of the atom is to a very good approximation given by its ground state |g, unless the atom becomes very hot. In the following, we therefore neglect its electronic degrees of freedom. Hence, the Hamiltonian *H* in Equation (19) equals

$$H\_{\perp} = -\hbar\nu \left( b^{\dagger}b + \frac{1}{2} \right) \,, \tag{20}$$

where *ν* and *b* denote again the frequency and the annihilation operator of a single phonon. Combining Equations (19) and (20), we find that [47]

$$Z\_{\perp} = \frac{\mathbf{e}^{-\frac{1}{2}\lambda}}{1 - \mathbf{e}^{-\lambda}}.\tag{21}$$

with *λ* = *βh*¯ *ν*. Here, we are especially interested in the expectation value of the thermal energy of the vibrational mode of the trapped atom which equals *H* = Tr(*Hρ*). Hence, using Equation (19), one can show that

$$\ln \langle H \rangle \quad = \quad \frac{1}{Z} \text{Tr} \left( H e^{-\beta H} \right) = -\frac{1}{Z} \frac{\partial}{\partial \beta} Z = -\frac{\partial}{\partial \beta} \ln Z \,. \tag{22}$$

Finally, combining this result with Equation (21), we find that

$$
\langle H \rangle\_{\!} = \; \hbar \nu \left( \frac{\mathbf{e}^{-\lambda}}{\mathbf{e}^{-\lambda} - 1} + \frac{1}{2} \right) \tag{23}
$$

which is Planck's expression for the average energy of a single quantum harmonic oscillator. Moreover,

$$m\_{\parallel} = \frac{\mathbf{e}^{-\lambda}}{\mathbf{e}^{-\lambda} - 1},\tag{24}$$

since the mean phonon number *<sup>m</sup>* <sup>=</sup> *b*†*b* relates to *H* via *<sup>m</sup>* <sup>=</sup> *H*/¯*h<sup>ν</sup>* <sup>−</sup> <sup>1</sup> 2 .

#### *3.2. The Thermal State of Many Atoms with Collisions*

Next we calculate the thermal state of a strongly confined atomic gas with strong elastic collisions. This situation has many similarities with the situation considered in the previous subsection. The atoms constantly collide with their respective neighbors which further increases the confinement of the individual particles. Hence, we assume in the following that the atoms no longer experience the phonon frequency *ν* but an increased phonon frequency *ν*eff. If all atoms experience approximately the same interaction, their Hamiltonian *H* equals

$$\begin{array}{rcl} H &=& \sum\_{i=1}^{N} \hbar \left( \nu\_{\rm eff} + \frac{1}{2} \right) b\_i^{\dagger} b\_i \end{array} \tag{25}$$

to a very good approximation. Here, *bi* denotes again the phonon annihilation operator of atom *i*. Comparing this Hamiltonian with the harmonic oscillator Hamiltonian in Equation (20) and substituting *H* in Equation (25) into Equation (19) to obtain the thermal state of many atoms, we find that this thermal state is simply the product of the thermal states of the individual atoms. All atoms have the same thermal state, their mean phonon number *mi* = *b*† *<sup>i</sup> bi* equals

$$m\_i \quad = \frac{\mathbf{e}^{-\lambda\_{\rm eff}}}{\mathbf{e}^{-\lambda\_{\rm eff}} - 1} \tag{26}$$

with *λ*eff = *h*¯ *ν*eff/*k*B*T*, in analogy to Equation (24). This equation shows that any previously depleted collective vibrational mode of the atoms becomes re-populated during thermalization stages.

#### **4. A Quantum Heat Exchanger with Cavitating Bubbles**

As pointed out in Section 1, the aim of this paper is to design a quantum heat exchanger for nanotechnology. The proposed experimental setup consists of a liquid on top of the device which we aim to keep cool, a transducer and a cooling laser (cf. Figure 1). The transducer generates cavitating bubbles which need to contain atomic particles and whose diameters need to change very rapidly in time. The purpose of the cooling laser is to stimulate the conversion of heat into light. The cooling of the atomic particles inside cavitating bubbles subsequently aids the cooling of the liquid which surrounds the bubbles and its environment via adiabatic heat transfers.

