Download Deep sub-Doppler cooling of Mg in MOT formed by light waves with

Deep sub-Doppler cooling of Mg
in MOT formed by light waves with
elliptical polarization
Abstract: We study magneto-optical trap of 24Mg atoms operating on the closed triplet 3P23D3 ( = 383.3 nm)
transition formed by the light waves with elliptical polarization (--* configuration). Compare to well-known trap
formed by light waves with circular polarization ( +-- configuration) the suggested configuration offer the lower
sub-Doppler temperature for trapped Mg atoms, that can’t be reached in conventional +-- MOT.
Oleg Prudnikov
D.V. Brazhnikov, A.V. Taichenachev, V.I. Yudin, A.N. Goncharov
Institute of Laser Physics, Novosibirsk
2016
Basic mechanism of laser cooling
1) Two-level model
radiation pressure for
moving atom
dipole force
F  k
Doppler cooling
k BTD   / 2
optical potential
sub-doppler laser cooling
2) polarization effects
   
lin  lin
Jg  1  Je  2
J.Dalibard and C.Cohen-Tannoudji, J.Opt.Soc. Am. B (1989).
3) light fields with elliptical polarization
"anomalous" sub-dopller cooling effects
   
rectification of dipole force
jg  1/ 2  je  3 / 2
   
F 0
 F  ( 0 = 0,  ) = 0
• O.N. Prudnikov, A. V. Taichenachev, A. M. Tumaikin and V. I. Yudin, Kinetics of atoms in the field produced by elliptically polarized waves, JETP vol.98, pp. 438-454, (2004)
• A.V. Bezverbnyi, O.N. Prudnikov, A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin The light pressure force and the friction and diffusion coefficients for atoms in a resonant nonuniformly
polarized laser field JETP v.96, pp. 383-401, (2003)
• O.N. Prudnikov, A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin Rectification of the dipole force in a monochromatic field created by elliptically polarized waves JETP vol.93, pp.63-70
(2001)
Kinetics of atoms in light fields
Two coordinate density matrix
example: optical transition 12
ˆ ( z1 , z 2 )
ˆ ( p1 , p2 )

ˆ (r , p )  
Coordinate representation
Momentum representation
d 3r
ˆ (r  q / 2, r  q / 2) exp(i p  q /  )
( 2 ) 3
d 3u

ˆ ( p  u / 2, p  u / 2) exp(i r  u / )
(2  )3
d
i
ˆ   Hˆ , ˆ  ˆ ˆ 
dt

 
d gg
i
ˆ   Hˆ eff , ˆ gg  ˆ ˆ gg
dt


Semiclassical approach
Fokker-Plank equation

p

   i  i W 
 t i m 

 
2
 
 
Fi ( r , p ) W  
Dij ( r , p ) W
i pi
ij pi p j
  
 ˆ ee
ˆ   ge
 ˆ
r1
r2
r = (r2 + r1)/2, q = r1 - r2
ˆ eg 

ˆ gg 
for S = ||2/(2/2+2) << 1
Quantum approaches
  /
- quantum Monte-Carlo wave-function method
[ J. Dalibard,et.al. PRL 68, 580 (1992) ]
- Band theory (cooling in optical potential)
[Y. Castin,et.al. Europhys. Lett.14, 761 (1991)]
- cooling in + - - field [A. Aspect,et.al. PRL 61,826 (1988)].
W (r , p)  Tr ˆ (r , p )
p- family approaches
Generalized continuous fraction method for
density matrix equation
Atom-laser interaction part of Hamiltonian can be expressed as sum of two parts from opposite ligth waves.
Vˆ  Vˆ1eikz  Vˆ2e  ikz
ˆ ( z , q)   ˆ ( n ) (q)e ikz
n
Density matrix equation in coordinate representation takes the following form for spatial harmonics (n):
d
i
ˆ   Hˆ , ˆ  ˆ ˆ 
dt

 
d (n)
i  ( n ) ˆ ( n ) ˆ ( n 1) ˆ ( n 1)
ˆ  n
ˆ  L0 ˆ  L ˆ
 L ˆ
dt
M q
 




1. We assume that the spatial coherence of density matrix is damped at enough large distance qmax we
consider the spatial interval [–qmax..qmax] and make a mesh with discrete points qi (total Nq points). On the
mesh we define derivative (n) in standard form:

1
ˆ q i 
ˆ q  ˆ q i 1
q
2q i 1


2. The equation can be written in Liouville representation with Liouvile operators L0, L+, L- and density matrix
in Liouville form:
.


 ee

   e ,  e/ ;qi 
  eg


e ,  g/ ; qi 

ˆ qi   qi  ge
 / 
 mggg ,  e ;qi 
  mg , mg/ ;qi 


.






