Long-Time Bit Storage and Retrieval without Cold Atom Technology

Abstract

We report computer studies showing how the duration of memory for storage and retrieval of a classical bit can be increased to 100 times the decay time of an isolated atom, with no use of high-tech cold-atom preparations recently developed in the light-matter field. We suggest that our low-tech procedure can greatly enlarge the number of experimenters able to enter this field. The role of symmetry in this procedure arises in a careful interplay of incoherent and coherent excitations of a large collection of “two-level” atoms, the level separation being matched by the dominant frequency of the electromagnetic fields (short pulses and continuing field) applied to the system.

1. Introduction

The subradiant regime of an ensemble of two-level atoms has been studied in [1,2,3,4,5], and the use of this system as an optical memory device has been explored inter alia in [6,7,8,9,10]. We study in this paper the state of this ensemble after excitation by a short pulse up to the edge of the subradiant regime; we call this dark state a “Nyxion” (Nyx was the goddess of the night in Greek mythology).

We emphasize that we are not trying to compete with the startling achievements in this field made possible by the controlled positioning of individual atoms in a cold-atom trap. In particular, our system does not aim at storage and retrieval of quantum information. We deal only with classical bit storage and retrieval. The value of our calculations depends on their applicability to a system under quite ordinary conditions. We hope that the results will stimulate significant work by experimenters who do not have access to the high-tech methods of cold-atom trapping and manipulation.

In a system under ordinary conditions, without cold-atom technology, a single Nyxion is stable only for a duration comparable to the longitudinal decay time ${\Gamma }_1^{-1}$ of an isolated atom. If a second Nyxion is produced by a new pulse within this duration, it can coalesce with the first to produce a flash of light; without such coalescence the system remains dark. This system can be used as an optical memory device in which the first Nyxion stores a (classical) bit and the second Nyxion reads it. However, the device as described above yields a memory lasting only for a time O(${\Gamma }_1^{-1}$).

We find, however, that the lifetime of the memory can be increased 100-fold by supplementing the Nyxion producing pulses with a pump of well-chosen constant strength (a “dc pump”).

The role of symmetry in this procedure arises in a careful interplay of incoherent and coherent excitations of a large collection of “two-level” atoms, the level separation being matched by the dominant frequency of the electromagnetic fields (short pulses and continuing field) applied to the system.

This is a theoretical paper. Our findings are based on quasinumerical simulations. Our calculations use the complete basis formed by the eigenfunctions of the Liénard–Wiechert one-dimensional Green function, details of which can be found in [11,12,13]. These eigenfunctions are found analytically; the sums over many eigenstates constitute the numerical calculation. The numerical part is by far the more computer intensive, as it must be done independently for each value of the time T during a simulation.

2. Theoretical Model (Methods)

Consider a sample of two-level atoms arranged in the so-called slab geometry, interacting with a classical electromagnetic field (see [11]). The atoms are taken as uniformly distributed in the “long” dimension (thickness of the slab) from $Z=-1$ to $Z=1$. The evolution in time T is given [11] by

$$\frac{\partial \chi (Z,T)}{\partial T}=-[{\Gamma }_2-i{\mathrm{\Omega }}_{L}n(Z,T)]\chi (Z,T)+\frac{i}{2}n(Z,T)\psi (Z,T)$$

(1)

$$\begin{array}{ccc}\hfill \frac{\partial n(Z,T)}{\partial T}& =& -i[{\chi }^{*}(Z,T)\psi (Z,T)-\chi (Z,T){\psi }^{*}(Z,T)]+{\Gamma }_1(1-n(Z,T))\hfill \\ & -& \frac{R\left(T\right)}{2}(1+n(Z,T)),\hfill \end{array}$$

(2)

$$\psi (Z,T)=i{u}_0{\int }_{-1}^{1}d{Z}^{\prime }\chi (Z^{\prime },T)exp(i{u}_0|Z-Z^{\prime }\left|\right),$$

(3)

where $\chi$ (complex) and n (real) are, respectively, the active medium polarization and the degree of excitation of the matter ($n=1$ if the atoms are all in the ground state and $n=-1$ if all in the excited state), and $\psi$ represents the normalized complex electric field.

