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Register a new userDouble degeneracy associated with hidden symmetries in the asymmetric two-photon Rabi model
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In this paper, we uncover the elusive level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Due to the absence of any explicit symmetry in the subspace, this double degeneracy implies the existence of the hidden symmetry. The non-degenerate exceptional points are also given completely. It is found that the number of the doubly degenerate crossing points in the asymmetric tpQRM is comparable to that in asymmetric one-photon QRM in terms of the same order of the constrained conditions. The bias parameter required for occurrence of level crossings in the asymmetric tpQRM is characteristically different from that at a multiple of the cavity frequency in the asymmetric one-photon QRM, suggesting the different hidden symmetries in the two asymmetric QRMs.
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In this paper, we derive the symmetry operators ($J$s) in the asymmetric two-photon quantum Rabi models in terms of Bogoliubov operator approaches. $ J^2$ can be expressed as a polynomial in terms of the Hamiltonian, which uncovers the $mathbb{Z}_{2}$ nature of the hidden symmetry in this two-photon model rigorously. The previous symmetry operators in the asymmetric one-photon quantum Rabi models are reproduced readily in terms of Bogoliubov operator approaches, and the obtained operators are expressed much more concisely. It is found that the polynomial degree of $J^2$ in the two-photon model is twice of that in the one-photon model.
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 textit{bare} polarons. Furthermore, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional textit{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 the 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 asymmetric quantum Rabi model (AQRM) has a broken $mathbb{Z}_2$ symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases where the asymmetric parameter is a multiple of the cavity frequency, stable level crossings typical of the $mathbb{Z}_2$-symmetric quantum Rabi model are recovered, however, without any obvious parity-like symmetry. This unknown symmetry has thus been referred to as hidden symmetry in the literature. Here we show that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models with an asymmetric qubit bias term. Conditions under which the hidden symmetry exists in these models are determined and discussed. By investigating tunnelling dynamics in the displaced oscillator basis, a strong connection is found between the hidden symmetry and selective tunnelling.
The hidden $mathbb{Z}_2$ symmetry of the asymmetric quantum Rabi model (AQRM) has recently been revealed via a systematic construction of the underlying symmetry operator. Based on the AQRM result, we propose an ansatz for the general form of the symmetry operators for AQRM-related models. Applying this ansatz we obtain the symmetry operator for three models: the anisotropic AQRM, the asymmetric Rabi-Stark model (ARSM) and the anisotropic ARSM.
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 $epsilon$ is smaller than the harmonic frequency $omega$, such an overweighted anti-polaron can occur beyond a critical value of the coupling strength $g$; if $epsilon$ 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.
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