Final Hamiltonians Research Articles

Anomaly in the dynamical quantum phase transition in a non-Hermitian system with extended gapless phases

The dynamical quantum phase transitions (DQPTs) and the associated winding numbers have been extensively studied in the context Hermitian system. We consider the non-Hermitian analogue of $p$-wave superconductor, supporting Hermitian gapless phase with complex hopping, in presence of on-site or superconducting loss term. This allows us to investigate the effect of non-Hermitian gapless phases on the DQPTs in addition to the Hermitian gapless phases. Our findings indicate that contour analysis of the underlying Hamiltonian, enclosing the origin and/or exceptional points, can predict the occurrences of DQPTs except the quench within the gapless phases. For the Hermitian case with initial and final Hamiltonians both being Hermitian, we find non-monotonic integer jump for the winding number as the hallmark signature of the gapless phase there. For the hybrid case with initial and final Hamiltonians being Hermitian and non-Hermitian respectively, winding number exhibits integer spike in addition to the non-monotonic integer jumps. For the non-Hermitian case with initial and final Hamiltonians both being non-Hermitian, the winding number show half-integer jumps for lossy superconcuctivity that does not have any Hermitian analogue. On the other hand, the integer jumps in winding number is observed for lossy chemical potential. We understand our findings by connecting them with the profile of Fisher zeros and number of exceptional points and/or origin.

Numerical renormalization group study of the Loschmidt echo in Kondo systems

We study the dynamical properties of the one-channel and two-channel spin-1/2 Kondo models after quenching in Hamiltonian variables. Eigen spectrum of the initial and final Hamiltonians is calculated by using the numerical renormalization group method implemented within the matrix product states formalism. We consider multiple quench protocols in the considered Kondo systems, also in the presence of external magnetic field of different intensities. The main emphasis is put on the analysis of the behavior of the Loschmidt echo L(t), which measures the ability of the system’s revival to its initial state after a quench. We show that the decay of the Loschmidt echo strongly depends on the type of quench and the ground state of the system. For the one-channel Kondo model, we show that L(t) decays as, L(t)sim (tcdot T_K)^{-1.4}, where T_K is the Kondo temperature, while for the two-channel Kondo model, we demonstrate that the decay is slower and given by L(t)sim (tcdot T_K)^{-0.7}. In addition, we also determine the dynamical behavior of the impurity’s magnetization, which sheds light on identification of the relevant time scales in the system’s dynamics.

On the quantum adiabatic evolution with the most general system Hamiltonian

In this paper, we study the problem that when quantum adiabatic evolution with the most general form of system Hamiltonian will get failed. Here the most general form means that the initial and final Hamiltonians are just designed according to the adiabatic theorem in quantum mechanics. As we will see, even in this most general model of quantum adiabatic evolution, it still exists the possibility that the quantum adiabatic computation can fail totally if some condition is satisfied, which implies the time complexity of the quantum algorithm is infinity. That is, here we propose a rather general criterion for judging whether a quantum adiabatic evolution is successful. This result largely extends the authors’ previous research on this topic, and it may be seen as a further important clue for us when designing quantum algorithms in the framework of adiabatic evolution for some practical problems.

Quantum work for sudden quenches in Gaussian random Hamiltonians.

In the context of nonequilibrium quantum thermodynamics, variables like work behave stochastically. A particular definition of the work probability density function (pdf) for coherent quantum processes allows the verification of the quantum version of the celebrated fluctuation theorems, due to Jarzynski and Crooks, that apply when the system is driven away from an initial equilibrium thermal state. Such a particular pdf depends basically on the details of the initial and final Hamiltonians, on the temperature of the initial thermal state, and on how some external parameter is changed during the coherent process. Using random matrix theory we derive a simple analytic expression that describes the general behavior of the work characteristic function G(u), associated with this particular work pdf for sudden quenches, valid for all the traditional Gaussian ensembles of Hamiltonians matrices. This formula well describes the general behavior of G(u) calculated from single draws of the initial and final Hamiltonians in all ranges of temperatures.

