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International Journal of Quantum Information

Detecting the work statistics through Ramsey-like interferometry

Laura Mazzola, Gabriele De Chiara, Mauro Paternostro

Published Mar 1, 2014• 17 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749914610073

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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting•Jan 1, 2013•0 citations
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.
We study an interferometric method for measuring the statistics of work performed on a driven quantum system, which has been put forward recently (Dorner et al 2013 Phys. Rev. Lett. 110 230601, Mazzola et al 2013 Phys. Rev. Lett. 110 230602). This method allows the replacement of two projective measurements of the energy of the driven system with qubit tomography of an ancilla that is appropriately coupled to it. We highlight that this method could be employed to obtain the work statistics of closed as well as open driven system, even in the strongly dissipative regime. We then illustrate an implementation of the method in a circuit QED setup, which allows one to experimentally obtain the work statistics of a parametrically driven harmonic oscillator. Our implementation is an extension of the original method, in which two ancilla-qubits are employed and the work statistics are retrieved through two-qubit state tomography. Our simulations demonstrate the experimental feasibility.
Very recently, interferometric methods have been proposed to measure the full statistics of work performed on a driven quantum system [Dorner et al., Phys. Rev. Lett. 110, 230601 (2013) and Mazzola et al., Phys. Rev. Lett. 110, 230602 (2013)]. The advantage of such schemes is that they replace the necessity to make projective measurements by performing phase estimation on an appropriately coupled ancilla qubit. These proposals are one possible route to the tangible experimental exploration of quantum thermodynamics, a subject which is the center of much current attention due to the current control of mesoscopic quantum systems. In this Rapid Communication we demonstrate that a modification of the phase estimation protocols can be used in order to measure the heat distribution of a quantum process. In addition, we demonstrate how our scheme maybe implemented using ion trap technology. Our scheme should pave the way for experimental explorations of the Landauer principle and hence the intricate energy to information conversion in mesoscopic quantum systems.
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.
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).