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arXiv:1106.0276 · Physical Review A

Full counting statistics of energy fluctuations in a driven quantum resonator

A. A. Clerk

Published Oct 14, 2011• 45 citations
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Abstract

We consider the statistics of time-integrated energy fluctuations of a driven bosonic single-mode resonator, as measured by a quantum nondemolition (QND) detector, using the standard Keldysh prescription to define higher moments. We find that, due to an effective cascading of fluctuations, these statistics are surprisingly nonclassical: the low-temperature, quantum probability distribution is not equivalent to the high-temperature classical distribution evaluated at some effective temperature. Moreover, for a sufficiently large drive detuning and low temperatures, the Keldysh-ordered quasiprobability distribution characterizing these fluctuations fails to be positive-definite; this is similar to the full counting statistics of charge in superconducting systems. We argue that this indicates a kind of nonclassical behavior akin to that tested by Leggett-Garg inequalities.

DOI
10.1103/physreva.84.043824

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We consider the statistics of time-integrated energy fluctuations of a driven bosonic single-mode resonator, as measured by a quantum nondemolition (QND) detector, using the standard Keldysh prescription to define higher moments. We find that, due to an effective cascading of fluctuations, these statistics are surprisingly nonclassical: the low-temperature, quantum probability distribution is not equivalent to the high-temperature classical distribution evaluated at some effective temperature. Moreover, for a sufficiently large drive detuning and low temperatures, the Keldysh-ordered quasiprobability distribution characterizing these fluctuations fails to be positive-definite; this is similar to the full counting statistics of charge in superconducting systems. We argue that this indicates a kind of nonclassical behavior akin to that tested by Leggett-Garg inequalities.
The Keldysh-ordered full counting statistics is a quasiprobability distribution describing the fluctuations of a time-integrated quantum observable. While it is well known that this distribution can fail to be positive, the interpretation and origin of this negativity has been somewhat unclear. Here, we show how the full counting statistics can be tied to trajectories through Hilbert space, and how this directly connects negative quasiprobabilities to an unusual interference effect. Our findings are illustrated with the example of energy fluctuations in a driven bosonic resonator; we discuss how negative quasiprobability here could be detected experimentally using superconducting microwave circuits.
We provide a full quantum mechanical analysis of a weak energy measurement of a driven mechanical resonator. We demonstrate that measurements too weak to resolve individual mechanical Fock states can nonetheless be used to detect the nonclassical energy fluctuations of the driven mechanical resonator, i.e., "phonon shot noise". We also show that the third moment of the oscillator's energy fluctuations provides a far more sensitive probe of quantum effects than the second moment, and that measuring the third moment via the phase shift of light in an optomechanical setup directly yields the type of operator ordering postulated in the theory of full-counting statistics.
We develop a general framework to investigate fluctuations of non-commuting observables. To this end, we consider the Keldysh quasi-probability distribution (KQPD). This distribution provides a measurement-independent description of the observables of interest and their time-evolution. Nevertheless, positive probability distributions for measurement outcomes can be obtained from the KQPD by taking into account the effect of measurement back-action and imprecision. Negativity in the KQPD can be linked to an interference effect and acts as an indicator for non-classical behavior. Notable examples of the KQPD are the Wigner function and the full counting statistics, both of which have been used extensively to describe systems in the absence as well as in the presence of a measurement apparatus. Here we discuss the KQPD and its moments in detail and connect it to various time-dependent problems including weak values, fluctuating work, and Leggett-Garg inequalities. Our results are illustrated using the simple example of two subsequent, non-commuting spin measurements.
We study the fluctuations of the work performed on a driven quantum system, defined as the difference between subsequent measurements of energy eigenvalues. These work fluctuations are governed by statistical theorems with similar expressions in classical and quantum physics. We show that we can distinguish quantum and classical work fluctuations, as the latter can be described by a macrorealistic theory and hence obey Leggett-Garg inequalities. We show that these inequalities are violated by quantum processes in a driven two-level system and in a harmonic oscillator subject to a squeezing transformation.
The Leggett-Garg inequalities serve to test whether or not quantum correlations in time can be explained within a classical macrorealistic framework. We apply this test to thermodynamics and derive a set of Leggett-Garg inequalities for the statistics of fluctuating work done on a quantum system unitarily driven in time. It is shown that these inequalities can be violated in a driven two-level system, thereby demonstrating that there exists no general macrorealistic description of quantum work. These violations are shown to emerge within the standard Two-Projective-Measurement scheme as well as for alternative definitions of fluctuating work that are based on weak measurement. Our results elucidate the influences of temporal correlations on work extraction in the quantum regime and highlight a key difference between quantum and classical thermodynamics.
At nonzero temperature classical systems exhibit statistical fluctuations of thermodynamic quantities arising from the variation of the system's initial conditions and its interaction with the environment. The fluctuating work, for example, is characterized by the ensemble of system trajectories in phase space and, by including the probabilities for various trajectories to occur, a work distribution can be constructed. However, without phase-space trajectories, the task of constructing a work probability distribution in the quantum regime has proven elusive. Here we use quantum trajectories in phase space and define fluctuating work as power integrated along the trajectories, in complete analogy to classical statistical physics. The resulting work probability distribution is valid for any quantum evolution, including cases with coherences in the energy basis. We demonstrate the quantum work probability distribution and its properties with an exactly solvable example of a driven quantum harmonic oscillator. An important feature of the work distribution is its dependence on the initial statistical mixture of pure states, which is reflected in higher moments of the work. The proposed approach introduces a fundamentally different perspective on quantum thermodynamics, allowing full thermodynamic characterization of the dynamics of quantum systems, including the measurement process.