Dynamics of edge currents in a linearly quenched Haldane model

In a finite time quantum quench of the Haldane model, the Chern number determining the topology of the bulk remains invariant, as long as the dynamics is unitary. Nonetheless, the corresponding boundary attribute, the edge current, displays interesting dynamics. For the case of sudden and adiabatic quenches the post quench edge current is solely determined by the initial and the final Hamiltonians, respectively. However for a finite time ($\tau$) linear quench in a Haldane nano ribbon, we show that the evolution of the edge current from the sudden to the adiabatic limit is not monotonic in $\tau$, and has a turning point at a characteristic time scale $\tau=\tau_c$. For small $\tau$, the excited states lead to a huge unidirectional surge in the edge current of both the edges. On the other hand, in the limit of large $\tau$, the edge current saturates to its expected equilibrium ground state value. This competition between the two limits lead to the observed non-monotonic behavior. Interestingly, $\tau_c$ seems to depend only on the Semenoff mass and the Haldane flux. A similar dynamics for the edge current is also expected in other systems with topological phases.

Merits and qualms of work fluctuations in classical fluctuation theorems

Work is one of the most basic notions in statistical mechanics, with work fluctuation theorems being one central topic in nanoscale thermodynamics. With Hamiltonian chaos commonly thought to provide a foundation for classical statistical mechanics, here we present general salient results regarding how (classical) Hamiltonian chaos generically impacts on nonequilibrium work fluctuations. For isolated chaotic systems prepared with a microcanonical distribution, work fluctuations are minimized and vanish altogether in adiabatic work protocols. For isolated chaotic systems prepared at an initial canonical distribution at inverse temperature β, work fluctuations depicted by the variance of e^{-βW} are also minimized by adiabatic work protocols. This general result indicates that, if the variance of e^{-βW} diverges for an adiabatic work protocol, it diverges for all nonadiabatic work protocols sharing the same initial and final Hamiltonians. Such divergence is hence not an isolated event and thus greatly impacts on the efficiency of using Jarzynski's equality to simulate free-energy differences. Theoretical results are illustrated in a Sinai model. Our general insights shall boost studies in nanoscale thermodynamics and are of fundamental importance in designing useful work protocols.

A Memory of Majorana Modes through Quantum Quench.

We study the sudden quench of a one-dimensional p-wave superconductor through its topological signature in the entanglement spectrum. We show that the long-time evolution of the system and its topological characterization depend on a pseudomagnetic field Reff(k). Furthermore, Reff(k) connects both the initial and the final Hamiltonians, hence exhibiting a memory effect. In particular, we explore the robustness of the Majorana zero-mode and identify the parameter space in which the Majorana zero-mode can revive in the infinite-time limit.

Schedule path optimization for adiabatic quantum computing and optimization

Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.

Quantum Quenches in Chern Insulators.

We explore the nonequilibrium response of Chern insulators. Focusing on the Haldane model, we study the dynamics induced by quantum quenches between topological and nontopological phases. A notable feature is that the Chern number, calculated for an infinite system, is unchanged under the dynamics following such a quench. However, in finite geometries, the initial and final Hamiltonians are distinguished by the presence or absence of edge modes. We study the edge excitations and describe their impact on the experimentally observable edge currents and magnetization. We show that, following a quantum quench, the edge currents relax towards new equilibrium values, and that there is light-cone spreading of the currents into the interior of the sample.

Interpolation approach to Hamiltonian-varying quantum systems and the adiabatic theorem

Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precisely estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example on which the applicability of the adiabatic theorem is questionable.

Dynamical Formation and Manipulation of Majorana Fermions in Driven Quantum Wires in Contact with a Superconductor

Controlling the dynamics of Majorana fermions (MF) subject to time-varying driving fields is of fundamental importance for the practical realization of topological quantum computing. In this work we study how it is possible to dynamically generate and maintain the topological phase in cold-atom nanowires after the temporal variation of the Hamiltonian parameters. Remarkably we show that for a sudden quench the system can never relax toward a state exhibiting fully developed MF, independently of the initial and final Hamiltonians. Only for sufficiently slow protocols the system behaves adiabatically, and the topological phase can be reached. Finally we address the crucial question of how "adiabatic" a protocol must be in order to manipulate the MF inside the topological phase without deteriorating their Majorana character.

