Time evolution of a quantum lattice system

It is shown that for a quantum lattice system associated with a Hamiltonian with a kinetic part and a potential sufficiently decreasing in the particle number, the time evolution can be described, under certain assumptions, by automorphisms of a suitable algebra.     

Metastable states of quantum lattice systems

We extend the characterisation of metastability, as given in [1] for classical systems, to show that quantal lattice systems with suitable long range forces can support metastable states.     

Markov processes in quantum lattice systems

In analogy with classical lattice systems [1], the existence of Markov processes is shown in quantum lattice systems with a class of finite range interactions. This result is then applied to show that the weak-clustering property and the ergodicity of translation-invariant state are preserved. The invariance of Gibbs states is also proved.     

Dynamics of fluctuations for quantum lattice systems

For short range interactions and forL ¹-space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations. This dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations.     

Density functional approach to quantum lattice systems

For quantum lattice systems, we consider the problem of characterizing the set of single-particle densities,, which come from the ground-state eigenspace of someN-particle Hamiltonian of the form whereH ₀ is a fixed, bounded operator representing the kinetic and interaction energies. We show that the conditions on are that it be strictly positive, properly normalized, and consistent with the Pauli principle. Our results are valid for both finite and infinite lattices and for either bosons or fermions. The Coulomb interaction may be included inH ₀ if the lattice dimension is 2. We also characterize those single-particle densities which come from the Gibbs states of such Hamiltonians at finite temperature. In addition to the conditions stated above, must satisfy a finite entropy condition.Research supported by the National Science Foundation under grant No. PHY-82-03669.Research supported by Office of Naval Research under grant No. 0014-80-G-0084.On leave from Department of Mathematics, University of Lowell, Massachusetts 01854.     

The property lattice of spatially separated quantum systems

We consider a quantum system composed of two subsystems. Among the properties of this system we study the set of those that can be tested when the subsystems are spatially separated. We show that not all properties satisfy this criterion, but that there are enough such properties to characterize any pure state of the composed system.     

Existence of phase transitions for quantum lattice systems

We prove that the following lattice systems:

1. anisotropic Heisenberg model,
2. Ising model with transverse magnetic field,
3. quantum lattice gas with hard cores extending over nearest neighbours,

exhibit phase transitions if the temperature is sufficiently low and the transverse (or kinetic) part of the interaction sufficiently small.     

Spontaneous translation symmetry breakdown for quantum lattice systems

We study the set of equilibrium states for quantum lattice states in the presence of a translation symmetry of the model. We derive a characterization of the spontaneous breaking of this symmetry, i.e., the decomposition of an invariant equilibrium state into a mixture of noninvariant equilibrium states, in terms of the separability in mean energy of these states for a class of perturbed dynamics.     

Linear response and relaxation in quantum lattice systems

For spin-lattice systems, the Kubo formula, expressing the relaxation function in terms of the linear response function, is found to be exact in the thermodynamic limit. In addition, analyticity properties are obtained.     

Signal propagation in lattice models of quantum many-body systems

Upper and lower bounds are proven for the speed of propagation of general classes of physical signals in particle lattice models.     

Equivalence of ensembles in quantum lattice systems: States

The general analysis of the equivalence of ensembles in quantum lattice systems, which was undertaken in paper I of this series, is continued.The properties of equilibrium states are considered in a variational sense. It is then shown that there exists a canonical as well as a microcanonical variational formulation of equilibrium both of which are equivalent to the grandcanonical formulation.Equilibrium states are constructed both in the canonical and in the microcanonical formalism by means of suitable limiting procedures.It is shown, in particular, that the invariant equilibrium states for a given energy and density are those for which the maximum of the mean entropy is reached. The mean entropy thus obtained coincides with the microcanonical entropy.     

Statistical mechanics of quantum lattice systems without translation invariance

The well-known results concerning the equilibrium of a translation invariant quantum lattice system — existence of the pressure and of the time automorphisms, variational principle for the pressure — are generalized to a large class of quantum lattice systems with potentials not exhibiting covariance under the group of lattice translations.     

Time optimal quantum control of two-qubit systems

We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms.We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state.As applications the minimal time to implement Controlled-NOT gate,SWAP gate and Controlled-U gate is calculated in detail.The experimental realizations of these quantum gates are explicitly presented.     

Time optimal control of two-level quantum systems

The objective of time-optimal control that helps to minimize relaxation losses, is the evolution of a quantum state from a given initial mixed state to a final target mixed state in minimum time. In this paper, we study a time-optimal control problem of the dynamic of a pure two-level system with unbounded control using Pontryagin's minimum principle and obtain the minimal time for some initial and final states. The results will apply to basically all qubit systems that one can consider such as NMR spectroscopy, trapped ions, superconducting qubits, etc. We also show that these results hold for pure states, and only the direction $\hat{n}$ is important in the evolution of a quantum state. In this work, the problem of computing minimum time to produce any unitary transformation $U_f\in SU(2)$ is reduced to finding the minimum time to steer the system from an initial to a final state.     

Quantum fluctuations in quantum lattice systems with continuous symmetry

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems atT=0. Our analysis is based on Pitaevskii and Stringari's idea that the uncertainty relation can be employed to show quantum fluctuations. For one-dimensional systems, it is shown that the ground state is invariant under a continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than|r| ^–d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice systems having not-too-long-range interactions.     

The cluster expansion for classical and quantum lattice systems

We develop a high-temperature expansion for general lattice systems which can be applied to classical as well as quantum systems. Applying the expansion we prove analyticity of correlation functions, uniqueness of equilibrium states, and cluster properties for classical and quantum lattice systems in the high-temperature region.     

On averaging of time evolution of quantum spin lattice systems

An investigation of a procedure of obtaining irreversible dynamics for systems in contact with surroundings is given.Research supported by M. Skodowska-Curie Fund Grant No. 0IP74-01416.     

Dynamic stimulation of quantum coherence in systems of lattice bosons

Thermal fluctuations tend to destroy long-range phase correlations. Consequently, bosons in a lattice will undergo a transition from a phase-coherent superfluid as the temperature rises. Contrary to common intuition, however, we show that nonequilibrium driving can be used to reverse this thermal decoherence. This is possible because the energy distribution at equilibrium is rarely optimal for the manifestation of a given quantum property. We demonstrate this in the Bose-Hubbard model by calculating the nonequilibrium spatial correlation function with periodic driving. We show that the nonequilibrium phase boundary between coherent and incoherent states at finite bath temperatures can be made qualitatively identical to the familiar zero-temperature phase diagram, and we discuss the experimental manifestation of this phenomenon in cold atoms.     

On uniqueness of KMS states of one-dimensional quantum lattice systems

We present a proof of the theorem on the uniqueness of KMS states of one-dimensional quantum lattice systems, which is based on some equicontinuity.