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arXiv:2405.12558 · SciPost Physics Core

Phase transitions from heating to non-heating in $SU(1,1)$ quantum dynamics: Applications to Bose-Einstein condensates and periodically driven coupled oscillators

Heng-Hsi Li, Po-Yao Chang

Published Feb 10, 2025• 0 citations
arXivHTMLPDF

Abstract

We study the entanglement properties in non-equilibrium quantum systems with the SU(1,1) S U ( 1 , 1 ) structure. Through Möbius transformation, we map the dynamics of these systems following a sudden quench or a periodic drive onto three distinct trajectories on the Poincaré disc, corresponding the heating, non-heating, and a phase boundary describing these non-equilibrium quantum states. We consider two experimentally feasible systems where their quantum dynamics exhibit the SU(1,1) S U ( 1 , 1 ) structure: the quench dynamics of the Bose-Einstein condensates and the periodically driven coupled oscillators. In both cases, the heating, non-heating phases, and their boundary manifest through distinct signatures in the phonon population where exponential, oscillatory, and linear growths classify these phases. Similarly, the entanglement entropy and negativity also exhibit distinct behaviors (linearly, oscillatory, and logarithmic growths) characterizing these phases, respectively. Notibly, for the periodically driven coupled oscillators, the non-equilibrium properties are characterized by two sets of SU(1,1) S U ( 1 , 1 ) generators. The corresponding two sets of the trajectories on two Poincaré discs lead to a more complex phase diagram. We identify two distinct phases within the heating region discernible solely by the growth rate of the entanglement entropy, where a discontinuity is observed when varying the parameters across the phase boundary within in heating region. This discontinuity is not observed in the phonon population.

