The quantum Fisher information (QFI) is widely used to characterize quantum phases and transitions, but its diagnostic power sometimes relies on selecting special generators based on prior knowledge of the underlying physics. We ask whether this requirement can be relaxed in nonequilibrium systems by studying the QFI of the long-time asymptotic state of a two-dimensional p+ip superfluid following an instantaneous quench of the coupling strength, using the particle number within a large subextensive subsystem as the generator. In equilibrium, the ground-state QFI is continuous across the topological transition between the weak-pairing BCS and the strong-pairing BEC phases, showing no direct signature of the transition. After the quench, however, the QFI associated with the same generator distinguishes the three dynamical phases and can encode the topology of the pre-quench state. In phase I, with a vanishing order parameter, the QFI Fourier spectrum consists of a single zero-frequency spike determined by the nonequilibrium distribution function. In phase II, with a constant nonzero order parameter, the QFI spectrum contains a zero-frequency spike and two continua separated by a gap set by the minimum asymptotic quasiparticle energy. The continuum edge behavior reveals whether this minimum occurs at zero or finite momentum. In the former case, the spectral weight vanishes at the edge, with the sign just inside the continuum encoding the pre-quench topology for large subsystem. In phase III, with a time-periodic order parameter, the QFI spectrum exhibits discrete peaks at integer multiples of the oscillation frequency, together with continua associated with the Floquet quasienergy spectrum. Our results show that driving the system out of equilibrium can enhance the diagnostic power of the QFI for a standard physical observable, revealing information inaccessible in equilibrium.
Quantum Fisher information in a quenched p+ip superfluid
The quantum Fisher information (QFI) is widely used to characterize quantum phases and transitions, but its diagnostic power sometimes relies on selecting special generators based on prior knowledge of the underlying physics. We ask whether this requirement can be relaxed in nonequilibrium systems by studying the QFI of the long-time asymptotic state of a two-dimensional p+ip superfluid following an instantaneous quench of the coupling strength, using the particle number within a large subextensive subsystem as the generator. In equilibrium, the ground-state QFI is continuous across the topological transition between the weak-pairing BCS and the strong-pairing BEC phases, showing no direct signature of the transition. After the quench, however, the QFI associated with the same generator distinguishes the three dynamical phases and can encode the topology of the pre-quench state. In phase I, with a vanishing order parameter, the QFI Fourier spectrum consists of a single zero-frequency spike determined by the nonequilibrium distribution function. In phase II, with a constant nonzero order parameter, the QFI spectrum contains a zero-frequency spike and two continua separated by a gap set by the minimum asymptotic quasiparticle energy. The continuum edge behavior reveals whether this minimum occurs at zero or finite momentum. In the former case, the spectral weight vanishes at the edge, with the sign just inside the continuum encoding the pre-quench topology for large subsystem. In phase III, with a time-periodic order parameter, the QFI spectrum exhibits discrete peaks at integer multiples of the oscillation frequency, together with continua associated with the Floquet quasienergy spectrum. Our results show that driving the system out of equilibrium can enhance the diagnostic power of the QFI for a standard physical observable, revealing information inaccessible in equilibrium.
