Quantum limits to superluminal pulse advancement
Michael D. Stenner and Daniel J. Gauthier
Duke University Department of Physics
Supported by the National Science Foundation
This is the text of the talk given by Michael D. Stenner at:
XXXI Winter Colloquium on the Physics of Quantum Electronics,
Snowbird, Utah, January 7-11, 2001.
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Title
Good morning. My name is Michael Stenner and I’ll be talking to you about
“Quantum limits to superluminal pulse advancement.”
This is based on work currently being done by myself and Dan Gauthier at
the Duke University Department of Physics.
This work is supported by the National Science Foundation.
We find this work interesting for a number of reasons:
• First, superluminal propagation is still a bit controversial. There is still
some disagreement about the interpretation of this process.
• Also, this process may find practical application in optical systems in order
to compensate for delays introduced elsewhere.
• More generally, the idea of tailoring dispersion has obvious applications.
Our present work focuses on exploring the limitations to this process.
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Some related work
Superluminal propagation is not a new topic. While I can’t list all of the work
done in this area, I’d like to mention a few areas of research and a few experimental techniques.
Very early in the 1900’s, Brillouin and Sommerfeld did a lot of theoretical
work in this area. This laid the foundation of our understanding of this topic
for most of the 20th century.
There have been a number of interesting microwave experiments recently.
Although, because we work in the optical regime, I only mention them here.
There has been a large body of recent theoretical work, with large and
notable contributions from Garret and McCumber, and also from Chiao and
co-workers.
Superluminal group velocities arise when a pulse propagates through an
anomalously dispersive medium. That is, when the slope of the index of refraction vs frequency is negative.
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One case in which this happens is in the center of an absorption line. This
was used by Chu and Wong in 1982. The only drawback of this technique is
that advancement is necessarily accompanied by absorption.
The frequency derivative of the index of refraction is also negative in the
wings of a gain line. While this does not have the absorption problem that the
previous method does, the slope is not linear, so there will always be some pulse
distortion.
A third technique, proposed by Chiao et al., and implemented last year
by Wang et al., uses two gain lines. Between the gain lines, the slope of the
index of refraction can be very linear, allowing for large advancement with little
distortion and no absorption.
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Limitations of these techniques
If you’re interested in large advancement, these first two techniques have clear
disadvantages: either the pulse is strongly absorbed or severely distorted.
What about the third method? It is certainly intriguing. It seems to get
around the problems of the first two but we find that it may ultimately be
limited by superfluorescence.
The remainder of the talk will deal with these limitations to this last method.
The limit that we’re interested in is a limit on what we call the relative
advancement, which is the advancement of a pulse compared to a pulse moving
through vacuum, measured in units of the pulse width.
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Maximizing advancement
In order to increase advancement, one must create a large negative slope in the
index of refraction (vs frequency). There are several parameters that one can
adjust to this end.
You can adjust the gain: it turns out that you simply want the gain as large
as possible.
The other parameters are: the width of the two gain lines, the separation
of the two gain lines, and the spectral width of the pulse. These three are all
optimized as ratios of each other, so setting one fixes the other two.
With these optimizations, we find that the relative advancement is roughly
g0 L, where g0 is the line-center gain coefficient, and L is the length of the
medium.
The pulse, whose frequency is between the lines, also experiences small gain:
in our case, gpulse ≈ 0.3g0 .
Note that gain and dispersion are intimately related; a direct result of the
Kramers-Kronig relations. In this scheme, advancement is necessarily accompanied by gain. You can reduce the gain that the pulse experiences, but that
means that the advancement will also be reduced.
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Limitations from superfluorescence
So how is this process limited?
Well, superfluorescence dominates for g0 L & 15. At this point, quantum
fluctuations can grow to become macroscopic fields which swamp other fields.
Based on the equation on the previous slide, this leads to a relative advancement limit of about two. This means that a pulse can be advanced by about
twice its width at most!
You may have noticed that I have been speaking entirely in terms of the
relative advancement, rather than the group velocity.
The reason is that in these extreme cases of fast or slow light, the group
velocity depends not only on the relative advancement, but also on the pulse
length and cell length. So, despite these limitations on the relative advancement,
you can get very small negative (i.e. fast) group velocities by using long pulses
or short cells.
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Large Raman gain
As I said earlier, in order to get large advancement, you need large gain. We
use a specific technique to get large Raman gain in 39 K vapor.
We apply a pump beam to our vapor cell that is slightly detuned to the blue
of the D1 line. Because it’s closer to resonance with the lower ground state
transitions, the population is preferentially pumped out of the lower ground
state and into the upper ground state.
Then, a probe beam, whose frequency is greater than the pump’s by the
ground state splitting, experiences Raman gain as the atoms go from the lower
ground state to the upper ground state via an intermediate virtual level.
Upon considering the degenerate Zeeman states, one finds that quantum
interference requires orthogonal polarizations for pump and probe.
Because we use a single beam for both optical and Raman pumping, we can
use very intense Raman pumps without fear of de-populating the upper ground
state. This allows us to achieve very high gain.
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Bichromatic pump field
In the actual experiment, we need gain at two frequencies, not just one. In order
to do this we simply apply two pump fields at slightly different frequencies.
