Thermodynamics of weakly measured quantum systems
We consider continuously monitored quantum systems and introduce definitions of work and heat along individual quantum trajectories that are valid for coherent superposition of energy eigenstates. We use these quantities to extend the first and second laws of stochastic thermodynamics to the quantum domain. We illustrate our results with the case of a weakly measured driven two-level system and show how to distinguish between quantum work and heat contributions. We finally employ quantum feedback control to suppress detector backaction and determine the work statistics.
Thermodynamics is, at its heart, a theory of work and heat. The first law is based on the realization that both quantities are two forms of energy and that their sum is conserved. At the same time, the fact that entropy, defined as the ratio of reversible heat and temperature, can only increase in an isolated system is an expression of the second law pip66 (); cal85 (). In classical thermodynamics, work is defined as the change of internal energy in an isolated system, , while heat is introduced as the difference, , in a nonisolated system. Thermal isolation is thus crucial to distinguish from . In the last decades, stochastic thermodynamics has successfully extended the definitions of work and heat to the level of single trajectories of microscopic systems sei12 (). In this regime, thermal fluctuations are no longer negligible and the laws of thermodynamics have to be adapted to fully include them. The second law has, for instance, been generalized in the form of fluctuation theorems that quantify the occurrence of negative entropy production jar11 (). A particular example is the Jarzynski equality, , that allows the determination of equilibrium free energy differences from the nonequilibrium work statistics in systems at initial inverse temperature jar97 (). The laws of stochastic thermodynamics have been verified in a large number of different experiments, see Refs. lip02 (); bli06 () and the review cil13 ().
The current challenge is to extend the principles of thermodynamics to include quantum effects which are expected to dominate at smaller scales and colder temperatures. Some of the unsolved key issues concern the correct definition of quantum work and heat, means to distinguish between the two quantities owing to the blurring effect of quantum fluctuations, and the proper clarification of the role of quantum coherence. A variety of approaches have been suggested to tackle these problems tal07 (); hub08 (); esp09 (); cam11 (); hor12 (); hek13 (); dor13 (); maz13 (); abe13 (); hal14 (); ron14 (); gal15 (), and quantum work statistics has been measured in isolated systems in two pioneering experiments using NMR bat14 () and trapped ions an14 (). A new approach may emerge from the possibility of weakly monitoring quantum systems. Recently, individual quantum trajectories of a superconducting qubit in a microwave cavity have been observed using weak measurements mur13 (); web14 (). These measurements only slightly disturb quantum systems owing to the weak coupling to the measuring device cle10 (). They hence allow to gain information about states without projecting them into eigenstates. They have been successfully employed to explain quantum paradoxes aha05 (), detect and amplify weak signals hos08 (); dix09 (), determine a quantum virtual state romito14 (), as well as directly measure a wave function lun11 (). Motivated by the two experiments mur13 (); web14 (), we here investigate the first and second law for continuously monitored quantum systems and aim at developing a quantum stochastic thermodynamics based on quantum trajectories. Such an extension faces several technical difficulties. First, since a weakly measured system can be in a coherent superposition of energy eigenstates, energy is not a well-defined concept along a single quantum trajectory. Furthermore, even in the absence of an external environment, a continuously monitored quantum system is not isolated and the detector backaction, albeit small, will perturb its dynamics cle10 (). As a result, its time evolution will be nonunitary and energy, in the form of heat, will be exchanged with the detector.
In the following, we introduce suitable and consistent definitions of work and heat contributions to the quantum stochastic evolution of a weakly measured system that is externally driven. We use these definitions to determine the distributions of quantum work and heat for a two-level system, and demonstrate the general validity of the Jarzynski equality, hence of the second law. We finally use the tools of quantum feedback control wis10 () to suppress detector backaction and thus effectively achieve thermal isolation of the system. This provides a practical scheme to experimentally test our definitions of work and heat along individual quantum trajectories.
Quantum work and heat.
We consider a system with time-dependent Hamiltonian that is initially in a thermal state at inverse temperature , , where is the partition function. The system is driven by an external parameter during a time . At the ensemble level, quantum work and heat are introduced by considering an infinitesimal variation of the mean energy, rei65 (); per93 ():
Heat is further related to entropy via rei65 (); per93 (). For an isolated system with unitary dynamics heat vanishes, since , and therefore in agreement with classical thermodynamics pip66 (); cal85 (). Heat therefore appears to be fundamentally associated with the nonunitary part of the dynamics.
