# Quantum coherence rather than quantum correlations reflect the effects of
reservoir on the system’s work capability

Hai Li1,2 Jian Zou1<EMAIL_ADDRESS>Wen-Li Yu3 Bao-Ming Xu1 Jun-Gang Li1
Bin Shao1 1School of Physics, Beijing Institute of Technology, Beijing 100081,
China  2School of Information and Electronic Engineering, Shandong Institute
of Business and Technology, Yantai 264000, China  3School of Computer Science
and Technology, Shandong Institute of Business and Technology, Yantai 264000,
China

###### Abstract

We consider a model of an optical cavity with a nonequilibrium reservoir
consisting of a beam of identical two-level atom pairs (TLAPs) in the general
X-state. We find that coherence of multiparticle nonequilibrium reservoir
plays a central role on the potential work capability of cavity. We show that
no matter whether there are quantum correlations in each TLAP (including
quantum entanglement and quantum discord) or not the coherence of the TLAPs
has an effect on the work capability of the cavity. Additionally, constructive
and destructive interferences could be induced to influence the work
capability of cavity only by adjusting the relative phase with which quantum
correlations have nothing to do. In this paper, the coherence of reservoir
rather than the quantum correlations effectively reflecting the effects of
reservoir on the system’s work capability is demonstrated clearly.

###### pacs:

05.70.-a 37.30.+i 42.50.Gy 64.10.+h

##  I Introduction

Quantum coherence as a physical resource, being at the heart of quantum
interference has a variety of manifestations in different areas of physics,
and arises in some form or other in almost all the phenomena of quantum
mechanics and its applications Ficek . The theoretical and experimental
exploration of quantum coherence has become a fascinating research topic.
Recently, many interesting investigations have been performed in various
systems and models, such as quantum optical systems including microwave
cavities Scully ; Quan2006 ; Sun ; Kim ; Pielawa ; Sarlette , ion traps
Poyatos ; Myatt , optical lattices Diehl ; Wolf , optomechanical systems Clerk
, and biological systems Engel ; Pani ; Huelga ; Wu1 ; Wu2 ; Kassal .
Meanwhile, it has been suggested that coherent quantum dynamics can play an
important role in the initial steps of photobiological processes Ishizaki ;
Lee1 ; Cheng1 ; Collini1 .

Recently, the thermodynamic effects of quantum coherence have attracted much
attention and have been investigated based on quantum thermodynamics cycles.
In this aspect, except for exploiting thermodynamics resource of quantum
mechanical working materials Scully ; Quan2006 ; Sun ; Hormoz ; Correa1 ;
Quan1 ; Wang1 ; Linden1 ; Grimsmo1 ; Ueda1 ; Kim1  with the help of quantum
engine and refrigerator models, the importance of reservoir manipulation has
also been very recently acknowledged in the context of quantum thermodynamics:
It has been recently demonstrated that superefficient operation of quantum
heat engines may be achieved, e.g., by reservoir squeezing Huang1 ; Correa2
and coherence Scully ; Quan2006 ; Sun  or using more general types of non-
equilibrium reservoirs Abah1 ; Mehta ; Lutz . Especially, the exploration of
reservoir’s coherence in quantum thermodynamics has provoked great interest
and the optical cavity model with a nonequilibrium coherent reservoir has been
considered. Compared with the situation of noncoherent reservoir some novel
features could be exhibited such as the improvement on work extraction and
efficiency in the thermodynamic cycle Scully ; Quan2006 , and heating and
cooling of cavity Sun .

However, previous investigations have mainly focused on the case of single-
particle reservoirs with coherence (e.g., the single two-level Sun  or three-
level Quan2006 ; Scully  coherent reservoirs). For two-particle or multi-
particle reservoirs the quantum effects of coherence on the work capability of
system have no related reports. Meanwhile, we also notice that based on a
photo-Carnot engine model similar to the one presented in Ref. Scully  the
thermodynamic effect of quantum correlations has been investigated in Ref.
Lutz . They considered a beam of thermally entangled pairs of two-level atoms
as a heat reservoir, and expressed the thermodynamic efficiency of the engine
in terms of quantum discord (QD) of the atomic pair. They also showed that
useful work could be extracted from quantum correlations, and believed that
quantum correlations of the atomic pair are a valuable resource in quantum
thermodynamics. However, for multiparticle systems the quantum correlations
including quantum entanglement (QE) and QD and quantum coherence may appear in
systems simultaneously, and they are closely related Plenio . Which one,
quantum coherence or quantum correlations from reservoir, is the good physical
quantity to effectively reflect the effects of reservoir on the system’s work
capability on earth? This is what we mainly concern about in this paper. It is
noted that most recently some interesting works have devoted to the
explorations of thermodynamic effects of quantum correlations Quan1 ; Wang1 ;
Linden1 ; Grimsmo1 ; Ueda1 ; Kim1  and quantum coherence Scully1 ; Scully2 ;
Nalbach ; Dorfman  where the quantum systems with quantum correlations or
coherence are generally served as the working substance of quantum engines or
thermodynamic cycles, i.e., the quantum correlations or the coherence come
from the working substance of quantum engines. However, in the present paper,
we are interested in the quantum effects of reservoir’s coherence and quantum
correlations in quantum thermodynamics which is very different from their
works.

In this paper, we choose the usual micromaser model (e.g., Refs. Scully3 ;
Meystre ; Scully4 ; Orszag ; Meschede ; Casagrande ; Cresser ; Cresser96 ;
Bergou ; Guerra ; Brune ; Rempe ), to illustrate our idea. In our model as
depicted in Sec. II we consider a series of two-level atomic pairs (TLAPs)
initially prepared in the general X-state passing through a cavity. Here, it
is emphasized that for choosing our model, two major reasons are taken into
account. Firstly, in contrast to the previous works Scully ; Quan2006 ; Sun
we take a series of TLAPs with coherence as a reservoir instead of a single
two-level Sun  or three-level Scully ; Quan2006  atom reservoir, and aim to
discuss the quantum effects of multi-particle reservoir’s coherence and
quantum correlations on the work capability of cavity field. Secondly, we want
to know which one, coherence or quantum correlation, plays the decisive role
on the thermodynamic properties of the cavity field although the quantum
correlations are closely related to the coherence in multi-particle systems.
Meanwhile, it is more meaningful for choosing the general X-state of the
injected TLAPs because they include a wide class of quantum states such as the
general W and GHZ states. In this paper, we find that no matter whether there
are quantum correlations or not the constructive and destructive interferences
could be induced to influence the thermodynamic properties (such as the
entropy and the average photon number) of cavity only via adjusting the
relative phase of the TLAPs. In this paper, we show that it is the reservoir’s
coherence rather than the quantum correlations that can be used to reflect the
effects of reservoir on the system’s work capability effectively. Furthermore,
we also notice that it is proper to measure the potential work capability of
the cavity by using the entropy of cavity rather than the average photon
number, except that the cavity is in thermal equilibrium, and in this case,
although the average photon number and the entropy are different physical
quantities, they have similar behavior and the average photon number could
also be used to describe the potential work capability of the cavity.

The paper is organized as follows. In Sec. II, we present our model of a
single-mode cavity field interacting with a series of TLAPs injected randomly.
A quantum master equation of the single-mode cavity field is derived. In Sec.
III, by considering the TLAPs being in a general X-state we investigate the
dynamics of cavity field. We analyze the role of reservoir’s coherence and
quantum correlations in detail numerically and analytically, and show that the
good physical quantity reflecting the effects of reservoir on the system’s
work capability is the quantum coherence not the quantum correlations in our
model. Finally, we summarize our paper with some discussions in Sec. IV.

##  II Cavity QED model and master equation

In this paper, we consider a QED model that contains a single-mode cavity
field and a nonequilibrium reservoir consisting of amount of TLAPs. We
respectively denote the two atoms in each TLAP as $A$ and $B$, and assume that
there is no interaction between them. When the TLAPs are sent through the
cavity at random as depicted in Fig.1, each atom $A$ ($B$) interacts with the
single mode cavity via a resonant Jaynes-Cummings (JC) coupling. The
Hamiltonian of system can be described as

$\begin{split}\hat{H}=&\hat{H}_{at}+\hat{H}_{ca}+\hat{H}_{int},\end{split}$
(1)

where $\hat{H}_{at}=\hbar\omega\sum_{j=1}^{2}\hat{\sigma}_{j}^{z}$,
$\hat{H}_{ca}=\hbar\omega\hat{a}^{{\dagger}}\hat{a}$ and
$\hat{H}_{int}=g\hbar\sum_{j=1}^{2}(\hat{a}\hat{\sigma}_{j}^{+}+\hat{\sigma}_{j}^{-}\hat{a}^{{\dagger}})$
with $\hat{\sigma}_{j}^{\pm}=\frac{1}{2}(\hat{\sigma}_{j}^{x}\pm
i\hat{\sigma}_{j}^{y})$ are independently the Hamiltonian of the TLAPs, cavity
field and interaction between the TLAPs and the cavity; $g$ and $\omega$ are
independently the coupling constant and the transition frequency between the
energy levels corresponding to excited state $|e\rangle$ and ground state
$|g\rangle$ of each two-level atom, $A$ and $B$; $\hat{a}$
$(\hat{a}^{{\dagger}})$ is the annihilation (creation) operator of the cavity
and satisfies the commutation relation $[\hat{a},\hat{a}^{{\dagger}}]=1$;
$\hat{\sigma}_{j}^{x,y,z}$ ($j=1,2$) are the usual Pauli operators.

