Bao-Ming Xu(徐宝明)^,1, Jian Zou(邹健)^,2, Zhan-Chun Tu(涂展春)³¹Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China
²School of Physics, Beijing Institute of Technology, Beijing 100081, China
³Department of Physics, Beijing Normal University, Beijing 100875, China

Received:2020-12-22Revised:2021-02-18Accepted:2021-02-24Online:2021-04-09


Abstract
To investigate the role of initial quantum coherence in work-probability distribution, it is necessary to consider an incomplete or partial measurement, in which the energy cannot be fully discriminated by the detector. In this paper, we use a harmonic oscillator with a coherent or squeezing state to realize this incomplete or partial measurement, and propose a unified framework of quantum work statistics for a closed system with an arbitrary initial state. We find that work is proportional to the change of the real part of the coherent state parameter, i.e., quantum work can be estimated by the coherent state parameter. The resulting work-probability distribution includes the initial quantum coherence, and can be reduced to the result of the traditional two projective energy measurement scheme (TPM) by squeezing the state of the harmonic oscillator. Our measurement scheme reveals the fundamental connections between measurement error and coherent work. By introducing a 'coherent work-to-noise ratio’, we find the optimal measurement error, which is determined by the energy difference between the superposed energy levels. As an application, we consider a driven two-level system and investigate the effects of driving velocity on work statistics. We find that only when the driving velocity matches the transition frequency of the system can initial quantum coherence play an important role.
Keywords： work distribution;quantum coherence;coherent state;squeezing state


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Bao-Ming Xu(徐宝明), Jian Zou(邹健), Zhan-Chun Tu(涂展春). A single-point measurement scheme for quantum work based on the squeezing state. Communications in Theoretical Physics, 2021, 73(6): 065102- doi:10.1088/1572-9494/abe931

1. Introduction

Triggered by recent advances in the coherent manipulation of elementary quantum systems [1-3], out-of-equilibrium quantum thermodynamics has recently aroused enormous research interest [4]. The concept of work is one of the cornerstones of thermodynamics, however it is very hard to define work in the quantum regime, because work is not an observable [5]. Traditionally, the two projective energy measurement (TPM) scheme is used to determine quantum work [6, 7]. Based on TPM, the quantum extension of fluctuation theorems has been obtained and experimentally verified [8-14] (see reviews [6, 7] for a detailed discussion). However, the initial quantum coherence is destroyed by the first measurement, and therefore the work fluctuation relation obtained by TPM is not 'quantum'to some extent. In addition, the change of the internal energy of the closed system (without the environment) is unequal to the average work obtained by TPM; this appears to be an avoidance of the first law of thermodynamics [23]. This is because the measurement inevitably impacts the system and also performs work on the system. In short, TPM cannot give a thermodynamically consistent description (without the influence of measurement) of the work distribution of an initial state with quantum coherence. Many attempts have been made to include the effects of initial quantum coherence, such as the full counting statistics method [15-18], the integral of the injected power [19-21] and quantum Hamilton-Jacobi theory [22]. Notably, all these work distributions were not able to satisfy the first law of thermodynamics and nonequilibrium fluctuation theorem at the same time. In fact, for a work distribution including initial coherence, the first law of thermodynamics and the nonequilibrium fluctuation theorem are mutually exclusive [23-25]. This incompatibility sheds light on the crucial roles of quantum measurement and quantum coherence, and the necessity of explicitly considering an auxiliary system to be a measurement apparatus rather than implicitly attempting to perform an energy measurement.

Several works have already suggested the use of ancillas as the measurement apparatus for extracting work statistics. In [26, 27] a Ramsey scheme using an auxiliary qubit was proposed to measure the work characteristic function. Its experimental realization with nuclear magnetic resonance was reported in [8]. In [28], a different approach was taken, in which a detector, for example the momentum of a quantum particle or a light mode [29], was coupled to the system to directly extract the work-probability distribution following a measurement of the detector position at the final time. This scheme was recently realized with cold atoms [13]. If the detector fully discriminates the different energies, the effects of the initial quantum coherence of the system cannot be detected. Conversely, if the detector cannot fully discriminate the different energies, for example, in the Gaussian measurement scheme [30, 31] in which the detector is initially prepared in a Gaussian distribution of the positional eigenstate [17, 13], some effects of initial quantum coherence can be observed. However, Gaussian measurement is not quantum because the quantum effect of the state of the detector is neglected. For this, Solinas et al considered that the detector was prepared in the coherent state, and the work value was estimated through the phase change of the coherent state [16]. Although the coherent state is also a Gaussian distribution of position, it is a pure quantum state and includes quantum features that Gaussian measurement does not possess. However, the fundamental connections between measurement error and quantum work distribution are still unclear. We note that measurement error can be modulated by squeezing the state of the detector. In this paper, we give a unified framework of quantum work statistics for an arbitrary initial state by using the squeezing state, and reveal the fundamental connections between measurement error and quantum work distribution. The coherent or squeezing state is the quantum extension of classical phase space [32-36]; the study of work distribution in the coherent or squeezing state has a direct correspondence with the classical case.

This paper is organized as follows: in the next section, we give a unified framework of quantum work statistics for an arbitrary initial state based on a coherent or squeezing state. As an application, we consider a driven two-level system in section 3. Finally, section 4 closes the paper with some concluding remarks. In the appendix, we give a brief review of the key concepts of the coherent state and the squeezing state.

2. A work measurement scheme using the coherent or squeezing state

First, we demonstrate the general external work protocol considered in this paper: consider a driven closed system described by a Hamiltonian
(1)$\begin{eqnarray}{H}_{s}({\lambda }_{t})=\displaystyle \sum _{n}{E}_{t}^{n}| {E}_{t}^{n}\rangle \langle {E}_{t}^{n}| ,\end{eqnarray}$
where λ_t is the external controlled parameter which is changed from its initial value λ₀ to the final value ${\lambda }_{t^{\prime} }$ during the time interval $t^{\prime} $. In the spectral decomposition, $| {E}_{t}^{n}\rangle $ is the nth eigenvector of ${\hat{H}}_{s}({\lambda }_{t})$ with the corresponding eigenvalue Eⁿ_t. In order to determine the work performed by the external protocol, it is necessary to explicitly consider a measurement apparatus, rather than implicitly appealing to the performance of an energy measurement. Recently, a single-point measurement scheme that directly samples the quantum work distribution was proposed in [28], where the momentum of a quantum particle (auxiliary detector) was coupled to the system and then the position of the particle was shifted by an amount that depended on the energy change of the system. This scheme was experimentally realized by a cloud of ⁸⁷Rb atoms [13], in which the system was represented by the Zeeman sublevels of an ⁸⁷Rb atom that behaved as a two-level system; the motional degree of freedom of the atom played the role of the detector [37]. We note that the initial motional state is a wave-packet localised in position; this makes this single-point measurement scheme equivalent to TMP in that the initial quantum coherence of the system is completely destroyed. To include the effects of initial quantum coherence, we consider that the auxiliary detector is supposed to be initially prepared in the squeezed vacuum state.

