Quantum limits to superluminal
pulse advancement
Michael D. Stenner
and
Daniel J. Gauthier
Duke University Department of Physics
Supported by the National Science Foundation
Some related work
Much theoretical work in the early 1900’s∗
Microwave experiments
Recent theoretical work†‡
Some experimental methods use:
• On-resonance
absorption
(used by Chu &
Wong, 1982)
• Gain wings
• Gain doublets
(used by Wang,
Kuzmich, & Dogariu,
2000)
∗ Brillouin, L. Wave Propagation and Group Velocity (Academic, New
York, 1960)
† Garret & McCumber, Phys. Rev. A 1, 305-313 (1970).
‡ see, for example, several papers by R.Y. Chiao and co-workers.
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Limitations of these techniques
• On-resonance absorption
large advancement comes with large absorption
• Gain wings
dispersion is not linear, so large advancement
comes with severe distortion
• Gain doublet
Intriguing! May ultimately be limited by superfluorescence
Consider the relative advancement
t
A ≡ adv
τp
L
t adv
vapor cell
vacuum
τp
3
Maximizing advancement
If the physical parameters
are optimized for maximal
advancement, then
tadv
A≡
≈ 0.13 g0L,
τp
dn
d∆
∆o
γ
where g0 = −α0 = line
center gain coefficient.
The pulse also experiences
small gain: gpulse ≈ .3 g0 in
our case.
pulse
spectrum
Note that gain and dispersion are intimately related; a direct result of the Kramers-Kronig
relations.
The pulse gain can be reduced, but then the
advancement is also reduced.
4
Limitations from superfluorescence
Superfluorescence dominates for g0L & 15, so
t
A ≡ adv . 2
τp
1
vg = 1
τp
−
A
c
L
⇒
vg ≈
1 L
A τp
small negative vg can be achieved by using
long pulses and/or short media!
5
Large Raman gain
probe
pump
39
K
4 2P1/2 F=2
F=1
ν
2
ν+∆g
ν
ν
F=2
4 S1/2
∆g
F=1
∆g = 462 MHz
;
c
λ = ≈ 770 nm
ν
• Upon considering the degenerate Zeeman states,
one finds that quantum interference requires
orthogonal polarizations for pump and probe∗
• Can achieve large gain because we can use very
strong beams for Raman pumping†
∗ M. Poelker and P. Kumar, Opt. Lett. 17, 399 (1992).
† H.M. Concannon et al., Phys. Rev. A 56, 1519 (1997).
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Bichromatic pump field
4 2P1/2 F=2
F=1
ν1+ ν2
+ ∆g
2
4 S1/2
F=2
F=1
ν2
probe
ν1
pumps ν
2
39
K
∆g
Gain
2
ν1
Frequency
7
Physical setup
39
K vapor cell
150 C n=1013cm-3
L = 20 cm
fast detector
probe
pumps
ao
ν+ 252 MHz
ν
Ti:Saph
λ = 770 nm
ao
ν- 200 MHz
intensity
probe 10-6 to 1 W/cm2
each pump 20 W/cm2
ao
ν- 220 MHz
power
spot size*
nW to µ W 100 µ m
25 mW
280 µ m
*(spot size = 1/e2 intensity radius)
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Pulse advancement
occurs in anomalous dispersion region
vacuum
Intensity (arb units)
advancement
0
400
Time (ns)
800
see pulse compression and ringing
both the peak and leading edge are advanced!
g0L = 9.5
τp = 184 ns
vg = −0.005 c
pulse gain ≈ 16
tadv = 130 ns
A = 0.7
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Pulse delay
in normal dispersion region at center of gain line
When we tune the pulse to a gain line, we see
pulse broadening and delay.
Intensity (arb units)
vacuum
delay
0
400
Time (ns)
g0L = 9.5
τp = 184 ns
vg = 0.032 c
800
pulse gain: saturated
tadv = −41 ns
A = −0.2
10
Experimental challenges
We have seen large advancement and delays and
hope to approach the limits of this process.
There are a number of issues that we must deal
with on the way!
11
Strong bichromatic fields
Cannot think of the two Raman gain features as
the incoherent combination of two gain lines!
Bichromatic fields
create complex
dressed level
structures.
0 0 N+1
δ
δ
δ
δ
N+1
1
0
1
0
1
N/2-1
N/2
N/2
N/2+1
N/2+1
N/2+1
N/2+1
N/2
N/2
N/2-1
0 N+1 0
ω0
level structurea for
a two level atom
and two fields
whose frequencies
are ω0 ± δ.
0 0 N
δ
δ
δ
δ
N
0
1
0
1
0
N/2-1
N/2-1
N/2
N/2
N/2+1
N/2+1
N/2
N/2
N/2-1
N/2-1
0 N 0
N = na + nb + nc
n a nb nc
a Y. Zhu et al., Phys. Rev.
A 41, 6574 (1990).
atomic state
photons at each
frequency
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New gain features from bichromatic field
• complicated function of pump beam powers
• exhibits nontrivial synergistic behavior
Transmitted Power (mW)
0.6
0.4
0.2
Red Pump
Blue Pump
Both Pumps
unexpected
peak
δ
expected
peaks
δ
0
-40
-20
0
20
Probe Detuning (MHz)
extra feature from dressed state structure?
have seen many more features with broader scan
13
Another issue: Polarization noise
Crossed pump-probe polarizations allow good
in-cell beam overlap
... but ...
the pump beams’ polarizations change, resulting
in noise!
39
50 mW
K
5 mW
After Cell
Before Cell
y
y y’
θ
x
x’
x
Ix
Iy
20
-4
10
φ(t)
Ix’
0.1
Iy’
φx’(t) φy’(t)
?
The phase difference
between the linear
components has
unknown
time-dependence!
This also depends on the
bichromatic nature of the
light, not just the total
intensity.
Transmitted Power (mW)
8
7
6
5
4
3
2
1
0
0
Mean = 4.6 mW
Standard Deviation = 0.7 mW
5
10
Time (ms)
15
14
20
Yet another issue: Pulse modulation
Output pulses are modulated at the beat frequency
of the two pumping beams
Intensity (arb units)
pulse through vacuum
single pulse
averaged pulse
0
400
Time (ns)
800
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Conclusions
We have seen:
• Theoretical prediction that pulse advancement
is quantum mechanically limited
• Large (70%) advancement from anomalous dispersion
• Large (20%) delay from normal dispersion
We want to further explore:
• Larger advancements for higher gain
• The effects of bichromatic fields
• The polarization effects in our system
• The modulation in the pulses
(slides and transcript will be available online at
http://www.phy.duke.edu/research/photon/qelectron/)
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