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arXiv:cond-mat/0403257 · Physical Review B

Cavity quantum electrodynamics in superconducting circuits: Susceptibility at elevated temperatures

I. G. Rau, Göran Johansson, Alexander Shnirman

Published Aug 31, 2004• 37 citations
arXivHTMLPDF

Abstract

We study the properties of superconducting electrical circuits, realizing cavity quantum electrodynamics. In particular we explore the limit of strong coupling, low dissipation, and elevated temperatures relevant for current and future experiments. We concentrate on the cavity susceptibility as it can be directly experimentally addressed, i.e., as the impedance or the reflection coefficient of the cavity. To this end we investigate the dissipative Jaynes-Cummings model in the strong coupling regime at high temperatures. The dynamics is investigated within the Bloch-Redfield formalism. At low temperatures, when only the few lowest levels are occupied, the susceptibility can be presented as a sum of contributions from independent level-to-level transitions. This corresponds to the secular (random phase) approximation in the Bloch-Redfield formalism. At temperatures comparable to and higher than the oscillator frequency, many transitions become important and a multiple-peak structure appears. We show that in this regime the secular approximation breaks down, as soon as the peaks start to overlap. In other words, the susceptibility is no longer a sum of contributions from independent transitions. We treat the dynamics of the system numerically by exact diagonalization of the Hamiltonian of the qubit plus up to 200 states of the oscillator. We compare the results obtained with and without the secular approximation and find a qualitative discrepancy already at moderate temperatures.

DOI
10.1103/physrevb.70.054521

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We study the properties of superconducting electrical circuits, realizing cavity quantum electrodynamics. In particular we explore the limit of strong coupling, low dissipation, and elevated temperatures relevant for current and future experiments. We concentrate on the cavity susceptibility as it can be directly experimentally addressed, i.e., as the impedance or the reflection coefficient of the cavity. To this end we investigate the dissipative Jaynes-Cummings model in the strong coupling regime at high temperatures. The dynamics is investigated within the Bloch-Redfield formalism. At low temperatures, when only the few lowest levels are occupied, the susceptibility can be presented as a sum of contributions from independent level-to-level transitions. This corresponds to the secular (random phase) approximation in the Bloch-Redfield formalism. At temperatures comparable to and higher than the oscillator frequency, many transitions become important and a multiple-peak structure appears. We show that in this regime the secular approximation breaks down, as soon as the peaks start to overlap. In other words, the susceptibility is no longer a sum of contributions from independent transitions. We treat the dynamics of the system numerically by exact diagonalization of the Hamiltonian of the qubit plus up to 200 states of the oscillator. We compare the results obtained with and without the secular approximation and find a qualitative discrepancy already at moderate temperatures.
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We propose a realizable architecture using one-dimensional transmission line resonators to reach the strong-coupling limit of cavity quantum electrodynamics in superconducting electrical circuits. The vacuum Rabi frequency for the coupling of cavity photons to quantized excitations of an adjacent electrical circuit (qubit) can easily exceed the damping rates of both the cavity and qubit. This architecture is attractive both as a macroscopic analog of atomic physics experiments and for quantum computing and control, since it provides strong inhibition of spontaneous emission, potentially leading to greatly enhanced qubit lifetimes, allows high-fidelity quantum nondemolition measurements of the state of multiple qubits, and has a natural mechanism for entanglement of qubits separated by centimeter distances. In addition it would allow production of microwave photon states of fundamental importance for quantum communication.
We present a multilevel quantum theory of decoherence for a general circuit realization of a superconducting qubit. Using electrical network graph theory, we derive a Hamiltonian for the circuit. The dissipative circuit elements (external impedances, shunt resistors) are described using the Caldeira-Leggett model. The master equation for the superconducting phases in the Born-Markov approximation is derived and brought into the Bloch-Redfield form in order to describe multilevel dissipative quantum dynamics of the circuit. The model takes into account leakage effects, i.e., transitions from the allowed qubit states to higher excited states of the system. As a special case, we truncate the Hilbert space and derive a two-level (Bloch) theory with characteristic relaxation (T1){(T}_{1})(T1​) and decoherence (T2){(T}_{2})(T2​) times. We apply our theory to the class of superconducting flux qubits; however, the formalism can be applied for both superconducting flux and charge qubits.
Critical slowing down of the time it takes a system to reach equilibrium is a key signature of bistability in dissipative first-order phase transitions. Understanding and characterizing this process can shed light on the underlying many-body dynamics that occur close to such a transition. Here, we explore the rich quantum activation dynamics and the appearance of critical slowing down in an engineered superconducting quantum circuit. Specifically, we investigate the intermediate bistable regime of the generalized Jaynes-Cummings Hamiltonian (GJC), realized by a circuit quantum electrodynamics (cQED) system consisting of a transmon qubit coupled to a microwave cavity. We find a previously unidentified regime of quantum activation in which the critical slowing down reaches saturation and, by comparing our experimental results with a range of models, we shed light on the fundamental role played by the qubit in this regime.
We examine a two-level system coupled to a quantum oscillator, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\ensuremath{\Omega}$ between coupling strength and oscillator frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the qubit tunneling matrix element $\ensuremath{\Delta}$ we are able to enlarge the regime of applicability of existing analytical treatments, including, in particular, also the finite-bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the qubit for an oscillator at low temperature. We consider the coupling strength ggg to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many frequencies contribute to the dynamics. They are gathered into groups whose spacing depends on the qubit-oscillator detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with ggg, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.