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.
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.
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.
We review the use of an external auxiliary detector for measuring the full distribution of the work performed on or extracted from a quantum system during a unitary thermodynamic process. We first illustrate two paradigmatic schemes that allow one to measure the work distribution: a Ramsey technique to measure the characteristic function and a positive operator valued measure (POVM) scheme to directly measure the work probability distribution. Then, we show that these two ideas can be understood in a unified framework for assessing work fluctuations through a generic quantum detector and describe two protocols that are able to yield complementary information. This allows us also to highlight how quantum work is affected by the presence of coherences in the system’s initial state. Finally, we describe physical implementations and experimental realisations of the first two schemes.
For a quantum system undergoing a unitary, process work is commonly defined based on the two projective measurement protocols, which measures the energies of the system before and after the process. However, it is well known that projective measurements disregard quantum coherences of the system with respect to the energy basis, thus removing potential quantum signatures in the work distribution. Here we consider weak measurements of the system's energy difference and establish corrections to work averages arising from initial system coherences. We discuss two weak measurement protocols that couple the system to a detector, prepared and measured either in the momentum or the position eigenstates. Work averages are derived for when the system starts in the proper thermal state versus when the initial system state is a pure state with thermal diagonal elements and coherences characterized by a set of phases. We show that by controlling only the phase differences between the energy eigenstate contributions in the system's initial pure state, the average work done during the same unitary process can be controlled. By changing the phases alone, one can toggle from regimes where the system absorbs energy, i.e., a work cost, to the ones where it emits energy, i.e., work can be drawn. This suggests that the coherences are additional resources that can be used to manipulate or store energy in a quantum system.
We consider the simultaneous measurement of two conjugate variables by coupling the system of interest to two independent probe modes. Our model consists of linearly coupled boson modes that can be realized by quantum optical fields in the rotating-wave approximation. We approach the setup both as a device to extract observable information and to prepare an emerging quantum state. The initial states of the probe modes and the coupling are utilized to optimize the operation in the various regimes. In contrast to the Arthurs and Kelly ideal scheme [Bell. Syst. Tech. J. 44, 725 (1965)], our more realistic coupling does not allow perfect operation but the ideal situations can be approximated closely. We discuss the conditions for maximum information transfer to the probe modes, information extraction with minimum disturbance of the system mode, and optimal state preparation for subsequent measurements. The minimum disturbance operation can be made to approximate a nondemolition measurement, especially when the information is carried in one quadrature component only. In the preparation mode, we find that the recording accuracy of the probe signals plays an essential role. We restrict the discussion to the first and second moments only, but the method can easily be generalized to any situation. Choosing all modes to be in squeezed coherent states originally, we can carry out analytical considerations; other cases can be treated numerically. The results are presented and discussed in detail as the paradigm of a class of realizable measurements.
We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical detector arises naturally and work and its statistics can be investigated in the most general case. In particular, we show that the quantum coherence of the initial state can lead to measurable effects on the moments of the work done on the system. At the same time, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our method can also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling the system to a detector, we can track the energy dissipated in the environment while accessing only the system degrees of freedom.
In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.