We study the nonadiabatic dynamics of a two-dimensional p+ip superfluid following an instantaneous quantum quench of the BCS coupling constant. The model describes a topological superconductor with a nontrivial BCS (trivial BEC) phase appearing at weak- (strong-) coupling strengths. We extract the exact long-time asymptotics of the order parameter $\ensuremath{\Delta}(t)$ by exploiting the integrability of the classical p-wave Hamiltonian, which we establish via a Lax construction. Three different types of asymptotic behavior can occur depending upon the strength and direction of the interaction quench. We refer to these as the nonequilibrium phases {I, II, III}, characterized as follows. In phase I, the order parameter asymptotes to zero due to dephasing. In phase II, $\ensuremath{\Delta}\ensuremath{\rightarrow}{\ensuremath{\Delta}}_{\ensuremath{\infty}}$, a nonzero constant. Phase III is characterized by persistent oscillations of $\ensuremath{\Delta}(t)$. For quenches within phases I and II, we determine the topological character of the asymptotic states. We show that two different formulations of the bulk topological winding number, although equivalent in the BCS or BEC ground states, must be regarded as independent out of equilibrium. The first winding number Q characterizes the Anderson pseudospin texture of the initial state; we show that Q is generically conserved. For $Q\ensuremath{\ne}0$, this leads to the prediction of a ``gapless topological'' state when $\ensuremath{\Delta}$ asymptotes to zero. The presence or absence of Majorana edge modes in a sample with a boundary is encoded in the second winding number W, which is formulated in terms of the retarded Green's function. We establish that W can change following a quench across the quantum critical point. When the order parameter asymptotes to a nonzero constant, the final value of W is well defined and quantized. We discuss the implications for the (dis)appearance of Majorana edge modes. Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooper-pair distribution function constitutes a Z2-valued quantum number, which is nonzero whenever $W\ensuremath{\ne}Q$. The pair distribution can in principle be measured using rf spectroscopy in an ultracold-atom realization, allowing direct experimental detection of the Z2 number. This has the following interesting implication: topological information that is experimentally inaccessible in the bulk ground state can be transferred to an observable distribution function when the system is driven far from equilibrium.
The quantum Fisher information has gained popularity due to its versatility and its multi-purposed nature. For example, in quantum metrology the quantum Fisher information is used to quantify the maximal accuracy for which a parameter may be estimated and in quantum information theory it is used as a witness of multipartite entanglement. In condensed matter physics the quantum Fisher information has been used to study phase transitions, primarily in equilibrium quantum many body systems. In this work we study the quantum Fisher information for a p+ip superfluid both in equilibrium and out of equilibrium. In equilibrium the p+ip superfluid exhibits a topological quantum phase transition between a topological weak pairing phase (BCS) and a topologically trivial strong pairing phase (BEC). After an instantaneous interaction quench, the long time asymptotic state of p+ip superfluid is characterized by three different dynamical phases and within each phase the behaviour of the asymptotic order parameter changes. In equilibrium, we find that the QFI calculated with respect to particle number using the BCS ground-state fails to probe the topological quantum phase transition. On the other hand, in our non-equilibrium treatment we find that by quenching the system out of equilibrium the quantum Fisher information can be used to extract topological information of the pre-quench state and the character of the energy dispersion of the post-quench state. Our results therefore suggest that by introducing a quantum quench the quantum Fisher information is able to extract useful information and does not rely on the choice of operator, as opposed to the equilibrium case.
We consider the dynamics of maximal quantum Fisher information (MQFI) after sudden quenches for the one-dimensional transverse-field Ising model. Our results show, the same as Loschmidt echo, that there is a universality for the revival times, i.e., they do not depend on the initial state and the size of the quench and are given by integer multiples of ${T}_{\text{rev}}\ensuremath{\simeq}\frac{N}{2{v}_{\text{max}}}$, where N is the system size and vmax is the maximal group velocity of quasiparticles. Critically enhanced and decreased at revival and decay times as ${T}_{\text{rev}}\ensuremath{\equiv}{T}_{\text{dec}}$ are characterized by quenching from the order and disorder phases into the quantum phase transition, respectively, that can be utilized to detect the quantum critical point (QCP). In some quenches crossed from the QCP, nonanalytic behaviors appear at some times due to the turning of the local observable from one direction to another because of identifying the maximum value. We name this phenomenon the dynamical MQFI transitions, occurring at the critical times tc. Interestingly, although no Fisher zero exists in the dynamics of MQFI, the first critical time emerged from the dynamical quantum phase transition is equal to the first time when the logarithm of MQFI is minimum. In addition, we unveil the long-time run of MQFI indicates a signature of a nonequilibrium quantum phase transition at the QCP. We also discuss the probability of the arising of macroscopic superpositions in the nonequilibrium dynamics of the system.