Figure 1: (Color online). Schematic diagram of the dynamic model of a series
of TLAPs initially prepared in a general X-state randomly passing through a
single-mode cavity. The two-level atoms in each TLAP respectively denoted as
$A$ (the left green solid ball) and $B$ (the right yellow one). When the TLAPs
pass through the cavity the coherence information in the TLAPs will transfer
into the cavity.

We suppose that the pairwise TLAPs are randomly sent through the cavity for a
fixed time interval $\tau$ and there is at most one TLAP in the cavity each
time, and then the dynamic evolution of the whole system (cavity + TLAP)
during each time interval is a unitary evolution and governed by the
interaction Hamiltonian, $\hat{H}_{int}$. The unitary evolution operator in
the interaction picture reads as

$\displaystyle\hat{U}(\tau)\equiv\exp(-i\hat{H}_{int}\tau)=\left({\begin{array}[]{*{20}c}{\hat{U}_{11}}&{\hat{U}_{12}}&{\hat{U}_{13}}&{\hat{U}_{14}}\\\
{\hat{U}_{21}}&{\hat{U}_{22}}&{\hat{U}_{23}}&{\hat{U}_{24}}\\\
{\hat{U}_{31}}&{\hat{U}_{32}}&{\hat{U}_{33}}&{\hat{U}_{34}}\\\
{\hat{U}_{41}}&{\hat{U}_{42}}&{\hat{U}_{43}}&{\hat{U}_{44}}\end{array}}\right),$
(6)

where the matrix elements are expressed as

$\displaystyle\hat{U}_{11}=$ $\displaystyle
1+2\hat{a}\frac{{\hat{A}-1}}{\hat{\Lambda}}\hat{a}^{{\dagger}},~{}~{}~{}~{}~{}\hat{U}_{44}=1+2\hat{a}^{{\dagger}}\frac{{\hat{A}-1}}{\hat{\Lambda}}\hat{a},$
$\displaystyle\hat{U}_{22}=$
$\displaystyle\hat{U}_{33}=\frac{1}{2}({\hat{A}+1}),~{}~{}~{}~{}\hat{U}_{23}=\hat{U}_{32}=\frac{1}{2}({\hat{A}-1}),$
$\displaystyle\hat{U}_{14}=$ $\displaystyle
2\hat{a}\frac{{\hat{A}-1}}{\hat{\Lambda}}\hat{a},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{U}_{41}=2\hat{a}^{{\dagger}}\frac{{\hat{A}-1}}{\hat{\Lambda}}\hat{a}^{{\dagger}},$
$\displaystyle\hat{U}_{12}=$
$\displaystyle\hat{U}_{13}=-i\hat{a}\frac{{\hat{B}}}{{\sqrt{\hat{\Lambda}}}},~{}~{}~{}~{}~{}\hat{U}_{21}=\hat{U}_{31}=-i\frac{{\hat{B}}}{{\sqrt{\hat{\Lambda}}}}\hat{a}^{{\dagger}}\hfill,$
$\displaystyle\hat{U}_{42}=$
$\displaystyle\hat{U}_{43}=-i\hat{a}^{{\dagger}}\frac{{\hat{B}}}{{\sqrt{\hat{\Lambda}}}},~{}~{}~{}\hat{U}_{24}=\hat{U}_{34}=-i\frac{{\hat{B}}}{{\sqrt{\hat{\Lambda}}}}\hat{a},\hfill$

and

$\displaystyle\hat{\Lambda}=2({2\hat{a}^{{\dagger}}\hat{a}}+1),~{}~{}\xi=g\tau,{}{}$
$\displaystyle\hat{A}=\cos(\xi\sqrt{\hat{\Lambda}}),$
$\displaystyle\hat{B}=\sin(\xi\sqrt{\hat{\Lambda}}).$

We assume that the $j$th TLAP is injected into the cavity at time $t_{j}$,
then, after a time interval $\tau$, the density matrix of the cavity field
becomes

$\displaystyle\hat{\rho}_{ca}(t_{j}+\tau)=\texttt{Tr}_{\textrm{AB}}[\hat{U}(\tau)\hat{\rho}_{\textrm{AB}}\otimes\hat{\rho}_{ca}(t_{j})\hat{U}^{\dagger}\left(\tau\right)]\equiv\mathcal{D}(\tau)\hat{\rho}_{ca}(t_{j}),$
(7)

where $\hat{\rho}_{\textrm{AB}}$ is the density matrix of the $j$th TLAP and
$\mathcal{D}(\tau)$ is a superoperator.

Since the TLAPs pass through the cavity randomly we assume that each one
arrives at the cavity with a probability $r$ per unit time. The probability of
a TLAP arrival, in a time interval of $(t,t+\delta t)$, is $r\delta t$, and
the probability without the TLAP passing is $1-r\delta t$. Hereafter, for
simplicity, we denote $\hat{\rho}_{ca}$ as $\hat{\rho}$. Then we can obtain
the density matrix of cavity field at time $t+\delta t$ Sun

$\displaystyle\hat{\rho}(t+\delta t)=(1-r\delta t)\hat{\rho}(t)+r\delta
t\mathcal{D}(\tau)\hat{\rho}(t).$ (8)

For $\delta t\rightarrow 0$, one obtains the master equation Scully3 ; Orszag
; Meschede ; Casagrande ; Cresser ; Cresser96 ; Bergou ; Guerra

$\displaystyle\dot{\hat{\rho}}(t)=r[\mathcal{D}(\tau)-1]\hat{\rho}(t),$ (9)

which describes the dynamics of the single-mode cavity field.

##  III dynamics of cavity field with a nonequilibrium reservoir

Here, we consider a nonequilibrium reservoir consisting of a beam of TLAPs in
a general X-state, and in the basis
$\\{|ee\rangle,|eg\rangle,|ge\rangle,|gg\rangle\\}$, the X-state is given by

$\displaystyle\hat{\rho}_{AB}=\left({\begin{array}[]{*{20}c}{{a}_{11}}&0&0&{{a}_{14}}\\\
0&{{a}_{22}}&{{a}_{23}}&0\\\ 0&{{a}_{32}}&{{a}_{33}}&0\\\
{{a}_{41}}&0&0&{{a}_{44}}\end{array}}\right),$ (14)

where $\hat{\rho}_{AB}$ is normalized $\sum_{i=1}^{4}{a}_{ii}=1$, and the
nondiagonal elements ${a}_{14}={a}_{41}^{*}$ and ${a}_{23}={a}_{32}^{*}$.
Inserting Eq. (14) into Eq. (7) and after some calculations, the superoperator
$\mathcal{D}(\tau)$ can be expressed as

$\displaystyle\mathcal{D}(\tau)\hat{\rho}(t)=\sum_{i,j=1}^{4}{a}_{ij}\sum_{m=1}^{4}\hat{U}_{mi}(\tau)\hat{\rho}(t)\hat{U}_{mj}^{\dagger}(\tau),$
(15)

where ${a}_{ij}$ $(i,j=1,2,3,4)$ are given in Eq. (14). Thus, for the X-state
one has

$\displaystyle\hat{\rho}(t_{j}+\tau)=\sum_{i,j=1}^{4}{a}_{ij}\sum_{m=1}^{4}\hat{U}_{mi}(\tau)\hat{\rho}(t)\hat{U}_{mj}^{\dagger}(\tau)$
(16)

describing the density matrix of the cavity field at time $t_{j}+\tau$, and
the master equation Eq. (9) can be rewritten as

$\displaystyle\dot{\hat{\rho}}(t)=r[\sum_{i,j=1}^{4}{a}_{ij}\sum_{m=1}^{4}\hat{U}_{mi}(\tau)\hat{\rho}(t)\hat{U}_{mj}^{\dagger}(\tau)-\hat{\rho}(t)].$
(17)

