Quantum self-contained refrigerator in terms of the cavity quantum electrodynamics in the weak internal-coupling regime

C. S. Yu; B. Q. Guo; T. Liu

doi:10.1364/OE.27.006863

1. Introduction

Quantum thermodynamics, as merging of thermodynamics and quantum mechanics, has been well developed in recent years [1, 2]. The wide and increasing interest has been paid for quantum heat engine [3–10], especially for quantum analogues of Carnot engines [11–15], whilst some other thermodynamical cycles such as Otto cycles [16–19] and Brownian motors [20] have also been covered with considerable progress. It was shown that some quantum heat engines [21–25] have the remarkable similarity to the classical engines well limited by the macroscopic thermodynamical laws and the Carnot efficiency. It provides a platform down to the quantum level to exploit the thermodynamical machines and to test the validity of the macroscopic thermodynamical laws in the small system.

How small can such a system be? The self-contained quantum refrigerator [26], as the smallest possible thermal machine, was shown to be able to reduce the complexity of a refrigerator to its extreme, and act as simple as a single logic gate [27]. It was shown that the efficiency of this refrigerator could get as close as the Carnot efficiency [28]. Recently the self-contained quantum thermal machines have also attracted wide attention in many aspects [29–41] such as the roles in the machines played by quantum information resources e.g. entanglement, quantum discord and coherence in this refrigerator [30,34,35] and a wide variety of other self-contained refrigerator and autonomous thermal engine models [42–47]. In particular, the strong internal coupling subject to the global master equation was considered [29,34,36,37,45,48–50] since the local master equation was shown to be thermodynamically inconsistent [50,51]. However, the global master equation is invalid for the case of weak internal coupling [51]. Furthermore, the master equation with cavities (except the double-cavity system) generally isn’t analytically solvable. Thus there is still no explicit evidence of the self-contained refrigerator on the cavity electrodynamics in the weak internal coupling regime.

In this paper, we consider such self-contained refrigerators with the weak internal coupling in the framework of the cavity quantum electrodynamics, which means we release the limitation of the size to the infinite dimensional Hilbert space. We design the self-contained quantum refrigerator by two models: (a) three resonant cavities interacting via a nonlinear optical (NLO) crystal and (b) two cavities simultaneously interacting with a three-level atom. The weak internal coupling means that both the interaction strength g of the three components and the leakage rates κ_i of each cavity approach to zero, but the ratio $\frac{g}{κ_{i}}$ is a non-vanishing constant. This should be distinguished from some previous similar contributions but in the different regimes [29,45,52,53] or the experimental realization [54]. We derive the master equation for these systems and develop the analytic techniques to reveal the thermal behaviors. It is found that such refrigerators can extract heat from the cold bath via the cavity or the atom in contact with the cold bath. The efficiency is completely determined by the frequencies of the two cavities in contact with the hotter baths, i.e. the intrinsic properties of the refrigerators. In particular, we demonstrate that the thermodynamical laws are obeyed even though the local master equations are employed. In addition, due to the well spatially separated components of our models, the cross coupling, namely, two or more baths simultaneously interacting with the same component can be properly avoided. Our second model could be experimentally feasible due to the well developed techniques for the cavity-atom interactions and the reservoir engineering.

2. The refrigerator based on three cavities

Our first model consists of three cavities denoted by H, R and C, respectively and in contact with three heat baths with different temperatures. The H, R, are referred to as the “hot” and the “room” heat baths with the temperature T_H, T_R and C denotes the cavity to be cooled and coupled with the “cold” bath with the temperature T_C. We suppose the three cavities interact with each other via the interaction of the type of the up and down parametric conversion. As is, in principle, sketched in Fig. 1(a), the three cavities are crossed through an NLO crystal (or other media depending on the practical experiment with the function is similar to the Beta barium borate (BBO) crystal [55]. The key role of the media is to realize the similar functions as the up and down parametric conversion among the three cavity modes). In this way, the cavities H and R associated with their contacted baths serve as a refrigerator which can cool the cavity C, i.e., guide the heat to flow out of the cold bath to the cavity C.

Fig. 1 Schematic illustration of our refrigerator made up of three cavities (a) and two cavities and one atom (b). Each cavity is labelled related to the natural reservoir which the cavity is coupled to. The corresponding leakage rate of the cavity is denoted by κ_i and the decay rates of the atom are denoted by Γ_i. The temperatures of the “hot” bath, the “room” bath and the “cold” bath are denoted by T_H, T_R and T_C, respectively. In (a), the three cavities interact with each other by an NLO crystal. In (b), the cavity H in contact with the hot bath is coupled to the transition |1〉 ↔ |2〉, the cavity R relating to the room bath is coupled to the transition |1〉 ↔ |3〉 and the atom denoted by C interacts with the cold bath.

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Now we will proceed our story in a rigorous way. If the three cavity modes do not interact with each other at the beginning, the free Hamiltonian for the whole three-cavity system is given by

H_{0} = H_{H} + H_{R} + H_{C},

where

H_{i} = ω_{i} a_{i}^{†} a_{i}

, with a_i representing the annihilators of cavity H, R and C, respectively and ω_i, i = H, R, C, denotes the frequency of cavity i. Here we set ħ = 1. Suppose every cavity be in the thermal state

τ_{i} = \frac{e^{- ω_{i} a_{i}^{†} a_{i} / k_{B} T_{i}}}{Z_{i}}

,

Z_{i} = Tr e^{- ω_{i} a_{i}^{†} a_{i} / k_{B} T_{i}}

, then the total system is in the state ϱ_H,R,C = ⊗_iτ_i, where k_B = 1 is the Boltzmann’s constant.