For clarity, we consider the auxiliary detector to be a harmonic oscillator and review the main idea of the single-point measurement scheme in [28]. The Hamiltonian of the auxiliary detector is
(2)$\begin{eqnarray}{\hat{H}}_{a}={\hslash }{\omega }_{a}({\hat{a}}^{\dagger }\hat{a}+1/2),\end{eqnarray}$
where ω_a is the oscillating frequency of the harmonic oscillator. To determine the work, one needs the following five steps (see figure 1): (1) at time t < − τ, the auxiliary detector a and the system s are prepared in a product state ${\hat{\rho }}_{a}(0)\otimes {\hat{\rho }}_{s}(0)$; (2) in order to know the initial system energy, at time t = − τ, a is coupled to s with the Hamiltonian ${\hat{H}}_{{sa}}({\lambda }_{0})=-g\hat{p}{\hat{H}}_{s}({\lambda }_{0})$, where g is the coupling strength. Notably, this interaction Hamiltonian does not influence the statistics of the initial system energy. The evolution of the total s + a system is described by ${\hat{U}}_{{sa}}({\lambda }_{0})=\exp \{-{\rm{i}}[{\hat{H}}_{s}({\lambda }_{0})+{\hat{H}}_{a}+{\hat{H}}_{{sa}}({\lambda }_{0})]\tau /{\hslash }\}$. We assume that the time interval τ is short enough, i.e., the time interval τ is much shorter than the oscillating period of the harmonic oscillator and the characteristic time of the system (satisfying ω_aτ ≈ 0 and ${\omega }_{s}^{\max }\tau \approx 0$, where ${\omega }_{s}^{\max }$ is the maximum transition frequency of the system). In this case, ${\hat{U}}_{{sa}}({\lambda }_{0})\approx \exp \{-{\rm{i}}{\hat{H}}_{{sa}}({\lambda }_{0})\tau /{\hslash }\}$. (3) Following the transient evolution of the total s + a system, the coupling is removed at time t = 0, and then a protocol is performed on s whereby the work parameter is changed from its initial value λ₀ to the final value ${\lambda }_{t^{\prime} }$. This process is governed by the unitary operator ${\hat{U}}_{s}(t^{\prime} )=\overleftarrow{T}\exp \left[-\tfrac{{\rm{i}}}{{\hslash }}{\int }_{0}^{t^{\prime} }{\hat{H}}_{s}({\lambda }_{t}){\rm{d}}t\right]$. (4) Subsequently, a is recoupled with s with the Hamiltonian ${\hat{H}}_{{sa}}({\lambda }_{t^{\prime} })=g\hat{p}{\hat{H}}_{s}({\lambda }_{t^{\prime} })$, and the transient evolution operator is ${\hat{U}}_{{sa}}({\lambda }_{t^{\prime} })\approx \exp \{-{\rm{i}}{\hat{H}}_{{sa}}({\lambda }_{t^{\prime} })\tau /{\hslash }\}$. (5) Perform the measurement (projective or weak) using the detector. The information of the work is recorded in the measured results of the detector, and one can obtain the work statistics through these measurement results.

Figure 1.

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Figure 1.Schematic representation of the work measurement scheme. Initially, the system and the detector are prepared in ${\hat{\rho }}_{a}(0)$ and ${\hat{\rho }}_{s}(0);$ They are then coupled to each other and evolved by U_sa(λ₀) over a short time τ; after the transient evolution, the coupling is removed; the work parameter has changed from its initial value λ₀ to the final value ${\lambda }_{t^{\prime} }$ and the system has evolved by ${U}_{s}(t^{\prime} );$ the system and the detector are recoupled with each other and evolved by ${U}_{{sa}}({\lambda }_{t^{\prime} })$ over a short time τ; finally, a measurement (triangle) is performed on the detector.


To include the effects of quantum coherence, the auxiliary detector is supposed to be initially prepared in the squeezed vacuum state (the basic concepts of the coherent state and the squeezing state are given in the appendix)
(3)$\begin{eqnarray}{\hat{\rho }}_{a}(0)={\hat{S}}_{a}(r)| 0{{ \rangle }_{aa}}_{} \langle 0| {\hat{S}}_{a}^{\dagger }(r),\end{eqnarray}$
where ∣0⟩_a is the vacuum state of the auxiliary detector, ${\hat{S}}_{a}(r)=\exp \left\{\tfrac{r}{2}\left[{\hat{a}}^{2}-{\left({\hat{a}}^{\dagger }\right)}^{2}\right]\right\}$ is the squeezing operator, and r ≥ 0 is the squeezing strength (here, the squeezing is performed only in real axis). Because $\hat{p}=-{\rm{i}}\sqrt{{\hslash }{\omega }_{a}/2}(\hat{a}-{\hat{a}}^{\dagger })$, ${\hat{U}}_{{sa}}({\lambda }_{t})\,\approx \exp \{-{\hat{H}}_{s}({\lambda }_{t}){\hat{a}}^{\dagger }+{\hat{H}}_{s}({\lambda }_{t})\hat{a}\}={\hat{D}}_{a}(-{\hat{H}}_{s}({\lambda }_{t}))$ (here, we let $g\tau /\sqrt{2}=1$, ℏ = 1, and ω_a = 1), which can be understood as meaning that the displacement operator acting on the vacuum state of harmonic oscillator ∣0⟩_a can generate the coherent state, depending on the energy level of the system, e.g. ${\hat{D}}_{a}(-{\hat{H}}_{s}({\lambda }_{t}))| 0{\rangle }_{a}\otimes | {E}_{t}^{n}\rangle =| -{E}_{t}^{n}{\rangle }_{a}\otimes | {E}_{t}^{n}\rangle $. When ${\hat{D}}_{a}[{\hat{H}}_{s}({\lambda }_{t})]$ acts on the squeezed vacuum state ${\hat{S}}_{a}(r)| 0{\rangle }_{a}$, it generates the squeezed state whose parameters depend on the energy level of the system, e.g. ${\hat{D}}_{a}[-{\hat{H}}_{s}({\lambda }_{t})]{\hat{S}}_{a}(r)| 0{\rangle }_{a}\otimes | {E}_{t}^{n}\rangle \,={\hat{S}}_{a}(r)D[-{{\rm{e}}}^{r}{\hat{H}}_{s}({\lambda }_{t})]| 0{\rangle }_{a}\,\otimes | {E}_{t}^{n}\rangle ={\hat{S}}_{a}(r)| -{{\rm{e}}}^{r}{E}_{t}^{n}{\rangle }_{a}\otimes | {E}_{t}^{n}\rangle \,=| -{{\rm{e}}}^{r}{E}_{t}^{n},r{\rangle }_{a}\otimes | {E}_{t}^{n}\rangle $. The squeezing state $| -{{\rm{e}}}^{r}{E}_{t}^{n},r{\rangle }_{a}$ can be considered as the coherent state $| -{{\rm{e}}}^{r}{E}_{t}^{n}{\rangle }_{b}$ in the representation $\hat{b}=\hat{a}\cosh r+{\hat{a}}^{\dagger }\sinh r$ [see equation (A17) of the appendix for a detailed discussion].

