Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum theory for two-photon-state generation by means of four-wave mixing in optical fiber

Jun Chen, Xiaoying Li, Prem Kumar

Published Oct 19, 2004• 5 citations
Publisher

Abstract

We present a quantum theory for generating two-photon states by means of four-wave mixing in optical fiber. We start with an interaction Hamiltonian that can correctly describe all nonlinear interactions among the four waves present in the fiber, namely, the frequency non-degenerate pumps, signal, and idler, including the terms responsible for self-phase modulation (SPM), cross-phase modulation (XPM), and four-photon scattering (FPS). The exact form of this Hamiltonian is obtained through comparison between the classical and quantum equations of motion. The two-photon state is then calculated by means of first-order perturbation theory. It turns out that only the FPM and the pump SPM terms contribute to the formation of the two-photon state. The entangled nature of this state is verified in a coincidence counting experiment. The results of the theoretical calculation agree well with experimental data.

DOI
10.1117/12.560665

Related papers

All-fiber photon-pair source for practical quantum communications

Prem Kumar, Marco Fiorentino B, Paul L. Voss +1 more

Q uantum entanglement refers to the nonclassical dependency of physically separated quantum systems. It is an essential resource that must be freely available for implementing many of the novel functions of quantum information processing, such as database searching, clock synchronization, teleportation, computing and cryptography.1 The efficient generation and transmission of quantum entanglement is therefore of prime importance. With the ubiquitous standard optical fiber serving as the transmission medium and the widespread availability of efficient active and passive fiber devices, technological synergy between the generation and propagation components of the overall quantum network can be achieved by deploying sources of entanglement that rely on the nonlinearity of the fiber itself. Such fiber-based sources of quantum entanglement would also have the advantage of modal purity over their crystal counterparts,2,3 which is very important for realizing complex networks that involve many quantum operations. In our quest to develop fiber-based devices for quantum information processing, we have recently demonstrated an allfiber source of quantum-correlated photon pairs.4 The operating physical mechanism in this source is nondegenerate fourphoton mixing, wherein two pump photons scatter through the Kerr nonlinearity of the fiber to create simultaneous signal and idler photons. Our experiment is conducted near the zero-dispersion wavelength of standard dispersion-shifted fiber, where such scattering is enhanced owing to phase matching of the photon wave functions. It must be pointed out here that in a conventional wavelength-divisionmultiplexed classical optical communication line, one strives to suppress the fourphoton or four-wave mixing process, which otherwise causes cross talk between the wavelength channels and sets limits on the total data capacity of the communication line. In our experiment, the four-photon mixing takes place in a nonlinear-fiber Sagnac interferometer (NFSI), which we previously used to generate quantum-correlated twin beams in the fiber.5 The pump is a mode-locked train of ~3-ps-long pulses. The pulsed operation serves two important purposes: the NFSI amplifier can be operated at low average powers and the production of the signal/ idler fluorescence photons is confined to well-defined temporal windows, allowing use of a gated detection scheme to increase the signal-to-noise ratio. To measure the nonclassical (i.e., quantum) correlations between the signal and the idler photons, one must effectively suppress the pump photons from reaching the detectors. Since a typical pump pulse contains ~108 photons and we are interested in detecting ~0.01 signal (idler) photons/pulse, a pump-to-signal (idler) rejection ratio in excess of 100 dB is required. We achieved this specification by sending the fluorescence photons through a free-space double-grating spectral filter that separates the signal and idler photons from each other and from the pump photons not rejected by the NFSI. To demonstrate the quantum nature of the four-photon scattering at the single photon level, we assembled a photonFigure 1.C oincidence rates as a function of the single-photon rates in two different cases: signalidler fluorescence produced by a pump pulse (diamonds) and signal-idler fluorescence produced by two consecutive pump pulses (triangles). T he line represents the calculated accidental counts. T he inset shows a plot of the detected idler photons as a function of the injected pump photons (hollow circles). A second-order polynomial is shown to fit the experimental data. T he contributions of the dark counts, linear scattering and quadratic scattering are plotted separately as well. counting apparatus for detecting the signal and idler photons in coincidence. This apparatus is based on commercial InGaAs avalanche photodiodes operating in the gated Geiger mode.4 In the inset in Fig. 1, we show the number of scattered photons, NS, versus the number of pump photons, NP, injected into the NFSI. We fit the experimental data with a second-order polynomial fit, including the number of dark counts during the gate interval, ND. The fit clearly shows that the quadratic scattering ( ∝NP) in the fiber can dominate over the residual linear scattering ( ∝NP) of the pump, owing to imperfect filtering. The main body of Fig. 1 shows the coincidence-counting results. The diamonds represent the rate of coincidence counts versus the rate of the signal (idler) photons generated during the same pump pulse. The triangles represent the measured coincidence rate versus the signalphoton count rate when the signal is delayed with respect to the idler by one pulse period. For two independent photon sources, each with a count rate RS <<1, the accidental coincidence rate RC is given by RC = RS 2, regardless of the photon statistics of the sources. This quadratic relation is plotted as the solid line in Fig. 1, which fits the delayed-coincidence data (triangles) very well. These measurements show that while the fluorescence photons produced by the adjacent pump pulses are independent, those coming from the same pump pulse show a strong correlation, which is a signature of their non-classical behavior. This work was supported by the Department of Defense Multidisciplinary University Research Initiative program administered by the Army Research Office under Grants DAAD19-00-1-0177 and DAAD19-00-1-0469, and by the U.S. Office of Naval Research under Grant N00014-91-J-1268.

