The Generalized Tavis—Cummings Model with Cavity Damping

Abstract

In this Communication, we consider a generalised Tavis–Cummings model when the damping process is taken into account. We show that the quantum dynamics governed by a non-Hermitian Hamiltonian is exactly solvable using the Quantum Inverse Scattering Method, and the Algebraic Bethe Ansatz. The leakage of photons is described by a Lindblad-type master equation. The non-Hermitian Hamiltonian is diagonalised by state vectors, which are elementary symmetric functions parametrised by the solutions of the Bethe equations. The time evolution of the photon annihilation operator is defined via a corresponding determinant representation.

1. Introduction

With the rapid advances both in theoretical understanding and in experimental implementation capability, quantum technology poses one of the most important and exciting challenges in modern science. Quantum dynamics in a realistic setup involving open environments, beyond unitarity, lies at the heart of any practical implementation of quantum protocols.

In this Communication, we apply a power mechanism of the Algebraic Bethe Ansatz to consider a non-Hermitian generalisation of the Tavis–Cummings model when the damping process is taken into account. We describe the diagonalisation of the non-Hermitian Hamiltonian via elementary symmetric functions (Bethe states) parametrised by the solutions of the relevant Bethe equations. The cavity problem with losses has been intensively investigated [1,2,3].

The conventional Tavis–Cummings model [4] is defined by the Hamiltonian

$${\mathbf{H}}_{TC}=\omega a^{†}a+{\omega }_0{S}^z+g(a^{†}{S}^{-}+a{S}^{+}),$$

(1)

where $\omega$ and ${\omega }_0$ are the frequencies of the cavity mode and the atomic system, and g is a coupling constant. The annihilation and creation operators of the field satisfy the Heisenberg–Weyl algebra: $[a,a^{†}]=1$. The collective N-atom Dicke operators (spin operators, for which total spin $S\le N/2$) $S^z={\sum }_{j=1}^N{\sigma }_j^z,S^{\pm }={\sum }_{j=1}^N{\sigma }_j^{\pm }$ satisfy the $su\left(2\right)$ Lie algebra

$$[S^{+},S^{-}]=2S^z,[S^z,S^{\pm }]=\pm S^{\pm }.$$

(2)

Here, ${\sigma }_i^{\pm }=\frac{1}{2}({\sigma }^x\pm i{\sigma }^y)$ and ${\sigma }_i^x,{\sigma }_i^y,{\sigma }_i^z$ are the Pauli matrices acting each on i-th site. This model becomes the JC model for $N=1$ ($S=1/2$).

The number of excitations $\mathbf{M}$ and the Casimir operator ${\mathbf{S}}^2$:

$$\mathbf{M}\equiv S^z+a^{†}a,$$

(3)

$${\mathbf{S}}^2\equiv S^{+}{S}^{-}+S^z(S^z-1),$$

(4)

are two nontrivial constants of motion $[{\mathbf{H}}_{TC},\mathbf{M}]=[{\mathbf{H}}_{TC},{\mathbf{S}}^2]=0$. In the next section, we consider a non-Hermitian extension of the Tavis–Cummings Hamiltonian, when the cavity damping is taken into account.

2. Non-Hermitian Hamiltonian

We will now consider an extension of the Tavis–Cummings Hamiltonian with a Stark shift term that describes the dependence of the atomic transitions on the photon number. The extended Hamiltonian reads

$${\mathbf{H}}_{STC}={\mathbf{H}}_{TC}+\gamma a^{†}a{S}^z,$$

(5)

To describe the time evolution of the photon annihilation operator of the model that is affected by the leakage of photons from the cavity, we can apply the quantum Langevin equations [5]:

$${\partial }_{t}a\left(t\right)=-i\left[a,{\mathbf{H}}_{STC}\right]+3\chi a.$$

(6)

In the explicit form

$${\partial }_{t}a\left(t\right)=-i\left(\tilde{\omega }a+g{S}^{-}+2ca{S}^z\right),$$

(7)

where $\chi$ is a constant, $\tilde{\omega }=\omega +i3\chi$, and $c=g^{-1}\gamma$.

The obtained Equation (7) may be represented as the quantum Heisenberg’s equation

$${\partial }_{t}a\left(t\right)=-i\left[a,{\mathbf{H}}_{LTC}\right],$$

(8)

with the Hamiltonian

$${\mathbf{H}}_{LTC}=\tilde{\omega }{a}^{†}a+{\omega }_0{S}^z+g(a^{†}{S}^{-}+a{S}^{+})+c{a}^{†}a{S}^z.$$

(9)

Since $\tilde{\omega }$ is a complex number, the introduced Hamiltonian is non-Hermitian: ${\mathbf{H}}_{LTC}\ne {\mathbf{H}}_{LTC}^{+}$.