2.1. Notation

All symbols in Equations (1)–(3) and their origin are detailed in [11]. However, in order to make this exposition as self-contained as possible, we summarize as follows the relations between the “normalized” (dimensionless) variables appearing in (1)–(3) and the physical quantities they represent:

The physical thickness of the slab is $L=2z_0$, and the position variable z has the range $-z_0\le z\le z_0$. The dimensionless variable corresponding to z is $Z=z/z_0$, which has the range $-Z_0\le Z\le Z_0$ (Similarly, $z^{\prime },Z^{\prime }$.).

The physical resonant wavelength is ${\lambda }_0$, and the corresponding wavenumber is $k_0=2\pi c/{\lambda }_0$ where c is the speed of light in vacuo. The dimensionless partner of $k_0$ is $u_0=k_0{z}_0$, so that $exp\left(i{u}_{0}Z\right)=exp\left(i{k}_{0}z\right)$.

Physical time variables $t, t_0$, etc., are converted to dimensionless times $T, T_0$, etc., by multiplication with the inverse time $C=4\pi {\mathcal{P}}^2\rho /\hslash$ where $\mathcal{P}$ is the reduced density matrix of the transition and $\rho$ is the number of atoms per volume, assumed to be constant throughout the slab.

Physical angular frequencies $\omega ,{\omega }_0$ have dimensionless partners $\mathrm{\Omega }=\omega /C,{\mathrm{\Omega }}_0={\omega }_0/C$, so that $exp(-i\mathrm{\Omega }T)=exp(-i\omega t)$, etc. Likewise, the longitudinal and transverse isolated-atom decay rates ${\gamma }_1,{\gamma }_2$ have dimensionless partners ${\Gamma }_1={\gamma }_1/C,{\Gamma }_2={\gamma }_2/C$.

The Lorentz shift in our units is ${\omega }_L=(4\pi /3){\mathcal{P}}^2\rho /\hslash =C/3$ so that its dimensionless partner is ${\mathrm{\Omega }}_L={\omega }_L/C=1/3.$

2.2. Skeleton Formulas

A “skeleton” version of (1) and (2) is

$$\begin{array}{ccc}\hfill \frac{\partial \chi (Z,T)}{\partial T}& =& +\frac{i}{2}n(Z,T)\psi (Z,T)\hfill \\ \hfill \frac{\partial n(Z,T)}{\partial T}& =& -i[{\chi }^{*}(Z,T)\psi (Z,T)-\chi (Z,T){\psi }^{*}(Z,T)]\hfill \end{array}$$

(4)

which resembles the familiar magnetic precession equation $\dot{\overrightarrow{B}}=\overrightarrow{H}\times \overrightarrow{B}$ if we regard $\Re \chi ,\Im \chi ,n/2$ as the $x, y, z$ components of the Bloch vector [14] and $\Re \psi ,\Im \psi$ as the $x, y$ components of a transverse applied magnetic field.

The skeletal version of (3) would be the same as the original:

$$\psi (Z,T)=i{u}_0{\int }_{-1}^{1}d{Z}^{\prime }\chi (Z^{\prime },T)exp(i{u}_0|Z-Z^{\prime }\left|\right).$$

(5)

in which $\psi (Z, T)$ is displayed as an instantaneous function of $\chi$ at other positions, neglecting retardation so that we write $\chi (Z^{\prime }, T)$ instead of $\chi (Z^{\prime }, T^{\prime })$. This can be justified only if the propagation time $L/c$ is too short to allow any significant evolution to take place in accordance with (1) and (2) while the light signal is traveling through the thickness of the slab. All our calculations throughout the paper are consequently contingent on this assumption.