To gain a better understanding of the experimental setup in Figure 1, Section 4.1 describes the main characteristics of single bubble sonoluminescence experiments [32–36]. Section 4.2 emphasizes that there are many similarities between sonoluminescence and quantum optics experiments [29,30]. From this, we conclude that sonoluminescence experiments naturally provide the main ingredients for the implementation of cavity-mediated collective laser cooling of an atomic gas [26–28]. Finally, in Sections 4.3 and 4.4, we have a closer look at the physics of the proposed quantum heat exchanger and estimate cooling rates.

#### *4.1. Single Bubble Sonoluminescence Experiments*

Sonoluminescence can be defined as a phenomenon of strong light emission from collapsing bubbles in a liquid, such as water [32–34]. These bubbles need to be filled with noble gas atoms, such as nitrogen atoms, which occur naturally in air. Alternatively, the bubbles can be filled with ions from ionic liquids, molten salts, and concentrated electrolyte solutions [48]. Moreover, the bubbles need to be acoustically confined and periodically driven by ultrasonic frequencies. As a result, the bubble radius changes periodically in time, as illustrated in Figure 5. The oscillation of the bubble radius regenerates itself with unusual precision.

At the beginning of every expansion phase, the bubble oscillates about its equilibrium radius until it returns to its fastness. During this process, the bubble temperature changes adiabatically and there is an exchange of thermal energy between the atoms inside the bubble and the surrounding liquid. During the collapse phase of a typical single-bubble sonoluminescence, i.e., when the bubble reaches its minimum radius, its inside becomes thermally isolated from the surrounding environment and the atomic gas inside the bubble becomes strongly confined. Usually, a strong light flash occurs at this point which is accompanied by a sharp increase of the temperature of the particles. Experiments have shown that increasing the concentration of atoms inside the bubble increases the intensity of the emitted light [35,36].

**Figure 5.** Schematic view of the time dependence of the bubble radius in a typical single-bubble sonoluminesence experiment. Most of the time, the bubble evolves adiabatically and exchanges thermal energy with its surroundings. However, at regular time intervals, the bubble radius suddenly collapses. At this point, the bubble becomes thermally isolated. When it reaches its minimum radius, the system usually emits a strong flash of light in the optical regime.

#### *4.2. A Quantum Optics Perspective on Sonoluminescence*

The above observations suggest many similarities between sonoluminescence and quantum optics experiments with trapped atomic particles [29,30]. When the bubble reaches its minimum radius, an atomic gas becomes very strongly confined [31]. The quantum character of the atomic motion can no longer be neglected and, as in ion trap experiments (cf. Section 2.1), the presence of phonons with different trapping frequencies *ν* has to be taken into account. Moreover, when the bubble reaches its minimum radius, its surface can become opaque and almost metallic [37]. When this happens, the bubble traps light inside and closely resembles an optical cavity which can be characterized by a frequency *ω*cav and a spontaneous decay rate *κ*. Since the confined particles have atomic dipole moments, they naturally couple to the quantized electromagnetic field inside the cavity. The result can be an exchange of energy between atomic dipoles and the cavity mode. The creation of photons inside the cavity is always accompanied by a change of the vibrational states of the atoms. Hence, the subsequent spontaneous emission of light in the optical regime results in a permanent change of the temperature of the atomic particles.

A main difference between sonoluminescence and cavity-mediated collective laser cooling is the absence and presence of external laser driving (cf. Section 2.3). However, even in the absence of external laser driving, there can be a non-negligible amount of population in the excited atomic states |e. This applies, for example, if the atomic gas inside the cavitating bubble is initially prepared in the thermal equilibrium state of a finite temperature *T*. Once surrounded by an optical cavity, as it occurs during bubble collapse phases, excited atoms can return into their ground state via the creation of a cavity photon (cf. Figure 6). Suddenly, an additional de-excitation channel has become available to them. As pointed out in Refs. [29,30], the creation a cavity photons is more likely accompanied by the creation of a phonon than the annihilation of a phonon since

$$\begin{array}{rcl}B^\dagger &=& \sum\_{m=0}^\infty \sqrt{m+1} \left| m+1 \right\rangle \langle m| \, , \\ B &=& \sum\_{m=0}^\infty \sqrt{m} \left| m-1 \right\rangle \langle m| \, . \end{array} \tag{27}$$

Here, *<sup>B</sup>* and *<sup>B</sup>*† denote the relevant phonon annihilation and creation operators, while |*m* denotes a state with exactly *m* phonons. As one can see from Equation (27), the normalization factor of *<sup>B</sup>*† |*m* is slightly larger than the normalization factor of the state *<sup>B</sup>* |*m*. When the cavity photon is subsequently lost via spontaneous photon emission, the newly-created phonon remains inside the bubble. Hence, the light emission during bubble collapse phases is usually accompanied by heating, until the sonoluminescing bubble reaches an equilibrium.

