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arXiv:1910.14246 · Physical Review A

Application of the polaron picture in the two-qubit quantum Rabi model

Xi-Mei Sun, Lei Cong, Hans-Peter Eckle, Zu‐Jian Ying, Hong‐Gang Luo

Published Jun 25, 2020• 11 citations
arXivHTMLPDF

Abstract

The polaron picture is employed to investigate and elucidate the physics of the two-qubit quantum Rabi model, which describes two identical qubits coupled to a common harmonic oscillator. This approach enables us to obtain the ground-state energy and some other simpler physical observables with high accuracy in all regimes of the coupling strengths $g$, while there is no constraint to the ration of tunneling frequency $\mathrm{\ensuremath{\Omega}}$ and field frequency $\ensuremath{\omega}$, which is not simultaneously possible using previous methods. In addition, we also discover a phenomenon that is not present in the one-qubit Rabi model: with the increase of coupling strength $g$, there is a transition of the ground state of the system from a multipolaron state to a bipolaron state. However, the tunneling frequency $\mathrm{\ensuremath{\Omega}}$ counteracts this process. Specifically, when the tunneling frequency $\mathrm{\ensuremath{\Omega}}=0$, the system always stays in the bipolaron state.

DOI
10.1103/physreva.101.063832

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The polaron picture is employed to investigate and elucidate the physics of the two-qubit quantum Rabi model, which describes two identical qubits coupled to a common harmonic oscillator. This approach enables us to obtain the ground-state energy and some other simpler physical observables with high accuracy in all regimes of the coupling strengths ggg, while there is no constraint to the ration of tunneling frequency \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ and field frequency \ensuremathω\ensuremath{\omega}\ensuremathω, which is not simultaneously possible using previous methods. In addition, we also discover a phenomenon that is not present in the one-qubit Rabi model: with the increase of coupling strength ggg, there is a transition of the ground state of the system from a multipolaron state to a bipolaron state. However, the tunneling frequency \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ counteracts this process. Specifically, when the tunneling frequency \ensuremathΩ=0\mathrm{\ensuremath{\Omega}}=0\ensuremathΩ=0, the system always stays in the bipolaron state.
We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two bare polarons. Importantly, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional induced polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of an incomplete spectral collapse behavior that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.
The experimental access to ultra-strong couplings in light-matter interactions has made the quantum phase transition (QPT) in the quantum Rabi model practically relevant, while the physics of the QPT has not yet been fully explored. The polaron picture is a method capable of analyzing in the entire coupling regime and extracting the essential physics behind the QPT. However, the asymmetric deformation of polarons is missing in the current polaron picture. In the present work we propose an improved variational method in asymmetric polaron picture (APP). Our APP not only increases the method accuracy but also reveals more underlying physics concerning the QPT. We find that in the ground state both the polarons and antipolarons are asymmetrically deformed to a large extent, which leads to a richer phase diagram. We also analyze the first excited state in which we unveil an asymmetry direction reversal for the polarons and an attraction/replusion transition differently from the ground state. Finally, we apply the APP in quantum Fisher information analysis and critical coupling extraction, the improvements indicate that the polaron asymmetry makes a considerable contribution to the quantum resource in quantum metrology and plays an unnegligible role in the QPT. Our results and mechanism clarifications expose more subtle energy competitions and abundant physics, and the method potentially might have broader applications in light-matter interactions.
Abstract The concept of the polaron in condensed matter physics has been extended to the Rabi model, where polarons resulting from the coupling between a two-level system and single-mode photons represent two oppositely displaced oscillators. Interestingly, tunneling between these two displaced oscillators can induce an anti-polaron, which has not been systematically explored in the literature, especially in the presence of an asymmetric term. In this paper, we present a systematic analysis of the competition between the polaron and anti-polaron under the interplay of the coupling strength and the asymmetric term. While intuitively the anti-polaron should be secondary owing to its higher potential energy, we find that, under certain conditions, the minor anti-polaron may gain a reversal in the weight over the major polaron. If the asymmetric amplitude ϵ is smaller than the harmonic frequency ω , such an overweighted anti-polaron can occur beyond a critical value of the coupling strength g ; if ϵ is larger, the anti-polaron can even be always overweighted at any g . We propose that the explicit occurrence of the overweighted anti-polaron can be monitored by a displacement transition from negative to positive values. This displacement is an experimentally accessible observable, which can be measured by quantum optical methods, such as balanced Homodyne detection.
The familiar Rabi model, comprising a two-level system coupled to a quantum harmonic oscillator, continues to produce rich and surprising physics when the coupling strength becomes comparable to the individual subsystem frequencies. We construct approximate solutions for the eigenstates and energies in the regime in which the oscillator frequency is small compared to that of the two-level system and the coupling strength matches or exceeds the oscillator frequency. The resulting oscillator dynamics closely resembles that of a particle tunneling in a classical double-well potential. Relating our calculation to an earlier semiclassical approximation in which coupling to the two-level system creates an effective potential for the oscillator, we examine the extent to which this picture is valid. We find that, for certain parameter regimes, the interpretation of the oscillator dynamics in terms of tunneling holds to a good approximation despite the fundamentally entangled nature of the joint system. We assess the prospects for observation of oscillator tunneling in the context of nano- or micromechanical experiments and find that it should be possible if suitably high coupling strengths can be engineered.
We present a frequency-renormalized multipolaron expansion method to explore the ground state of the quantum Rabi model (QRM). The main idea is to take polaron as starting point to expand the ground state of QRM. The polarons are deformed and displaced oscillator states with variationally determined frequency-renormalization and displacement parameters. This method is an extension of the previously proposed polaron concept and the coherent state expansion used in the literature, which shows high efficiency in describing the physics of the QRM. The proposed method is expected to be useful for solving other more complicated light-matter interaction models.
We revisit the analytic solution of the two-qubit quantum Rabi model with inhomogeneous coupling and transition frequencies using a displaced oscillator basis. This approach enables us to apply the same truncation rules and techniques adopted in the Rabi model to the two qubits system. The derived analytical spectra match perfectly with the numerical solutions in the parameter regime where the qubits’ transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultrastrong coupling regime. We further explore the dynamical behavior of the two qubits as well as the evolution of entanglement. The analytical methods provide unexpectedly accurate results in describing the dynamics of the two qubits in the present experimentally accessible coupling regime. The time evolutions of the probability for the qubits show that the collapse-revival phenomena emerge, survive and finally disappear when one coupling strength increases from weak to strong coupling regimes and the other coupling strength is well into the ultrastrong coupling regime. The inhomogeneous coupling system exhibits new dynamics, which are different from the homogeneous coupling case.