Quantum ‘bath’ Breaks the Rules of Heat and Work for New Devices

Researchers have long sought to understand how stochastic dynamics emerge when classical systems couple with dynamic quantum environments. Pedro V. Paraguassú from Departamento de Física, PUC-Rio, along with collaborators, now present a novel thermodynamic framework to describe this interplay, defining heat, work, and entropy production in such semi-classical regimes. Their work significantly advances the field by deriving a modified Second Law of Thermodynamics that accounts for non-equilibrium effects like quantum squeezing, offering a crucial tool for analysing systems where energy exchange isn’t simply about heat flow. This framework is demonstrated using an optomechanical example, allowing characterisation of the non-stationary noise induced by cavity fields and providing insights into a range of nanoscale systems.

Unlike standard thermal reservoirs, this environment acts as a dynamic bath, capable of simultaneously exchanging heat and performing work. They define heat, work, and entropy production within this framework. A modified Second Law is derived that accounts for non-equilibrium quantum features.

Characterising non-stationary noise in an optomechanical system is crucial for sensitive measurements

Scientists are squeezing the limits of quantum dynamics by investigating systems where classical and quantum behaviours intertwine. The framework is exemplified by an optomechanical setup, where researchers characterize the thermodynamics of the non-stationary noise induced by the cavity field. To understand behaviour at microscopic scales, quantum dynamics was developed.
While this framework successfully predicts the behaviour of atoms, electrons, and other fundamental constituents, the theory itself does not explicitly prescribe the limits of its validity. Current experiments aim to probe the boundaries where quantum dynamics can be observed, including on levitated nanoparticles [1, 4] and clamped optomechanical systems [5, 8].

Although these systems are typically described by classical mechanics, they exhibit quantum behaviours due to their high degree of isolation or their interaction with inherently quantum systems. In such scenarios, a classical stochastic dynamics can emerge from the interaction with a quantum system. Recently, this regime has been investigated in optomechanics [9, 10], the Jaynes-Cummings interaction, and even in gravitational wave detection [12, 13].
Researchers refer to this as Quantum-Induced Stochastic Dynamics (QISD), where stochasticity arises solely from the interaction with a quantum degree of freedom [14, 17]. Here, the effect of the quantum system manifests as fluctuation and dissipation, effectively acting as a reservoir that exchanges energy with the classical system.

By considering a strong decoherence limit on the system of interest, the dynamics becomes a generalized Langevin equation characterized by a Wiener measure. It is different from quantum Brownian motion, since the Brownian particle is classical [14, 19]. Given the stochastic nature of the dynamics, the energy exchange is inherently stochastic, making the framework of stochastic thermodynamics well-suited to investigate this exchange from the perspective of the system’s dynamics.

Stochastic thermodynamics treats thermodynamic functionals, such as heat, work, and entropy, as random variables. The success of the theory relies on extending the second law of thermodynamics through fluctuation theorems [20, 24], which find applications ranging from RNA measurements [25, 27] to Brownian machines [28, 31].

Originally established by the works of Sekimoto, Seifert, and Jarzynski almost three decades ago, the framework has since seen various generalizations, some of them include special relativistic [35, 37] and general relativistic extensions [38, 42], active matter [43, 46], computing [47, 49], opinion dynamics [50, 51], quantum systems [52, 55], and quantum fields. In this work, researchers propose to understand the exchange of energy of the QISD dynamics, thereby incorporating these dynamics into the broader context of stochastic thermodynamics.

Their objective is to employ the framework of stochastic thermodynamics to characterize the energy exchanges within Quantum-Induced Stochastic Dynamics. They seek to elucidate, from a semiclassical perspective, the fundamental mechanisms of energy transfer between a probe and its quantum environment. Comprehending these exchanges at the classical-quantum interface is essential for the development of protocols designed to harvest or utilize quantum-induced energy.

Researchers find that the quantum reservoir acts as a dynamic bath, capable of simultaneously exchanging heat and performing work, thereby challenging the standard tripartite separation in stochastic thermodynamics. In Section 3, they discuss the definitions of work and heat, addressing the distinction between them given the ultimately quantum nature of the forces involved.

In Section 4, they propose a formulation for entropy production based on path entropy to quantify the system’s irreversibility; they show that if the fluctuation-dissipation theorem (FDT) holds, irreversibility can be directly connected to heat exchange. Finally, in Section 5, they apply this framework to the example of a trapped nanoparticle exhibiting stochastic behaviour due to its interaction with squeezed light modes in a cavity.

They conclude with a Discussion of the Results and future perspectives. Researchers start by considering a bipartite system composed of subsystems A and B. In their framework, they treat A as the classical system of interest, while B represents the quantum degrees of freedom.

The time evolution of the joint density matrix is given by ρtot(t) = U(t)ρ0U†(t). They obtain the reduced density matrix of system A by taking the partial trace over the degrees of freedom of B, assuming it is prepared in a separable quantum state, and representing the result in the position basis {|q⟩} of system A.

By expressing the time-evolution operator in terms of path integrals, they arrive at the Feynman-Vernon functional formalism ρ(q, q′, t) = ∫ dq0dq′0 J[q, q′; q0, q′0] ρ0(q0, q′0). The central object of their analysis is the propagator of the reduced density matrix, J, which accounts for the free evolution of the system modified by the coupling to the environment via the influence functional J[q, q′; q0, q′0] = ∫ Dq Dq′ e^(i ħ(SA[q]−SA[q′])F[q, q′]).