During each bubble collapse phase, cavitating bubbles are thermally isolated from their surroundings. However, during the subsequent expansion phase, system parameters change adiabatically and there is a constant exchange of thermal energy between atomic gas inside the bubble and the surrounding liquid (cf. Figure 5). Eventually, the atoms reach an equilibrium between heating during bubble collapse phases and the loss of energy during subsequent expansion phases. Experiments have shown that the atomic gas in side the cavitating bubble can reaches temperature of the order of 104 K which strongly supports the hypothesis that there is a very strong coupling between the vibrational and the electronic states of the confined particles [35,36].

**Figure 6.** (**a**) From a quantum optics point of view, one of the main characteristics of sonoluminescence experiments is that cavitating bubbles provide a very strong confinement for atomic particles. This means that the quantum character of their motional degrees of freedom has to be taken into account. As in ion trap experiments, we denote the corresponding phonon frequency in this paper by *ν*. Moreover, during its collapse phase, the surface of the bubble becomes opaque and confines light, thereby forming an optical cavity with frequency *ω*cav and a spontaneous decay rate *κ*. (**b**) Even in the absence of external laser driving, some of the atoms are initially in their excited state |e due to being prepared in a thermal equilibrium state at a finite temperature *T*. When returning into their ground state via the creation of a cavity photon, which is only possible during the bubble collapse phase, most likely a phonon is created. This creation of phonons implies heating. Indeed, sonoluminescence experiments often reach relatively high temperatures [35,36].

#### *4.3. Cavity-Mediated Collective Laser Cooling of Cavitating Bubbles*

The previous subsection shows that, during each collapse phase, the dynamics of the cavitating bubbles in Figure 1 is essentially the same as the dynamics of the experimental setup in Figure 3 but with the single atom replaced by an atomic gas. When the bubble reaches its minimum diameter *d*min, it forms an optical cavity which supports a discrete set of frequencies *ω*cav,

$$
\omega\_{\text{cav}} = \beta \times \frac{\pi c}{d\_{\text{min}}},\tag{28}
$$

where *c* denotes the speed of light in air and *j* = 1, 2, ... is an integer. As illustrated in Figure 7, the case *j* = 1 corresponds to a cavity photon wavelength *λ*cav = 2*d*min. Moreover, *j* = 2 corresponds to *λ*cav = *d*min, and so on. Under realistic conditions, the cavitating bubbles are not all of the same size which is why every *j* is usually associated with a range of frequencies *ω*cav (cf. Figure 7). Here, we are especially interested in the parameter *j*, where the relevant cavity frequencies lie in the optical regime. All other parameters *j* can be neglected, once a laser field with an optical frequency *ω*<sup>L</sup> is applied, if neighboring frequency bands are sufficiently detuned.

In addition, we know that the phonon frequency *ν* of the collective phonon mode *B* assumes its maximum *ν*max during the bubble collapse phase. Suppose the cavity detuning Δcav = *ω*<sup>L</sup> − *ω*cav of the applied laser field is chosen such that

$$
\Delta\_{\text{cIV}} \sim \nu\_{\text{max}} \qquad \text{and} \qquad \nu\_{\text{max}} \ge \kappa \,, \tag{29}
$$

in analogy to Equation (2). As we have seen in Section 2.3, in this case, the two-step transition which results in the simultaneous annihilation of a phonon and the creation of a cavity photon becomes resonant and dominates the system dynamics. If the creation of a cavity photon is followed by a spontaneous emission, the previously annihilated phonon cannot be restored and is permanently lost. Overall, we expect this cooling process to be very efficient, since the atoms are strongly confined and cavity cooling rates are collectively enhanced (cf. Equation (17)).

