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We investigate the singularities of the trace of the half-wave group, $\mathrm{Tr} \, e^{-it\sqrt\Delta}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author \cite{Hil} and the two-dimensional case of the work of the first author and Wunsch \cite{ForWun} as well as the seminal result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann \cite{MelUhl}., Comment: 45 pages, 2 figures. Comments welcome

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times \left(\mathbb{R} \big/ 2\pi\rho \mathbb{Z}\right)$ of radius $\rho > 0$ equipped with the metric $h(r,\theta) = d r^2 + r^2 \, d\theta^2$. Using explicit oscillatory integrals and relying on the fundamental solution to the wave equation in geometric regions related to flat wave propagation and diffraction by the cone point, we can prove spectral cluster estimates equivalent to those in works on smooth Riemannian manifolds., Comment: 21 pages, comments welcome. Updated references to sharpness and wave kernel formulations, typos fixed thanks to anonymous referee

Let $(X,g)$ be a compact manifold with conic singularities. Taking $\Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{ \smash[b]{\Delta_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points., Comment: 46 pages; 3 figures. New version reflects correction to trace formula (factor of primitive length rather than length), as well as some corrections to discussions of Jacobi fields in section 1.3

Here the problem considered is that of a pair of oscillators coupled to a common heat bath. Many, if not most, discussions of a single operator coupled to a bath have used the independent oscillator model of the bath. However, that model has no notion of separation, so the question of phenomena when the oscillators are near one another compared with when they are widely separated cannot be addressed. Here the Lamb model of an oscillator attached to a stretched string is generalized to illustrate some of these questions. The coupled Langevin equations for a pair of oscillators attached to the string at different points are derived and their limits for large and small separations obtained. Finally, as an illustration of a different phenomenon, the fluctuation force between a pair of masses attached to the string is calculated, with closed form expressions for the force at small and large separations.

We present a simple calculation of the Lorentz transformation of the spectral distribution of blackbody radiation at temperature T. Here we emphasize that T is the temperature in the blackbody rest frame and does not change. We thus avoid the confused and confusing question of how temperature transforms. We show by explicit calculation that at zero temperature the spectral distribution is invariant. At finite temperature we find the well known result familiar in discussions of the the 2.7! K cosmic radiation., Comment: 6 pages

We present an analysis of the dust and gas in Andromeda, using Herschel images sampling the entire far-infrared peak. We fit a modified-blackbody model to ~4000 quasi-independent pixels with spatial resolution of ~140pc and find that a variable dust-emissivity index (beta) is required to fit the data. We find no significant long-wavelength excess above this model suggesting there is no cold dust component. We show that the gas-to-dust ratio varies radially, increasing from ~20 in the center to ~70 in the star-forming ring at 10kpc, consistent with the metallicity gradient. In the 10kpc ring the average beta is ~1.9, in good agreement with values determined for the Milky Way (MW). However, in contrast to the MW, we find significant radial variations in beta, which increases from 1.9 at 10kpc to ~2.5 at a radius of 3.1kpc and then decreases to 1.7 in the center. The dust temperature is fairly constant in the 10kpc ring (ranging from 17-20K), but increases strongly in the bulge to ~30K. Within 3.1kpc we find the dust temperature is highly correlated with the 3.6 micron flux, suggesting the general stellar population in the bulge is the dominant source of dust heating there. At larger radii, there is a weak correlation between the star formation rate and dust temperature. We find no evidence for 'dark gas' in M31 in contrast to recent results for the MW. Finally, we obtained an estimate of the CO X-factor by minimising the dispersion in the gas-to-dust ratio, obtaining a value of (1.9+/-0.4)x10^20 cm^-2 [K kms^-1]^-1., Comment: 19 pages, 18 figures. Submitted to ApJ April 2012; Accepted July 2012

