Photon Wave Function Research Articles

Multidimensional pump-probe spectroscopy with entangled twin-photon states.

We show that entangled photons may be used in coherent multidimensional nonlinear spectroscopy to provide information on matter by scanning photon wave function parameters (entanglement time and delay of twin photons), rather than frequencies and time delays, as is commonly done with classical pulses. Signals are expressed and interpreted intuitively in terms of products of matter and field correlation functions using a diagrammatic close time path loop formalism which reveals the entangled quantum pathways of the fields and matter. The pump-probe signal measured when the pump and the probe are in a twin entangled state shows two-photon resonant contributions which scale linearly rather than quadratically with the incident beam intensity and reveal frequencies of off-resonant transitions. Two-dimensional spectrograms obtained by double Fourier transform of the signal with respect to the entanglement time and delay of the twins could provide detailed information on correlations among states and dynamical processes with high temporal resolution. The analogy with multidimensional time-domain optical techniques which use sequences of short classical pulses and pulse shaping algorithms is pointed out.

Why photons cannot be sharply localized

Physicists have pondered over the problem of photon localization and the related problem of the photon wave function for almost 80 years now, beginning with the work of Landau and Peierls 1. An extensive review of the photon localization problem was recently presented by Keller 2. The problem of photon localization is closely related to the widely studied problem of the photon position operator. In this paper we introduce an operational definition of partial localization based on the measurements of correlation functions for electric or magnetic fields. For a want of better names, we shall use the terms electric localization and magnetic localization even though this might erroneously suggest the presence of some electric or magnetic devices that confine the photons. Since a sharp localization of photons according to our operational definition of localization is not possible, a photon position operator compatible with this definition does not exist. Earlier studies of the photon localization emphasized the mathematical aspects see, for example, 3‐5. In this paper we emphasize the physical properties of the electromagnetic field. In particular, we exhibit the role of the photon helicity and the symmetry between the electric and magnetic fields. We proceed in the footsteps of Glauber 6‐9 who was the first to recognize the significance of the space-dependent creation and annihilation operators. This approach was recently summarized and expanded in an extensive paper by Smith and Raymer 10. In our work we concentrate on the analysis of the photon localization in terms of the electric and magnetic field operators. We treat these fields after smearing over space-time regions as bona fide observables. We put emphasis on the field aspect that is complementary to the particle aspect. It might have a weaker connection with experiments usually based on photon counting as highlighted by Glauber, but it is more precise as was explained in detail by Bohr and Rosenfeld 11,12 The standard method of quantization of the free electromagnetic field based on the decomposition into monochromatic modes is not well suited for the discussion of localizability because the monochromatic mode functions are not localized. To overcome this problem we further developed an alternative method of quantization that does not require a mode decomposition. The essential mathematical tools in our analysis are the Riemann-Silberstein RS vector and the helicity operator. This formulation has some merits of its own, and it can also be used to study other general properties of

Deeply virtual Compton scattering via color dipoles: Nonperturbative effects

We study the deeply virtual Compton scattering amplitude within the color dipole approach. The light-cone wave function of a real photon is evaluated in the instanton vacuum model. Our parameter-free calculations are able to describe H1 data, both the absolute values and the $t$ dependences, at medium-high values of ${Q}^{2}$. The ${Q}^{2}$ dependence is found to be sensitive to the choice of the phenomenological cross section fitted to deep-inelastic scattering data.

Down-conversion of a single photon with unit efficiency

The down-conversion dynamics of a single photon in a single three-level system are investigated theoretically. The wave functions of the output photons are obtained analytically as functions of the input photon profile, and the down-conversion probability is evaluated from these output wave functions. It is demonstrated that down-conversion of a single photon with unit efficiency is possible and the necessary conditions are clarified.

Multiphoton scattering in a one-dimensional waveguide with resonant atoms

We study multiphoton scattering inside a one-dimensional waveguide caused by a number of resonant two-level atoms located in a region small compared to the photon wavelength. Applying powerful methods of the theory of integrable quantum systems, we analyze photon correlations caused by scattering. In particular, for the important case of two-photon scattering, we describe explicitly the dependence of the scattering parameters on the number of two-level atoms. Also, we outline the method of describing scattering of an arbitrary number of photons and present explicit expressions for the scattering of three photons in the effective chiral one-dimensional model. The photon wave function is represented as a sum of terms of a different degree of ``entanglement.'' We discuss qualitative distinctions in photon correlations for the single two-level atom case and the case of several atoms.

Photon wave function: A covariant formulation and equivalence with QED

We discuss the limits of the photon wave function (PWF) formalism, which is experiencing a revival these days as a result of new practical applications in photonics and quantum optics. We build a Dirac-like equation for the PWF written in a manifestly covariant form and show that, in the presence of charged matter fields, it reproduces the standard formulation of (classical) electrodynamics. This shows that attempts to construct a full quantum theory of interacting photons (mutually or interacting with matter) based on the so-called photon wave function approach can lead only to results already described by standard quantum electrodynamics (QED). The PWF formalism can then be used only to provide an easier description of some particular situations---for example, the propagation of free photons or photons propagating in a medium as described in Bialynicki-Birula [in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 1996), pp. 245--294] especially when the photon number remains fixed in time but not to replace QED in toto.

