Time-independent Hamiltonian Research Articles

Debugging quantum processes using monitoring measurements

Since observation on a quantum system may cause the system state collapse, it is usually hard to find a way to monitor a quantum process, which is a quantum system that continuously evolves. We propose a protocol that can debug a quantum process by monitoring, but not disturb the evolution of the system. This protocol consists of an error detector and a debugging strategy. The detector is a projection operator that is orthogonal to the anticipated system state at a sequence of time points, and the strategy is used to specify these time points. As an example, we show how to debug the computational process of quantum search using this protocol. By applying the Skolem--Mahler--Lech theorem in algebraic number theory, we find an algorithm to construct all of the debugging protocols for quantum processes of time independent Hamiltonians.

Topological gaps without masses in driven graphene-like systems

We illustrate the possibility of realizing band gaps in graphene-like systems that fall outside the existing classification of gapped Dirac Hamiltonians in terms of masses. As our primary example we consider a band gap arising due to time-dependent distortions of the honeycomb lattice. By means of an exact, invertible, and transport-preserving mapping to a time-independent Hamiltonian, we show that the system exhibits Chern-insulating phases with quantized Hall conductivities $\ifmmode\pm\else\textpm\fi{}{e}^{2}/h$. The chirality of the corresponding gapless edge modes is controllable by both the frequency of the driving and the manner in which sublattice symmetry is broken by the dynamical lattice modulations. Finally, we discuss a promising possible realization of this physics in photonic lattices.

Numerical Langevin Simulations: Equilibrium Dynamics and Nonequilibrium Thermodynamics

Stochastic differential equations of motion (e.g., Langevin dynamics) provide a popular framework for simulating molecular systems. Any computational algorithm must discretize these equations, yet the resulting finite time step integration schemes suffer from several practical shortcomings: for example, equilibrium dynamical properties diverge from those of the continuous equations of motion; path-sampling applications are impeded by a path action that is either unknown or cumbersome; and thermodynamic statistics of driven systems are skewed, leading to biased inference using recently-developed nonequilibrium fluctuation theorems. We show how any finite time step Langevin integrator, even with a time-independent Hamiltonian, can be thought of as a driven, nonequilibrium physical process. Once an appropriate work-like quantity (the shadow work) is defined, nonequilibrium fluctuation theorems can characterize or correct for the errors introduced by the use of finite time steps. In particular, we demonstrate that amending estimators based on nonequilibrium work theorems to include this shadow work removes the time step dependent error from estimates of free energies. We also quantify, for the first time, the magnitude of deviations between the sampled stationary distribution and the desired equilibrium distribution for equilibrium Langevin simulations of solvated systems of varying size. What is more, we show that the incorporation of a novel time step rescaling in the deterministic updates of position and velocity can correct a number of dynamical defects in these integrators. Finally, we identify a particular splitting that has essentially universally appropriate properties for the simulation of Langevin dynamics for molecular systems in equilibrium, nonequilibrium, and path sampling contexts.

Strongly anisotropic Dirac quasiparticles in irradiated graphene

We study quasiparticle dynamics in graphene exposed to a linearly polarized electromagnetic wave of very large intensity. We demonstrate that low-energy transport in such system can be described by an effective time-independent Hamiltonian, characterized by multiple Dirac points in the first Brillouin zone. Around each Dirac point the spectrum is anisotropic: the velocity along the polarization of the radiation significantly exceeds the velocity in the perpendicular direction. Moreover, in some of the points the transverse velocity oscillates as a function of the radiation intensity. We find that the conductance of a graphene p-n junction in the regime of strong irradiation depends on the polarization as $G(\ensuremath{\theta})\ensuremath{\propto}{|sin\ensuremath{\theta}|}^{3/2}$, where $\ensuremath{\theta}$ is the angle between the polarization and the p-n interface, and oscillates as a function of the radiation intensity.

Quantum speed limit and optimal evolution time in a two-level system

Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent Hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation between optimal times found in quantum control and the QSL bound, in the (relevant) time-dependent regime, by discussing the ubiquitous two-level Landau-Zener–type Hamiltonian.