After step (4), the state of the auxiliary detector is
(4)$\begin{eqnarray}{\hat{\rho }}_{a}(t\mbox{'})=\mathop{}\limits_{{lmn}}{\rho }_{s}^{{mn}}(0){U}_{{lm}}{U}_{{ln}}^{* }| {\beta }_{r}^{{lm}}{{ \rangle }_{bb}}_{} \langle {\beta }_{r}^{{ln}}| ,\end{eqnarray}$
where ${\rho }_{s}^{{mn}}(0)=\langle {E}_{0}^{m}| {\hat{\rho }}_{s}(0)| {E}_{0}^{n}\rangle $, ${U}_{{lm}}=\langle {E}_{t^{\prime} }^{l}| {\hat{U}}_{s}(t^{\prime} )| {E}_{0}^{m}\rangle $, and ${\beta }_{r}^{{lm}}={{\rm{e}}}^{r}({E}_{t^{\prime} }^{l}-{E}_{0}^{m})$. For a closed system, the change of internal energy corresponds to work, i.e., $W={E}_{t^{\prime} }^{l}-{E}_{0}^{m}$. In other words, the work value is proportional to the real part of the parameter of coherent state ∣$\beta$_r⟩_b, thus the work can be estimated by
(5)$\begin{eqnarray}W=\mathrm{Re}({\beta }_{r})/{{\rm{e}}}^{r}.\end{eqnarray}$
According to $\overline{{[\mathrm{Re}({\beta }_{r})]}^{n}}={\sum }_{m=0}^{m=n}{C}_{n}^{m}{2}^{-n}{\pi }^{-1}\int \langle {\beta }_{r}| {\left({\hat{b}}^{\dagger }\right)}^{n-m}{\hat{\rho }}_{a}(t^{\prime} ){\hat{b}}^{m}| {\beta }_{r}\rangle {{\rm{d}}}^{2}{\beta }_{r}={2}^{-n}{\sum }_{m=0}^{m=n}{C}_{n}^{m}\mathrm{Tr}\{{\hat{b}}^{m}{\left({\hat{b}}^{\dagger }\right)}^{n-m}{\hat{\rho }}_{a}(t^{\prime} )\},$ the nth moment of work is
(6)$\begin{eqnarray}\begin{array}{l}\langle {W}^{n}\rangle \equiv \displaystyle \int { \mathcal P }(W){W}^{n}{\rm{d}}W\\ \quad =\ \displaystyle \frac{{{\rm{e}}}^{-{nr}}}{{2}^{n}}\displaystyle \sum _{m=0}^{m=n}{C}_{n}^{m}\mathrm{Tr}\{{\hat{b}}^{m}{\left({\hat{b}}^{\dagger }\right)}^{n-m}{\hat{\rho }}_{a}(t^{\prime} )\},\end{array}\end{eqnarray}$
where ${C}_{n}^{m}=\tfrac{n!}{m!(n-m)!}$. The first line of equation (6) is the definition of the nth moment of work using the quantum work distribution ${ \mathcal P }(W)$, and the second line of equation (6) is its calculation using equation (5), which can be used to estimate quantum work distribution ${ \mathcal P }(W)$. Because ${\hat{b}}^{m}{\left({\hat{b}}^{\dagger }\right)}^{n-m}$ is the antinormal ordering operator, $\mathrm{Tr}\{{\hat{b}}^{m}{\left({\hat{b}}^{\dagger }\right)}^{n-m}{\hat{\rho }}_{a}{(t^{\prime} )\}=\int {{\rm{d}}}^{2}{\beta }_{r}Q({\beta }_{r},{\beta }_{r}^{* }){\beta }_{r}^{m}({\beta }_{r}^{* })}^{n-m}$. In other words,
(7)$\begin{eqnarray}\int { \mathcal P }(W){W}^{n}{\rm{d}}W=\int \int \mathrm{dIm}({\beta }_{r}){{\rm{e}}}^{r}Q({\beta }_{r},{\beta }_{r}^{* }){W}^{n}{\rm{d}}W,\end{eqnarray}$
so the quantum work distribution is
(8)$\begin{eqnarray}{ \mathcal P }(W)=\int \mathrm{dIm}({\beta }_{r}){{\rm{e}}}^{r}Q({\beta }_{r},{\beta }_{r}^{* }).\end{eqnarray}$
According to equation (4), the quantum work distribution is
(9)$\begin{eqnarray}\begin{array}{l}{ \mathcal P }(W)=\displaystyle \sum _{{ln}}{P}_{n}{P}_{l| n}{ \mathcal N }\left(W| {E}_{t^{\prime} }^{l}-{E}_{0}^{n},\sigma \right)\\ \quad +\ \displaystyle \sum _{{{lmn}}_{m\ne n}}{\varrho }_{s}^{{mn}}(0){U}_{{ln}}{U}_{{lm}}^{* }{ \mathcal N }\left(W| {E}_{t^{\prime} }^{l}-\displaystyle \frac{{E}_{0}^{m}+{E}_{0}^{n}}{2},\sigma \right),\end{array}\end{eqnarray}$
where ${P}_{n}={\rho }_{s}^{{nn}}(0)$ is the initial energy distribution, and ${ \mathcal N }{(W| \mu ,\sigma )\equiv \exp \{-(W-\mu )}^{2}/(2{\sigma }^{2})\}/(\sqrt{2\pi }\sigma )$ is the normal distribution of W, where $\mu$ is the the average value, $\sigma =\sqrt{2}{\rm{\Delta }}q$ is the variance or the measurement error, and ${\rm{\Delta }}q=\sqrt{\mathrm{Tr}{[{\hat{\rho }}_{a}(t^{\prime} ){\hat{q}}^{2}]-\mathrm{Tr}[{\hat{\rho }}_{a}(t^{\prime} )\hat{q}]}^{2}}={{\rm{e}}}^{-r}/2$ is the standard deviation of the position of the detector. ${\varrho }_{s}^{{mn}}{(0)={\rho }_{s}^{{mn}}(0)\exp \{-({E}_{0}^{m}-{E}_{0}^{n})}^{2}/(4{\sigma }^{2})\}$ is the off-diagonal element of the system density matrix after removing the coupling in step (3). The first line of equation (9) can be understood as incoherent work distribution, i.e., ${{ \mathcal P }}_{{in}}(W)={\sum }_{{ln}}{P}_{n}{P}_{l| n}{ \mathcal N }\left(W| {E}_{t^{\prime} }^{l}-{E}_{0}^{n},\sigma \right)$, which stems from the incoherent part of the initial state ${\rho }_{{in}}(0)\,\equiv {\rho }_{s}^{{nn}}(0)| {E}_{0}^{n}\rangle \langle {E}_{0}^{n}| $. The second line of equation (9) can be understood as coherent work distribution, i.e., ${{ \mathcal P }}_{c}(W)\,={\sum }_{{{lmn}}_{m\ne n}}{\varrho }_{s}^{{mn}}(0){U}_{{ln}}{U}_{{lm}}^{* }{ \mathcal N }\left(W| {E}_{t^{\prime} }^{l}-\tfrac{{E}_{0}^{m}+{E}_{0}^{n}}{2},\sigma \right)$, which stems from the coherent part of the initial state ${\rho }_{c}(0)\,\equiv {\rho }_{s}^{{mn}}(0)| {E}_{0}^{m}\rangle \langle {E}_{0}^{n}| $. The incoherent work value W not only depends on the corresponding energy level transition with ${E}_{m}^{t^{\prime} }-{E}_{n}^{0}=W$, but depends on all the transitions of the energy levels according to the Gaussian distribution form. In other words, the incoherent part of the work-probability distribution (9) is a coarse-grained version of the TPM result, where the Dirac delta functions have been replaced by Gaussians with measurement error σ. The transitions with the energy difference ${E}_{t^{\prime} }^{m}-{E}_{0}^{n}=W$ gives the greatest contribution, and the more detuning between the work value W and the energy level difference ${E}_{t^{\prime} }^{m}-{E}_{0}^{n}$, the less the contribution of this energy level transition. It should be noted that although both are Gaussian, our quantum work distribution differs significantly from those in Gaussian measurements [17, 13, 30, 31]. Gaussian measurement is essentially classical and the quantum resource of the detector is not being exploited, but our measurement scheme is quantum and takes advantage of the quantum properties of the detector. It is also worth noting that Solinas et al used the phase change of the coherent state to estimate work value [16], but our measurement scheme can reveal the fundamental connections between measurement error and quantum work distribution, which will be shown in the following.

