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Aspects of work in quantum thermodynamics

Cormac Browne

Published Jan 1, 2017• 0 citations
Publisher

Abstract

Landauer’s principle states that it costs at least kBT ln 2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of kBT ln 2 is achieved in the unphysical infinite-time limit. Here we consider two different finite-time protocols - one with discretised time and the second in the continuous limit. We prove analytically that the discrete time protocol enables one to reset a bit with a work cost close to kBT ln 2 in a finite time. We construct an explicit protocol that achieves this, which involves thermalising and changing the system’s Hamiltonian so as to avoid quantum coherences. Using concepts and techniques pertaining to single-shot statistical mechanics, we furthermore prove that the heat dissipated is exponentially close to the minimal amount possible not just on average, but guaranteed with high confidence in every run. Moreover we exploit the protocol to design a quantum heat engine that works near the Carnot efficiency in finite time. We further contrast this to a continuous time version of the protocol which is substantially less energy sufficient. We also consider the fluctuations in the work cost, and calculate how their magnitude is suppressed by a factor depending on the length of the protocol. We demonstrate with an experiment how molecules are a natural test-bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. By considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states, we demonstrate that the quantum Jarzynski equality is satisfied in this set-up. This relates the heat dissipated into the environment to the free energy difference between the initial and final state. We demonstrate also how utilizing the quantum Jarzynski equality allows for the detection of energy shifts within a molecule, beyond the relative shift.

DOI
10.5287/ora-z6o8oknow

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•Apr 28, 2016•0 citations
Landauer’s principle states that it costs at least kBT ln 2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of kBT ln 2 is achieved in the unphysical infinite-time limit. Here we consider two different finite-time protocols - one with discretised time and the second in the continuous limit. We prove analytically that the discrete time protocol enables one to reset a bit with a work cost close to kBT ln 2 in a finite time. We construct an explicit protocol that achieves this, which involves thermalising and changing the system’s Hamiltonian so as to avoid quantum coherences. Using concepts and techniques pertaining to single-shot statistical mechanics, we furthermore prove that the heat dissipated is exponentially close to the minimal amount possible not just on average, but guaranteed with high confidence in every run. Moreover we exploit the protocol to design a quantum heat engine that works near the Carnot efficiency in finite time. We further contrast this to a continuous time version of the protocol which is substantially less energy sufficient. We also consider the fluctuations in the work cost, and calculate how their magnitude is suppressed by a factor depending on the length of the protocol. We demonstrate with an experiment how molecules are a natural test-bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. By considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states, we demonstrate that the quantum Jarzynski equality is satisfied in this set-up. This relates the heat dissipated into the environment to the free energy difference between the initial and final state. We demonstrate also how utilizing the quantum Jarzynski equality allows for the detection of energy shifts within a molecule, beyond the relative shift.
Landauer's principle states that it costs at least kBTln2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of kBTln2 is achieved in the unphysical infinite-time limit. Here we ask what is possible if one is restricted to finite-time protocols. We prove analytically that it is possible to reset a bit with a work cost close to kBTln2 in a finite time. We construct an explicit protocol that achieves this, which involves thermalizing and changing the system's Hamiltonian so as to avoid quantum coherences. Using concepts and techniques pertaining to single-shot statistical mechanics, we furthermore prove that the heat dissipated is exponentially close to the minimal amount possible not just on average, but guaranteed with high confidence in every run. Moreover, we exploit the protocol to design a quantum heat engine that works near the Carnot efficiency in finite time.
We demonstrate with an experiment how molecules are a natural test bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. We demonstrate that the quantum Jarzynski equality for heat is satisfied in this set-up by considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states. This relates the heat dissipated into the environment to the free energy difference between the initial and final state. We demonstrate also how utilizing the quantum Jarzynski equality allows for the detection of energy shifts within a molecule, beyond the relative shift.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
The thermodynamic cost of resetting an arbitrary initial state to a particular desired state is lower bounded by Landauer's bound. However, here we demonstrate that this lower bound is necessarily unachievable for every initial state (except possibly the single minimally dissipative input) for any reliable reset mechanism. Since local heating threatens rapid decoherence, this issue is of substantial importance beyond mere energy efficiency. For the case of qubit reset, we find the minimally dissipative state analytically for any reliable reset protocol, in terms of the entropy-flow vector introduced here. This allows us to verify a recent theorem about initial-state dependence of entropy production for any finite-time transformation, as it pertains to quantum state preparation.
A nonstochastic quantum engine is one that operates in a cycle of transformations in which no sources of stochasticity, such as thermal baths and projective measurements, are present and, therefore, no entropy is generated in the driven system. Defining work and heat as the energy corresponding to different types of transformations between pure states, we arrive at an expression similar to the first law of thermodynamics and prove a version of the Kelvin-Planck statement for the second law of thermodynamics. Essentially, the first law can be obtained thanks to the normalization condition of a quantum state and the second law can be obtained thanks to the orthogonalization condition between energy eigenstates. For nonstochastic engines that operate between two given energy gaps, we prove a version of Carnot's theorem. Regarding operationalization, we present a protocol that leads the system through a cycle in which heat exchange occurs by performing two quantum quenches separated by a precise time interval and involving an energy-level anticrossing. Furthermore, with this protocol it is possible to make the engine's efficiency as close to 1 as one wants; however, efficiency equal to 1 is a case prohibited by the version of the Kelvin-Planck statement that we proved. Finally, we illustrate these results in an exactly solvable single-qubit model.
Recent developments of the analysis of the second law of thermodynamics by using the information theoretic quantities has led to the discovery of the so-called nonequilibrium free energy and has allowed us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states.By using the nonequilibrium free energies, we can quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines in a unified way.In those general situations, reducing dissipation allows us to increase the extractable work from the system (or to reduce the work cost), and reducing work fluctuation allows us to narrow the work probability distribution and to extract a fluctuation-free work from the system.Suppressing both work fluctuation and dissipation as much as possible is vital to control mesoscopic and nano systems that work at the level of thermal fluctuations.Previous studies have searched for a vanishing work fluctuation protocol by using techniques of quantum information theory, known as the single-shot statistical mechanics (and sometimes called the deterministic work extraction protocol) and the vanishing dissipation protocol, known as the thermodynamically reversible protocol by using the second law of thermodynamics.In this thesis, we address issues concerning a method to control nonequilibrium systems by using fluctuation theorems and trade-off relation between work fluctuation and fluctuation in dissipation.We derive quantum fluctuation theorems under measurement and feedback control, which allow us to study how to suppress dissipation during a general quantum measurement and a feedback control.We then consider a method to suppress work fluctuation and fluctuation in dissipation simultaneously as much as possible by showing the trade-off relation between work and dissipation in terms of their standard deviations and derive an explicit protocol that achieves the lower bound of the trade-off relation.We show that the explicit protocol reproduces the deterministic work extraction protocol in the limit of vanishing work fluctuation and the thermodynamically reversible protocol in the limit of vanishing fluctuation in dissipation.