Quantum Stochastic Equations Research Articles

Quantum model for supercontinuum generation process

A quantum theory is established for the propagation of electromagnetic waves in highly nonlinear dispersive optical fibers. By applying the method recently presented dispersion terms and retarded response of the medium are included for the propagation of light in a fiber in this work. A coupled stochastic generalized nonlinear Schrödinger equation (GNLSE) is obtained via the coherent positive-P representation to describe the supercontinuum generation process. This coupled quantum-stochastic equation is applied to obtain the linearized fluctuation equation for studying quantum noise and the fluctuation in the vicinity of the formed solitons in the supercontinuum generation process in the region of anomalous dispersion. Also, these equations can be used to study the soliton self-frequency shift quantum mechanically. Finally, we simulate the obtained coupled stochastic generalized nonlinear Schrödinger in the mean case and compare our simulation results with experimental results.

A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state.

Quantum Stochastic Cable Equation Acting on Functionals of Discrete-Time Normal Martingales

Let M be a discrete-time normal martingale satisfying some mild conditions. Then Gel’fand triple S(M)⊂L2(M)⊂S⁎(M) can be constructed of functionals of M, where elements of S(M) are called testing functionals of M, while elements of S⁎(M) are called generalized functionals of M. In this paper, we consider a quantum stochastic cable equation in terms of operators from S(M) to S⁎(M). Mainly with the 2D-Fock transform as the tool, we establish the existence and uniqueness of a solution to the equation. We also examine the continuity of the solution and its continuous dependence on initial values.

Quantum Stochastic Equations for an Opto-Mechanical Oscillator with Radiation Pressure Interaction and Non-Markovian Effects

The quantum stochastic Schroedinger equation or Hudson-Parthasareathy (HP) equation is a powerful tool to construct unitary dilations of quantum dynamical semigroups and to develop the theory of measurements in continuous time via the construction of output fields. An important feature of such an equation is that it allows to treat not only absorption and emission of quanta, but also scattering processes, which however had very few applications in physical modelling. Moreover, recent developments have shown that also some non-Markovian dynamics can be generated by suitable choices of the state of the quantum noises involved in the HP-equation. This paper is devoted to an application involving these two features, non-Markovianity and scattering process. We consider a micro-mirror mounted on a vibrating structure and reflecting a laser beam, a process giving rise to a radiation-pressure force on the mirror. We show that this process needs the scattering part of the HP-equation to be described. On the other side, non-Markovianity is introduced by the dissipation due to the interaction with some thermal environment which we represent by a phonon field, with a nearly arbitrary excitation spectrum, and by the introduction of phase noise in the laser beam. Finally, we study the full power spectrum of the reflected light and we show how the laser beam can be used as a temperature probe.

A Theoretical Study of the Photoisomerization of Glycolaldehyde and Subsequent OH Radical-Initiated Oxidation of 1,2-Ethenediol.

It has recently been discovered that carbonyl compounds can undergo UV-induced isomerization to their enol counterparts under atmospheric conditions. This study investigates the photoisomerization of glycolaldehyde (HOCH2CHO) to 1,2-ethenediol (HOCH═CHOH) and the subsequent (•)OH-initiated oxidation chemistry of the latter using quantum chemical calculations and stochastic master equation simulations. The keto-enol tautomerization of glycolaldehyde to 1,2-ethenediol is associated with a barrier of 66 kcal mol(-1) and involves a double-hydrogen shift mechanism to give the lower-energy Z isomer. This barrier lies below the energy of the UV/vis absorption band of glycolaldehyde and is also considerably below the energy of the products resulting from photolytic decomposition. The subsequent atmospheric oxidation of 1,2-ethenediol by (•)OH is initiated by addition of the radical to the π system to give the (•)CH(OH)CH(OH)2 radical, which is subsequently trapped by O2 to form the peroxyl radical (•)O2CH(OH)CH(OH)2. According to kinetic simulations, collisional deactivation of the latter is negligible and cannot compete with rapid fragmentation reactions, which lead to (i) formation of glyoxal hydrate [CH(OH)2CHO] and HO2(•) through an α-hydroxyl mechanism (96%) and (ii) two molecules of formic acid with release of (•)OH through a β-hydroxyl pathway (4%). Phenomenological rate coefficients for these two reaction channels were obtained for use in atmospheric chemical modeling. At tropospheric (•)OH concentrations, the lifetime of 1,2-ethenediol toward reaction with (•)OH is calculated to be 68 h.

