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

arXiv:1910.12124 · Physical Review A

Phase effects in coherently stimulated down-conversion with a quantized pump field

Richard J. Birrittella, Paul M. Alsing, Christopher C. Gerry

Published Jan 14, 2020• 7 citations
arXivHTMLPDF

Abstract

We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}=i\ensuremath{\hbar}\ensuremath{\kappa}\left(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}c\right)$, wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by nondegenerate coherently stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two-mode state have been shown to depend greatly on the cumulative phase $\mathrm{\ensuremath{\Phi}}={\ensuremath{\theta}}_{s}+{\ensuremath{\theta}}_{i}\ensuremath{-}2\ensuremath{\phi}$ where ${\ensuremath{\theta}}_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2\ensuremath{\phi}$ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the near-thermal-like nature of the steady state valid for long times. Finally, we examine the effect of the cumulative phase $\mathrm{\ensuremath{\Phi}}$ on the correlations between the bipartite system of pump and signal/idler modes, and the effect of the quantized pump on the correlations between signal and idler modes.

DOI
10.1103/physreva.101.013813

Related papers

Coherently stimulated parametric down-conversion, phase effects, and quantum-optical interferometry

Richard J. Birrittella, Anna Gura, Christopher C. Gerry

We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.

Physical Review A•May 5, 2015•14 citations

Effect of an incoherent pump on two-mode entanglement in optical parametric generation

Stephen Vintskevich, D. A. Grigoriev, Sergey N. Filippov

Pumping a nonlinear crystal by an intense radiation results in the optical parametric generation of photons in two modes (the signal and the idler). The quantized electromagnetic field in these modes is described by a continuous-variable quantum state, which is entangled if the pump is a coherent state produced by a laser. The signal and the idler modes remain populated by photons even if the pump becomes incoherent (dephased by a medium, superposed with a thermal state, or produced by an alternative source such as the superluminescent diode). However, the incoherent pump does effect the entanglement and purity of the signal and the idler modes, which is of vital importance for quantum information applications and interferometry. Here we develop an approach to infer the signal-idler entanglement and purity for a general quantum incoherent pump with the given Glauber-Sudarshan function. We show that the signal-idler entanglement is extremely sensitive to the phase distribution of the pump and illustrate our findings by physically relevant examples of the incoherent pump: the noisy coherent state, slightly dephased and phase-averaged coherent states, the thermal state, and states modulated by the Kerr medium. The effect of an incoherent pump on the combined quadratures is discussed as well.

arXiv:1905.05756•Physical Review A•Nov 6, 2019•9 citations

Nondegenerate parametric interactions and nonclassical effects

A. Bandilla, G. Drobný, Igor Jex

We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Jan 1, 1996•29 citations

Mode entanglement in nondegenerate down-conversion with quantized pump

G. Drobný, Igor Jex, Vladimír Bužek

We study entanglement between field modes in the process of nondegenerate two-photon down-conversion with quantized pump. We show that due to the quantum nature of the dynamics, strong entanglement between the pump and the signal-idler subsystems can be observed. We find that the higher the initial intensity of the pump mode the stronger the entanglement between the pump and the signal-idler subsystem is established during the first instants of the time evolution. We also show that the signal and the idler modes are strongly entangled (correlated). This entanglement is much stronger than the entanglement between the pump and the signal-idler subsystem. Correlation between the signal and the idler modes leads to a high degree of two-mode squeezing, which can be observed during the first instants of the time evolution when the pump mode is still approximately in a pure state. On the other hand, the back action of the signal-idler subsystem on the pump mode leads to a strong single-mode squeezing of the pump mode. At the time interval during which squeezing of the pump mode can be observed the pump mode is far from being in the minimum uncertainty state. We also analyze the long-time behavior of the quantum-optical system under consideration and we show that the interesting collapse-revival effect in the time evolution of the mean photon number and of the purity parameters of field modes can be observed. Finally, we show that the degree of entanglement between modes in the nondegenerate quantum-optical down-conversion strongly depends on the initial state of the system.

Physical Review A•Jul 1, 1993•54 citations

Stability and squeezing of the three-photon degenerate parametric down-conversion

Vinícius V. Seco, Alencar J. de Faria

Nonlinear multiphoton conversion processes are theoretically and experimentally challenging systems in quantum optics and related areas. These phenomena bring remarkable quantum properties and can be of fundamental relevance in quantum information and computation applications. Based on recent experimental advances and the possible uses in generalized three-photon squeezing, we investigate degenerate three-photon parametric down-conversion generated inside a cavity fed by a classical pump. In addition, we consider that the process is stimulated by a coherent driving mode and subjected to dissipation by spontaneous emission of the cavity. A treatment using phase-space methods and stochastic differential equations is applied and the stability conditions of its steady states are found. We find that the system does not exhibit multistability, instead it has a limited parameter region leading to a single stable steady branch, alongside multiple unstable branches. Focusing on the stable steady states and linearizing the dynamics around them, we calculate the spectral densities of the quadrature variables. We identify and characterize the squeezing of three-photon down-conversion. Moreover, since the quasi-probability differential equations of the higher-order multiphoton processes have higher-order derivatives, hindering their mapping into stochastic differential equations, we make use of a positive Wigner function approach to calculate two-dimensional spectral densities associated with three-time correlation functions, with the goal of studying non-Gaussian properties of the system.

quant-pharXiv:2608.19494•Aug 21, 2026•0 citations

Dynamics of parametric processes with a trilinear hamiltonian

Govind Prasad Agrawal, C. L. Mehta

The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.