Accurate Effective Hamiltonians via Unitary Flow in Floquet Space

We present a systematic construction of effective Hamiltonians of periodically driven quantum systems. Because of an equivalence between the time dependence of a Hamiltonian and an interaction in its Floquet operator, flow equations, that permit us to decouple interacting quantum systems, allow us to identify time-independent Hamiltonians for driven systems. With this approach, we explain the experimentally observed deviation of expected suppression of tunneling in ultracold atoms.

Bulk-boundary correspondence for chiral symmetric quantum walks

Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary timestep operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk ZxZ topological invariant that controls the number of topologically protected 0 and pi energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk--boundary correspondence for DTQW's on the example of the "4-step quantum walk", where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.

PT symmetry as a necessary and sufficient condition for unitary time evolution

While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that PT symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any PT-symmetric Hamiltonian H, there always exists an operator V that relates H to its Hermitian adjoint according to VHV(-1)=H(†). When the energy spectrum of H is complete, Hilbert space norms <ψ(1)|V|ψ(2)> constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with PT-symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but PT-symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein-Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through PT symmetry, the fourth-order derivative Pais-Uhlenbeck theory can equally be stabilized.

Speakable and unspeakable in physics of time

In order to discriminate speakable and unspeakable in physics of time, the single ion optical atomic clock and the double time-slit experiments are considered. Their description clarifies that temporal quantities such as time and frequency are fundamentally unspeakable, while speakable is only related to ratios between wavelengths and to boundary conditions of stationary light beams. In addition, time intervals lower than the minimum period realized by the experimental apparatus are the results of an extrapolation out of the domain of definition of measurement operations. A toy model is employed to isolate the hidden source of self-recursivity in the definition of clocks and time, which resides in the abuse of language generated by terms such as duration and repetitive event, instead of interval extension and multiplicity. The timeless scenario which emerges from the experimental picture is theoretically supported by fundamental timelessness of the Hamiltonian formalism employed to define dynamics. Clock metrology, employed to parametrize dynamics in experiments, works successfully as it returns a discrete time variable, which approximates the parametrization of the trajectories in the phase space induced by the invariance of the action and of the time independent Hamiltonian.

Time-fractional extensions of the Liouville and Zwanzig equations

AbstractThis paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.

Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator

We carry out an exact quantization of a PT-symmetric (reversible) Liénard-type one-dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time-independent classical Hamiltonian is of nonstandard type and is invariant under a combined coordinate reflection and time reversal transformation. We use the von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schrödinger equation therein. We explicitly obtain the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr–Sommerfeld quantization rule, while the eigenfunctions are completely different.

Orbital Josephson effect and interactions in driven atom condensates on a ring

In a system of ac-driven condensed bosons we study a new type of Josephson effect occurring between states sharing the same region of space and the same internal atom structure. We first develop a technique to calculate the long-time dynamics of a driven interacting many-body system. For resonant frequencies, this dynamics can be shown to derive from an effective time-independent Hamiltonian which is expressed in terms of standard creation and annihilation operators. Within the subspace of resonant states, and if the undriven states are plane waves, a locally repulsive interaction between bosons translates into an effective attraction. We apply the method to study the effect of interactions on the coherent ratchet current of an asymmetrically driven boson system. We find a wealth of dynamical regimes which includes Rabi oscillations, self-trapping and chaotic behavior. In the latter case, a full many-body calculation deviates from the mean-field results by predicting large quantum fluctuations of the relative particle number.

Quasi-equilibria in reduced Liouville spaces

The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.

Entanglement dynamics via semiclassical propagators in systems of two spins

We analyze the dynamical generation of entanglement in systems of two interacting spins initially prepared in a product of spin coherent states. For arbitrary time-independent Hamiltonians, we derive a semiclassical expression for the purity of the reduced density matrix as a function of time. The final formula, subsidiary to the linear entropy, shows that the short-time dynamics of entanglement depends exclusively on the stability of trajectories governed by the underlying classical Hamiltonian. In addition, this semiclassical measure is shown to reproduce the general properties of its quantum counterpart and give the expected result in the large spin limit. The accuracy of the semiclassical formula is further illustrated in a problem of phase exchange for two particles of spin $j$.

