Velocity of detectable information in faster-than-light pulses

The velocity of detectable information (signal velocity) in a medium capable of supporting abnormal (superluminal or negative) group velocities is calculated. This is carried out by tracking the time instant at which the signal-to-noise ratio (SNR) at the detector output reaches a predetermined threshold while considering the total classical and quantum noise of the channel in addition to the detector noise. Furthermore, the method of steepest descent is incorporated to systematically study various forms of pulse reshaping associated with superluminal propagation and its effect on SNR. By studying the behavior of SNR as a function of both space and time, the present analysis predicts the existence of a cutoff distance beyond which signal velocity of a superluminal pulse is delayed as compared to a companion pulse traveling the same distance in vacuum. Finally, the interplay between the relative strength of the medium-generated noise and the detector noise and its effect on signal velocity is discussed.

Figure 1

Index of refraction, $n\left(\omega \right)$, and group delay as a function of detuning frequency (the blue dashed horizontal line denotes the luminal group delay in vacuum). The vertical arrows mark the spectral width of the input pulse.

Figure 2

Path of the saddle points of $\left[\varphi (\omega ,{\theta }^{\prime })\right]$ as a function of ${\theta }^{\prime }$ at a distance $L=25$ cm inside the ammonia vapor. The colored contours illustrate the $\text{Re}\left[\varphi (\omega ,{\theta }^{\prime })\right]$ evaluated at ${\theta }^{\prime }=-0.45$ (which corresponds to time $t=L/c$). The first and second resonances are denoted by $\left({\omega }_{0,1}\right)$ and $\left({\omega }_{0,2}\right)$, respectively; $({{\omega }_{+}}^{\left(0\right)},{{\omega }_{+}}^{\left(1\right)})$ and $({{\omega }_{+}}^{\left(2\right)},{{\omega }_{+}}^{\left(3\right)})$ signify the branch cuts for the resonances ${\omega }_{0,1}$ and ${\omega }_{0,2}$.

Figure 3

Normalized medium response at propagation distance $L=25$ cm. The inset depicts the arrival of the pulse front after duration $L/c$ in compliance with Einstein's causality.

Figure 4

Relative signal delay in ammonia vapor cells with respect to a vacuum channel as a function of propagation distance in units of $c/\delta$. The relative signal delay is compared with the corresponding relative group delay (denoted by the blue line). This comparison is carried out for three different noise levels at the detector side.

Figure 5

Signal-to-noise ratio as a function of time at a propagation depth of 25 cm in the ammonia vapor. The inset shows the information window at vicinity of the detection threshold as compared to a vacuum channel of the same length.

Figure 6

$\text{SNR}(z,t)$ at different propagation depths $L=2.083 c/\delta ,2.5 c/\delta$, and $2.917 c/\delta$ (which corresponds to 25, 30, and 35 cm, respectively) in an AGD medium and compared to a vacuum channel. Information velocity is retarded in an AGD medium despite the exhibited superluminal behavior indicated by the center-of-mass calculations.