Biphoton Wave Function Research Articles

Quality of spatial entanglement propagation

We explore, both experimentally and theoretically, the propagation dynamics of spatially entangled photon pairs (biphotons). Characterization of entanglement is done via the Schmidt number, which is a universal measurement of the degree of entanglement directly related to the non-separability of the state into its subsystems. We develop expressions for the terms of the Schmidt number that depend on the amplitude and phase of the commonly used double-Gaussian approximation for the biphoton wave function, and demonstrate migration of entanglement between amplitude and phase upon propagation. We then extend this analysis to incorporate both phase curvature in the pump beam and higher spatial frequency content of more realistic non-Gaussian wave functions. Specifically, we generalize the classical beam quality parameter M2 to the biphotons, allowing the description of more information-rich beams and more complex dynamics. Agreement is found with experimental measurements using direct imaging and Fourier optics.

Parametric fluorescence in a sequence of resonators: An analogy with Dicke superradiance

We investigate spontaneous four wave mixing (SFWM) in a single-channel side-coupled integrated spaced sequence of resonators (SCISSOR). Analytic expressions for the number of photon pairs generated, as well as the biphoton wave function (joint spectral amplitude) describing the pairs, are derived and numerically computed for different pump pulse durations and numbers of ring resonators. In the limit of a long input pump pulse, we show a strong analogy between super-linear scaling of generation efficiency with respect to the number of rings in the structure and Dicke superradiance. More generally, we discuss in detail the factors that influence the shape of the biphoton wave function, as well as the conditions for observing super-SFWM.

Implementation of quantum and classical discrete fractional Fourier transforms.

Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.

Numerical solution of Schrödinger equation for biphoton wave function in twisted waveguide arrays

We consider the generation of entangled biphoton states with orbital-angular-momentum in triangular quadratic waveguide arrays with twisted geometry. For this purpose, we derive the Shrodinger equation for biphoton wave function and equation for pump field profile. We describe numerically the process of biphoton generation through spontaneous four-wave mixing. We suggest that biphoton correlations can be controlled by the amount of twist and pump field profile.

Quantum walks of photon pairs in twisted waveguide arrays

We consider an array of closely spaced optical waveguides, which are twisted around a central axis along the propagation direction. We derive Schrodinger-type equation of the biphoton wavefunction, taking into account the waveguide bending through the appearance of additional phase in the coupling coefficients. We present an example of the evolution of quantum photon-pair state.

Interferometric measurement of the biphoton wave function.

Interference between an unknown two-photon state (a "biphoton") and the two-photon component of a reference state gives a phase-sensitive arrival-time distribution containing full information about the biphoton temporal wave function. Using a coherent state as a reference, we observe this interference and reconstruct the wave function of single-mode biphotons from a low-intensity narrow band squeezed vacuum state.

Effect of loss on photon-pair generation in nonlinear waveguide arrays

We describe theoretically the process of spontaneous parametric down-conversion in quadratic nonlinear waveguide arrays in the presence of linear loss. We derive a set of discrete Schrodinger-type equations for the biphoton wave function, and the wave function of one photon when the other photon in a pair is lost. We demonstrate effects arising from loss-affected interference between the generated photon pairs and show that nonlinear waveguide arrays can serve as a robust loss-tolerant integrated platform for the generation of entangled photon states with non-classical spatial correlations.

Experimental observation of double-peak structure of coincidence spectra in ultrafast spontaneous parametric down-conversion

The requirement for symmetry of the biphoton wave function results in a double-peak structure of the coincidence spectra of spontaneous parametric down-conversion (SPDC) [Fedorov, Mikhailova, and Volkov, Phys. Scr. 85, 058105 (2012)]. In this work we report the experimental observation of this effect by carefully tailoring parameters of the SPDC crystal, the ultrafast pump laser, and the measurement setup. The results are shown to be relevant to quantum state engineering of ultrafast polarization or frequency-entangled mixed states.

Entangled photon generation using four-wave mixing in azimuthally symmetric microresonators

A novel quantum mechanical formulation of the bi-photon wavefunction and spectra resulting from four-wave mixing is developed for azimuthally symmetric systems. Numerical calculations are performed verifying the use of the angular group velocity and angular group velocity dispersion in such systems, as opposed their commonly used linear counterparts. The dispersion profile and bi-photon spectra of two illustrative examples are given, emphasizing the physical origin of the effects leading to the conditions for angular momentum and energy conservation. A scheme is proposed in which widely spaced narrowband entangled photons may be produced through a four-wave mixing process in a chip-scale ring resonator. The entangled photon pairs are found to conserve energy and momentum in the four-wave mixing interaction, even though both photon modes lie in spectral regions of steep angular group velocity dispersion.