In order to let our model act as a refrigerator, we place an NLO crystal in point of intersection of the cavity as in Fig. 1(a). In this way, the photons in the cavity R can be regarded as “pump” beam. Most of photons will pass through the crystal directly. However, with small probability, the spontaneous parameter down-conversion and up-conversion processes will happen, which is in accordance with the law of energy conversion. So we have ω_R = ω_H + ω_C. The Hamiltonian governing the interaction of the three cavities can be given by

H_{I} = g (a_{H} a_{R}^{†} a_{C} + a_{H}^{†} a_{R} a_{C}^{†}) .

Here we say ω_i ≫ g → 0, i.e., the weak interaction between the three cavities corresponding to the small probability of conversion, such that each cavity evolves approximately governed by its free Hamiltonian.

Based on the spirit of the self-contained refrigerator, we let the cavity H be in contact with the hot bath, the cavity R interact with the room bath and the cavity C be coupled with the cold bath. It is especially noted that we suppose all the baths satisfy the canonical equilibrium distribution, i.e., the quantum states of them are described by the Gibbs states. The master equation that governs the evolution of the total system including the three cavities can be given by (cf. Appendix A)

\dot{ρ} = - i [H_{0} + H_{I}, ρ] + ℒ_{H} [ρ] + ℒ_{R} [ρ] + ℒ_{C} [ρ],

where ρ is the density matrix of the composite system of the three cavities and ℒ_j[ρ], j = H, R, C is the dissipator defined by

\begin{matrix} ℒ_{j} [ρ] = - \frac{κ_{j}}{2} {\bar{n}}_{j} (a_{j} a_{j}^{†} ρ + ρ a_{j} a_{j}^{†} - 2 a_{j}^{†} ρ a_{j}) \\ - \frac{κ_{j}}{2} ({\bar{n}}_{j} + 1) (a_{j}^{†} a_{j} ρ + ρ a_{j}^{†} a_{j} - 2 a_{j} ρ a_{j}^{†}), \end{matrix}

with κ_j representing the leakage rate, and

{\bar{n}}_{j} = \frac{1}{e^{ω_{j} / T_{j}} - 1}

denoting the average photon number of the bath corresponding to the jth cavity at frequency ω_j. From the Appendix A, one can find that, besides the Born-Markovian and the secular approximations, the above master equation is valid if the coupling between the cavities is as weak as the coupling between the cavity and the bath. It is obvious that we have to solve Eq. (3) in the steady-state case in order to reveal the thermal behavior of such a system. Next we will address this question in two cases.

Without the cold bath—As mentioned in the previous part, the cavity C initially interacts with the cold bath with the temperature T_C, so the cavity C will own the same temperature T_C when it reaches the heat equilibrium with its bath. Now let’s cut off the connection with the cavity C and its bath. The cavity C becomes isolated and its temperature T_C isn’t changed. At this moment, we switch on the interaction between the three cavities by insetting the NLO crystal. Thus the dynamics of such a system is governed by the master equation (3) with ℒ_C[ρ] = 0. Suppose the steady state is denoted by ρ^S, then ρ^S will subject to the following equation:

- i [H_{0} + H_{I}, ρ^{S}] + ℒ_{H} [ρ^{S}] + ℒ_{R} [ρ^{S}] = 0 .

From Eq. (6), one can find that if there were no the interaction term H_I, both cavities H and R would be at the thermally equilibrium states and the Gibbs state of Cavity C with any temperature

T_{C}^{*}

would be able to satisfy the equation. In order to compensate for the interaction terms H_I, one can easily find

\frac{ω_{H}}{T_{H}} - \frac{ω_{R}}{T_{R}} + \frac{ω_{C}}{T_{C}^{*}} = 0

. Thus the solution to Eq. (6) can be directly given by

ρ^{S} = e^{- \frac{H_{H}}{T_{H}}} \otimes e^{- \frac{H_{R}}{T_{R}}} \otimes e^{- \frac{H_{C}}{T_{C}^{*}}} / Z,

with Z being the partition function and

T_{C}^{*} = \frac{ω_{R} - ω_{H}}{\frac{ω_{R}}{T_{R}} - \frac{ω_{H}}{T_{H}}} .

It is apparent from Eq. (7) that after the system reaches the steady state, the cavity C is isolated again. But it is worth noting that the cavity C arrives at a new Gibbs state with a different temperature

T_{C}^{*}

. Compared with its initial temperature T_C, the cavity C can be cooled down if

T_{C}^{*} < T_{C}

which means

\frac{1 - \frac{T_{R}}{T_{H}}}{\frac{T_{R}}{T_{C}} - 1} > \frac{ω_{C}}{ω_{H}} .

This can also be realized by our properly designing the temperatures of the hot and the room baths and the frequencies of the three cavities. In addition, one can find that this condition happens to be the same as that for the original self-contained refrigerator with three qubits and form a fundamental design condition of a refrigerator.

With the cold bath— Now, we allow the cavity C to interact with its cold bath with the leakage rate κ_C. This means that the dynamic evolution of the total system is described by the entire master Eq. (3). Instead of an exact temperature for the cavity C in this case, we have to find out the heat currents between the cavities and their corresponding baths. As we know, when a system interacts with several baths through the system’s Hamiltonian H_S = H₀ + H_I and the dissipators ℒ_j, the steady-state heat current between the system and the jth bath is defined as [3, 56, 57] Q̇_j = Tr {H_Sℒ_j [ϱ]}, where ϱ denotes the steady-state solution of the dissipative dynamics. Note that Q̇_j > 0 means the heat flowing from the bath j to the system, and Q̇_j < 0 means the heat flowing from the system into the bath j. With our current system taken into account, one can easily find the heat currents related to the three baths are given by

{\dot{Q}}_{j} = κ_{j} {ω_{j} ({\bar{n}}_{j} - {〈 {\hat{n}}_{j} 〉}_{ϱ}) - {〈 H_{I} 〉}_{ϱ}}

with 〈A〉_ϱ = Tr(Aϱ), and

{\hat{n}}_{j} = a_{j}^{†} a_{j}

, j = H, R, C. So the remaining task is actually to determine the two expectation values in Eq. (10).