•Jan 1, 2002•5 citations

Photon pair generation in birefringent optical fibers

Brian J. Smith, Pierre Mahou, Offir Cohen +2 more

We study both experimentally and theoretically the generation of photon pairs by spontaneous four-wave mixing (SFWM) in standard birefringent optical fibers. The ability to produce a range of two-photon spectral states, from highly correlated (entangled) to completely factorable, by means of cross-polarized birefringent phase matching, is explored. A simple model is developed to predict the spectral state of the photon pair which shows how this can be adjusted by choosing the appropriate pump bandwidth, fiber length and birefringence. Spontaneous Raman scattering is modeled to determine the tradeoff between SFWM and background Raman noise, and the predicted results are shown to agree with experimental data.

arXiv:0909.4319•Optics Express•Dec 9, 2009•156 citations

Four-wave mixing with non-degenerate pumps: steady states and squeezing in the presence of phase modulation

Gagik Yu. Kryuchkyan, Karen V. Kheruntsyan

We present a quantum theory of an optical four-wave mixing oscillator under pumping by two laser fields of different frequencies. Our analysis includes all the physical processes that are relevant in the parametric limit of interaction in a medium with cubic non-linearity: four-wave mixing, self- and cross-phase modulation. Dynamical aspects of stable above-threshold generation of an intracavity signal mode, the bistability and squeezing phenomena are studied.

Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Aug 1, 1995•6 citations

Two-photon-state generation via four-wave mixing in optical fibers

Jun Chen, Xiaoying Li, Prem Kumar

A quantum theory of two-photon-state generation via four-wave mixing in optical fibers is studied, with emphasis on the case where the pump is a classical, narrow (picosecond-duration) pulse. One of the experiments performed in our lab is discussed and analyzed. Numerical predictions from the theory are shown to be in good agreement with the experimental results.

Physical Review A•Sep 2, 2005•99 citations

Optical Amplification in a Nonlinear Fiber Interferometer

M.E. Marhic, S.-H. Hsia, J.-M. Jeong

Optical amplification in optical fibers by means of degenerate four-wave mixing (DFWM) is studied by classical equations. A balanced interferometer is used to mix and separate the signal and the pump. Exact solutions are obtained even with pump depletion. If the cross-phase modulation is small, the signal power gain varies as the square of the pump self-phase shift. This gain can be sizable for moderate fiber lengths and laser powers, before the onset of stimulated Raman scattering. We have experimentally verified these calculations with a Sagnac interferometer, comprising a 300 m long polarization-preserving fiber, emitting pulses with 100-200 W peak power and 100-200 nsec duration. We have measured a signal power gain of 22 (13.3 dB); this value correlated well with a pump self-phase shift of 3π/2 measured in a separate two-wavelength experiment. This system can be used for squeezing fluctuations of the signal quadrature which is not amplified. We have performed a classical derivation of squeezing, based upon a geometrical study of the transformation of the signal phasor. The results are in agreement with those of more elaborate quantum-mechanical derivations.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations

