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Physical Review A

Limits on tradeoffs between third-order optical nonlinearity, absorption loss, and pulse duration in self-induced transparency and real excitation

Kimitaka Watanabe, Y. Yamamoto

Published Aug 1, 1990• 12 citations
Publisher

Abstract

Self-induced transparency in a resonant two-level system creates a 2\ensuremath{\pi}-soliton pulse, which realizes large optical nonlinearities with a small absorption loss \ensuremath{\alpha} for a short pulse duration \ensuremath{\tau}. Quantum-mechanical zero-point field fluctuations introduce an ultimate dissipation loss in such a resonant and coherent process and place fundamental limits on the ${\mathrm{\ensuremath{\chi}}}^{(3)}$/\ensuremath{\alpha}\ensuremath{\tau} value. This value is independent of the dipole moment, atomic density, and pulse duration and is uniquely determined by an optical wavelength, e.g., ${\mathrm{\ensuremath{\chi}}}^{(3)}$/\ensuremath{\alpha}\ensuremath{\tau}\ensuremath{\sim}1.2\ifmmode\times\else\texttimes\fi{}${10}^{21}$${\ensuremath{\lambda}}^{4}$ esu cm/s. The similar limit on ${\mathrm{\ensuremath{\chi}}}^{(3)}$/\ensuremath{\alpha}\ensuremath{\tau} value is obtained in a usual operation mode, when the pulse duration becomes much longer than the atomic decay constants and the real excitation of the atoms occurs instead of the virtual excitation in self-induced transparency. The implication of these limits on optical squeezed-state generation, quantum nondemolition measurement, and reversible logic is discussed.

