Quantum Thermodynamics
Quantum thermodynamics is the study of the interplay between two independent physical theories: Thermodynamics and quantum mechanics. Both theories address the same physical phenomena of light and matter. Learning from example has been the motto of thermodynamics since its inception by Carnot [1]. Comparing the predictions of the two theories on a specific example has been a source of insight. Einstein in 1905, while studying radiation in a cavity, postulated that the requirement of consistency between thermodynamics and electromagnetism require light to be quantised [2].
Thermodynamics is notorious in reducing the physical description to only a few variables. Systems at equilibrium can be fully characterised by one variable: their energy. The opposite is true in quantum mechanics where complexity of a complete description scales exponentially with the number of degrees of freedom or the number of particles. All operating quantum devices that convert heat to work or pump heat out of a cold bath operate far from equilibrium. This is reflected by the fundamental tradeoff between efficiency and power. A dynamical description of such devices require additional variables. Quantum theory can in principle supply a dynamical framework able to predict the evolution of systems far from equilibrium. Nevertheless the complexity of the treatment makes direct predictions impossible[3].
The field of quantum thermodynamics aims to bridge the two viewpoints [4, 5]. Our approach to the problem was always based on learning from an example; constructing a quantum model of a heat device and comparing the two theories [6–8]. A fundamental theoretical tool of quantum thermodynamics is the system-bath partition. The idea is toseek a reduced description of the system - meaning an explicit equation of motion for the system alone, where the bath is described implicitly. This is the turf of the theory of open quantum systems [9]. A major construction is the Lindblad Gorini-Kosakowsku-Sudarshan (LGKS) equation of motion known as the quantum Master equation [10, 11]. The equation describes Markovian dynamics (no memory) generated from a completely positive map [12].
Under this dynamical description at all times the system and bath are uncorrelated i.e. have a tensor product structure [4]. As a result system variables are well defined and can be directly related to thermodynamical predictions.
Recently we have shown by analysing an example of heat transport through a quantum wire that solutions based on local L-GKS violates the second law of thermodynamics [13]. We identified conditions where heat was flowing from the cold to the hot bath. This analysis is a typical learning from an example: Comparing the predictions of quantum theory and thermodynamics. The remedy to the discrepancy could be traced to the local construction. Employing a global construction of the master equation known as Davis construction, restored the consistency [14]. This method is based on the weak coupling limit between the system and bath. It requires a pre-diagonalization of the global system. For example the whole chain composing the quantum wire. The outcome of the procedure is a more complex L-GKS quantum master equation.
An important outcome of the analysis is that the popular approach to assemble Lego-like models of quantum devices is bound to violate the second law of thermodynamics. Quantum devices are global and therefore require an integrated approach.
Quantum heat engines and quantum refrigerators are the next level of complexity. A quantum model of such a device is composed of a system, a hot and cold baths and an external driving mechanism. If a wire has two leads, a quantum heat engine has three external leads referred to as a quantum tricycle [15]. Naive constructions of L-GKS equations for time dependent driving represented by a time dependent Hamiltonian violate the second law [16]. A thermodynamical consistent treatment has been worked out only for the case of periodic driving. An L-GKS master equation is derived by Floquet analysis separating the driving to its Fourier components [17, 18]. The result is a separate Master equation for each frequency component. Such a theory has been employed for models of heat engine and refrigerators [16, 19–21]. These models have been employed to explore the tradeoff between power and efficiency in thermal devices showing remarkable consistency with the phenomenological theory of finite time thermodynamics [22–26]. Due to the complexity of the Floquet procedure only very simple model system are amenable to analysis.
Can we construct a quantum dynamical theory consistent with thermodynamics which goes beyond the weak coupling Markovian limit? Can this model deal with a general time dependent external driving that could be the outcome of quantum control?
