Time-evolved State Research Articles

Orthogonality Catastrophe in Dissipative Quantum Many-Body Systems.

We present an analog of the phenomenon of orthogonality catastrophe in quantum many-body systems subject to a local dissipative impurity. We show that the fidelity F(t), giving a measure for distance of the time-evolved state from the initial one, displays a universal scaling form F(t)∝t^{θ}e^{-γt}, when the system supports long-range correlations, in a fashion reminiscent of traditional instances of orthogonality catastrophe in condensed matter. An exponential falloff at rate γ signals the onset of environmental decoherence, which is critically slowed down by the additional algebraic contribution to the fidelity. This picture is derived within a second-order cumulant expansion suited for Liouvillian dynamics, and substantiated for the one-dimensional transverse field quantum Ising model subject to a local dephasing jump operator, as well as for XY and XX quantum spin chains, and for the two-dimensional Bose gas deep in the superfluid phase with local particle heating. Our results hint that local sources of dissipation can be used to inspect real-time correlations and to induce a delay of decoherence in open quantum many-body systems.

Differentiating resting brain states using ordinal symbolic analysis.

Symbolic methods of analysis are valuable tools for investigating complex time-dependent signals. In particular, the ordinal method defines sequences of symbols according to the ordering in which values appear in a time series. This method has been shown to yield useful information, even when applied to signals with large noise contamination. Here, we use ordinal analysis to investigate the transition between eyes closed (EC) and eyes open (EO) resting states. We analyze two electroencephalography datasets (with 71 and 109 healthy subjects) with different recording conditions (sampling rates and the number of electrodes in the scalp). Using as diagnostic tools the permutation entropy, the entropy computed from symbolic transition probabilities, and an asymmetry coefficient (that measures the asymmetry of the likelihood of the transitions between symbols), we show that the ordinal analysis applied to the raw data distinguishes the two brain states. In both datasets, we find that, during the EC-EO transition, the EO state is characterized by higher entropies and lower asymmetry coefficient, as compared to the EC state. Our results thus show that these diagnostic tools have the potential for detecting and characterizing changes in time-evolving brain states.

Quantum transitions and quantum entanglement from Dirac-like dynamics simulated by trapped ions

Quantum transition probabilities and quantum entanglement for two-qubit states of a four-level trapped ion quantum system are computed for time-evolving ionic states driven by Jaynes-Cummings Hamiltonians with interactions mapped onto a $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ group structure. Using the correspondence of the method of simulating a $3+1$ dimensional Dirac-like Hamiltonian for bispinor particles into a single trapped ion, one preliminarily obtains the analytical tools for describing ionic state transition probabilities as a typical quantum oscillation feature. For Dirac-like structures driven by generalized Poincar\'e classes of coupling potentials, one also identifies the $\text{SU}(2)\ensuremath{\bigotimes}\text{SU}(2)$ internal degrees of freedom corresponding to intrinsic parity and spin polarization as an adaptive platform for computing the quantum entanglement between the internal quantum subsystems which define two-qubit ionic states. The obtained quantum correlational content is then translated into the quantum entanglement of two-qubit ionic states with quantum numbers related to the total angular momentum and to its projection onto the direction of the trapping magnetic field. Experimentally, the controllable parameters simulated by ion traps can be mapped into a Dirac-like system in the presence of an electrostatic field which, in this case, is associated to ionic carrier interactions. Besides exhibiting a complete analytical profile for ionic quantum transitions and quantum entanglement, our results indicate that carrier interactions actively drive an overall suppression of the quantum entanglement.

Exact quantum decay of an interacting many-particle system: the Calogero–Sutherland model

The exact quantum decay of a one-dimensional Bose gas with inverse-square interactions is presented. The system is equivalent to a gas of particles obeying generalized exclusion statistics. We consider the expansion dynamics of a cloud initially confined in a harmonic trap that is suddenly switched off. The decay is characterized by analyzing the fidelity between the initial and the time-evolving states, also known as the survival probability. It exhibits early on a quadratic dependence on time that turns into a power-law decay, during the course of the evolution. It is shown that the particle number and the strength of interactions determine the power-law exponent in the latter regime, as recently conjectured. The nonexponential character of the decay is linked to the many-particle reconstruction of the initial state from the decaying products.

