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arXiv:1009.4382 · Journal of Physics A Mathematical and Theoretical

Effective thermodynamics of strongly coupled qubits

N S Williams, Karyn Le Hur, A N Jordan

Published Sep 2, 2011• 24 citations
arXivHTMLPDF

Abstract

Interactions between a quantum system and its environment at low temperatures can lead to apparent violations of thermal laws for the system. The source of these violations is the coupling between the system and environment, which prevents the system from entering into a thermal state, a state of Gibb's form. On the other hand, for two-state systems, we show that one can define an effective temperature, placing the system into a 'pseudo-thermal' state where effective thermal laws can be applied. We then numerically explore these assertions for an n -state environment inspired by the spin-boson environment.

DOI
10.1088/1751-8113/44/38/385003

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Interactions between a quantum system and its environment at low temperatures can lead to apparent violations of thermal laws for the system. The source of these violations is the coupling between the system and environment, which prevents the system from entering into a thermal state, a state of Gibb's form. On the other hand, for two-state systems, we show that one can define an effective temperature, placing the system into a 'pseudo-thermal' state where effective thermal laws can be applied. We then numerically explore these assertions for an n -state environment inspired by the spin-boson environment.
Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.
A standard theory of thermodynamics states that a quantum system in contact with a thermal environment relaxes to the equilibrium state known as the Gibbs state wherein decoherence occurs in the system's energy eigenbasis. When the interaction between the system and environment is strong, a different equilibrium state can be reached that is not diagonal in the system energy eigenbasis. Zurek's theory of einselection predicts that the decoherence takes place in the so-called pointer basis under the strong coupling regime, which can be viewed as continuous measurement of the system by the environment. The thermal state under the strong coupling regime is thus expected to be diagonal in the pointer states rather than energy eigenstates. We have postulated that the thermals state in the strong coupling limit is a Gibbs state projected onto the pointer basis and have demonstrated this with a simple model of single qubit strongly interacting with a bosonic environment.
Pseudomode approaches allow for an exact and unapproximated description of a quantum system interacting arbitrarily strongly with a bath. In general, a system coupled to pseudomodes will not thermalize to the system's Gibbs state: This is to be expected when the system-bath coupling is non-perturbative, but conflicts with common thermodynamic intuition when the system-bath coupling is asymptotically weak. We explore under which circumstances pseudomode models satisfy detailed balance (and subsequently thermalize to the system Gibbs state) and how specific choices of parameters can force "weak" detailed balance that is restricted to a limited frequency range. A combination of Hermitian and non-Hermitian pseudomodes that yields a flat effective-temperature profile is also considered. The results and criteria established here are relevant for the construction of pseudomode models in contexts where thermodynamic consistency is required.
Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.
Two qubits strongly coupled to a common bosonic reservoir can become entangled with each other, despite having no direct interaction. In equilibrium, such coupling-induced coherences can be described by the mean-force Gibbs state. Here we derive approximate, analytic expressions for the two-qubit mean-force Gibbs state, and use these to characterize equilibrium qubit-qubit entanglement mediated by a thermal reservoir. Entanglement, which is highest at lowest temperatures, is a non-monotonic function of the system-reservoir coupling strength. Moreover, we find that broadening the reservoir spectral density beyond a single mode, as is realistic for typical baths, can enhance the qubit entanglement. Our results provide a comprehensive understanding of reservoir-mediated two-qubit entanglement in thermal equilibrium and provide a benchmark to compare with numerical methods, as well as demonstrating the utility of strong system-reservoir coupling as a resource.