To do so, we would like to suppose the leakage of the cavity C is very weak, i. e., g² ≪ κ_C ≪ g, κ_i. Thus to the first order approximation with respect to κ_C, we can let the solution of the master equation (3) be

ϱ = ρ^{S} + κ_{C} γ

with Trγ = 0 and ρ^S given by Eq. (7). Substitute ϱ into Eq. (3), one can find that the evolution equation for γ is

\dot{γ} = - i [H_{0} + H_{I,} γ] + ℒ_{H} [γ] + ℒ_{R} [γ] + \frac{1}{κ_{C}} ℒ_{C} [ρ^{S}],

where ℒ_H and ℒ_R have the same form as those in Eq. (4) and

\begin{matrix} ℒ_{C} [ρ^{S}] & = & - \frac{{\bar{n}}_{C}}{2} (a_{C} a_{C}^{†} ρ^{S} + ρ^{S} a_{C} a_{C}^{†} - 2 a_{C}^{†} ρ^{S} a_{C}) \\ - \frac{({\bar{n}}_{C} + 1)}{2} (a_{C}^{†} a_{C} ρ^{S} + ρ^{S} a_{C}^{†} a_{C} - 2 a_{C} ρ^{S} a_{C}^{†}) . \end{matrix}

From Eq. (12), one can get the the evolution equation for the expectation values of the operators of our interest as

\frac{d {〈 a_{H}^{†} a_{H} 〉}_{γ}}{d t} = {〈 \hat{S} 〉}_{γ} - κ_{H} {〈 {\hat{n}}_{H} 〉}_{γ} = 0,

\frac{d {〈 a_{R}^{†} a_{R} 〉}_{γ}}{d t} = - {〈 \hat{S} 〉}_{γ} - κ_{R} {〈 {\hat{n}}_{R} 〉}_{γ} = 0,

\frac{d {〈 a_{C}^{†} a_{C} 〉}_{γ}}{d t} = {〈 \hat{S} 〉}_{γ} + {\bar{n}}_{C} - {\bar{n}}_{S} = 0,

\begin{array}{l} \frac{d {〈 a_{H}^{†} a_{R} a_{C}^{†} 〉}_{γ}}{d t} = - \frac{κ_{H} + κ_{R}}{2} {〈 a_{H}^{†} a_{R} a_{C}^{†} 〉}_{γ} \\ + ig ({〈 {\hat{n}}_{H} {\hat{n}}_{R} 〉}_{γ} + {〈 {\hat{n}}_{C} {\hat{n}}_{R} 〉}_{γ} - {〈 {\hat{n}}_{H} {\hat{n}}_{C} 〉}_{γ} + {〈 {\hat{n}}_{R} 〉}_{γ}) = 0, \end{array}

where the operator Ŝ is defined by

\hat{S} = ig (a_{H} a_{R}^{†} a_{C} - a_{H}^{†} a_{R} a_{C}^{†}),

and we use

Tr (a_{H}^{†} a_{R} a_{C}^{†} ρ^{S}) = 0

and

{\bar{n}}_{S} = Tr ({\hat{n}}_{C} ρ^{S}) = \frac{{\bar{n}}_{R} (1 + {\bar{n}}_{H})}{{\bar{n}}_{H} - {\bar{n}}_{R}}

. Solving Eqs. (14)–(17), one can immediately find

{〈 {\hat{n}}_{H} 〉}_{γ} = \frac{{\bar{n}}_{C} - {\bar{n}}_{S}}{κ_{H}},

{〈 {\hat{n}}_{R} 〉}_{γ} = \frac{{\bar{n}}_{C} - {\bar{n}}_{S}}{κ_{R}},

{〈 H_{I} 〉}_{γ} = 0 .

Thus the heat current Q̇_H in such a weak leakage regime, can be directly calculated in terms of Eqs. (19)–(21) as

\begin{matrix} {\dot{Q}}_{H} & = & κ_{H} ω_{H} ({\bar{n}}_{H} - {〈 {\hat{n}}_{H} 〉}_{ρ^{S}} - κ_{C} {〈 {\hat{n}}_{H} 〉}_{γ}) \\ - & κ_{H} ({〈 H_{I} 〉}_{ρ^{S}} + κ_{C} {〈 H_{I} 〉}_{γ}) \\ = & ω_{H} κ_{C} ({\bar{n}}_{C} - {\bar{n}}_{S}), \end{matrix}

where we use the fact 〈H_I〉_ρ^S = 0 and 〈n̂_H〉 _ρ^S = n̄_H. Similary, we have

{\dot{Q}}_{R} = - ω_{R} κ_{C} ({\bar{n}}_{C} - {\bar{n}}_{S})

and

\begin{array}{l} {\dot{Q}}_{C} & = & κ_{C} {ω_{C} ({\bar{n}}_{C} - {\bar{n}}_{S} - κ_{C} {〈 {\hat{n}}_{C} 〉}_{γ}) \\ = & ω_{C} κ_{C} ({\bar{n}}_{C} - {\bar{n}}_{S}), \end{array}

where we neglect the term proportional to

κ_{C}^{2}

. We plot the approximate analytic heat currents and the numerical results based on Eq. (3) in Fig. 2, which shows the good consistency.