At the level of individual realizations, energy is a stochastic quantity owing to thermal and quantum fluctuations. The distribution ) of the total energy change may be determined by performing projective measurements and , with outcomes and , at the beginning and at the end of the driving protocol tal07 (); kaf12 (),
Here denotes the probability of the eigenvalue , the transition probability from state to , with the time evolved projected density operator , and the energy difference. For unitary dynamics, reduces to the work distribution , but, in general, Eq. (2) does not allow to distinguish work from heat. In the following, we generalize Eq. (2) and identify work and heat for a weakly measured system.
A quantum system continuously monitored by a quantum limited detector may be assigned, for each individual trajectory, a conditional density operator that reduces to the usual density operator when averaged over all the trajectories, wis10 (); jac14 (). The evolution of is commonly described by a stochastic master equation that contains a random parameter that accounts for the detector shot noise, see Eqs. (9)-(Application to a monitored qubit.) below for an example. An important observation is that such master equation has a unitary component, corresponding to the dynamics generated by the system’s Hamiltonian, and a nonunitary part that stems from the continuous coupling to the detector. For an infinitesimal time step, these two contributions are additive and may be written as,
where and are operators associated with the respective unitary and nonunitary parts of the dynamics. We identify them as corresponding to work and heat at the level of an infinitesimal quantum trajectory. This separation cannot be directly extended to the entire (time integrated) trajectory, since the stochastic master equation is generally a nonlinear function of the operator . However, when averaged over quantum fluctuations, Eq. (3) allows to extend the first law (1) to single realizations of the stochastic measurement outcome,
where in the second line and the middle term since is unitary unitary (). In the last line, and , indicating that work is related to a change of the Hamiltonian, as expected, and heat to the nonunitary . Equation (Quantum work and heat.) is a direct extension of stochastic thermodynamics to the quantum domain. The first law (1) is recovered when Eq. (3) is averaged over both stochastic and quantum fluctuations. The integrated work and heat contributions to the changes in transition probabilities, , with the initial transition probability, can be further obtained from Eq. (3) by carefully adding all the different terms (see Ref. suppl ()). We find, for each individual quantum trajectory,
with the two quantities,
These expressions depend explicitly on the quantum trajectory which we stress by using the notation instead of . They provide an unambiguous way to distinguish between work and heat at the level of a single trajectory. Remarkably, they are valid even if the system remains in a coherent superposition of energy eigenstates, that is, when its energy is ill-defined. Equation (5) holds for the trajectory averaged quantities , with , . This averaged distinction between work and heat contributions to transition probabilities requires access to single trajectories, thus cannot be established directly at the ensemble level.
The second law in the form of the Jarzynski equality, , immediately follows from Eq. (2) for an isolated system tal07 (). However, the equality is not satisfied for an open system with nonunitary dynamics owing to the heat term kaf12 (). The second law may be restored by replacing by , that is, by setting to zero at each time step, see Fig. 3. We next show how quantum work and heat may be identified theoretically, by numerically analyzing a weakly measured two-level system, and experimentally, by means of quantum feedback control.
Application to a monitored qubit.
In order to illustrate our approach, we consider a driven two-level system S with Hamiltonian, , where is the external driving and the usual Pauli matrices. The system is continuously coupled to a quantum limited detector D via the interaction Hamiltonian :
where, without loss of generality, we identify as the system’s observable that is monitored by the detector. The effect of the detector is fully characterized by the averaged signals, (, ), and Gaussian noises, (, ), measured when the qubit is in the two eigenstates, (, ), of the measured observable. We assume to be in the weak measurement regime, i.e. at time scales smaller than the measurement time . For concreteness and simplicity, we will interpret the qubit in Hamiltonian (8) as describing a double quantum dot sharing a single electron and interacting with a quantum point contact (QPC), but it can also be applied to a qubit coupled to a microwave resonator kor11 () in a circuit QED set-up as in the experiments mur13 (); web14 (). We accordingly identify the configurations where the electron occupies only one dot by . Coherent superpositions of the two are possible. The detector monitoring the occupation of the dots is a voltage biased QPC with Hamiltonian gurvitz97 (); korotkov99 (); korotkov01 (); korotkov02 () and the interaction term reads . The signal in the detector is the current across the QPC, with averages and noises . Here are the densities of states in the left and right electrodes, and are the dimensionless transmission probabilities across the QPC.
