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arXiv:2006.04056 · Physical Review A

Power-law growth of time and strength of squeezing near a quantum critical point

Deepti Sharma, Brijesh Kumar

Published Sep 2, 2020• 3 citations
arXivHTMLPDF

Abstract

The dynamics of squeezing across a quantum phase transition in two basic models, viz., the one-axis twisting model in a transverse field and the Dicke model, is investigated using Holstein-Primakoff representation in the large spin limit. Near the phase boundary between the disordered (normal) and the ordered (superradiant) phases, the strength of spin and photon squeezing and the duration of time for which the system stays in the highly squeezed state are found to exhibit strong power-law growth with distance from the quantum critical point. The critical exponent for squeezing time is found to be 1/2 in both of the models, and for squeezing strength, it is shown to be 1/2 in the one-axis twisting model and 1 for the Dicke model, which in the limit of extreme detuning also becomes 1/2.

DOI
10.1103/physreva.102.033702

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Scientific Reports•Dec 6, 2018•10 citations
The dynamics of squeezing across a quantum phase transition in two basic models, viz., the one-axis twisting model in a transverse field and the Dicke model, is investigated using Holstein-Primakoff representation in the large spin limit. Near the phase boundary between the disordered (normal) and the ordered (superradiant) phases, the strength of spin and photon squeezing and the duration of time for which the system stays in the highly squeezed state are found to exhibit strong power-law growth with distance from the quantum critical point. The critical exponent for squeezing time is found to be 1/2 in both of the models, and for squeezing strength, it is shown to be 1/2 in the one-axis twisting model and 1 for the Dicke model, which in the limit of extreme detuning also becomes 1/2.
In this study, the energy spectrum and thermal equilibrium states of the finite-size Dicke-Stark model were numerically obtained within the extended coherent state space by solving the dressed master equation for strongly coupled light-atom systems. The critical point of the superradiant phase transition in the infinite-size Dicke-Stark model was analytically derived using the mean-field approach and confirmed with numerical calculation. Under thermal equilibrium conditions, analyses of the negativity, zero-time-delay two-photon correlation function, and atom-spin squeezing parameters in the finite-size Dicke-Stark model reveal that as the coupling strength increases, the light field undergoes a transition from photon bunching to anti-bunching and then back to bunching. The Stark field can modulate both the maximum and minimum values of the two-photon correlation function and their corresponding coupling strengths. At low temperatures, the system exhibits entanglement and spin squeezing. As temperature rises, entanglement gradually diminishes, while strong coupling facilitates the preservation of entanglement in the system state. Atom-spin squeezing spin squeezing is highly sensitive to temperature and vanishes rapidly with increasing temperature. This work contributes to the fundamental understanding of quantum phenomena in Dicke-Stark systems.
In view of recent proposals for the realization of anisotropic light-matter interaction in such platforms as (i) nonstationary or inductively and capacitively coupled superconducting qubits, (ii) atoms in crossed fields, and (iii) semiconductor heterostructures with spin-orbital interaction, the concept of a generalized Dicke model, where coupling strengths of rotating wave and counter-rotating wave terms are unequal, has attracted great interest. For this model we study photon fluctuations in the critical region of normal-to-superradiant phase transition when both the temperatures and numbers of two-level systems are finite. In this case, the superradiant quantum phase transition is changed to a fluctuational region in the phase diagram that reveals two types of critical behaviors. These are regimes of Dicke model (with discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry), and that of anti-Tavis-Cummings and Tavis-Cummings U(1)U(1)U(1) models. We show that squeezing parameters of photon condensate in these regimes show distinct temperature scalings. Besides, relative fluctuations of a photon number take universal values. We also find a temperature scale below which one approaches a zero-temperature quantum phase transition where quantum fluctuations dominate. Our effective theory is provided by a non-Goldstone functional for condensate mode and by Majorana representation of Pauli operators. We also discuss the Bethe ansatz solution for integrable U(1)U(1)U(1) limits.
We analyze the properties across steady-state phase transitions of two all-to-all driven-dissipative spin models that describe possible dynamics of NNN two-level systems inside an optical cavity: squeezed superradiance and driven superradiance. We show that the finite size behavior around the critical points can be captured correctly by carefully identifying the relevant nonlinearities in the Holstein-Primakoff representation of spin operators in terms of bosonic variables. With these tools, we calculate analytically various observables across the phase transitions and obtain their finite size scalings, including numerical prefactors. In particular, we look at the amount of spin squeezing carried by the steady states, of relevance for quantum metrology applications, and describe in analytical detail the mechanism by which the optimal spin squeezing acquires logarithmic corrections that depend on the system size. We also demonstrate that the logarithmic nature of these corrections is difficult to characterize through numerical procedures for any experimentally realistic and/or simulable values of particle number. We complement all of our analytical arguments with numerical benchmarks.
doi:10.1088/1367-2630/16/6/063039 The single-mode Dicke model is well known to undergo a quantum phase transition from the so-called normal phase to the superradiant phase (hereinafter called the ‘superradiant quantum phase transition’). Normally, quantum phase transitions can be identified by the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study the role of the quantum Fisher information (QFI) of both the field mode and the atoms in the ground state of the Dicke Hamiltonian. For a finite but large number of atoms, our numerical results show that near the critical atom-field coupling, the QFI of the atomic and the field subsystems can surpass their classical limits, due to the appearance of nonclassical quadrature squeezing. As the coupling increases far beyond the critical point, each subsystem becomes a highly mixed state, which degrades the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present the analytical results of the QFI and their relationship with the reduced variances of the field mode and the atoms. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of the quantum criticality in the Dicke model.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
By employing quantum renormalization group (QRG) method, we investigate quantum phase transitions (QPT) in the Ising transverse field (ITF) model and in the XXZ Heisenberg model, with and without Dzyaloshinskii Moriya (DM) interaction, on a periodic chain of N lattice sites. We adopt a new approach called spin squeezing as an indicator of QPT. Spin squeezing, through analytical expression of a spin squeezing parameter, is calculated after each step of QRG. As the scale of the system becomes larger, (after many QRG steps), the ground state (GS) spin squeezing parameters show an abrupt change at a quantum critical point (QCP). Moreover, in all of the studied models, the first derivative of the spin squeezing parameter with respect to the control parameter is discontinuous, which is a signature of QPT. The spin squeezing parameters develop their saturated values after enough QRG iterations. The divergence exponent of the first derivative of the spin squeezing parameter in the near vicinity of the QCP is associated with the critical exponent of the correlation length.