From equation (9), it can be seen that quantum work not only depends on the initial energy distribution P_m but also depends on the off-diagonal element of the density matrix ${\rho }_{s}^{{mn}}(0)$, or quantum coherence. The off-diagonal element of the density matrix ${\rho }_{s}^{{mn}}(0)$, or quantum coherence, can be considered as the information that can be used to perform the work. Depending on the energy difference (${E}_{0}^{m}-{E}_{0}^{n}$) between the initial energy levels, $| {E}_{0}^{m}\rangle $ and $| {E}_{0}^{n}\rangle $, their coherence also contributes to the work according to the Gaussian distribution form. The less the energy difference (${E}_{0}^{m}-{E}_{0}^{n}$) between energy levels, the more the contribution of their coherence. For the same energy difference, the quantum coherence between the energy levels $| {E}_{0}^{m}\rangle $ and $| {E}_{0}^{n}\rangle $ with ${E}_{t^{\prime} }^{l}-({E}_{0}^{m}+{E}_{0}^{n})/2\,=\,W$ gives the greatest contribution. The uncertainty of the position of the detector makes the work distribution imprecise (it cannot recover the result of TMP), but can survive some effects of initial quantum coherence from TPM. For the precise measurement, i.e., σ = 0, the quantum work-probability distribution is reduced to the result of TPM, i.e., ${\mathrm{lim}}_{r\to \infty }{ \mathcal P }(W)={\sum }_{{mn}}{P}_{n}{P}_{m| n}\delta (W\,-({E}_{t^{\prime} }^{m}-{E}_{0}^{n}))$, in which only the transitions with the energy difference ${E}_{t^{\prime} }^{m}-{E}_{0}^{n}=W$ contribute to the work with the value W.

After the Fourier transformation of the quantum work distribution $\chi =\int {\rm{d}}W{ \mathcal P }(W)\exp ({\rm{i}}\kappa W)$, the characteristic function of the quantum work distribution can be expressed as
(10)$\begin{eqnarray}\begin{array}{l}\chi ={{\rm{e}}}^{-{\kappa }^{2}{\sigma }^{2}/2}\mathrm{Tr}\left[{{\rm{e}}}^{{\rm{i}}\kappa {\hat{{ \mathcal H }}}_{s}({\lambda }_{t^{\prime} })}{{\rm{e}}}^{-{\rm{i}}\kappa {\hat{{ \mathcal H }}}_{s}({\lambda }_{0})/2}{\hat{\varrho }}_{s}(0){{\rm{e}}}^{-{\rm{i}}\kappa {\hat{{ \mathcal H }}}_{s}({\lambda }_{0})/2}\right],\end{array}\end{eqnarray}$
where ${\hat{{ \mathcal H }}}_{s}({\lambda }_{t})={\hat{U}}_{s}^{\dagger }(t){\hat{H}}_{s}({\lambda }_{t}){\hat{U}}_{s}(t)$ is the system Hamiltonian at time t in the Heisenberg picture. If the system is initially in the thermal equilibrium state ${\hat{\rho }}_{G}=\exp \{-\beta {\hat{H}}_{s}({\lambda }_{0})\}/Z({\lambda }_{0})$, where $\beta$ = 1/(k_BT) is the inverse of the temperature, $Z({\lambda }_{0})=\mathrm{Tr}[\exp \{-\beta {\hat{H}}_{s}({\lambda }_{0})\}]$ is the partition function, and k_B is the Boltzmann constant, we can obtain the modified Jarzynski equality (the quantum fluctuation relation) by letting κ = i$\beta$:
(11)$\begin{eqnarray}\langle \exp \{-\beta W\}\rangle =\exp \{-\beta {\rm{\Delta }}F\}\exp \left\{\displaystyle \frac{{\beta }^{2}{\sigma }^{2}}{2}\right\},\end{eqnarray}$
where ${\rm{\Delta }}F={k}_{B}T\mathrm{ln}[Z({\lambda }_{t^{\prime} })/Z({\lambda }_{0})]$ is the variation of the Helmholtz free energy. This modified Jarzynski equality is consistent with the result of [30].

All the moments of the work done can be obtained by ⟨Wⁿ⟩ = ( − i)ⁿ∂ⁿχ/∂κⁿ∣_κ=0. The average work is
(12)$\begin{eqnarray}\langle W\rangle =\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{t^{\prime} }){\hat{\varrho }}_{s}(t^{\prime} )]-\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{0}){\hat{\varrho }}_{s}(0)],\end{eqnarray}$
where ${\hat{\varrho }}_{s}(t^{\prime} )={\hat{U}}_{s}(t^{\prime} ){\hat{\varrho }}_{s}(0){\hat{U}}_{s}^{\dagger }(t^{\prime} )$ is the evolution of the state after the first interaction with the detector. The average work can be divided into incoherent work and coherent work ⟨W⟩ = ⟨W_in⟩ + ⟨W_c⟩, where $\langle {W}_{{in}}\rangle \equiv \int W{{ \mathcal P }}_{{in}}(W){\rm{d}}W=\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{t^{\prime} }){\hat{\varrho }}_{s}^{{in}}(t^{\prime} )]\,-\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{0}){\hat{\varrho }}_{s}^{{in}}(0)]$ is the incoherent work and $\langle {W}_{c}\rangle \,\equiv \int W{{ \mathcal P }}_{c}(W){\rm{d}}W=\langle {W}_{c}\rangle =\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{t^{\prime} }){\hat{\varrho }}_{s}^{c}(t^{\prime} )]$ is the coherent work, where ${\hat{\varrho }}_{s}^{{in}}(t^{\prime} )={\hat{U}}_{s}(t^{\prime} ){\hat{\varrho }}_{s}^{{in}}(0){\hat{U}}_{s}^{\dagger }(t^{\prime} )$, ${\hat{\varrho }}_{s}^{c}(t^{\prime} )\,={\hat{U}}_{s}(t^{\prime} ){\hat{\varrho }}_{s}^{c}(0){\hat{U}}_{s}^{\dagger }(t^{\prime} )$, ${\hat{\varrho }}_{s}^{{in}}(0)={\hat{\rho }}_{s}^{{in}}(0)={\sum }_{m}{\rho }_{s}^{{mm}}(0)| {E}_{0}^{m}\rangle \langle {E}_{0}^{m}| $ and ${\hat{\varrho }}_{s}^{c}(0)\,={{\sum }_{m,n\ne m}{\rho }_{s}^{{mn}}(0)\exp \{-({E}_{0}^{m}-{E}_{0}^{n})}^{2}/(4{\sigma }^{2})\}| {E}_{0}^{m}\rangle \langle {E}_{0}^{n}| $. We can see that only coherent work is influenced by measurement, i.e., if the system is in the coherent state, it will be inevitably affected by measurement and the average work is not equal to the internal energy, i.e., $\langle W\rangle \ne {\rm{\Delta }}U\,\equiv \mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{t^{\prime} }){\hat{\rho }}_{s}(t^{\prime} )]-\mathrm{Tr}[{\hat{H}}_{s}({\lambda }_{0}){\hat{\rho }}_{s}(0)]$, where ${\hat{\rho }}_{s}(t^{\prime} )={\hat{U}}_{s}(t^{\prime} ){\hat{\rho }}_{s}(0){\hat{U}}_{s}^{\dagger }(t^{\prime} )$. From ${\varrho }_{s}^{{mn}}(0)={\rho }_{s}^{{mn}}(0)\exp {\{-({E}_{0}^{m}-{E}_{0}^{n})}^{2}/(4{\sigma }^{2})\}$, we can see that if σ = 0, ${\varrho }_{s}^{{mn}}(0)=0$; the initial quantum coherence is completely destroyed by measurement. If σ ≠ 0, ${\varrho }_{s}^{{mn}}(0)\ne 0$, this means that the measurement error protects the quantum coherence. If σ → ∞ , ${\varrho }_{s}^{{mn}}(0)\,\to {\rho }_{s}^{{mn}}(0)$. At first glance, it seems that the greater the error, the more the quantum coherence is protected. However, it should be noted that a measurement with excessive measurement error is meaningless, because you cannot obtain any information about quantum work. To describe this tradeoff, we define the 'coherent work-noise ratio'∣⟨W_c⟩∣/σ by analogy with the 'information-noise ratio’. For both σ → 0 and σ → ∞ , ∣⟨W_c⟩∣/σ → 0. If the system is a two-level system, the optimal measurement error that maximizes the 'coherent work-noise ratio'∣⟨W_c⟩∣/σ is $\sigma \,=\tfrac{\sqrt{2}}{2}{\rm{\Delta }}E$, where ΔE is the energy difference between the two levels. This means that the optimal measurement error is determined by the energy difference between the superposed energy levels.