Eq (17) is the quantum master equation of the cavity field for each TLAP in
the X-state. In order to obtain the dynamics of the cavity field we consider
two cases of the TLAPs passing through the cavity: ${1)}$ passing through
instantly corresponding to $\xi\rightarrow 0$, and ${2)}$ passing through at a
low speed corresponding to a finite $\xi$. In case 1, Eqs. (16) and (17) can
be further expressed as

$\displaystyle\hat{\rho}({t_{j}+\tau})\approx$
$\displaystyle\xi^{2}\\{{{({2a_{11}+a_{22}+a_{33}+a_{23}+a_{32}})[{\hat{a}^{\dagger}\hat{\rho}({t_{j}})\hat{a}-\hat{\rho}({t_{j}})}]}}$
(18) $\displaystyle+$
$\displaystyle({2a_{44}+a_{22}+a_{33}+a_{23}+a_{32}})\hat{a}\hat{\rho}({t_{j}})\hat{a}^{\dagger}$
$\displaystyle-$
$\displaystyle({a_{11}+a_{44}+a_{23}+a_{32}})\hat{a}^{\dagger}\hat{a}\hat{\rho}({t_{j}})$
$\displaystyle-$
$\displaystyle({1+a_{22}+a_{33}+a_{23}+a_{32}})\hat{\rho}({t_{j}})\hat{a}^{\dagger}\hat{a}$
$\displaystyle+$ $\displaystyle
a_{14}(2\hat{a}^{{\dagger}}\hat{\rho}\hat{a}^{{\dagger}}-\hat{a}^{{\dagger}}\hat{a}^{{\dagger}}\hat{\rho}-\hat{\rho}\hat{a}^{{\dagger}}\hat{a}^{{\dagger}})$
$\displaystyle+$ $\displaystyle
a_{41}(2\hat{a}\hat{\rho}\hat{a}-\hat{a}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}\hat{a})\\}+\hat{\rho}({t_{j}})$

and

$\displaystyle\dot{\hat{\rho}}\approx$ $\displaystyle
r\xi^{2}\\{a_{11}(2\hat{a}^{{\dagger}}\hat{\rho}\hat{a}-\hat{\rho}\hat{a}\hat{a}^{{\dagger}}-\hat{a}\hat{a}^{{\dagger}}\hat{\rho})$
(19) $\displaystyle+$
$\displaystyle{a_{44}(2\hat{a}\hat{\rho}\hat{a}^{{\dagger}}-\hat{\rho}\hat{a}^{{\dagger}}\hat{a}-\hat{a}^{{\dagger}}\hat{a}\hat{\rho})}$
$\displaystyle-$
$\displaystyle(a_{22}+a_{33})[(2\hat{\rho}\hat{a}^{{\dagger}}\hat{a}-\hat{a}\hat{\rho}\hat{a}^{{\dagger}}-\hat{a}^{{\dagger}}\hat{\rho}\hat{a})+\hat{\rho}]$
$\displaystyle+$ $\displaystyle
a_{14}(2\hat{a}^{{\dagger}}\hat{\rho}\hat{a}^{{\dagger}}-\hat{a}^{{\dagger}}\hat{a}^{{\dagger}}\hat{\rho}-\hat{\rho}\hat{a}^{{\dagger}}\hat{a}^{{\dagger}})$
$\displaystyle+$ $\displaystyle
a_{41}(2\hat{a}\hat{\rho}\hat{a}-\hat{a}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}\hat{a})$
$\displaystyle+$
$\displaystyle(a_{23}+a_{32})(\hat{a}\hat{\rho}\hat{a}^{{\dagger}}+\hat{a}^{{\dagger}}\hat{\rho}\hat{a}-\hat{a}^{{\dagger}}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{{\dagger}}\hat{a}-\hat{\rho})\\},$

where we have made the approximation
$\hat{A}=\cos(\xi\sqrt{{\hat{\Lambda}}})\approx 1-\xi^{2}{\hat{\Lambda}}/2$
and $\hat{B}=\sin(\xi\sqrt{{\hat{\Lambda}}})\approx\xi\sqrt{{\hat{\Lambda}}}$,
and kept $\xi$ up to the second order. For simplicity, we denote
$\hat{\rho(t)}$ in Eq. (19) as $\hat{\rho}$. In case 2, the above
approximation is not valid any more and from Eq. (16) the density matrix of
cavity field at time $t_{j}+\tau$ can be rewritten as

$\displaystyle\hat{\rho}(t_{j}+\tau)=$
$\displaystyle\sum_{m,n=0}^{\infty}\\{\rho_{m,n}f_{1}+\rho_{m+1,n+1}f_{2}+\rho_{m-1,n-1}f_{3}$
(20) $\displaystyle+$
$\displaystyle\rho_{m-2,n-2}f_{4}+\rho_{m+2,n+2}f_{5}+\rho_{m,n+2}f_{6}$
$\displaystyle+$
$\displaystyle\rho_{m+2,n}f_{7}+\rho_{m+1,n-1}f_{8}+\rho_{m-1,n+1}f_{9}$
$\displaystyle+$
$\displaystyle\rho_{m,n-2}f_{10}+\rho_{m-2,n}f_{11}\\}|m\rangle\langle n|,$

where $\rho_{m,-2}=\rho_{m,-1}=\rho_{-2,n}=\rho_{-1,n}=0$, $(m,n=0,1,2,...)$,
and during the derivation we have assumed that the state of cavity field
$\hat{\rho}(t_{j})=\sum_{m,n=0}^{\infty}\rho_{m,n}(t_{j})|m\rangle\langle n|$
and denoted $\rho_{m,n}(t_{j})$ as $\rho_{m,n}$. For simplicity, $f_{i}$
($i=1,2,3,...,11$) are given in the appendix A. Though the expression of Eq.
(20) is complex, it, in the limit $\xi\rightarrow 0$, is in consistency with
Eq. (18).

Next, we first consider the dynamics of the cavity field in case 1. For
simplicity, throughout this paper we choose the vacuum state as the initial
state of the cavity field. From Eq. (18) it can be seen that the density
matrices $\hat{\rho}(t_{j}+\tau)$ and $\hat{\rho}(t_{j})$ possess the same
form of structure. When considering the cavity initially prepared in vacuum
state (being in the diagonal distribution), the density matrix of cavity
field, $\hat{\rho}(t_{j}+\tau)$, will keep the diagonal distribution.
Moreover, from Eq. (18) we can obtain that the average photon number of the
cavity field at time ${t_{j}+\tau}$ reads as

$\displaystyle\langle\hat{n}(t_{j}+\tau)\rangle=$
$\displaystyle\texttt{Tr}[{\hat{\rho}}(t_{j}+\tau)\hat{n}]$ (21)
$\displaystyle=$
$\displaystyle[2\xi^{2}(a_{11}-a_{44})+1]\langle\hat{n}(t_{j})\rangle$
$\displaystyle+$ $\displaystyle\xi^{2}(2a_{11}+a_{22}+a_{33}+a_{23}+a_{32}).$

From Eq. (21) it can be seen that the increment of average photon number
between two neighboring passings
$\Delta\langle\hat{n}(t_{j}+\tau)\rangle=\langle\hat{n}(t_{j}+\tau)\rangle-\langle\hat{n}(t_{j})\rangle$,
satisfies

$\displaystyle\Delta\langle\hat{n}(t_{j}+\tau)\rangle=[1-2\xi^{2}(a_{44}-a_{11})]\Delta\langle\hat{n}(t_{j})\rangle.$
(22)

The increment ratio, $k$, for the two neighboring passings at time $t_{j}$ and
$t_{j}+\tau$ is directly obtained as

$\displaystyle
k=\frac{\Delta\langle\hat{n}(t_{j}+\tau)\rangle}{\Delta\langle\hat{n}(t_{j})\rangle}=1-2\xi^{2}(a_{44}-a_{11}).$
(23)

Since the initial state of the cavity field is supposed to be the vacuum
state, i.e., $\langle\hat{n}(0)\rangle=0$, from Eq. (21) the average photon
number after the first passing is

$\displaystyle\langle\hat{n}(\tau)\rangle=\xi^{2}(2a_{11}+a_{22}+a_{33}+a_{23}+a_{32}).$
(24)

From Eqs. (23) and (24) the average photon number, after the $j$th time
passing (i.e., at time $t_{j}$), can be expressed as

$\displaystyle\langle\hat{n}(t_{j})\rangle=\sum_{i=1}^{j}k^{i-1}\langle\hat{n}(\tau)\rangle.$
(25)

