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Acta Physica Sinica

Photon squeezing of the Rabi model

Yu Li-Xian, Qifeng Liang, Wang Li-Rong, Zhu Shi-Qun

Published Jan 1, 2013• 0 citations
Publisher

Abstract

Recent experiments about the circuit cavity quantum electrodynamics have realized the ultrastrong couplings between the artificial atom and the photon, in which the coupling strengths have the same order of the photon frequency. In such a regime, the well-known rotating wave approximation is invalid, and the system dynamics is thus governed by the Rabi model. In this paper, we investigate the photon squeezing of the Rabi model. We find numerically that with the increase of the atom-photon coupling strength, the photon squeezing does not increase linearly, but displays a maximum in the ultrastrong coupling regime. In addition, we also reveal that the photonsqueezing can be enhanced by the counter-rotating terms of the Rabi model. Our results are of benefit to preparing the required squeezing state of the photon in experiment.

DOI
10.7498/aps.62.160301

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•May 16, 2018•1 citation
Recent experiments about the circuit cavity quantum electrodynamics have realized the ultrastrong couplings between the artificial atom and the photon, in which the coupling strengths have the same order of the photon frequency. In such a regime, the well-known rotating wave approximation is invalid, and the system dynamics is thus governed by the Rabi model. In this paper, we investigate the photon squeezing of the Rabi model. We find numerically that with the increase of the atom-photon coupling strength, the photon squeezing does not increase linearly, but displays a maximum in the ultrastrong coupling regime. In addition, we also reveal that the photonsqueezing can be enhanced by the counter-rotating terms of the Rabi model. Our results are of benefit to preparing the required squeezing state of the photon in experiment.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.
Dynamics of quantum entanglement of two qubits in two identical quantum Rabi models is studied analytically in the framework of corrections to the rotating-wave approximations. A closed-form expression for the entanglement dynamics initiated from the well-known Bell states is derived, which is very close to the numerical exact results up to the ultrastrong coupling regime. It is found that the vanishing entanglement can be purely induced by the counter-rotating terms, and can be enhanced with the atom–cavity coupling.
In recent years, quantum Rabi model has aroused considerable interest because of its fundamental importance and potential applications in quantum technologies. For a conventional cavity-quantum-electrodynamic (cavity-QED) system involving the interaction between an atom and photons in a cavity, the atom-photon coupling frequency is much smaller than the transition frequency of the atom and the frequency of the cavity mode. This cavity-QED system is usually described by the Jaynes-Cummings model in which the rotating-wave approximation can be adopted by neglecting the counter-rotating coupling terms in the Hamiltonian of the system. However, by designing the unique structure of the superconducting circuit, the ultrastrong-coupling regime can be achieved in a circuit-QED system in which the counter-rotating coupling terms become as important as the rotating terms. Thus, the rotating-wave approximation cannot be used in the ultrastrongly coupled circuit-QED system. Owing to the ultrastrong coupling, this circuit-QED system is described by the standard quantum Rabi model when a superconducting qubit is coupled only to a single resonator mode. In this work, we experimentally study an ultrastrongly coupled circuit-QED system consisting of a four-junction superconducting flux qubit and a muti-mode coplanar-waveguide resonator. The transmission-spectrum measurement and numerical simulations show that the system is in the ultrastrong-coupling regime. By changing the photon number in the resonator, we observe the frequency shift of the flux qubit via the spectroscopic measurement. This frequency shift contains the contributions from not only the rotating-coupling terms but also the counter-rotating terms, which is in good agreement with the theory. The result indicates that this ultrastrongly-coupled quantum system can be used as a good platform to investigate the quantum Rabi model and has potential applications in various aspects of quantum technology, such as quantum simulation, ultrafast quantum gates, entangled-state preparation and protected qubits.
We study the ground state of the two-qubit squeezed Rabi model. Two special transformations are found to diagonalize the system Hamiltonian when each qubit’s frequency is close to the field frequency, where both the squeezing and counterrotating-wave interactions are removed, leading to an effective integrable Hamiltonian. The analytical ground state is determined and matches with numerical solutions well for a range of squeezing strengths and qubit-field detunings in the ultrastrong-coupling regime. We demonstrate that the ground state exhibits a first-order quantum phase transition at a phase boundary linearly induced by the squeezed light. We characterize the two-qubit negativity analytically and find that its two-qubit entanglement increases with the increasing squeezing strength nonlinearly. The average photon numbers of the field mode and variances of position and momentum quadratures are also analyzed and discovered to have a nonlinear relation with the squeezing strength. Finally, we discuss the experimental scheme and realization possibility of the predicted results.
When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED was reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit-QED systems can undergo two- and three-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits.
Ultrastrong-coupling between an effective two-level system and radiation offers immense potential for advancing both fundamental and applied quantum electrodynamics (QED). Such regimes cannot be treated using the rotating-wave approximation, which applied to the quantum Rabi model yields the apparently less fundamental Jaynes-Cummings model. However, the implications of gauge-freedom for ultrastrong-coupling QED are yet to be recognised. Here we show that when truncating the material system to only two levels, each gauge gives a different effective description whose predictions can vary significantly for ultrastrong-coupling. Rabi models are obtained through specific gauge choices, but so too is a Jaynes-Cummings model without needing the rotating-wave approximation. Analysing a circuit QED setup, we find that this Jaynes-Cummings model is not only valid well beyond the rotating-wave approximation, it is often more accurate than the Rabi model. Among the many implications of this finding is that the system's ground state is not necessarily highly entangled, which is usually considered a hallmark of ultrastrong-coupling light-matter physics.