Fig. 2 The heat currents or the entropy production rate vs the temperature T_H. Q_H (the red dashed line), Q_R (the yellow solid line) and Q_C (the blue dash-dotted line) are obtained directly by the numerical solutions of Eq. (3) and the ’x’ labeled points correspond to Eqs. (22), (23) and (24). Here we set ω_R = 75, ω_H = 30, T_R = 20, T_C = 15, g = κ_H = κ_R = 10⁻⁴ and κ_C = 10⁻⁷ in (a), but κ_C = 10⁻⁴ in (b) and (c). To numerically solve the master Eq. (3), the photon numbers are truncated at 2, 3, 4, respectively for Cavities C, R and H.

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Equation (24) shows that if Q̇_C > 0, i.e., n̄_C > n̄_S, the heat will flow out of the cold bath to the cavity C, namely, the bath C will be cooled down. One can find that n̄_C > n̄_S is equivalent to $\frac{1 - \frac{T_{R}}{T_{H}}}{\frac{T_{R}}{T_{C}} - 1} > \frac{ω_{C}}{ω_{H}}$ which in fact is the same as Eq. (9). In this sense, so long as we can carefully select the frequencies of the cavities and the temperatures of the baths. This requirement can be easily satisfied. That is, our current system can safely work as a valid refrigerator. In other words, $T_{C}^{*} = \frac{ω_{C}}{\frac{ω_{R}}{T_{R}} - \frac{ω_{H}}{T_{H}}}$ (i.e., Eq. (8)) forms the critical temperature (the lower bound) which limits the cooling ability of the self-contained refrigerator once the frequencies of the cavities and the temperature T_H and T_R are fixed. To further find out the cooling ability, we would like to focus the efficiency of this refrigerator. In fact, the working essence of this refrigerator system can be understood as the refrigerator extracts the heat from the cold bath by consuming the heat of the hot bath. Thus the efficiency can be naturally given by

η = \frac{{\dot{Q}}_{C}}{{\dot{Q}}_{H}} = \frac{ω_{C}}{ω_{H}} = \frac{ω_{R}}{ω_{H}} - 1,

which is only determined by the frequencies of the cavities (i.e. the intrinsic properties of the refrigerator) and again the same as the efficiency of the original self-contained refrigerator using three qubits. In addition, Eq. (9) is also an upper bound on the efficiency η. One can find that this upper bound just equals the Carnot efficiency given by [28].

Even though the local master equation could lead to the thermodynamical inconsistency [51]. One will find that in our case, the thermodynamical laws are actually obeyed to some good approximations. From the heat currents given in Eqs. (22),(23) and (24), one can easily find that with the current approximation,

{\dot{Q}}_{C} + {\dot{Q}}_{R} + {\dot{Q}}_{H} = 0

which indicates that the total heat flowing into the three-cavity system just equals the total heat flowing out of the system. This implies that the current thermodynamics satisfies the law of the conservation of energy. In addition, we would also like to say that this cooling process obeys the second law of thermodynamics. To see this, let’s consider the entropy production rate in our system. It is shown in [58] that the second law requires the non-equilibrium thermodynamics obeying

σ = \frac{d S}{d t} + J > 0,

where S is the von Neumann entropy of the system and J is defined by

J = - \frac{1}{T} {\frac{d}{d t} |}_{diss} Tr {H_{S} ϱ} = - \frac{1}{T} Tr {H_{S} ℒ (ϱ)}

denoting the entropy flux due to the changes of the system’s internal energy resulting from the dissipation ℒ(·) with the subscript corresponding to the entropy change due to the dissipative effects. Since ϱ is the steady state which is independent of time, so

\frac{d S}{d t} = 0

and

Tr {H_{S} ℒ (ϱ)} = \sum_{j} \frac{{\dot{Q}}_{j}}{T_{j}}

. Considering the heat currents Eq. (22), Eq. (23), and Eq. (24), one can easily obtain that

σ = \frac{{\dot{Q}}_{H}}{T_{H}} - \frac{{\dot{Q}}_{C}}{T_{C}} - \frac{{\dot{Q}}_{R}}{T_{R}} > 0

, which coincides with the second law of thermodynamics. Finally, one can also check that the system to the g order can be well thermalized if the bath C doesn’t exist.

In order to give an illustration of the quantum refrigerator in a general case, i.e., g ∼ κ_i, which hasn’t an analytic way up to now, we only numerically show the heat currents in Fig. 2(b) and the entropy rate σ in Fig. 2(c). From Fig. 2(b), one can see that the thermodynamic behaviors are quite similar to [26,36] and Fig. 2(c) shows the second law isn’t violated either.