The second-order moment of the quantum work is
(13)$\begin{eqnarray}\langle {W}^{2}\rangle =\mathrm{Tr}\{{\hat{{ \mathcal H }}}_{s}({\lambda }_{t^{\prime} })-{\hat{{ \mathcal H }}}_{s}({\lambda }_{0}){]}^{2}{\hat{\varrho }}_{s}(0)\}+{\sigma }^{2}.\end{eqnarray}$
The work fluctuation can be expressed as
(14)$\begin{eqnarray}\delta {W}^{2}=\delta {\left({\rm{\Delta }}{{ \mathcal H }}_{s}\right)}^{2}+{\sigma }^{2},\end{eqnarray}$
where $\delta {({\rm{\Delta }}{{ \mathcal H }}_{s})}^{2}=\mathrm{Tr}\{{\hat{{ \mathcal H }}}_{s}({\lambda }_{t^{\prime} })-{\hat{{ \mathcal H }}}_{s}({\lambda }_{0}){]}^{2}{\hat{\varrho }}_{s}(0)\}-\mathrm{Tr}\{{\hat{{ \mathcal H }}}_{s}({\lambda }_{t^{\prime} })\,-{\hat{{ \mathcal H }}}_{s}({\lambda }_{0})]{\hat{\varrho }}_{s}(0)\}{}^{2}$ is the variance of the change of the internal energy under the influence of the measurement. From equation (14), it can be seen that the fluctuation of work for our measurement scheme is the variance of the change of the internal energy under the influence of the measurement, plus the measurement error.

3. A driven two-level system

As an application, in this section, we consider a nuclear spin system modulated by a radio frequency (rf) field in the transverse (x and y) directions and investigate the effects of initial quantum coherence on work distribution. This nuclear spin system can be realized in a liquid-state nuclear magnetic resonance setup, and is widely used to experimentally investigate the quantum work distribution and fluctuation relation [8]. The Hamiltonian of the nuclear spin system is (ℏ = 1)
(15)$\begin{eqnarray}{\hat{H}}_{s}(t)=\nu (t)\left({\hat{\sigma }}_{x}\cos \displaystyle \frac{\pi t}{2t^{\prime} }+{\hat{\sigma }}_{y}\sin \displaystyle \frac{\pi t}{2t^{\prime} }\right),\end{eqnarray}$
where ${\hat{\sigma }}_{x}=| 1\rangle \langle 0| +| 0\rangle \langle 1| $ and ${\hat{\sigma }}_{y}=-{\rm{i}}| 1\rangle \langle 0| +{\rm{i}}| 0\rangle \langle 1| $ are the Pauli operators, and $\nu (t)={\nu }_{0}(1-t/t^{\prime} )+{\nu }_{t^{\prime} }t/t^{\prime} $ is the linear ramp of the rf field frequency over time $t^{\prime} $, from ν₀ to ${\nu }_{t^{\prime} }$, $t\in [0,t^{\prime} ]$. To investigate the effects of quantum coherence, we consider that the nuclear spin system is initially in the so-called coherent Gibbs state
(16)$\begin{eqnarray}| \psi (0)\rangle =\sqrt{{{\rm{e}}}^{-\beta {\nu }_{0}}/{Z}_{0}}| +\rangle +\sqrt{{{\rm{e}}}^{\beta {\nu }_{0}}/{Z}_{0}}| -\rangle ,\end{eqnarray}$
where $| \pm \rangle =\tfrac{1}{\sqrt{2}}(| 1\rangle \pm | 0\rangle )$ is the eigenvector of ${\hat{\sigma }}_{x}$ with the corresponding eigenvalue±1, and ${Z}_{0}={{\rm{e}}}^{-\beta {\nu }_{0}}+{{\rm{e}}}^{\beta {\nu }_{0}}$. Z₀ is the partition function of Gibbs state ${\hat{\rho }}_{G}({\nu }_{0})={{\rm{e}}}^{-\beta {\hat{H}}_{s}(0)}/{Z}_{0}$ with a temperature of 1/$\beta$. It should be noted that ∣ψ(0)⟩ and ${\hat{\rho }}_{G}({\nu }_{0})$ are energetically indistinguishable because they have the same diagonal elements; in this sense, we call parameter $\beta$ in ∣ψ(0)⟩ the 'temperature'or the 'effective temperature’.

