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Transcript
Electromagnetically Induced Transparency with Broadband Laser Pulses
D. D. Yavuz
Department of Physics, University of Wisconsin, Madison, WI, 53706
Time-delay-bandwidth Product
Over the last decade, counterintuitive optical effects using Electromagnetically
Induced Transparency (EIT) have gained considerable attention [1-3]. The
essence of EIT is to create a narrow transparency window in an otherwise
opaque medium using quantum interference. Noting Fig. 1, in a three state
atomic medium the quantum interference is achieved with a coupling laser
beam that couples states |2 and |3.
In the perturbative limit where the probe laser beam is much weaker than the
coupling laser beam, the susceptibility for the probe wave is:
13 : dipole matrix
N 13

 ( p )  4
 0 22(   )  j  c 2
2
1

c
p c
|2
2
|1
/
0
j
2
   p N 13 c1c3*
z

c
j
2
  c N 23 c2c3*
z

1
In Fig. 1, the imaginary part of the susceptibility is plotted for Ωc=Г/3, and
Δ=- Γ, Δ=0, and Δ=Γ respectively. In all cases perfect transparency is obtained
at exact two-photon resonance. Furthermore, even though the lineshape
becomes asymmetric for Δ≠0, the steep dispersion is maintained at the point
of vanishing absorption.
frequency
The scheme of Fig. 2 is identical to the EIT with matched pulses as suggested
by Harris et. al. [6]. In this work, we extend the suggestion of Harris, and show
that using matched frequencies for the two transitions, one can obtain a large
group delay for a broadband probe pulse. One can therefore obtain a large
time-delay-bandwidth product.
(a)
We proceed with a perturbative analytical solution of Eqs. (1) and (2) to get an
insight into the results of Figs. 3 and 4. We follow closely the formalism of Eberly
and colleagues [7]. We proceed perturbatively and take the probe beam to be much
weaker when compared with the coupling beam. With counter-intuitive pulse
sequence the medium can be prepared in the dark state when the following
two-field adiabatic condition is satisfied:
 p
c
3
p
 c
 c


z=0
z=2.5 mm
 p 0
1
c 0
400
time (s)
0
1
time (s)
2
(b)
z=0
z=2.5 mm
2

 f ( )
(5)
We note that the adiabatic criteria of Eq. (5) is independent of the bandwidth of the
time function f ( ) . This is the key reason why the dynamics of the medium is largely
independent of the time variation common to both fields, f ( ) , as long as Eq. (5) is
satisfied. With the medium prepared adiabatically, the solution of the Schrodinger
equation [Eq. (1)] for the probability amplitudes, including the first non-adiabatic
correction to c3 is:
*
2    p 
c1  1, c2   * , c3  j
c   *c 
c
*p
100
200
N : atomic density
   / 0
(2)
300
 p 0 , c 0 : long envelopes
(3)
f ( )  q f q exp( jq )
0
1
time (s)
In Eq. (3),  p 0 , c0 are long envelopes that allow adiabatic preparation of the
medium and the function f ( ) defines the broad set of frequencies that are
considered, f ( )  q f q exp( jq ) . The function, f ( ), is dimensionless and is
2
normalized such that q | f q |  1 . As we will show, the dynamics of the EIT medium
will be determined by the long envelopes  p 0 , c0 and will largely be independent
of the time variation that is common to both fields f ( ).
2
0
1
time (s)
2
z=0
z=5 mm
0
50
100
150
200
time (s)
Figure 3: Slowing of broadband light pulses in 87Rb vapor with an atomic density of N=1012 /cm3. At the
beginning of the cell the probe beam is assumed to be  p ( z  0,  )   p 0 ( ) f ( ) where  p 0 ( ) is a long
envelope with a Gaussian width of 12 µs and f ( ) is a rapidly varying time waveform. The bottom plot
shows the envelopes of the probe beam at z=0 and z=5 mm respectively. The probe beam is delayed by
84 µs. The insets zoom in on the central portion of the waveform to display the rapidly varying square wave.
At the output, the structure of the square waveform is almost perfectly preserved. The time-delay-bandwidth
product that is achieved in this simulation is ≈103.
Figure 3 shows the envelope of the broadband probe pulse at z=0 and z=5 mm
respectively. The probe pulse propagates with a group velocity of vg=58 m/s and
is delayed by 84 µs at the end of the medium. The shape of the square-wave
temporal waveform is almost perfectly preserved demonstrating that all Fourier
components of the input pulse propagate without significant loss and phase shift.
The time-delay-bandwidth product that is achieved in this simulation is ≈103.
400
time (s)
(6)
With the solution of Eq. (6), the propagation equation for the probe beam becomes:
Figure 4: Stopping of broadband light pulses using EIT. The probe pulse is stored for 100 µs in part
(a) and 250 µs in part (b) and then released. Stopping of the probe pulse is achieved by smoothly
turning down the intensity of the coupling laser beam. In both plots, the normalized intensity of the
probe pulse at z=0 and z=2.5 mm is plotted. The dotted line is the intensity envelope of the coupling
laser beam. The inset in (b) is a zoom in on the central portion of the released pulse.
 p