3. The refrigerator based on two cavities and one atom

Our second model is made up of two cavities and a three-level atom in contact with the cold bath. The setup is sketched in Fig. 1(b). Similar to our first model, considering the “hot”, “room” and “cold” baths, we would like to, respectively, denote the two cavities by H and R and the atom by C if it does not cause confusion with our first model. The free Hamiltonian of the system reads

H_{0} = \sum_{i = 1}^{3} E_{i} | i 〉 〈 i | + \sum_{k = H, R} ω_{k} a_{k}^{†} a_{k}

with E_i corresponding to the energy of each energy level |i〉, i = 1, 2, 3, and ω_k denoting the frequency of the cavity. We suppose that the cavity mode a_H is resonantly coupled with the transition |1〉 ↔ |2〉 and the cavity mode a_R is resonantly coupled with the transition |1〉 ↔ |3〉, so we have ω_H = E₂ − E₁, ω_R = E₃ − E₁. In addition, we let ω_C = E₃ − E₂ for convenience. Thus the interaction Hamiltonian between the two cavities and the atom can be given by

H_{I} = g (σ_{R}^{-} a_{R}^{†} + σ_{R}^{+} a_{R}) + g (σ_{H}^{-} a_{H}^{†} + σ_{H}^{+} a_{H}),

where

σ_{R}^{-} = | 1 〉 〈 3 |

,

σ_{H}^{-} = | 1 〉 〈 2 |

and

σ_{k}^{+} = {(σ_{k}^{-})}^{†}

and we set the same coupling strength for simplicity. We also define

σ_{C}^{-} = | 2 〉 〈 3 |

. Considering the weak internal coupling as our first model with the total Hamiltonian H_S = H₀ + H_I, we can also obtain the master equation based on the Appendix A as

\dot{ρ} = - i [H_{S}, ρ] + ℒ_{C} (ρ) + ℒ_{R} (ρ) + ℒ_{H} (ρ)

where

\begin{matrix} ℒ_{C} (ρ) = - \sum_{k = C, H, R} {\frac{Γ_{k}}{2} {\tilde{n}}_{k} (σ_{k}^{-} σ_{k}^{+} ρ + ρ σ_{k}^{-} σ_{k}^{+} - 2 σ_{k}^{+} ρ σ_{k}^{-}) \\ + \frac{Γ_{k}}{2} ({\tilde{n}}_{k} + 1) (σ_{k}^{+} σ_{k}^{-} ρ + ρ σ_{k}^{+} σ_{k}^{-} - 2 σ_{k}^{-} ρ σ_{k}^{+})}, \end{matrix}

with

{\tilde{n}}_{k} = \frac{1}{e^{\frac{ω_{k}}{T_{C}}} - 1}

which should be distinguished from

{\tilde{n}}_{k} = \frac{1}{e^{\frac{ω_{k}}{T_{k}}} - 1}

. Furthermore, ℒ_R(ρ) ℒ_H (ρ) and defined as the same form as Eq. (4). Although the qutrit is also used as that in [26], our model can well separate the three different baths and the interaction can be better established between the atom and the cavities. In addition, such a model can also be used to demonstrate the effective and analytic approximation method as mentioned above.

Analogously, we will also proceed in two cases. If no cold bath interacts with the atom, which means ℒ_C (ρ) = 0. In this case, one can easily find that the steady-state solution to the master equation reads

ρ^{S} = \frac{e^{- \frac{H_{H}}{T_{H}}}}{Z_{H}} \otimes \frac{e^{- \frac{H_{R}}{T_{R}}}}{Z_{R}} \otimes \sum_{i = 1}^{3} P_{i} | i 〉 〈 i |,

where

Z_{A} P_{3} = \frac{{\bar{n}}_{R}}{{\bar{n}}_{R} + 1}

,

Z_{A} P_{2} = \frac{{\bar{n}}_{H}}{{\bar{n}}_{H} + 1}

, Z_AP₁ = 1 with

Z_{A} = (1 + \frac{{\bar{n}}_{R}}{{\bar{n}}_{R} + 1} + \frac{{\bar{n}}_{H}}{{\bar{n}}_{H} + 1})

and Z_H\R is again supposed to be the partition function (normalization factor). One will see that ρ^S plays the key role in the following analytic procedure with the presence of the cold bath.

Now let’s turn to the case with the cold bath present. In order to give an analytic understanding of the cooling mechanism, we suppose that the cold bath is so weakly coupled with the atom C only by the transition |3〉 ↔ |2〉 with the spontaneous decay rates Γ_C ≪ 1. In the meanwhile, the two cavities are supposed to have the same leakage rates, i.e., κ_H = κ_R. Thus the master equation has the full form as Eq. (31) with Γ_H\R = 0. Similar to the first model, in order to solve the heat currents, we can expand the steady state ϱ to the first order of Γ_C as ϱ = ρ^S + Γ_Cγ with ρ^S given in Eq. (33) and γ denoting a traceless Hermitian matrix. Thus one can obtain an evolution equation for γ with the same form as Eq. (12) if replacing κ_C by Γ_C. This concrete expression of this equation is given by Eq. (43) in Appendix B. Considering the very weak decay rate Γ_C, omitting the terms with $Γ_{C}^{2}$ will lead to the heat currents subject to different thermal baths as

{\dot{Q}}_{C} = ω_{C} Γ_{C} [{\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3}],

{\dot{Q}}_{H} = \frac{ω_{H}}{ω_{C}} {\dot{Q}}_{C}, {\dot{Q}}_{R} = - \frac{ω_{R}}{ω_{C}} {\dot{Q}}_{C},

where all the details of the derivation can be found in the Appendix B. The comparison between the approximate analytic heat currents and the numerical results based on Eq. (31) in Fig. 3(a), which again shows the good consistency. It is obvious that Q̇_C > 0 in Eq. (34), which is just equivalent to Eq. (9) and shows that the heat is flowing out of the cold bath. This is what we expect initially. In addition, all the heat currents have the same form as those of our first model, so all the same conclusions can be drawn parallel with the first model. Here we won’t repeat the analysis again.

