Unitary Time Evolution Research Articles

Multiple-event probability in general-relativistic quantum mechanics. II. A discrete model

We introduce a simple quantum mechanical model in which time and space are discrete and periodic. These features avoid the complications related to continuous-spectrum operators and infinite-norm states. The model provides a tool for discussing the probabilistic interpretation of generally covariant quantum systems, without the confusion generated by spurious infinities. We use the model to illustrate the formalism of general-relativistic quantum mechanics, and to test the definition of multiple-event probability introduced in a companion paper [Phys. Rev. D 75, 084033 (2007)]. We consider a version of the model with unitary time evolution and a version without unitary time evolution.

Classical trajectories for complex Hamiltonians

It has been found that complex non-Hermitian quantum-mechanical Hamiltonians may have entirely real spectra and generate unitary time evolution if they possess an unbroken $\cP\cT$ symmetry. A well-studied class of such Hamiltonians is $H= p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$). This paper examines the underlying classical theory. Specifically, it explores the possible trajectories of a classical particle that is governed by this class of Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate structure that depends sensitively on the value of the parameter $\epsilon$ and on the initial conditions. A system for classifying complex orbits is presented.

Unitary time evolution in the Gowdy T3 model

The polarized Gowdy T3 model is a classically solvable cosmological model with local degrees of freedom. This explains the increasing interest in its quantization. After gauge fixing, the system can be described in terms of a point particle and a massless scalar field propagating in an expanding torus. A Fock space representation has been proposed for the quantization of the model, but the classical dynamics for the scalar field cannot be unitarily implemented in this quantum theory. We show that, nevertheless, unitarity can be regained by slightly modifying the dynamics with quantum corrections. Moreover, a time-dependent rescaling of the scalar field leads to new evolution equations that can be unitarily implemented.

Can chemistry be derived from quantum mechanics? Chemical dynamics and structure

Following an early suggestion by Dirac, contemporary quantum mechanics (QM) and measurement theory are used to stepwise construct a conceptual framework to chemistry. In QM arbitrary physical quantum states are represented by linear superpositions. Base sets and quantum states are clearly distinguished; the latter are sets of ordered complex numbers. The generator of time evolution $$\fancyscript{H}$$ is shown to have besides the molecular hamiltonian H=H C+K N, the electron–phonon and spin–orbit operators.H is shown diagonal in a product base set formed with generalized electronic diabatic (GED) and nuclear wave functions; the latter represented in a plane wave base set and unique inertial frame; these functions form the GEDM base set. Chemical species are assigned to particular GED base functions and a chemical quantum state is given as a linear superposition in the GEDM base set. Thus, given a fixed number of electrons and nuclei, the GEDM base set includes all electronuclear quantum states from clusters, supermolecule, molecular aggregates, asymptotic states, and ionized states (continuum electron states) to all plasma states. Experimentally measurable quantum states are given by linear superpositions. A chemical process is sensed by amplitude changes involving different electronic states in the linear superposition; vibration states intervene in excitation–relaxation processes; a time evolution of a global quantum 1-system is the feature. If sufficient time is allowed to an isolated system, with a given total energy E, unitary time evolution with $$\fancyscript{H}$$ (but not H) ensures that the system will evolve from the initial state to get amplitudes different from zero for all energy conserving final states. Response towards external dynamic couplings, e.g. electromagnetic fields, is given by changes of amplitudes and expressed in correlation functions. For a quantum state, the amplitude at a given base function reflects the possibility for the system submitted to excitation sources to respond with its spectral signatures; if the experiment is done, and the amplitude is different from zero, then, there will be a response. Algorithms permitting to relate Hilbert space to real space events are discussed. A chemical compound (analyte) is identified to (i) a fixed electronic quantum state; (ii) linear nuclear base states superpositions; (iii) zero amplitude for all other electronic base states. To get this implies either the action of a chemist introducing constraints to bottle the analyte or an accidental change of external (to the box) conditions producing spontaneous separations.

Relaxation in Time-Dependent Current-Density-Functional Theory

We apply the time-dependent current-density-functional theory to the study of the relaxation of a closed many-electron system evolving from a nonequilibrium initial state. We show that the self-consistent unitary time evolution generated by the exchange-correlation vector potential irreversibly drives the system to equilibrium. We also show that the energy dissipated in the Kohn-Sham system, i.e., the noninteracting system whose particle and current densities coincide with those of the physical system under study, is related to the entropy production in the real system.

Quantum hydrodynamics: Capturing a reactive scattering resonance

The hydrodynamic equations of motion associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a meshless method based upon a moving least-squares approach. An arbitrary Lagrangian-Eulerian frame of reference and a regridding algorithm which adds and deletes computational points are used to maintain a uniform and nearly constant interparticle spacing. The methodology also uses averaged fields to maintain unitary time evolution. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. A new and more robust artificial viscosity algorithm is presented which gives accurate scattering results and is capable of capturing quantum resonances. The methodology is applied to a one-dimensional model chemical reaction that is known to exhibit a quantum resonance. The correlation function approach is used to compute the reactive scattering matrix, reaction probability, and time delay as a function of energy. Excellent agreement is obtained between the scattering results based upon the quantum hydrodynamic approach and those based upon standard quantum mechanics. This is the first clear demonstration of the ability of moving grid approaches to accurately and robustly reproduce resonance structures in a scattering system.

Evolution of entanglement entropy in one-dimensional systems

We study the unitary time evolution of the entropy of entanglement of a one-dimensionalsystem between the degrees of freedom in an interval of length and its complement, starting from a pure state which is not an eigenstate of theHamiltonian. We use path integral methods of quantum field theory as well as explicitcomputations for the transverse Ising spin chain. In both cases, there is a maximum speedv of propagation of signals. In general the entanglement entropy increases linearly with timet up to , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as aconsequence of causality.