According to Eqs. (23) and (25) for $a_{11}\geq a_{44}$ the ratio $k\geq 1$
holds which means that the average photon number,
$\langle\hat{n}(t_{j})\rangle$, is divergent. On the contrary, for $k<1$
$(a_{11}<a_{44})$, $\langle\hat{n}(t_{j})\rangle$ is convergent, and in the
limit $j\rightarrow\infty$ (i.e., $t_{j}\rightarrow\infty$)

$\displaystyle\langle\hat{n}(t_{j})\rangle|_{j\rightarrow\infty}=\frac{\langle\hat{n}(\tau)\rangle}{1-k}=\frac{{2a_{11}+a_{22}+a_{33}+a_{23}+{a}_{32}}}{{2\left({a_{44}-a_{11}}\right)}}.$
(26)

Eq. (26) shows that for fixed X-state with $a_{11}<a_{44}$ the cavity field,
in the limit $t_{j}\rightarrow\infty$, can reach a steady state. Besides, this
result can also be verified by the master equation Eq. (19) from which, after
some calculations, we have

$\displaystyle\langle\dot{\hat{n}}\rangle=r\xi^{2}\left[{2\left({a_{11}-a_{44}}\right)\left\langle{\hat{n}}\right\rangle+2a_{11}+a_{22}+a_{33}+a_{23}+a_{32}}\right]=0,$
(27)

and

$\displaystyle\left\langle{{\hat{n}}}\right\rangle_{ss}=\frac{{2a_{11}+a_{22}+a_{33}+a_{23}+{a}_{32}}}{{2\left({a_{44}-a_{11}}\right)}},$
(28)

where
$\langle\hat{n}\rangle_{ss}=\langle\hat{n}(t_{j}\rightarrow\infty)\rangle$
represents the average photon number of the cavity in the steady state.

Here, we consider two special noncoherent states of TLAPs,
$\hat{\rho}_{AB}^{(1)}$ with classical correlation and $\hat{\rho}_{AB}^{(2)}$
without any correlation (i.e., product state), as follows

$\displaystyle{\hat{\rho}_{AB}^{(1)}}=$ $\displaystyle a_{11}|ee\rangle\langle
ee|+a_{22}|eg\rangle\langle eg|+a_{33}|ge\rangle\langle
ge|+a_{44}|gg\rangle\langle gg|,$ (29)

and

$\displaystyle{\hat{\rho}_{AB}^{(2)}}=\hat{\rho}_{A}\otimes\hat{\rho}_{B},$
(30)

where $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$ respectively read as

$\displaystyle\hat{\rho}_{A}=\left({\begin{array}[]{*{20}c}{a}_{11}+{a}_{22}&0\\\
0&{a}_{33}+{a}_{44}\end{array}}\right)$ (33a)
$\displaystyle\hat{\rho}_{B}=\left({\begin{array}[]{*{20}c}{a}_{11}+{a}_{33}&0\\\
0&{a}_{22}+{a}_{44}\end{array}}\right).$ (35b)

It is noted that the choice of above two states enables the three density
matrices $\hat{\rho}_{AB}$ and $\hat{\rho}_{AB}^{(m)}$ (${m=1,2}$) to possess
the same reduced density matrices $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$. In
terms of Eq. (28) when the TLAPs are initially prepared in the states
$\hat{\rho}_{AB}^{(1)}$ and $\hat{\rho}_{AB}^{(2)}$, respectively, one has

$\displaystyle\langle\hat{n}\rangle_{ss}^{(m)}=\frac{{2a_{11}+a_{22}+a_{33}}}{{2\left({a_{44}-a_{11}}\right)}},$
(36)

where $\langle\hat{n}\rangle_{ss}^{(m)}$ represents the average photon number
of the cavity in the steady state for $\hat{\rho}_{AB}^{(m)}$ (${m=1,2}$).

Especially, if we consider $\hat{\rho}_{A}=\hat{\rho}_{B}$ (i.e.,
$a_{22}=a_{33}$) and $a_{11}+a_{22}<a_{33}+a_{44}$ the temperature of
reservoir consisting of TLAPs can be defined well by the two-level atoms. For
simplicity, we denote ${p}_{e}=a_{11}+a_{22}$ and ${p}_{g}=a_{22}+a_{44}$ the
inverse temperature of reservoir $\beta_{\textrm{eff}}$
($\beta_{\textrm{eff}}=1/(k_{B}T)$, $k_{B}$ is the Boltzmann constant) is
expressed as

$\displaystyle\frac{p_{e}}{p_{g}}=e^{-\beta_{\textrm{eff}}\omega}\Rightarrow\beta_{\textrm{eff}}=-\frac{1}{\omega}\ln\frac{p_{e}}{p_{g}},$
(37)

where we let $\hbar=1$. In this case, the asymptotic solution of the master
equation Eq. (19) is the thermal state. From Eqs. (28) and (36) the thermal
average photon numbers
$\langle\hat{n}\rangle_{th}=\frac{a_{11}+a_{22}+\texttt{Re}[a_{23}]}{a_{44}-a_{11}}$
for ${\hat{\rho}_{AB}}$ corresponds to the inverse temperatures of cavity
field

$\displaystyle{\beta}_{coh}=-\frac{1}{\omega}\ln\frac{\langle\hat{n}\rangle_{th}}{1+\langle\hat{n}\rangle_{th}}=-\frac{1}{\omega}\ln\frac{p_{e}+\texttt{Re}[a_{23}]}{p_{g}+\texttt{Re}[a_{23}]},$
(38)

and $\langle\hat{n}\rangle_{th}^{(m)}=\frac{a_{11}+a_{22}}{a_{44}-a_{11}}$ for
$\hat{\rho}_{AB}^{(m)}$ (m=1,2) with

$\displaystyle{\beta}_{non}^{(m)}=-\frac{1}{\omega}\ln\frac{\langle\hat{n}\rangle_{th}^{(m)}}{1+\langle\hat{n}\rangle_{th}^{(m)}}=-\frac{1}{\omega}\ln\frac{p_{e}}{p_{g}}.$
(39)

From Eqs. (37-39) we can see that for the noncoherent reservoir with
${\hat{\rho}_{AB}^{(m)}}$ the cavity field is thermalized and reaches the same
temperature as the reservoir, ${\beta}_{non}^{(m)}=\beta_{\textrm{eff}}$, but
for the coherent reservoir with ${\hat{\rho}_{AB}}$ the temperature of cavity
field after thermalization does not coincide with the reservoir’s any more due
to the reservoir’s coherence, i.e., ${\beta}_{coh}\neq\beta_{\textrm{eff}}$.
It is similar to the model of a single atom coherent reservoir in Ref. Scully
where Scully _et al_. showed that the detailed balance between photon
absorption and emission could be broken with the help of the coherent
superposition of the two (nearly degenerate) lower levels of a three-level
atom. In our model, the TLAP has four energy levels including a higher level
with double excitation $|ee\rangle$, a lower level with no excitation
$|gg\rangle$ and two degenerate intermediate levels respectively corresponding
to single excitation $|eg\rangle$ and $|ge\rangle$. Here, we can also use the
coherent superposition of the two degenerate intermediate levels of the TLAP
to break the detailed balance. Meanwhile, the deviation away from thermal
equilibrium is completely determined by the real part of the coherence term
(see Eq. (38)), i.e., $\texttt{Re}[a_{23}]=|a_{23}|\cos\phi$ where we denote
$\phi$ as the relative phase between the two degenerate intermediate levels in
the TLAP. It is clear that the reservoir’s coherence in our model also plays
an important role in the thermalization of the cavity field.

In addition, from Eq. (36) it can be seen that the cavity field possesses the
same average photon number for $\hat{\rho}_{AB}^{(1)}$ and
$\hat{\rho}_{AB}^{(2)}$ which implies that the classical correlation of the
TLAP has no contribution to the work capability of cavity field. Actually,
even for the quantum correlations including the QE and QD they are not always
good quantities to effectively reflect the contributions of the TLAPs to the
work capability of cavity field. Using the density matrix Eq. (14), the
quantum correlations between the two atoms in each TLAP as measured by QE and
QD can be calculated. We adopt Wootter’s concurrence Wootters  as entanglement
measure. For the density matrix Eq. (14), the concurrence is given by

$C(\hat{\rho}_{AB})=2\max(0,|a_{23}|-\sqrt{a_{11}a_{44}},|a_{14}|-\sqrt{a_{22}a_{33}}).$
(40)

On the other hand, quantum discord captures all nonclassical correlations
between two two-level atoms Ollivier . For the X state described by the
density matrix Eq. (14), the analytic expression of QD has been reported Wang
and expressed by