d  (n)
i
  n G   ( n )  L   ( n1)  L0   ( n)  L   ( n 1)
dt
M
Example: For the case of optical transition 1/23/2 the vector  contains 18 x Nq elements,
for the case of 1 2 it contains 34 x Nq elements.
O.N. Prudnikov, A.V. Taichenachev, A.V. Tumaikin, V.I. Yudin, JETP v. 104, p839, (2007)
O. N. Prudnikov, R.Ya. Ilenkov, A. V. Taichenachev, A. M. Tumikin, and V. I. Yudin, JETP v.112, p.939 (2011)
Solution of density matrix equation
Example 2
Example 1
spatial distribution
 
E  Tr Ĥ ̂
(b)
density matrix spatial harmonics
(c)
Energy
(a)
momentum distribution
spatial coherence
(d)
Energy of 85Rb atoms in linlin field
(5S1/2 (F = 3)5P3/2 (F'=4)).
The black and white dots represents the temperature
measurements results [P. S. Jessen, et.al., Phys. Rev.
Lett. 69, 49 (1992)]
Density matrix spatial (а) and momentum (b) distributions for
atoms with Jg=1/2Je=3/2 optical transition in standing wave
with linear polarization (= - /2,  = , R = 0.1).
Density matrix spatial harmonics of the ground R{g (n)} and
excited state R{e (n)} (c).

 Tr ˆ

(q  0)
R e ( n )  Tr ˆ ee ( n ) (q  0)
R g (n)
gg ( n )
Spatial coherence functions for the
ground Qg and excited state Qe (d).

( q )  Tr ˆ

( q )
Q e (q )  Tr ˆ ee ( n 0 ) ( q)
Qg
gg ( n  0 )
max. of U
min. of U
Momentum distribution (solid line), and Gaussian
approximation (dashed line).
p2

I/Is = 1.4,  = -8
F ( p )  e 2 mT
Sub-Doppler cooling with taking into accounts all quantum recoil effects
Sub-Doppler laser cooling takes place at low saturation S = ||2/(2 + 2/4) << 1.
In this limit authors use reduced form of density matrix equation that allows to reduce the numbers of elements
matrix gg is function of two parameters only!
reduced equation
1) detuning /
2) optical potential depth u = S/R
 gg
i
ˆ   Hˆ eff , ˆ gg  ˆ ˆ gg
t


  
( or intensity I )
• In band theory gg is function of only one parameter
(U0/ER) [Y. Castin Europhysics Lett.v.14, 761, (1991) ]
pˆ 2
Hˆ eff 
  S Vˆ Vˆ
2M
Example 1: atom with jg=1/2je=3/2 in lin lin
Band theory
approximation:

U0

ER

2
Ek  prms
/ 2M
Ek/  pe2 / 2 M
Fig. Kinetic energy for different detunings
[Y. Castin, et al.
Europhysics Lett. v.14, 761, (1991)]
approximation assumes vibration
levels splitting large the levels
width caused by spontaneous
relaxation and tunneling effects
- it do not takes into account above
barier atoms
- is not valid for deep optical
potential (small detunings)
We see significant difference for field parameters out of secular
approximation.
O. N. Prudnikov, et.al. "Steady state of a low-density ensemble of atoms in a monochromatic field taking into account recoil effects", JETP v.112, pp.939-945 (2011)
Sub-Doppler cooling with taking into accounts all quantum recoil effects
reduced equation
VS
 gg
i
ˆ   Hˆ eff , ˆ gg  ˆ ˆ gg
t


  
base equation
d
i
ˆ   Hˆ , ˆ  ˆ ˆ 
dt

 
 ˆ ee
ˆ   ge
 ˆ
ˆ eg 

ˆ gg 
pˆ 2
Hˆ eff 
  S Vˆ Vˆ
2M
matrix gg is function of two parameters only!
1) detuning /
2) optical potential depth u = S/R
matrix gg is function of 3 parameters!
( or intensity I )
3) semiclassical parameters R.
R 
k 2
2 M
Example 2: Solution on a base of general equation
R = 0.05
Results:
1) density matrix is function of 3 parameters
(additional parameter is R)
2) reduced equation is valid for only
extremely low R , i.e. in conditions of
semiclassical approach!
R = 0.01
reduced equation
R 
R = 0.005
k 2
2 M
i.e. when we restrict consideration by
reduced equation with effective Hamiltonian
we stay in conditions of semiclassical
approach.
Fig. Kinetic energy of atoms with jg=1/2  je=3/2
optical transition in linlin field for  = -5.
O. N. Prudnikov, et.al. "Steady state of a low-density ensemble of atoms in a monochromatic field taking into account recoil effects", JETP v.112, pp.939-945 (2011)
Sub-Doppler cooling of 24Mg in +-- field:
semiclassical and quantum approaches
   
