This report shows that the electromagnetically-induced transparency (EIT) window can be used as a time-to-frequency filter for the probe pulse whose spectrum is much broader than the width of the EIT window. This study demonstrates that the narrow spectral region of the pulse, coincident with the EIT window, develops in a long Gaussian pulse with slow group velocity. An analytical theory is developed describing the propagation of a small amplitude pulse through an optically dense absorptive medium with a narrow EIT window.
This report shows that the electromagnetically-induced transparency (EIT) window can be used as a time-to-frequency filter for the probe pulse whose spectrum is much broader than the width of the EIT window. This study demonstrates that the narrow spectral region of the pulse, coincident with the EIT window, develops in a long Gaussian pulse with slow group velocity. An analytical theory is developed describing the propagation of a small amplitude pulse through an optically dense absorptive medium with a narrow EIT window.
A simple analytical solution for the propagation of a weak Gaussian pulse in a dense absorptive medium with electromagnetically induced transparency is found. This solution is applied to the analysis of three regimes: (1) and (2) the pulse spectrum is narrower than the transparency window [which is narrow (1) or wide (2) with respect to the width of the absorption line] and (3) the pulse spectrum is broader than the transparency window. It is shown that the pulse maintains its area in all three regimes and maintains its Gaussian shape but narrows in spectrum in regime 1. In regime 2, the pulse begins to distort after a certain distance. In regime 3, the pulse is split into two parts. One part is an adiabatic part with a spectrum defined by the effective width of the transparency window for a thick medium and the other is an oscillating nonadiabatic part of short duration. The adiabatic part propagates slowly and the nonadiabatic part propagates with a velocity close to the speed of light. Thus in regime 3, the medium acts as a time-frequency filter, separating the narrow and wide spectrum components of the pulse in time at the output of the absorber.
We consider the propagation of a Gaussian probe pulse in an absorptive, optically dense, two-level medium if a deep, persistent hole is created in advance by another pump field in the inhomogeneously broadened absorption spectrum of this medium. Both fields are well separated in time and the lifetime of the hole is assumed to be long with respect to the delay time between the pump and probe pulses. We show that the group velocity of the Gaussian probe pulse reduces several orders in magnitude, similarly to the reduction of the group velocity for the probe field in electromagnetically induced transparency (EIT) phenomena. In contrast to EIT, the width of the transparency window can be made very wide because of the saturation broadening of the hole.
The propagation of broadband pulse in a medium with electromagnetically induced transparency (EIT) was studied. The results show that if the carrier frequency of pulse is exactly tuned to the center of the transparency window, and the pulse spectrum is broader than transparency window, the pulse is temporally broadened and the delay effect is significant. If the carrier frequency of pulse the is exactly tuned to the region of absorption, and the pulse spectrum is wide with respect to the width of absorption line, the pulse begins to distort after a certain distance and is split into two parts at last. The peak value depends not only on the frequency and intensity of the coupling field, but also on the length of the sample. Under appropriate conditions, strong and time domain narrowed pulse can be obtained in the output light.
Multi-level atomic coherence induced bycoupling andprobe fields canproduce manyinteresting effects suchas electromagnetically induced transparency (EIT), lasing without inversion, electromagnetically induced absorption, slow light, stopped light, nonlinear conversion withsubstantially reduced phase matching conditions, etc. EITisobserved asanarrow holeinabroadabsorption line.Steep dispersion within theEITwindowresults ina substantial reduction ofthegroupvelocity oftheprobe pulse. Onecanexpect that theabsorption andthegroupvelocity ofthepulse aresubstantially reduced ifthecarrier frequency andthespectrum oftheinput pulse areconfined inside the EITwindow.Inthis report we showthat theEITwindowcanbeusedasatime-to-frequency filter fortheprobe pulse whosespectrum ismuchbroader thanthewidth oftheEITwindow. Wedemonstrate that thenarrowspectral part ofthe pulse, coincident withtheEITwindow, develops inalong Gaussian pulse withslowgroupvelocity. Thatpartofthe pulse spectrum, whichcoincides withthemaximumoftheabsorption line, isstrongly absorbed. Sidebands ofthepulse interacting withthewings oftheabsorption line develop ina0-tpulse. Thelatter hasanoscillatory shape andthetime integral ofitsenvelope (pulse area) isclose tozero, henceits nameof0-it pulse. Thegroupvelocity ofa0-tpulse is close tothephase velocity oflight. Inanoptically denseabsorptive medium, theslowGaussian pulse andthefast 0-it pulse arewellseparated intimeandthey donotoverlap attheoutput oftheabsorber. Timegating ofthepromptpart of thepulse allows separating thenarrowspectrum, theslowpartofthepulse attheoutput. Suchafiltering isnotsensitive tothespectrum oftheinput pulse. Theparameters ofthenarrowspectrum pulse, i.e., thewidth, theshape andthetime of arrival, depend only ontheparameters ofthefilter, i.e., theeffective length oftheabsorber andthewidth oftheEIT window.
Electromagnetically induced transparency (EIT) has been proposed as a way to greatly enhance cross-phase modulation, with the possibility of leading to few-photon-level optical nonlinearities [Schmidt and Imamoglu, Opt. Lett. 21, 1936 (1996)]. This enhancement grows as the transparency window width, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{EIT}}$, is narrowed. Decreasing ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{EIT}}$, however, has been shown to increase the response time of the nonlinear medium. This suggests that, for a given pulse duration, the nonlinearity would diminish once the window width became narrower than this pulse bandwidth. We show that this is not the case: the peak phase shift saturates but does not decrease. We show that in the regimes of most practical interest---narrow EIT windows perturbed by short signal pulses---the enhancement offered by EIT is not only in the magnitude of the nonlinear phase shift but also in its increased duration. That is, for the case of signal pulses much shorter (temporally) than the inverse EIT bandwidth, the narrow window serves to prolong the effect of the passing signal pulse, leading to an integrated phase shift that grows linearly with $1/{\mathrm{\ensuremath{\Delta}}}_{\mathrm{EIT}}$; this continued growth of the integrated phase shift improves the detectability of the phase shift, in principle, without bound. For many purposes, it is this detectability which is of more interest than the absolute magnitude of the peak phase shift. We present analytical expressions based on a linear time-invariant model that accounts for the temporal behavior of the cross-phase modulation for several parameter ranges of interest. We conclude that in order to optimize the detectability of the EIT-based cross-phase shift, one should use the narrowest possible EIT window and a signal pulse that is as broadband as the excited-state linewidth and detuned by half a linewidth.
We theoretically study classical step-modulated pulse propagation in an optically thick medium with electromagnetically induced transparency (EIT) using a fast Fourier transform and a proposed hybrid-asymptotic analysis. The 100% transmitted leading edge traveling at c consists of Sommerfeld-Brillouin precursors. The main signal, falling inside of the narrow EIT window, experiences a steep dispersion and slow group velocity. We suggest active control of the precursors and main signal separately by adjusting the parameters of the medium and incident pulse.