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Journal of Experimental and Theoretical Physics Letters

Schmidt Modes and the Time-Dependent Wavefunction of a Broadband Biphoton Field

Pavel A. Prudkovskii

Published Nov 1, 2022• 0 citations

Abstract

The wavefunction of a broadband biphoton field arising in parametric down-conversion in an aperiodically polarized nonlinear crystal with a linear chirp under the action of a short pump pulse has been studied theoretically. The form of Schmidt modes has been determined in both the spectral and temporal representations. The dependence of the number of Schmidt modes on the duration of the pump pulse has been found and it has been shown that it is minimal at a quite short duration of the pump pulse of about 100 fs. The form of the wavefunction of biphotons in the time representation has been determined and it has been examined how the region of maximum correlation between the detection times of the down-converted signal and idler photons changes under the variation of the duration of the pump pulse. It has been shown that the maximum of the time correlation function of the signal and idler radiation intensities is significantly asymmetric. Its sharper edge in the case of sufficiently long pump pulses has a small width shorter than 30 fs and can be used to sy-nchronize events even without the compensation of the dispersion spreading of wave packets of the biphoton field.

DOI: 10.1134/s0021364022602354

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The wavefunction of a broadband biphoton field arising in parametric down-conversion in an aperiodically polarized nonlinear crystal with a linear chirp under the action of a short pump pulse has been studied theoretically. The form of Schmidt modes has been determined in both the spectral and temporal representations. The dependence of the number of Schmidt modes on the duration of the pump pulse has been found and it has been shown that it is minimal at a quite short duration of the pump pulse of about 100 fs. The form of the wavefunction of biphotons in the time representation has been determined and it has been examined how the region of maximum correlation between the detection times of the down-converted signal and idler photons changes under the variation of the duration of the pump pulse. It has been shown that the maximum of the time correlation function of the signal and idler radiation intensities is significantly asymmetric. Its sharper edge in the case of sufficiently long pump pulses has a small width shorter than 30 fs and can be used to sy-nchronize events even without the compensation of the dispersion spreading of wave packets of the biphoton field.

We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter

R

defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\ensuremath{\tau}$. The function $R(\ensuremath{\tau})$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high

({R}_{\mathrm{min}}=73)

. The Schmidt number

K

is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of

R

and

K

are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter

{R}_{t}

(defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number

K

.

Spectral and temporal coincidence and single-particle photon wave packets are described and their widths and durations are found. The degree of entanglement is characterized by the experimentally measurable parameter R defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration. This function is shown to have a minimum and even in the minimum, at rather natural conditions, the parameter R is found to be very high (R=73>>1). The Schmidt number K is found for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of R and K are found to be identical and numerical difference between them is shown to be not exceeding 20%. Two-time temporal wave function of a biphoton state is investigated in details, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter R is defined as the ratio of durations of the single-particle and coincidence signals, and the defined in such a way parameter R is shown to be very close to the Schmidt number K.

Abstract Properties of intense twin beams generated in parametric down-conversion in periodically poled LiNbO3 crystals and their chirped variants by intense pump fields are analyzed along the model of intense parametric down-conversion that allows for pump depletion and uses the dual spatio-spectral Schmidt modes. Spectral and spatial intensity auto- and cross-correlation functions are determined as they depend on the pump power. Temporal correlations in intense twin beams at the fs time scale are investigated using the Hong-Ou-Mandel interferometer and the process of sum-frequency generation.

We study the theory of linearly chirped biphoton wave-packets produced in two basic quasi-phase-matching configurations: chirped photonic-like crystals and aperiodically poled crystals. The novelty is that these structures are considered as definite assembles of nonlinear layers that leads to detailed description of spontaneous parametric down-conversion (SPDC) processes through the discrete Gauss sums. We demonstrate that biphoton spectra for chirped photonic crystals involving a small number of layers consist from definite well-resolved spectral lines. We also discuss the forming of broadband spectra of signal (idler) waves in SPDC for both configurations as number of layers increases as well as in dependence of chirping parameters .

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.