Journal of Physics A Mathematical Nuclear and General•Mar 21, 1974•50 citations
We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian H\ifmmode^\elseˆ\fiI=i\ensuremathℏ\ensuremathκ(a\ifmmode^\elseˆ\fib\ifmmode^\elseˆ\fic\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi\ensuremath−a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fib\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fic){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}=i\ensuremath{\hbar}\ensuremath{\kappa}\left(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}c\right)H\ifmmode^\elseˆ\fi​I​=i\ensuremathℏ\ensuremathκ(a\ifmmode^\elseˆ\fib\ifmmode^\elseˆ\fi​c\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fi\ensuremath−a\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fib\ifmmode^\elseˆ\fi​\ifmmode†\else\textdagger\fic), wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by nondegenerate coherently stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two-mode state have been shown to depend greatly on the cumulative phase \ensuremathΦ=\ensuremathθs+\ensuremathθi\ensuremath−2\ensuremathϕ\mathrm{\ensuremath{\Phi}}={\ensuremath{\theta}}_{s}+{\ensuremath{\theta}}_{i}\ensuremath{-}2\ensuremath{\phi}\ensuremathΦ=\ensuremathθs​+\ensuremathθi​\ensuremath−2\ensuremathϕ where \ensuremathθs(i){\ensuremath{\theta}}_{s\left(i\right)}\ensuremathθs(i)​ are the signal(idler) coherent state phases and 2\ensuremathϕ2\ensuremath{\phi}2\ensuremathϕ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the near-thermal-like nature of the steady state valid for long times. Finally, we examine the effect of the cumulative phase \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ on the correlations between the bipartite system of pump and signal/idler modes, and the effect of the quantized pump on the correlations between signal and idler modes.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
Pumping a nonlinear crystal by an intense radiation results in the optical parametric generation of photons in two modes (the signal and the idler). The quantized electromagnetic field in these modes is described by a continuous-variable quantum state, which is entangled if the pump is a coherent state produced by a laser. The signal and the idler modes remain populated by photons even if the pump becomes incoherent (dephased by a medium, superposed with a thermal state, or produced by an alternative source such as the superluminescent diode). However, the incoherent pump does effect the entanglement and purity of the signal and the idler modes, which is of vital importance for quantum information applications and interferometry. Here we develop an approach to infer the signal-idler entanglement and purity for a general quantum incoherent pump with the given Glauber-Sudarshan function. We show that the signal-idler entanglement is extremely sensitive to the phase distribution of the pump and illustrate our findings by physically relevant examples of the incoherent pump: the noisy coherent state, slightly dephased and phase-averaged coherent states, the thermal state, and states modulated by the Kerr medium. The effect of an incoherent pump on the combined quadratures is discussed as well.
We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.
We study entanglement between field modes in the process of nondegenerate two-photon down-conversion with quantized pump. We show that due to the quantum nature of the dynamics, strong entanglement between the pump and the signal-idler subsystems can be observed. We find that the higher the initial intensity of the pump mode the stronger the entanglement between the pump and the signal-idler subsystem is established during the first instants of the time evolution. We also show that the signal and the idler modes are strongly entangled (correlated). This entanglement is much stronger than the entanglement between the pump and the signal-idler subsystem. Correlation between the signal and the idler modes leads to a high degree of two-mode squeezing, which can be observed during the first instants of the time evolution when the pump mode is still approximately in a pure state. On the other hand, the back action of the signal-idler subsystem on the pump mode leads to a strong single-mode squeezing of the pump mode. At the time interval during which squeezing of the pump mode can be observed the pump mode is far from being in the minimum uncertainty state. We also analyze the long-time behavior of the quantum-optical system under consideration and we show that the interesting collapse-revival effect in the time evolution of the mean photon number and of the purity parameters of field modes can be observed. Finally, we show that the degree of entanglement between modes in the nondegenerate quantum-optical down-conversion strongly depends on the initial state of the system.
Nonlinear multiphoton conversion processes are theoretically and experimentally challenging systems in quantum optics and related areas. These phenomena bring remarkable quantum properties and can be of fundamental relevance in quantum information and computation applications. Based on recent experimental advances and the possible uses in generalized three-photon squeezing, we investigate degenerate three-photon parametric down-conversion generated inside a cavity fed by a classical pump. In addition, we consider that the process is stimulated by a coherent driving mode and subjected to dissipation by spontaneous emission of the cavity. A treatment using phase-space methods and stochastic differential equations is applied and the stability conditions of its steady states are found. We find that the system does not exhibit multistability, instead it has a limited parameter region leading to a single stable steady branch, alongside multiple unstable branches. Focusing on the stable steady states and linearizing the dynamics around them, we calculate the spectral densities of the quadrature variables. We identify and characterize the squeezing of three-photon down-conversion. Moreover, since the quasi-probability differential equations of the higher-order multiphoton processes have higher-order derivatives, hindering their mapping into stochastic differential equations, we make use of a positive Wigner function approach to calculate two-dimensional spectral densities associated with three-time correlation functions, with the goal of studying non-Gaussian properties of the system.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.