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arXiv:1006.1989 · Physical Review A

Variational study of a two-level system coupled to a harmonic oscillator in an ultrastrong-coupling regime

Myung-Joong Hwang, Mahn‐Soo Choi

Published Aug 30, 2010• 57 citations
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Abstract

The nonclassical behavior of a two-level system coupled to a harmonic oscillator is investigated in the ultrastrong coupling regime. We revisit the variational solution of the ground state and find that the existing solutions do not account accurately for nonclassical effects such as squeezing. We suggest a trial wave function and demonstrate that it has an excellent accuracy for the quantum correlation effects as well as for the energy.

DOI
10.1103/physreva.82.025802

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The nonclassical behavior of a two-level system coupled to a harmonic oscillator is investigated in the ultrastrong coupling regime. We revisit the variational solution of the ground state and find that the existing solutions do not account accurately for nonclassical effects such as squeezing. We suggest a trial wave function and demonstrate that it has an excellent accuracy for the quantum correlation effects as well as for the energy.
We consider a system composed of a two-level system (i.e., a qubit) and a harmonic oscillator in the ultrastrong-coupling regime, where the coupling strength is comparable to the qubit and oscillator energy scales. Special emphasis is placed on the possibility of preparing nonclassical states in this system. These nonclassical states include squeezed states, Schr\"odinger-cat states, and entangled states. We start by comparing the predictions of a number of analytical methods that can be used to describe the system under different assumptions, thus analyzing the properties of the system in various parameter regimes. We then examine the ground state of the system and analyze its nonclassical properties. We finally discuss some questions related to the possible experimental observation of the nonclassical states and the effect of decoherence.
When the coupling rate between two quantum systems becomes as large as their characteristic frequencies, it induces dramatic effects on their dynamics and even on the nature of their ground state. The case of a qubit coupled to a harmonic oscillator in this ultrastrong coupling regime has been investigated theoretically and experimentally. Here, we explore the case of two harmonic oscillators in the ultrastrong coupling regime. Probing the properties of their ground state remains out of reach in natural implementations. Therefore, we have realized an analog quantum simulation of this coupled system by dual frequency pumping a nonlinear superconducting circuit. The pump amplitudes directly tune the effective coupling rate. We observe spectroscopic signature of a mode hybridization that is characteristic of the ultrastrong coupling. We experimentally demonstrate a key property of the ground state of this simulated ultrastrong coupling between modes by observing simultaneous single- and two-mode squeezing of the radiated field below vacuum fluctuations.
We study the system involving mutual interaction between three qubits and an oscillator within the ultrastrong coupling regime. We apply the adiabatic approximation approach to explore two extreme regimes: (i) the oscillator's frequency is far larger than each qubit's frequency and (ii) the qubit's frequency is far larger than the oscillator's frequency, and analyze the energy-level spectrum and the ground-state property of the qubit-oscillator system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low-energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum qubit-oscillator coupling strength needed for generating the nonclassical states of the Schr\"odinger-cat type in the oscillator is just one half of that in the Rabi model. We find that the qubit-qubit entanglement in the ground state vanishes if the qubit-oscillator coupling strength is strong enough, for which the entropy of three qubits remains larger than 1. We also observe the phase-transition-like behavior in the regime where the qubit's frequency is far larger than the oscillator's frequency.
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.
We consider two coupled quantum harmonic oscillators with different free frequencies. Here the interaction between the two modes does not involve the Rotating Wave Approximation (RWA). The Heisenberg equations of motion are solved analytically using an approximate technique and the solutions are used to measure nonclassicalities associated with the system. The nonclassicality criteria chosen in this study are experimentally measurable. The analytical solutions are matched with numerical simulations with the help of a numerical toolbox. It is evident from the time developments of the operators that nonclasities, namely squeezing and quantum entanglement are present in the system. The counting statistics for the oscillator with lower free energy and the coupled mode are shown to be sub-Poissonian for a particular set of parameters. Consequently, the system may exhibit quantum mechanical antibunching.
Abstract In order to obtain the ground state of the biased quantum Rabi model in the ultrastrong coupling regime, a generalized squeezing rotating-wave approximation method including the displacement and the squeezing transformations is proposed in this work. While considering the deformation of the oscillator state, our method evidently improves the generalized variational method. The analyses of a few physical quantities show that our method works well especially in the ultrastrong coupling and large atomic frequency regime.