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Quantum transition in nonequilibrium environments

Tero Tapio Heikkilä, Teemu Ojanen, Peter Samuelsson, Jukka P Pekola

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

In an open quantum system, the approach to some steady (equilibrium) state is controlled by the coupling of the system to its environment. In a system with a fixed number of particles, this coupling typically makes the steady-state occupation numbers for the energy eigenstates of the system follow the Maxwell-Boltzmann distribution, where the temperature is set by the properties of the environment. Along with the approach to the equilibrium, the coupling projects the system to a diagonal state in its energy eigenbasis. This dephasing makes the system forget all the information of the quantum coherences between the different energy states.

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In an open quantum system, the approach to some steady (equilibrium) state is controlled by the coupling of the system to its environment. In a system with a fixed number of particles, this coupling typically makes the steady-state occupation numbers for the energy eigenstates of the system follow the Maxwell-Boltzmann distribution, where the temperature is set by the properties of the environment. Along with the approach to the equilibrium, the coupling projects the system to a diagonal state in its energy eigenbasis. This dephasing makes the system forget all the information of the quantum coherences between the different energy states.
Relaxation dynamics of complex quantum systems with strong interactions towards the steady state is a fundamental problem in statistical mechanics. The steady state of subsystems weakly interacting with their environment is described by the canonical ensemble which assumes the probability distribution for energy to be of the Boltzmann form. The emergence of this probability distribution is ensured by the detailed balance of the transitions induced by the interaction with the environment. Here we consider relaxation of an open correlated quantum system brought into contact with a reservoir in the vacuum state. We refer to such a system as emissive since particles irreversibly evaporate into the vacuum. The steady state of the system is a statistical mixture of the stable eigenstates. We found that, despite the absence of the detailed balance, the stationary probability distribution over these eigenstates is of the Boltzmann form in each N-particle sector. A quantum statistical ensemble corresponding to the steady state is characterized by different temperatures in the different sectors, in contrast to the Gibbs ensemble. We investigate the transition rates between the eigenstates to understand the emergence of the Boltzmann distribution and find their exponential dependence on the transition energy. We argue that this property of transition rates is generic for a wide class of emissive quantum many-body systems.
We analytically and numerically investigate the steady-state entanglement and coherence of two coupled qubits, each interacting with a local boson or fermion reservoir, based on the Bloch-Redfield master equation beyond the secular approximation. We find that there is nonvanishing steady-state coherence in the nonequilibrium scenario, which grows monotonically with the nonequilibrium condition quantified by the temperature difference or chemical potential difference of the two baths. The steady-state entanglement, in general, is a nonmonotonic function of the nonequilibrium condition as well as the bath parameters in the equilibrium setting. We also discover that weak interqubit coupling and high base temperature or chemical potential of the baths can strongly suppress the steady-state entanglement and coherence, regardless of the strength of the nonequilibrium condition. On the other hand, the energy detuning of the two qubits, when used in a compensatory way with the nonequilibrium condition, can lead to significant enhancement of the steady-state entanglement in some parameter regimes. In addition, the qubits typically have a stronger steady-state entanglement when coupled to fermion baths exchanging particles with the system than boson baths exchanging energy with the system, under similar conditions. We also identify a close connection between the energy current flowing through the system and the steady-state coherence. Preliminary investigations suggest that these results are insensitive to the form of the reservoir spectral densities in the Markovian regime. Feasible experimental realization of measuring the steady-state entanglement and coherence is discussed for the coupled qubit system in nonequilibrium environments. Our findings offer some general guidelines for optimizing the steady-state entanglement and coherence in the coupled qubit system and may find potential applications in quantum information technology.
Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.
Abstract In the interactions of electromagnetic radiation with matter which we have considered so far, we have almost invariably known accurately the state of the system before interaction. As a result the notion of an open system with an external environment did not have to be systematically discussed. In the quantum electrodynamics of elementary particles, the scattering events are predominantly affected by only few-body interactions and the concepts of temperature and environment are irrelevant. (The resolution of the problem of infra-red divergences is an exception where the environment of soft photons is crucial.) For very small numbers of atoms it may be possible to know the initial state of the system, but in a macroscopic device this is essentially impossible. Although for transitions at optical frequencies the temperature of the environment may be negligible, this will not necessarily be the case for microwave transitions. We have already seen how atomic linewidths can be produced by interactions with the bath of photons at zero temperature. In the general case a quantum-mechanical system will show not only fluctuations due to the non-commutative nature of quantum theory but also thermal fluctuations and dissipation due to interactions with the environment. For complicated systems it will not be possible to know with certainty the initial state of the system, and so the approach of studying the evolution of a state vector through the Schrodinger equation has to be replaced by a consideration of the time evolution of the density operator introduced in Chapter 4. We will formulate a systematic way of treating the dynamics of a quantum-mechanical open system interacting with a heat bath, which will be illustrated with important examples.
We formulate a geometric framework for quasistatic thermodynamics in open quantum systems by parameterizing the dynamics on a control manifold. In the quasistatic limit, the system follows a manifold of stationary states, and the work performed over a cycle is given by the flux of a curvature two-form, W∼∫ΩW \sim \int ΩW∼∫Ω, defined by the parametric response of the stationary state, establishing an open-system analog of classical thermodynamic area laws. \erbedit{For thermal stationary states at fixed temperature, the curvature vanishes, reflecting the integrability of the work one-form.} Beyond this limit, nonequilibrium stationary states can retain coherence in the energy representation; using a fixed-basis Lindblad model, we show that this coherence reshapes the curvature, making it anisotropic and sign-changing, so that work depends sensitively on the placement and orientation of the cycle. Quantum coherence, therefore, partitions the control manifold into regions of opposite curvature, producing geometric cancellation of work and allowing the net work over a cycle to be reduced or reversed despite dissipative dynamics. Thermodynamic work thus emerges as a curvature flux whose structure is set by thermodynamic response in classical systems and by basis misalignment between the Hamiltonian eigenbasis and the environment-selected pointer basis in open quantum systems.