Scientists are increasingly focused on understanding the interplay between quantum and classical descriptions of light-matter interactions within cavity electrodynamics. Marcelo Janovitch, University of Basel, Sander Stammbach, Universität Siegen, and Matteo Brunelli, Collège de France, with co-authors including Patrick P. Potts, demonstrate a rigorous semi-classical limit of cavity QED, revealing potential qualitative differences from fully quantised models. This research is significant because it clarifies how seemingly valid semi-classical approaches can violate fundamental uncertainty relations unless they correctly account for the re-use of photons as an energy source. By examining a driven three-level system coupled to a cavity, the team provides crucial insight into bridging the gap between quantum and semi-classical thermodynamics in these systems.
Reconciling semi-classical and quantum thermodynamics in cavity quantum electrodynamics
Scientists have uncovered a crucial link between different thermodynamic frameworks used to analyse cavity quantum electrodynamics (QED) systems. This work addresses a long-standing question regarding how semi-classical descriptions of quantum thermal machines relate to fully quantum mechanical treatments.
Researchers rigorously formulated a semi-classical limit of cavity QED, revealing that standard thermodynamic approaches fail to align with this limit, incorrectly predicting continued entropy production from the light field. Conversely, the input-output (IO) thermodynamic framework successfully recovers the semi-classical thermodynamic description, demonstrating that the coherent portion of emitted photons can be legitimately considered a power source.
This breakthrough stems from the unique behaviour of photons within cavity QED, where they can either be irreversibly lost or repurposed as a power source, leading to two distinct, yet valid, thermodynamic interpretations. The study demonstrates that violations of thermodynamic uncertainty relations (TURs) , bounds on the fluctuations of currents, are only maintained in the semi-classical limit when employing the IO framework.
This suggests that the standard approach overestimates entropy production due to an inaccurate assessment of light dissipation. The findings are illustrated using a three-level maser, a thermally driven three-level atom coupled to a cavity, which is known to violate TURs in semi-classical descriptions.
The research begins with a fully quantised model, establishing a Hamiltonian encompassing the system, a driven cavity, and their interaction. Dynamics are governed by a master equation accounting for both coherent evolution and dissipative processes affecting the cavity and system. By carefully considering the parameter regime where the average cavity field dominates over fluctuations, researchers derived a semi-classical limit, effectively treating the cavity field as a classical coherent drive.
This process involved defining a displaced mode and identifying the conditions under which quantum noise becomes negligible, ultimately leading to a reduced density matrix for the intra-cavity system. This work clarifies the physical origin of power output in semi-classical models of quantum thermal machines, confirming that it originates from the coherent component of the emitted field.
The implications extend to the understanding and potential enhancement of TUR violations, paving the way for improved designs of quantum devices and thermal machines operating within cavity QED platforms. The study’s rigorous formulation and benchmarking of thermodynamic approaches provide a solid foundation for future investigations into non-equilibrium statistical physics and quantum thermodynamics.
Cavity Hamiltonian and Dissipative Master Equation Formulation
A driven, dissipative cavity forms the basis of this research, with the Hamiltonian described as H(t) = H′ + H0(t) + V. Here, H′ represents the bare system Hamiltonian, while H0(t) = Ωa†a + iE a†eiωdt −ae−iωdt details a cavity with frequency Ωlaser-driven with amplitude E and frequency ωd. The interaction between the system and the cavity is defined by V = g(aO† + a†O), where O is a system operator and g is the coupling constant.
The composite cavity-system dynamics are governed by a master equation, dρ dt = −i[H(t), ρ] + D[a]ρ + D′ρ, which accounts for both coherent and incoherent processes. Dissipation from the cavity is modelled using D[a] = κ( n + 1)D[a] + κ nD[a†], where κ is the decay rate, and n = [eωd/T −1]−1 represents the Bose, Einstein occupation of the cavity bath at temperature T.
System dissipation is captured in D′, defined as D′ = r X j=1 ΓjD[Lj], with jump operators Lj and rates Γj specific to the system under consideration. To formulate a semi-classical limit, the researchers considered a regime where the average cavity field dominates over its fluctuations, expressing the field as a = α(t) + a, with α(t) representing the classical amplitude.
This amplitude is calculated as α(t) = −2E κχe−iωdt, where χ = 1 1 + 2i∆/κ and ∆= Ω−ωd is the detuning. The semi-classical limit is defined by the conditions |α| →∞, g κ →0, and g κ|α| = const., ensuring a large coherent field with negligible quantum fluctuations. Tracing out the cavity mode under these conditions yields the semi-classical model, dρ′ dt = −i[Hsc(t), ρ′] + D′ρ′, where the cavity field acts as a purely external, time-dependent drive.
Thermodynamic consistency in semi-classical cavity quantum electrodynamics necessitates a power source interpretation of photon flux
Researchers demonstrate that violations of the uncertainty relations are recovered in the semi-classical limit only when treating part of the photon flux as a power source. This work rigorously formulates a semi-classical limit of cavity quantum electrodynamics and benchmarks two thermodynamic approaches against this limit.
The study reveals that the standard thermodynamic approach fails to recover the semi-classical thermodynamic description, as light continues to produce entropy in this limit. Instead, the input-output framework successfully recovers the thermodynamic description of the semi-classical model. This finding clarifies that the power output in semi-classical descriptions of quantum thermal machines originates from the coherent part of the emitted field.
The research focuses on a three-level maser, a thermally driven three-level atom coupled to a cavity, revealing that thermodynamic uncertainty relation violations persist in a semi-classical description only when employing the input-output thermodynamic approach. The composite cavity-system dynamics is governed by a master equation, with the cavity described by a Hamiltonian including a driven term and detuning of laser.
Dissipative processes acting on the system are quantified by jump operators and rates, and the semi-classical limit is defined by the condition that the average cavity field dominates over its fluctuations, specifically with parameters satisfying |α| →∞, g/κ →0, and g/κ|α| = const. In this regime, the cavity hosts a large coherent field, while noise from thermal fluctuations and system back-action is negligible.
Upon tracing out the cavity mode in the semi-classical limit, the effective drive is directly associated with the injected power, calculated as −iωdg ⟨α(t)O† −α∗(t)O⟩. The semi-classical model obeys both the first and second laws of thermodynamics, with entropy production quantified by σsc = − X j ⟨Jj⟩ Tj.
Importantly, the heat current from the cavity field is zero in the semi-classical model, while the standard framework predicts a non-zero contribution. The average power in the standard framework is given by −ωdE ⟨ae−iωdt + a†eiωdt⟩.
Entropy production defines thermodynamic limits in cavity quantum electrodynamics
Researchers have demonstrated that thermodynamic uncertainty relations in cavity quantum electrodynamics depend critically on how entropy production is defined. The investigation reveals that standard formulations of entropy production fail to accurately represent the semi-classical limit of cavity QED, whereas an input, output thermodynamic approach successfully recovers these predictions.
This alternative formulation correctly describes the behaviour of the system and captures the suppression of current fluctuations expected in non-classical regimes. The findings extend beyond specific three-level maser systems, offering a broadly applicable tool for characterising thermodynamic uncertainty and non-classicality in various cavity-QED architectures.
By appropriately bounding current fluctuations, the study establishes a framework for understanding energetic fluctuations and uncertainty relations in driven quantum systems. The authors acknowledge a limitation in the current formalism, specifically the lack of a full-counting-statistics description for coherent power and cavity heat currents within their input, output framework. Future work should address this gap to provide a more complete and unified description of energetic fluctuations and potentially reveal additional thermodynamic uncertainty relations applicable to work-like observables.