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arXiv:1904.10178 · Physical Review A

Generalized coherent-squeezed-state expansion for the super-radiant phase transition

Xiang-You Chen, Yuyu Zhang, Libin Fu, Hang Zheng

Published Mar 18, 2020• 26 citations
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Abstract

We develop a coherent-squeezed-state expansion for the ground state of the quantum phase transition in the quantum Rabi model. The wave function undergoes a transition from a localized state in the normal phase to a two-peaks delocalized state in the super-radiant phase. Interestingly, the two displaced wave packets are squeezed in the momentum space, and overlap strongly to form a correlated state. We develop a superposition of coherent-squeezed states to describe the phase transition correctly, which includes the higher-order excitations at the critical value. The corresponding ground-state energy and the average photon number are in agreement with numerical ones, exhibiting an improvement over previous coherent-state expansions. Moreover, our method successfully captures the reduced quantum fluctuations in momentum variable in the phase transition.

DOI
10.1103/physreva.101.033827

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We develop a coherent-squeezed-state expansion for the ground state of the quantum phase transition in the quantum Rabi model. The wave function undergoes a transition from a localized state in the normal phase to a two-peaks delocalized state in the super-radiant phase. Interestingly, the two displaced wave packets are squeezed in the momentum space, and overlap strongly to form a correlated state. We develop a superposition of coherent-squeezed states to describe the phase transition correctly, which includes the higher-order excitations at the critical value. The corresponding ground-state energy and the average photon number are in agreement with numerical ones, exhibiting an improvement over previous coherent-state expansions. Moreover, our method successfully captures the reduced quantum fluctuations in momentum variable in the phase transition.
By analytically solving the quantum Rabi model, we investigate the photonic properties of its ground eigenstate. In particular, we find that in the deep strong coupling regime, where the coupling strength ggg exceeds the mode frequency ωωω, the photonic state is effectively squeezed in one of its quadratures. The squeezing reaches its maximum at the curve corresponding to the quantum phase transition of the quantum Rabi system, and decreases rapidly on both sides of the phase transition. Notably, for g/ω≈3g/ω\approx 3g/ω≈3, which is experimentally testable in existing trapped-ion platforms, the achievable squeezing parameter can reach approximately r≈0.8r\approx 0.8r≈0.8. Intriguingly, the photonic state is squeezed while its number distribution follows a super-Poissonian distribution, with the largest deviation from Poissonian behavior occurring at the phase transition between the normal and superradiant phases. In other words, the ground state of the quantum Rabi model contains super-Poissonian quantum squeezed photons.
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.
The Dicke model exhibits a variety of phase transitions. The quantum phase transition from the normal phase to the super-radiant phase is marked by a dramatic change in the scaling of the participation ratio. We find that the ground state in the super-radiant phase exhibits multifractality manifest in the participation ratio scaling as the square root of the full Hilbert-space dimension. The thermal phase transition temperature, for which an exact analytical expression may be obtained using the partition function, is strikingly captured by the mutual information between two spins. In the excited-state quantum phase transition within the super-radiant phase, there are two cutoff energies; the central energy bands between the lower and upper cutoff energies show distinctly different behaviors. While the level statistics of the central band is Wigner-Dyson, the lower and upper bands show mixed behavior that is closer to Poisson than Wigner-Dyson. This finding is corroborated with the aid of several eigenvector properties: von Neumann entanglement entropy between spins and bosons, the mean photon number, concurrence between two spins, and participation ratio.
Abstract Quantum phase transition and entanglement in the Rabi model with a squeezed light are investigated. We find a special unitary-transformation method that removes the nonintegrable squeezing and counterrotating-wave interactions when the qubit frequency is close to the field frequency. The analytical ground state agrees well with the numerical solution. We demonstrate that the ground state exhibits a first-order quantum phase transition at a critical point induced linearly by the squeezed light. This quantum phase transition requires neither multiple qubits nor an infinite ratio of qubit frequency to field frequency, which solves a critical problem for the theory and experiment in Rabi model. As the qubit-field coupling strength increases, the ground-state entanglement reaches its maximum value at the critical point.
The Rabi model, a two-level atom coupled to a harmonic oscillator, can undergo a second-order quantum phase transition (QPT) [M.-J. Hwang et al., Phys. Rev. Lett. 115, 180404 (2015)]. Here we show that the Rabi QPT accompanies critical behavior in the higher-energy excited states, i.e., the excited-state QPT (ESQPT). We derive analytic expressions for the semiclassical density of states, which show a logarithmic divergence at a critical energy eigenvalue in the broken symmetry (superradiant) phase. Moreover, we find that the logarithmic singularities in the density of states lead to singularities in the relevant observables in the system such as photon number and atomic polarization. We corroborate our analytical semiclassical prediction of the ESQPT in the Rabi model with its numerically exact quantum mechanical solution.
The ground state of the quantum Rabi model (QRM) exhibits rich nonclassical states including the squeezed state, cat state, and entangled state in different parameter regimes. In this paper, we first use the polaron picture with high precision and fidelity to figure out the process of the superradiant phase transition (SPT) in view of the nonclassical states. The squeezing effect is considered to be a result of the higher-order excitations and important in order to understand the fundamental quantum physics of the SPT in the QRM. According to the polaron wave functions, the squeezed state extends across the entire range of parameters. Based on the squeezing and superradiance, a classification of the coupling strength for the nonclassical states containing the ground state is presented; the system sequentially goes through the vacuum states, the squeezed vacuum states, the squeezed cat states, and the cat states as the coupling strength increases. The process of the SPT is also demonstrated by the photon number distribution in Fock space, and then the novel behavior of parity chains for the SPT is revealed. Additionally, the nonclassical states can be intuitively characterized by the Wigner functions with analytical expressions in the polaron picture, and the Wigner negativity is evaluated to measure the nonclassicality. We find that a small but nonzero weighted antipolaron is necessary to form novel semicat states. Our paper provides guidance for preparing nonclassical states in experiments based on the light-matter interaction.