${Q}(\hat{\rho}_{AB})=\min({Q_{1}},{Q_{2}}),$ (41)

where
$Q_{j}=H(a_{11}+a_{33})+\sum_{i=1}^{4}\lambda_{i}\log_{2}\lambda_{i}+D_{j}$
with $\lambda_{i}$ being the four eigenvalues of $\hat{\rho}_{AB}$,
$D_{1}(\tau)=H(\tau)$,
$D_{2}(\tau)=-\sum_{i=1}^{4}a_{ii}\log_{2}a_{ii}-H(a_{11}+a_{33})$ with
$\tau=(1+\sqrt{[1-2(a_{33}+a_{44})]^{2}+4(|a_{14}|+|a_{23}|)^{2}})/{2}$ and
$H(\tau)=-\tau\log_{2}\tau-(1-\tau)\log_{2}(1-\tau)$. For the convenience of
discussion we give our construction of the general X-state. As we know, the
space of a general X-state is composed of two independent subspaces which are
spanned by the base vectors $\\{|ee\rangle,|gg\rangle\\}$ and
$\\{|eg\rangle,|ge\rangle\\}$, respectively. We can choose the arbitrary state
in each subspace to construct the X-state via the direct sum as follows

$\displaystyle\hat{\rho}_{AB}=$
$\displaystyle\cos^{2}\alpha\hat{\rho}_{1}^{\prime}\oplus\sin^{2}\alpha\hat{\rho}_{2}^{\prime}=$
$\displaystyle\left({\begin{array}[]{*{20}c}{\frac{1}{2}{\cos^{2}\alpha}(1+{r}_{1}\cos\theta_{1})}&0&0&{\frac{1}{2}r_{1}{\cos^{2}\alpha}\sin\theta_{1}e^{-i\varphi}}\\\
0&{\frac{1}{2}{\sin^{2}\alpha}(1+{r}_{2}\cos\theta_{2})}&{\frac{1}{2}r_{2}{\sin^{2}\alpha}\sin\theta_{2}e^{-i\phi}}&0\\\
0&{\frac{1}{2}r_{2}{\sin^{2}\alpha}\sin\theta_{2}e^{i\phi}}&{\frac{1}{2}{\sin^{2}\alpha}(1-{r}_{2}\cos\theta_{2})}&0\\\
{\frac{1}{2}r_{1}{\cos^{2}\alpha}\sin\theta_{1}e^{i\varphi}}&0&0&{\frac{1}{2}{\cos^{2}\alpha}(1-r_{1}\cos\theta_{1})}\end{array}}\right),$
(46)

in which the density matrices
$\hat{\rho}_{1,2}^{\prime}=\frac{1}{2}(\mathbb{I}_{1,2}+\vec{r}_{1,2}\cdot\vec{\sigma}_{1,2})$,
respectively, represent the arbitrary state in each subspace where
$\mathbb{I}_{1}(\mathbb{I}_{2})$ is a unit matrix in the state space of
$\hat{\rho}_{1}^{\prime}(\hat{\rho}_{2}^{\prime})$, $\vec{r}_{1,2}$ are the
Bloch sphere vectors, and $\vec{\sigma}_{1,2}$ are the pauli matrices with
$\hat{\sigma}_{1z}=|ee\rangle\langle ee|-|gg\rangle\langle gg|$ and
$\hat{\sigma}_{2z}=|eg\rangle\langle eg|-|ge\rangle\langle ge|$. And we assume
that the probability of the TLAP in each subspace is respectively
$\cos^{2}\alpha$ $(\sin^{2}\alpha)$. The parameters in the X-state of Eq. (46)
satisfy $\alpha\in[0,\pi/2]$, $\theta_{1},\theta_{2}\in[0,\pi]$,
$\varphi,\phi\in[0,2\pi]$ and $r_{1},r_{2}\in[0,1]$ which guarantee the
positivity, normalization and trace preservation of $\hat{\rho}_{AB}$. By
choosing proper parameters of the X-state in Eq. (46) we can find some states
which only possess coherence but no quantum correlations. That is, these
states have nondiagonal elements but their concurrence (quantum entanglement)
and quantum discord are zero. As an example, we choose the parameters:
$r_{1}=r_{2}=2/3$, $\alpha=\sqrt{\pi}/4$, $\theta_{1}=\theta_{2}=11\pi/20$,
$\varphi=\pi/3$, $\phi_{2}=0$, and the X-state becomes

$\displaystyle\hat{\rho}_{1}=\left({\begin{array}[]{*{20}c}{0.223928}&0&0&{0.0823074-0.142561i}\\\
0&{0.223928}&{0.164615}&0\\\ 0&{0.164615}&{0.276072}&0\\\
{0.0823074+0.142561i}&0&0&{0.276072}\end{array}}\right).$ (51)

From Eqs. (40) and (41) we can obtain that
$C(\hat{\rho}_{1})=Q(\hat{\rho}_{1})=0$. It is noted that although the state
$\hat{\rho}^{\texttt{dia}}_{1}$, only preserving the diagonal elements of
density matrix in Eq. (51), has no quantum correlations like $\hat{\rho}_{1}$
the average photon numbers of cavity field in the two cases are different.
This mean that quantum coherence can be the work resource even in the absence
of quantum correlations.

Meanwhile, although the states of the TLAPs, $\hat{\rho}_{AB}$ and
$\hat{\rho}_{AB}^{(m)}$ (m=1,2) have the same reduced density matrices as
mentioned before, they lead to different average photon numbers of cavity
field $\langle\hat{n}\rangle_{ss}$ and $\langle\hat{n}\rangle_{ss}^{(m)}$
given in Eqs. (28) and (36), respectively. It is noted that from Eqs. (28) and
(36) $\langle\hat{n}\rangle_{ss}\geq\langle\hat{n}\rangle_{ss}^{(m)}$ for
$a_{23}+a_{32}\geq 0$;
$\langle\hat{n}\rangle_{ss}<\langle\hat{n}\rangle_{ss}^{(m)}$ for
$a_{23}+a_{32}<0$. So, it means that due to the coherence of the TLAPs not
only the constructive quantum interference but also the destructive quantum
interference can be induced and cause the average photon number of cavity
field away from $\langle\hat{n}\rangle_{ss}^{(m)}$ corresponding to
$\hat{\rho}_{AB}^{(m)}$ (m=1,2) without any coherence. This is an interesting
and meaningful thing, and by using it we can easily perform a thermodynamic
cycle with a single nonequilibrium reservoir only by controlling an external
parameter, i.e., the relative phase. Furthermore, in terms of the definitions
of quantum correlations in Eqs. (40) and (41), QE and QD are only related to
the amplitude of the nondiagonal terms in the X-state and independent of the
relative phase. It demonstrates that quantum correlations could not always
reflect the effects of reservoir on the system’s work capability completely,
and the relative phase of the TLAP also plays an important role.

Moreover, we also notice that Dillenschneider _et al_. in Ref. Lutz  also
considered the same model where a series of TLAPs pass through a cavity. They
showed that the quantum correlations act as the resource of system’s work
capability which seems to be contradictory with our results (the reservoir’s
coherence acting as the resource rather than the quantum correlations). In
fact, it is not that case. This can be explained as follows. The state of TLAP
considered in Ref. Lutz  is a thermal entangled state which belongs to a very
special X-state with $a_{22}=a_{33}$, $a_{14}=a_{41}=0$ and
$a_{23}=a_{32}=-|a_{23}|$. For this state we can obtain the average photon
number of the cavity field in the final steady state as

$\langle\hat{n}_{1}\rangle_{ss}=\frac{a_{11}+a_{22}-|a_{23}|}{a_{44}-a_{11}}=\langle\hat{n}\rangle_{det}+\Delta,$
(52)

where $\langle\hat{n}\rangle_{det}=\frac{a_{11}+a_{22}}{a_{44}-a_{11}}$
represents the average photon number for the system being in the detailed
balance, and $\Delta=-\frac{|a_{23}|}{a_{44}-a_{11}}$ denotes the deviation
from the detailed balance. Since $|a_{23}|$ can effectively reflect the
quantum correlations of the thermal entangled X-state the deviation term,
$\Delta$, can not only be understood as the thermodynamic effects of quantum
coherence on cavity field but also the contribution of quantum correlations.
In order to demonstrate that quantum correlations are not good physical
quantities to effectively reflect the thermodynamic effects of nonequilibrium
reservoir, we could implement a quantum phase gate operation on one of the
atoms in the TLAP to make the coherence term $a_{23}$ $(a_{32})$ of the
thermal entangled state add a relative phase factor, i.e., the transition
$a_{23}$ $(a_{32})\rightarrow e^{-i\phi}a_{23}$
$(e^{i\phi}a_{32})=-e^{-i\phi}|a_{23}|$ $(-e^{i\phi}|a_{32}|)$,
$\phi\in[0,\pi]$. It is noted that this operation only adds a relative phase
and does not change the quantum correlations (concurrence and QD) of the TLAP.
For the state after the operation the average photon number of the cavity
field in the steady state can be expressed as

$\langle\hat{n}_{2}\rangle_{ss}=\langle\hat{n}\rangle_{det}+\Delta\cos\phi.$
(53)

From Eq. (53) it is clear that, for the average photon number of the cavity
field, the deviation from the detailed balance not only depends on the quantum
correlations but also on the relative phase, and via modifying the relative
phase the instructive and the destructive interferences could be introduced.
Comparing Eqs. (52) and (53) we can see that even though the TLAPs have the
same quantum correlations they could correspond to different average photon
number, i.e.,
$\langle\hat{n}_{1}\rangle_{ss}\neq\langle\hat{n}_{2}\rangle_{ss}$ for
$\phi\neq 0$. This demonstrates that, in general, the reservoir’s quantum
correlations could not completely reflect the influence of a nonequilibrium
reservoir on the work capability of the cavity field, and the relative phase
also plays an important role. And by modifying the relative phase the
deviation from the detailed balance for the average photon number may be
positive, negative or zero.

