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![Page 1: Long coherence times with dense trapped atoms collisional narrowing and dynamical decoupling Nir Davidson Yoav Sagi, Ido Almog, Rami Pugatch, Miri Brook.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d575503460f94a36c70/html5/thumbnails/1.jpg)
Long coherence times with dense trapped atoms collisional narrowing and dynamical decoupling
Nir Davidson
Yoav Sagi, Ido Almog, Rami Pugatch, Miri Brook(Kurizki group, Michael Aizenman)
Weizmann Institute of Science, Israel
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• Efficiency of quantum memories depends on optical depth
• Strong nonlinearity per photon
• Collective coupling to SC circuits
• Unique model system!
Why dense atomic ensembles?
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Quantum memories
2010 : - Us, Kuzmich, Porto, Rosenbusch, Bloch.…
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Experimental setup
• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling
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Experimental setup
• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling
KT 51
1100 scol
Hzrosc )640285(2,
100OD
5103N
05.0
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Experimental setup
• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling
WMW
52S½ ,F=1
B=3.2G
d
m=-1
WRF
m=1
52S½ ,F=2
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Experimental setup
• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling
WMW
52S½ ,F=1
B=3.2G
d
m=-1
WRF
m=1
52S½ ,F=2
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• Collisional narrowing • Spectrum with discrete fluctuations
• Motional broadening • Dynamical decoupling
• Bath spectral characterization
Outline
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Motional narrowing
“
”
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Collisional narrowing
2
2
Control field detuning is dc
212
10 21
2
1)( tiet d
133 scol
13 scol
0
<0
x
t
tGaussian
Exponent
)()( tietR 2/)(2 te
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Experimental results
Collisional narrowed decay time
Inhomogeneous decay time
22 1
col
Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)
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Experimental results
Data collapse!
2
Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)
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Mott insulator suppresses collisions
• Mott-Insulator with exactly one atom per site
• ~80 Hz EIT lines
• ~250 msec storage time for light
U. Schnorrberger, J. D. Thompson, S. Trotzky, R. Pugatch, N. Davidson, S. Kuhr, and I. Bloch, PRL 2010
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Time
Randomizing event
dP
d
De
tun
ing
Discrete Vs continuous fluctuations
• Kubo-Anderson model
22)(0 tH d
Y. Sagi, R. Pugatch, I. Almog and N. Davidson, Phys. Rev. Lett. 104, 253003 (2010)
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Time
Randomizing event
dP
d
De
tun
ing
Discrete Vs continuous fluctuations
• Cold collisions in atomic ensembles
Time
Randomizing event
dP
d
De
tun
ing
• Kubo-Anderson model
22)(0 tH d
Y. Sagi, R. Pugatch, I. Almog and N. Davidson, Phys. Rev. Lett. 104, 253003 (2010)
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• Telegraph noise in semiconductors
• Single molecule spectroscopy
Discrete fluctuations
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Solution of the discrete model
)()( tietR
Without collisions:tt d )(
With collisions:
)(~
1
)(~
)(~
0
0
sR
sRsR
A. Brissaud and U. Frisch, J. Math. Phys. 15, 524 (1974).
Time
Randomizing event
dP
d
Det
unin
g
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Atoms in 3D harmonic trap
0
<0
x
2
1
kT
C
ePd
dd
20 )(
Density of states for 3D
harmonic trap
Boltzmann factor
2
32
10
00 1)()(
dd d tdePtR ti
1),1(
2),1(
2)(
~11
210 sYsHssR
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How do we measure the parameters?
• 1 is measured in low density with
2
32
10 /1)(
ttR
0
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• is measured by inducing oscillations in the waist of the atomic cloud and observing their decay:
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Comparing theory to experiment
)(~
1
)(~
)(~
0
0
sR
sRsR
1),1(
2),1(
2)(
~ 20 sYsHssR )(tR)(0 dP
Y. Sagi, R. Pugatch, I. Almog and N. Davidson, Phys. Rev. Lett. 104, 253003 (2010)
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Comparison to Kubo’s model
Bloembergen et al, PRA 1984
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Can fluctuations broaden the spectrum ?
2
)1(
20
21
0 )1()(
r
rPdddExample: Student’s t-distribution
Motional narrowing
d
A. Burnstein, Chem. Phys. Lett. 83, 335 (1981).
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Can fluctuations broaden the spectrum ?
2
)1(
20
21
0 )1()(
r
rPdddExample: Student’s t-distribution
Motional narrowingMotional broadening
ddd ,
A. Burnstein, Chem. Phys. Lett. 83, 335 (1981).
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Can fluctuations broaden the spectrum ?
Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)
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Mathematical proof for stable distributions
α - characteristic exponent of a stable distributionGaussian: α=2, Cauchy: α=1, Levi: α=1/2
Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)
d TT )(0
where
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Motional broadening: exponential decay
Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)
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Effect of cutoff
Motional broadening persists until cutoff is sampled
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Relation to Zeno and anti Zeno
Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)
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Suppression of collisional decoherence by dynamical decoupling
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Echo fails at high densities
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Dynamical Decoupling
Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)
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Process tomography of DD
Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)
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Process tomography of non-linear Hamiltonian“twist” of the Bloch sphere
Rubidium 87: a11+a22-2*a12 = 0.3% of a11 and a22
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Continuous Rabi pulse
Measuring the bath spectrum
tSetR )()( W
)(),( 2 ttttF Dirac W d
0
),()(
)(tFSd
etR
S()F(t)
W
The decay rate is
G. Gordon et. al., J. Phys. B: At. Mol. Opt. Phys. 42, 223001 (2009)
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Measured collisional bath spectrum
Trap oscillation frequency
Lorentzian
I. Almog et. al., submitted (2011)
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Measured decay vs predictions from bath spectrum
I. Almog et. al., submitted (2011)
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Anomalous diffusion of atoms in a 1D dissipative lattice
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Motional broadening in real space
vx d
Q=1.0
Q=1.57
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Measurements of 1D anomalous diffusion
Ballistic
Diffusion
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Self similarity
)()( /1/1 txtxt
-2 -1 0 1 2 3
0
0.5
1
1.5
2
2.5
3
3.5
x 10-3
Position [mm]
Spa
cial
dis
trib
utio
n
=1.25 t=60 msec
t=52 msec
t=44 msec
t=36 msec
t=28 msec
t=20 msect=12 msec
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x 107
0
2
4
6
8
10
12
x 10-5
Position t-1/ [mm sec-1/]
Spa
cial
dis
trib
utio
n t
1/ [
sec1/
]
t=60 msec
t=52 msec
t=44 msec
t=36 msect=28 msec
t=20 msec
t=12 msec
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Collisional narrowing PRL 105 093001 (2010)
Discrete fluctuationsPRL 104, 253003 (2010)
Dynamical decoupling PRL 105 053201 (2010)
Collisional broadening PRA, in press (2011)
Time
Randomizing event
dP
d
Det
unin
g
Bath characterization submitted (2011)
Anomalous diffusion in preparation (2011)
Summary
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• Collisional narrowing Y. Sagi, I. Almog and ND, PRL 105 093001 (2010)
• Spectrum with discrete fluctuations Y. Sagi, I. Almog, R. Pugatch and ND, PRL 104, 253003 (2010)
• Motional broadening Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and ND, submitted (2010)
• Dynamical decoupling Y.Sagi, I. Almog and ND, PRL 105 053201 (2010)
• Bath spectral charecterizationI. Almog et. al., submitted (2011)
Outline
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How to create a Power-law velocity distribution?
• Don’t be in thermal equilibrium !• Sisyphus cooling scheme:
Y. Castin, J. Dalibrad, C. Cohen-Tannoudji (1990)
rE
U
v
vvP 44
20
2
0 )1()(
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Measurements of 1D anomalous diffusion
Ballistic
Diffusion
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Measurements of 1D anomalous diffusion
It is possible to measure both the spatial atomic
distribution and the velocity distribution (by a
time of flight method).
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Direct observation of anomalous diffusion
tFWHM 2
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1D anomalous diffusion
2
2
tFWHMBallistic
Normal diffusion
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Self similarity in the experiment
)()( /1/1 txtxt
-2 -1 0 1 2 3
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
-3
Position [mm]
Spa
cial
dis
trib
utio
n
=1.8 t=60 msec
t=52 msec
t=44 msec
t=36 msec
t=28 msec
t=20 msect=12 msec
-6 -4 -2 0 2 4 6
x 106
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10-4
Position t-1/ [mm sec-1/]S
paci
al d
istr
ibut
ion
t1/
[
sec1/
]
t=60 msec
t=52 msec
t=44 msec
t=36 msect=28 msec
t=20 msec
t=12 msec
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Self similarity in the experiment (2)
)()( /1/1 txtxt
-2 -1 0 1 2 3
0
0.5
1
1.5
2
2.5
3
3.5
x 10-3
Position [mm]
Spa
cial
dis
trib
utio
n
=1.25 t=60 msec
t=52 msec
t=44 msec
t=36 msec
t=28 msec
t=20 msect=12 msec
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x 107
0
2
4
6
8
10
12
x 10-5
Position t-1/ [mm sec-1/]S
paci
al d
istr
ibut
ion
t1/
[
sec1/
]
t=60 msec
t=52 msec
t=44 msec
t=36 msect=28 msec
t=20 msec
t=12 msec
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Effect of cutoff
Motional broadening persists until cutoff is sampled
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Optimal DD sequence for a Lorentzian bath
G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).
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Process tomography of non-linear Hamiltonian
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Mott insulator suppresses collisions
• Mott-Insulator with exactly one atom per site
• ~80 Hz EIT lines
• ~250 msec storage time for light
U. Schnorrberger, J. D. Thompson, S. Trotzky, R. Pugatch, N. Davidson, S. Kuhr, and I. Bloch, PRL 2010
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Measured collisional bath spectrum
Axial oscillation frequency
Radial oscillation frequency
Lorentzian part
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• An ensemble of oscillators with a distribution of resonant frequencies.
• If is a Gaussian process, the dephasing is given in terms of the correlation function
by: • For a Poissonian fluctuations,
we obtain:
Gaussian theory: Kubo’s model
)()(1
)(2
dd
d
tt
)(td
])(exp[)(0
2 t
dttR d
e)(
122
)(
te t
etR d
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The solution of the model)()( tietR
Without collisions:tt d )(
With collisions:
)(~
1
)(~
)(~
0
0
sR
sRsR
Where the tilde stands for the Laplace transform.
)(~
)(~
)( iRiRS The spectrum can be calculated by:
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Measuring the bath spectrum
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B
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Dephasing of optically trapped atoms
0
<0
x
2)(2 xU
1)(1 xU
2
21 )(
0
tie
td
12
)()(
xIxU
2112 )()()( xUxUxU
5106 In our experiment
MHz2.0Hz10
)()( tietR 2/)(2 te For Gaussian phase
distribution