 p 0
z
Following Eberly and colleagues [7], we make the following change of variable in

Eq. (7),  ( )   f ( ) d . With this transformation, the analytical solution of Eq. (7) is:
2
2
2 1   p 


   p N 13
*
z

c   c 
2
1  p 0
2
2
f ( )    p N 13
2

c 0 
[7] R. Grobe, F. T. Hioe, and J. H. Eberly, Phys. Rev. Lett. 73, 3183 (1994).
(7)
Conclusions
0
~
 p 0 ( z , )   p 0 (  z / v~g ),
1 2 p N 13 ~

,  p 0 ( )   p 0 (0, )
2
v~g
 c 0
2
(8)
(4)
With  p ( z, )   p0 ( z, ) f ( ) and c ( z, )  c0 f ( ) ( c 0 independent of space and time),
the two field adiabatic condition of Eq. (4) reduces to:
300
c ( z  0, )  c 0 ( ) f ( )
Below, we present a numerical simulation that demonstrate slowing of broadband
light pulses in 87Rb vapor. In this simulation, we assume an atomic density of
N=1012 /cm3 and take the two Raman states to be |1|F=1,mF=0 and
|2|F=2,mF=0 hyperfine states of the ground electronic state 5S1/2. The excited
state is chosen to be |3|F΄=2,mF΄=1 of 5P3/2. We take the two laser beams to
have the same circular polarization. We take f ( ) to consist of 31 equally spaced
frequencies and choose the amplitudes and the phases of the Fourier components
such that the synthesized time function is a square wave with a period of 0.66 µs.
The total spectral content of the probe pulse is therefore 45 MHz. We assume a
Gaussian envelope,  p 0 , for the probe laser beam with a Gaussian width of 12 µs.
The envelope for the coupling laser beam, c 0 , smoothly turns on to its peak value
of c0, peak   / 3 and stays constant (not shown in Fig. 3). To assure adiabatic
preparation of the medium, the coupling laser beam is turned on before the probe
laser beam (counter-intuitive pulse sequence).
Perturbative Analytical Solution
Figure 4 is a numerical simulation that demonstrates stopping of broadband
light pulses. Here, we take the coupling beam envelope to smoothly turn-off
for a duration of 100 µs in (a) and 250 µs in (b) and then turn back on again.
The probe beam is therefore stored in the medium for a controllable amount
of time and then released. For both cases, the released pulse contains all
the Fourier components of the input pulse with relative phases and
amplitudes preserved. The inset in (b) is a zoom in on the central portion of
the probe envelope that shows the square-wave temporal structure.
0
(1)
 p ( z  0, )   p 0 ( ) f ( )
Figure 2: The suggestion of our scheme. We consider the propagation of a broad set of frequencies for the
probe laser beam where each frequency has a matching component in the coupling laser beam such that the
two-photon resonance condition is maintained. EIT is achieved in parallel channels for each of the frequency
components.
Numerical Simulation
200
 : excited state decay rate
[4] Z. Dutton, M. Bashkansy, M. Steiner, and J. Reintjes, Proc. SPIE 5735, 115 (2005).
[5] Q. Sun, Y. V. Rostovtsev, J. P. Dowling, M. O. Scully, and M. S. Zubairy, Phys. Rev. A 72, 031802(R) (2005).
[6] S. E. Harris, Phys. Rev. Lett. 70, 552 (1993).
[1] M. O . Scully and M. S. Zubairy, “Quantum Optics” (Cambridge University Press, 1997).
[2] S. E. Harris, Phys. Today 50, No. 7, 36 (1997).
[3] O. Kocharovskaya and P. Mandel, Phys. Rev. A 42, 523 (1990).
100
c : Rabi frequency of the coupling laser beam
|1
Figure 1: The susceptibility for the probe laser beam as a function of two-photon detuning, for
Δ=-Γ,Δ=0, and Δ=Γ respectively.
0
 p : Rabi frequency of the probe laser beam
We analyze the propagation of a broad set of frequencies for the probe and coupling
laser beams through an atomic system defined by the above coupled equations. We
solve Eqs. (1) and (2) with the initial condition that all atoms start in the ground state
|1 and the following boundary condition at the beginning of the cell (z=0) for the
two laser beams:
|2
=
-3 -2 -1
c1 , c2 , c3 : probabilit y amplitudes
The decay processes are assumed to be to states outside the system. With the
probability amplitudes calculated by Eq. (1), the slowly varying envelope Maxwell’s
equations for the two laser beams are:
…
power spectral density
0
|3
p
=0
-2 -1