According to equation (9), the work distribution performed by the rf field on the nuclear spin system is
(17)$\begin{eqnarray}{ \mathcal P }(W)={{ \mathcal P }}_{+}(W)+{{ \mathcal P }}_{-}(W)+{{ \mathcal P }}_{c}(W),\end{eqnarray}$
where
(18)$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal P }}_{\pm }(W) & = & \displaystyle \frac{{{\rm{e}}}^{\mp \beta {\nu }_{0}}}{{Z}_{0}}| {U}_{{\rm{i}},\pm }{| }^{2}{ \mathcal N }\left(W| {\nu }_{t^{\prime} }\mp {\nu }_{0},\sigma \right)\\ & & +\ \displaystyle \frac{{{\rm{e}}}^{\mp \beta {\nu }_{0}}}{{Z}_{0}}| {U}_{-{\rm{i}},\pm }{| }^{2}{ \mathcal N }\left(W| -{\nu }_{t^{\prime} }\mp {\nu }_{0},\sigma \right),\end{array}\end{eqnarray}$
is the work distribution for the energy level ∣ ± ⟩, and
(19)$\begin{eqnarray}{{ \mathcal P }}_{c}(W)=\tfrac{2{{\rm{e}}}^{-\tfrac{{\nu }_{0}^{2}}{{\sigma }^{2}}} \quad \mathrm{Re}({U}_{{\rm{i}},+}{U}_{{\rm{i}},-}^{* })}{{Z}_{0}}\left[{ \mathcal N }(W| {\nu }_{t^{\prime} },\sigma )-{ \mathcal N }(W| -{\nu }_{t^{\prime} },\sigma )\right],\end{eqnarray}$
is the work distribution induced by the initial quantum coherence. In the above equations, ${U}_{\pm {\rm{i}},\pm }=\langle \pm {\rm{i}}| {\hat{U}}_{s}(t^{\prime} )| \pm \rangle $, $| \pm {\rm{i}}\rangle \,=\tfrac{1}{\sqrt{2}}(| 1\rangle \pm {\rm{i}}| 0\rangle )$ is the eigenvector of ${\hat{\sigma }}_{y}$ with the corresponding eigenvalue±1, and ${\hat{U}}_{s}(t^{\prime} )=\overleftarrow{T}\exp \left[-\tfrac{{\rm{i}}}{{\hslash }}{\int }_{0}^{t^{\prime} }{\hat{H}}_{s}(t){\rm{d}}t\right]$ is the time evolution operator, which needs to be calculated numerically. Figure 2 shows the work distribution for different evolution periods $t^{\prime} $. From figure 2, we can see that for the quench process $t^{\prime} =0.01{\nu }_{0}^{-1}$ and the adiabatic process $t^{\prime} =100{\nu }_{0}^{-1}$, the initial quantum coherence has no effect on the work distribution, but for the finite process $t^{\prime} ={\nu }_{0}^{-1}$, the initial quantum coherence can make a significant contribution to work distribution. Besides, we also find that the work distribution performed by an adiabatic process is Gaussian (see figure 2(c)). Although [38, 39] also pointed this result out very recently, their conditions were quite different from ours. In [38, 39], the system is continuous and open, but our system is finite and closed. At first glance, our Gaussian distribution is induced by the squeezing state of the detector, but it should be noted that any adiabatic process will minimize the fluctuations, so that all higher cumulants beyond σ disappear. In this sense, our Gaussian result for adiabatic process might be universal.

Figure 2.

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Figure 2.Work distribution ${ \mathcal P }(W)$ (red dashed curve) and the corresponding incoherent part ${{ \mathcal P }}_{+}(W)+{{ \mathcal P }}_{-}(W)$ (black solid curve) for (a) $t^{\prime} =0.01{\nu }_{0}^{-1}$, (b) $t^{\prime} ={\nu }_{0}^{-1}$ and (c) $t^{\prime} =100{\nu }_{0}^{-1}$. For panels (a) and (c), ${ \mathcal P }(W)$ and ${{ \mathcal P }}_{+}(W)+{{ \mathcal P }}_{-}(W)$ are coincident with each other. For all panels, ${\nu }_{t^{\prime} }=1.8{\nu }_{0}$, σ = ν₀, $\beta =0.01{\nu }_{0}^{-1}$ and ν₀ = 1.


For the Gaussian work distribution, the work fluctuation is completely determined by the modified fluctuation-dissipation theorem:
(20)$\begin{eqnarray}\langle {W}_{{irr}}\rangle =\tfrac{1}{2}\beta {(\delta {W}^{2}-{\sigma }^{2})=\tfrac{1}{2}\beta \delta ({\rm{\Delta }}{{ \mathcal H }}_{s})}^{2},\end{eqnarray}$
where ⟨W_irr⟩ = ⟨W⟩ − ΔF is the irreversible work and ${\rm{\Delta }}F=-\tfrac{1}{\beta }\mathrm{ln}\tfrac{{Z}_{t^{\prime} }}{{Z}_{0}}$ is the difference of free energy with ${Z}_{t^{\prime} }=\exp [-\beta {\nu }_{t^{\prime} }]+\exp [\beta {\nu }_{t^{\prime} }]$. It should be noted that our fluctuation-dissipation theorem connects the irreversible work and the fluctuation of the internal energy change after the work is measured, which is different from the traditional fluctuation-dissipation theorem $\langle {W}_{{irr}}\rangle =\tfrac{1}{2}\beta \delta {W}^{2}$ in [40]. The traditional fluctuation-dissipation theorem is obtained by TPM, in which initial quantum coherence is destroyed; our fluctuation-dissipation theorem is derived by the single-point measurement scheme, in which the initial quantum coherence is partially preserved by introducing the measurement error. The measurement error is finally removed in our modified fluctuation-dissipation theorem (20).

We now investigate the average work (the first moment of work) and the work fluctuation (the second moment of work). The average work can be expressed as
(21)$\begin{eqnarray}\langle W\rangle =\langle {W}_{+}\rangle +\langle {W}_{-}\rangle +\langle {W}_{c}\rangle ,\end{eqnarray}$
where
(22)$\begin{eqnarray}\langle {W}_{\pm }\rangle =\displaystyle \frac{{{\rm{e}}}^{\mp \beta {\nu }_{0}}}{{Z}_{0}}\left[| {U}_{{\rm{i}},\pm }{| }^{2}({\nu }_{t^{\prime} }\mp {\nu }_{0})-| {U}_{-{\rm{i}},\pm }{| }^{2}({\nu }_{t^{\prime} }\pm {\nu }_{0})\right],\end{eqnarray}$
is the average work for the energy level ∣ ± ⟩, and
(23)$\begin{eqnarray}\langle {W}_{c}\rangle =\displaystyle \frac{4}{{Z}_{0}}{{\rm{e}}}^{-{\nu }_{0}^{2}/{\sigma }^{2}}\mathrm{Re}({U}_{{\rm{i}},+}{U}_{{\rm{i}},-}^{* }){\nu }_{t{\prime} }.\end{eqnarray}$


The incoherent part of average work ⟨W₊⟩ + ⟨W₋⟩ is shown in figure 3(a). We can see that the amount of incoherent work is related to the driving velocity $1/t^{\prime} $ or the driving period $t^{\prime} $. If ${\nu }_{0}t^{\prime} \lt 1$, the rf field performs positive work, but if ${\nu }_{0}t^{\prime} \gt 1$, the rf field performs negative work on the nuclear spin system. We also plot the fluctuation of the difference of internal energy $\tfrac{1}{2}\beta \delta {({\rm{\Delta }}{{ \mathcal H }}_{s})}^{2}$ (see red curves). The fluctuation of the difference of internal energy is greatest for the quench process, and it is reduced when the driving velocity is decreased, until it reaches a minimum for the adiabatic process. This can be understood as follows: besides internal energy, the quench process can also provide extra energy that causes the system to fluctuate significantly; on the contrary, the adiabatic process only provides internal energy, and has no extra energy to fluctuate the system. Interestingly, the behavior of ⟨W₊⟩ + ⟨W₋⟩ is similar to that of the fluctuation of the difference of internal energy $\tfrac{1}{2}\beta \delta {({\rm{\Delta }}{{ \mathcal H }}_{s})}^{2}$, which means that the fluctuation of the change of internal energy is mainly determined by incoherent work.

Figure 3.

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Figure 3.(a) Incoherent work ⟨W₊⟩ + ⟨W₋⟩ (black curve) and the fluctuation of the difference of internal energy $\tfrac{1}{2}\beta \delta {({\rm{\Delta }}{{ \mathcal H }}_{s})}^{2}$ (red curve) and (b) coherent work ⟨W_c⟩ (olive curve) as functions of the evolution period $t^{\prime} $. ${\nu }_{t^{\prime} }=1.8{\nu }_{0}$, σ = ν₀, $\beta ={10}^{-2}{\nu }_{0}^{-1}$ and ν₀ = 1.