DOI
10.21468/scipostphyscore.8.1.018

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Phase transitions from heating to non-heating in SU(1,1)SU(1,1)SU(1,1) quantum dynamics: Applications to Bose-Einstein condensates and periodically driven coupled oscillators
We study the entanglement properties in non-equilibrium quantum systems with the SU(1,1) S U ( 1 , 1 ) structure. Through Möbius transformation, we map the dynamics of these systems following a sudden quench or a periodic drive onto three distinct trajectories on the Poincaré disc, corresponding the heating, non-heating, and a phase boundary describing these non-equilibrium quantum states. We consider two experimentally feasible systems where their quantum dynamics exhibit the SU(1,1) S U ( 1 , 1 ) structure: the quench dynamics of the Bose-Einstein condensates and the periodically driven coupled oscillators. In both cases, the heating, non-heating phases, and their boundary manifest through distinct signatures in the phonon population where exponential, oscillatory, and linear growths classify these phases. Similarly, the entanglement entropy and negativity also exhibit distinct behaviors (linearly, oscillatory, and logarithmic growths) characterizing these phases, respectively. Notibly, for the periodically driven coupled oscillators, the non-equilibrium properties are characterized by two sets of SU(1,1) S U ( 1 , 1 ) generators. The corresponding two sets of the trajectories on two Poincaré discs lead to a more complex phase diagram. We identify two distinct phases within the heating region discernible solely by the growth rate of the entanglement entropy, where a discontinuity is observed when varying the parameters across the phase boundary within in heating region. This discontinuity is not observed in the phonon population.
We study the quantum dynamics of Bose-Einstein condensates when the scattering length is modulated periodically or quasi-periodically in time within the Bogoliubov framework. For the periodically driven case, we consider two protocols where the modulation is a square-wave or a sine-wave. In both protocols for each fixed momentum, there are heating and non-heating phases, and a phase boundary between them. The two phases are distinguished by whether the number of excited particles grows exponentially or not. For the quasi-periodically driven case, we again consider two protocols: the square-wave quasi-periodicity, where the excitations are generated for almost all parameters as an analog of the Fibonacci-type quasi-crystal; and the sine-wave quasi-periodicity, where there is a finite measure parameter regime for the non-heating phase. We also plot the analogs of the Hofstadter butterfly for both protocols.
In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we study randomly driven (1+1) ( 1 + 1 ) dimensional conformal field theories (CFTs), a family of quantum many-body systems with soluble non-equilibrium quantum dynamics. The sequence of driving Hamiltonians is drawn from an independent and identically distributed random ensemble. At each driving step, the deformed Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength and therefore induces a Möbius transformation on the complex coordinates. The non-equilibrium dynamics is then determined by the corresponding sequence of Möbius transformations, from which the Lyapunov exponent \lambda_L λ L is defined. We use Furstenberg’s theorem to classify the dynamical phases and show that except for a few exceptional points that do not satisfy Furstenberg’s criteria, the random drivings always lead to a heating phase with the total energy growing exponentially in the number of driving steps n n and the subsystem entanglement entropy growing linearly in n n with a slope proportional to central charge c c and the Lyapunov exponent \lambda_L λ L . On the contrary, the subsystem entanglement entropy at an exceptional point could grow as \sqrt{n} n while the total energy remains to grow exponentially. In addition, we show that the distributions of the operator evolution and the energy density peaks are also useful characterizations to distinguish the heating phase from the exceptional points: the heating phase has both distributions to be continuous, while the exceptional points could support finite convex combinations of Dirac measures depending on their specific type. In the end, we compare the field theory results with the lattice model calculations for both the entanglement and energy evolution and find remarkably good agreement.
We study the equilibrium and dynamical properties of a ferromagnetic spinor atomic Bose-Einstein condensate. In the vicinity of the critical point for a continuous quantum phase transition, universal behaviors are observed both in the equilibrium state and in the dynamics when the quadratic Zeeman shift is swept linearly. Three distinct dynamical regions are identified for different sweeping time scales ($\ensuremath{\tau}$) when compared to the time scale ${\ensuremath{\tau}}_{\mathrm{KZ}}\ensuremath{\sim}{N}^{(1+\ensuremath{\nu}z)/\ensuremath{\nu}d}$ decided by external driving in a system with finite size NNN ($\ensuremath{\nu},z$ are critical exponents and ddd the dimensionality). They are manifested by the excitation probability P\mathcal{P}P and the excess heat density Q\mathcal{Q}Q. The adiabatic region of $\mathcal{P}\ensuremath{\sim}\mathcal{Q}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2}$ follows from the adiabatic perturbation theory when $\ensuremath{\tau}>{\ensuremath{\tau}}_{\mathrm{KZ}}$, while the nonadiabatic universal region of $\mathcal{P}\ensuremath{\sim}\mathcal{Q}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}$ in the thermodynamic limit is described by the Kibble-Zurek mechanism when ${\ensuremath{\tau}}_{\mathrm{KZ}}>\ensuremath{\tau}>$ the time scale given by initial gap. The Kibble-Zurek scaling hypothesis is augmented with finite-size scaling in the latter region and several experimentally falsifiable features for the finite system we consider are predicted. The region of the fastest sweeping is found to be nonuniversal and far from equilibrium, with P\mathcal{P}P and Q\mathcal{Q}Q essentially being constants independent of $\ensuremath{\tau}$.
Quantum states beyond thermodynamic equilibrium represent fascinating and cutting-edge research. However, the behavior of dynamical quantum phase transitions in complex open quantum systems remains poorly understood. Here, using state-of-the-art numerical approaches, we show that by quenching the qubits-oscillator coupling in a dissipative two-qubit Rabi model, the system undergoes dynamical quantum phase transitions. These transitions are characterized by kinks in the Loschmidt echo rate function at parameter values close to a thermodynamic quantum phase transition and are associated with distinct entanglement features. The two classes of critical phenomena depend on qubit interactions and entanglement, revealing different behaviors of the critical exponent of the first kink of the Loschmidt echo for interacting versus non-interacting qubits. This research enhances our understanding of non-equilibrium quantum systems and offers potential applications in quantum sensing and metrology, as it examines how dynamical transitions can enhance the sensitivity of the Loschmidt echo to the quench parameters. Dynamical quantum phase transitions can be observed when time is treated as a control parameter in non-equilibrium quantum systems. The authors show that quenching the qubits-oscillator coupling in a dissipative two-qubit system leads to different transitions depending on interactions and entanglement, with promising applications in quantum sensing and metrology.
In this thesis we explore different phenomena occurring in open quantum many-body systems. This is a research field which is becoming increasingly more important due to the experimental progress towards realization of quantum-simulators and quantum computers using a variety of platforms, from ultracold atoms to superconducting qubit arrays. These systems are inherently driven and open, and it is an ongoing effort to develop theoretical tools and approaches to study their non-equilibrium physics. In the first part of this thesis we explore the different non-equilibrium condensate phases of resonantly interacting bosons in the presence of coherent pump and incoherent losses. This study might be of relevance to existing Rydberg-polariton setups. We derive an effective non-equilibrium field-theory of this model and study its resulting phase diagram. We find a rich phase diagram in- cluding a phase where particles form a condensate of tightly bound molecules. By changing the detuning from the scattering resonance it is possible to drive an Ising phase transition from the molecule condensate phase to a more standard atom condensate phase. In the second part of this thesis we explore the many-body localization (MBL) transition in an open-driven system. Typically, coupling to a bath is expected to destroy localization and transform the sharp MBL transition to a crossover. Here we show how one can use the couplings to non- equilibrium baths in order to detect sharp signatures of the transition, including the divergence of the dynamical exponent in the Griffith regime of the ergodic phase. This is done by solving for the steady-state of the Lindblad quantum-master equation using matrix-product operator tech- niques. Our work here suggests a new scalable numerical approach to study the MBL transition. In the third part of this thesis we study the entanglement properties of measurement trajectories in a free-fermion system subject to dephasing noise. These trajectories describe the evolution of an open system when photons exiting the system are continuously monitored. We are inspired by recent works which studied the entanglement dynamics in hybrid quantum circuits consisting of random-unitary gates and projective measurements. There it was found that a phase transition between a phase with volume-law scaling of entanglement and an area-law phase occurs at a critical value of the measurement rate. In our case we find a new regime, where at weak noise rate the system exhibits logarithmic entanglement scaling, similar to that of (1 + 1)d conformal field- theories. For some measurement protocols, we find a transition to an area-law phase at a critical noise strength.
Bose-Einstein condensates of cold atoms in optical lattices are interesting because the dynamics of the transition from a Mott insulator to a superfluid driven by a quench into the superfluid phase exemplifies non-equilibrium evolutions of gauge-symmetry broken quantum phases. Here we have actually visualised features in real space, which have domain structures that depend on the stage of the transition dynamics. Namely, the system after the quench relaxes in three processes: (i) an exponential growth of the superfluid density, (ii) collapse of the domain structure, and (iii) vortex-antivortex pair-annihilation as the long-range superfluid order is formed. Topological picture due to Kibble-Zurek mechanism and Zurek's freeze-out scenario for slow quenches are confirmed.