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arXiv:1301.7021 · Physical Review Letters

Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry

Ross Dorner, Stephen R. L. Clark, Liam Gabriel Heaney, Rosario Fazio, John Goold, Vlatko Vedral

Published Jun 7, 2013• 311 citations
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Abstract

We propose an experimental scheme to verify the quantum nonequilibrium fluctuation relations using current technology. Specifically, we show that the characteristic function of the work distribution for a nonequilibrium quench of a general quantum system can be extracted by Ramsey interferometry of a single probe qubit. Our scheme paves the way for the full characterization of nonequilibrium processes in a variety of quantum systems, ranging from single particles to many-body atomic systems and spin chains. We demonstrate our idea using a time-dependent quench of the motional state of a trapped ion, where the internal pseudospin provides a convenient probe qubit.

DOI
10.1103/physrevlett.110.230601

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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.

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We propose an experimental scheme to verify the quantum nonequilibrium fluctuation relations using current technology. Specifically, we show that the characteristic function of the work distribution for a nonequilibrium quench of a general quantum system can be extracted by Ramsey interferometry of a single probe qubit. Our scheme paves the way for the full characterization of nonequilibrium processes in a variety of quantum systems, ranging from single particles to many-body atomic systems and spin chains. We demonstrate our idea using a time-dependent quench of the motional state of a trapped ion, where the internal pseudospin provides a convenient probe qubit.
We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4].We first derive the Jarzynski relation [1] between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. We then comment on the information-theoretic underpinning of Jarzynski’s reasoning which helps explain why the Jarzynski relation holds identically both quantumly and classically [2]. We then present a scheme to verify the quantum non-equilibrium fluctuation relations as encapsulated by Jarzynski. We show that the characteristic function of the work distribution of a quantum system (which is basically equal to the Wick rotated average exponential of the thermodynamical work) can be extracted from Ramsey interferometry of a single probe qubit (which need not itself be pure, though it must not be fully depolarized) [3,4]. An interesting fact is that while the quantum version of the Jarzynski equality remains satisfied even in the presence of quantum correlations, the individual thermodynamical work moments in the expansion of the free energy are, in fact, sensitive to the genuine quantum correlations [5]. Whether this is a fortuitous coincidence remains to be seen, but it certainly goes towards explaining why the laws of thermodynamics happen to be so robust as to be independent of the underlying micro-physics.We intend to elucidate the subtle connection between Jarzynski’s relation and the “quantum arrow of time”. Finally we comment on the fact that our scheme for measuring the quantum work characteristic function belongs to the computational class known as DCQ1 [6], namely all computations that can be performed with only one pure qubit (and N maximally mixed ones).
We present an experimental scheme to measure the full distribution of many-body observables in spin systems, both in and out of equilibrium, using an auxiliary qubit as a probe. We focus on the determination of the magnetization and the kink number statistics at thermal equilibrium. The corresponding characteristic functions are related to the analytically continued partition function. Thus, both distributions can be directly extracted from experimental measurements of the coherence of a probe qubit that is coupled to an Ising-type bath, as reported in [X. Peng et al., Phys. Rev. Lett. 114, 010601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.010601] for the detection of Lee-Yang zeros.
Out-of-equilibrium statistical mechanics is attracting considerable interest due to the recent advances in the control and manipulations of systems at the quantum level. Recently, an interferometric scheme for the detection of the characteristic function of the work distribution following a time-dependent process has been proposed [L. Mazzola et al., Phys. Rev. Lett.110 (2013) 230602]. There, it was demonstrated that the work statistics of a quantum system undergoing a process can be reconstructed by effectively mapping the characteristic function of work on the state of an ancillary qubit. Here, we expand that work in two important directions. We first apply the protocol to an interesting specific physical example consisting of a superconducting qubit dispersively coupled to the field of a microwave resonator, thus enlarging the class of situations for which our scheme would be key in the task highlighted above. We then account for the interaction of the system with an additional one (which might embody an environment), and generalize the protocol accordingly.
Out-of-equilibrium statistical mechanics is attracting considerable interest due to the recent advances in the control and manipulations of systems at the quantum level. Recently, an interferometric scheme for the detection of the characteristic function of the work distribution following a time-dependent process has been proposed [L. Mazzola {\it et al}, Phys. Rev. Lett. {\bf 110} 230602 (2013)]. There, it was demonstrated that the work statistics of a quantum system undergoing a process can be reconstructed by effectively mapping the characteristic function of work on the state of an ancillary qubit. Here, we expand that work in two important directions. We first apply the protocol to an interesting specific physical example consisting of a superconducting qubit dispersively coupled to the field of a microwave resonator, thus enlarging the class of situations for which our scheme would be key in the task highlighted above. We then account for the interaction of the system with an additional one (which might embody an environment), and generalise the protocol accordingly.
The quantum fluctuation theorem can be regarded as the first principle of quantum nonequilibrium thermodynamics. However, many different formulations have been proposed, which depend on how quantum work is defined. In this context, we have seen that for some situations the one-time measurement (OTM) scheme can be more informative than the two-time measurement (TTM) scheme. Yet, so far the focus of OTM has been on integral fluctuation theorems, since, the work distribution of the backward process has been lacking. To this end, we prove that the OTM scheme is the nondestructive TTM scheme, in which the final state is a pointer state of the second measurement whose Hamiltonian is conditioned on the first measurement outcome. This insight leads to a derivation of the detailed fluctuation theorem for the characteristic functions of the forward and backward work distributions. Finally, our conceptual findings are experimentally verified with the IBM quantum computer.
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.