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arXiv:1203.5394 · Physical Review A

Dynamics of two-photon paired superradiance

M. Yoshimura, N. Sasao, Minoru Tanaka

Published Jul 9, 2012• 39 citations
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Abstract

We develop for dipole-forbidden transition a dynamical theory of two-photon paired superradiance (PSR). This is a cooperative process characterized by two photons emitted back to back with equal energies. By irradiating the trigger laser from two target ends, with its frequency tuned at the half energy between two levels, a macroscopically coherent state of medium and fields dynamically emerges as time evolves, and a large signal of amplified output occurs with a time delay. The basic semiclassical equations in $1+1$ space-time dimensions are derived for the field-plus-medium system to describe the space-time evolution of the entire system and are numerically solved to demonstrate the existence of both explosive and weak PSR phenomena in the presence of relaxation terms. The explosive PSR event terminates accompanying a sudden release of most of the energy stored in the target. Our numerical simulations are performed using the vibrational transition ${X}^{1}{\ensuremath{\Sigma}}_{g}^{+}v=1\ensuremath{\rightarrow}0$ of a para-H${}_{2}$ molecule and taking many different excited atom number densities and different initial coherences between the metastable and the ground states. In an example with a number density close to $O({10}^{21}$ cm${}^{\ensuremath{-}3}$) and a high initial coherence, the explosive event terminates several nanoseconds after the trigger irradiation, when the phase relaxation time larger than $O$(10 ns) is taken. After PSR events the system is expected to follow a steady-state solution which is obtained by analytic means and is made of many objects of field condensates endowed with a topological stability.

