Effects of Noise-Induced Coherence on the Performance of Quantum Absorption Refrigerators

Abstract

We study two models of quantum absorption refrigerators with the main focus on discerning the role of noise-induced coherence on their thermodynamic performance. Analogously to the previous studies on quantum heat engines, we find the increase in the cooling power due to the mechanism of noise-induced coherence. We formulate conditions imposed on the microscopic parameters of the models under which they can be equivalently described by classical stochastic processes and compare the performance of the two classes of fridges (effectively classical vs. truly quantum). We find that the enhanced performance is observed already for the effectively classical systems, with no significant qualitative change in the quantum cases, which suggests that the noise-induced-coherence-enhancement mechanism is caused by static interference phenomena.

Leggett–Garg inequalities

If the system dynamics cannot be described by a classical stochastic process, this is revealed by the breaking of the so-called Leggett–Garg inequalities [25] for the two-time correlation function

$$\begin{aligned} C_{ij} = \left\langle Q(t_i)Q(t_j)\right\rangle \end{aligned}$$

(49)

of a bounded observable Q with possible outcomes lying in the interval $[-1,1]$, i.e., $|Q|\le 1$. Let

$$\begin{aligned} K_n = C_{21} + C_{32} + \cdots + C_{n(n-1)} - C_{n1} \end{aligned}$$

(50)

denote the n-measurement Leggett–Garg string with $t_1< t_2< \dots < t_n$. Then the following inequalities for $n\ge 1$ must be fulfilled if the process in question is a classical stochastic one:

$$\begin{aligned} - (2n+1) \le K_{2n+1} \le 2n-1, \end{aligned}$$

(51)

$$\begin{aligned} - \,2n \le K_{2(n+1)} \le 2n. \end{aligned}$$

(52)

The values of the Leggett–Garg strings in inequalities (51)–(52) depend on the state of the system at the initial time $t_1$. By choosing a quantum superposition as the initial state, these transient Leggett–Garg inequalities can be easily broken for the simplest systems. For example, considering a two-level system the maximum value of $K_3 = 1.5 > 1$ can be obtained [39].

If one chooses as the initial condition the stationary state, the inequalities (51)–(52) reduce for equidistant measurement times to the condition

$$\begin{aligned} (n-1)\left\langle Q(t)Q(0)\right\rangle - \left\langle Q((n-1)t)Q(0)\right\rangle \le n-2\ . \end{aligned}$$

(53)

Thus the breaking of the stationary Leggett–Garg inequalities (53) is proof for the fact that the system dynamics can not be described by a classical stochastic process even in the steady state into which the system eventually spontaneously relaxes regardless of the initial condition.

Fig. 7

V-type system used for testing quantumness of noise-induced coherence using the Leggett–Garg inequalities (Color figure online)

We have tested whether the dynamics of systems containing noise-induced coherence can result in breaking the inequalities (51) and (53) using the specific system depicted in Fig. 7. In this system, the transitions from the lower level to both degenerate upper levels is caused by a reservoir at temperature $T_1$. In addition to this, the transitions between the levels $\left| 2\right\rangle$ and $\left| 3\right\rangle$ can be also caused by another reservoir at temperature $T_2 \ne T_1$. We assume that the magnitudes of the dipole moment matrix elements corresponding to both transition channels equal to $\gamma$ and that the coefficient $\gamma _{12}$ which couples populations and coherences assumes its maximum value $\gamma _{12} = \gamma$. Using the procedure described around Eqs. (12)–(15), we find that this system is described by the set of dynamical equations

$$\begin{aligned} \dot{\rho }_{11}= & {} n_1 \rho _{33} - (n_1+1)\rho _{11} - (n_1+1) \rho _R , \end{aligned}$$

(54)

$$\begin{aligned} \dot{\rho }_{22}= & {} (n_1+n_2)\rho _{33} - (n_1+n_2+2)\rho _{22} - (n_1+1) \rho _R, \end{aligned}$$

(55)

$$\begin{aligned} \dot{\rho }_{33}= & {} (n_1+1)\rho _{11} + (n_1+n_2+2)\rho _{22} - (2n_1+n_2)\rho _{33} + 2(n_1+1)\rho _R, \end{aligned}$$

(56)

$$\begin{aligned} \dot{\rho }_R= & {} n_1\rho _{33} - \frac{n_1+1}{2}(\rho _{11} + \rho _{22})- \left[ 2(n_1+1) + \frac{1}{2}(n_2+1)\right] \rho _R, \end{aligned}$$

(57)

with $\dot{\rho }\equiv d\rho (t)/{\tilde{\gamma }} dt$ and the same meaning of the coefficients $n_{1,2}$ and ${\tilde{\gamma }}$ as in Eqs. (16)–(20). Note that the bath at $T_2$ which couples only with one transition channel does not increase the coupling between populations and coherences, but it causes faster decay of coherences. At long times, the system described by Eqs. (54)–(57) reaches a non-equilibrium steady state with nonzero noise-induced coherence. As can be checked by substituting parameters of the present model ($\gamma _{1h}=\gamma _{2h}=\gamma _{12h}=\gamma _{2m}=\gamma$, $\gamma _{1m}=\gamma _{12h}=0$) to Eqs. (22)–(24), the coupling to the two reservoirs is chosen in such a way that there exists no basis in the degenerate subspace $\{|1\rangle ,|2\rangle \}$ which would lead to decoupling of coherences and populations in the master equation.

Using the quantum regression theorem [40], we have calculated both the Leggett–Garg string $K_3$ and the stationary time correlation function $2\left\langle Q(t)Q(0)\right\rangle - \left\langle Q(2t)Q(0)\right\rangle$ for the set of observables of the type $Q_1 = (\cos \theta \left| 1\right\rangle + \sin {\theta } \left| 2\right\rangle )(\cos \theta \left\langle 1\right| + \sin {\theta } \left\langle 2\right| )$ and $Q_2 = |1\rangle \langle 1|+|2\rangle \langle 2|+|3\rangle \langle 3| - Q_1$ exploring a large part of the model parameter space. While the transient Leggett–Garg inequality (51) can be indeed broken by the present system if one chose a suitable initial condition, we were not able to break the stationary Leggett–Garg inequality (53). This suggests that the effect of noise-induced coherence in the steady state can be mimicked by a classical stochastic dynamics. This is actually in accord with the results of the study [16] where performance of a heat engine similar to that found in noise-induced-coherence works [6,7,8,9] has been achieved using diagonal elements of the density matrix only. For an example of a quantum engine which breaks the Legget–Garg inequalities, we refer to Ref. [32].