True Single-Photon Stimulated Four-Wave Mixing

Shuai Dong, Xin Yao, Wei Zhang +5 more

Single-photon stimulated four-wave mixing (StFWM) is an important nonlinear optical phenomenon at the single-photon level. In this process, signal and idler photons are stimulated by seeded single photons when two pump photons are annihilated. The generated photons are correlated with each other, and more importantly, one of them is in the same mode with the seeded photons. Hence, StFWM has great potential to realize complex functions of photonic quantum information processing, such as quantum state manipulation and quantum cloning. In this paper, we provide the first experimental demonstration of single-photon StFWM in a piece of optical fiber. By using heralded single photons as the seeded photons, a true single-photon StFWM process is realized at the telecom band. It is confirmed by time-resolved four-photon coincidence measurement and variation of four-photon coincidence counts under different seed–pump delays. According to the experiment results, the potential performance of a quantum cloning machine based on this process is estimated.

ACS Photonics•Mar 25, 2017•12 citations
We present a quantum theory for generating two-photon states by means of four-wave mixing in optical fiber. We start with an interaction Hamiltonian that can correctly describe all nonlinear interactions among the four waves present in the fiber, namely, the frequency non-degenerate pumps, signal, and idler, including the terms responsible for self-phase modulation (SPM), cross-phase modulation (XPM), and four-photon scattering (FPS). The exact form of this Hamiltonian is obtained through comparison between the classical and quantum equations of motion. The two-photon state is then calculated by means of first-order perturbation theory. It turns out that only the FPM and the pump SPM terms contribute to the formation of the two-photon state. The entangled nature of this state is verified in a coincidence counting experiment. The results of the theoretical calculation agree well with experimental data.
Q uantum entanglement refers to the nonclassical dependency of physically separated quantum systems. It is an essential resource that must be freely available for implementing many of the novel functions of quantum information processing, such as database searching, clock synchronization, teleportation, computing and cryptography.1 The efficient generation and transmission of quantum entanglement is therefore of prime importance. With the ubiquitous standard optical fiber serving as the transmission medium and the widespread availability of efficient active and passive fiber devices, technological synergy between the generation and propagation components of the overall quantum network can be achieved by deploying sources of entanglement that rely on the nonlinearity of the fiber itself. Such fiber-based sources of quantum entanglement would also have the advantage of modal purity over their crystal counterparts,2,3 which is very important for realizing complex networks that involve many quantum operations. In our quest to develop fiber-based devices for quantum information processing, we have recently demonstrated an allfiber source of quantum-correlated photon pairs.4 The operating physical mechanism in this source is nondegenerate fourphoton mixing, wherein two pump photons scatter through the Kerr nonlinearity of the fiber to create simultaneous signal and idler photons. Our experiment is conducted near the zero-dispersion wavelength of standard dispersion-shifted fiber, where such scattering is enhanced owing to phase matching of the photon wave functions. It must be pointed out here that in a conventional wavelength-divisionmultiplexed classical optical communication line, one strives to suppress the fourphoton or four-wave mixing process, which otherwise causes cross talk between the wavelength channels and sets limits on the total data capacity of the communication line. In our experiment, the four-photon mixing takes place in a nonlinear-fiber Sagnac interferometer (NFSI), which we previously used to generate quantum-correlated twin beams in the fiber.5 The pump is a mode-locked train of ~3-ps-long pulses. The pulsed operation serves two important purposes: the NFSI amplifier can be operated at low average powers and the production of the signal/ idler fluorescence photons is confined to well-defined temporal windows, allowing use of a gated detection scheme to increase the signal-to-noise ratio. To measure the nonclassical (i.e., quantum) correlations between the signal and the idler photons, one must effectively suppress the pump photons from reaching the detectors. Since a typical pump pulse contains ~108 photons and we are interested in detecting ~0.01 signal (idler) photons/pulse, a pump-to-signal (idler) rejection ratio in excess of 100 dB is required. We achieved this specification by sending the fluorescence photons through a free-space double-grating spectral filter that separates the signal and idler photons from each other and from the pump photons not rejected by the NFSI. To demonstrate the quantum nature of the four-photon scattering at the single photon level, we assembled a photonFigure 1.C oincidence rates as a