Fig. 3 The heat currents vs the temperature T_H. Q_H (the red dashed line), Q_R (the yellow solid line) and Q_C (the blue dash-dotted line) are obtained directly by the numerical solutions of Eq. (31) with Γ_H = Γ_R = 0 in (a) and with Γ_H = Γ_R = 0.2 × Γ_C in (b). The ’x’ labeled points correspond to Eqs. (34) and (35) in (a) and Eqs. (37), (36) and (38) in (b). The other parameters in both figures are taken as ω_R = 75, ω_H = 45, T_R = 40, T_C = 35, g = κ_H = κ_R = 10⁻⁵ and Γ_C = 5 × 10⁻⁸. The photon numbers in the numerical procedure are truncated at 3 for both cavities R and H.

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Now we would like to consider how the imperfect coupling affects the cooling, namely, we will study a more general case with the cold bath weakly coupled to all the transitions in the three-level atom. That is, we are restricted to the conditions κ_H = κ_R and Γ_k = λ_kΓ_C ≠ 0 with Γ_C ≪ 1. Here λ_k is a constant and k = C, R, H. Following the same procedure given in Appendix B, one will find the heat currents can be given by

\begin{array}{l} {\dot{Q}}_{C}^{'} = ω_{C} Γ_{C} ({\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3}) \\ + ω_{H} Γ_{H} ({\tilde{n}}_{H} P_{1} - ({\tilde{n}}_{H} + 1) P_{2}) \\ + ω_{R} Γ_{R} ({\tilde{n}}_{R} P_{1} - ({\tilde{n}}_{R} + 1) P_{3}), \end{array}

\begin{array}{l} {\dot{Q}}_{R}^{'} = - ω_{R} Γ_{C} ({\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3}) \\ - ω_{R} Γ_{R} ({\tilde{n}}_{R} P_{1} - ({\tilde{n}}_{R} + 1) P_{3}), \end{array}

\begin{array}{l} {\dot{Q}}_{H}^{'} = ω_{H} Γ_{C} ({\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3}) \\ - ω_{H} Γ_{H} ({\tilde{n}}_{H} P_{1} - ({\tilde{n}}_{H} + 1) P_{2}) . \end{array}

An intuitive comparison between Eqs. (36), (37) and (38) and the numerical solutions of Eq. (31) is given in Fig. 3(b), which shows the validity of our analytic approximation. It is obvious that Q̇′_k = Q̇_k for Γ_R = Γ_H = 0, which shows that Eqs. (34) and (35) are just the special case of Eqs. (36),(37) and (38). However, one can immediately find that ñ_H P₁ − (ñ_H + 1)P₂ < 0 and ñ_R P₁ − (ñ_R + 1)P₃ < 0 (which actually corresponds to

\frac{ω_{H \ R}}{T_{C}} > \frac{ω_{H \ R}}{T_{H \ R}}

, so Q̇′_C in this case is always less than Q̇_C given in Eq. (35) and the cooling ability is greatly reduced, which can be explicitly found by comparing Fig. 3(a) with Fig. 3(b). In particular, Q̇_C could become negative with the bad design (e.g. the last two terms in Q̇′_C play the dominant role), which means that the heat isn’t extracted from the cold bath, on the contrary, it is transferred to the cold bath. Such a phenomenon is illustrated in Fig. 4(a) for the large ω_R.

We also consider the general case with g ∼ Γ_C. The heat currents and the entropy production rate are plotted in Fig. 4(b) and 4(c), which again shows the quantum self-contained refrigerator can be established similar to the original one, and at the same time, the second law of the thermodynamics is not violated. In addition, we also check many cases with the non-vanishing Γ_H and Γ_R, the figures aren’t given for concision. Although the heat could fail to be extracted from the cold bath with the bad design as mentioned above, the second law is always satisfied.

Fig. 4 The heat currents vs ω_R in (a) and vs the temperature T_H in (b), and the entropy production rate σ vs T_H in (c). Q_H (the red dashed line), Q_R (the yellow solid line) and Q_C (the blue dash-dotted line) are obtained directly by the numerical solutions of Eq. (31) and the ’x’ labeled points correspond to Eqs. (37), (36) and (38). In (a), we set ω_H = 40, T_H = 90 and all the other parameters are taken the same as Fig. 3(b). In (b) and (c), we let Γ_C = 10⁻⁵ and all the other parameters are taken the same as Fig. 3(a).

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4. Discussion and conclusion

Before the end, we would like to emphasize that the NLO crystal has been widely and successfully used in the optical experiments and the atom can be well trapped inside a cavity. For some basic order-of-magnitude estimates on experiments, one can use the optical cavities with the frequency of the order 10¹⁴Hz and the quality factor larger than 10⁶ [59, 60]. The temperature of the baths are of the order 10³K which can be achieved by the sunlight or the spectral lamp [61,62]. Although our first model is reasonable in theory and the setups can be easily built in the practical scenario, the nonlinear response of an NLO crystal such as BBO [63] that enables the effective three-body interaction usually requires the intense laser field, which implies that our first model could be hard to implement in the current experiment. It could be feasible if the weak-light nonlinearity could be successfully induced with the future advance or the similar functions to the NLO crystal (or even the total setup of our first model) could be replaced by some other media such as the artificial atoms and so on. In fact, here we just borrow such a setup to show the interaction mechanism of three boson modes. One can use different system to mimic this interaction such as the recent experiment [54] where the similar working regime is considered since the weak or strong internal coupling is relative to the system frequencies. In contrast, our second model could be much more feasible due to well-developed techniques of the atom-cavity coupling in experiment. We would like to emphasize that the three-level atom was also employed in [22, 26]. For example, one can use cesium atomic levels 6S1/2, 6P3/2 and 6D5/2 as the trapped qutrit [64], and the optical cavities and the baths can also be chosen the same as the first model. The atom-cavity coupling strength is of the order 102MHz and of the same order of the decay/leakage rate [59,60]. One can also use the LC resonators and select some artificial atoms in order to work in the low-temperature regimes [60]. The reservoir engineering allowing to mimic various baths and to select the particular coupling has been used in many cases [22,65–70]. The cooling effects can be easily detected by the average photon number or the population of the atom.