DOI
10.1103/physreva.42.1699

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Nonlinear Optics: Materials, Fundamentals, and Applications. Technical Digest. Postconference Edition. TOPS Vol.46 (IEEE Cat. No.00CH37174)•Nov 7, 2002•1 citation
Self-induced transparency in a resonant two-level system creates a 2\ensuremath{\pi}-soliton pulse, which realizes large optical nonlinearities with a small absorption loss \ensuremath{\alpha} for a short pulse duration \ensuremath{\tau}. Quantum-mechanical zero-point field fluctuations introduce an ultimate dissipation loss in such a resonant and coherent process and place fundamental limits on the \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3)/\ensuremath{\alpha}\ensuremath{\tau} value. This value is independent of the dipole moment, atomic density, and pulse duration and is uniquely determined by an optical wavelength, e.g., \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3)/\ensuremath{\alpha}\ensuremath{\tau}\ensuremath{\sim}1.2\ifmmode\times\else\texttimes\fi{}{10}^{21}$${\ensuremath{\lambda}}^{4} esu cm/s. The similar limit on \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3)/\ensuremath{\alpha}\ensuremath{\tau} value is obtained in a usual operation mode, when the pulse duration becomes much longer than the atomic decay constants and the real excitation of the atoms occurs instead of the virtual excitation in self-induced transparency. The implication of these limits on optical squeezed-state generation, quantum nondemolition measurement, and reversible logic is discussed.
Self-induced transparency in a resonant two-level system creates a 2\ensuremath{\pi}-soliton pulse, which produces large optical nonlinearities with a very fast response time and an extremely small loss. The self-phase modulation of a 2\ensuremath{\pi} soliton and the mutual-phase modulation by the collision of two 2\ensuremath{\pi} solitons in the excitonic range of the spectrum in CdS can achieve the effective ${\ensuremath{\chi}}^{(3)}$ coefficient of ${10}^{\mathrm{\ensuremath{-}}6}$ to ${10}^{\mathrm{\ensuremath{-}}4.5}$ esu for a pulse duration of 1 to 3 ps. The quantum theory of 2\ensuremath{\pi} solitons predicts this new nonlinear process can realize squeezed states and quantum nondemolition measurement 20 dB below the standard quantum limit.
Above a critical power threshold for a given pulse width, a short pulse of coherent traveling-wave optical radiation is observed to propagate with anomalously low energy loss while at resonance with a two-quantum-level system of absorbers. The line shape of the resonant system is determined by inhomogeneous broadening, and the pulse width is short compared to dissipative relaxation times. A new mechanism of self-induced transparency, which accounts for the low energy loss, is analyzed in the ideal limit of a plane wave which excites a resonant medium with no damping present. The stable condition of transparency results after the traversal of the pulse through a few classical absorption lengths into the medium. This condition exists when the initial pulse has evolved into a symmetric hyperbolic-secant pulse function of time and distance, and has the area characteristic of a "$2\ensuremath{\pi}$ pulse." Ideal transparency then persists when coherent induced absorption of pulse energy during the first half of the pulse is followed by coherent induced emission of the same amount of energy back into the beam direction during the second half of the pulse. The effects of dissipative relaxation times upon pulse energy, pulse area, and pulse delay time are analyzed to first order in the ratio of short pulse width to long damping time. The analysis shows that the $2\ensuremath{\pi}$ pulse condition can be maintained if losses caused by damping are compensated by beam focusing. In an amplifying inhomogeneously broadened medium an analytic "$\ensuremath{\pi}$ pulse area" solution is presented in the limit of a sharp leading edge of the pulse. The dynamics of self-induced transparency are studied for the particular effects of Doppler velocities upon a resonant gas. The analysis of transparency for random orientations of dipole moments associated with degenerate rotational states yields modified forms of self-induced transparency behavior, which indicates a finite pulse energy loss as a function of distance in some cases. The effect of self-induced transparency on the photon echo is considered. Experimental observations of self-induced transparency have been made in a ruby sample at resonance with a pulsed ruby-laser beam. Single and multiple $2\ensuremath{\pi}$ pulse outputs have been observed, and pulse areas measured in the range of $2\ensuremath{\pi}$. The experimental results are compared with the predictions of the ideal-plane-wave theory. Deviations from the ideal-plane-wave theory are discussed. An analysis is made of the effect of a transverse mode of the propagating beam upon the transparency properties of the pulse.
Summary form only given. The field of quantum optics was greatly enriched with the discovery of dark states and electromagnetically induced transparency (EIT). Although these two processes give similar effects, they are distinctly different. Dark states (Gray et al, 1978) are quantum states created in a three-level lambda system that is decoupled from the excited state. Some of the main features are zero linear absorption, no population in the excited state and no spontaneous emission. One limiting factor of a dark state system is that self-phase modulation disappears together with absorption. EIT is a nonlinear effect that can provide transparency to a probe field in a variety of systems. Much work has been done on two-level, cascaded-three-level and even four-level systems. It has also been shown that a large third-order nonlinear susceptibility is present in the absence of absorption. However, spontaneous emission cannot be eliminated in such system, thereby limiting their applications, particularly, to squeezed light generation. Here, we combine the effects of both dark state and EIT to create a system that is suitable for squeezed light generation via self phase modulation.
Quantum coherence and interference in atomic systems can lead to interesting optical phenomena such as nonabsorbing resonances, lasing without inversion, generating an ultrahigh index of refraction without absorption, propagation of light pulses without losses. In the last-named is given the title transparency. Up to now it is known some types of transparency in multi-level resonant media. One of them is electromagnetically induced transparency (EIT). The phenomenon of EIT in the propagation of laser radiation through an absorbing medium has been recently investigated and explained in terms of quantum coherence and interference for multi-level atomic system. For instance in the (Lambda) -system, if the atomic medium is prepared in a quantum superposition of states, matched pulses, i.e., a pair of optical pulses whose amplitude and phase have a well defined relation, propagate without absorption. The mechanism of coherent population trapping is responsible for EIT in a (Lambda) -system. There is another, well known mechanism for creating transparency in a two-state atom: self-induced transparency, or SIT. Here, a single pulse whose area (the integral of its Rabi frequency over time) is 2(pi) is applied to the medium. Such a pulse causes the atoms to cycle smoothly from the ground to the excited state and then back again. In principle, such a pulse can propagate over long distances. SIT can be generalized to multi-level systems and multiple quasi-monochromatic fields. Generalization of SIT in a (Lambda) -system was obtained in reference 2. In this report we present a new type of transparency (combined transparency). This type describes propagation of laser pulse through a (Lambda) -type medium where the initial atomic state is prepared in the form of spatially varying coherent superposition of low-energetic states (initial coherency gradient). The common envelope behaves as usual 2(pi) hyperbolic secant pulse, but the ratio between the components of frequencies transitions 1> yields 3> and 2> yields 3> changes due to coherent Raman scattering so (phi) equals atan(E1/E2) equals (pi) /2-Gz, where z is the coordinate, G is the constant determined by both the incident pulse parameters and the initial coherence between the atomic states. Our solution demonstrates a direct connection of EIT in adiabatic regime to the SIT effects. This type of a transparency is given the title screw-type transparency, because for the given form of incident pulse (pitch of screw) medium is transparent, if medium's the initial coherence gradient is correspondingly prepared (thread on a engine nut). Besides a transparency, this phenomenon contains a transformation of the incident pulse frequency during the process of the propagation. This property of the phenomena can be result in a complete conversion of the frequency.
The optical solitons that play the role of information carriers, are moving, requiring careful experimental timing to achieve the desired logic operation. In contrast, self-induced transparency (SIT) is a strong resonant nonlinear effect, supporting the propagation of optical solitons and are usually associated with a two-level system. By the appropriate application of light fields, the optical soliton, resonant with the SIT transition, can generate an excited region with ground state coherence in the medium, allowing logic operations to be performed on this coherence.