The coherent part of the work is shown in figure 3(b). It can be seen that the coherent work for an adiabatic process ${\nu }_{0}t^{\prime} \sim \infty $ is zero. That is because the adiabatic process cannot induce an energy level transition, thus initial quantum coherence cannot be used to do work. The coherent work for the quench process ${\nu }_{0}t^{\prime} \sim 0$ is zero as well; because the time required for the quench process is much shorter than the energy level transition time, the initial quantum coherence has no time to contribute work. Only when the driving time matches the transition time of the energy level (for the nuclear spin system we consider it is $t^{\prime} \approx {\nu }_{0}^{-1}$), does initial quantum coherence contribute significant work.

4. Conclusions

In this paper, we extended the traditional TPM by proposing a unified framework of quantum work statistics for an arbitrary initial state (including quantum coherence). Specifically, we considered a harmonic oscillator with a coherent state or a squeezing state as a detector. The momentum of the detector is coupled to the system Hamiltonian, and then the real part of the coherent state parameter of the detector is linearly changed by the change of system energy; therefore, the work can be directly estimated by a single-point quantum measurement of the coherent state of the detector at the final time. The resulting work-probability distribution is positive and can be reduced to the result of the TPM. To be specific, the incoherent part of our work-probability distribution is the coarse-grained version of the TPM result, where the Dirac delta functions have been replaced by Gaussians. The uncertainty of our measurement scheme protects some of the effects of the initial quantum coherence. Through our measurement scheme, the fundamental connection between measurement error and coherent work is revealed and the optimal measurement error is given by defining a 'coherent work-noise ratio’. Physically, the optimal measurement error is determined by the energy difference between superposed energy levels. Finally, we also considered a driven two-level system as an example, and found that only when the driving velocity matched the transition frequency of the system did initial quantum coherence play an important role.

Appendix. Coherent state and squeezing state

We begin by reviewing some key concepts of the coherent state and the squeezing state, allowing us to define the formalism that is used in the rest of our study. The details of the coherent state and the squeezing state can be found in the seminal papers [32-36]. The coherent state is an eigenstate of the annihilation operator $\hat{a}$ with the eigenvalue $\alpha$, i.e.,
(A1)$\begin{eqnarray}\hat{a}| \alpha {\rangle }_{a}=\alpha | \alpha {\rangle }_{a}.\end{eqnarray}$
Because $\hat{a}$ is not a Hermitian operator, $\alpha$ is a complex number. An expression of ∣$\alpha$⟩ in terms of the number state ∣n⟩_a is given by
(A2)$\begin{eqnarray}| \alpha {\rangle }_{a}={{\rm{e}}}^{-| \alpha {| }^{2}/2}\displaystyle \sum _{n}\displaystyle \frac{{\alpha }^{n}}{\sqrt{n!}}| n{\rangle }_{a}.\end{eqnarray}$
Since $| n{\rangle }_{a}=\tfrac{{\hat{a}}^{\dagger }}{\sqrt{n!}}| 0{\rangle }_{a}$ and $\exp (-{\alpha }^{* }\hat{a})| 0{\rangle }_{a}=| 0{\rangle }_{a}$, the coherent state can then be expressed as
(A3)$\begin{eqnarray}| \alpha {\rangle }_{a}={\hat{D}}_{a}(\alpha )| 0{\rangle }_{a},\end{eqnarray}$
where ∣0⟩_a is the vacuum state with zero photons and
(A4)$\begin{eqnarray}{\hat{D}}_{a}(\alpha )=\exp (\alpha {\hat{a}}^{\dagger }-{\alpha }^{* }\hat{a}),\end{eqnarray}$
is the displacement operator. The displacement operator ${\hat{D}}_{a}(\alpha )$ is a unitary operator, i.e., ${\hat{D}}_{a}^{\dagger }(\alpha )={\hat{D}}_{a}(-\alpha )={\hat{D}}_{a}^{-1}(\alpha )$, and has the properties ${\hat{D}}_{a}^{\dagger }(\alpha )\hat{a}{\hat{D}}_{a}(\alpha )=\hat{a}+\alpha $ and ${\hat{D}}_{a}^{\dagger }(\alpha ){\hat{a}}^{\dagger }{\hat{D}}_{a}(\alpha )={\hat{a}}^{\dagger }+{\alpha }^{* }$. Different coherent states are not orthogonal to each other, i.e.,
(A5)$\begin{eqnarray}{}_{a}\langle \alpha | \alpha ^{\prime} {\rangle }_{a}=\exp \left(-\displaystyle \frac{1}{2}| \alpha {| }^{2}+\alpha ^{\prime} {\alpha }^{* }-\displaystyle \frac{1}{2}| \alpha ^{\prime} {| }^{2}\right),\end{eqnarray}$
from which it follows that ${| }_{a}\langle \alpha | \alpha ^{\prime} {\rangle }_{a}{| }^{2}=\exp (-| \alpha -\alpha ^{\prime} {| }^{2})$. If $\alpha$ and $\alpha ^{\prime} $ are quite different, i.e., $| \alpha -\alpha ^{\prime} | \gg 1$, then $| \alpha ^{\prime} {\rangle }_{a}$ and ∣$\alpha$⟩_a are nearly orthogonal. Another consequence of their non-orthogonality is that the coherent states form an overcomplete basis, i.e.,
(A6)$\begin{eqnarray}\frac{1}{\pi }\int {{\rm{d}}}^{2}\alpha | \alpha {{ \rangle }_{aa}}_{} \langle \alpha | ={\mathbb{I}}\end{eqnarray}$
where ${\mathbb{I}}$ is the identity matrix and ${{\rm{d}}}^{2}\alpha =\mathrm{dRe}(\alpha )\mathrm{dIm}(\alpha )$. One of the applications of the overcompleteness relation of the coherent state is to calculate the trace, i.e.,
(A7)$\begin{eqnarray}\mathrm{Tr}[\hat{A}]=\displaystyle \frac{1}{\pi }\int {}_{a}\langle \alpha | \hat{A}| \alpha {\rangle }_{a}{{\rm{d}}}^{2}\alpha ,\end{eqnarray}$
where $\hat{A}$ is an arbitrary operator.