function of the single-photon rates in two different cases: signalidler fluorescence produced by a pump pulse (diamonds) and signal-idler fluorescence produced by two consecutive pump pulses (triangles). T he line represents the calculated accidental counts. T he inset shows a plot of the detected idler photons as a function of the injected pump photons (hollow circles). A second-order polynomial is shown to fit the experimental data. T he contributions of the dark counts, linear scattering and quadratic scattering are plotted separately as well. counting apparatus for detecting the signal and idler photons in coincidence. This apparatus is based on commercial InGaAs avalanche photodiodes operating in the gated Geiger mode.4 In the inset in Fig. 1, we show the number of scattered photons, NS, versus the number of pump photons, NP, injected into the NFSI. We fit the experimental data with a second-order polynomial fit, including the number of dark counts during the gate interval, ND. The fit clearly shows that the quadratic scattering ( ∝NP) in the fiber can dominate over the residual linear scattering ( ∝NP) of the pump, owing to imperfect filtering. The main body of Fig. 1 shows the coincidence-counting results. The diamonds represent the rate of coincidence counts versus the rate of the signal (idler) photons generated during the same pump pulse. The triangles represent the measured coincidence rate versus the signalphoton count rate when the signal is delayed with respect to the idler by one pulse period. For two independent photon sources, each with a count rate RS <<1, the accidental coincidence rate RC is given by RC = RS 2, regardless of the photon statistics of the sources. This quadratic relation is plotted as the solid line in Fig. 1, which fits the delayed-coincidence data (triangles) very well. These measurements show that while the fluorescence photons produced by the adjacent pump pulses are independent, those coming from the same pump pulse show a strong correlation, which is a signature of their non-classical behavior. This work was supported by the Department of Defense Multidisciplinary University Research Initiative program administered by the Army Research Office under Grants DAAD19-00-1-0177 and DAAD19-00-1-0469, and by the U.S. Office of Naval Research under Grant N00014-91-J-1268.
We study both experimentally and theoretically the generation of photon pairs by spontaneous four-wave mixing (SFWM) in standard birefringent optical fibers. The ability to produce a range of two-photon spectral states, from highly correlated (entangled) to completely factorable, by means of cross-polarized birefringent phase matching, is explored. A simple model is developed to predict the spectral state of the photon pair which shows how this can be adjusted by choosing the appropriate pump bandwidth, fiber length and birefringence. Spontaneous Raman scattering is modeled to determine the tradeoff between SFWM and background Raman noise, and the predicted results are shown to agree with experimental data.
Optical amplification in optical fibers by means of degenerate four-wave mixing (DFWM) is studied by classical equations. A balanced interferometer is used to mix and separate the signal and the pump. Exact solutions are obtained even with pump depletion. If the cross-phase modulation is small, the signal power gain varies as the square of the pump self-phase shift. This gain can be sizable for moderate fiber lengths and laser powers, before the onset of stimulated Raman scattering. We have experimentally verified these calculations with a Sagnac interferometer, comprising a 300 m long polarization-preserving fiber, emitting pulses with 100-200 W peak power and 100-200 nsec duration. We have measured a signal power gain of 22 (13.3 dB); this value correlated well with a pump self-phase shift of 3π/2 measured in a separate two-wavelength experiment. This system can be used for squeezing fluctuations of the signal quadrature which is not amplified. We have performed a classical derivation of squeezing, based upon a geometrical study of the transformation of the signal phasor. The results are in agreement with those of more elaborate quantum-mechanical derivations.
Single-photon stimulated four-wave mixing (StFWM) is an important nonlinear optical phenomenon at the single-photon level. In this process, signal and idler photons are stimulated by seeded single photons when two pump photons are annihilated. The generated photons are correlated with each other, and more importantly, one of them is in the same mode with the seeded photons. Hence, StFWM has great potential to realize complex functions of photonic quantum information processing, such as quantum state manipulation and quantum cloning. In this paper, we provide the first experimental demonstration of single-photon StFWM in a piece of optical fiber. By using heralded single photons as the seeded photons, a true single-photon StFWM process is realized at the telecom band. It is confirmed by time-resolved four-photon coincidence measurement and variation of four-photon coincidence counts under different seed–pump delays. According to the experiment results, the potential performance of a quantum cloning machine based on this process is estimated.