On the other hand, we can express Equation (7) in the Lindblad form [6]. For the discussed model with the leakage at the rate $\chi$ the Lindblad master equation in the Heisenberg picture is given as

$${\partial }_{t}Q\left(t\right)=-i\left[Q,{\mathbf{H}}_{STC}\right]+\chi \left(2aQ{a}^{†}-a^{†}aQ-Q{a}^{†}a\right)$$

(10)

for any given observable operator Q. Formally substituting $Q\equiv a$ into (10), we obtain Equation (7).

To solve Hamiltonian Equation (9) for its eigenstates and eigenvalues, it is convenient to transform this in a convenient form. The number operator Equation (3) commutes with ${\mathbf{H}}_{LTC}$: $[{\mathbf{H}}_{LTC},\mathbf{M}]=0$. We can therefore set $\mathbf{H}=g^{-1}({\mathbf{H}}_{LTC}-\tilde{\omega }\mathbf{M})$ and $[{\mathbf{H}}_K,\mathbf{H}]=0$. We can then write

$$\mathbf{H}=\Delta S^z+(a^{†}{S}^{-}+a{S}^{+})+c{a}^{†}a{S}^z,$$

(11)

where $\Delta =g^{-1}({\omega }_0-\omega -i3\chi )$.

To find the eigenstates of the model, we use the approach of Refs. [7,8], developed for the Tavis–Cummings model. For the non-Hermitian case applying the Algebraic Bethe Ansatz technique (cf. Ref. [9] ), it can be proved that the right and the left M-particle state vectors of the Hamiltonian (11) have to satisfy equations:

$$\begin{array}{c}\hfill \mathbf{H}|{\mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle =E_{S,M}|{\mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle ,\\ \hfill \langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|\mathbf{H}=E_{S,M}\langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|.\end{array}$$

(12)

For the right state vectors, they satisfy

$$\begin{array}{c}\hfill {\mathbf{S}}^2|{\mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle =S(S+1){\mid \mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle ,\end{array}$$

(13)

$$\begin{array}{c}\hfill \mathbf{M}{\mid \mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle =(M-S){\mid \mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle ,\end{array}$$

(14)

and the same for the left ones

$$\begin{array}{c}\hfill \langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|{\mathbf{S}}^2=S(S+1)\langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|,\end{array}$$

(15)

$$\begin{array}{c}\hfill \langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|\mathbf{M}=(M-S)\langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)|.\end{array}$$

(16)

Here, $0\le S\le N/2$ and M ($0\le M<\infty$) is the number of excitations in the system. Sets of parameters $\{\lambda \}\equiv {\lambda }_1,{\lambda }_2,\dots ,{\lambda }_M$.

The M-particle state vectors are constructed in the usual fashion for the Quantum Inverse Scattering Method [9]. The right state vectors are of the form

$${\mid \mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle =\sum _{m=0}^M{(-1)}^{M-m}{e}_m{\mathbf{X}}^m{\mathbf{Y}}^{M-m}\mid {\Omega }_S\rangle ,$$

(17)

while the left state vectors are

$$\langle {\mathbf{\Phi }}_{S,M}\left(\{\lambda \}\right)\mid =\langle {\Omega }_S\mid \sum _{m=0}^M{(-1)}^{M-m}{e}_m{a}^m{\mathbf{Z}}^{M-m}.$$

(18)

The vacuum state $|{\Omega }_S\rangle =|0\rangle |S,-S\rangle$ ($a|0\rangle =0;S^{-}|S,-S\rangle =0$, with ${\mathbf{S}}^2|S,-S\rangle =S(S+1)|S,-S\rangle$, and $S^z|S,-S\rangle =-S|S,-S\rangle$). Operators $\mathbf{X},\mathbf{Y}$ and $\mathbf{Z}$ are equal to

$$\begin{array}{cc}\mathbf{X}=a^{†}+c{S}^{+},\hfill & \\ & \mathbf{Y}=(1+c\Delta )S^{+}-c{a}^{†}{S}^z+c^2{a}^{†}a{S}^{+},& \\ & & \hfill \mathbf{Z}=S^{-}+ca{S}^z.\end{array}$$

The introduced state vectors are elementary symmetric functions [10] of their arguments $\{\lambda \}$:

$$e_m=\sum _{i_1<i_2<\dots <i_m}{\lambda }_{i_1}{\lambda }_{i_2}\dots {\lambda }_{i_m}.$$

