Laser linewidth effects in quantum state discrimination by electromagnetically induced transparency

We discuss the use of electromagnetically modified absorption to achieve selective excitation in atoms: that is, the laser excitation of one transition while avoiding simultaneously exciting another transition whose frequency is the same as or close to that of the first. The selectivity which can be achieved in the presence of coherent population trapping (CPT) is limited by the decoherence rate of the dark state. We present exact analytical expressions for this effect, and also physical models and approximate expressions which give useful insights into the phenomena. When the laser frequencies are near-resonant with the single-photon atomic transitions, CPT is essential for achieving discrimination. When the laser frequencies are far detuned, the “bright” two-photon Raman resonance is important for achieving selective excitation, while the “dark” resonance (CPT) need not be. The application to laser cooling of a trapped atom is also discussed.

Figure 1

Atomic level schemes considered in the text. (a) $\mid I⟩$ is part of a two-level manifold; $\mid S⟩$ is part of a three-level manifold. (b) $\mid I⟩$ and $\mid S⟩$ are each part of separate three-level manifolds. In either case, the atom is illuminated by a single pair of laser beams which drive both manifolds; the single photon transition 1-3 in the $D$ manifold is either degenerate with or close to the single-photon transition out of $\mid I⟩$ in the $B$ manifold. ${\Omega }_1,{\Omega }_2$ are Rabi frequencies, ${\Gamma }_1,{\Gamma }_2$ spontaneous decay rates, and $\gamma$ is the rate of decay of coherence between levels 2 and 1. Both types of energy level structure are common in groups of atomic levels with $J\ne 0$. Case (b) also occurs in the combination of internal and vibrational states of a trapped atom.

Figure 2

Example fluorescence profiles for a set of values of laser linewidth. The parameter values are (in units where $\Gamma =1$) ${\Omega }_1=0.1,{\Omega }_2=1,{\Delta }_2=3,{\Gamma }_1={\Gamma }_2=0.5$, and $\gamma =0$, 0.05, 0.1 for full, dashed, dotted curves, respectively. Note that at finite $\gamma$ the absorption minimum is displaced with respect to $\delta =0$, as remarked by Kofman [16].

Figure 3

Example of state discrimation for atomic structure of the form shown in Fig. 1a. The curves show the steady-state value of the excited state population for an atom prepared in $\mid S⟩$ (full curve) and $\mid I⟩$ (dashed curve), respectively, as a function of detuning $\delta$. The $B$ manifold shows the standard “2-level atom” Lorentzian profile, while the $D$ manifold shows a dark resonance at ${\Delta }_1=0$ in between two peaks at $\pm {\Omega }_2∕2$ (these show the positions of the dressed states created by the pump laser). By choosing $\delta =0$ the ratio of excitation rates is maximized. The parameter values for the three level system are (in units where $\Gamma =1$) ${\Omega }_1=0.2,{\Omega }_2=4,{\Gamma }_1={\Gamma }_2=0.5$, and $\gamma =0.1$. For the two level system ${\Omega }_1=0.2\sqrt{2}$ and ${\Omega }_2=0$.

Figure 4

Physical models of the bright resonance. (a) The pump laser dresses the atom; the probe laser excites the atom from $\mid 1⟩$ to the dressed states. These give a broad resonance displaced by $-{\Delta }^{\prime }$ and a narrow resonance displaced by ${\Delta }_2+{\Delta }^{\prime }$ from the position of the undressed excited state $\mid 3⟩$. (b) The two lasers together drive Rabi oscillations between (undressed) levels 1 and 2 by a Raman transition, and the pump laser off-resonantly excites transitions from $\mid 2⟩$ to $\mid 3⟩$. The Rabi oscillation is resonant when the laser frequencies match the energy difference between 1 and the Stark-shifted level 2.

Figure 5

Physical model of the effect of decoherence on a dark resonance. The atom is analyzed in the basis $\mid 3⟩,\mid +⟩,\mid -⟩$, where $\mid -⟩$ is the dark state. A simple rate equation picture, with rates as shown, suffices to give the main features of the behavior.

Figure 6

Discrimination ratio $r$ for the case of two $\Lambda$ systems with the same coupling constants, and 2-photon resonance conditions of frequency separation $Z$. The surface shows $r$ as a function of ${\Delta }_2$ and ${\Omega }_2$ for the case $Z=0.2,\gamma =0.001$, and small ${\Omega }_1$, in units where $\Gamma =1$. All scales are logarithmic, marked in powers of 10. Note that the range of validity of the approximate equation (30) is such that it gives the same results (i.e., no discernible difference in this surface plot) as the exact equation (14), even where ${\Delta }_1$ is small.

Figure 7

Cooling/heating ratio $1∕q$ for the case of laser cooling of a trapped three-level atom using the bright resonance. The surface shows $1∕q$ as a function of ${\Delta }_2$ and ${\Omega }_2$ for the case ${\omega }_z=0.2,\gamma =0.001$, and small ${\Omega }_1,{\eta }_1,{\eta }_2$, in units where $\Gamma =1$. All scales are logarithmic, marked in powers of 10. (The small irregular ripples at large $1∕q$ are a numerical artifact.)