Effect of Injected Noise on
Electromagnetically Induced Transparency
and Slow Light
Abstract
We have examined theoretically the phenomenon of Electromagnetically Induced Transparency (EIT) in a three–level system operating in the –configuration in the presence of an externally injected noise coupling the ground level to the intermediate (metastable) level. The changes in the depth and the width of the induced transparency and the slowing down of the probe light have been calculated as function of the probe detuning and the strength of the injected noise. The calculations are within the rotatingwave approximation (RWA). Our main results are the reduction and the broadening of the EIT with increasing strength of the injected noise, and a reduction in the slowing down of group velocity of the probelaser beam. Thus, the injected semiclassical noise, unlike the quantumdynamical noise associated with the spontaneous emission, is not effectively cancelled by the EIT mechanism.
Keywords
EIT, group velocity, Lindblad, Novikov’s theorem, gaussian white noise.
Introduction
Electromagnetically Induced Transparency (EIT) is a coherent quantum optical phenomenon in which the absorption of a weak probe beam of light vanishes, opening thereby a window of transparency narrower than the natural linewidth in the center of the otherwise much broader resonance absorption peak. This transparency is due to the destructive quantum interference of the transition amplitudes along the allowed alternative paths, much as in the case of the wellknown Fano resonance/antiresonance. This phenomenon is known to give rise to several other interesting effects, e.g., the slow light. Remarkably, this interference effectively cancels the spontaneous downtransitions (quantum noise) as well, giving a subnatural linewidth as noted above. In contrast to this, as the present work shows, the EIT is indeed affected/degraded by injecting a ’classical’ noise into one of the alternative paths.
The purpose of this work is to analyze theoretically the nature and the magnitude of the effect of the injected noise on the depth and the width of the transparency (EIT), as well as the associated change in the group velocity of the probe light as function of the strength of noise and the probe detuning. The calculations have been done using the densitymatrix formalism wherein the various natural linewidths (spontaneous decays) involved are introduced via the appropriately chosen Lindblad superoperators, while the injected noise is introduced explicitly in the Hamiltonian. The latter is treated within the Rotating Wave Approximation (RWA). We compute the reduced densitymatrix averaged over the noise . This could be carried out here in a closed form by assuming the injected noise to be a Gaussian White Noise (GWN), and using the wellknown Novikov theorem. The latter holds for averaging an arbitrary functional of the gaussian white noise. The physical quantities of interest are then calculated in terms of this noiseaveraged density matrix. Our main results are a quantitative reduction and broadening of the EIT with increasing strength of the injected noise, and a reduction in the slowing down of the group velocity of the probe light.
The atomiclevel scheme
The atomiclevel scheme considered here is as in a typical EIT experiment shown schematically in Figure 1. It involves a manifold of three energy levels with the respective energies , , and , connected in the –configuration with . Such a –EIT scheme is possible, e.g., in the D2line transitions of vapor with , , and . As follows from the selection rules, in a scheme, the transition (here ) between the ground and the intermediate state is necessarily an electric–dipole forbidden transition, since the other two transitions involved are electric–dipole allowed. It can, however, be, e.g., an electricquadrupole allowed transition, much as in the case of a controlling microwave field used in the recent EIT–experiment in rubidium vapor system by Hebin Li., et al.. A novel feature of the present work is a noise field injected externally at the transition .
In a standard EIT experiment, the absorption of the probe beam is studied as function of its detuning while the coupling laser is kept at resonance. Accordingly, here the probe laser is detuned off–resonance by an amount from the transition which is being probed, i.e., , where and are, respectively, the frequency and the associated Rabifrequency of the probe laser. A strong coupling laser beam is applied and kept at resonance to the transition , with , where and are, respectively, the frequency and the associated Rabi frequency of the coupling laser. Further, and denote the rates of spontaneous decay of the excited level to the ground level and to the intermediate level respectively.
