Unitary Evolution Operator Research Articles

Supersymmetry versus ghosts

We consider the simplest nontrivial supersymmetric quantum mechanical system involving higher derivatives. We unravel the existence of additional bosonic and fermionic integrals of motion forming a nontrivial algebra. This allows one to obtain the exact solution both in the classical and quantum cases. The supercharges Q and Q¯ are not Hermitially conjugate to each other anymore, which allows for the presence of negative energies in the spectrum. We show that the spectrum of the Hamiltonian is unbounded from below. It is discrete and infinitely degenerate in the free oscillatorlike case and becomes continuous running from −∞ to ∞ when interactions are added. Notwithstanding the absence of the ground state, there is no collapse, which suggests that a unitary evolution operator may be defined.

Quantum walks on quotient graphs

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup used to construct it. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting time can be infinite. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with fast hitting times correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.

Quantization of linearly polarized cosmological models with two Killing vector fields

We discuss the quantization of two-Killing symmetry reductions of general relativity with cosmological interpretation. In particular we will focus on the obtention of unitary evolution operators in closed form for the class of time-dependent quadratic Hamiltonians that appear in this context. This will be done by extending to field theoretical models the techniques developed for the exact quantization of the harmonic oscillator with time-dependent frequency. The abstract operators that we will derive are useful both to discuss different field representations and the problem of unitary implementation of dynamics. Finally, we consider in some detail the applications of the formalism developed to the Gowdy and Schmidt cosmological models.

Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition

Recent research in generalizing quantum computation from 2-valued qudits to d-valued qudits has shown practical advantages for scaling up a quantum computer. A further generalization leads to quantum computing with hybrid qudits where two or more qudits have different finite dimensions. Advantages of hybrid and d-valued gates (circuits) and their physical realizations have been studied in detail by Muthukrishnan and Stroud [Multi-valued logic gates for quantum computation, Phys. Rev. A 62 (2000) 052309. [10]], Daboul et al. [Quantum gates on hybrid qudits, J. Phys. A Math. Gen. 36 (2003) 2525–2536. [5]], and Bartlett et al. [Quantum encodings in spin systems and harmonic oscillators, Phys. Rev. A 65 (2002) 052316. [17]]. In both cases, a quantum computation is performed when a unitary evolution operator, acting as a quantum logic gate, transforms the state of qudits in a quantum system. Unitary operators can be represented by square unitary matrices. If the system consists of a single qudit, then Tilma et al. [Generalized Euler angle parameterization for SU(N), J. Phys. A Math. Gen. 35 (2002) 10467–10501. [15]] have shown that the unitary evolution matrix (gate) can be synthesized in terms of its Euler angle parametrization. However, if the quantum system consists of multiple qudits, then a gate may be synthesized by matrix decomposition techniques such as QR factorization and the cosine–sine decomposition (CSD). In this article, we present a CSD based synthesis method for n qudit hybrid quantum gates, and as a consequence, derive a CSD based synthesis method for n qudit gates where all the qudits have the same dimension.

Chiral conical diffraction

The geometrical and wave optics are explored for light emerging from a slab of transparent biaxial crystal with optical activity (chirality), for an incident beam directed along the optic axis. The geometrical optics, here derived from Hamilton's principle, is dominated by a circularly symmetric cusped caustic surface ('spun cusp') threaded by an axial focal line. In wave optics, formulated exactly in the paraxial approximation in terms of integrals previously obtained by Belsky and Stepanov and here derived using a unitary evolution operator, the field is determined by two dimensionless parameters. The geometrical features are decorated by interference, here explored in the focal image plane (where the Poggendorff rings of the non-chiral case are in sharpest focus) and along the axis. Asymptotic approximations are derived in terms of the geometrical optics rays (including interference and evanescent waves), near the spun cusp, and uniformly across the caustic surface far from the cusp.