Under the assumption of a weakly coupled detector, the detector signal is a random variable, and the evolution of the system depends on the specific realization of the stochastic process. This is captured by a well-established Bayesian formalism korotkov01 (); korotkov02 () which describes the evolution of the system conditional to the detector’s outcome in terms of a nonlinear stochastic differential equation for the system’s density matrix . In the Ito formulation, we have korotkov01 (); korotkov02 (),
where and is the white noise of the detector’s signal with , and . The detector current is further,
For each realization of the measurement outcome, Eqs. (9) and (Application to a monitored qubit.) allow to identify the unitary and nonunitary contributions to the time evolution, since the nonunitary part is proportional to . We rewrite Eq. (3) as , and identify with the work done by the driving along an infinitesimal trajectory and as the heat associated with the detector backaction of the detector. Due to the nonlinearity of the stochastic master equation, we can only determine the distributions of work and heat numerically. We specify the driving as , where is the duration of the experiment, and reformulate equations (9) and (Application to a monitored qubit.) in the Stratonovich form korotkov01 (); korotkov02 (). We solve them numerically by the Monte-Carlo method for an ensemble of 300 realizations of the random signal in the interval using a time step . The results for work, heat and energy along a given quantum trajectory, Eq. (Quantum work and heat.), are shown in Fig. 1, while those for the work and heat contributions to the transition amplitudes, Eqs. (6)-(7), are presented in Fig. 2 (see Ref. suppl () for details).
Figure 1a) demonstrates the reconstruction of quantum averaged work and heat changes, and , along a single quantum trajectory, based on the definitions given in Eq. (4). The corresponding signal in the detector is displayed in Fig. 1b). Contrary to the case of an isolated system for which , the heat contribution is here clearly visible. Equation(Quantum work and heat.) holds for each individual realization and thus extends the first law of stochastic thermodynamics to the quantum regime. Figure 2b) shows the unambiguous distinction of the work and heat contributions, and , evaluated via Eqs. (6)-(7), to the final transition probability . We stress that, although is always positive, as a proper probability should be, the work and heat contributions need not be: the probability to go from state to at time can, for instance, be smaller than the initial transition probability rem (). Note that a quantity, , that only depends on initial and final times, cannot be defined, reflecting the fact that there are no heat or work operators.
Quantum feedback control.
In classical thermodynamics work is associated with the variation of the internal energy of the isolated system pip66 (); cal85 (). After having shown above how heat can be theoretically identified, we next take advantage of a feedback loop protocol to suppress the detector backaction wis10 (), offering a scheme to reach isolation experimentally. Quantum feedback has recently been demonstrated experimentally for a superconducting qubit mur12 (). Specifically we control the amplitude, , of the system’s driving depending on the continuos detector outcome, i.e. , where is the feedback strength and the phase difference between the actual vector (with backaction) and desired vector (without backaction) in the Bloch sphere of the qubit (see Refs. korotkov01 (); suppl () for details). This allows to operationally counter the effects induced by the continuous monitoring. From a thermodynamic point of view, the feedback adds an extra amount of work that exactly cancels the heat contribution to the transition probabilities.
Figure 3 shows the numerically simulated final transition probabilities for the weakly measured qubit with (orange) and without (yellow) quantum feedback for two driving times. We observe in both cases that the feedback process effectively suppresses the heat contributions (identified in Fig. 2) and that the transition probabilities agree with those of the isolated system with unitary dynamics (brown). Quantum feedback control thus appears as a powerful tool to determine the statistics of the work done by the external driving in a continuously monitored system. The heat statistics can be further easily obtained by measuring the undriven system, that is, when no work is performed and .
The above findings can be directly used to verify the quantum Jarzynski equality for the driven qubit. Since any measurement induced heating is prevented by the feedback, only the initial inverse temperature of the system matters. Determining the quantum work statistics via Eq. (2), we find and with feedback control for and and in the unitary case (for ). The excellent agreement demonstrates the correctness of the definitions of work and heat, and confirms the second law for a weakly measured quantum system.
We have extended the laws of stochastic thermodynamics along individual quantum trajectories of a weakly measured system. We have shown how to distinguish work and heat contributions to both the energy changes and the transition probabilities. We have further demonstrated the usefulness of our approach with the analysis of a driven qubit and introduced methods to identify work from heat numerically as well as experimentally with the help of quantum feedback control.
This work was partially supported by DFG under Grants No. RO 4710/1-1 and LU 1382/4-1, the EU Collaborative Project TherMiQ (Grant Agreement 618074) and the COST Action MP1209.
While completing this manuscript, we became aware of a preprint by Elouard et al. Elouard15 () that also discusses quantum stochastic thermodynamics.
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