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arXiv:2006.13908 · Communications in Theoretical Physics

A single-point measurement scheme for quantum work based on the squeezing state

Bao-Ming Xu, Jian Zou, Z. C. Tu

Published Feb 23, 2021• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1572-9494/abe931

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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.
An open question of fundamental importance in thermodynamics is how to describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to simultaneously satisfy requirements (i) and (ii). Nevertheless, improvements do appear, and in particular, we develop a measurement scheme that acts simultaneously on two copies of the state and allows us to describe a whole class of coherent transformations.
What is the role of coherence in determining the distribution of work done on a quantum system? We approach this question from an operational perspective and consider a setup in which the internal energy of a closed system is recorded by a quantum detector before and after the system is acted upon by an external drive. We find that the resulting work distribution depends on the initial state of the detector as well as on the choice of the final measurement. We consider two complementary measurement schemes, both of which show clear signatures of quantum interference. We specifically discuss how to implement these schemes in the circuit QED architecture, using an artificial atom as the system and a quantized mode of the electromagnetic field as the detector. Different measurement schemes can be realized by preparing the field either in a superposition of Fock states or in a coherent state and exploiting state-of-the art techniques for the characterization of microwave radiation at the quantum level. More generally, the single bosonic mode we utilize is arguably the minimal quantum detector capable of capturing the complementary aspects of the work distribution discussed here.
The work is a concept of fundamental importance in thermodynamics. An open question is how to describe the work fluctuation for quantum coherent processes in the presence of initial quantum coherence in the energy basis. With the aim of giving a unified description, here we introduce and study a class of quasiprobability distributions of work, which give an average work equal to the average energy change of the system and reduce to the two-projective measurement scheme for an initial incoherent state. Moreover, we characterize the work with the help of fluctuation relations. In particular, by considering the joint distribution of work and initial quantum coherence, we find a fluctuation theorem involving quantum coherence, from which follows a second law of thermodynamics in the case of initial thermal populations. Furthermore, we propose a way to measure the characteristic function of work and we discuss the negativity of the quasiprobability. The effects of coherence are also investigated for a simple system of a qubit.
Quantum work statistics differ from classical ones because initial energy coherence matters. The standard two-point measurement (TPM) gives a positive distribution but erases phase information. Coherence-retaining endpoint-work quasistatistics provide a compact probe of shortcut-to-adiabaticity performance. For work defined with respect to a reference Hamiltonian, an exact counterdiabatic shortcut pulls the final reference Hamiltonian back to an operator diagonal in the initial energy basis. Endpoint Kirkwood-Dirac or Margenau-Hill quasistatistics then lose sensitivity to initial coherence and reduce to the TPM result. Imperfect shortcuts restore this sensitivity: a non-commuting control error produces off-diagonal pulled-back Hamiltonian elements at first order in the error amplitude, whereas population-only transition probabilities change only at second order. Harmonic-oscillator and qubit benchmarks confirm this linear-versus-quadratic contrast. The result complements inclusive work-cost analyses: it does not measure the auxiliary field's energetic cost, but provides a phase-sensitive endpoint diagnostic of residual nonadiabaticity.
An open question of fundamental importance in quantum thermodynamics is how to describe the statistics of work for the initial state with quantum coherence. In this paper, work statistics is considered from the perspective of ``wave-particle'' duality. Based on the generalized quantum work measurement, predictability of energy levels DW{\mathcal{D}}_{W}DW​ and effectiveness of coherence VW{\mathcal{V}}_{W}VW​ are defined, and they obey inequality ${\mathcal{D}}_{W}^{2}+{\mathcal{V}}_{W}^{2}\ensuremath{\le}1$, which is the fundamental tradeoff relations between the contributions of population and coherence to quantum work distribution. As an application, we consider a driven two-level system and discuss the condition of the bound of above tradeoff relation. These results shed light on the effects of quantum coherence in quantum thermodynamics.
A central challenge in quantum thermodynamics is to access work fluctuations in coherent processes without distorting the energetics of the unmeasured evolution. In standard two-point schemes, the initial energy measurement dephases coherent inputs, causing the measured average work to differ from that of the unmeasured evolution. Here, we develop an operational scheme for accessing work statistics for closed quantum systems based on the abstract notion of variation in the Heisenberg picture Hamiltonian. This scheme preserves energetically relevant coherences, thereby faithfully reproducing unmeasured work, while still producing positive probabilities. We derive modified Jarzynski and Crooks relations, as well as a thermodynamic uncertainty relation, identifying coherence-induced correction terms. Furthermore, we show that this scheme can reliably quantify the performance of a coherent engine in situations where the two-point energy measurement would suppress work output. In addition, the scheme requires only a single measurement and can predict the work associated with a subsequent unitary transformation. We exploit this feature to construct a Maxwell-demon protocol that can outperform energy-based feedback engines for coherent work extraction. Our results establish this scheme as a framework for accessing coherent work fluctuations without erasing the coherence that drives quantum thermodynamic performance.