From Eq. (28) we can see that the coherence elements $a_{14}$ and $a_{41}$ do
not like $a_{23}$ and $a_{32}$ contribute to the average photon number
$\langle\hat{n}\rangle_{ss}$. They seem to have no effect on the average
photon number. In fact, it is not that case. This can be explained that
$a_{14}$ ($a_{41}$) corresponds to the process of the double excitation during
the evolution of the cavity field, and appears in the higher order terms than
the second order term $\xi^{2}$ of the average photon number. In the limit
$\xi\rightarrow 0$, these high order terms can be omitted, however, they can
not be ignored any more in case 2 with finite $\xi$. In case 2 the effects of
coherence $a_{14}$ $(a_{41})$ on the dynamics of the cavity field will be
fully demonstrated.

Figure 2: (Color online). (a) The variations of the average photon number,
$\langle\hat{n}\rangle$, and (b) the variations of the entropy, $S^{non}$, of
the cavity field with passing times, $j$, for state $\hat{\rho}_{2a}$ (black
squares) and $\hat{\rho}_{2b}$ (red dots).

From Eq. (20) we can see that the expression of the density matrix of the
cavity after the $(j+1)$th passing of the TLAPs is very complicated, and the
nondiagonal elements appear in the density matrix which is very different from
that in case 1. Thus, it is very difficult to obtain general analytical
expressions of the average photon number and the entropy of the cavity field
during the evolution. Next, we will make use of numerical calculations to
explore the dynamics of the cavity field. In order to demonstrate the effects
of coherence terms $a_{14}$ and $a_{41}$ in the X-state given in Eq. (14) we
consider the TLAPs being respectively prepared in the following two states

$\displaystyle\hat{\rho}_{2a}=\left({\begin{array}[]{*{20}c}{0.142864}&0&0&{0}\\\
0&{0.0122355}&{0.0012236}&0\\\ 0&{0.0012236}&{0.0122355}&0\\\
{0}&0&0&{0.832665}\end{array}}\right)$ (58)

and

$\displaystyle\hat{\rho}_{2b}=\left({\begin{array}[]{*{20}c}{0.142864}&0&0&{0.344901}\\\
0&{0.0122355}&{0.0012236}&0\\\ 0&{0.0012236}&{0.0122355}&0\\\
{0.344901}&0&0&{0.832665}\end{array}}\right),$ (63)

where from Eq. (46) we choose the parameters
$\\{r_{1}=0.7071,r_{2}=0.1,\alpha=\pi/20,\theta_{1}=\pi,\theta_{2}=\pi/2,\varphi=\phi=0\\}$
for $\hat{\rho}_{2a}$ and
$\\{r_{1}=1,r_{2}=0.1,\alpha=\pi/20,\theta_{1}=3\pi/4,\theta_{2}=\pi/2,\varphi=\phi=0\\}$
for $\hat{\rho}_{2b}$. It is noted that $a_{14}=0$ in $\hat{\rho}_{2a}$ and
$a_{14}=0.344901$ in $\hat{\rho}_{2b}$ and the other density matrix elements
are the same. Hereafter, we set $\xi=0.5$. In terms of Eq. (20) we plot the
variations of the average photon number, $\langle\hat{n}\rangle$, and the
entropy, $S^{non}$, of the cavity field with passing times, $j$, for
$\hat{\rho}_{2a}$ and $\hat{\rho}_{2b}$ in Fig. 2. Throughout this paper, the
entropy of the cavity field is defined by von Neumann entropy
$S(\rho)=-k_{B}\texttt{Tr}(\rho\log_{2}\rho)$, and for simplicity, we set
$k_{B}=1$. From Fig. 2 it can be seen that the nondiagonal element $a_{14}$
($a_{41}$) has an obvious effects on the average photon number shown in Fig.
2(a) and the entropy of the cavity field in Fig. 2(b), and the values of
$\langle\hat{n}\rangle$ and $S^{non}$, for $\hat{\rho}_{2a}$ are always larger
than those for $\hat{\rho}_{2b}$. Moreover, we also see that the changes of
the entropy and the average photon number of cavity field are obvious for the
first few passings and for $j>10$ the entropy and the average photon number
gradually approach their steady values, respectively.

On the other hand, $\xi$ could also affect the dynamic behavior of the cavity
field, such as the average photon number and the entropy. Different from case
1 $\xi\rightarrow 0$ $(\xi=g\tau)$ the cavity field could exhibit very
different and complicated dynamic behaviors in case 2 with finite $\xi$ even
though we choose the same nonequilibrium reservoir. The finite $\xi$ could be
understood as a strong coupling or a long interaction time interval between
the TLAP and the cavity field, and in this case the double excitation process
corresponding to the coherent term $a_{14}$ $(a_{41})$ is involved in the
dynamic evolution of the cavity field in case 2, while for $\xi\rightarrow 0$
in case 1 this can not occur. According to the unitary operator
$\hat{U}(\tau)$ in Eq. (6) we can see that each element of $\hat{U}(\tau)$ in
case 2 is a nonlinear function of $\xi$ that could lead to a very complicated
evolution of the cavity field, $\hat{\rho}(t_{j}+\tau)$ in Eq. (20). From the
density matrix $\hat{\rho}(t_{j}+\tau)$ it can be seen that the single
excitation process corresponding to the coherent term $a_{23}$ $(a_{32})$, the
double excitation process with coherence $a_{14}$ $(a_{41})$ and the parameter
$\xi$ are all involved in the cavity evolution in a very complicated way.
Naturally, from the definitions of the average photon number and the entropy
of the cavity field, we can infer that during the cavity evolution they are
also closely related to the single excitation, double excitation processes,
and the parameter $\xi$. Since the expressions of the average photon number
and the entropy of the cavity field are very complicated we can only
demonstrate how the parameter $\xi$ and reservoir’s coherence $a_{23}$ and
$a_{14}$ influence the dynamics of the cavity by numerical calculations. Only
as an example, we plot Fig. 2 to demonstrate the thermodynamic effects of the
coherent term $a_{14}$ (or the double excitation process of the TLAP) on the
dynamics of the cavity field for specific states of the TLAP $\hat{\rho}_{2a}$
and $\hat{\rho}_{2b}$, and $\xi$ ($\xi=0.5$). From a lot of numerical
calculations we find that the dynamic behaviors of the cavity field are
determined by $a_{23}$, $a_{14}$, and $\xi$ altogether. For example, suppose
that the state of the TLAP is $\hat{\rho}_{AB}$ when we keep $\hat{\rho}_{AB}$
fixed and only change the value of $\xi$, or keep $\xi$ fixed and only change
the values of the non-diagonal elements of $\hat{\rho}_{AB}$ the average
photon number and the entropy of the cavity field usually could exhibit
different variation curves with the passing times $j$, i.e., the entropy and
the average photon number of the cavity field can exhibit monotonous behaviors
for some specific $\xi$ and specific X-states of the TLAP, and non-monotonous
behaviors for the others. For example, if we choose the state of the TLAP as
$\hat{\rho}_{2b}$ given in Eq. (63) and $\xi=0.2$ instead of $\xi=0.5$ in Fig.
2 the entropy of the cavity will be monotonously increasing with $j$ which is
very different from the non-monotonous behavior as shown in Fig. 2(b).
Similarly, the monotonous increasing behavior of the entropy can also appear
only by choosing the proper state of the TLAP and keeping $\xi$ unchanged such
as $\xi=0.5$ and changing $\hat{\rho}_{2b}$ in Eq. (63) into the state
$\hat{\tilde{\rho}}_{2b}$ with
$\\{r_{1}=0.7071,r_{2}=0.1,\alpha=\pi/4,\theta_{1}=3\pi/4,\theta_{2}=\pi/2,\varphi=\phi=0\\}$
in Eq. (46). In Fig. 2(a) although the average photon number $\langle
n\rangle$ exhibits a monotonous increasing behavior for $\hat{\rho}_{2b}$ and
$\xi=0.5$ the curve of $\langle n\rangle$ can also exhibit the non-monotonous
behavior for other value of $\xi$ and other state of the TLAP, such as
$\xi=1.2$ and $\hat{\rho}_{AB}$ in Eq. (46) with
$\\{r_{1}=1,r_{2}=0.5,\alpha=\pi/20,\theta_{1}=3\pi/4,\theta_{2}=\pi/2,\varphi=\phi=0\\}$.
In addition, from numerical calculations we also find that if $a_{14}=0$ both
the average photon number and the entropy of the cavity always monotonously
increase with $j$ which indicates that the coherent term $a_{14}$ may
effectively influence the dynamic behaviors of the cavity field in case 2 with
finite $\xi$. This means that the non-monotonous behavior of the cavity field
might be caused by the double excitation process corresponding to the coherent
term $a_{14}$ for finite $\xi$.