The Hamiltonian and RWA
The full Hamiltonian of the EIT system here corresponds essentially to the case of a three–level system being acted upon by two optical fields in the socalled semiclassical approximation, but now with an extra feature, namely, a term corresponding to the injected noise. More explicitly, we have
(1) 
Let
with
(2) 
and
(3) 
We now proceed to the interaction(I) picture with the corresponding Hamiltonian given by
(4) 
In the Rotating Wave Approximation (RWA) (i.e., neglecting the rapidly oscillating terms proportional to ), we obtain
(5) 
with .
Hereinafter, we will use the interactionpicture Hamiltonian , but drop the subscript for convenience.
Lindblad EoMs
With this, the master equation of motion for the density matrix is
(6) 
where we have introduced the Linblad superoperator , given in the diagonal form as
(7) 
Here and are the rates of spontaneous decays and respectively. The Lindblad operators and are chosen appropriately so as to describe the spontaneous decays as:
(8) 
(9) 
As is known well, a Lindblad master equation describes the nonunitary evolution of the density matrix preserving the trace condition without violating its complete positivity and hermiticity for all initial conditions.
Substituting from eqs.(5,7) into eq.(6), we obtain
(10) 
(11) 
(12) 
(13) 
(14) 
along with the trace condition and the hermiticity condition .
Averaging the EoMs over noise
Now, the density matrix is a functional of the noise occurring in eqs. (1014), and, therefore, it must be averaged over all the realizations of . For this, we have taken the noise to be a Guassian White Noise (GWN), i.e.,
(15) 
where , having the dimension of frequency, is a measure of the strength of the noise (much like the Rabi frequency , which is a measure of the strength of the a laser field). For this, we make use of the Novikov theorem giving
(16) 
Straightforward functional differentiation of eqs.(1014) w.r.t. and using eq.(16), we obtain, within RWA (this time neglecting the rapidly oscillating terms proportional to ), a closed set of equations for the noiseaveraged density matrix elements for typographic convenience) as follows : (
(17) 
(18) 
(19) 
(20) 
(21) 
These linear firstorder differential equations, along with the trace condition and the hermiticity condition , are now solved numerically in the steady state (i.e., ) so as to calculate the physical quantities of interest, as described below.
Results
The absorption coefficient of the probe light in a dilute gaseous medium can be expressed in terms of the density matrix as
(22) 
where, is the atomic number density of the medium, is the wavelength of the probe light at a detuning , and is the wavelength at the corresponding resonance. Also, we have set (which is in fact a good approximation for most of the practical type EIT systems), and . The corresponding real part of the refractive index can be expressed in terms of the various parameters involved as
(23) 
The associated group velocity of the probe light in the medium concerned can be conveniently expressed as:
(24) 
In the following Figures 26, we have plotted these various physical quantities of interest (
=  atoms per ,  

=  
=  
=  
=  
=  . 
From Figure 2, it can be readily seen, as expected for pure EIT (without noise), that the absorption of the probe light beam increases with detuning (), for small values of the detuning.
With the increasing strength of the noise (), however, the absorption as well as its spectral width increases. Overall, the effect of noise is more pronounced within the EIT window, and much less so outside the window, as clearly seen in Figure 3.
Figure 4 gives specifically the variation of the real part of the refractive index () as function of detuning in the anomalous regime of dispersion, within the EIT window. There is a pronounced decrease in the variation of across the EIT window with the increasing strength of noise.
Now we come to the effect of noise on the group velocity of the probe light beam across the EIT window. This has an important bearing on the phenomenon of slow light associated with the EIT.
The overall effect of the noise is to reduce the slowing down of light beam as can be seen from the Figure 5. Thus, e.g., for zero noise and zero detuning). Figure 6 resolves the finer features of the effect of the noise on the group velocity at and near the resonance (i.e., ).
Finally, a few words are in order to explain how the injected noise enters the physics of the EIT. Basically, the injected noise is transfered by the coupling laser beam into the excited level , giving rise to two related effects : First, it reduces (dephases) the interference effects responsible for the EIT itself; and secondly, it broadens the excited energy level. These effects in turn reduce the depth of the EIT window while enhancing its width. This transfer effect is seen to be reflected in Figure 7.
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Acknowledgements: We would like to thank Andal Narayanan for fruitful discussions. One of us (RV) thanks the Raman Research Institute for support during the course of this work.