In summary, we have presented two models to realize the self-contained quantum refrigerator based on the cavity quantum electrodynamics. The analytic procedure is developed to reveal the cooling mechanism. It is shown that both models serve as a quantum self-contained refrigerator in the same way as the original self-contained refrigerator even though it works in the regime of infinite degrees of freedom. The enabled interaction between the three components and the well separated energy levels make our models friendly for the practical experiment.

A. The weak internal coupling master equation

We would like to give a brief introduction to how to derive the master equation Eq. (3). Let’s begin with the unitary evolution of the whole system plus environment. The total interaction Hamiltonian is given by H_T (t) = H_I (t) + H_SB(t) with H_I denoting the internal interaction in the system and H_SB representing the interaction between the system and the environments. Thus the evolution of the total system is governed by the von Neumann equation as

\begin{array}{l} \frac{d ρ (t)}{d t} & = & - i [H_{T} (t), ρ (t)] \\ = & - i [H_{I} (t), ρ (t)] - i [H_{S B} (t), ρ (t)] \end{array}

with ρ(t) denoting the density matrix of the whole system. This equation can be formally solved by the integral as

ρ (t) = ρ (0) - i \int_{0}^{t} d s [H_{T} (s), ρ (s)] .

Substitute Eq. (40) into Eq. (39), one will arrive at

\begin{matrix} \frac{d ρ (t)}{d t} = - i [H_{I} (t), ρ (t)] - i [H_{S B} (t), ρ (0)] \\ - \int_{0}^{t} d s [H_{S B} (t), [H_{T} (s), ρ (s)]] . \end{matrix}

Considering the Born approximation which assumes that the weak couplings between the system and the environments don’t influence the state of the environment, we have that the state of the total system can be approximately expressed by ρ(t) = ρ_S(t) ⊗ ρ_B with ρ_B being the Gibbs state of the environments. Thus one can easily get the reasonable assumptions Tr_B [H_SB, ρ(0)] = 0 and Tr_B [H_SB, [H_I (s), ρ(s)]] = 0. Hence Eq. (41) becomes

\begin{matrix} \frac{d ρ_{S} (t)}{d t} = - i [H_{I} (t), ρ_{S} (t)] \\ - \int_{0}^{t} d s [H_{S B} (t), [H_{S B} (s), ρ_{S} (s)]] . \end{matrix}

Following the standard procedure (SP) given in [58] with the Markovian approximation and secular approximation, one will arrive at the master Eq. (3). Compared with the SP in [58], it is implied that the eigenfrequencies of the system in Eq. (42) are the frequencies ω_j which just corresponds to the free Hamiltonian of the system of interest. So a good secular approximation requires that the interaction strength of H_I(t) should be much less than |ω_j − ω_k| for j ≠ k. That is, the master equation given in Eq. (3) is usually thought to be valid for the weak internal coupling regime.

B. The heat currents with the cold bath in the second model

Under the condition, Γ_C ≪ 1, we can expand the steady-state density matrix ϱ = ρ^S + Γ_Cγ with ρ^S given in Eq. (33). Similarly, we can also find the equation governing the evolution of the matrix γ as

\dot{γ} = - i [H, γ] + \frac{1}{Γ_{C}} ℒ_{C} (ρ^{S}) + ℒ_{R} (γ) + ℒ_{H} (γ)

with

\begin{matrix} \frac{ℒ_{C} (ρ^{S})}{Γ_{C}} = - \sum_{k = C, R, H} {\frac{λ_{k} {\tilde{n}}_{k}}{2} (σ_{k}^{-} σ_{k}^{+} ρ^{S} \\ {+ ρ^{S} σ_{k}^{-} σ_{k}^{+} - 2 σ_{k}^{+} ρ^{S} σ)}_{k}^{-} + \frac{λ_{k} ({\tilde{n}}_{k} + 1)}{2} \\ \times (σ_{k}^{+} σ_{k}^{-} ρ^{S} + ρ^{S} σ_{k}^{+} σ_{k}^{-} - 2 σ_{k}^{-} ρ^{S} σ_{k}^{+})} . \end{matrix}

Here, we assume κ_R = κ_H. Based on the above equations, one can obtain the evolution equations for the average values of all the operators of interest. So we have

\begin{array}{l} \frac{d {〈 | 1 〉 〈 1 | 〉}_{γ}}{d t} = - {〈 {\tilde{S}}_{R} 〉}_{γ} - λ_{R} {\tilde{n}}_{R} {〈 ρ_{S} 〉}_{11} + λ_{R} ({\tilde{n}}_{R} + 1) {〈 ρ_{S} 〉}_{33} \\ - {〈 {\tilde{S}}_{H} 〉}_{γ} - λ_{H} {\tilde{n}}_{H} {〈 ρ_{S} 〉}_{11} + λ_{H} ({\tilde{n}}_{H} + 1) {〈 ρ_{S} 〉}_{22}, \end{array}

\begin{matrix} \frac{d {〈 | 2 〉 〈 2 | 〉}_{γ}}{d t} & = & {〈 {\tilde{S}}_{H} 〉}_{γ} - {\tilde{n}}_{C} {〈 ρ_{S} 〉}_{22} + ({\tilde{n}}_{C} + 1) {〈 ρ_{S} 〉}_{33} \\ + λ_{H} {\tilde{n}}_{H} {〈 ρ_{S} 〉}_{11} - λ_{H} ({\tilde{n}}_{H} + 1) {〈 ρ_{S} 〉}_{22}, \end{matrix}

\begin{array}{l} \frac{d {〈 | 3 〉 〈 3 | 〉}_{γ}}{d t} = {〈 {\tilde{S}}_{R} 〉}_{γ} + {\tilde{n}}_{C} {〈 ρ_{S} 〉}_{22} - ({\tilde{n}}_{C} + 1) {〈 ρ_{S} 〉}_{33} \\ + λ_{R} {\tilde{n}}_{R} {〈 ρ_{S} 〉}_{11} - λ_{R} ({\tilde{n}}_{R} + 1) {〈 ρ_{S} 〉}_{33}, \end{array}