Dissipation evidence for the quantum damped harmonic oscillator via pseudo-bosons

It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator. We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable. This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.

Merging and alignment of Dirac points in a shaken honeycomb optical lattice

Inspired by the recent creation of the honeycomb optical lattice and the realization of the Mott insulating state in a square lattice by shaking, we study here the shaken honeycomb optical lattice. For a periodic shaking of the lattice, a Floquet theory may be applied to derive a time-independent Hamiltonian. In this effective description, the hopping parameters are renormalized by a Bessel function, which depends on the shaking direction, amplitude and frequency. Consequently, the hopping parameters can vanish and even change sign, in an anisotropic manner, thus yielding different band structures. Here, we study the merging and the alignment of Dirac points and dimensional crossovers from the two dimensional system to one dimensional chains and zero dimensional dimers. We also consider next-nearest-neighbor hopping, which breaks the particle-hole symmetry and leads to a metallic phase when it becomes dominant over the nearest-neighbor hopping. Furthermore, we include weak repulsive on-site interactions and find the density profiles for different values of the hopping parameters and interactions, both in a homogeneous system and in the presence of a trapping potential. Our results may be experimentally observed by using momentum-resolved Raman spectroscopy.

On Products of Skew Rotations

Let $H_1(p,q)$, $H_2(p,q)$ be two time-independent Hamiltonians with one degree of freedom and $\{S_1^t\}$, $\{S_2^t\}$ be the one-parametric groups of shifts along the orbits of Hamiltonian systems generated by $H_1$, $H_2$. In some problems of population genetics there appear the transformations of the plane having the form $T^{(h_1,h_2)}=S^{h_2}_2\cdot S_1^{h_1}$ under some conditions on $H_1$, $H_2$. We study in this paper asymptotical properties of trajectories of $T^{(h_1,h_2)}$.

Controlled Dynamics at an Avoided Crossing Interpreted in Terms of Dynamically Fluctuating Potential Energy Curves

The nonadiabatic nuclear wavepacket dynamics on the coupled two lowest (1)Σ(+) states of the LiF molecule under the action of a control pulse is investigated. The control is achieved by a modulation of the characteristics of the potential energy curves using an infrared field with a cycle duration comparable to the time scale of nuclear dynamics. The transition of population between the states is interpreted on the basis of the coupled nuclear wavepacket dynamics on the effective potential curves, which are transformed from the adiabatic potential curves with use of a diabatic representation that diagonalizes the dipole-moment matrix of the relevant electronic states. The basic feature of the transition dynamics is characterized in terms of the notion of the collision between the dynamical crossing point and nuclear wavepackets running on such modulated potential curves, and the transition amplitude is mainly dominated by the off-diagonal matrix element of the time-independent electronic Hamiltonian in the present diabatic representation. The importance of the geometry dependence of the intrinsic dipole moments as well as of the diabatic coupling potential is illustrated both theoretically and numerically.

Simulating typical entanglement with many-body Hamiltonian dynamics

We study the time evolution of the amount of entanglement generated by one-dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribution obtained by a time-independent Hamiltonian can simulate the average and standard deviation of the typical entanglement, if the Hamiltonian contains suitable many-body interactions. We also show that the time required to achieve such a distribution is polynomial in the system size for certain types of Hamiltonians.

Effective quantum Brownian dynamics in presence of a rapidly oscillating space-dependent time-periodic field

We explore the Brownian dynamics in the quantum regime (by investigating the quantum Langevin and Smoluchowski equations) in terms of an effective time-independent Hamiltonian in the presence of a rapidly oscillating field. We achieve this by systematically expanding the time-dependent system-reservoir Hamiltonian in the inverse of driving frequency with a systematic time-scale separation and invoking a quantum gauge transformation within the framework of Floquet theorem.