Here, SA is the classical action of the system A, defined by SA[q] = ∫t0 dτ( 1/2m q2 − V(x)). Assuming the potential energy is V(x). Moreover, they define the influence functional as F[q, q′].

Researchers assume the quantum environment plays a threefold role: it acts simultaneously as the source of noise, dissipation, and the deterministic drive. To resolve this ambiguity, they distinguish the forces based on their capacity to induce ordered versus disordered motion. The noise term (η), arising from decoherence, induces random fluctuations.

The dissipation term (Fdiss), while unitary in origin, acts as a friction force that removes energy from the system. In contrast, Fdet provides a controlled, ordered energy input. Therefore, they associate heat with the energy exchange mediated by the non-conservative and stochastic forces Q[x] = ∫τ0 dτ x(t) [Fdiss(t) + η(t)].

A direct consequence of this definition is the recovery of the first law of stochastic thermodynamics. By substituting the equation of motion m x + ∂xV = Fdet + Fdiss + η into the previous equation, they obtain Q[x] = ∫τ0 dt d/dt (m x2/2 + V(x)) − ∫τ0 dt Fdetx = ∆K + ∆V − [Fdetx]τ0 − ∫τ0 dt Fdetx = ∆H −W[x].

Here, H = K + V −Fdetx represents the total effective energy of the system, including the potential energy contribution from the driving force. Note that the variables ∆K and ∆V are also random variables having a probability distribution [59, 60]. This formulation highlights a crucial physical insight: the quantum environment operates as a dynamic bath.

Unlike a passive reservoir that merely absorbs energy, it actively exchanges work with the system. This behaviour is not exclusive to the quantum nature of the bath but rather to its specific preparation. For instance, if the quantum system is in a coherent state, work will be made.

An analogy can be drawn to classical experiments involving active feedback traps or electrical noise circuits [1, 28, 61, 63], where the same apparatus can provide both random fluctuations (heat exchange) and deterministic control forces. To ensure the thermodynamic consistency of these definitions, they must next examine the entropy production and its relation to the influence functional.

To validate the thermodynamic quantities derived in the previous section, researchers establish the connection between energetics and irreversibility. A standard approach involves computing the stochastic entropy production, defined as the logarithmic ratio of the forward and backward path probability densities.

For standard thermal baths, this ratio recovers the usual entropy production, which is directly proportional to the dissipated heat. However, in more complex scenarios, this ratio needs further investigation.

Semi-classical dynamics via functional expansion of the reduced density matrix propagator offers a computationally efficient approach to open quantum systems

Researchers detail a framework for quantum-induced stochastic dynamics, revealing how a classical system couples with a dynamic quantum environment. The study establishes definitions for heat, work, and entropy production within this semi-classical regime, deriving a modified Second Law that accounts for non-equilibrium features like squeezing.

Characterization of the non-stationary noise induced by a cavity field exemplifies the nature of the forces involved in this interaction. The work centers on the propagator of the reduced density matrix, represented by the functional J, which describes the free evolution of the system modified by coupling to the environment.

Assuming quadratic terms in the functional expansion, a Gaussian influence functional is obtained, incorporating a noise kernel K and a dissipation kernel D. Analysis using center-of-mass and coherence coordinates simplifies the expression, leading to a stochastic Wiener path integral and ultimately, a phase-space probability distribution evolution described by equation (5).

This formulation culminates in a Generalized Langevin Equation (GLE) governing the system’s dynamics, expressed as m x(t) + ∂xV (x) = Fdiss(t) + η(t) + Fdet(t). The correlation function of the Gaussian colored noise is defined as ⟨η(t)η(t′)⟩= K(t, t′), while the dissipation force is given by Fdiss(t) = − Z τ 0 D(t, t′) xdt′.

The stochastic work functional is defined as W[x] = − Z τ 0 dt Fdet(t)x(t), aligning with the thermodynamic interpretation of energy change due to external control. Heat is defined as the energy exchange mediated by non-conservative and stochastic forces, calculated as Q[x] = Z τ 0 dτ x(t) [Fdiss(t) + η(t)].

Application of the derived equation of motion recovers the first law of stochastic thermodynamics, demonstrating that ∆K + ∆V − [Fdetx]τ 0 accurately represents the energy balance within the system. This detailed energetic analysis is essential for understanding the system’s irreversibility and potential applications in areas like optomechanical systems and nanoparticle manipulation.

Quantum Thermodynamics of Classical Systems via Dynamic Environmental Interactions reveals emergent behavior

Scientists have developed a framework describing how classical systems interact with quantum environments, termed Quantum Induced Stochastic Dynamics. This approach details the exchange of heat and work between these systems, defining concepts like entropy production within this semi-classical regime.

The research demonstrates that a quantum environment can act as a dynamic bath, simultaneously exchanging heat and performing work on a classical system, a departure from traditional stochastic thermodynamics which typically separates these functions. This formalism reveals that the quantum environment induces unique noise characteristics in the classical system, dependent on the quantum state involved.

Importantly, the study establishes a modified Second Law of Thermodynamics to account for non-equilibrium features, such as squeezing, and identifies that the consumption of quantum resources can lead to a temporary reduction in entropy. The framework successfully connects quantum resources with classical stochastic dynamics and offers potential applications in areas like gravitational wave detection and levitated optomechanics, where engineered noise properties could optimise thermodynamic processes.

Acknowledging limitations, the authors note the complexity of extending this model to even more intricate systems remains a challenge. Future research should focus on exploring multiplicative noise arising from interactions and further investigating the potential for engineering specific noise properties in levitated optomechanical systems to enhance thermodynamic protocols and heat engine designs.