Besides, for a cavity field coupled to a non-equilibrium atomic reservoir with
effective temperature well defined by the two-level atom if the detailed
balance is not broken the temperature of the cavity field being in a steady
state will be the same as the effective temperature of the reservoir, and the
cavity has the average photon number defined by the effective temperature of
the reservoir. For the cavity field with the TLAPs being initially in the
X-state $\hat{\rho}_{AB}$ in Eq. (14) if the detailed balance is not broken
the average photon number of cavity field will only depend on the diagonal
elements of $\hat{\rho}_{AB}$. In another word, if other parameters such as
the coherence of the non-equilibrium reservoir, i.e., the non-diagonal
elements of $\hat{\rho}_{AB}$, or the parameter $\xi$ enter into the average
photon number of the cavity field the detailed balance in general could not be
reached. From the density matrix of the cavity evolution
$\hat{\rho}(t_{j}+\tau)$ in Eq. (20) we can see that the density matrix of the
cavity field in case 2 always has nonzero nondiagonal elements which means
that the cavity field always stays in a nonequilibrium state even if it
reaches a steady state. In addition, we also notice that in case 2 all the
coherence of the TLAP $a_{23}$ and $a_{14}$ have been involved in the diagonal
elements of $\hat{\rho}(t_{j}+\tau)$ which means that the average photon
number during the cavity evolution will always carry the reservoir’s coherence
information. Thus, the detailed balance in case 2 is usually broken by the
reservoir’s coherence $a_{23}$ and $a_{14}$ cooperatively, which is different
from case 1 where only the reservoir’s coherence $a_{23}$ corresponding to the
single excitation process breaks the detailed balance.

Figure 3: (Color online). The variations of the entropy of the cavity field,
$S^{non}$, with passing times of the TLAPs, $j$, for different relative
phases, $\phi=\\{0,\pi/2,3\pi/5,4\pi/5,\pi\\}$, and the other parameters of
the state in Eq. (46) are $r_{1}=0.2$, $r_{2}=1$, $\alpha=\pi/3$,
$\theta_{1}=3\pi/4$, $\theta_{2}=\pi/2$, $\varphi=0$.

Next, let us consider the influence of the relative phase in the coherent term
$a_{23}$ of the TLAP on the work capability of the cavity field. As mentioned
before, although the quantum correlations, QE and QD, have nothing to do with
the relative phase in $a_{23}$ $(a_{32})$, it is very important to determine
the constructive or the destructive interference in the work capability of
cavity field as shown in Eq. (28) in the limit $\xi\rightarrow 0$. For case 2
with finite $\xi$ it is difficult to obtain explicit expressions of the
average photon number and the entropy of the cavity, but we can demonstrate
the effects of the relative phase on the entropy of the cavity field via
numerical calculations. In terms of Eqs. (20) and (46) we plot the variations
of the entropy of the cavity field, $S^{non}$, with passing times, $j$, for
different relative phases, $\phi=\\{0,\pi/2,3\pi/5,4\pi/5,\pi\\}$ in Fig. 3
where the other parameters are: $r_{1}=0.2$, $r_{2}=1$, $\alpha=\pi/3$,
$\theta_{1}=3\pi/4$, $\theta_{2}=\pi/2$, $\varphi=0$. From Fig. 3 we can see
that for fixed passing time, $j$, the entropy of cavity field decreases with
the relative phase $\phi$ for $\phi\in[0,\pi]$. In fact, when the relative
phase ranges from 0 to $2\pi$ we can find that for fixed passing time, $j$,
the entropy of cavity field $S^{non}$ is symmetric about $\phi=\pi$, i.e.,
$S^{non}(\pi-\phi)=S^{non}(\pi+\phi)$ with $\phi\in[0,\pi]$. Moreover, Fig. 3
also shows that for an arbitrary $\phi$ the entropy of the cavity field always
increases with the passing times, $j$, and its increment for the neighboring
twice passings decreases which means that the entropy, $S^{non}$, is
convergent as expected.

Based on the above analysis, we can see that no matter whether there exist
quantum correlations or not the reservoir’s coherence could have an effect on
the work capability of the cavity. Moreover, it has been shown that although
the relative phase is independent of the quantum correlations it has an
important effect on the dynamics of cavity field. The constructive and
destructive interferences could be induced to change the thermodynamic
features of cavity field, such as the entropy and the average photon number of
cavity, via controlling the relative phase. It is obvious that the reservoir’s
coherence plays a central role in system’s work capability in our model, and
it could be taken as an effective source of system’s work capability even in
the absence of quantum correlations. This is our major result in this paper.

It is emphasized that the cavity is always in a nonequilibrium state with the
TLAPs passing. As mentioned before, only for certain conditions
$\hat{\rho}_{A}=\hat{\rho}_{B}$ and $a_{11}+a_{22}<a_{33}+a_{44}$ the steady
state of the cavity in case 1 becomes a thermal equilibrium state. In general,
the work capability refers to the entropy of the cavity field not to the
average photon number. Roughly speaking, when the density matrix of the cavity
keeps in diagonal distribution the average photon number of cavity has similar
behavior as the entropy of cavity (i.e., when the average photon number of
cavity increases the corresponding entropy of cavity will also increase), and
it can also be used to describe the work capability of cavity. Strictly
speaking, it is not precise to use the average photon number describing the
potential work capability of cavity because it can not effectively determine
the actual degree of work capability of cavity in a nonequilibrium state.

Especially, when the density matrix of the cavity has nondiagonal elements the
change tendency of the average photon number of the cavity field can not
always coincide with that of the entropy of the cavity field as shown in Fig.
2. Only for the cavity being in a thermal state although the average photon
number denoted as $\bar{n}$ and the entropy denoted as $S$ are different
physical quantities they may have similar behavior which can be explained as
follows. As we know that the probabilities $P_{m}$ for the cavity field being
in a thermal state with $m=0,1,2,...$ photons can be expressed as

$P_{m}=\frac{\bar{n}^{m}}{(\bar{n}+1)^{m+1}},$ (64)

which satisfy the normalization condition $\sum_{m}P_{m}=1$. Inserting Eq.
(64) into the entropy expression of the cavity field
$S=-\sum_{m}P_{m}\log_{2}P_{m}$ and after some calculations one obtains

$S=(\bar{n}+1)\log_{2}(\bar{n}+1)-\bar{n}\log_{2}\bar{n}.$ (65)

From Eq. (65) it is easy to verify that ${dS}/{d\bar{n}}>0$ which means that
the entropy of the cavity field in a thermal state, $S$, is a monotonous
increasing function of the average photon number, $\bar{n}$. So, both the
average photon number and the entropy of the cavity field in the thermal state
can be used to describe the work capability of cavity.

Figure 4: (Color online). The variations of the entropy difference of cavity
field, $\triangle S$, with passing times of the TLAPs, $j$, for different
relative phases, $\phi=\\{0,\pi/2,3\pi/5,4\pi/5,\pi\\}$, and the other
parameters of the state in Eq. (46) are $r_{1}=0.2$, $r_{2}=1$,
$\alpha=\pi/3$, $\theta_{1}=3\pi/4$, $\theta_{2}=\pi/2$, $\varphi=0$.