We have obtained Herschel images at five wavelengths from 100 to 500 micron of a ~5.5x2.5 degree area centred on the local galaxy M31 (Andromeda), our nearest neighbour spiral galaxy, as part of the Herschel guaranteed time project "HELGA". The main goals of HELGA are to study the characteristics of the extended dust emission, focusing on larger scales than studied in previous observations of Andromeda at an increased spatial resolution, and the obscured star formation. In this paper we present data reduction and Herschel maps, and provide a description of the far-infrared morphology, comparing it with features seen at other wavelengths. We use high--resolution maps of the atomic hydrogen, fully covering our fields, to identify dust emission features that can be associated to M31 with confidence, distinguishing them from emission coming from the foreground Galactic cirrus. Thanks to the very large extension of our maps we detect, for the first time at far-infrared wavelengths, three arc-like structures extending out to ~21, ~26 and ~31 kpc respectively, in the south-western part of M31. The presence of these features, hosting ~2.2e6 Msol of dust, is safely confirmed by their detection in HI maps. Overall, we estimate a total dust mass of ~5.8e7 Msol, about 78% of which is contained in the two main ring-like structures at 10 and 15 kpc, at an average temperature of 16.5 K. We find that the gas-to-dust ratio declines exponentially as a function of the galacto-centric distance, in agreement with the known metallicity gradient, with values ranging from 66 in the nucleus to ~275 in the outermost region. [Abridged], Comment: 15 Pages, 9 Figures. Accepted for publication in Astronomy and Astrophysics. A high resolution version of the paper can be found at http://wazn.ugent.be/jfritz/HelgaI_final.pdf

The Abraham-Lorentz-Dirac equation for a point electron, while suffering from runaway solutions and an acausal response to external forces, is compatible with the optical theorem. We show that a theory of radiative reaction that allows for a finite charge distribution is not only causal and free of runaway solutions, but is also consistent with the optical theorem and the standard formula for the Rayleigh scattering cross section., Comment: 4 pages, 2 figures

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $\rho > 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 \pi \rho \mathbb{Z}$ equipped with the metric $\g(r,\theta) = dr^2 + r^2 d\theta^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated., Comment: 24 pages, 1 figure. Submitted

Decoherence is well understood, in contrast to disentanglement. According to common lore, irreversible coupling to a dissipative environment is the mechanism for loss of entanglement. Here, we show that, on the contrary, disentanglement can in fact occur at large enough temperatures $T$ even for vanishingly small dissipation (as we have shown previously for decoherence). However, whereas the effect of $T$ on decoherence increases exponentially with time, the effect of $T$ on disentanglement is constant for all times, reflecting a fundamental difference between the two phenomena. Also, the possibility of disentanglement at a particular $T$ increases with decreasing initial entanglement., Comment: 3 pages

We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. We calculate \emph{exactly} the time development of this non-Gaussian state due to interaction with an \emph{arbitrary} heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.

We outline an exact approach to decoherence and entanglement problems for continuous variable systems. The method is based on a construction of quantum distribution functions introduced by Ford and Lewis \cite{ford86} in which a system in thermal equilibrium is placed in an initial state by a measurement and then sampled by subsequent measurements. With the Langevin equation describing quantum Brownian motion, this method has proved to be a powerful tool for discussing such problems. After reviewing our previous work on decoherence and our recent work on disentanglement, we apply the method to the problem of a pair of particles in a correlated Gaussian state. The initial state and its time development are explicitly exhibited. For a single relaxation time bath at zero temperature exact numerical results are given. The criterion of Duan et al. \cite{duan00} for such states is used to prove that the state is initially entangled and becomes separable after a finite time (entanglement sudden death).

Entanglement, which is an essential characteristic of quantum mechanics, is the key element in potential practical quantum information and quantum communication systems. However, there are many open and fundamental questions (relating to entanglement measures, sudden death, etc.) that require a deeper understanding. Thus, we are motivated to investigate a simple but non-trivial correlated two-body continuous variable system in the absence of a heat bath, which facilitates an \underline{exact} measure of the entanglement at all times. In particular, we find that the results obtained from all well-known existing entanglement measures agree with each other but that, in practice, some are more straightforward to use than others.

We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schr\"odinger equation on the Euclidean cone., Comment: 12 pages

We study the Schr\"odinger equation on a flat euclidean cone $\mathbb{R}_+ \times \mathbb{S}^1_\rho$ of cross-sectional radius $\rho > 0$, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the "geometric front", the part of the solution coming from formal application of the method of images, and the "diffractive front" emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr\"odinger propagator on this class of cones., Comment: 21 pages, 4 figures. Minor typos corrected. To be published in Comm. Math. Phys

The quantum analog of the joint probability distributions describing a classical stochastic process is introduced. A prescription is given for constructing the quantum distribution associated with a sequence of measurements. For the case of quantum Brownian motion this prescription is illustrated with a number of explicit examples. In particular it is shown how the prescription can be extended in the form of a general formula for the Wigner function of a Brownian particle entangled with a heat bath., Comment: Phys. Rev. A, in press

