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Qubit decay in circuit quantum thermodynamics

Jukka P Pekola, Bayan Karimi

Published Oct 21, 2020• 4 citations
OpenAlex

Abstract

We describe a qubit linearly coupled to a heat bath, either directly or via a cavity. The bath is formed of oscillators with a distribution of energies and coupling strengths, both for qubit-oscillator and oscillator-oscillator interaction. A direct numerical solution of the Schrodinger equation for the full system including up to $10^6$ oscillators in the bath and analytic solutions are given, verifying quantum decay in short time quadratic (Zeno), long time exponential and eventually power law relaxation regimes. The main new results of the paper deal with applications and implications in quantum thermodynamics setups. We start by providing a correspondence of the oscillator bath to a resistor in a circuit. With the presented techniques we can then shed light on two topical questions of open quantum systems. First, splitting a quantum to uncoupled baths is presented as an opportunity for detection of low energy photons. Second, we address quantitatively the question of separation between a quantum system and its classical environment.

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We describe a qubit linearly coupled to a heat bath, either directly or via a cavity. The bath is formed of oscillators with a distribution of energies and coupling strengths, both for qubit-oscillator and oscillator-oscillator interaction. A direct numerical solution of the Schrodinger equation for the full system including up to 10610^6106 oscillators in the bath and analytic solutions are given, verifying quantum decay in short time quadratic (Zeno), long time exponential and eventually power law relaxation regimes. The main new results of the paper deal with applications and implications in quantum thermodynamics setups. We start by providing a correspondence of the oscillator bath to a resistor in a circuit. With the presented techniques we can then shed light on two topical questions of open quantum systems. First, splitting a quantum to uncoupled baths is presented as an opportunity for detection of low energy photons. Second, we address quantitatively the question of separation between a quantum system and its classical environment.
We describe a qubit linearly coupled to a heat bath, either directly or via a cavity.The main focus of the paper is on calorimetric detection in a realistic circuit, specifically a solid-state qubit coupled to a resistor as an absorber.The bath in the model is formed of oscillators initially in the ground state with a distribution of energies and coupling strengths.A direct numerical solution of the Schrödinger equation for the full system including up to 10 6 oscillators in the bath verifies the expected decay process.We address quantitatively the question of separation of the qubit and bath by adding a cavity in between which by detuning allows one to adjust the decay rate into a convenient regime for detection purposes.Most importantly, we propose splitting a quantum to two uncoupled baths and performing a cross-correlation measurement of their temperatures.This technique enhances significantly the signal-to-noise ratio of the calorimeter.
The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.
Recent developments in experimental methods allow for the study of thermodynamic properties of quantum systems. In quantum integrated circuits, quantum systems are elements in an electric circuit that can straightforwardly be coupled to other elements. This manipulability allows one to construct quantum heat engines, Maxwell demons etc. Quantum integrated circuits also are one of the main potential settings to realise a working quantum computer. Calorimetric measurements in integrated circuits serve as a promising technique to probe thermodynamic laws of the quantum regime and to study the inner workings of quantum devices. Due to this experimental accessibility, the theoretical study of open quantum systems in the context of quantum integrated circuits is highly relevant. Open quantum systems are typically small systems, e.g. qubits or oscillators, in contact with one or more reservoirs. The research on which this thesis is based can roughly be divided into two parts. The first part is concerned with the thermodynamics of a driven qubit in contact with a thermal bath. This system is the archetype of a quantum out-ofequilibrium system. In one case the qubit is strongly driven by a semiclassical driving field. Building on earlier works, the thermodynamic relations of the system are found by proving the equivalence with an easier to study qubit-oscillator system. In the other, case the qubit is driven by being in contact with two baths with a temperature gradient. The full generating function is derived in a proper approximative scheme and a fluctuation-dissipation relation is found. The second part focusses on a specific experimental scheme to perform calorimetric measurements. The scheme relies on coupling a quantum system to a finite reservoir and performing fast temperature measurements on the reservoir. Doing so allows one to infer energy changes in the reservoir and therefore to obtain the heat exchanged with the system. The dynamics of this system are modelled for weak system-reservoir coupling and concrete experimental predictions are made. In new work, the dynamics for a toy model of a system interacting with a finite reservoir are derived from first principles for a specific model. The first principle derivation matches with the earlier modelled dynamics in the weak coupling and allows to consider strong system-reservoir coupling as well.
We discuss the concept and realization of a heat bath in solid state quantum systems. We demonstrate that, unlike a true resistor, a finite one-dimensional Josephson junction array or analogously a transmission line with non-vanishing frequency spacing, commonly considered as a reservoir of a quantum circuit, does not strictly qualify as a Caldeira-Leggett type dissipative environment. We then consider a set of quantum two-level systems as a bath, which can be realized as a collection of qubits. We show that only a dense and wide distribution of energies of the two-level systems can secure long Poincare recurrence times characteristic of a proper heat bath. An alternative for this bath is a collection of harmonic oscillators, for instance, in the form of superconducting resonators.
We compare two popular models applicable to analyzing heat transport by thermal microwave photons in quantum circuits. The first model is derived from a weak-coupling Lindblad master equation, with transition rates determined by Fermi's golden rule induced by thermal dissipation sources. The second approach employs a circuit model, where thermal Johnson-Nyquist noise generated by dissipative elements introduces currents, and consequently Joule power, in other parts of the circuit. This leads to a Landauer type expression of heat transport where the transmission coefficient is proportional to the transconductance in the circuit. We find that the two models yield identical results in a linear circuit in the weak coupling limit with an analytic expression of power in an archetypal circuit of a cavity mediating heat between two baths. Our analysis yields a quantitative assessment of the range of validity of the weak coupling assumption in a circuit. Due to the correspondence of the two results, we feel confident in applying the weak coupling Lindblad model also for analyzing heat transport in quantum circuits consisting, e.g. of qubits and/or non-linear resonators.
We discuss the long-time relaxation of a qubit linearly coupled to a finite bath of [Formula: see text] spins (two-level systems, TLSs), with the interaction Hamiltonian in rotating wave approximation. We focus on the regime [Formula: see text], assuming that the qubit–bath coupling is weak, that the range of spin frequencies is sufficiently broad, and that all the spins are initialized in the ground state. Despite the model being perfectly integrable, we make two interesting observations about the effective system relaxation. First, as one would expect, the qubit relaxes exponentially towards its zero-temperature state at a well characterized rate. Second, the bath spins, even when mutually coupled, do not relax towards a thermal distribution, but rather form a Lorentzian distribution peaked at the frequency of the initially excited qubit. This behaviour is captured by an analytical approximation that makes use of the property [Formula: see text] to treat the TLS frequencies as a continuum and is confirmed by our numerical simulations.