Thus, we argue that in general the proper physical quantity to measure the
work capability of cavity is the entropy of cavity rather than the average
photon number. In order to make this point more clearer we perform a
thermalization on the cavity when the $j$th TLAP passes through the cavity,
and keep the average photon number of the cavity unchanged, i.e., the energy
of the cavity remains constant during the thermalization. Denote the average
photon number and the density matrix of the cavity, after the $j$th passing,
as ${\bar{n}_{j}}$ and ${\hat{\rho}_{j}}$ satisfying Eq. (20) and the state
after thermalization as ${\hat{\rho}_{j}^{th}}$. Then, the thermal state of
cavity, ${\hat{\rho}_{j}^{th}}$, can be expressed as

${\hat{\rho}_{j}^{th}}=Z^{-1}\sum_{n^{\prime}=0}^{\infty}e^{-n^{\prime}/\bar{n}_{j}}|n{{}^{\prime}}\rangle\langle
n{{}^{\prime}}|,$ (66)

where $Z=1-e^{-1/\bar{n}_{j}}$ and $\bar{n}_{j}=\sum_{n=0}^{2j+1}n\rho_{nn}$
with $n$ and $\rho_{nn}$ respectively being the energy level and the
corresponding probability distribution of the state $\hat{\rho}_{j}$. The
entropy difference of the cavity, ${\triangle S}$, between the states before
and after thermalization is given by

${\triangle S}=S^{non}-S^{th},$ (67)

where $S^{non}$ and $S^{th}$ independently correspond to the von Neumann
entropy $S(\hat{\rho}_{j})$ and $S(\hat{\rho}_{j}^{th})$. Here, the entropy
difference ${\triangle S}$ can effectively describe the deviation away from
the corresponding thermal equilibrium. As an example, we choose the same
parameters as that in Fig. 3, and plot the variations of the entropy
difference of the cavity with the passing times $j$ for different relative
phases $\phi=\\{0,\pi/2,3\pi/5,4\pi/5,\pi\\}$ in Fig. 4. From Fig. 4 we can
see that during the evolution the entropy difference of the cavity may be
larger or smaller than zero which depends on the passing times of the TLAPs,
$j$, and the relative phase, $\phi$. That implies that the cavity might absorb
heat from the reservoir $(\triangle S>0)$ or release heat to the reservoir
$(\triangle S<0)$ during the thermalization process. Thus, it is the entropy
of cavity actually describing the potential work capability of cavity not the
average photon number except that the cavity reaches thermal equilibrium, and
in this case, although the average photon number and the entropy are different
physical quantities but they have similar behavior as shown in Eq. (65). This
is consistent with the spirit demonstrated in another model Esposito  most
recently reported where the authors proposed a entropic motor by exploiting
entropy to fuel an engine and showed that the generation of the entropic
forces is surprisingly robust to local changes in kinetic and topological
parameters.

##  IV summary and conclusions

In conclusion, we have studied the dynamics of cavity with a nonequilibrium
reservoir consisting of a beam of identical TLAPs initially preparing in the
general X-state, and have derived a quantum master equation. We have found
that the coherence of a nonequilibrium reservoir consisting of TLAPs in the
X-state plays a central role in the dynamics of cavity field. It has been
shown that not only the constructive interference but also the destructive
interference could be induced only by adjusting the relative phase with which
the quantum correlations have nothing to do. Using this property a
thermodynamic cycle with a single reservoir can be implemented only via
controlling one external parameter, the relative phase. Meanwhile, we have
also found that no matter whether the quantum correlations exist or not the
coherence of reservoir could have contributions to the work capability of
cavity. We, in the present paper, have clearly demonstrated that quantum
coherence rather than quantum correlations can reflect the effects of
reservoir on the system’s work capability effectively. In addition, we have
also shown that the proper physical quantity to measure the potential work
capability of cavity field is the entropy of cavity field rather than the
average photon number except that the cavity arrives at thermal equilibrium,
and in this case both of them can be used to describe the work capability of
cavity field. This work might prompt further studies on how to use coherence
as a thermodynamic resource, such as the study of coherence in the heat
dissipation of atomic-scale junctions in most recent experiments Lee .
Finally, it is also interesting to extend our present work to the Dicke model
and some new results might be revealed which will be considered in the future
work.

## ACKNOWLEDGMENTS

This work is financially supported by National Science Foundation of China
(Grants Nos. 11274043, 11375025 and 61307041), the National Science Foundation
of Shandong Province, China (Grants No. ZR2011FL009, ZR2013AQ013), the Science
and Technology Project of University in Shandong Province, China (Grant No.
J12LJ01) and the Youth Foundation of Shandong Institute of Business and
Technology (Grant No. 2013QN059).

##  Appendix A The expressions of $f_{i}$

The parameters $f_{i}$, ($i=1,2,3,...,11$), in Eq. (20) are given by

$\displaystyle f_{1}=$ $\displaystyle
a_{11}U_{11}(m)U_{11}(n)+a_{44}U_{44}(m)U_{44}(n)$ (68) $\displaystyle+$
$\displaystyle(a_{22}+a_{33})[U_{22}(m)U_{22}(n)+U_{23}(m)U_{23}(n)]$
$\displaystyle+$
$\displaystyle(a_{23}+a_{32})[U_{22}(m)U_{23}(n)+U_{23}(m)U_{22}(n)],$
$\displaystyle f_{2}=$
$\displaystyle(a_{22}+a_{33}+a_{23}+a_{32})U_{12}(m+1)U_{12}^{*}(n+1)$
$\displaystyle+$ $\displaystyle 2a_{44}U_{24}(m+1)U_{24}^{*}(n+1),$
$\displaystyle f_{3}=$
$\displaystyle(a_{22}+a_{33}+a_{23}+a_{32})U_{42}(m-1)U_{42}^{*}(n-1)$
$\displaystyle+$ $\displaystyle 2a_{11}U_{21}(m-1)U_{21}^{*}(n-1),$
$\displaystyle f_{4}=$ $\displaystyle a_{11}U_{41}(m-2)U_{41}(n-2),$
$\displaystyle f_{5}=$ $\displaystyle a_{44}U_{14}(m+2)U_{14}(n+2),$
$\displaystyle f_{6}=$ $\displaystyle a_{14}U_{11}(m)U_{14}(n+2),$
$\displaystyle f_{7}=$ $\displaystyle a_{41}U_{14}(m+2)U_{11}(n),$
$\displaystyle f_{8}=$ $\displaystyle 2a_{41}U_{24}(m+1)U_{42}^{*}(n-1),$
$\displaystyle f_{9}=$ $\displaystyle 2a_{14}U_{24}(m-1)U_{42}^{*}(n+1),$
$\displaystyle f_{10}=$ $\displaystyle 2a_{41}U_{44}(m)U_{41}(n-2),$
$\displaystyle f_{11}=$ $\displaystyle 2a_{14}U_{44}(m-2)U_{41}(n),$

where $U_{ij}(x)$, $(i,j=1,2,3,4)$ ($x$ is the nonnegative integer ) are
expressed as

$\displaystyle U_{11}(x)=$ $\displaystyle 1+\frac{(x+1)[\cos
g\tau\sqrt{2(2x+3)}-1]}{2x+3}$ (69) $\displaystyle U_{44}(x)=$ $\displaystyle
1+\frac{x[\cos g\tau\sqrt{2(2x-1)}-1]}{2x-1}$ $\displaystyle U_{22}(x)=$
$\displaystyle U_{33}(x)=\frac{1}{2}{[\cos g\tau\sqrt{2(2x+1)}+1]}$
$\displaystyle U_{23}(x)=$ $\displaystyle U_{32}(x)=\frac{1}{2}{[\cos
g\tau\sqrt{2(2x+1)}-1]}$ $\displaystyle U_{14}(x)=$
$\displaystyle\frac{\sqrt{x(x-1)}[\cos g\tau\sqrt{2(2x-1)}-1]}{2x-1}$
$\displaystyle U_{41}(x)=$ $\displaystyle\frac{\sqrt{(x+1)(x+2)}[\cos
g\tau\sqrt{2(2x+3)}-1]}{2x+3}$ $\displaystyle U_{12}(x)=$
$\displaystyle=U_{13}(x)=-i\sqrt{x}\frac{\sin
g\tau\sqrt{2(2x+1)}}{\sqrt{2(2x+1)}}$ $\displaystyle U_{21}(x)=$
$\displaystyle=U_{31}(x)=-i\sqrt{x+1}\frac{\sin
g\tau\sqrt{2(2x+3)}}{\sqrt{2(2x+3)}}$ $\displaystyle U_{24}(x)=$
$\displaystyle=U_{34}(x)=-i\sqrt{x}\frac{\sin
g\tau\sqrt{2(2x-1)}}{\sqrt{2(2x-1)}}$ $\displaystyle U_{42}(x)=$
$\displaystyle=U_{43}(x)=-i\sqrt{x+1}\frac{\sin
g\tau\sqrt{2(2x+1)}}{\sqrt{2(2x+1)}}.$

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