Recently [27] we have developed a quantum non-Markovian scheme for general system bath dynamics which is consistent with thermodynamics. The method is termed Stochastic Surrogate Hamiltonian (SSH). The main idea is to partition the bath to a primary and secondary part. The dynamics of the primary bath and the system is generated by a combined Hamiltonian. A wavefunction description is employed for the combined system. For this purpose we have employed a spin bath. The typical size of the combined Hilbert space is 108 − 109 states.
Quantum large and complex systems have a remarkable property termed quantum typicality. The expectation of almost any local property converges to a value which is independent of the details of the initial state [28–32]. We have observed that the system observables in SSH method converge extremely rapidly to typical values when the number of bath modes increases. As a result an extremely small number of stochastic realisations of wave-function calculations are sufficient to evaluate the systems properties.
The role of the secondary bath is to establish an outgoing thermodynamical boundary condition. We achieve this goal by stochastically applying partial or full swap between a pair of spins of the primary and secondary baths. The bath temperature is imposed by choosing the population of the secondary bath to conform to Boltzmann statistics. In addition the phase of these spins is random.
Since the spins that are swapped are in resonance the energy transfer between the primary and secondary bath is pure heat [33]. We have put the scheme to test in a heat transport model based on a molecular system coupled at each side to a hot and cold bath. The SSH, independent of the system construction, always showed the correct energy flow from the hot to the cold bath. In addition we discovered a heat rectifying effect; asymmetry in the heat current when the connection to the leads was reversed [34, 35].
We have thus established a fully quantum system-bath scheme which is not based on perturbation theory. At all times the system and bath are entangled which eliminates a basic assumption in the weak coupling theory. The system dynamics is non-Markovian and can be strongly driven.
Convergence can be verified by increasing the number of bath modes. The SSH reliance on wavefunctions has significant computational advantages. The scaling of the basic operation φ = Hˆ ψ is semi-linear with the size of Hilbert space (∝ N log N ). Time propagation is carried out by a
Chebychev polynomial expansion of the evolution equation which computation effort scales linearly with the propagation time [36, 37]. Recently we developed a new Chebychev propagator specifically for time dependent Hamiltonains [38] overcoming the time ordering issue. The stochastic surrogate Hamiltonian is the best candidate to generate quantum simulations of mesoscopic systems. Since the method is wavefunction based we anticipate Hilbert space sizes of 235 ∼ 1011.
The power of the stochastic surrogate Hamiltonian was next put on test on a model of a molecular refrigerator. A periodic electric field was used to drive the system through the molecular dipole. With sufficient driving the natural heat current was reversed pumping heat from the cold to hot bath. Sufficient driving power was required to overcome the natural heat leak flow from the hot to cold bath [39]. Effects of strong coupling were observed: the cooling power saturated and then was suppressed when the driving amplitude increased. This led to a performance graph of efficiency vs power mimicking known result from macroscopic refrigerators [40]. This means that a single quantum device composed for example from a molecule has performance characteristics of a macroscopic air conditioner.
FIG. 1: The quantum tricycle, a three lead quantum device coupled simultaneously to a hot, cold, and power reservoir. A reversal of the heat currents constructs a quantum refrigerator. The tricycle topology is the template of the autonomous quantum heat device. At steady state the first and second laws of thermodynamics are indicated.
The combined detailed analysis carried out by many groups of quantum heat devices [8, 41–46] has led to a the surprising conclusion: The devices closely mimic their macroscopic counterparts. The efficiency is restricted by the Carnot efficiency. The hallmark of the tradeoff between efficiency and power, the Novikov-Curzon-Ahlborn efficiency at maximum power, is obtained in the high temperature limit of both continuous and reciprocating quantum models [47, 48]. Exceptions are either the result of erroneous analysis neglecting the global structure of quantum mechanics or the use of baths that can deliver both heat and work. Such baths, are nicknamed quantum fuels [49–52], include coherence which can be directly transferred to work.
