In this paper we analyze some important characteristics of the spatial Wigner function of entangled photon pairs. We show that the Wigner description of quantum states that live in the infinite-dimensional space-momentum degree of freedom proves to be particularly useful. We propose an experimental configuration that can be used to retrieve the Wigner function of paired photons entangled in the spatial degree of freedom. In particular, it allows the full characterization of the paired photons emitted from a spontaneous parametric down-conversion source.
Spontaneous emission of a photon by an atom is described theoretically in three dimensions with the initial wave function of a finite-mass atom taken in the form of a finite-size wave packet. Recoil and wave-packet spreading are taken into account. The total atom-photon wave function is found in the momentum and coordinate representations as the solution of an initial-value problem. The atom-photon entanglement arising in such a process is shown to be closely related to the structure of atom and photon wave packets which can be measured in the coincidence and single-particle schemes of measurements. Two predicted effects, arising under the conditions of high entanglement, are anomalous narrowing of the coincidence wave packets and, under different conditions, anomalous broadening of the single-particle wave packets. Fundamental symmetry relations between the photon and atom single-particle and coincidence wave-packet widths are established. The relationship with the famous scenario of Einstein-Podolsky-Rosen is discussed.
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
Einstein–Podolsky–Rosen (EPR) correlations produced for two-photon entangled wave packets are analyzed by use of optical Wigner functions. In particular, type-II optical parametric downconversion systems are used, and the effects of the length of the crystal and the filter bandwidth on the Wigner functions are studied. The relation between the optical Wigner functions and the interference experiment with a beam splitter is studied. We find that, whereas in first-order correlation experiments (typified, for example, by Young’s interference experiment) the “photon interferes with itself” [ P. A. M. Dirac , The Principles of Quantum Mechanics (Clarendon, Oxford, UK, 1958), p. 9], in the EPR correlations the two-photon amplitude function interferes with itself. Such interference is analyzed by use of the optical Wigner functions.
A thorough unified treatment is given of the quantum-mechanical operators and wave functions for a molecular system (composed of
N moving charged particles) in static uniform electric and magnetic fields
E and
B. The treatment is rigorous within the nonrelativistic approximation. The system may either be neutral or charged. The fields may have arbitrary intensities and orientations. Close correspondence is maintained between the classical and quantum-mechanical treatments. The wave functions are expressed both in the time-independent energy representation and in time-dependent wave packets. Three types of momentum play important roles. For single-particle systems they are the canonical momentum
P, the mechanical momentum $\mathit{\ensuremath{\Pi}}=\mathit{P}\ensuremath{-}(\frac{e}{c})\mathit{A}=M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, and the pseudomomentum $\mathcal{K}=\mathit{\ensuremath{\Pi}}\ensuremath{-}(\frac{e}{c})\mathit{R}\ifmmode\times\else\texttimes\fi{}\mathit{B}\ensuremath{-}e\mathit{E}t$. The pseudomomentum is a constant of the motion. Except in the absence of magnetic fields, not all of the components of either $\mathit{\ensuremath{\Pi}}$ or
K commute. This complicates the quantum-mechanical formalism. The components of the pseudomomentum have simple classical interpretations, and in quantum mechanics they are related to the operator which performs a boost to a reference frame moving with constant velocity
v. In this moving frame, the electric field intensity is ${\mathit{E}}^{\ensuremath{'}}=\mathit{E}+(\frac{\mathit{v}}{c})\ifmmode\times\else\texttimes\fi{}\mathit{B}$. Thus, in a frame moving with the drift velocity ${\mathit{v}}_{d}=(\frac{c}{{B}^{2}})\mathit{E}\ifmmode\times\else\texttimes\fi{}\mathit{B}$, the components of the electric field intensity perpendicular to
B vanish. In this paper, the velocity boost operators are used to show the relationships between wave functions expressed in reference frames moving with repsect to each other. The dynamics of the
N-particle systems are simplified by making the Power-Zienau-Woolley transformation (which reduces as a special case to a unitary transformation used by Lamb) and by using center-of-mass and internal coordinates. Generalizing previous works, it is shown how the pseudomomentum is involved in separating these degrees of freedom for
N particles in both
E and
B. For neutral molecules, the Schr\"odinger equation is "pseudoseparated" and the internal degrees of freedom are coupled to the center of mass motion only by the "motional Stark Effect," which involves the constant of motion
K. For ionic systems, only one component of the center of mass is coupled to the internal motion. Quantitative estimates of the weak center-of-mass coupling are made for both neutral and ionic
n=1 and
n=2 quantum states of two-body systems by perturbation expansions in powers of the field strengths. In the usual nonrigorous treatments of systems in magnetic fields, no distinction is made between $\mathit{\ensuremath{\Pi}}$ and
K and both are approximated by $M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, where
M is the mass of the molecule and
R is taken to be the classical orbital position of a single particle having the same mass and charge. This is an excellent approximation for ground-state molecules and ions in weak fields. However, the resulting errors can be much larger if the system is in either intense fields or high-lying states (e.g., Rydberg levels). These are the conditions under which the rigorous formalism should be most useful. The determination of the wave functions and energy levels for many-particle ionic systems is complicated due to the coupling with the center of mass. The suggestion is made that it might be useful to boost the reference frame so that it moves with the velocity $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$ of a single classical particle. The Schr\"odinger equation in this frame could serve as a more suitable basis for perturbative or variational treatments.