On the Role of Density Matrices in Bohmian Mechanics

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (``statistical mixture'') or a system that is entangled with another system (``reduced density matrix''). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the ``conditional density matrix,'' conditional on the configuration of the environment. A precise definition can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system.Comment: 16 pages LaTeX, no figure

Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system $S$ weakly interacting with a very large but finite system $B$ called the heat bath, and suppose that the composite $S\cup B$ is in a pure state $\Psi$ with participating energies between $E$ and $E+\delta$ with small $\delta$. Then, it is known that for most $\Psi$ the reduced density matrix of $S$ is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on $S$'s Hamiltonian and the temperature but not on $B$'s Hamiltonian, on the interaction Hamiltonian, or on the details of $\Psi$. It has also been pointed out that $S$ can also be attributed a random wave function $\psi$ whose probability distribution is universal in the same sense. This distribution is known as the "Scrooge measure" or "Gaussian adjusted projected (GAP) measure"; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the "conditional wave function". In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the "conditional density matrix", which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most $\Psi$ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.Comment: 13 pages, no figures; v2 minor improvement

Large scale correlations in galaxy clustering from the Two degree Field Galaxy Redshift Survey

We study galaxy correlations from samples extracted from the 2dFGRS final release. Statistical properties are characterized by studying the nearest neighbor probability density, the conditional density and the reduced two-point correlation function. The result is that the conditional density has a power-law behavior in redshift space described by an exponent \gamma=0.8 \pm 0.2 in the interval from about 1 Mpc/h, the average distance between nearest galaxies, up to about 40 Mpc/h, corresponding to radius of the largest sphere contained in the samples. These results are consistent with other studies of the conditional density and are useful to clarify the subtle role of finite-size effects on the determination of the two-point correlation function in redshift and real spaceComment: 11 pages, 14 figures. Accepted for publication in Astronomy and Astrophysic

Time evolution and squeezing of the field amplitude in cavity QED

We present the conditional time evolution of the electromagnetic field produced by a cavity QED system in the strongly coupled regime. We obtain the conditional evolution through a wave-particle correlation function that measures the time evolution of the field after the detection of a photon. A connection exists between this correlation function and the spectrum of squeezing which permits the study of squeezed states in the time domain. We calculate the spectrum of squeezing from the master equation for the reduced density matrix using both the quantum regression theorem and quantum trajectories. Our calculations not only show that spontaneous emission degrades the squeezing signal, but they also point to the dynamical processes that cause this degradation.Comment: 12 pages. Submitted to JOSA

Joint scalar PDF simulations of a bluff-body stabilised flame with the REDIM approach

Transported joint scalar probability density function (PDF) results are presented for ‘Sydney Flame HM3’, a jet type turbulent flame with strong turbulence – chemistry interaction, stabilized behind a bluff body. We apply the novel Reaction-Diffusion Manifold (REDIM) technique, by which a detailed chemistry mechanism is reduced, including diffusion effects. Only N2 and CO2 mass fractions are used as reduced coordinates. A second-moment closure RANS turbulence model is applied. As micro-mixing model, the modified Curl’s coalescence/dispersion (CD) and the Euclidean Minimum Spanning Tree (EMST) models are used. In physical space, agreement between experimental data and simulation results is good up to the neck zone, for the unconditional mean values of velocity, mixture fraction, major and some minor chemical species. Conditional mean profiles in mixture fraction space are also in reasonable agreement with experiments up to the neck zone, though conditional fluctuations tend to be under-predicted. CD generally yields better predictions for the level of fluctuations in mixture fraction space than EMST, but this is partly due to unrealistic particle evolution in composition space. In general, simulations using the REDIM approach for reduction of detailed C2-chemistry confirm earlier findings for micro-mixing model behaviour, obtained with C1-chemistry

Asymptotic entanglement in a two-dimensional quantum walk

The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the coin and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.Comment: Major revision. Improved structure. Theoretical results are now separated from specific examples. Most figures have been replaced by new versions. The paper is now significantly reduced in size: 11 pages, 7 figure

Statistics of cosmic density profiles from perturbation theory

The joint probability distribution function (PDF) of the density within multiple concentric spherical cells is considered. It is shown how its cumulant generating function can be obtained at tree order in perturbation theory as the Legendre transform of a function directly built in terms of the initial moments. In the context of the upcoming generation of large-scale structure surveys, it is conjectured that this result correctly models such a function for finite values of the variance. Detailed consequences of this assumption are explored. In particular the corresponding one-cell density probability distribution at finite variance is computed for realistic power spectra, taking into account its scale variation. It is found to be in agreement with $\Lambda$-CDM simulations at the few percent level for a wide range of density values and parameters. Related explicit analytic expansions at the low and high density tails are given. The conditional (at fixed density) and marginal probability of the slope -- the density difference between adjacent cells -- and its fluctuations is also computed from the two-cells joint PDF; it also compares very well to simulations, in particular in under-dense regions, with a significant reduced cosmic scatter compared to over-dense regions. It is emphasized that this could prove useful when studying the statistical properties of voids as it can serve as a statistical indicator to test gravity models and/or probe key cosmological parameters.Comment: 22 pages, 15 figures, submitted to PR