It should be understood that the quantities $\chi ,\psi$ in (1)–(3) have the rapid temporal oscillation $exp(-i{\mathrm{\Omega }}_{0}T)=exp(-i{\omega }_{0}t)$ factored out. (This is equivalent to writing Bloch’s Equation [14] in the rotating coordinate system.) On the other hand, the rapid spatial oscillation is retained and appears in (3) and (5) through the prefactor $u_0=k_0{z}_0$ and the exponential factor $exp(i{u}_0|Z-Z^{\prime }\left|\right)=exp(i{k}_0|z-z^{\prime }\left|\right)$.

2.3. Limits on n

The reader may wonder how it is certain, from (1) and (2), that n always remains within the interval from −1 to 1. This can be reasoned out as follows.

Resolve $\chi ={\chi }_1+i{\chi }_2$ and $\psi ={\psi }_1+i{\psi }_2$. Define the “Bloch vector” $\overrightarrow{B}$ to be $({\chi }_1,{\chi }_2,n/2)$. Consider the skeletal forms (4). We wish to interpret them as $\dot{\overrightarrow{B}}=\overrightarrow{H}\times \overrightarrow{B}$. Multiplying out the expression for $dn/dT$, we find

$$(d/dT)\frac{n}{2}={\chi }_1{\psi }_2-{\chi }_2{\psi }_1$$

(6)

which is exactly in the desired form, with $\overrightarrow{H}=({\psi }_1,{\psi }_2)$ (the fact that $\overrightarrow{H}$ changes with time makes no difference to the argument). Turning to the skeletal expression for $d\chi /dT$, we have

$$\begin{array}{c}\hfill (d/dT){\chi }_1=-\frac{n}{2}{\psi }_2\\ \hfill (d/dT){\chi }_2=+\frac{n}{2}{\psi }_1\end{array}$$

(7)

which is good as far as it goes, but the full analogue to Bloch’s equation would be

$$\begin{array}{c}\hfill (d/dT){\chi }_1=-\frac{n}{2}{\psi }_2+{\chi }_2{\psi }_3\\ \hfill (d/dT){\chi }_2=+\frac{n}{2}{\psi }_1-{\chi }_1{\psi }_3\end{array}$$

(8)

and so far, we have introduced no quantity ${\psi }_3$. What should we do?

The missing component ${\psi }_3$ is in fact the large frequency $\mathrm{\Omega }$ that has not appeared because $exp(-i\mathrm{\Omega }T)$ has been factored out of $\chi$, and hence out of ${\psi }_1,{\psi }_2$, as remarked above. If we restore it, we obtain exactly the desired form (8). This shows that $\dot{\overrightarrow{B}}=\overrightarrow{H}\times \overrightarrow{B}$ is instantaneously satisfied and the skeletal forms (4) will not cause $(\chi ,n)$ to grow in magnitude. Since n starts out between −1 and 1, it will remain so. This conclusion about n remains true if the fast precession of $\chi$ and of ${\psi }_1,{\psi }_2$ is factored out as in (7).

The third component of H in NMR is usually a very strong unchanging magnetic field which creates an energy difference between the “up” and “down” states of the dipole. The corresponding quantity ${\psi }_3=\mathrm{\Omega }$ is likewise (apart from the factor ℏ) the energy difference between the two resonant levels. (Physically, it is the incorporation of the frequency $\mathrm{\Omega }$ into all of the exciting pulses that selects the resonant levels 1 and 2 and renders all other atomic levels insignificant [15,16] except for the rôle of level 3 in realizing the incoherent pump, see Section 2.4 below).

Now, we must consider the effect of the “nonskeletal” terms. In (1), we have the terms in ${\Gamma }_2$ and ${\mathrm{\Omega }}_L$. The first of these simply causes the real decay of ${\chi }_1$ and ${\chi }_2$ independently. The second is just a correction to the frequency $\mathrm{\Omega }$ that was factored out. Being imaginary, it does not change the magnitude of $\chi$.

