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arXiv:2307.08792 · Physical Review A

Quantum coherence and the principle of microscopic reversibility

K. Khan, W. F. Magalhães, J. S. Araújo, Bertúlio de Lima Bernardo, G. H. Aguilar

Published Nov 15, 2023• 3 citations
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Abstract

The principle of microscopic reversibility is a fundamental element in the formulation of fluctuation relations and Onsager reciprocal relations. As such, a clear description of whether and how this principle is adapted to the quantum-mechanical scenario might be essential for a better understanding of nonequilibrium quantum processes. Here, we propose a quantum generalization of this principle which highlights the role played by coherence in the symmetry relations involving the probability of observing a quantum transition and that of the corresponding time-reversed process. We study the implications of our findings in the framework of a qubit system interacting with a thermal reservoir and implement an optical experiment that simulates the dynamics. Our theoretical and experimental results show that the influence of coherence is more decisive at low temperatures and that the maximum departure from the classical case does not take place for maximally coherent states. Classical predictions are recovered in the appropriate limits.

DOI
10.1103/physreva.108.052215

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The principle of microscopic reversibility is a fundamental element in the formulation of fluctuation relations and Onsager reciprocal relations. As such, a clear description of whether and how this principle is adapted to the quantum-mechanical scenario might be essential for a better understanding of nonequilibrium quantum processes. Here, we propose a quantum generalization of this principle which highlights the role played by coherence in the symmetry relations involving the probability of observing a quantum transition and that of the corresponding time-reversed process. We study the implications of our findings in the framework of a qubit system interacting with a thermal reservoir and implement an optical experiment that simulates the dynamics. Our theoretical and experimental results show that the influence of coherence is more decisive at low temperatures and that the maximum departure from the classical case does not take place for maximally coherent states. Classical predictions are recovered in the appropriate limits.
The principle of microscopic reversibility lies at the core of fluctuation theorems, which have extended our understanding of the second law of thermodynamics to the statistical level. In the quantum regime, however, this elementary principle should be amended as the system energy cannot be sharply determined at a given quantum phase space point. In this Letter, we propose and experimentally test a quantum generalization of the microscopic reversibility when a quantum system interacts with a heat bath through energy-preserving unitary dynamics. Quantum effects can be identified by noting that the backward process is less likely to happen in the existence of quantum coherence between the system's energy eigenstates. The experimental demonstration has been realized by mixing coherent and thermal states in a beam splitter, followed by heterodyne detection in an optical setup. We verify that the quantum modification for the principle of microscopic reversibility is critical in the low-temperature limit, while the quantum-to-classical transition is observed as the temperature of the thermal field gets higher.
Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises as Bayesian retrodiction, both in classical and quantum theories. Fluctuation relations then become expressions of the statistical divergence between the forward (i.e., predictive) and the reverse (i.e., retrodictive) process. With the reverse process constructed on consistent logical inference, rather than on particular balancing mechanisms in microscopic physics, fluctuation relations acquire a much broader form and scope. Previous paradigmatic results like Jarzynski's and Crooks' fluctuation relations are recovered as special cases.
A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.
Symmetry principles have played a crucial role in the development of modern physics and underpin our most fundamental theories of nature. The present thesis is concerned with the analysis of symmetries in the context of quantum information theory. Whenever a quantum mechanical system interacts with its environment it is subjected to decoherence, a process that is typically irreversible and most generally treated abstractly using the concept of quantum operations. If there is an underlying symmetry principle, additional structures emerge. The central question we address is, What are the consequences of global or local gauge symmetry on the structure of many-body quantum processes?This leads us to a diagrammatic framework of decomposing quantum operations into terms that respond to the symmetry principle in particular ways and respect the causal structures involved. We present two core applications. First, we address the interplay between irreversibility and repeatable use of coherent resources under symmetry constraints. Second, we give an information-theoretic perspective on gauging globally symmetric dynamics to a local symmetry applicable even in the presence of irreversibility and thus it goes beyond the usual Lagrangian formulation. Finally, we analyse the departure from conservation laws under symmetric dynamics subject to decoherence.
The first half of this thesis is dedicated to the study of generalizations of probability theory which include quantum theory and many of its alternatives. This research program seeks to understand quantum theory by exploring it ‘from the outside’, where it sits within a landscape of all operationally defined theories. We approach this problem with a new perspective: generalized decoherence. Quantum theory recovers classical theory via decoherence, where a ‘classical limit’ is achieved by a quantum system interacting with its environment, losing its quantum coherence. Typically it is the emergence of the classical world from the quantum that is studied. We reverse this paradigm and ask - how does the existence of a classical limit define the structure of quantum theory, or any other post-classical theory? We derive the existence of entangled states as a necessary feature of any theory that allows for the emergence of the classical world in this way. We then use our framework for generalized decoherence to explore the properties of theories that can ‘hyperdecohere’ to quantum theory. We find that any such theory must be very exotic compared to quantum theory, violating either the postulate of tomographic locality, reversibility, causality or purity. In the second half of this thesis we turn our attention to the second law of thermodynamics. We present a framework for deriving the second law under general constraints and explore two pertinent examples, constraining the fluctuations in work and constraining the size of the thermal bath and deriving the second law in these cases. Bounding fluctuations results in a unified free energy that contains the single-shot and Helmholtz free energies as limiting cases. Bounding the size of the thermal bath results in a finite-bath second law that is more general that previous attempts and draws connections between non-asymptotic thermodynamics and second-order information theory.
We establish the general framework of quantum fluctuation theorems by finding the symmetry between the forward and backward transitions of any given quantum channel. The Petz recovery map is adopted as the reverse quantum channel, and the notion of entropy production in thermodynamics is extended to the quantum regime. Our result shows that the fluctuation theorems, which are normally considered for thermodynamic processes, can be a powerful tool to study the detailed statistics of quantum systems as well as the effect of coherence transfer in an arbitrary non-equilibrium quantum process. We introduce a complex-valued entropy production to fully understand the relation between the forward and backward processes through the quantum channel. We find the physical meaning of the imaginary part of entropy production to witness the broken symmetry of the quantum channel. We also show that the imaginary part plays a crucial role in deriving the second law from the quantum fluctuation theorem. The dissipation and fluctuation of various quantum resources including quantum free energy, asymmetry and entanglement can be coherently understood in our unified framework. Our fluctuation theorem can be applied to a wide range of physical systems and dynamics to query the reversibility of a quantum state for the given quantum processing channel involving coherence and entanglement.