The localization transition in the Hermitian Aubry-André model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-André case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.
The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i) the initial Husimi distribution evaluated along phase-space trajectories and (ii) the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.
We study the effects of non-Hermitian perturbations on a quantum kicked model exhibiting a localization transition. Using an exact renormalization scheme, we show that the critical line separating the extended and localized phases approaches its semiclassical limit as the imaginary part of the kicking parameter is steadily increased. Further, the metastability of the quantum states appears to be directly correlated with the deviation between the semiclassical and quantum results. This direct evidence of quantum-classical correspondence suggests that decoherence may be usefully modeled by non-Hermitian perturbations.
The exploration of non-Hermitian quantum systems, particularly those governed by parity-time ([Formula: see text])-symmetry, has revealed a rich landscape of unconventional phase transitions and localization phenomena. Spin chains with quasiperiodic potentials and competing Heisenberg interactions provide a versatile framework for probing these effects. In this study, we investigate the dynamical and spectral signatures of [Formula: see text]-symmetry breaking and localization-delocalization transitions in spin chains subjected to a non-Hermitian Aubry-André potential and tunable Heisenberg couplings across nearest ([Formula: see text]) and next-nearest neighbors ([Formula: see text]). Our analysis reveals that increasing [Formula: see text] monotonically shifts the critical point [Formula: see text], while [Formula: see text] induces a non-linear response, generating a distinct minimum in [Formula: see text]. Notably, [Formula: see text] leads to the emergence of partially imaginary eigenvalues beyond [Formula: see text], decoupling the onset of [Formula: see text]-symmetry breaking from the localization transition. Using inverse and normalized participation ratios, we identify three distinct phases: extended, localized, and an intermediate hybrid regime characterized by a mobility edge. To capture the dynamical fingerprints of these phases, we employ time-dependent density distribution, long-time survival probability [Formula: see text], analysis. Extended phases exhibit ballistic spreading and algebraic decay, while localized regimes show spatial confinement and exponential decay. These observables serve as robust indicators of non-Hermitian phase transitions, offering insights beyond static spectral measures. Overall, our findings underscore the critical role of dynamical metrics in characterizing phase structure in non-Hermitian spin systems. As interest in non-equilibrium quantum dynamics continues to grow, such models provide a compelling platform for understanding the interplay between [Formula: see text]-symmetry, localization, and quantum coherence in complex many-body systems.
The critical exponents and universality classes of localization transitions in quasiperiodic systems are of fundamental importance for understanding critical phenomena in aperiodic systems. Here we show that the dynamical critical behavior can be tuned without adding new terms or changing the form of the Hamiltonian, but solely by varying the incommensurate frequency of the quasiperiodic onsite potential. We construct a family of incommensurate frequencies from the limiting ratios of generalized Fibonacci sequences controlled by the parameters
(m,n), and use them to define the quasiperiodic onsite potential. By combining generalized fidelity susceptibility, localization-length scaling, and finite-size gap analysis, we find that the correlation-length exponent is insensitive to the choice of the incommensurate frequency and remains consistent with the correlation-length critical exponent,
ν≃1, in the localization transition of the standard Aubry--Andr'e--Harper model. In contrast, the dynamical exponent extracted from the low-energy gap scaling varies systematically with the incommensurate frequency. Our results show that changing the incommensurate frequency provides a simple way to tune dynamical critical scaling in deterministic aperiodic systems. Our results suggest instead that the arithmetic structure of an irrational number can serve as a control parameter for nonequilibrium quantum dynamics, enabling the tuning of dynamical critical behavior without changing the microscopic Hamiltonian or the physical spatial dimension.