Finally, we look at the terms in ${\Gamma }_1$ and R appearing in (2). The term in ${\Gamma }_1$ drives n toward the ground state $n=1$. However, the ground state is never quite reached under the influence of this term, because ${\Gamma }_1$ is multiplied by the factor $1-n$, which vanishes at that point. Hence, this term can cause only exponential decay toward $n=1$, not beyond it.

Likewise, we study the term in R. R is never negative because it always consists of a sum of positive quantities $\alpha$ (see below). Hence, it can only drive n toward the totally inverted state $n=-1$. However, this state is not reached because R is multiplied by $1+n$. Hence, at most, n can only approach the inverted state exponentially. (Actually, such an approach would trigger superradiance, which also would drive n away from total inversion; however, the foregoing argument shows independently that the inverted limit cannot be reached).

2.4. Pump

The pump (by which we mean the two pulses as well as the continuous (dc) pump) is described by the last term in (2), in which R is a real function of T to be chosen at the start of each simulation. This term creates only an incoherent excitation of the two-level system, since n carries no phase. The incoherent pump may be achieved by adding a third level, higher than the first two, which is driven by a resonant interaction from the first (ground) level, but which decays with an extremely fast time constant to the second (middle) level, much faster than any other time parameter in the problem. This rapid decay destroys the initial 1–3 coherence, leaving the system in an incoherent mixture of levels 1 and 2. The mixture is parametrized by the population difference n ($n=1$ for all atoms in level 1, $n=-1$ for all in level 2).

If the incoherence is perfect, the system is left in a state with $\chi =0$, from which (1)–(3) provide no escape. This standstill, however, is unstable, and it requires only a minuscule extra pulse at the 1–2 resonance frequency to activate a positive-feedback loop between the polarization and the internal field, which stabilizes at a magnitude determined by n. We do not give the stabilized value of $\chi$ analytically, as its calculation requires the nonlinear methods of [11]. Thus, the only effect of level 3 is to produce the incoherent pumping effect modulated by $R\left(T\right)$; the Equations (1)–(3) are written only in terms of levels 1 and 2 (however, see the issues raised in our Conclusion).

In Equation (1), the transverse decay rate ${\Gamma }_2$ (called ${\Gamma }_T$ in [11]) has been calculated [12,13,17] to be $2.33/4+{\Gamma }_1/2$, where the first (large) term is the normalized resonance width in a gas; this is its value for the $(J=1)\to (J=0)$ transition in Helium-like atoms, in the statistical approximation [13]. This value must be adjusted by the individual experimenter depending on the multiplicities in the atomic transition under study. For the much smaller longitudinal decay rate ${\Gamma }_1$ appearing in ${\Gamma }_2$ and in Equation (2), we take a convenient multiple .002 of ${\Gamma }_2$ that lies in the range often studied by experimenters on gas lasers at ordinary conditions. Thus, ${\Gamma }_2=1.001\times 2.33/4$. In this paper, we fix the physical slab thickness to be $L=2.25{\lambda }_0$ where ${\lambda }_0$ is the resonance wavelength between levels 1 and 2.

The function $R\left(T\right)$ is parametrized by six quantities ${\alpha }_0, {\alpha }_1, {\alpha }_2, \beta , R_0$ and $\delta$. The $\alpha$’s are each either 1 or 0 to indicate whether the corresponding input is to be applied or not. The parameter $\beta$ determines the delay between the time $T_0=5$ when the dc pump, if ${\alpha }_0=1$, and the first Nyxion, if ${\alpha }_1=1$, are turned on, and the time $\beta T_0$ when the second Nyxion, if ${\alpha }_2=1$, is turned on. (This means that the actual delay is only $(\beta -1)T_0$, but in this paper, we shall always have $\beta >>1$.) The reason for making $T_0>0$ is only to show the rapid excitation from the ground state in our graphs, for ${\alpha }_1=1$. The parameter $\delta$ controls the strength of the dc pump, when ${\alpha }_0=1$. (The purpose of the tanh function multiplying $\delta$ is explained in Section 3 below).

