First-principles quantum dynamics in interacting Bose gases: I. The positive P representation

The performance of the positive P phase-space representation for exact many- body quantum dynamics is investigated. Gases of interacting bosons are considered, where the full quantum equations to simulate are of a Gross-Pitaevskii form with added Gaussian noise. This method gives tractable simulations of many-body systems because the number of variables scales linearly with the spatial lattice size. An expression for the useful simulation time is obtained, and checked in numerical simulations. The dynamics of first-, second- and third-order spatial correlations are calculated for a uniform interacting 1D Bose gas subjected to a change in scattering length. Propagation of correlations is seen. A comparison is made with other recent methods. The positive P method is particularly well suited to open systems as no conservation laws are hard-wired into the calculation. It also differs from most other recent approaches in that there is no truncation of any kind.

Positive-P and Wigner representations for quantum-optical systems with nonorthogonal modes

We generalize the basic concepts of the positive-P and Wigner representations to unstable quantum-optical systems that are based on nonorthogonal quasimodes. This lays the foundation for a quantum description of such systems, such as, for example an unstable cavity laser. We compare both representations by calculating the tunneling times for an unstable resonator optical parametric oscillator.

Gaussian quantum operator representation for bosons

We introduce a Gaussian quantum operator representation, using the most general possible multimode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.

Dynamical quantum noise in trapped Bose-Einstein condensates

We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-P and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-P approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial stare for the condensate and compare results to those for single-mode models. [S1050-2947(98)06612-8].

Dynamics of trapped Bose-Einstein condensates: Beyond the mean-field

Since dilute Bose gas condensates were first experimentally produced, the Gross-Pitaevskii equation has been successfully used as a descriptive tool. As a mean-field equation, it cannot by definition predict anything about the many-body quantum statistics of condensate. We show here that there are a class of dynamical systems where it cannot even make successful predictions about the mean-field behavior, starting with the process of evaporative cooling by which condensates are formed. Among others are parametric processes, such as photoassociation and dissociation of atomic and molecular condensates.

Quantum phase-space analysis of the pendular cavity

We perform a quantum-mechanical analysis of the pendular cavity, using the positive-P representation, showing that the quantum state of the moving mirror, a macroscopic object, has noticeable effects on the dynamics. This system has previously been proposed as a candidate for the quantum-limited measurement of small displacements of the mirror due to radiation pressure, for the production of states with entanglement between the mirror and the field, and even for superposition states of the mirror. However, when we treat the oscillating mirror quantum mechanically, we find that it always oscillates, has no stationary steady state, and exhibits uncertainties in position and momentum which are typically larger than the mean values. This means that previous linearized fluctuation analyses which have been used to predict these highly quantum states are of limited use. We find that the achievable accuracy in measurement is fat, worse than the standard quantum limit due to thermal noise, which, for typical experimental parameters, is overwhelming even at 2 mK

Quantum analysis of the nondegenerate optical parametric oscillator with injected signal

In this paper we study the nondegenerate optical parametric oscillator with injected signal, both analytically and numerically. We develop a perturbation approach which allows us to find approximate analytical solutions, starting from the full equations of motion in the positive-P representation. We demonstrate the regimes of validity of our approximations via comparison with the full stochastic results. We find that, with reasonably low levels of injected signal, the system allows for demonstrations of quantum entanglement and the Einstein-Podolsky-Rosen paradox. In contrast to the normal optical parametric oscillator operating below threshold, these features are demonstrated with relatively intense fields.

Quantum criticality

We investigate the theory of quantum fluctuations in non-equilibrium systems having large critical fluctuations. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction, and also to envisage future tests of quantum theory in regions of macroscopic quantum fluctuations. A long-term objective of this research is to identify suitable physical systems in which macroscopic 'Schrodinger cat'-like behaviour may be observed. We investigate two systems in particular of much current experimental interest, namely the degenerate parametric oscillator near threshold, and the evaporatively cooled (BEC). We compare the results obtained in the positive-P representation, as a fully quantum mechanical calculation, with the truncated Wigner phase space equation, also known as semi-classical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. In the region where the largest quantum fluctuations and Schrodinger cat-like behaviour might be expected, we find that the quantum predictions correspond very closely to the semi-classical theory. Nature abhors observing a Schrodinger car.

