Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions

We expand the standard thermodynamic framework of a system coupled to a thermal reservoir by considering a stream of independently prepared units repeatedly put into contact with the system. These units can be in any nonequilibrium state and interact with the system with an arbitrary strength and duration. We show that this stream constitutes an effective resource of nonequilibrium free energy, and we identify the conditions under which it behaves as a heat, work, or information reservoir. We also show that this setup provides a natural framework to analyze information erasure (“Landauer’s principle”) and feedback-controlled systems (“Maxwell’s demon”). In the limit of a short system-unit interaction time, we further demonstrate that this setup can be used to provide a thermodynamically sound interpretation to many effective master equations. We discuss how nonautonomously driven systems, micromasers, lasing without inversion and the electronic Maxwell demon can be thermodynamically analyzed within our framework. While the present framework accounts for quantum features (e.g., squeezing, entanglement, coherence), we also show that quantum resources do not offer any advantage compared to classical ones in terms of the maximum extractable work.

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Popular Summary

Thermodynamic machines in the macroscopic world—such as steam and car engines—make use of very simple resources, for example, heat from a reservoir or work sources exerting pressure on the device. In the microscopic domain, where fluctuations and quantum effects become manifest, new types of resources become available, such as small detectors measuring fluctuations and using those fluctuations to influence the device of interest. In an age of biological and synthetic nanotechnologies, it becomes important to understand these resources, which are usually out of equilibrium and can themselves be microscopically small. Significant progress has been made in recent years to include them in the traditional thermodynamic framework. However, up to now, this was done with case-by-case studies that produced seemingly unrelated results. Here, we present a unifying framework where the nanoscopic device, apart from being coupled to an ideal heat and work reservoir, is allowed to repeatedly interact with a stream of systems that have been identically prepared in an arbitrary state.

Under simple assumptions (such as an independent preparation of the individual external systems), it is possible to establish generalized laws of thermodynamics demonstrating that these external systems act as a resource of nonequilibrium free energy. Our framework naturally combines energy and information transduction and can be used to study many different and seemingly unrelated setups from a thermodynamic perspective. This includes, for example, information-processing and feedback control, the micromaser (a tiny laser that produces microwave light), and a description at the level of an effective quantum master equation.

Our work paves the way towards a unified thermodynamic theory at the nanoscale. While the resource content of a stream of independently prepared systems is now understood, extending our theory to correlated streams remains an outstanding challenge.

Figure 1

Overview of the structure of the article covering its main results and applications. Here and in the following, $E_{S,U}$, $S_{S,U}$, and $F_{S,U}$ denote the energy, entropy, and nonequilibrium free energy of the system ($S$) or unit ($U$). $Q$ is the heat flowing from the reservoir at inverse temperature $\beta =T^{-1}$ ($k_B\equiv 1$), and $W$ is the work done on the system. Furthermore, ${\mathrm{\Sigma }}_S$ denotes the entropy production and $I_{S:U}$ the mutual information between the system and unit. The picture of the Mandal-Jarzynski engine was taken from Ref. [35].

Figure 2

Sketch of a stream of units interacting with a system in contact with a heat reservoir at inverse temperature $\beta$. The lower panel shows the switching on and off of the system-unit interaction as a function of time. Note that $\tau$ denotes the full interaction period, whereas the system and unit are only physically coupled during a time ${\tau }^{\prime }$.

Figure 3

Venn diagram of the thermodynamic role of the stream of units. In general, the interaction can be arbitrary, but if the initial state of the units is thermal, they can mimic an ideal heat reservoir when they fulfill the Clausius equality (55). In the limiting case where $T^{\prime }\to \infty$, we obtain a work reservoir (Sec. 4b). The converse is not true; i.e., not every work reservoir can be obtained as a limiting case of a heat reservoir. Similarly, we can obtain an information reservoir (Sec. 4c) out of a heat reservoir for $T^{\prime }\to 0$, but again the converse is not true. We note that some setups do not fit in any of the three categories.

Figure 4

Overview of the various quantities involved during the measurement and feedback step in the steady-state regime.

Figure 5

Sketch of the Poisson-distributed regime: The system evolves freely most of the time but, once in a while, undergoes a short (${\tau }^{\prime }\to 0$) and strong ($H_{SU}\sim {{\tau }^{\prime }}^{-1}$) interaction with a unit where ${\tau }^{\prime }$ denotes the duration of the interaction as in Sec. 3. The duration $\tau$ between successive system-unit interactions fluctuates according to an exponential time distribution with average duration ${\gamma }^{-1}$.

Figure 6

Sketch of a micromaser setup. The units are produced in an atomic beam oven and are initialized with a velocity selector and laser excitation. They then interact with a microwave (“maser”) cavity (the system) before they are finally detected and their state is read out. The figure is taken from Ref. [174], reporting on experimental observation of sub-Poissonian photon statistics in a micromaser.

Figure 7

Sketch of the electronic Maxwell demon. The system consists of a single-level quantum dot tunnel coupled to two electronic reservoirs with chemical potentials ${\mu }_L\ge {\mu }_R$ and inverse temperature $\beta$. The demon mechanism is implemented by a stream of units which monitors the state of the dot and, depending on its state, changes the tunneling barriers ${\mathrm{\Gamma }}_{\nu }$ to make electronic transport against the bias possible. In the absence of the demon mechanism, the tunneling barriers would not depend on the state of the dot, and electrons would flow from left to right. The picture of the demon was provided courtesy of Ania Brzezinska.

Figure 8

Plot of the rate of chemical work $\beta ({\mu }_L-{\mu }_R)I_L$ (thick and solid black line) and of three different dissipations: the total one generated by the repeated interaction feedback mechanism $\dot{\mathrm{\Sigma }}$ [dotted blue line, Eq. (170)], the total one generated by the capacitive feedback mechanism ${\dot{\mathrm{\Sigma }}}^{\mathrm{cap}}$ [dashed green line, Refs. 155, 157], and the “effective” estimate obtained in these two cases if the demon mechanisms were not known ${\dot{\mathrm{\Sigma }}}_S^{\mathrm{eff}}$ [dash-dotted red line, Eq. (172) and Ref. 151]. The top row shows these quantities as a function of the bias voltage $V\equiv {\mu }_L-{\mu }_R$ for two different measurement errors $\epsilon$. The bottom row shows them as a function of the measurement error $\epsilon \in [0,1/2)$ for two different voltages $V$. The region in which we extract work is shaded in gray. Other parameters are $\mathrm{\Gamma }=1$, $\delta =\ln 2$, $\beta =0.1$, and $U=0.1$, and we choose a symmetric configuration of the bias, ${\mu }_L={\epsilon }_S+V/2$ and ${\mu }_R={\epsilon }_S-V/2$, effectively eliminating the dependence on ${\epsilon }_S$ in the equations.

Figure 9

Sketch of the setup to demonstrate the Kelvin-Planck statement of the second law: The outgoing units $U^{\prime }$ after the interaction with system $S$ are reset to the state $U$ via interaction with a second system $S^{\prime }$ coupled to the same thermal reservoir as $S$.