Transcript Slide 1

Absolute absorption on the rubidium D lines:
1
comparison between theory and experiment
P. Siddons, C.S. Adams, C. Ge and I.G. Hughes
Department of Physics, Durham University, Durham DH1 3LE, UK
Having a theoretical model which predicts the absorption and refractive index of a medium is useful, for example, in predicting the magnitude of pulse propagation effects [1,2]. We study the
Doppler-broadened absorption of a weak monochromatic probe beam in a thermal rubidium vapour cell on the D lines. A detailed model of the susceptibility is developed which takes into
account the absolute linestrengths of the allowed electric dipole transitions and the motion of the atoms parallel to the probe beam. All transitions from both hyperfine levels of the ground
term of both isotopes are incorporated. The absorption and refractive index as a function of frequency are expressed in terms of the complementary error function. The absolute absorption
profiles are compared with experiment, and are found to be in excellent agreement provided a sufficiently weak probe beam with an intensity under one thousandth of the saturation intensity
is used. The importance of hyperfine pumping for open transitions is discussed in the context of achieving the weak-probe limit. Theory and experiment show excellent agreement, with an
rms error better than 0.2% for the D2 line at 16.5oC.
Why rubidium and caesium?
Doppler-broadened resonance lines
Theoretical weak probe transmission
The transmission, T, of a weak laser beam after traversing a cell of length L is
The absorptive and dispersive properties of a medium are encapsulated in the electric
susceptibility,  . In a Doppler-broadened medium, we integrate the contribution of atoms
with longitudinal velocity , v, over the Maxwell-Boltzmann distribution, g(v).
1
 ()  CF2 d 2 N
 0



f (  kv)  g (v)dv
1
i  i 
1 ( v / u ) 2
f ()   1    , g (v) 
e
u 
2 
2 
T  exp[ L  i ]
i
Im[ i ( y )]
 i ()  k
2(2I  1)
ku
2
2
where CF , d , N , are the transition strength, dipole moment and atomic
number density.     0 is detuning from resonance
f is a lineshape factor, of FWHM , from
the steady state solution to optical Bloch
equations of a two-level atom in the
weak-probe limit. See [3]
1
u  2kBT / M is the typical
velocity of an atom of mass
M at temperature T
Re[ ( y )]  i
Erfc[ z ]  1 
e
2
2