This analysis has raised the alarming question: what is quantum in quantum thermo- dynamics? Does each engine type require a separate analysis or can we identify universal behaviour? Recently we addressed this challenge [53]. The key was an analysis of reciprocating heat engines; the four stroke Otto engine. The next step was to establish the limit of zero cycle time obtained by allocating less time for each of the strokes. Obviously we obtain less work per cycle, nevertheless in the limit of zero cycle time finite power is obtained. Formally this is the limit of small action on each stroke, where action can be measured in units of planks constant. We found that the limit of small action engines is universal. Four-stroke two-stroke and continuous cycles have the same power and efficiency. A necessary condition for power production could be traced to quantum coherence maintained throughout the cycle. Using this result we could define a quantum signature of operation using only thermodynamical performance characteristics
The role of coherence in quantum reciprocating engines can be explored by partitioned the propagator into distinct strokes. For example the quantum Otto cycle is composed of four strokes: Two adiabats where the scale of energy of the working system is modified externally, and two isochores where heat is transferred from the hot or cold bath. Each of these strokes can be described by a CP map; a propagator. The total cycle propagator is the product of the stroke propagators. A necessary condition for extracting power or for refrigeration is that the stroke propagators do not commute.
Two extreme possibilities can be identified. The field which is termed the stochastic limit, the cycle is characterised by population of the energy levels. The only quantum feature in this limit is the discrete spectrum of the Hamiltonian. In this case the non- commutativity is the result of permutation and thermalisation propagators. Coherence allows another source of non-commutative phenomena. The coherence represents operators which do not commute with the Hamiltonian. When the Hamiltonian does not commute with itself at different times fast dynamics generates coherence on the adiabats. As a result the adiabatic propagator will not commute with the thermalisation propagator. In the limit of small action only the coherence can generate a sustainable finite power or finite cooling rate [53, 54].
A related question is the role of entanglement in quantum heat engines. It was suggested that entanglement can enhance the performance [55] in a model of a three qubit refrigerator. This analysis has been criticised using a more refined model [20] claiming that in the point of maximum cooling power no entanglement exists. At this point I would state that the role of entanglement in heat devices is inconclusive.
An integral part of a heat engine is an energy storage device, either temporary such as a flywheel or more permanent such as a rechargeable battery. Quantum treatments address this issue by adding an explicit term to the Hamiltonian of the system. This term should be able to store large amounts of energy therefore a typical choice for a model is an unbounded Hamiltonian. The common choice is an harmonic oscillator [56, 57]. The immediate consequence is that the storage device carries entropy as well as energy. Now if the fluctuations grow out of control the storage device becomes useless [58]. In addition when large amplitude of stored energy is accumulated it causes back reaction on the system bath coupling. As a result the energy storage operation is stopped. We analysed a model of a two qubit engine coupled to a harmonic oscillator representing a flywheel.
The function of the flywheel is to store work and to release it on demand increasing the instantaneous power beyond the power of the engine. At certain operating conditions we could reduce the dynamics of the flywheel to an oscillator driven by an inverse temperature L-GKS equation. Under these conditions the fluctuations diverge. To correct for this problem we added a quantum controller composed of weak measurement and feedback [58]. When the rate of energy current from the heat engine matches the weak measurement strength then the extraction efficiency is maximised.
The most studied model of a quantum heat bath is composed of a set of uncoupled harmonic oscillators linearly coupled to the system. The framework is of normal modes and the linear coupling cannot generate entanglement between the modes. The model has been employed extensively in the study of thermalisation [9]. The linear harmonic model is consistent with the second [59, 60] and third law of thermodynamics [61]. Nevertheless the linear model has been criticised for too weak ergodic properties [62]. One obvious aspect is that the model does not contain a mechanism of internal thermalisation of the different bath modes. What is missing conceptually is a heat exchanger. A device that filters out a thermal distribution in a small frequency range for example a two level system [63]. By incorporating such a device we were able demonstrate quantum equivalence in the strong coupling regime.
Collective effects of quantum heat engines have only recently been addressed [64]. Is there an advantage of operating such engines in series or in parallel? An attractive idea is to combine such engines in series and to transfer coherence from one engine to another. As a result one can obtain an effect of feeding the engine with quantum fuels enhancing the overall efficiency [65].
