Bao-Ming Xu(徐寶明),Jian Zou(鄒健) and Zhan-Chun Tu(涂展春)

1 Shandong Key Laboratory of Biophysics,Institute of Biophysics,Dezhou University,Dezhou 253023,China

2 School of Physics,Beijing Institute of Technology,Beijing 100081,China

3 Department of Physics,Beijing Normal University,Beijing 100875,China

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

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

where λtis the external controlled parameter which is changed from its initial value λ0to the final valueλt′during the time intervalt′.In the spectral decomposition,is the nth eigenvector ofwith the corresponding eigenvalueIn 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 of87Rb atoms[13],in which the system was represented by the Zeeman sublevels of an87Rb atom that behaved as a twolevel 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 singlepoint measurement scheme in[28].The Hamiltonian of the auxiliary detector is

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)

After step(4),the state of the auxiliary detector is

so the quantum work distribution is

According to equation(4),the quantum work distribution is

All the moments of the work done can be obtained byThe average work is

The second-order moment of the quantum work is

The work fluctuation can be expressed as

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(?=1)

According to equation(9),the work distribution performed by the rf field on the nuclear spin system is

where

is the work distribution for the energy level|±〉,and

For the Gaussian work distribution,the work fluctuation is completely determined by the modified fluctuation–dissipation theorem:

where〈Wirr〉=〈W〉-ΔF is the irreversible work andis the difference of free energy withIt 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 theoremin[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

where

is the average work for the energy level|±〉,and

The coherent part of the work is shown in figure 3(b).It can be seen that the coherent work for an adiabatic processν0t′～∞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 processν0t′～0is 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 ist′≈),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.

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.

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 operatorwith the eigenvalue α,i.e.,

where|0〉ais the vacuum state with zero photons and

whereI is the identity matrix andd2α=dRe (α)dIm(α).One of the applications of the overcompleteness relation of the coherent state is to calculate the trace,i.e.,

where

is the P representation.In general,P(α,α*)is an extremely singular function.The average of the antinormal ordering operatorcan be expressed as

where

where

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

where