Small systems (of interest in the areas of nanophysics, quantum information, etc.) are particularly vulnerable to environmental effects. Thus, we determine various thermodynamic functions for an oscillator in an arbitrary heat bath at arbitrary temperatures. Explicit results are presented for the most commonly discussed heat bath models: Ohmic, single relaxation time and blackbody radiation., Comment: Phys. Rev. B, in press

Anomalous diffusion is discussed in the context of quantum Brownian motion with colored noise. It is shown that earlier results follow simply and directly from the fluctuation-dissipation theorem. The limits on the long-time dependence of anomalous diffusion are shown to be a consequence of the second law of thermodynamics. The special case of an electron interacting with the radiation field is discussed in detail. We apply our results to wave-packet spreading.

An apparent violation of the second law of thermodynamics occurs when an atom coupled to a zero-temperature bath, being necessarily in an excited state, is used to extract work from the bath. Here the fallacy is that it takes work to couple the atom to the bath and this work must exceed that obtained from the atom. For the example of an oscillator coupled to a bath described by the single relaxation time model, the mean oscillator energy and the minimum work required to couple the oscillator to the bath are both calculated explicitly and in closed form. It is shown that the minimum work always exceeds the mean oscillator energy, so there is no violation of the second law.

The free energy of a quantum oscillator in an arbitrary heat bath at a temperature T is given by a "remarkable formula" which involves only a single integral. This leads to a corresponding simple result for the entropy. The low temperature limit is examined in detail and we obtain explicit results both for the case of an Ohmic heat bath and a radiation heat bath. More general heat bath models are also examined. This enables us to determine the entropy at zero temperature in order to check the third law of thermodynamics in the quantum regime, Comment: International Conference on "Frontiers of Quantum and Mesoscopic Thermodynamics"

It is generally accepted that a system undergoing uniform acceleration with respect to zero-temperature vacuum will thermalize at a finite temperature (the so-called Unruh temperature) that is proportional to the acceleration. However, the question of whether or not the system actually radiates is highly controversial. Thus, we are motivated to present an exact calculation using a generalized quantum Langevin equation to describe an oscillator (the detector) moving under a constant force and coupled to a one-dimensional scalar field (scalar electrodynamics). Moreover, our analysis is simplified by using the oscillator as a detector. We show that this system does not radiate despite the fact that it does in fact thermalize at the Unruh temperature. We remark upon a differing opinion expressed regarding a system coupled to the electromagnetic field.

The low temperature solution of the exact master equation for an oscillator coupled to a linear passive heat bath is known to give rise to serious divergences. We now show that, even in the high temperature regime, problems also exist, notably the fact that the density matrix is not necessarily positive.

Motivated by recent experiments, we consider a Schr\"{o}dinger cat superposition of two widely separated coherent states in thermal equilibrium. The time development of our system is obtained using Wigner distribution functions. In contrast to our discussion for a two-Gaussian wave packet [Phys. Lett. A 286 (2001) 87], we find that, in the absence of dissipation, the interference term does not decay rapidly in time, but in common with the other two terms, it oscillates in time and persists for all times, Comment: Proc. of Wigner Centennial Conference

While agreeing with our exact expression for the time dependence of the motion of a free particle in an initial superposition state, corresponding to two identical Gaussians separated by a distance $d$, at temperature $T$, Gobert et al., in the preceding Comment [Phys. Rev. A xxx], dispute our conclusions on decoherence time scales. However, the parameters they used to generate their figures are outside the regime of validity of our interpretation of the results and, moreover, are not of physical interest in that they correspond to $T\approx 0$. The point is that in their figures they have chosen the thermal de Broglie wavelength $\lambda_{th}=\hbar \sqrt{mkT}$ to be equal to slit spacing \emph{d}, whereas we have clearly stated [in the paragraph preceding Eq. (21) of our paper] that decoherence occurs and that our expression for the decoherence time applies only in the limit where \emph{d} is large compared not only with the slit width $\sigma $ but also with the thermal de Broglie wavelength, $d\gg \lambda_{th},\sigma $., Comment: This is a Reply to quant-ph/0306019 "Decoherence without dissipation? " by Dominique Gobert, Jan von Delft, Vinay Ambegaokar The Fig. associated with our Reply appears in the PRA journal paper l