We take for the complete form of the R-function

$$R\left(T\right)=R_0[{\alpha }_1{sech}^2((T-T_0)/T_s)+{\alpha }_2{sech}^2((T-\beta T_0)/T_s)+{\alpha }_0\delta tanh(2T/T_0)].$$

(9)

The values $T_0=5$ and $T_s=2$ are fixed throughout the whole paper. Besides these, there are four parameters that remain fixed during a simulation as T increases from $T_0$ to its value at the end: the three $\alpha$’s and the overall coefficient $R_0$ controlling the pump strength $R\left(T\right)$. To this we may add a fifth constant $\beta$, which controls the delay of the second Nyxion pulse if ${\alpha }_2=1$, and a sixth, $\delta$, which gives the relative strength of the dc pump if ${\alpha }_0=1$.

For our slab thickness $2.25{\lambda }_0$, we find a pump strength $R_0=3.6{\Gamma }_2$ to be suitable. It is small enough so that the system remains dark after a single Nyxion generating pulse (${\alpha }_1=1$, ${\alpha }_2={\alpha }_0=0$) but large enough so that the noise coming from machine rounding for a 32-bit word can be disregarded.

For our main results, we set the relative pump strength at $\delta =0.001686$. This choice will be explained in Section 3.

3. Physics (Preliminary Results)

Suppose (${\alpha }_1=1$, ${\alpha }_2={\alpha }_0=0$) that a single pulse is applied at $T=T_0$ with no continuous (dc) pump. With $R_0=3.6$, the system is rapidly excited (Figure 1a) from the ground state $n=1$ to the superradiant threshold $n=-0.2$. (The reason that the threshold is not at $n=0$ is that when n becomes negative the transverse decay rate ${\Gamma }_2$ must be overcome to keep the system on the Bloch sphere; the laser amplification for a thin slab such as we are studying is not strong enough to do this easily.) Afterward, the system relaxes to ground (Figure 1b) at the rate ${\Gamma }_1$. (Figure 1c shows the tiny emission coming from the dark state shortly after reaching threshold).

By adding a second Nyxion pulse, also of subradiant strength in itself (Figure 2a,b) within this time (${\alpha }_1={\alpha }_2=1$, ${\alpha }_0=0$, $5\beta <1000$), we can cause the second pulse to “read” the bit that has been “stored” by the first pulse. The reading produces a superradiant flash shown in Figure 2c.

However, suppose that instead of the first pulse, we apply the dc pump at $T=T_0$ (${\alpha }_0=1$, ${\alpha }_2={\alpha }_1=0$). The dc pump is much weaker ($\delta <<1$) than the pulse would have been, but it continues to operate. (The tanh function multiplying $\delta$ in (9) serves only the purpose of softening the discontinuity which might generate irregularities in the calculations. For practical purposes, we may think of it as a step function.) Under this continuous influence, n decreases exponentially (Figure 3a) at the rate ${\Gamma }_1$ toward the threshold level −0.2, at which the dc pump is just strong enough to prevent the decay toward ground. By maintaining the dc pump, one can prolong the excited state indefinitely in the ideal slab environment.

The parameter $\delta$ has a peculiar influence on what happens if, after using the dc pump alone as in the preceding paragraph to reach the state with $n=-0.2$, we apply a “second” pulse at $5\beta$, which now may be be taken as O($100{\Gamma }_1^{-1})$ (${\alpha }_0={\alpha }_2=1$, ${\alpha }_1=0$, $5\beta$ = 100,000). If $\delta$ is sufficiently high, the second pulse may produce a flash (the “Aurora”, Figure 3b,c) although there was no input pulse at $T=T_0$.

