Feynman's Path Integral Theory Research Articles

Theoretical analysis of a polarized two-photon Michelson interferometer with broadband chaotic light

Two-photon interference of broadband chaotic light in a Michelson interferometer is theoretically and numerically studied with a two-photon-absorption detector when polarizations are taken into account. Under the frame of quantum optics, the theoretical analysis is based on two-photon interference and Feynman path integral theory. A two-photon coherence matrix is introduced to calculate the second-order interference pattern. Our research shows that polarization is another dimension, as well as time and space, to tune the second-order interference pattern in the two-photon interference process. Polarizations can act as switches to manipulate the interference process and may open the door to some new experimental schemes. The quantum optical vector model is also helpful to understand the physics of second-order interference when the light source is quantum.

Correlation interference demultiplexer controlled by phase transition for Pr3+/Eu3+: YPO4

We propose a model of optical demultiplexer based on second- and third-order temporal interference of intensity-noise correlation, which is based on the superposition principle in Feynman's path integral theory. The switch from two-mode bunching to anti-bunching is controlled by gate position for pure hexagonal-phase of Pr3+/Eu3+: YPO4 crystal. We report, robust side correlation peaks caused by the destructive interference between two spontaneous parametric four-wave mixing processes from mixed (hexagonal and tetragonal)-phase of Pr3+/Eu3+: YPO4. By adjusting power of laser beam, the transition from asymmetric correlation side peaks to symmetric correlation side peaks are investigated. The robust center peak comes from the interference between two-mode and three-mode bunching. The switching from single correlation peak to multi-peaks is controlled by laser power. The multi-sharp peaks are translated into multi-dips through gate position. Such results may be used for the design of the optical multiplexer and demultiplexer. Our experimental results provide a technique to achieve larger linewidth change rate about 93% and higher channel equalization ratio about 70%.

Fractionally charged anyons generated by topological path fusion of propagating electron in magnetic flux lattice

Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.

Philippine Sea observations of acoustic transmission phase and intensity fluctuations and comparisons to path-integral theory

From April 2010 to March 2011, low frequency acoustic transmissions were carried out on a 330-km radius pentagonal acoustic array with a sixth transceiver in the middle in the Philippine Sea. Herein, we describe the fluctuation statistics of the signals originating on the six transceivers and recorded on a water column spanning receiver array located slightly off center of the pentagon. The signals analyzed here are interpretable in terms of specific acoustic ray paths clearly separable in time. Acoustic field statistics treated include (1) variances of phase and intensity, (2) vertical coherence and intensity covariance, (3) glinting and fade out rates, and (4) intensity probability density functions. Several of these observables are compared to predictions from Feynman path integral theory assuming the Garrett-Munk internal wave model suitably modified using Philippine Sea oceanographic observations. Data and theory differ by at most a factor of 2 and reveal the wave propagation regimes of unsaturated, and partially saturated.

Observations of phase and intensity fluctuations for low-frequency, long-range transmissions in the Philippine Sea and comparisons to path-integral theory.

In the Philippine Sea, from April 2010 to March 2011, a 330-km radius pentagonal acoustic transceiver array with a sixth transceiver in the center transmitted broadband signals with center frequencies between 172 and 275 Hz and 100 Hz bandwidth eight times a day every other day. The signals were recorded on a large-aperture vertical-line array located near the center of the pentagon at ranges of 129, 210, 224, 379, 396, and 450 km. The acoustic arrival structures are interpretable in terms of ray paths. Depth and time variability of the acoustic observations are analyzed for six ray paths (one from each transceiver) with similar vertical sampling properties in the main thermocline. Acoustic-field statistics treated include: (1) variances of phase and intensity, (2) vertical coherence and intensity covariance, (3) glinting and fadeout rates, and (4) intensity probability density functions. Several observed statistics are compared to predictions using Feynman path-integral theory assuming the Garrett-Munk internal-wave spectrum. In situ oceanographic observations support this assumption and are used to estimate spectral parameters. Data and theory differ at most by a factor of two and reveal the wave propagation regimes of unsaturated, partially saturated, and fully saturated. Improvements to the evaluation of path-integral quantities are discussed.

