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arXiv:1702.06658 · Journal of Physics A Mathematical and Theoretical

The asymmetric quantum Rabi model in the polaron picture

Maoxin Liu, Zu‐Jian Ying, Jun‐Hong An, Hong‐Gang Luo, Haiqin Lin

Published Jan 5, 2017• 11 citations
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Abstract

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.

DOI
10.1088/1751-8121/aa56f6

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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.
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.
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 $\mathrm{\ensuremath{\Omega}}$ 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 ggg, 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.
We investigate the entanglement features in the interacting system of a quantized optical field and a two-level system which is statically driven, known as the asymmetric quantum Rabi model (AsymQRM). Intriguing entanglement resonance valleys with the increase of the photon-atom coupling strength and peaks with the increase of the driving amplitude are found. It is revealed that both of these two kinds of entanglement resonance are caused by the avoided level crossing of the associated eigenenergies. In sharp contrast to the quantum Rabi model, the entanglement of the AsymQRM collapses to zero in the strong coupling regime except when the driving amplitude is equal to $m\ensuremath{\omega}/2$, with mmm being an integer and $\ensuremath{\omega}$ being the photon frequency. Our analysis demonstrates that such entanglement reappearance is induced by the hidden symmetry of the AsymQRM. Supplying an insightful understanding of the AsymQRM, our results will be helpful in exploring the hidden symmetry and in preparing photon-atom entanglement in light-matter coupled systems.
The introduction of an asymmetric term into the quantum Rabi model generally lifts energy-level degeneracies. However, when the asymmetry parameter takes specific multiples of the bosonic mode frequency, level degeneracies reappear−-−a phenomenon referred to as the hidden symmetry in the asymmetric quantum Rabi model. Identifying the origin of this hidden symmetry and its explicit operator form constitutes two central tasks in studying this system. Here, we investigate the origin of this hidden symmetry using the method of two successive diagonalizations, with a focus on physics in the regime where the ratio between the two-level splitting ΔΔΔ and the mode frequency ωωω satisfies Δ/ω≫1Δ/ω\gg 1Δ/ω≫1. We find that the hidden symmetry stems from energy-level matching within the asymmetric double-well potential, a picture strongly supported by the wavefunctions of both the ground and excited states. Moreover, the emergence of an excited-state quantum phase transition is identified and qualitatively discussed, which arises from the breaking and restoration of this hidden symmetry across different coupling regimes. Our results provide deeper insight into the physics of the asymmetric quantum Rabi model, particularly in the previously less-explored strong-coupling regime where Δ/ω≫1Δ/ω\gg 1Δ/ω≫1.
The quantum Rabi model (QRM) is a fundamental model for light-matter interactions. A fascinating feature of the QRM is that it manifests a quantum phase transition which is applicable for critical quantum metrology (CQM). Effective application for CQM needs the exact location of the transition point, however the conventional expression for transition coupling is only valid in the extreme limit of low frequency, while apart from zero frequency an accurate location is still lacking. In the present work we conversely use the quantum Fisher information (QFI) in the CQM to identify the transition coupling, which reveals that transition coupling indeed deviates greatly from the conventional one once a finite frequency is turned on. Polaron picture is applied to analytically reproduce the numeric QFI. An accurate expression for the transition coupling is obtained by inspiration from the revealed fractional-power-law effect of polaron frequency renormalization. From the QFI in the polaron picture we find that the transition occurs around a point where the effective velocity and the susceptibility of the expected value of mode quadrature reach maximum. Our result provides an accurate reference of transition couplings for quantum metrology at nonzero frequencies. The formulation of the QFI in the polaron picture also prepares an analytic method with an accurate compensation for the parameter regime difficult to access for the numerics. Besides the integer/fractional-power-law analysis to extract the underlying physics of transition, the QFI/velocity relation may also add some insight in bridging the QFI and transition observables.