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arXiv:1007.5437 · Physical Review A

Qubit-oscillator system: An analytical treatment of the ultrastrong coupling regime

Johannes Hausinger, Milena Grifoni

Published Dec 21, 2010• 75 citations
arXivHTMLPDF

Abstract

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 $g$ 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 $g$, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.

DOI
10.1103/physreva.82.062320

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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/\ensuremathΩg/\ensuremath{\Omega}g/\ensuremathΩ 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 \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ 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.
We introduce an approach to studying a driven qubit-oscillator system in the ultrastrong coupling regime, where the ratio $g/\ensuremath{\Omega}$ between coupling strength and oscillator frequency approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the qubit energy splitting (extreme driving). Both qubit-oscillator coupling and external driving lead to a dressing of the qubit tunneling matrix element of different nature: the former can be used to suppress selectively certain oscillator modes in the spectrum, while the latter can bring the qubit's dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit—oscillator systems analytically in a unified way. Many previous analytical treatments, such as generalized rotating-wave approximation (GRWA) [ Phys. Rev. Lett. 99 , 173601 (2007) ] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [ Phys. Rev. Lett. 105 , 263603 (2010) ] can be recovered straightforwardly within the present scheme. Moreover, further improving GRWA and the extension to the finite-bias case are implemented easily. The algebraic formulae for the eigensolutions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengths, detunings, and static bias including the recent experimentally accessible parameters. The dynamics of the qubit for an oscillator in the ground state is also studied. At the experimentally accessible coupling regime, GRWA can always work well. When the coupling is enhanced to the intermediate regime, only the improving GRWA can give the correct description, while the result of GRWA shows strong deviations. The previous Van Vleck perturbation theory is not valid to describe the dynamics in the present-day experimentally accessible regime, except for the strongly biased cases.
Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically in a regime where deviations from the exactly solvable Jaynes–Cummings model are important. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equation. Using the derived master equation, we discuss a spectroscopic technique which can be used to probe the dressed energy-level structure of the qubit-oscillator system.
Abstract This work aims to provide an alternative approach to modeling a two-state system (qubit) coupled to a nonlinear oscillator. Within a single algebraic scheme based upon the f -deformed oscillator description, hard and soft nonlinearities are proposed to be simulated by making use of fitting algebraic models extracted from the trigonometric and modified Pöschl–Teller potentials, respectively. In the regime where the strength of the coupling is considered to be moderate, this approach allows for an analytic, albeit approximate, diagonalization process of the proposed Hamiltonian through using the Van Vleck perturbation theory and embracing the two types of nonlinear features. In the ultrastrong-coupling regime, the effect of such nonlinearities upon the squeezing and phase space properties of the ground state of the composite system is also explored by numerical means.
We propose a reliable scheme to realize the ultrastrong Jaynes-Cummings (JC) model by simultaneously modulating the resonance frequencies of the two-level system and the bosonic mode in the ultrastrong quantum Rabi model. We find that in both the high- and low-frequency modulation regimes, the counter-rotating terms can be completely suppressed without reducing the coupling strength of the rotating-wave terms, and hence the ultrastrong JC Hamiltonian is achieved. The ultrastrong JC interaction can not only be used to implement ultrafast quantum operations such as ultrafast quantum state transfer and quantum logic gates but also will open up a new route to the demonstration of quantum phase transition associated with the JC Hamiltonian across the deep-strong-coupling point. Some discussions on the experimental implementation of this scheme with circuit-QED systems are presented.