Electro-Optic Modulation of Single Photons

We use the Stokes photon of a biphoton pair to set the time origin for electro-optic modulation of the wave function of the anti-Stokes photon thereby allowing arbitrary phase and amplitude modulation. We demonstrate conditional single-photon wave functions composed of several pulses, or instead, having Gaussian or exponential shapes.

Lorentz-invariant photon number density

A Lorentz-invariant positive definite expression for photon number density is derived as the absolute square of the invariant scalar product of a polarization-sensitive position eigenvector and the photon wave function. It is found that this scalar product is independent of the form chosen for the wave function and that the normalized positive frequency vector potential--electric field pair is a convenient choice of wave function in the presence of matter. The number amplitude describing a localized state is a $\ensuremath{\delta}$ function at the instant at which localization and detection are seen as simultaneous.

A high speed, postprocessing free, quantum random number generator

A quantum random number generator (QRNG) based on gated single photon detection of an In–GaAs photodiode at gigahertz frequency is demonstrated. Owing to the extremely long coherence time of each photon, each photons’ wave function extends over many gating cycles of the photodiode. The collapse of the photon wave function on random gating cycles as well as photon random arrival time detection events are used to generate sequences of random bits at a rate of 4.01Mbit∕s. Importantly, the random outputs are intrinsically biasfree and require no postprocessing procedure to pass random number statistical tests, making this QRNG an extremely simple device.

Michelson-Morley Experiment Within the Quantum Mechanics Framework

It is revisited the Michelson-Morley experiment within the quantum mechanics framework. One can define the wave function of photon in the whole space at a given moment of time. The phase difference between the source and receiver is a distance between the source and receiver at the time of reception hence it does not depend on the velocity of the frame. Then one can explain the null result of the Michelson-Morley experiment within the quantum mechanics framework.

Quantum computation with Kerr-nonlinear photonic crystals

In this paper, we consider a method for implementing a quantum logic gate with photons whose wave function propagates in a one-dimensional Kerr-nonlinear photonic crystal. The photonic crystal causes the incident photons to undergo Bragg reflection by its periodic structure of dielectric materials and forms the photonic band structure, namely, the light dispersion relation. This dispersion relation reduces the group velocity of the wave function of the photons, so that it enhances nonlinear interaction of the photons. (Because variation of the group velocity against the wave vector is very steep, we have to tune up the wavelength of injected photons precisely, however.) If the photonic crystal includes layers of a Kerr medium, we can rotate the phase of the wave function of the incident photons by a large angle efficiently. We show that we can construct the nonlinear sign-shift (NS) gate proposed by Knill, Laflamme and Milburn (KLM) by this method. Thus, we can construct the conditional sign-flip gate for two qubits, which is crucial for quantum computation. Our NS gate works with probability unity in principle while KLM's original one is a nondeterministic gate conditioned on the detection of an auxiliary photon.

Diffractive excitation in DIS andppcollisions

We have in earlier papers presented an extension of Mueller's dipole cascade model, which includes subleading effects from energy conservation and running coupling as well as colour suppressed effects from pomeron loops via a ``dipole swing''. The model was applied to describe the total cross sections in pp and gamma*p collisions. In this paper we present a number of improvements of the model, in particular related to the confinement mechanism. A consistent treatment of dipole evolution and dipole--dipole interactions is achieved by replacing the infinite range Coulomb potential by a screened potential, which further improves the frame-independence of the model. We then apply the model to elastic scattering and diffractive excitation, where we specifically study the effects of different sources for fluctuations. In our formalism we can take into account contributions from all different sources, from the dipole cascade evolution, the dipole--dipole scattering, from the impact-parameter dependence, and from the initial photon and proton wavefunctions. Good agreement is obtained with data from the Tevatron and from HERA, and we also present some predictions for the LHC.

Photon wave functions, wave-packet quantization of light, and coherence theory

The monochromatic Dirac and polychromatic Titulaer–Glauber quantized field theories (QFTs) of electromagnetism are derived from a photon-energy wave function in much the same way that one derives QFT for electrons, i.e., by quantization of a single-particle wave function. The photon wave function and its equation of motion are established from the Einstein energy–momentum–mass relation, assuming a local energy density. This yields a theory of photon wave mechanics (PWM). The proper Lorentz-invariant single-photon scalar product is found to be non-local in coordinate space, and is shown to correspond to orthogonalization of the Titulaer–Glauber wave-packet modes. The wave functions of PWM and mode functions of QFT are shown to be equivalent, evolving via identical equations of motion, and completely describe photonic states. We generalize PWM to two or more photons, and show how to switch between the PWM and QFT viewpoints. The second-order coherence tensors of classical coherence theory and the two-photon wave functions are shown to propagate equivalently. We give examples of beam-like states, which can be used as photon wave functions in PWM, or modes in QFT. We propose a practical mode converter based on spectral filtering to convert between wave packets and their corresponding biorthogonal dual wave packets.