**Figure 7.** When the cavitating bubbles inside the liquid reach their minimum diameters *d*min, their walls become opaque and trap light on the inside. To a very good approximation, they form cavities which can be described by spontaneous decay rates *κ* and cavity frequencies *ω*cav (cf. Equation (28)). Suppose the diameters of the bubbles inside the liquid occupy a relatively small range of values. Then, every integer number *j* in Equation (28) corresponds to a relatively narrow range of cavity frequencies *ω*cav. Here, we are especially interested in the parameter *j* for which the *ω*cav's lie in the optical regime. When this applies, we can apply a cooling laser with an optical frequency *ω*<sup>L</sup> which can cool the atoms in all bubbles. Some bubbles will be cooled more efficiently than others. However, as long as the relevant frequency bands are relatively narrow, none of the bubbles will be heated.

To cool not only very tiny but larger volumes, the experimental setup in Figure 1 should contain a relatively large number of cavitating bubbles. Depending on the quality of the applied transducer, the minimum diameters *d*min of these bubbles might vary in size. Consequently, the collection of bubbles supports a finite range of cavity frequencies *ω*cav (cf. Figure 7 so that it becomes impossible to realize the ideal cooling condition Δcav ∼ *ν*max in Equation (29) for all bubbles. However, as long as the frequency *ω*<sup>L</sup> of the cooling laser is smaller than all optical cavity frequencies *ω*cav, the system dynamics will be dominated by cooling and not by heating. In general, it is important that the diameters of the bubbles does not vary by too much.

Section 2.3 also shows that cavity-mediated collective laser cooling only removes thermal energy from a single collective vibrational mode *B* of the atoms. Once this mode is depleted, the cooling process stops. To efficiently cool an entire atomic gas, a mechanism is needed which rapidly re-distributes energy between different vibrational degrees of freedom, for example, via thermalization based on elastic collisions (cf. Section 3). As shown above, between cooling stages, cavitating bubbles evolve essentially adiabatically and the atoms experience strong collisions. In other words, the expansion phase of cavitating bubbles automatically implements the intermittent thermalization stages of cavity-mediated collective laser cooling.

Finally, let us point out that it does not matter whether the cooling laser is turned on or off during thermalization stages, i.e., during bubble expansion phases. As long as optical cavities only form during the bubble collapse phases, the above-described conversion of heat into light only happens when the bubble reaches its minimum diameter. The reason for this is that noble gas atoms, such as nitrogen, have very large transition frequencies *ω*0. The direct laser excitation of atomic particles is therefore relatively unlikely, even when the cooling laser is turned on. If we could excite the atoms directly by laser driving, we could cool them even more efficiently (cf. Section 2.1).

#### *4.4. Cooling of the Surroundings via Heat Transfer*

The purpose of the heat exchanger which we propose here is to constantly remove thermal energy from the liquid surrounding the cavitating bubbles and device on which the liquid is placed (cf. Figure 1). As described in the previous subsection, the atomic gas inside the bubbles is cooled by very rapidly converting heat into light during each collapse phase. In between collapse phases, the cavitating bubbles evolve adiabatically and naturally cool their immediate environment via heat transfer. As illustrated in Figure 8, alternating cooling and thermalization stages (or collapse and expansion phases) is expected to implement a quantum heat exchanger, which does not require the actual transport of particles from one place to another.

**Figure 8.** Schematic view of the expected dynamics of the temperature of a confined atomic gas during bubble collapse stages (blue) and expansion stages (pink). During expansion stages, heat is transferred from the outside into the inside of the bubble, thereby increasing the temperature of the atoms. During bubble collapse stages, heat is converted into light, thereby resulting in the cooling of the system in Figure 1. Eventually, both processes balance each other out and the temperature of the system remains constant on a coarse grained time scale.