Symbolic algorithm for factorization of the evolution operator of the time-dependent Schrödinger equation

A symbolic algorithm for the decomposition of the unitary evolution operator is developed. This algorithm allows one to generate multilayer implicit schemes for solution of the time-dependent Schrodinger equation. Some additional gauge transformations are also implemented in the algorithm. This allows one to distinguish symmetric operators, which are required for constructing efficient evolutionary schemes. The efficiency of the generated schemes is demonstrated by integrable models.

Engineering transient spectrum for a two-level system interacting with two-mode intelligent states

The problem of a single two-level atom in its excited state, placed in a bimodal cavity in which an arbitrary form of the intensity-dependent atom–field coupling is present, is solved exactly. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time-dependent unitary evolution operator of the system is given, by means of which we analyze the analytic form of the fluorescence spectrum using the transitions among the dressed states of the system. The influence of parameter misfits on the quantum system is discussed. An interesting relation between the fluorescence spectrum and the dynamical evolution is found when the initial field states are prepared in properly intelligent states. By comparison with exact numerical results, we demonstrate that features of the fluorescence spectrum are influenced significantly by the different kinds of the nonlinearity of the intensity-dependent atom–field coupling. We provide numerous examples to illustrate this correspondence, and provide evidence of its usefulness. The present work considerably extends earlier results for two-level systems.

Quantized alternating integrable mappings

A new alternating quantum integrable map is presented (i.e. a non-autonomous quantum integrable map whose sole time-dependence is of the form ). For a certain parameter reduction, the map becomes autonomous, and is, in fact, the quantum version of a recently introduced close-to-the-identity integrable map. The quantum version of the recently introduced alternating McMillan map is also given. The unitary evolution operator is given explicitly for both maps, hence, both maps give a true quantum mechanical time evolution.

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.

Entanglement Distribution and Entangling Power of Quantum Gates

Quantum gates, that play a fundamental role in quantum computation and other quantum information processes, are unitary evolution operators $\hat U$ that act on a composite system changing its entanglement. In the present contribution we study some aspects of these entanglement changes. By recourse of a Monte Carlo procedure, we compute the so called "entangling power" for several paradigmatic quantum gates and discuss results concerning the action of the CNOT gate. We pay special attention to the distribution of entanglement among the several parties involved.

Entangled photon states in consecutive nonlinear optical interactions

Quantum theory of two consecutive light-wave parametric interactions with aliquant frequencies produced by a common pump wave in a crystal is developed. Using the differentiation method, the unitary evolution operator of the system is reduced to an ordered form that allows the calculation of the field state and the statistical characteristics of interacting waves. It is shown that, for the initial vacuum field state, the created photons obey the super-Poisson statistics at the interacting frequencies and are in a multiparticle entangled state.

Mixed-state entanglement of a three-level system with an intensity-dependent interaction

In this communication, the entanglement degree due to quasi-mutual entropy of a three-level atom interacting with a single cavity field is investigated. We consider the situation for which the three-level system is initially in a mixed state, whereas the field may start from either a coherent or a squeezed state. We present a derivation of the unitary evolution operator on the basis of the dressed-state formalism taking into account an arbitrary form of nonlinearity of the intensity-dependent coupling, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. Most interestingly, it is shown that features of the degree of entanglement are influenced significantly by different forms of the nonlinearity. The atom and radiation subsystems exhibit alternating sets of collapses and revivals due to the initially mixed states of the atom and radiation employed here.

Control of the Evolution of Heisenberg Spin Systems

In this paper, we give an algorithm for the control of the unitary evolution operator for the system of two spin 1 2 ’s interacting through Heisenberg interaction. This algorithm allows for arbitrarily bounded control. The results extend previous ones that only concerned the control of a pure state. A Lie group decomposition technique is used to this goal and the problem is reduced to a two level problem with constraints. We also show how the control result allows to obtain information on the initial state through a series of evolutions and measurements. The study of the Heisenberg spin model is motivated by applications in electron paramagnetic resonance and in molecular magnetism. For these systems, the drift which models the interaction between the spins is typically large and cannot be neglected in the control design but it has to be used to reach the objective.