The III-law of thermodynamics is where one can expect quantum features. Two independent formulations of the III-law of thermodynamics exist, both originally stated by Nernst [66–68]. The first is a purely static (equilibrium) one, also known as the ”Nernst heat theorem” : phrased:
• The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.
The second formulation is dynamical, known as the unattainability principle:
• It is impossible by any procedure, no matter how idealised, to reduce any assembly to absolute zero temperature in a finite number of operations [68].
There is an ongoing debate on the relations between the two formulations and their relation to the II-law regarding which and if at all, one of these formulations implies the other [69–72]. Quantum considerations can illuminate these issues. Insight has been obtain by the analysis of specific cases. The quantum refrigerator models differ in their operational mode being either continuous or reciprocating. When optimised for maximum cooling rate the energy gap of the receiving mode should scale linearly with temperature ωc ∼ Tc [15, 19, 73–75]. Once optimised the cooling power of all refrigerators studied have the same universal dependence on the coupling to the cold bath. This means that the III-law depends on the scaling properties of the heat conductivity γc(Tc) and the heat capacity cV (Tc) as Tc → 0.
Theoretical challenges to the III-law have been proposed [76] based on anomalous spectral function. We find this spectral function to violate the stability criterion of the combine system-bath leading to imaginary frequencies [77]. A recent paper claims that higher order terms in the system bath coupling will lead to residual heating of the baths thus stopping the cooling process at finite temperature [61]. The quantum nature of the III-law needs further analysis connecting different approaches.
Experimental realisations of single component quantum heat devices
The template for an experimental realisation is a single quantum component coupled through leads to heat baths performing a thermodynamical task of converting heat to work or pumping heat against a thermal gradient. The minimal requirement is quantum systems with three leads or coupling wires to a hot cold and work reservoir [78]. The field of quantum information has motivated the development of highly controllable quantum experimental platforms. They include ion traps [79], Josephson junction based devises [80], quantum dots [81], NV centres in diamonds [82] and NMR spectrometers [83]. All these devices can be reengineered to operate as thermal machines
with the advantage that the requirement for high fidelity operations can be relaxed. These realisations can be either continuous or reciprocating based on quantum gates. The first example of this new era of experiments is a realisation of a quantum Otto cycle implemented by a single ion in a trap [84]. Other realisations based on other platforms will be realised soon. For example, a suggestion of a Josephson based device was published [85].
Laser cooling is a crucial technology for implementing quantum devices. When the temperature is decreased, degrees of freedom freeze out and systems reveal their quantum character. Inspired by the mechanism of solid state lasers and their analogy with Carnot engines [86] it was realised that inverting the operation of the laser at the proper conditions will lead to refrigeration [87, 88]. A few years later a different approach for laser cooling was initiated based on the doppler shift [89, 90].
In this scheme, translational degrees of freedom of atoms or ions were cooled by laser light detuned to the red of the atomic transition. Unfortunately the link to thermodynamics was forgotten.
Currently laser cooling is among the basic enablers of quantum technology. It can operate to the level of a single atom or single ion in a trap [91–94]. These are templates of quantum heat devices, where entropy is carried away by scattering light. Other non- local quantum entropy generation methods should be explored. The methods of laser cooling should be unified in a thermodynamical framework.
FIG. 2: A heat machine scheme with heat exchangers (gears). Various engine types can be implemented in this scheme by controlling the coupling function to the engine (ellipse). In each cycle, the gears turn and the work repository shifts so that new particles enter the interaction zone (gray shaded area). The heat exchangers enable the use of Markovian baths while having non-Markovian engine dynamics. This includes strong coupling and/or short time evolution. In this model, the work is stored in many batteries (work qubit in green); (b) the engine level diagram. This machine is based on two-body energy conserving unitaries. This is in contrast to other machines that employ three-body interaction.
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