Speedup in adiabatic evolution based quantum algorithms

In this context, we study three different strategies to improve the time complexity of the widely used adiabatic evolution algorithms when solving a particular class of quantum search problems where both the initial and final Hamiltonians are one-dimensional projector Hamiltonians on the corresponding ground state. After some simple analysis, we find the time complexity improvement is always accompanied by the increase of some other “complexities” that should be considered. But this just gives the implication that more feasibilities can be achieved in adiabatic evolution based quantum algorithms over the circuit model, even though the equivalence between the two has been shown. In addition, we also give a rough comparison between these different models for the speedup of the problem.

Testing quantum adiabaticity with quench echo

Adiabaticity of quantum evolution is important in many settings; one example is adiabatic quantum computation (AQC). Nevertheless, to date, there is no effective method available for testing the adiabaticity of the evolution for the case where the eigenenergies of the driven Hamiltonian are not known. We propose a simple method for checking the adiabaticity of a quantum process for an arbitrary quantum system. We further propose an operational method for finding more efficient protocols that approximate adiabaticity, and suggest a ‘uniformly adiabatic’ quench scheme based on the Kibble–Zurek mechanism for the case where the initial and final Hamiltonians are given. This method should help in implementing AQC and other tasks where preserving the system in the ground state of a time-dependent Hamiltonian is desired.

Adiabatic quantum computation with a one-dimensional projector Hamiltonian

Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.

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.

Microcanonical quantum fluctuation theorems

Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states, explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the corresponding canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time-reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.

Theory of polarization of crystalline solids.

We consider the change in polarization \ensuremath{\Delta}P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for \ensuremath{\Delta}P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically \ensuremath{\Delta}P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation.

Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian

If an integrable classical Hamiltonian H describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables I of the motion are conserved. Here the fate of the angle variables is analysed. Because of the unavoidable arbitrariness in their definition, angle variables belonging to distinct initial and final Hamiltonians cannot generally be compared. However, they can be compared if the Hamiltonian is taken on a closed excursion in parameter space so that initial and final Hamiltonians are the same. The result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega =dH/dI, but differs from it by a definite extra angle which depends only on the circuit in parameter space, not on the duration of the process. The 2-form which describes this angle variable holonomy is calculated.

Variational derivation of a first-order charge transfer theory

It is shown that a first-order charge transfer theory recently introduced in the literature can be derived from the Schwinger variational principle. The formal properties of the Schwinger principle are also used to establish a hierarchy of higher-order approxima- tions. An asymmetric charge transfer process refers to a collision in which an electron is captured from a target of charge ZT to a projectile of charge Zp, whereby the asymmetry is governed by the relative size of ZT compared with Zp. The theory of such processes has received a new impulse of late. Following recent papers by Shakeshaft (1980), Macek and Shakeshaft (1980), Macek and Taulbjerg (1981), and Macek and Alston (1982) on inner-shell capture (Zp/ZT<< l), a first-order charge transfer theory has been formulated (Briggs et al 1982) which provides the starting point for the derivation of several existing electron capture approaches. For reasons which will become clear shortly, the theory is called the strong potential Born (SPB) approximation. In this letter, we give further support to this theory by showing how it emerges as the lowest-order approximation compatible with the Schwinger variational principle. Both asymmetric cases, ZT/Z~<< 1 and Zp/Z~<< 1, are treated and the respective expressions are developed in parallel. We first establish a notation and then retrace the formal steps leading to the first-order theory. Let us decompose the full Hamiltonian, H, in terms of the initial and final unperturbed Hamiltonians, HT and Hp,

Examination of direct radiative capture formalism

Within the direct reaction framework the formalism and approximations used to describe particle radiative capture processes at low to medium energies are reviewed. Survey calculations documenting sizeable cross-section sensitivity to different forms of the electromagnetic interaction (Siegert's theorem versus J · A ) that preserve overall gauge invariance are reported. The source of this large difference is the inconsistent use of different initial and final state model hamiltonians. This defect, common to most reaction theories using optical-model wave functions, generates wave function nonorthogonality effects which are found to be significant for M1 transitions. The importance of the spin-flip contribution to both electric and magnetic transitions is also investigated.