TheCoperator inPT-symmetric quantum field theory transforms as a Lorentz scalar

A non-Hermitian Hamiltonian has a real positive spectrum and exhibits unitary time evolution if the Hamiltonian possesses an unbroken PT (space-time reflection) symmetry. The proof of unitarity requires the construction of a linear operator called C. It is shown here that C is the complex extension of the intrinsic parity operator and that the C operator transforms under the Lorentz group as a scalar.

Pure point spectrum for the time evolution of a periodically rank-N kicked Hamiltonian

We find the conditions under which the spectrum of the unitary time evolution operator for a periodically rank-N kicked system remains pure point. This stability result allows one to analyze the onset of, or lack of chaos in this class of quantum mechanical systems, extending the results for rank-1 systems produced by Combescure and others. This work includes a number of unitary theorems equivalent to those well known and used in the self-adjoint theory.

The Script C operator in Script PScript T-symmetric quantum theories

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT-symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.

Connection between decoherence and fidelity decay in echo dynamics

Entanglement between a quantum system and its environment leads to loss of coherence in the former. In general, the temporal fate of coherences is complicated. Here, we establish the connection between decoherence of a central system and fidelity decay in the environment for a variety of situations, including both, energy conserving and dissipative couplings. We show how properties of unitary time evolution of the environment can be inferred from the non-unitary evolution of coherences in the central system. This opens up promising ways for measuring Loschmidt echoes in a variety of situations.

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p 2+ix 3 and H=p 2-x 4. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.

A new method for solving the quantum hydrodynamic equations of motion: application to two-dimensional reactive scattering.

The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method.

Must a Hamiltonian be Hermitian?

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space–time reflection symmetry (𝒫𝒯 symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if 𝒫𝒯 symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any 𝒫𝒯-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.

Decoherence in chaotic and integrable systems: a random matrix approach

We study the influence of chaos and order on entanglement and decoherence. In view of applications in quantum computing and teleportation which should be able to work with arbitrarily complicated states, we pay particular attention to the behavior of random states. While studies with coherent states indicate that chaos accelerates decoherence and entanglement, we find that there is practically no difference between the chaotic and the integrable case, as far as random states are concerned. In the present studies we use unitary time evolution of the total system, and partial traces to emulate decoherence. Random matrix models are a natural choice to describe the dynamics of random states. The invariant aspects of chaos and order are then reflected in the different spectral statistics. We develop random matrix models for the evolution of entanglement for a large variety of situations, discussing the strong coupling case in full detail.

Space-adiabatic perturbation theory

We study approximate solutions to the Schrodinger equation $i\epsi\partial\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x) \psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\Hi_{\rm f}$ of fast ``internal'' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \cite{NS} we prove that interband transitions are suppressed to any order in $\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\mathbb{R}^d,\Hi _{\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.

Complex extension of quantum mechanics.

Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.

Quantum fast Fourier transform using multilevel atoms

We propose an implementation of the quantum fast Fourier transform algorithm in an entangled system of multilevel atoms. The Fourier transform occurs naturally in the unitary time evolution of energy eigenstates and is used to define an alternative wave-packet basis for quantum information in the atom. A change of basis from energy levels to wave packets amounts to a discrete quantum Fourier transform within each atom. The algorithm then reduces to a series of conditional phase transforms between two entangled atoms in mixed energy and wave-packet bases. We show how to implement such transforms using wave-packet control of the internal states of the ions in the linear ion-trap scheme for quantum computing.

Dephasing by two-level systems at zero temperature by unitary evolution

We analyze the unitary time evolution of a conduction electron, described by a two-level system, interacting with two-level systems (spins) through a spin–spin interaction and prove that coherent spin states of the conduction electron are obtained in the strong coupling regime, when the number of the spins is taken to be, formally, infinitely large (thermodynamic limit). This model describes a spin interacting with a spin bath in a strong coupling regime and gives a dephasing time at zero temperature that agrees with the recent experimental results for quantum dots. Dephasing is proved to occur as the conduction electron oscillates between the two states with frequency going to infinity in the thermodynamic limit, that is, increasing the number of spins. Then, it is shown that the only meaning that can be attached to such oscillations with infinite frequency is by an average in time, eliminating the off-diagonal terms of the density matrix. This model is in agreement with a recent proposal of appearance of classical states in quantum mechanics due to the large number of components of a quantum system, if properly prepared for the states of each component part. The strong coupling study of the model is accomplished through the principle of duality as recently introduced in the perturbation theory (M. Frasca, Phys. Rev. A 58 (1998) 3439.

Critical points in edge tunneling between generic fractional quantum Hall states

A general description of weak and strong tunneling fixed points is developed in the chiral-Luttinger-liquid model of quantum Hall edge states. Tunneling fixed points are a subset of `termination' fixed points, which describe boundary conditions on a multicomponent edge. The requirement of unitary time evolution at the boundary gives a nontrivial consistency condition for possible low-energy boundary conditions. The effect of interactions and random hopping on fixed points is studied through a perturbative RG approach which generalizes the Giamarchi-Schulz RG for disordered Luttinger liquids to broken left-right symmetry and multiple modes. The allowed termination points of a multicomponent edge are classified by a B-matrix with rational matrix elements. We apply our approach to a number of examples, such as tunneling between a quantum Hall edge and a superconductor and tunneling between two quantum Hall edges in the presence of interactions. Interactions are shown to induce a continuous renormalization of effective tunneling charge for the integrable case of tunneling between two Laughlin states. The correlation functions of electronlike operators across a junction are found from the B matrix using a simple image-charge description, along with the induced lattice of boundary operators. Many of the results obtained are also relevant to ordinary Luttinger liquids.