Critical quantum fluctuations in the degenerate parametric oscillator

We develop a systematic theory of critical quantum fluctuations in the driven parametric oscillator. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-P representation, as a fully quantum-mechanical calculation, with the truncated Wigner phase-space equation, also known as the semiclassical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. We find that the optimal broadband noise reduction occurs just above threshold. In this region where there are large quantum fluctuations in the conjugate variance and macroscopic quantum superposition states might be expected, we find that the quantum predictions correspond very closely to the semiclassical theory.

Limits to squeezing in the degenerate optical parametric oscillator

We develop a systematic theory of quantum fluctuations in the driven optical parametric oscillator, including the region near threshold. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction in this nonequilibrium quantum phase transition. In particular, we compute the squeezing spectrum near threshold and calculate the optimum value. We find that the optimal noise reduction occurs at different driving fields, depending on the ratio of damping rates. The largest spectral noise reductions are predicted to occur with a very high-Q second-harmonic cavity. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-P representation, as a fully quantum-mechanical calculation, with the truncated Wigner phase-space equation, also known as the semiclassical theory.

Quantum dynamics with stochastic gauge simulations

The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite stochastic time-evolution equations, equivalent to master equations, for many systems including quantum time evolution. The method is illustrated with a variety of simple examples ranging from astrophysical molecular hydrogen production, through to the topical problem of Bose-Einstein condensation in an optical trap and the resulting quantum dynamics.

Critical fluctuations and entanglement in the nondegenerate parametric oscillator

We present a fully quantum mechanical treatment of the nondegenerate optical parametric oscillator both below and near threshold. This is a nonequilibrium quantum system with a critical point phase transition, that is also known to exhibit strong yet easily observed squeezing and quantum entanglement. Our treatment makes use of the positive P representation and goes beyond the usual linearized theory. We compare our analytical results with numerical simulations and find excellent agreement. We also carry out a detailed comparison of our results with those obtained from stochastic electrodynamics, a theory obtained by truncating the equation of motion for the Wigner function, with a view to locating regions of agreement and disagreement between the two. We calculate commonly used measures of quantum behavior including entanglement, squeezing, and Einstein-Podolsky-Rosen (EPR) correlations as well as higher order tripartite correlations, and show how these are modified as the critical point is approached. These results are compared with those obtained using two degenerate parametric oscillators, and we find that in the near-critical region the nondegenerate oscillator has stronger EPR correlations. In general, the critical fluctuations represent an ultimate limit to the possible entanglement that can be achieved in a nondegenerate parametric oscillator.

Gauge Poisson representations for birth/death master equations

Poisson representation techniques provide a powerful method for mapping master equations for birth/death processes -- found in many fields of physics, chemistry and biology -- into more tractable stochastic differential equations. However, the usual expansion is not exact in the presence of boundary terms, which commonly occur when the differential equations are nonlinear. In this paper, a gauge Poisson technique is introduced that eliminates boundary terms, to give an exact representation as a weighted rate equation with stochastic terms. These methods provide novel techniques for calculating and understanding the effects of number correlations in systems that have a master equation description. As examples, correlations induced by strong mutations in genetics, and the astrophysical problem of molecule formation on microscopic grain surfaces are analyzed. Exact analytic results are obtained that can be compared with numerical simulations, demonstrating that stochastic gauge techniques can give exact results where standard Poisson expansions are not able to.

Phase-space analysis of bosonic spontaneous emission

We present phase-space techniques for the modelling of spontaneous emission in two-level bosonic atoms. The positive-P representation is shown to give a full and complete description within the limits of our model. The Wigner representation, even when truncated at second order, is shown to need a doubling of the phase-space to allow for a positive-definite diffusion matrix in the appropriate Fokker-Planck equation and still fails to agree with the full quantum results of the positive-P representation. We show that quantum statistics and correlations between the ground and excited states affect the dynamics of the emission process, so that it is in general non-exponential. (c) 2005 Elsevier B.V. All rights reserved.

First-principles quantum dynamics in interacting Bose gases II: stochastic gauges

First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.