We investigate the dynamical quantum Fisher information of a two-level system coupled to a bosonic environment, focusing on the estimation of the qubit gap. We combine analytical calculations with numerically controlled matrix-product-state simulations. In the exactly solvable pure-dephasing Ohmic regime at zero temperature, the long-time quantum Fisher information displays a coupling-dependent algebraic behavior, leading to a dynamical crossover: it grows without bound at weak coupling, approaches a finite asymptotic value at the crossover coupling, and vanishes at strong coupling. At finite temperature, thermal fluctuations suppress the long-time growth and generate a finite-time maximum, whose dependence on the dephasing coupling retains a clear signa- ture of the zero-temperature crossover. We show that this crossover is absent at zero temperature in non-Ohmic baths: the long-time quantum Fisher information vanishes in the sub-Ohmic and diverges in the super-Ohmic regimes. At non-zero temperature instead, the crossover appears for a super-Ohmic quadratic bath and the crossover coupling becomes temperature-dependent. Moreover, in the zero-temperature Ohmic regime, we introduce an additional amplitude-damping system-bath coupling that induces energy relaxation. This relaxation channel replaces the unbounded long-time growth with a finite asymptotic quantum Fisher information associated with the reduced interacting ground state, while a signature of the pure-dephasing crossover persists in the early-time dynamics. These results establish a direct connection between the low-frequency structure of the bath and the asymptotic metrological behavior of dynamical gap sensing, and show how thermal fluctuations and energy relaxation regularize the ideal pure-dephasing dynamical crossover of the quantum Fisher information.
Quantum Fisher Information (QFI) is a ubiquitous quantity with applications ranging from quantum metrology and resource theories to condensed matter physics. In equilibrium local quantum many-body systems, the QFI of a subsystem with respect to an extensive observable is typically proportional to the subsystem's volume. Specifically, in large subsystems at equilibrium, the QFI per unit volume squared becomes negligible. We reveal a natural mechanism that amplifies the QFI in a quantum spin chain with a zero-temperature ordered phase. At zero or sufficiently low temperatures, a transient localized perturbation enhances the QFI, causing it to scale quadratically with the subsystem's length. Furthermore, this enhancement can be controlled through more general localized kicking protocols. We also revisit the behavior of the quantum Fisher information after a global quench in the thermodynamic limit, focusing on the generation of localized -- confined within compact subsystems -- multipartite entanglement. We show that the density of localized multipartite entanglement approaches zero at late times, but there is an optimal time frame proportional to the length in which subsystems fall into macroscopic quantum states. We test our predictions against numerical data obtained using a novel technique, based on a remarkable identity between quantum Fisher information and Wigner-Yanase-Dyson skew information, that allows one to compute the quantum Fisher information with respect to the order parameter in noninteracting spin chains.
The topological order of the Kitaev toric code is insensitive to disorder in its couplings since all star and plaquette operators commute; consequently, the stabilizer ground-state manifold is independent of the individual coupling strengths, and the phase remains stable against weak local perturbations. This insensitivity, however, is a property of the eigenstates and not of the dynamics they generate, and we show that the disorder sensitivity of a probe prepared outside that eigenbasis can be a metrological resource. In particular, we investigate the estimation of the strength of three topological-order-breaking perturbations, the nonlinear Castelnovo-Chamon deformation, nearest-neighbor Ising interactions, and a magnetic field along horizontal edges, encoded on an initially separable probe through unitary evolution under the disordered toric code Hamiltonian. We demonstrate that the quenched average quantum Fisher information (QFI), a figure of merit for metrological precision, exceeds that of its ordered counterpart in suitable regimes of perturbation strength and encoding time. For the ordered case, we further identify optimal product probes, for which the quadratic and cubic terms of the short-time expansion vanish identically, leaving a transient quartic scaling of QFI. Interestingly, this window persists as long as the multipartite entanglement generated by the encoding continues to increase, and it survives under local dephasing, bit-flip, and amplitude damping noise acting independently on each site, with the non-unital channel being the least detrimental.