 
3
|3
…
normalized absorption
2
c1 j
  p c3
 2
c2 j
 c c3
 2
c3 
j
j
 c3  *p c1  *c c2
 2
2
2
 p
…
1
In this work, motivated by the results of Fig. 1, we consider a broad set of
frequencies for the probe laser beam where each frequency has a matching
component in the coupling laser beam such that the two-photon resonance
condition is maintained. Noting Fig. 2, EIT and therefore slow light is achieved in
parallel channels for each of the frequencies of the probe laser beam.
…
0
We proceed with the analysis of schematic of Fig. 2. Working in local time, the
Schrodinger equation for the probability amplitudes of the three states in the
interaction picture are:
element
 : excited state decay rate
 : one photon detuning
 : two - photon detuning
=-
-1
EIT provides a unique way to controllably delay and coherently store light pulses.
However, narrow transparency window of EIT puts stringent limitations on the
bandwidth of the light pulses that can be slowed and stopped inside the medium.
A key figure of merit that is usually discussed in this context is the
time-delay-bandwidth product which is obtained by multiplying the bandwidth of the
optical pulse with the delay time of the pulse while propagating through the EIT
medium. The largest time-delay-bandwidth product that has been experimentally
demonstrated using EIT is ≈5. Recently Dutton and colleagues [4] and Zubairy and
colleagues [5] have suggested schemes to overcome this limitation.
Numerical Simulation
Formalism
normalized probe intensity
Introduction
In agreement with the numerical results of Figs 3 and 4, Eq. (8) shows that the
probe pulse propagates without attenuation and with a group velocity determined
by the intensity of the coupling laser beam, | c 0 |2. Remarkably, this group velocity,
and therefore the time delay obtained while propagating through the EIT medium
is independent of the bandwidth of f ( ) assuming infinite dephasing time of the
Raman transition. As a result it becomes possible to obtain large group delays for
large bandwidth optical pulses.
One important practical application of EIT with large bandwidth pulses is to optical
information processing. Our scheme may provide a unique way to controllably
delay and coherently store large bandwidth optical pulses. However, a significant
disadvantage of our approach is that it requires a time varying coupling laser beam
with Fourier components exactly matched to the probe laser beam.
Our approach may also find applications in achieving giant nonlinearities effective at
single photon levels using EIT [8-12]. When compared with the traditional narrow
bandwidth EIT schemes, it may be advantageous to use larger bandwidth, therefore
higher peak power, single photon pulses.
[8] H. Scmidt and A. Imamoglu, Opt. Lett. 21, 1936 (1996).
[9] S. E. Harris and Y. Yamamoto, Phys. Rev. Lett. 81, 3611 (1998).
[10] M. D. Lukin and A. Imamoglu, Phys. Rev. Lett. 84, 1419 (2000).
[11] H. Kang and Y. Zhu, Phys. Rev. Lett. 91, 093601 (2003).
[12] D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 93, 183601 (2004).
We have extended the suggestion of Harris et. al. [6], and demonstrated that when
matched Fourier components are used, the dynamics of an EIT system decouple
from the time variation that is common to both fields. As a result, it becomes possible
to obtain slow light and large group delays with large bandwidth optical pulses.
Our numerical simulations show that time-delay-bandwidth products exceeding 103
is readily realizable with current experimental parameters.
We expect possible applications to optical information processing using EIT and to
achieving giant nonlinearities effective at the single photon levels.
Acknowledgements
I would like to thank Brett Unks and Nick Proite for helpful discussions. This work
was supported by a start-up grant from the department of Physics at University
of Wisconsin-Madison.