z
0
(i )
1
4
a i 2 y 2
e t dt
2
Erfc[  iy ]  e
a
2
1
4
 a i 2 y 2

Erfc[ a2  iy ]
87
(ii )
2
1
4

2(2I + 1) is the ground state degeneracy of the atom, assuming magnetic
sublevels are degenerate and equally populated. For 85Rb (I = 5/2)
degeneracy = 12, for 87Rb (I = 3/2) degeneracy = 8
Cell of length 75 mm with rubidium
vapour at its natural composition of
72% 85Rb, 28% 87Rb
The integral can be solved analytically to give the following results for the real and
imaginary components of the susceptibility (with substitutions y   / ku , a   / ku )
The imaginary part characterises absorption, the
  a i 2 y 
i 2 ay
a
a
Erfc[ 2  iy]  e Erfc[ 2  iy]
Im[ ( y )] 
e
real part dispersion.
2
where the sum is taken over all absorption coefficients,  , of
contributing transitions
D2
20oC
The real part can be differentiated to arbitrary
powers of y, to evaluate e.g. the group refractive
index
D1
30oC
Rb Fg  2  Fe
85
Rb Fg  3  Fe
(iii)
85
(iv)
87
Rb Fg  2  Fe
Rb Fg  1  Fe
Colour lines show individual hyperfine
transitions (12 for D2, 8 for D1)
Erfc[z] is the complementary error function. Most mathematical packages can evaluate
this rapidly and to arbitrary precision
Transition frequencies for D2 [4,5,6] and D1 [7]
Effects of hyperfine pumping
The weak-probe limit
Absolute absorption
Lines marked * (known as ‘laser cooling transitions’) are
dominated by closed transitions and are hence relatively
unaffected by optical pumping. The other transitions are
open and show reduced transmission due to population
redistribution among ground states
Line-centre transmission as a function of I / Isat. The data
points converge to the transmission expected by theory (black
lines) towards I / Isat = 0.001, corresponding to I ~ 20 nW/mm2.
Note that optical pumping also depends on beam size [9], so
this weak-probe limit only applies to a beam width of ~2 mm.
Normalised absorption coefficient  ( I ) /  (0) for the
open transitions of D2 and D1. The black line is the
predicted Doppler-broadened two-level atom
saturation behaviour of 1 / 1  I / I sat
Measured at I / I sat  0.1, where Isat is the saturation intensity [8]
D2
25.4oC
D1
35.6oC
For a multilevel
atom with more
than one ground
state,
hyperfine
pumping tends to
dominates
over
saturation effects
[10]
*
*
D2
25.4oC
*
D1
35.6oC
1.6 W/mm 2
D2 & D1
1.5 W/mm2
*
To avoid saturation, I < Isat
Experimental method
To avoid pumping, I << Isat
Experimental comparison
Experimental Apparatus
• External cavity diode lasers were the source of light (Toptica DL100
at 780.2 nm and 795.0 nm for D2 and D1 respectively).
Telescope
With knowledge of how weak the probe beam had to be, we recorded a series of D2 spectra
at different temperatures. The probe intensity was 32 nW/mm2 i.e. I / Isat = 0.002.
• Before the cell the beam had a radius of (2.00 +/- 0.05) mm.
Heater
Top: 16.5oC
Middle: 25.0oC
Bottom: 36.6oC
Frequency Linearization
The laser frequency scan from the piezo-electric drivers is nonlinear
(shown for D2, left) and needs to be calibrated.
• Sub-Doppler spectroscopy was performed to provide reference
frequencies. The crossing angle between probe and counterpropagating pump within the vapour cell was 6 mrad.
• The Fabry-Perot etalon was used to assist with calibrating and
linearizing the frequency scan. A plane-plane cavity was used, with
a separation of the mirrors of 250 mm, with a free-spectral range of
0.60 GHz.
D2
(a) The reference frequencies (marked X) show deviation from their expected positions by up to 50 MHz
(b) After calibration, the points lie close to a straight line. The inset shows the residual nonlinearity to be less than 5 MHz
Transmission
difference for the
16.5oC spectrum
Future work
We will use the predicted absorption and
dispersion to find the optimum experimental
conditions to produce slow light pulses.
Using a hot atomic vapour it is possible to
delay nanosecond pulses by many times
their width, whilst still having relatively high
transmission and low pulse dispersion
D1
150oC
Predicted group refractive index and absorption coefficient.
For large delay/transmission, we detune from resonance by
several GHz
Conclusion
We have developed a model of the Doppler-broadened electric susceptibility for the Rb D
lines which uses tabulated coefficients and the complementary error functions for fast and
accurate evaluation of atomic spectra.
We have shown the model accurately predicts laser transmission provided the light intensity
is in the weak-probe limit, giving an rms error of 0.2% for the D2 line at 16.5oC.
From our verification of the absorptive properties of the medium, we can be confident that the
model is also able to accurately predict dispersion caused by the medium.
References
Acknowledgements
[1] Camacho R M, Pack M V, Howell J C, Schweinsberg A and Boyd R W 2007 Phys. Rev. Lett. 98 153601.
[7]
Banerjee A, Das D, and Natarajan V 2004 Europhysics Letters 65(2) 172.
[2] Vanner M R, McLean R J, Hannaford P and Akulshin A M 2008 J. Phys. B 41 051004.
[8]
Corney A Atomic and Laser Spectroscopy (Oxford University Press, Oxford, 1977).
[3] Loudon R The Quantum Theory of Light (Oxford University Press, Oxford, Third edition, 2000).
[9]
Sherlock B E and Hughes I G 2008 (in preparation).
[4] Arimondo E, Inguscio M, and Violino P 1977 Rev. Mod. Phys. 49 31.
[10] Smith D A and Hughes I G 2004 Am. J. Phys. 72 631
[5] Gustafsson J, Rojas D, and Axner O 1997 Spectrochimica Acta Part B 52 1937.
[6] Rapol UD, Krishna A, and Natarajan V 2003 Euro. Phys. J. D 23 185
1 P. Siddons, C.S. Adams, C. Ge and I.G. Hughes J. Phys. B 41 155004 (2008)
This work is funded by EPSRC