We have found by changing $\delta$ in steps of ${10}^{-7}$ that the Aurora never takes place when $\delta <{\delta }_{crit}=0.0016866$. Therefore, we have set $\delta =0.0016860$ in our main calculations, so as not to provoke an Aurora when ${\alpha }_1=0$. (See Figure 4: ${\alpha }_0={\alpha }_2=1, {\alpha }_1=0,5\beta$ = 100,000 and $\delta =0.0016860$).

4. Delayed Bit Retrieval (Final Results and Discussion)

For our main calculation, we set all three ${\alpha }_0={\alpha }_1={\alpha }_2=1$ and $\beta$ = 20,000, $\delta =0.0016860$. The first input pulse at $T=T_0$ rapidly excites the system (Figure 5a) to $n=-0.2$. Meanwhile, the continuous (dc) pump has started to operate and maintains the excitation at this level, which is its equilibrium point. At $T=5\beta$ = 100,000, a second Nyxion pulse is applied (Figure 5b), and a flash is observed (Figure 5c), qualitatively resembling the Aurora seen (Figure 3) without an input pulse when $\delta$ is set too high.

During the long delay while the dc is maintaining the excitation at $n=-0.2$, there is no sign that the first pulse has had any effect. Everything looks just as though we had set ${\alpha }_1=0$ as in Figure 4. However, there is a hidden effect, revealed (Figure 5) when the second pulse is applied. Thus, we may consider the first pulse as the “input” of a bit which is “read” by the second pulse. In the presence of the dc pump, the delay between storing and reading is 100 times the isolated atom decay time ${\Gamma }_1^{-1}$, and the output signal (Figure 5c) is 4–5 orders of magnitude greater than that produced (too small to see in Figure 4c but shown by the underlying data from the simulation) in the absence of an input.

Thus, the simulations indicate that by including a dc pump in the function $R\left(T\right)$, a bit of information (pulse or no pulse at $T=T_0$) can be read with essentially no error after a delay as long as 100 times the isolated atom decay rate. What is more, the multiple 100 has been chosen arbitrarily; there is no reason, within the framework of our calculations, that it could not be made arbitrarily large. Perhaps the practical limit will have to do with the possibility of experimentally achieving the ideal slab picture, or with the assumption (“pressure broadening”) that ${\Gamma }_2$ is so large that all broadening can be regarded as homogeneous.

5. Conclusions

We repeat that these theoretical predictions can in principle be tested experimentally without cold-atom technology, and do not bear on the storage of quantum bits. We hope that the present paper will encourage research in classical bit storage and retrieval on the part of experimental scientists who seek to achieve a significant result with low-tech methods.

At the same time, we warn that the realization of an incoherent pump as described in Section 2 may pose challenging problems in practice. Although first observed [4] in 1985, the production of subradiant states has always gained far less attention than that of superradiant states.

Perhaps the most troubling possibility is the production of local pockets of quasi-superradiance inherited from the initial $1\to 3$ excitation. If such pockets are distributed unevenly through the lateral extent of the slab, they will destroy the crucial assumption of Section 2, that the value of $\chi$ at any time depends spatially only on Z. Perhaps this could be prevented by selecting a substance for which the $3\to 2$ transition is strongly inhomogeneously broadened (in contrast to the $2\to 1$ transition which is assumed essentially homogeneous).

On the other hand, a slight deviation from total incoherence in noninteracting atoms is probably harmless—in fact, it may obviate the requirement of a $1\to 2$ micropulse to kick $\chi$ out of the null state.

It must be admitted that our scheme appears quite naive in comparison with some recent studies of incoherent pumping [18,19]. On the other hand, we aim for less. We are not trying to use subradiant pumping to target a particular sublevel of level 2. On the contrary, we assume that ${\Gamma }_2$ is large enough so that any such sublevels will merge effectively into a single spectral line.

One can also hope that experimenters interested in our proposal will think of improvements that would never have occurred to the present authors. We can only claim that if the configuration described in (1)–(3) can be produced, our calculations indicate that some interesting long-life advances in classical bit storage and retrieval will be possible.