Measuring Hanbury Brown and Twiss Effect of Multi-Spatial-Mode Thermal Light at Ultrashort Timescale by Two-Photon Absorption

Hanbury Brown and Twiss effect of multi-spatial-mode thermal light is obtained at ultrashort timescale by two-photon-absorption (TPA) in a semiconductor photon-counting module. A characteristic spectrum, due to the interference in multi-mode fiber, is observed directly in TPA detector for the first time. To analyze the output of such measurement, a theoretical model based on the Feynman's path-integral theory is built, which makes the contribution of related item and the corresponding spectral component match very well. In addition, by introducing an all-optical method into the current HBT apparatus, one can obtain the concerned second-order coherence spectrum directly without any postprocessing. This result may be helpful to develop ultrafast detection for the intensity interferometer with natural light. Meanwhile, the observation of HBT effect of multi-mode interference offers a new detection method for the optical fiber sensors.

Observations and Feynman path integral analysis of acoustic fluctuations from the 2010-2011 Philippine Sea experiment

Six acoustic sources arranged in a pentagon with a central point transmitted broadband signals in the frequency range 200–300 Hz to a water column spanning vertical array in the Philippine Sea from Spring 2010 to Spring 2011. The propagation ranges were 128, 210, 224, 379, 396, and 449 km. Observations of phase and intensity variance as well as vertical coherence were obtained for several ray identified wavefronts. These observations are compared to Feynman path integral theory predictions utilizing Garrett-Munk internal wave spectral fits to in-situ measurements of temperature, salinity, and current. The observations and theory suggest that the three shortest ranges are in the unsaturated regime while the three longer ranges are in the partially saturated regime. Vertical coherence lengths are well predicted by theory. Implications for future experiments designed to observe internal wave spectra acoustically will be discussed. Six acoustic sources arranged in a pentagon with a central point transmitted broadband signals in the frequency range 200–300 Hz to a water column spanning vertical array in the Philippine Sea from Spring 2010 to Spring 2011. The propagation ranges were 128, 210, 224, 379, 396, and 449 km. Observations of phase and intensity variance as well as vertical coherence were obtained for several ray identified wavefronts. These observations are compared to Feynman path integral theory predictions utilizing Garrett-Munk internal wave spectral fits to in-situ measurements of temperature, salinity, and current. The observations and theory suggest that the three shortest ranges are in the unsaturated regime while the three longer ranges are in the partially saturated regime. Vertical coherence lengths are well predicted by theory. Implications for future experiments designed to observe internal wave spectra acoustically will be discussed.

Studying fermionic ghost imaging with independent photons.

Ghost imaging with thermal fermions is calculated via two-particle interference based on the superposition principle for different alternatives in Feynman's path integral theory. It is found that ghost imaging with fully polarized thermal fermions can be simulated by ghost imaging with fully polarized thermal bosons and classical particles. Photons in pseudothermal light are employed to experimentally study fermionic ghost imaging. Ghost imaging with thermal bosons and fermions is discussed based on the point-to-point (spot) correlation between the object and image planes. The employed method offers an efficient guidance for future ghost imaging with real thermal fermions, which may also be generalized to study other second-order interference phenomena with fermions.

Studying the optical second-order interference pattern formation process with classical light in the photon counting regime.

The formation process of the second-order interference pattern is studied experimentally in the photon counting regime by superposing two independent single-mode continuous-wave lasers. Two-photon interference based on the superposition principle in Feynman's path integral theory is employed to interpret the experimental results. The second-order interference pattern of classical light can be formulated when, with high probability, there are only two photons in the interferometer at one time. The studies are helpful in understanding the second-order interference of classical light in the language of photons. The method and conclusions can be generalized to the third- and higher-order interference of light and interference of massive particles.

Two-photon interference with non-identical photons

Two-photon interference with non-identical photons is studied based on the superposition principle in Feynman's path integral theory. The second-order temporal interference pattern is observed by superposing laser and pseudothermal light beams with different spectra. The reason why there is two-photon interference for photons of different spectra is that non-identical photons can be indistinguishable for the detection system when Heisenberg's uncertainty principle is taken into account. These studies are helpful to understand the second-order interference of light in the language of photons.

The first- and second-order temporal interference between thermal and laser light.

The first- and second-order temporal interference between two independent thermal and laser light beams is discussed by employing the superposition principle in Feynman's path integral theory. It is concluded that the first-order temporal interference pattern can not be observed by superposing two independent thermal and laser light beams, while the second-order temporal interference pattern can be observed in the same condition. These predictions are experimentally verified by employing pseudothermal light to simulate thermal light. The relationship between the indistinguishability of alternatives and photons is analyzed. The conclusions are helpful to understand the interference of different kinds of light and the difference between the coherence properties of thermal and laser light.