DOI
10.1103/physreva.86.013812

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We develop for dipole-forbidden transition a dynamical theory of two-photon paired superradiance (PSR). This is a cooperative process characterized by two photons emitted back to back with equal energies. By irradiating the trigger laser from two target ends, with its frequency tuned at the half energy between two levels, a macroscopically coherent state of medium and fields dynamically emerges as time evolves, and a large signal of amplified output occurs with a time delay. The basic semiclassical equations in 1+11+11+1 space-time dimensions are derived for the field-plus-medium system to describe the space-time evolution of the entire system and are numerically solved to demonstrate the existence of both explosive and weak PSR phenomena in the presence of relaxation terms. The explosive PSR event terminates accompanying a sudden release of most of the energy stored in the target. Our numerical simulations are performed using the vibrational transition X1\ensuremathΣg+v=1\ensuremath→0{X}^{1}{\ensuremath{\Sigma}}_{g}^{+}v=1\ensuremath{\rightarrow}0X1\ensuremathΣg+​v=1\ensuremath→0 of a para-H2{}_{2}2​ molecule and taking many different excited atom number densities and different initial coherences between the metastable and the ground states. In an example with a number density close to O(1021O({10}^{21}O(1021 cm\ensuremath−3{}^{\ensuremath{-}3}\ensuremath−3) and a high initial coherence, the explosive event terminates several nanoseconds after the trigger irradiation, when the phase relaxation time larger than OOO(10 ns) is taken. After PSR events the system is expected to follow a steady-state solution which is obtained by analytic means and is made of many objects of field condensates endowed with a topological stability.
The problem of light propagation of frequency corresponding to half of the energy difference between a metastable excited state and the ground state of atoms is examined, and solved for coherent medium by analytic means. We demonstrate that the non-linear system of Maxwell-Bloch equation for the effective model of the Λ -type three levels is integrable in the mathematical sense. Analytic solutions thus obtained describe pulse splitting accompanied by compression, indicating a kind of non-linear instability of propagating pulses. The instability is eventually terminated by coherent two photon emission (called paired superradiance or PSR in short). These results are displayed by numerical outputs for visual understanding, as well. It is further shown that the integrable system allows a new class of soliton solutions. Solitons, implying the phenomenon of self-induced transparancy at non-resonant frequencies, are stable against PSR. One of our goals of the present work is construction of a calculable theoretical framework for PSR rates associated with a trigger pulse propagation, which is achieved by combining analytic results with perturbative methods. PSR photon spectrum and its rate ∝(target number density) 2 , along with their time structure, are clarified this way. These results may open a new path for interesting technological applications such as quantum entanglement and for solving the remaining problems of the still mysterious neutrino. Some basic strategy for realistic experiments of PSR detection and soliton production is also outlined.
Paired superradiance (PSR) is a macro-coherent two-photon process capable of very large gain, making it promising for detecting ultra-weak signals induced by neutrinos or dark matter. A major goal has been to increase the system volume VVV and density nnn, since the signal intensity scales as (nV)2(nV)^2(nV)2. We recast finite PSR as a parametric amplifier driven by the electromagnetic vacuum. The usual zero-field semiclassical initial condition is replaced by vacuum inputs fixed by the quantum two-point function. Combining this formulation with Maxwell--Bloch evolution and finite-length stability analysis, we find that PSR produces an irreducible vacuum background that can develop into macroscopic bursts once the gain-length product exceeds ΓL=π/2ΓL=π/2ΓL=π/2 for a sufficient coherence time. These results, together with a closed-form formula for estimating the vacuum-seeded photon yield, establish a previously overlooked constraint for high-gain PSR, with direct implications for proposed neutrino and dark-matter studies.
When atoms are coupled to a common electromagnetic environment, the exchange of photons through dipole-dipole interactions leads to the emergence of cooperative effects. As a particular example, superradiance arises from spontaneous emission when this exchange leads to constructive interference of the emitted photons. Here, we introduce an integrated method for studying cooperative radiation in many-body systems. This method, which allows to study extended systems with arbitrarily large number of particles, can be formulated by an effective, nonlinear, two-atom master equation that describes the dynamics using a closed form which treats single- and many-body terms on an equal footing. We apply this method to a homogeneous gas of initially inverted two-level atoms, and demonstrate the appearance of both superradiance and subradiance, identifying a many-body coherence term as the source of these cooperative effects. We describe the many-body induced broadening -- which is analytically found to scale with the optical depth of the system -- and light shifts, and distinguish spontaneous effects from induced ones. In addition, we theoretically predict the time-dependence of subradiance, and the phase change of the radiated field during the cooperative decay.
We discuss the cooperative decay of initial atomic excitation for a pencil-shaped active volume filled with two-level atoms. As long as the length of the sample is much smaller than a certain maximal cooperation length, the atom-field interaction producing the superradiant pulse can be treated in terms of the simplest possible laser model (single mode). The basic laser master equation turns out to be exactly solvable if specified for the superradiance limit which is characterized by two conditions: (i) The photons escape from the low-QQQ cavity so fast that they cannot feed themselves back into atomic excitation to any appreciable amount (this no-feedback condition is equivalent to the above-mentioned requirement for the length of the sample). (ii) The incoherent atomic decay due to natural relaxation is so slow that the individual atomic dipoles do not dephase before engaging themselves cooperatively in the interaction with the electromagnetic field. The present paper presents the derivation and general discussion of the equations describing the statistical properties of atoms and field in a superradiant pulse. Analytical and numerical solutions will be presented in a subsequent paper.
We have developed a novel method to describe superradiance and related cooperative and collective effects in a closed form. Using the method we derive a two-atom master equation in which any complexity of atomic levels, semiclassical coupling fields and quantum fluctuations in the fields can be included, at least in principle. As an example, we consider the dynamics of an initially inverted two-level system and show how even such in a simple system phenomena such as the initial radiation burst or broadening due to dipole-dipole interactions occur, but it is also possible to estimate the population of the subradiant state during the radiative decay. Finally, we find that correlation only, not entanglement is responsible for superradiance.
The dynamical evolution of an excited-state population in a macroscopic volume of three-level molecules is considered where the population is driven optically between the initial ground state and the excited states by an externally applied (ccc-number) coherent pump. The intermediate and ground states are taken as nonradiatively coupled whereas the excited-to-intermediate-state transition is coupled to the radiation field which is treated quantum mechanically. It is found that stimulated Raman processes can significantly influence the dynamical evolution of population inversion and macroscopic polarization by producing coherence effects among the populations. We examine the evolution of collective relaxation between the excited and intermediate levels in the time regime of the pump pulse duration. It is shown, using the mean-field approximation, that for uniform pumping and conditions such that the pump Rabi rate ${\ensuremath{\omega}}_{R}$, the characteristic collective radiation time ${\ensuremath{\tau}}_{R}$, and the pump pulse duration ${\ensuremath{\tau}}_{P}$ satisfy the inequalities, ${\ensuremath{\omega}}_{R}>\frac{1}{{\ensuremath{\tau}}_{R}}>\ensuremath{\gamma}$ and ${\ensuremath{\tau}}_{P}\ensuremath{\gtrsim}{\ensuremath{\tau}}_{R}$ ($\ensuremath{\gamma}$ is a characteristic internal dephasing rate for the molecules), the system is left in a state with classical transverse polarization when the pump pulse terminates. The system can evolve collectively from such a state only as superradiant (classical) evolution. For superfluorescent evolution which requires a state of preparation of complete inversion (zero transverse polarization), it is shown that the pump pulse must be effectively of area $\ensuremath{\pi}$ and that ${\ensuremath{\tau}}_{P}<{\ensuremath{\tau}}_{R}$. Our results show that when the former conditions are satisfied the delay time ${\ensuremath{\tau}}_{D}$ for collective free pulse evolution is a function of both ${\ensuremath{\tau}}_{P}$ and ${\ensuremath{\tau}}_{R}$. It is shown further that the pump pulse shape as well as temporal duration significantly affect the final state of preparation. The results of this work are interpreted in connection with recent reported results of experiments in superfluorescence and superradiance.