33P2 → 33D3
[*] M. Riedmann, et. al. PRA v.86,
043416 (2012). (T  1 mK)
M  238 k 2 / 
  476 Erec
  1.28 mK
Semiclassical approach:
  2 26.3 MHz
k / p  1
ultracold fraction with |p|<3hk
  5
  5
momentum distribution
  5
I  20 mW / cm 2
T
p
2
M
k 2
R 
 1
2 M
  / 2
Here the solid lines represent quantum simulation results, and the dashed lines - semiclassical approach results
O. N. Prudnikov, D. V. Brazhnikov, A. V. Taichenachev, V. I. Yudin, A. E. Bonert, R. Ya. Il'enkov, and A. N. Goncharov,
Quantum treatment of two-stage sub-Doppler laser cooling of magnesium atoms, Phys. Rev. A 92, 063413 (2015)
Sub-Doppler cooling conditions
for S = ||2/(2/2+2) << 1
Sub-Doppler temperature
T
p2
M

D ( 0)
s
z
k 2
 ( /  )  S
M
   ( /  ) S
z
slow atom approach:
p  M S/k
negligible
in semiclassical
approach
k 2
R 
 2M
What happens for non negligible R ?
optical transition
Is [mW/cm2]
(/) function for atoms
with 23 optical transition
R 
1
~
2M
 [nm]
p  5 hk
.
24Mg
33P2 → 33D3
61.7
1) violation of slow atom approach
2) range of sub-Doppler force might be few recoil
momentum.
383.9
238
atom may not "see" sub-Doppler force!
Sub-Doppler cooling of 24Mg: quantum approach
laser cooling in +-- field
T    1.28 mK
I (mW/cm2)
   
Doppler temperature
.
I (mW/cm2)
I (mW/cm2)
Fig. Optimal parameters of intensity and
minimum temperature for cooling in +-- field
laser cooling in linlin field
I (mW/cm2)
lin  lin
Doppler temperature
I (mW/cm2)
I (mW/cm2)
Fig. Optimal parameters of intensity and
minimum temperature for cooling in linlin
field
Sub-Doppler cooling of 24Mg in MOT
No trapping force in linlin filed!
  1.28 mK
[deg.]
[deg.]
Fig. Temperature of laser cooling of
24
Mg in lin--lin field ( = -/4)
[deg.]
[deg.]
-- * field
I (mW/cm2)
Fig. Temperature of laser cooling of 24Mg in --* field
( = -/4,  = -) and fraction of atoms with |p|<3hk.
I [mW/cm2]
100
200
300
0 [deg.]
-0.28
-0.86
-1.15
T [h]
0.097
0.118
0.157
-2
-5.16
-5.16
53.7
48.2
43.4
0 [deg.]
N|p|<3hk [%]
Fig. Optimal intensity and minimum
temperature in lin--lin field ( = -/4)
Fig. Optimal ellipticity and minimum temperature for
different intensity in -- field ( = -/4).
Magneto-optical potential for 24Mg in in --* MOT
assuming linear growth of magnetic field in trapping zone
U (H )
RW

 H ( RW )
H/ = 1 ( H = 12.7 Gs )
 H ( RW )

F ( H ) (v  0,  H ) d H
0
   
Fig. Force in h units and magneto-optical potential in hRW/
units (RW is beam radius) as function of Zeeman splitting on MOT
edge in +-- MOT. (I = 100mW/cm2,  = -)
Rw is radius of light beams forming the MOT
--*
Fig. Force in h units and magneto-optical potential in hRW/
units (RW is beam radius) as function of Zeeman splitting on MOT
edge in --* MOT. (I = 100mW/cm2,  = -)
Estimation: RW = 0.5 cm, 
for magnetic field gradient z H = 12.7 Gs/cm
( H/ = 0.5 Gs at edge),
the MOT depth U(H) = 0.094 hRW/ 1.56K that much
exceed sub-Doppler cooling temperature T125 K.
Magneto-optical force for trapping in --* MOT
Trap should be stable for moving atom as well.
Magneto-optical force as function of atoms velocity and
magnetic field.
(lin--lin field configuration with  = -/4,  = -, I = 100 mW/cm2)
here force push the
atoms out of the trap
H/ = 1 ( H = 12.7 Gs )
Magneto-optical force trapping zone for  --*
MOT ( = -/4,  = -, I = 100 mW/cm2)
Number of trapped atom Ncvc4
[C. Monroe, et.al. PRL. 65, 1571 (1990)]
vc> 3.5k/ results Nc107 - 108 at.
Critical magnetic field for stable --* MOT
Results
1. Quantum recoil effects make sub-Doppler cooling of 24Mg on 3P2-3D3 not
efficient in +-- field configuration.
2. For deep sub-Doppler cooling linlin configuration can be used.
(Can't be used for MOT!)
3. For deep sup-Doppler cooling in MOT --* light field configuration can
be used.
- enough deep magneto-optical potential
- for stable MOT magnetic field should not exceed critical values in
trapping zone.
- positive small ellipticities ( < 5 deg.) of light waves forming the MOT do
not much increases the cooling temperatures but increase much the critical
values for magnetic field.