Efficient quantum filtering for quantum feedback control

We discuss an efficient numerical scheme for the recursive filtering of diffusive quantum stochastic master equations. We show that the resultant quantum trajectory is robust and may be used for feedback based on inefficient measurements. The proposed numerical scheme is amenable to approximation, which can be used to further reduce the computational burden associated with calculating quantum trajectories and may allow real-time quantum filtering. We provide a two-qubit example where feedback control of entanglement may be within the scope of current experimental systems.

Canonical ensemble of an interacting Bose gas: Stochastic matter fields and their coherence

We present a novel quantum stochastic evolution equation for a matter field describing the canonical state of a weakly interacting ultracold Bose gas. In the ideal gas limit our approach is exact. This numerically very stable equation suppresses high-energy fluctuations exponentially, which enables us to describe condensed and thermal atoms within the same formalism. We present applications to ground-state occupation and fluctuations, density profile of ground-state and thermal cloud, and ground-state number statistics. Our main aim are spatial coherence properties which we investigate through the determination of interference contrast and spatial density correlations. Parameters are taken from actual experiments.

Resonant cascaded down-conversion

We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO presents two ${\ensuremath{\chi}}^{(2)}$-level oscillation thresholds that are therefore much lower than for a ${\ensuremath{\chi}}^{(3)}$ OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such a ${\ensuremath{\chi}}^{(2)}$ cascaded device also has potential applications in frequency conversion to far-infrared regimes. But, most importantly, it can generate novel multipartite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian and therefore not describable by any semiclassical model. In this paper, we derive quantum stochastic equations in the positive-$P$ representation and undertake an analysis of steady-state and dynamical properties of this system.

Norm preserving stochastic field equation for an ideal Bose gas in a trap: numerical implementation and applications

Stochastic field equations represent a powerful tool to describe the thermal state of a trapped Bose gas. Often, such approaches are confronted with the old problem of an ultraviolet catastrophe, which demands a cutoff at high energies. In Heller and Strunz (2009 J. Phys. B: At. Mol. Opt. Phys. 42 081001) we introduce a quantum stochastic field equation, avoiding the cutoff problem through a fully quantum approach based on the Glauber–Sudarshan P-function. For a close link to actual experimental setups, the theory is formulated for a fixed particle number and thus based on the canonical ensemble. In this work the derivation and the non-trivial numerical implementation of the equation is explained in detail. We present applications for finite Bose gases trapped in a variety of potentials and show results for ground state occupation numbers and their equilibrium fluctuations. Moreover, we investigate spatial coherence properties by studying correlation functions of various orders.

Noncommutative dynamics and generalized master equations

We review the basic concepts of quantum probability and stochastics using the universal Ito B*-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Levy process is defined in terms of the modular B*-Ito algebra, and the corresponding quantum stochastic master equation on the predual space of theW*-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.

Quantum stochastic walks: A generalization of classical random walks and quantum walks

We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on a line, the QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum algorithms as well as of quantum walks with environmental effects.

Continuous-Variable Hyperentanglement in a Parametric Oscillator with Orbital Angular Momentum

We describe continuous-variable hyperentanglement in polarization and orbital angular momentum modes of an optical parametric oscillator. The quantum stochastic equations for the multimode parametric interaction are derived and solved allowing for calculation of the quadrature noise spectra that characterize continuous-variable entanglement. As a main result, we predict simultaneous entanglement between different combinations of amplitude and phase quadratures of the interacting modes. We also propose experimental setups to access the different kinds of entanglement.

On the quantum master equation under feedback control

The nature of the quantum trajectories, described by stochastic master equations, may be jump-like or diffusive, depending upon different measurement processes. There are many different unravelings corresponding to different types of stochastic master equations for a given master equation. In this paper, we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control. We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement. It is due to the fact that the information gained by the measurement plays an important role in the control process. The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term. By establishing a fundamental limit on performance of the master equation with feedback control, we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation, and show the superiority of the stochastic master equation for feedback control.