Quench dynamics and parity blocking in Majorana wires

We theoretically explore quench dynamics in a finite-sized topological fermionic p-wave superconducting wire with the goal of demonstrating that topological order can have marked effects on such non-equilibrium dynamics. In the case studied here, topological order is reflected in the presence of two (nearly) isolated Majorana fermionic end bound modes together forming an electronic state that can be occupied or not, leading to two (nearly) degenerate ground states characterized by fermion parity. Our study begins with a characterization of the static properties of the finite-sized wire, including the behavior of the Majorana end modes and the form of the tunnel coupling between them; a transfer matrix approach to analytically determine the locations of the zero energy contours where this coupling vanishes; and a Pfaffian approach to map the ground state parity in the associated phase diagram. We next study the quench dynamics resulting from initializing the system in a topological ground state and then dynamically tuning one of the parameters of the Hamiltonian. For this, we develop a dynamic quantum many-body technique that invokes a Wick’s theorem for Majorana fermions, vastly reducing the numerical effort given the exponentially large Hilbert space. We investigate the salient and detailed features of two dynamic quantities—the overlap between the time-evolved state and the instantaneous ground state (adiabatic fidelity) and the residual energy. When the parity of the instantaneous ground state flips successively with time, we find that the time-evolved state can dramatically switch back and forth between this state and an excited state even when the quenching is very slow, a phenomenon that we term ‘parity blocking’. This parity blocking becomes prominently manifest as non-analytic jumps as a function of time in both dynamic quantities.

Stationary and uniformly accelerated states in nonlinear quantum mechanics

We consider two kinds of solutions of a recently proposed field theory leading to a nonlinear Schr\odinger equation exhibiting solitonlike solutions of the power-law form ${e}_{q}^{i(kx\ensuremath{-}wt)}$, involving the $q$ exponential function naturally arising within nonextensive thermostatistics [${e}_{q}^{z}\ensuremath{\equiv}{[1+(1\ensuremath{-}q)z]}^{1/(1\ensuremath{-}q)}$, with ${e}_{1}^{z}={e}^{z}$]. These fundamental solutions behave like free particles, satisfying $p=\ensuremath{\hbar}k$, $E=\ensuremath{\hbar}\ensuremath{\omega}$, and $E={p}^{2}/2m$ $(1\ensuremath{\le}ql2)$. Here we introduce two additional types of exact, analytical solutions of the aforementioned field theory. As a first step we extend the theory to situations involving a potential energy term, thus going beyond the previous treatment concerning solely the free-particle dynamics. Then we consider both bound, stationary states associated with a confining potential and also time-evolving states corresponding to a linear potential function. These types of solutions might be relevant for physical applications of the present nonlinear generalized Schr\odinger equation. In particular, the stationary solution obtained shows an increase in the probability for finding the particle localized around a certain position of the well as one increases $q$ in the interval $1\ensuremath{\le}ql2$, which should be appropriate for physical systems where one finds a low-energy particle localized inside a confining potential.

Unification of the State with the Dynamical Law

We address the question of why particular laws were selected for the universe, by proposing a mechanism for laws to evolve. Normally in physical theories, timeless laws act on time-evolving states. We propose that this is an approximation, good on time scales shorter than cosmological scales, beyond which laws and states are merged into a single entity that evolves in time. Furthermore the approximate distinction between laws and states, when it does emerge, is dependent on the initial conditions. These ideas are illustrated in a simple matrix model.

Topological blocking in quantum quench dynamics

We study the non-equilibrium dynamics of quenching through a quantum critical point in topological systems, focusing on one of their defining features: ground state degeneracies and associated topological sectors. We present the notion of 'topological blocking', experienced by the dynamics due to a mismatch in degeneracies between two phases and we argue that the dynamic evolution of the quench depends strongly on the topological sector being probed. We demonstrate this interplay between quench and topology in models stemming from two extensively studied systems, the transverse Ising chain and the Kitaev honeycomb model. Through non-local maps of each of these systems, we effectively study spinless fermionic $p$-wave paired superconductors. Confining the systems to ring and toroidal geometries, respectively, enables us to cleanly address degeneracies, subtle issues of fermion occupation and parity, and mismatches between topological sectors. We show that various features of the quench, which are related to Kibble-Zurek physics, are sensitive to the topological sector being probed, in particular, the overlap between the time-evolved initial ground state and an appropriate low-energy state of the final Hamiltonian. While most of our study is confined to translationally invariant systems, where momentum is a convenient quantum number, we briefly consider the effect of disorder and illustrate how this can influence the quench in a qualitatively different way depending on the topological sector considered.

Monotonically convergent optimization in quantum control using Krotov's method

The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.

Strong Electronic Correlation Effects in Coherent Multidimensional Nonlinear Optical Spectroscopy

We discuss a many-body theory of the coherent ultrafast nonlinear optical response of systems with a strongly correlated electronic ground state that responds unadiabatically to photoexcitation. We introduce a truncation of quantum kinetic density matrix equations of motion that does not rely on an expansion in terms of the interactions and thus applies to strongly correlated systems. For this we expand in terms of the optical field, separate out contributions to the time-evolved many-body state due to correlated and uncorrelated multiple optical transitions, and use "Hubbard operator" density matrices to describe the exact dynamics of the individual contributions within a subspace of strongly coupled states, including "pure dephasing". Our purpose is to develop a quantum mechanical tool capable of exploring how, by coherently photoexciting selected modes, one can trigger nonlinear dynamics of strongly coupled degrees of freedom. Such dynamics could lead to photoinduced phase transitions. We apply our theory to the nonlinear response of a two-dimensional electron gas (2DEG) in a magnetic field. We coherently photoexcite the two lowest Landau level (LL) excitations using three time-delayed optical pulses. We identify some striking temporal and spectral features due to dynamical coupling of the two LLs facilitated by inter-Landau-level magnetoplasmon and magnetoroton excitations and compare to three-pulse four-wave-mixing (FWM) experiments. We show that these features depend sensitively on the dynamics of four-particle correlations between an electron-hole pair and a magnetoplasmon/magnetoroton, reminiscent of exciton-exciton correlations in undoped semiconductors. Our results shed light into unexplored coherent dynamics and relaxation of the quantum Hall system (QHS) and can provide new insight into non-equilibrium co-operative phenomena in strongly correlated systems.

Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap.

Rigorous bounds on the performance of a hybrid dynamical-decoupling quantum-computing scheme

We study dynamical decoupling in a multiqubit setting, where it is combined with quantum logic gates. This is illustrated in terms of computation using Heisenberg interactions only, where global decoupling pulses commute with the computation. We derive a rigorous error bound on the trace distance or fidelity between the desired computational state and the actual time-evolved state, for a system subject to coupling to a bounded-strength bath. The bound is expressed in terms of the operator norm of the effective Hamiltonian generating the evolution in the presence of decoupling and logic operations. We apply the bound to the case of periodic pulse sequences and find that in order to maintain a constant trace distance or fidelity, the number of cycles---at fixed pulse interval and width---should scale in inverse proportion to the square of the number of qubits. This sets a scalability limit on the protection of quantum computation using periodic dynamical decoupling.

Transport of entanglement through a Heisenberg–XY spin chain

The entanglement dynamics of spin chains is investigated using Heisenberg–XY spin Hamiltonian dynamics. The various measures of two-qubit entanglement are calculated analytically in the time-evolved state starting from initial states with no entanglement and exactly one pair of maximally entangled qubits. The localizable entanglement between a pair of qubits at the end of chain captures the essential features of entanglement transport across the chain, and it displays the difference between an initial state with no entanglement and an initial state with one pair of maximally-entangled qubits.

Entanglement dynamics in a two-mode nonlinear bosonic Hamiltonian

An exactly solvable case of an interacting Hamiltonian of two bosonic modes is considered to study fundamental properties of the entanglement dynamics for coupled nonlinear oscillators. Such an interaction is of physical importance, either in a two-species Bose–Einstein condensate or in the case of two modes of electromagnetic fields interacting in Kerr media. The time-evolved state is obtained analytically for initial products of two Fock and two coherent states, and the purification times of the subsystems are determined. The possibility of dynamical generation of a quantum superposition state is discussed at such purification times. We also identify the existence of two regimes: the short time, phase spread regime where subsystem entropy rises monotonically and the self-interference regime where it oscillates and a purification phenomenon can be observed. Our results also show that the break time from the first regime to the second one becomes longer, as well as the purification and reversibility times, as the Planck constant becomes much smaller than a typical action in phase space.

Entangling power of quantized chaotic systems.

We study the quantum entanglement caused by unitary operators that have classical limits that can range from the near integrable to the completely chaotic. Entanglement in the eigenstates and time-evolving arbitrary states is studied through the von Neumann entropy of the reduced density matrices. We demonstrate that classical chaos can lead to substantially enhanced entanglement. Conversely, entanglement provides a useful characterization of quantum states in higher-dimensional chaotic or complex systems. Information about eigenfunction localization is stored in a graded manner in the Schmidt vectors, and the principal Schmidt vectors can be scarred by the projections of classical periodic orbits onto subspaces. The eigenvalues of the reduced density matrices is sensitive to the degree of wave-function localization, and is roughly exponentially arranged. We also point out the analogy with decoherence, as reduced density matrices corresponding to subsystems of fully chaotic systems, are diagonally dominant.

Discrimination theory of physical processes

Discrimination problems of stochastic processes and time-evolving physical states are treated on the basis of an information theoretical method. From a set of received data we can select a suitably assumed probability distribution. The basic method is applied to discriminate colored stochastic processes of the Brownian motion and laser system with nonlinearity and fluctuations. Dynamical state change is clearly specified: This is quite difficult by a direct observation of the data. The methodology is further extended to include quantum mechanical systems.

Unitarity and electron-positron pairs created by strong external fields.

Lowest-order perturbative calculations of the electron-positron production probability in relativistic heavy-ion collisions exceed unitarity bounds for the heaviest collision systems at extreme relativistic energies and sufficiently small impact parameters. Starting with the exponential representation of the time-evolution operator in the Furry picture, we derive manifestly unitary and gauge-invariant expressions for transition amplitudes and probabilities associated with the created electron-positron pairs by employing the Magnus expansion to first order. The time-evolved ground state of the electron-positron field around the heavy nuclei is expressed as a superposition of the unperturbed vacuum state and virtual excitation modes consisting of electron-positron pairs.