Roles of wave functions in the electroproduction of vector mesons

Elastic vector meson electroproduction is calculated through integrals of the overlap product of photon and vector meson wave functions, multiplied by the amplitude for the scattering of $q \bar{q}$ dipole pairs off the proton. In this nonperturbative QCD calculation, for sizes of the overlap functions that are smaller than the typical ranges of the interaction of the dipoles with the proton, the amplitudes factorize, with overlap strengths (integration extended over the light front coordinates describing the $q \bar{q}$ dipoles) containing all Q² dependence of the observables. This factorization is important in the description of the experimental data for all S-wave vector mesons.

Photon wave mechanics and position eigenvectors

One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.

Unified treatment of electroproduction for all vector mesons

We present results of a treatment for elastic photo- and electroproduction of ρ , ω , ϕ , J / ψ and ϒ vector mesons based on a framework in which the photon and vector meson wave functions and properties of the QCD vacuum field play prominent roles. The model, which has been applied and tested before, is used, without free parameters, in a successful description of the existing data and for predictions in extended kinematical ranges.

Novel Method for Solving the Quantum Nonlinear Dynamics of Photons: Use of a Classical Input

In order to investigate the nonlinear dynamics of photons, one must in principle solve a quantum many-particle problem, which usually requires intensive computation. In this study, we show that the spatial wave function of photons after nonlinear interaction can be obtained with less computation by assuming a classical input pulse and calculating a correlation function in the output field. This method is particularly useful when nonlinear optical media have many mechanical degrees of freedom, where quantum many-particle problems become extremely difficult.

Towards a nonperturbative foundation of the dipole picture: II. High energy limit

This is the second of two papers in which we study real and virtual photon–proton scattering in a nonperturbative framework. In the first paper we have identified the leading contributions to this process at high energies and have derived expressions for them which take into account the renormalisation of the photon–quark–antiquark vertex. In the present paper we investigate the approximations and assumptions that are necessary to obtain the dipole model of high energy scattering from the results derived in the first paper. We discuss the gauge invariance of different contributions to the scattering amplitude and point out some subtleties related to gauge invariance in the correct definition of a perturbative photon wave function. As a phenomenological consequence of the dipole picture we derive a bound on the ratio of the cross sections for longitudinally and transversely polarised photons. This bound is independent of any particular model for the dipole–proton cross section and allows one to test the validity of the assumptions leading to the dipole picture in particular at low photon virtualities. We conclude that the naive dipole model formula should be supplemented by two additional terms which can potentially become large at small photon virtualities.

Photon states from propagating complex electromagnetic fields

A wave function for single- and many-photon states is defined by associating photons with different momenta with different spectral and polarization components of the classical, generally complex, electromagnetic field that propagates in a definite direction. When each spectral component of the classical field is scaled to the square root of the photon energy, the appropriately normalized photon wave function acquires the desired interpretation of probability density amplitude, in contradistinction to the Riemann-Silberstein wave function that can be considered the amplitude of the photon probability energy density.

Quark-antiquark states and their radiative transitions in terms of the spectral integral equation: Charmonia

Earlier by the authors (Yad. Fiz. 70, 68 (2007)), the bƃ states were treated in the framework of the spectral integral equation, together with simultaneous calculations of radiative decays of the considered bottomonia. In the present paper, such a study is carried out for the charmonium % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaeiikaGIaem4yamMafm4yamMbaebacqGGPaqkaaa!3F33! $$(c\bar c)$$ states. We reconstruct the interaction in the c-c sector on the basis of the data for the charmonium levels with J PC = 0−+, 1−−, 0++, 1++, 2++, 1+− and radiative transitions ψ(2S) → γχ c0(1P), γχ c1(1P), γχ c2(1P), γχ c(1S) and χ c0(1P), χ c1(1P), χ c2(1P) → γJ/ψ. The c-c levels and their wave functions are calculated for the radial excitations with n ≤ 6. Also, we determine the c-c component of the photon wave function using the e + e −-annihilation data: e + e − → J/ψ(3097), ψ(3686), ψ(3770), ψ(4040), ψ(4160), ψ(4415) and perform the calculations of the partial widths of the two-photon decays for the n = 1 states η c0(1S), χ c0(1P), χ c2(1P) → γγ and n = 2 states η c0(2S) → γγ, χ c0(2P) → γγ. We discuss the status of the recently observed c-c states X(3872) and Y(3941): according to our results, the X(3872) can be either χ c1(2P) or η c2(1D), while Y(3941) is χ c2(2P).