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born–Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.

Monte Carlo method for a quantum measurement process by a single-electron transistor

We derive the quantum trajectory or stochastic (conditional) master equation for a single superconducting Cooper-pair box (SCB) charge qubit measured by a single-electron transistor (SET) detector. This stochastic master equation describes the random evolution of the measured SCB qubit density matrix which both conditions and is conditioned on a particular realization of the measured electron tunneling events through the SET junctions. Hence it can be regarded as a Monte Carlo method that allows us to simulate the continuous quantum measurement process. We show that the master equation for the ``partially'' reduced density matrix [Y. Makhlin et al., Phys. Rev. Lett. 85, 4578 (2000)] can be obtained when a ``partial'' average is taken on the stochastic master equation over the fine grained measurement records of the tunneling events in the SET. Finally, we present some Monte Carlo simulation results for the SCB/SET measurement process. We also analyze the probability distribution $P(m,t)$ of finding $m$ electrons that have tunneled into the drain of the SET in time $t$ to demonstrate the connection between the quantum trajectory approach and the ``partially'' reduced density matrix approach.

Quantum stochastic equation for a test particle interacting with a dilute Bose gas

We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity S−1, where S is a one-particle S matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a Fock–anti-Fock (or Gel’fand–Naimark–Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state.

QUANTUM MULTIPOLE NOISE AND GENERALIZED QUANTUM STOCHASTIC EQUATIONS

A notion of quantum multipole (in particular, dipole) noise is considered. Quantum dipole noise is an analogue of quantum white noise but it acts in a Fock space with indefinite metric. Quantum white noise describes the leading term in the stochastic limit approximation to quantum dynamics while quantum multipole noise describes the corrections to the leading term. We obtain and study the generalized quantum stochastic equations describing corrections to the stochastic limit which include quantum dipole noise.

Quantum stochastic equation for the low density limit

A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.

A quantum stochastic equation describing evolution of the transverse energy of a channeled particle

A stochastic equation describing evolution of the nth quantum energy level of a relativistic electron (positron) moving in planar channels of a crystal is derived using the condition of conservation of the transverse motion energy comprising a sum of the energies of interaction with atomic electrons and crystal lattice nuclei.

Photoemissive sources and quantum stochastic calculus

Just at the beginning of quantum stochastic calculus (QSC), Hudson and Parthasarathy proposed a quantum stochastic Schrodinger equation linked to dilations of quantum dynamical semigroups [7, 8, 11]. Such an equation has found applications in physics, mainly in quantum optics, but not in its full generality [6, 1, 2]. It has been used to give, at least approximately, the dynamics of photoemissive sources such as an atom absorbing and emitting light or matter in an optical cavity, which exchanges light with the surrounding free space. But in these cases the possibility of introducing the gauge (or number) process in the dynamical equation has not been considered. In this paper we want to show, in the case of the simplest photoemissive source, namely a two–level atom stimulated by a laser, how the full Hudson– Parthasarathy equation allows to describe in a consistent way not only absorption and emission, but also the scattering of the light by the atom. Let us recall the Hudson–Parthasarathy equation; this is just to fix our notations, while for the proper mathematical definitions and the rules of QSC we refer to the book by Parthasarathy [11]. We denote by F := F(X ) the Boson Fock space over the Hilbert space X := Z ⊗ L(R+) ≃ L(R+;Z), where Z is another separable complex Hilbert space. Let {ei, i ≥ 1} be a c.o.n.s. in Z and let us denote by Ai(t), A†i (t), Λij(t) the annihilation, creation and gauge processes associated with such a c.o.n.s. We denote by E(h), h ∈ X , the exponential vectors in F with normalization ‖E(h)‖2 = exp{‖h‖2}; E(0) is the Fock vacuum. We shall also use the Boson Fock spaces Ft := F ( L([0, t];Z) ) and F t := F ( L2((t,∞);Z) ) , for which we have F = Ft ⊗F , and the Weyl operators