In this thesis, we study phase-space approximations to quantum dynamics for both Hermitian and non-Hermitian quantum systems, motivated by the need to efficiently describe quantum effects in models that are analytically intractable but admit classical analogues offering insight into their complex behaviour. Such systems arise in diverse areas, include but are not limited to quantum optics, condensed matter, open quantum systems, and quantum technologies. We begin by exploring phase-space approximations for quantum systems governed by the Heisenberg-Weyl algebra. Building on established coherent-state methods, we develop a semiclassical Husimi approximation for non-Hermitian Hamiltonians. This approach reveals that the resulting dynamics comprise the transport of initial Husimi distributions along classical trajectories, overlaid with a time-dependent norm landscape reflecting non-unitary evolution. We demonstrate the effectiveness of this method through several examples, including damped Kerr oscillators and
PT-symmetric models, providing a superior approximation for observables compared to standard coherent-state techniques. To generalise these ideas, we develop a unified framework for phase-space approximations that applies across a wide range of algebraic structures. By identifying a set of general criteria, we enable systematic application of the approximation scheme to any quantum system satisfying these conditions. This provides a cohesive theoretical foundation that connects previously distinct results under a common approach. We then apply this framework to both the
su2 algebra and non-Lie algebraic bosonic conversion systems. For
su2, we analyse two-mode Bose-Hubbard models with various Hamiltonians, demonstrating quantum-classical correspondence and exploring the effects of
PT-symmetry. For bosonic conversion systems, we propose coherent states for these non-Lie algebraic structures and derive their associated phase-space geometry, characterised as quantum Kummer shapes with classical counterparts as orbifolds. Our results advance the understanding of quantum-classical correspondence in multi-boson systems beyond traditional Hermitian frameworks and offer practical tools for simulating quantum dynamics in experimentally relevant models.
In this work, we investigate the Stark localization near the Aubry-Andr\'e (AA) critical point. We perform careful studies for reporting system-dependent parameters, such as localization length, inverse participation ratio (IPR), and energy gap between the ground and first excited state, for characterizing the localization-delocalization transition. We show that the scaling exponents possessed by these key descriptors of localization are quite different from that of a pure AA model or Stark model. Near the critical point of the AA model, in the presence of Stark field of strength
h, the localization length $\ensuremath{\zeta}$ scales as $\ensuremath{\zeta}\ensuremath{\propto}{h}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.29$, which is different from both the pure AA model $(\ensuremath{\nu}=1)$ and Stark model $(\ensuremath{\nu}\ensuremath{\approx}0.33)$. The IPR in this case scales as IPR $\ensuremath{\propto}{h}^{s}$ with $s\ensuremath{\approx}0.096$, which is again significantly different from both the pure AA model $(s\ensuremath{\approx}0.33)$ and Stark model $(s\ensuremath{\approx}0.33)$. The energy gap, $\mathrm{\ensuremath{\Delta}}$, scales as $E\ensuremath{\propto}{h}^{\ensuremath{\nu}z}$, where $z\ensuremath{\approx}2.37$, which is however the same as the pure AA model. Finally, we discuss how invoking a criticality-inducing additional control parameter may help in designing better many-body quantum sensors. Quantum critical sensors exploit the venerability of the wave function near the quantum critical point against small parameter shifts. By incorporating a control parameter in the form of the quasiperiodic field, i.e., the AA potential, we show a significant advantage can be drawn in estimating an unknown parameter, which is considered here to be the Stark weak field strength, with high precision. We compute and extract the scaling exponent associated with the quantum Fisher information,
FQ, which provides a bound on the uncertainty associated with measuring an unknown parameter. We demonstrate that by a control field in the form of the quasiperiodic potential, and setting it at AA criticality,
FQ scales with the system size as ${F}_{Q}\ensuremath{\propto}{L}^{\ensuremath{\beta}}$, where $\ensuremath{\beta}\ensuremath{\approx}6.7$, which is greater in comparison to the pure Stark case, for which $\ensuremath{\beta}$ is known to be 5.8. We, thus, pitch a generic idea of invoking quantum-criticality-inducing additional control parameters for obtaining enhanced quantum advantage in many-body critical sensors.