Qutrits and ququarts in spontaneous parametric down-conversion, correlations and entanglement

We investigate features of biphoton qutrits and ququarts with the symmetry of biphoton wavefunctions fully taken into account. Entanglement quantifiers of such states are found as functions of qutrit and ququart parameters. For biphoton qutrits, a general relation between degrees of entanglement and polarization is established. In the case of biphoton ququarts, photon frequencies are considered as variables independent of polarization variables, with both polarization and frequency degrees of freedom equally contributing to the total ququart entanglement. Averaging over one of these two degrees of freedom is shown to reduce pure ququart states to mixed two-qubit states. The degree of polarization and degrees of correlations in mixed polarization states are investigated.

Spontaneous four-wave mixing in microring resonators

We consider spontaneous four-wave mixing in a microring resonator, presenting photon-pair generation rates and biphoton wave functions. We show how generation rates can be simply predicted from the performance of the device in the classical regime and that a wide variety of biphoton wave functions can be achieved by varying the pump pulse duration.

Generation and Compression of Chirped Biphotons

We describe an experiment demonstrating the radarlike technique of "chirp and compress." Chirped biphotons are generated using a quasi-phase-matched nonlinear crystal where the phase-matched frequency varies linearly with position. Sum frequency generation is used to measure the amplitude of the biphoton wave function. As compared to a nonchirped crystal, compression and an increase in summing efficiency of a factor of 5 is observed.

Gaussian modelling and Schmidt modes of SPDC biphoton states

A double-Gaussian model and the Schmidt modes are found for the biphoton wavefunction characterizing spontaneous parametric down-conversion with the degenerate collinear phase-matching of type I and with a pulsed pump. The obtained results are valid for all durations of the pump pulses: short, long and intermediately long.

Proposal for in-fiber generation of telecom-band polarization-entangled photon pairs using a periodically poled fiber

We treat spontaneous parametric downconversion in a periodically poled fiber, quasi-phase matched to allow for the generation of photon pairs at wavelengths within the low-loss telecommunications window. For an appropriate pump polarization, the unusual properties of such a fiber's effective chi(2) result in a biphoton wave function that is symmetric upon simultaneous exchange of downconverted photon frequencies and polarizations and that is nonzero over a wide range of downconverted frequencies. This could lead to a significant technical simplification of sources of in-fiber telecom-band polarization-entangled photons.

BIPHOTON WAVE PACKETS GENERATED IN APERIODICALLY POLED CRYSTALS

The influence of disordering of quasi-phase-matched structures on the properties of biphoton fields generated by spontaneous parametric down-conversion in periodically poled crystals is studied in the general form, taking into consideration arbitrary one-dimensional spatial distribution of the crystal second-order susceptibility. It is shown that the spectral amplitudes of the biphoton wave function become complex and gain frequency- and angle-dependent phase shifts in the structures with unequal lengths of domains. Effects of the spectrum broadening and the phase shifting are expressed in terms of the crystal non-linear transfer function. Spectrum of the first-order correlation function is found to be strictly dependent on the transfer function module, but insensitive to the phase shifts. Availability of the phase shifts makes the second-order temporal correlation function of the broadened as-generated biphoton field to be almost the same as in case of a single-domain crystal. At the same time, possibility of strong temporal biphoton compression appears further if an additional phase-compensating element is placed after the aperiodically poled crystal.

Nonlocal modulation of entangled photons

We consider ramifications of the use of high-speed light modulators to questions of correlation and measurement of time-energy entangled photons. Using phase modulators, we find that temporal modulation of one photon of an entangled pair, as measured by correlation in the frequency domain, may be negated or enhanced by modulation of the second photon. Using amplitude modulators, we describe a Fourier technique for measurement of biphoton wave functions with slow detectors.

Theory of biphoton generation in a single-resonant optical parametric oscillator far below threshold

We present a quantum-theoretical treatment of biphoton generation in single-resonant type-II parametric down-conversion. The nonlinear medium is continuously pumped and is placed inside a cavity which is resonant for the signal field, but nonresonant for the idler deflected by an intracavity polarizing beam splitter. The intensity of the classical pump is assumed to be sufficiently low in order to yield a biphoton production rate that is small compared to the cavity loss rate. Explicit expressions are derived for the rate of biphoton generation and for the biphoton wave function. The output spectra of the signal and idler field are determined, as well as the second-order signal-idler cross-correlation function which is shown to be asymmetric with respect to the time delay. Due to frequency entanglement in the signal-idler photon pair, the idler spectrum is found to reveal the longitudinal mode structure of the cavity, even though the idler field is not resonant.