Finally, let us have a closer look at achievable cooling rates for micro- and nanotechnology devices with length dimensions in the nano- and micrometer regime. Unfortunately, we do not know how rapidly heat can be transferred from the nanotechnology device to the liquid and from there to the atomic gas inside the cavitating bubbles. However, any thermal energy which is taken from the atoms comes eventually from the environment which we aim to cool. Suppose the relevant phonon frequencies *ν*max are sufficiently large to ensure that every emitted photon indicates the loss of one phonon, i.e., the loss of one energy quantum *h*¯ *ν*max. Moreover, suppose our quantum heat exchanger contains a certain amount of liquid, let us say water, of mass *m*water and heat capacity *c*water(*T*) at an initial temperature *T*0. Then, we can ask the question: How many photons *N*photons do we need to create in order to cool the water by a certain temperature Δ*T*?

From thermodynamics, we know that the change in the thermal energy of the water equals

$$
\Delta Q \quad = \ c\_{\text{water}}(T\_0) \, m\_{\text{water}} \Delta T \tag{30}
$$

in this case. Moreover, we know that

$$
\Delta Q \quad = \quad N\_{\text{photors}} \hbar \nu\_{\text{max}} \,. \tag{31}
$$

Hence, the number of photons that needs to be produced is given by

$$N\_{\text{photors}} = \frac{c\_{\text{water}} \left(T\_0\right) m\_{\text{water}} \Delta T}{\hbar \nu\_{\text{max}}} \,. \tag{32}$$

The time *t*cool it would take to create this number of photons equals

$$t\_{\rm cool} = \frac{N\_{\rm photons}}{N\_{\rm atoms}I} \, ^{\prime} \tag{33}$$

where *I* denotes the average single-atom photon emission rate and *N*atoms is the number of atoms involved in the cooling process. When combining the above equations, we find that the cooling rate *γ*cool = *t*cool/Δ*T* of the proposed cooling process equals

$$\gamma\_{\rm cool} = \frac{c\_{\rm water}(T) \, m\_{\rm water}}{N\_{\rm atoms} \, I \, \hbar \nu\_{\rm max}} \tag{34}$$

to a very good approximations.

As an example, suppose we want to cool one cubic micrometer of water (*V*water = 1 μm3) at room temperature (*T*<sup>0</sup> = 20 ◦C). In this case, *m*water = 10−<sup>15</sup> g and *c*water(*T*0) = 4.18 J/gK to a very good approximation. Suppose *ν* = 100 MHz (a typical frequency in ion trap experiments is *ν* = 10 MHz), *I* = 106/s and *N*atoms = 10<sup>8</sup> (a typical bubble in single bubble sonoluminescence contains about 10<sup>8</sup> atoms). Substituting these numbers into Equation (33) yields a cooling rate of

$$
\gamma\_{\rm cool} = \text{--} 3.81 \,\text{ms/K}.\tag{35}
$$

Achieving cooling rates of the order of Kelvin temperatures per millisecond seems therefore experimentally feasible. As one can see from Equation (33), to reduce cooling rates further, one can either reduce the volume that requires cooling, increase the number of atoms involved in the cooling process or increase the trapping frequency *ν*max of the atomic gas inside collapsing bubbles. All of this is, at least in principle, possible.

#### **5. Conclusions**

In this paper, we point out similarities between quantum optics experiments with strongly confined atomic particles and single bubble sonoluminescence experiments [29,30]. In both situations, interactions are present, which can be used to convert thermal energy very efficiently into light. When applying an external cooling laser to cavitating bubbles, as illustrated in Figure 1, we therefore expect a rapid transfer of heat into light which can eventually result in the cooling of relatively small devices. Our estimates show that it might be possible to achieve cooling rates of the order of milliseconds per Kelvin temperatures for cubic micrometers of water. The proposed quantum heat exchanger is expected to find applications in research experiments and in micro- and nanotechnology. A closely related cooling technique, namely laser cooling of individually trapped ions, already has a wide range of applications in quantum technology [9–16].

**Author Contributions:** A.A. and A.B. are quantum opticians and contributed the design of the proposed quantum heat exchanger and its formal analysis. S.A.P. helped to improve the initial design and provided additional insight into micro- and nanoscale physics with cavitating bubbles. All authors have read and agreed to the published version of the manuscript.

**Funding:** A.A. acknowledges funding from The Ministry of Education in Saudi Arabia. In addition, this work was supported by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) from the Oxford Quantum Technology Hub NQIT (grant number EP/M013243/1). Statement of compliance with EPSRC policy framework on research data: This publication is theoretical work that does not require supporting research data.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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