The second-order interference of two independent single-mode He–Ne lasers

The second-order spatial and temporal interference patterns with two independent single-mode continuous-wave He–Ne lasers are observed when these two lasers are incident to two adjacent input ports of a 1:1 non-polarizing beam splitter, respectively. Two-photon interference based on the superposition principle in Feynman's path integral theory is employed to interpret the experimental results. The conditions to observe the second-order interference pattern with two independent single-mode continuous-wave lasers are discussed. It is concluded that frequency stability is important to observe the second-order interference pattern with two independent light beams.

A NEW CLASS OF AMPLITUDE FUNCTIONS FOR THE STATIONARY PHASE METHOD ON AN ABSTRACT WIENER SPACE

A stochastic oscillatory integral is a probabilistic counterpart to a Feynman path integral, and the stationary phase method for stochastic oscillatory integrals relates to the semi-classical limits in the Feynman path integral theory. A complete stationary phase method for stochastic oscillatory integrals has not been achieved yet. In this paper, a new class of amplitude functions of stochastic oscillatory integrals is introduced, and it is shown that the stationary phase method works for the class.

Few-body semiclassical approach to nucleon transfer and emission reactions

A three-body semiclassical model is proposed to describe the nucleon transfer and emission reactions in a heavy-ion collision. In this model the two heavy particles, i.e. nuclear cores A$_1(Z_{A_1}, M_{A_1})$ and A$_2(Z_{A_2}, M_{A_2})$, move along classical trajectories $\vec R_1(t)$ and $\vec R_2(t)$ respectively, while the dynamics of the lighter neutron, n, is considered from a quantum mechanical point of view. Here, $M_i$ are the nucleon masses and $Z_i$ are the Coulomb charges of the heavy nuclei ($i=1,2$). A Faddeev-type semiclassical formulation using realistic paired nuclear-nuclear potentials is applied so that all three channels (elastic, rearrangement and break-up) are described in an unified manner. In order to solve these time-dependent equations the Faddeev components of the total three-body wave-function are expanded in terms of the input and output channel target eigenfunctions. In the special case when the nuclear cores are identical (A$_1 \equiv$ A$_2$) and the two-level approximation in the expansion over target functions the time-dependent semiclassical Faddeev equations are resolved in an explicit way. To determine the realistic $\vec R_1(t)$ and $\vec R_2(t)$ trajectories of the nuclear cores a self-consistent approach based on the Feynman path integral theory is applied.

Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer

The spatial second-order interference of two independent pseudothermal light beams in a Hong-Ou-Mandel interferometer is studied experimentally and theoretically. The similar cosine modulation in the second-order coherence function as the one with entangled-photon pairs in a Hong-Ou-Mandel interferometer is observed. Two-photon interference based on Feynman's path integral theory is employed to interpret the results. The experimental results and theoretical simulations agree with each other very well.

Bose–Einstein condensate in a double-well potential: Feynman path integral variational approach

A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.

Observation on the incompatibility between the first-order and second-order interferences with laser beams

We observed experimentally that the first-order and second-order interference patterns of laser beams in a modified Michelson interferometer are complementary and the conditions for seeing one preclude the possibility to see the other. One- and two-photon interferences based on Feynman's path integral theory are employed to interpret the experimental results.

Unified interpretation for second-order subwavelength interference based on Feynman’s path-integral theory

The second-order spatial subwavelength interference pattern is observed in a modified Michelson interferometer with single-mode continuous-wave laser beams. By analyzing our subwavelength interference experiment based on Feynman's path integral theory, a unified interpretation for all the second-order subwavelength interference is suggested.

Bose–Einstein condensation of atomic hydrogen Feynman path integral approach

We study the ground state properties of Bose–Einstein condensation of atomic hydrogen in the Ioffe–Pritchard magnetic trap using many-body Feynman path integral theory, which leads to the calculation of the ground state energy and the wavefunction. We also calculate the size, peak condensate density and the value of the ground state energy, which are in good agreement with the experimental results obtained by laser spectroscopy of the 1S–2S transition.

Resonance patterns of an antidot cluster: From classical to quantum ballistics

We explain the experimentally observed Aharonov-Bohm (AB) resonance patterns of an antidot cluster by means of quantum and classical simulations and Feynman path integral theory. We demonstrate that the observed behavior of the AB period signals the crossover from a low B regime which can be understood in terms of electrons following classical orbits to an inherently quantum high B regime where this classical picture and semiclassical theories based on it do not apply.