Subrecoil cooling and Lévy-flights

In both subrecoil schemes had excitation probability P(v) with dip at v=0
p=0: Dark state
p$\not=$0: Momentum diffusion which transfers atoms into dark state
Fluorescence rate dependence on momentum:
(=excitation probability)

The smaller p the longer the delay between successive spontaneous emissions

Estimate efficiency of cooling process: Define trapping zone p<p_trap.
Atoms remain trapped for time $\tau(p)$ of order 1/R_F(p).
$R_{F}\propto p^{\alpha}$ for small p ($\alpha=2$ for VSCPT)
Evolution of an atom:
successive trapping periods $\tau_{1},\tau_{2},\ldots$
diffusion periods $\hat{\tau_{1}},\hat{\tau_{2}},\ldots$ (first return times)
Consider 2N alternating trapping and diffusion periods
Total trapping time = T(N) = $\sum_{i=1}^{N}\tau_{i}$
Total escape time = $\hat{T}(N)$ = $\sum_{i=1}^{N}\hat{\tau_{i}}$
How do T(N), $\hat{T}(N)$ grow with N?
$\tau_{i}$, $\hat{\tau_{i}}$ independent random variables
$\Rightarrow$ need to find probability distributions P($\tau$), $\hat{P}(\hat{\tau}$)
Introduce distribution $\Pi(p)$: Distribution of p for atoms that fall into the trap.
Atoms trapped if p<p_trap, $p_{trap}<<\hbar k$, but p-distribution of recoil has width of order $\hbar k$
$\Rightarrow \Pi(p)=constant=1/2p_{trap}$
Calculate P($\tau$) asymptotically for large $\tau$:
$\Pi(p)dp=P(\tau)d\tau$
(Trapping probability, $\tau\rightarrow\infty$ for $p\rightarrow 0$)
$\Rightarrow P(\tau)=\Pi(p)\frac{dp}{d\tau}\propto \tau^{-(1+1/\alpha)}(\propto\tau^{-3/2}$ for $R_{F}(p)\propto p^{2}$)
Broad distribution of trapping times, $<\tau>=\infty$
T(N) is Lévy-sum: T(N) $\propto N^{\alpha}$( $\propto N^{2}$ for $R_{F}(p)\propto p^{2}$) for large N.
Sum dominated by largest term, i.e. longest trapping time is of the order of total observation (or interaction) time
How about escape times?
Different models to consider:

1.

atomic momentum confined to |p|<p_max (e.g. friction mechanism at large p)
$\Rightarrow$ have mean value for first return times to the trap
$\hat{\tau_{i}}$ of the order of diffusion time from 0 to p_max
$\Rightarrow \hat{T}(N)\propto N$

2.

constant fluorescence rate for large p (in real situation: p_e large )
$\Rightarrow \hat{\tau_{i}}$ coincide with first return times of Brownian motion
$\Rightarrow \hat{T}(N)\propto N^{2}$

3.

realistic R_F(p) as in figure (probed for long interaction times)
$\Rightarrow \hat{P}(\hat{\tau})$ even broader, $\hat{T}(N)\propto N^{4}$

Conclusion for cooling efficiency in the three models:

1.

T(N) dominates over $\hat{T}(N)$ for large N, all atoms get trapped

2.

T(N) and $\hat{T}(N)$ both $\propto N^{2}$ (for $R_{F}(p)\propto p^{2}$), finite fraction of atoms get trapped.

3.

$\hat{T}(N)$ dominates over T(N), atoms lost from trap for long interaction times

Cooling in 2 and 3 dimensions:
$P(\tau)$ becomes narrower
$\hat{P}(\hat{\tau})$ becomes broader
$\rightarrow$ cooling efficiency decreases
For $R_{F}(p)\propto p^{2}$ get
d=2: $P(\tau)\propto\tau^{-2}\rightarrow T(N)\propto N log{N}$
d=3: $P(\tau)\propto\tau^{-5/2}\rightarrow T(N)\propto N^{2/3}$
Raman cooling:
Can tailor excitation probability, i.e. manipulate $\alpha$ in $R_{F}(p)\propto p^{\alpha}$ (by choice of pulse shapes)
Which $\alpha$ leads to most effective cooling (both accumulation of atoms in trap and lowest achievable temperature)?
Got effective friction mechanism for large p
$\rightarrow \hat{P}(\hat{\tau})$ is narrow distribution
$\rightarrow \hat{T}(N)\propto N$
$\rightarrow$ need $\alpha > 1$ to accumulate atoms into trap
On the other hand:
Found that longest trapping time $\tau$ is of the order of total interaction time $\Theta$ and the corresponding momentum of the atom is a measure for width of the resulting momentum distribution $\delta p_{\Theta}$, i.e. the temperature of the atoms:
$\tau(\delta p_{\Theta})\approx \Theta$
$\rightarrow R_{F}(\delta p_{\Theta})\Theta\approx 1$
$\rightarrow (\delta p_{\Theta})^{\alpha}\Theta\propto 1$
$\rightarrow \delta p_{\Theta}\propto \Theta^{\alpha}$
For given interaction time:
The smaller $\alpha$ is chosen the colder the atoms get. Together with efficiency considerations: Want $\alpha$ close to but greater than 1.
First experiments used so-called Blackman pulses that have $\alpha\approx 4$ (chosen to minimize side bands).
Square pulses (in time domain) lead to $\alpha\approx 2$.
$\Rightarrow$ New temperature record of 3 nK (compared to 100 nK before) was achieved with help of Lévy-flight analysis of the problem.
Conclusions:

• Introduced basic theory of Doppler cooling
• Explained two methods of sub-recoil cooling
• Showed how the cooling process can be described in terms of Lévy-flight statistics
• Drew conclusions about efficiency of the cooling process

References:
Doppler cooling: '' Optical molasses'', P.D. Lett et al., J. Opt. Soc. Am. B, 6 (1989) 2084
VSCPT: ''Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping'', A. Aspect et al., Phys. Rev. Lett. 61 (1988) 826
Raman cooling: ''Laser cooling below a photon recoil with three-level atoms'', M. Kasevich and S. Chu, Phys. Rev. Lett. 69 (1992) 1741
Lévy flights: ''Subrecoil laser cooling and Lévy flights'', F. Bardou et al., Phys. Rev. Lett. 72 (1994) 203
''Raman cooling of Cesium below 3 nK: New approach inspired by Lévy flights statistics'', J. Reichel et al., Phys. Rev. Lett. 75 (1995) 4575
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