The coherent state is closest to the classical state because it is a minimum-uncertainty state, in which the uncertainties of momentum $\hat{p}=-{\rm{i}}\sqrt{{\hslash }{\omega }_{a}/2}(\hat{a}-{\hat{a}}^{\dagger })$ and coordinate $\hat{q}=\sqrt{{\hslash }/(2{\omega }_{a})}(\hat{a}+{\hat{a}}^{\dagger })$ satisfy ΔqΔp = ℏ/2 (in fact, ${\rm{\Delta }}q={\rm{\Delta }}p=\sqrt{{\hslash }/2}$), where ω_a is the oscillation frequency of the harmonic oscillator. In this sense, the coherent state can be used to establish the quantum extension of phase space. To be specific, the complex plane of $\alpha$ corresponds to phase space, and $\mathrm{Re}(\alpha )$ and $\mathrm{Im}(\alpha )$ correspond to the classical coordinates and momentum, respectively. We should note that $\mathrm{Re}(\alpha )$ and $\mathrm{Im}(\alpha )$ are not the eigenvalues of $\hat{q}$ and $\hat{p}$, in fact $\mathrm{Re}(\alpha )=\langle \hat{q}\rangle $ and $\mathrm{Im}(\alpha )=\langle \hat{p}\rangle $ with $\langle \hat{q}\rangle $ and $\langle \hat{p}\rangle $ are the average of $\hat{q}$ and $\hat{p}$. In the quantum extension of phase space, the average of a microscopic observable $\hat{F}(\hat{a},{\hat{a}}^{\dagger })$ can be written as an integral of the product of a weight function w($\alpha$, $\alpha$^*) and a function F($\alpha$, $\alpha$^*) which refers to the operator $\hat{F}(\hat{a},{\hat{a}}^{\dagger })$,
(A8)$\begin{eqnarray}\langle \hat{F}(\hat{a},{\hat{a}}^{\dagger })\rangle \equiv \mathrm{Tr}[\hat{\rho }\hat{F}(\hat{a},{\hat{a}}^{\dagger })]=\int w(\alpha ,{\alpha }^{* })F(\alpha ,{\alpha }^{* }){{\rm{d}}}^{2}\alpha ,\end{eqnarray}$
where $\hat{\rho }$ is the density matrix. This expression is similar to the phase space integrals in classical statistical mechanics. The average of the normal ordering operator ${\hat{F}}_{N}(\hat{a},{\hat{a}}^{\dagger })\,={\sum }_{{mn}}{c}_{{mn}}{\left({\hat{a}}^{\dagger }\right)}^{m}{\hat{a}}^{n}$ can be expressed as
(A9)$\begin{eqnarray}\langle {\hat{F}}_{N}(\hat{a},{\hat{a}}^{\dagger })\rangle =\int P(\alpha ,{\alpha }^{* }){F}_{N}(\alpha ,{\alpha }^{* }){{\rm{d}}}^{2}\alpha ,\end{eqnarray}$
where
(A10)$\begin{eqnarray}P(\alpha ,{\alpha }^{* })=\mathrm{Tr}[\hat{\rho }\delta ({\alpha }^{* }-{\hat{a}}^{\dagger })\delta (\alpha -\hat{a})],\end{eqnarray}$
is the P representation. In general, P($\alpha$, $\alpha$^*) is an extremely singular function. The average of the antinormal ordering operator ${\hat{F}}_{A}(\hat{a},{\hat{a}}^{\dagger })={\sum }_{{mn}}{c}_{{mn}}{\hat{a}}^{m}{\left({\hat{a}}^{\dagger }\right)}^{n}$ can be expressed as
(A11)$\begin{eqnarray}\langle {\hat{F}}_{A}(\hat{a},{\hat{a}}^{\dagger })\rangle =\int Q(\alpha ,{\alpha }^{* }){F}_{A}(\alpha ,{\alpha }^{* }){{\rm{d}}}^{2}\alpha ,\end{eqnarray}$
where
(A12)$\begin{eqnarray}Q(\alpha ,{\alpha }^{* })=\frac{1}{\pi }{}_{a}\langle \alpha | \hat{\rho }| \alpha {\rangle }_{a}\end{eqnarray}$
is the Q representation. The average of the symmetric ordering operator ${\hat{F}}_{S}(\hat{a},{\hat{a}}^{\dagger })={\sum }_{{mn}}{c}_{{mn}}[{\hat{a}}^{m}{\left({\hat{a}}^{\dagger }\right)}^{n}+{\left({\hat{a}}^{\dagger }\right)}^{m}{\hat{a}}^{n}]$ can be expressed as
(A13)$\begin{eqnarray}\langle {\hat{F}}_{S}(\hat{a},{\hat{a}}^{\dagger })\rangle =\int W(\alpha ,{\alpha }^{* }){F}_{S}(\alpha ,{\alpha }^{* }){{\rm{d}}}^{2}\alpha ,\end{eqnarray}$
where
(A14)$\begin{eqnarray}W(\alpha ,{\alpha }^{* })=\displaystyle \frac{1}{{\pi }^{2}}\int {{\rm{d}}}^{2}\nu \mathrm{Tr}\left[\hat{\rho }\exp [\nu ({\hat{a}}^{\dagger }-{\alpha }^{* })-{\nu }^{* }(\hat{a}-\alpha )]\right],\end{eqnarray}$
is the Wigner-Weyl distribution. The Wigner-Weyl distribution is always a smooth function, but it can take negative values.

The squeezing state is defined as
(A15)$\begin{eqnarray}| \alpha ,\xi {\rangle }_{a}={\hat{S}}_{a}(\xi )| \alpha {\rangle }_{a},\end{eqnarray}$
where
(A16)$\begin{eqnarray}{\hat{S}}_{a}(\xi )=\exp \left[\displaystyle \frac{1}{2}\left({\xi }^{* }{\hat{a}}^{2}-\xi {\left({\hat{a}}^{\dagger }\right)}^{2}\right)\right],\end{eqnarray}$
is the squeezing operator and ξ = re^iθ is an arbitrary complex number. The squeezing operator ${\hat{S}}_{a}(\xi )$ is a unitary operator, i.e., ${\hat{S}}_{a}^{\dagger }(\xi )={\hat{S}}_{a}^{-1}(\xi )={\hat{S}}_{a}(-\xi )$. The squeezing state ∣$\alpha$, ξ⟩_a is also a minimum-uncertainty state such that ΔqΔp = ℏ/2, but the uncertainty of $\hat{q}$ or $\hat{p}$ can be ${\rm{\Delta }}q\lt \sqrt{{\hslash }}/2$ or ${\rm{\Delta }}p\,\lt \sqrt{{\hslash }}/2$, (depending on ξ), and this is the meaning of squeezing. From equation (A3), the squeezing state can be rewritten as
(A17)$\begin{eqnarray}| \alpha ,\xi {\rangle }_{a}={\hat{D}}_{b}(\alpha )| 0{\rangle }_{b}=| \alpha {\rangle }_{b},\end{eqnarray}$
where ${\hat{D}}_{b}(\alpha )=\exp (\alpha {\hat{b}}^{\dagger }-{\alpha }^{* }\hat{b})$ is the displacement operator and $| 0{\rangle }_{b}={\hat{S}}_{a}(\xi )| 0{\rangle }_{a}$ is the vacuum state in the new representation $\hat{b}=\hat{S}(\xi )\hat{a}{\hat{S}}^{\dagger }(\xi )=\hat{a}\cosh r+{\hat{a}}^{\dagger }{{\rm{e}}}^{{\rm{i}}\theta }\sinh r$ and ${\hat{b}}^{\dagger }=\hat{S}(\xi ){\hat{a}}^{\dagger }{\hat{S}}^{\dagger }(\xi )={\hat{a}}^{\dagger }\cosh r+\hat{a}{{\rm{e}}}^{-{\rm{i}}\theta }\sinh r$. In this sense, the squeezing state in the representation $\hat{a}$ is the coherent state in the representation $\hat{b}$. The vacuum state in the representation $\hat{b}$ is also called the squeezing vacuum state in the representation $\hat{a}$.

Acknowledgments

B.-M.X. acknowledges the support of the National Natural Science Foundation of China through Grant No. 11 705 099 and the Talent Introduction Project of Dezhou University of China through Grant No. 30 101 437. J.Z. acknowledges the support of the National Natural Science Foundation of China through Grant No. 11 675 017. Z.-C.T. acknowledges the support of the National Natural Science Foundation of China through Grant No. 11 775 019.

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