By tuning the frequencies of these pump fields, we can control the frequency
separation between the two gain lines.
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Physical setup
This is a simplified diagram of our physical setup.
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We use a single Titanium:Sapphire laser to generate all of our beams. The
beam out of the Ti:Saph is split into orthogonal linear polarizations by a polarizing beamsplitter.
One leg of this is shifted up in frequency by 252 MHz by an acousto-optic
modulator (AO). This will be the probe beam.
The other leg goes through two AOs, producing two beams, shifted down in
frequency by 220 and 200 MHz. These are the pump beams.
The probe and pumps are then recombined by a second polarizing beamsplitter. The three beams together travel through our 39 K vapor cell. At the
cell output, the pumps are picked off by a third polarizing beamsplitter and the
probe is measured on a fast detector.
Using the probe AO, we can both pulse the beam and scan its frequency.
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Pulse advancement
When we send a pulse through our system, with its frequency set to the anomalous dispersion region between the gain lines, we see significant advancement.
Here is a plot of intensity on our fast detector vs. time. The black curve is
a pulse that traveled through vacuum. The red curve is a pulse that traveled
through our cell. Note that the red pulse arrived much earlier than the vacuum
pulse.
Although the pulse experiences some compression and ringing, both the peak
and leading edge of the pulse are advanced.
In this case, g0 L = 9.5, the pulse gain was 16 (the intensity was rescaled for
the plot), the pulse had a width of 184 ns, and was advanced by 130 ns. This
leads to a group velocity of about −0.005c and a relative advancement of about
70%.
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Pulse delay
Now, if we tune the pulse frequency to the center of one of the gain lines, we
see that the pulse is broadened and delayed. Again, the black line is a pulse
through vacuum, and the red line is a pulse through our cell.
Here, g0 L is again 9.5, the pulse gain is strongly saturated here (the pulse is
again rescaled in the plot), the pulse has a width of 184 ns, was delayed by 41
ns, leading to a group velocity of 0.032c and a relative delay of 20%.
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Experimental challenges
We have seen some very interesting results so far. We have observed large
advancement and delays and hope to approach the limits of this process in the
future.
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This is, however, a talk about a work-in-progress, and so there are a number
of interesting challenges that we are currently dealing with. I would like to share
some of those with you.
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Strong bichromatic fields
One such challenge is the fact that our Raman pumps are effectively a strong
bichromatic field. We cannot think of these two Raman gain features as the
incoherent combination of two gain lines. The reason is that bichromatic fields
create complex dressed level structures.
For example, here on the right is part of the level structure for a two level
atom and two fields whose frequencies are ω0 ± δ. Here, ω0 is the frequency of
the natural atomic resonance.
This actually creates a comb of absorption frequencies spaced by the detuning between the two fields.
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New gain features from bichromatic field
We see a very similar effect in our system. Here is a plot of transmitted probe
power as a function of probe frequency. When we have only one pump on (the
red line), we see gain at only one frequency, as we would expect. When we turn
only the other pump on (the blue line) we again see gain at only one frequency,
as expected. Now, if we turn on both pumps, we see gain at both the expected
frequencies, but also at some unexpected frequencies. This may be because of
some more complex dressed state structure.
Note that our scan range is limited by our probe AO here. Earlier, when we
were using a diode laser for our probe beam, we saw as many as 15 or 20 such
“unexpected peaks.”
Also note that the total pump power was the same for all three plots: the
gain is a complex function of the pump powers as well.
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Another issue: Polarization noise
As you saw earlier, our pump and probe beams have orthogonal polarizations.
This allows us to have excellent beam overlap in the cell and separate them with
a polarizing beamsplitter.
However, if our pump beams’ polarizations change, then pump light gets
through the polarizing beamsplitter and results in noise on the detector.
In our case, we put in about 50 mW of total pump power and have about
5 mW make it to the detector. If we send in pump beams that are linearly
polarized to a part in 104 , what comes out is only linear to a part in 10. The
outgoing light has some ellipticity, and the major axis is rotated by about 20 ◦ .
The worst part is that the phase difference between the linear components has
unknown time-dependence.
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This effect also depends on the bichromatic nature of the light in some
complex way.
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Yet another issue: Pulse modulation
Yet another issue that we’re dealing with is that our pulses are actually modulated at the beat frequency of the two pumping beams.
This is a plot of intensity on the detector vs. time. The black line is the
pulse through vacuum as before. The blue line is an averaged pulse through
the medium (like ones on earlier slides). The red line is a single pulse with no
averaging. Note that the oscillation is at about 20 MHz, which is the separation
between the two pump beam frequencies.
We have a number of ideas about why this is happening, but are not yet
convinced that we completely understand it.
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Conclusions
We have predicted that this process for superluminal pulse advancement will
be limited to advancing the pulse by twice its width. So far, we have seen
advancements of 70% and delays of 20%.
In the future, we hope to achieve larger advancements as we increase the
gain. Along the way, we must deal with the effects of our bichromatic pump
field, the polarization noise in our system, and the pulse modulation.
Thank you for your attention.
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