Non-commutative solitons in finite quantum mechanics

We construct the unitary evolution operators that realize the quantization of linear maps of SL(2,R) over phase spaces of arbitrary integer discretization N and show the non-trivial dependence on the arithmetic nature of N. We discuss the corresponding uncertainty principle and construct the corresponding coherent states, that may be interpreted as non-commutative solitons.

Schrödinger equation for a noninteracting Bose gas on a spatiotemporal lattice

We propose a spatiotemporal discretization of the Schr\"odinger equation for field operators of noninteracting bosons. The scheme, being based on the associated space of the creation and annihilation operators and on a discretization of the Schr\"odinger equation, preserves equal time commutation relations, possesses Hermitian action, unitary evolution operator, and all relevant integrals which are counterparts of the respective conservation quantities in the continuum limit. Although the scheme is implicit, it allows a reduction to a nonlocal but explicit scheme. In the limit of small time steps the scheme is reduced to a model studied earlier. Some particular examples of the discrete-time Schr\"odinger equation in the presence of external potentials are given.

Chemical Shift Anisotropy and Offset Effects in Cross Polarization Solid-State NMR Spectroscopy

The effect of an offset term in the cross-polarization (CP) Hamiltonian of a heteronuclear spin- 1 2 pair due to off-resonant radio frequency (rf) irradiation and/or chemical shift anisotropy on one of the rf channels is investigated. Analytical solutions, simulations, and experimental results are presented. Formulating the CP spin dynamics in terms of an explicit unitary evolution operator enables the CP period to be inserted as a module in a given pulse scheme regardless of the initial density matrix present. The outcome of post-CP manipulation via pulses can be calculated on the resulting density matrix as the phases and amplitudes of all coherence modes are available. Using these tools it is shown that the offset can be used to reduce the rf power on that channel and the performance is further improved by a post-CP pulse whose flip angle matches and compensates the tilt of the effective field on the offset channel. Experimental investigations on single crystalline and polycrystalline samples of peptides confirm the oscillatory nature of CP dynamics and prove the slowing down of the dynamics under offset and/or mismatch conditions.

The unitary evolution operator for cross-polarization schemes in NMR

The evolution operator for cross-polarization (CP) schemes in NMR is presented. The operator is applied to investigate the evolution of all possible coherences for a heteronuclear spin- 1 2 pair during CP under the conditions of Hartman–Hahn matching, antimatching or mismatching. This enables to utilize the CP unit as a module in pulse sequences by permitting pre- and post-pulses to manipulate the density matrix prior to and after evolution under CP. This formalism is applicable to analyze CP schemes for a heteronuclear spin- 1 2 pair with dipolar coupling and/or scalar coupled solid-state (powder or crystal form) or partially ordered or isotropic solution systems.

Strongly Coupled Quantum Discrete Liouville Theory.¶I: Algebraic Approach and Duality

The quantum discrete Liouville model in the strongly coupled regime, 1<c<25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

Time Evolution in an External Field: The Unitarity Paradox

One of the axioms of quantum field theory is the property of unitarity of the evolution operator. However, if one considers the quantum electrodynamics in the external field in the leading order of perturbation theory, one will find that the evolution transformation is a nonunitary canonical transformation of creation and annihilation operators. This observation was one of the arguments for the hypothesis that one should choose different representations of the canonical commutation relations at different moments of time in the exact quantum field theory. In this paper the contradiction is analyzed for the case of a simple quantum mechanical model being an analog of the leading order of the large-N field theory. On the one hand, this model is renormalized with the help of the constructive field theory methods; the Hilbert space and unitary evolution operator are constructed. On the other hand, the leading order of the evolution transformation in the strong external field is shown to be nonunitary. Thus, unitarity of evolution in the exact theory is not in contradiction with the nonunitarity of the approximate theory.

Unitary Evolution and Quantum Fluctuation in SQUID

A circular superconductor with a weak-link junction is considered in the presence of a dc voltage bias. Using a time-dependent perturbed method, we obtain a unitary evolution operator of the driven SQUID system. We investigate the step structure of the screening current in the system and show that, due to the Josephson nonlinear term, the flux and charge can exhibit reduced quantum fluctuation behavior.