Dispersion relations for dipolar modes propagating along a chain of metal nanoparticles are calculated by solving the full Maxwell equations, including radiation damping. The nanoparticles are treated as point dipoles, which means the results are valid only for a/d <= 1/3, where a is the particle radius and d the spacing. The discrete modes for a finite chain are first calculated, then these are mapped onto the dispersion relations appropriate for the infinite chain. Computed results are given for a chain of 50-nm diameter Ag spheres spaced by 75 nm. We find large deviations from previous quasistatic results: Transverse modes interact strongly with the light line. Longitudinal modes develop a bandwidth more than twice as large, resulting in a group velocity that is more than doubled. All modes for which k_mode <= w/c show strongly enhanced decay due to radiation damping., Comment: 26 pages, 7 figures, 2 tables. to appear in Phys. Rev. B

Most discussions of decoherence in the literature consider the high-temperature regime but it is also known that, in the presence of dissipation, decoherence can occur even at zero temperature. Whereas most previous investigations all assumed initial decoupling of the quantum system and bath, we consider that the system and environment are entangled at all times. Here, we discuss decoherence for a free particle in an initial Schr\"{o}dinger cat state. Memory effects are incorporated by use of the single relaxation time model (since the oft-used Ohmic model does not give physically correct results).

In a letter to Nature (Ford G W and O'Connell R F 1996 Nature 380 113) we presented a formula for the derivative of the hyperbolic cotangent that differs from the standard one in the literature by an additional term proportional to the Dirac delta function. Since our letter was necessarily brief, shortly after its appearance we prepared a more extensive unpublished note giving a detailed explanation of our argument. Since this note has been referenced in a recent article (Estrada R and Fulling S A 2002 J. Phys. A: Math. Gen. 35 3079) we think it appropriate that it now appear in print. We have made no alteration to the original note.

The prototypical Schr\"{o}dinger cat state, i.e., an initial state corresponding to two widely separated Gaussian wave packets, is considered. The decoherence time is calculated solely within the framework of elementary quantum mechanics and equilibrium statistical mechanics. This is at variance with common lore that irreversible coupling to a dissipative environment is the mechanism of decoherence. Here, we show that, on the contrary, decoherence can in fact occur at high temperature even for vanishingly small dissipation.

The Hu-Paz-Zhang equation is a master equation for an oscillator coupled to a linear passive bath. It is exact within the assumption that the oscillator and bath are initially uncoupled . Here an exact general solution is obtained in the form of an expression for the Wigner function at time t in terms of the initial Wigner function. The result is applied to the motion of a Gaussian wave packet and to that of a pair of such wave packets. A serious divergence arising from the assumption of an initially uncoupled state is found to be due to the zero-point oscillations of the bath and not removed in a cutoff model. As a consequence, worthwhile results for the equation can only be obtained in the high temperature limit, where zero-point oscillations are neglected. In that limit closed form expressions for wave packet spreading and attenuation of coherence are obtained. These results agree within a numerical factor with those appearing in the literature, which apply for the case of a particle at zero temperature that is suddenly coupled to a bath at high temperature. On the other hand very different results are obtained for the physically consistent case in which the initial particle temperature is arranged to coincide with that of the bath.

Distribution functions defined in accord with the quantum theory of measurement are combined with results obtained from the quantum Langevin equation to discuss decoherence in quantum Brownian motion. Closed form expressions for wave packet spreading and the attenuation of coherence of a pair of wave packets are obtained. The results are exact within the context of linear passive dissipation. It is shown that, contrary to widely accepted current belief, decoherence can occur at high temperature in the absence of dissipation. Expressions for the decoherence time with and without dissipation are obtained that differ from those appearing in earlier discussions.

A localized free particle is represented by a wave packet and its motion is discussed in most quantum mechanics textbooks. Implicit in these discussions is the assumption of zero temperature. We discuss how the effects of finite temperature and squeezing can be incorporated in an elementary manner. The results show how the introduction of simple tools and ideas can bring the reader into contact with topics at the frontiers of research in quantum mechanics. We discuss the standard quantum limit, which is of interest in the measurement of small forces, and decoherence of a mixed (``Schrodinger cat'') state, which has implications for current research in quantum computation, entanglement, and the quantum-classical interface.

Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schr\"{o}dinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks., Comment: in press in Laser Phys.13(2003)