\frac{d {〈 a_{i}^{†} a_{i} 〉}_{γ}}{d t} = - {〈 {\tilde{S}}_{i} 〉}_{γ} - κ_{i} {〈 a_{i}^{†} a_{i} 〉}_{γ}, i = H, R,

\begin{array}{l} \frac{d {〈 a_{R} | 3 〉 〈 1 | 〉}_{γ}}{d t} = - \frac{κ_{R}}{2} {〈 a_{R} | 3 〉 〈 1 | 〉}_{γ} \\ - i g ({〈 a_{R} a_{R}^{†} | 3 〉 〈 3 | 〉}_{γ} - {〈 a_{R}^{†} a_{R} | 1 〉 〈 1 | 〉}_{γ} + {〈 a_{R} a_{H}^{†} | 3 〉 〈 2 | 〉}_{γ}), \end{array}

\begin{array}{l} \frac{d {〈 a_{H} | 3 〉 〈 1 | 〉}_{γ}}{d t} = - \frac{κ_{H}}{2} {〈 a_{H} | 2 〉 〈 1 | 〉}_{γ} \\ - i g ({〈 a_{H} a_{H}^{†} | 2 〉 〈 2 | 〉}_{γ} - {〈 a_{H}^{†} a_{H} | 1 〉 〈 1 | 〉}_{γ} + {〈 a_{H} a_{R}^{†} | 2 〉 〈 3 | 〉}_{γ}), \end{array}

where

{〈 {\tilde{S}}_{R} 〉}_{γ} = Tr {i (a_{R}^{†} | 1 〉 〈 3 | - a_{R} | 3 〉 〈 1 |) γ}

,

{〈 {\tilde{S}}_{H} 〉}_{γ} = Tr {i (a_{H}^{†} | 1 〉 〈 2 | - a_{H} | 2 〉 〈 1 |) γ}

and 〈ρ_S〉_kk =Tr{ρ_S |k 〉〈k|}. The steady-state solutions is hence obtained as

\begin{array}{l} {〈 {\tilde{S}}_{H} 〉}_{γ} = {{\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3} \\ - λ_{H} ({\tilde{n}}_{H} P_{1} - ({\tilde{n}}_{H} + 1) P_{2})}, \end{array}

\begin{array}{l} {〈 {\tilde{S}}_{R} 〉}_{γ} = - {{\tilde{n}}_{C} P_{2} - ({\tilde{n}}_{C} + 1) P_{3}} \\ - λ_{R} ({\tilde{n}}_{R} P_{1} - ({\tilde{n}}_{R} + 1) P_{3})}, \end{array}

κ_{i} {〈 a_{i}^{†} a_{i} 〉}_{γ} = - {〈 {\tilde{S}}_{i} 〉}_{γ}, i = H, R,

{〈 H_{I} 〉}_{γ} = 0 .

Thus, neglecting the terms

Γ_{C}^{2}

, one can find that the heat currents are given by

\begin{array}{l} {\dot{Q}}_{C}^{'} = Tr H_{S} ℒ_{C} (ρ^{S}) = \\ - Γ_{C} (ω_{H} {〈 {\tilde{S}}_{H} 〉}_{γ} + ω_{R} {〈 {\tilde{S}}_{R} 〉}_{γ}), \end{array}

{\dot{Q}}_{H}^{'} = Γ_{C} Tr H_{S} ℒ_{H} (γ) = ω_{H} Γ_{C} {〈 {\tilde{S}}_{H} 〉}_{γ},

{\dot{Q}}_{R}^{'} = Γ_{C} Tr H_{S} ℒ_{R} (γ) = ω_{R} Γ_{C} {〈 {\tilde{S}}_{R} 〉}_{γ} .

In the ideal case, we can assume the cold bath is only allowed to couple with the transition |3〉 ↔ |2〉 which corresponds to λ_R = λ_H = 0. Based on the previous calculation, one can immediately find that

{\dot{Q}}_{C} = ω_{C} Γ_{C} {〈 {\tilde{S}}_{R} 〉}_{γ},

{\dot{Q}}_{R} = - \frac{ω_{R}}{ω_{C}} {\dot{Q}}_{C},

{\dot{Q}}_{H} = \frac{ω_{H}}{ω_{C}} {\dot{Q}}_{C} .

Funding

National Natural Science Foundation of China (11775040, 11375036); Xinghai Scholar Cultivation Plan; Fundamental Research Fund for the Central Universities (DUT18LK45).

Acknowledgments

Yu thanks Sandu Popescu and Paul Skrzypczyk for their help and valuable discussions.

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Quantum self-contained refrigerator in terms of the cavity quantum electrodynamics in the weak internal-coupling regime

Abstract

1. Introduction

2. The refrigerator based on three cavities

3. The refrigerator based on two cavities and one atom

4. Discussion and conclusion

A. The weak internal coupling master equation

B. The heat currents with the cold bath in the second model

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (60)

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