The state vectors are the eigenstates Equation (12) of the Hamiltonian (11) if the sets $\{\lambda \}$ are the roots of the Bethe equations, for $n=1,2,\dots ,M$,

$$(1+\Delta c-c{\lambda }_n)\frac{{\lambda }_n+cS}{{\lambda }_n-cS}=\prod _{j=1,j\ne n}^M\frac{{\lambda }_n-{\lambda }_j+c}{{\lambda }_n-{\lambda }_j-c}.$$

(19)

The spectrum of the model is given by the equation

$$E_{S,M}=\frac{S}{c}\prod _{j=1}^M\left(1-{\frac{c}{\lambda }}_j\right)-\left(S\Delta +\frac{S}{c}\right)\prod _{j=1}^M\left(1+{\frac{c}{\lambda }}_j\right).$$

(20)

Knowing the eigenstates of the Hamiltonian Equation (11), we can study the time evolution of the photon annihilation operator.

The transition element of the photon annihilation operator is defined by

$$A\equiv \langle {\mathbf{\Phi }}_{S,M-1}\left(\{\lambda \}\right)\left|a\right|{\mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle ,$$

(21)

where $\{\lambda \}$ and $\{\lambda \}$ are the solutions to the Bethe Equation (19) for M and $M-1$ particles, respectively. In Ref. [8], it was proved that the transition element A is represented in the determinant form:

$$\begin{array}{cc}\langle {\mathbf{\Phi }}_{S,M-1}\left(\{\lambda \}\right)\mid a\mid \hfill & {\mathbf{\Psi }}_{S,M}\left(\{\lambda \}\right)\rangle \\ & =\prod _{j=1}^M(cS-{\lambda }_j)\frac{{\prod }_{j=1}^{M-1}{\prod }_{\alpha =1}^M({\mu }_j-{\lambda }_{\alpha })}{{\prod }_{j>k>1}({\mu }_k-{\mu }_j){\prod }_{\alpha <\beta }({\lambda }_{\beta }-{\lambda }_{\alpha })}detT(\{\lambda \},\{\lambda \}).\hfill \end{array}$$

(22)

The entries of $M\times M$ matrix are

$$T(\left\{{\lambda }^{\prime }\right\},\{\lambda \})=\left(\begin{array}{ccccc}-1& -1& -1& \dots & -1\\ T_{21}& T_{22}& T_{23}& \dots & T_{2M}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ T_{M1}& T_{M2}& T_{M3}& \dots & T_{MM}\end{array}\right),$$

where

$$\begin{array}{ccc}\hfill T_{ab}& \equiv T({\mu }_a,{\lambda }_b)\hfill & \\ \hfill =& \frac{1}{{({\lambda }_b-{\mu }_a)}^2}[({\mu }_a-\Delta -c^{-1})({\mu }_a+cS)\prod _{j=1,j\ne b}^M\left(1-\frac{c}{{\mu }_a-{\lambda }_j}\right)\hfill \\ \hfill +& c^{-1}({\mu }_a-cS)\prod _{j=1,j\ne b}^M\left(1+\frac{c}{{\mu }_a-{\lambda }_j}\right)].\hfill \end{array}$$

(23)

From Equations (8) and (11), it follows that

$${\partial }_{t}a\left(t\right)=-i[a,g\mathbf{H}+\tilde{\omega }\mathbf{M}]=-ig[a,\mathbf{H}]-i\tilde{\omega }a.$$

(24)

This equation allows for the time evolution of the transition element to be found:

$${\partial }_{t}A\left(t\right)=-ig(E_{S,M}-E_{S,M-1})A-i\tilde{\omega }A.$$

(25)

Knowing the projection of the initial coherent state of the model on the state vectors (17) and (18), one can obtain the following answers for the correspondent dynamical correlation functions that are affected by the leakage of photons from the cavity.

3. Conclusions

In this Communication, we have described the process of diagonalisation of a non-Hermitian extension of the Tavis–Cummings Hamiltonian when the cavity damping is taken into account. The set of eigenstates and eigenvalues allows us to build the evolution operator and study the dynamics of quantum states for the non-Hermitian Tavis–Cummings model. Notice that the non-Hermitian Hamiltonian is diagonalised by the elementary symmetric functions (Bethe states), which are parametrised by the solutions of the Bethe equations. Our research paves the way to consideration of the quantum navigation problem as an important class of controlled quantum dynamics, whereby the objective is to transport one quantum state into another or to generate quantum gates.

In a forthcoming publications, we intend to tackle the problem of finding the time-optimal Hamiltonian that transports one quantum state to another under the influence of an open environment, characterised by a given set of Lindblad operators. This problem can be considered as an open counterpart to the quantum navigation problem [11,12,13].