Long coherence times with dense trapped atoms collisional narrowing and dynamical decoupling Nir Davidson Yoav Sagi, Ido Almog, Rami Pugatch, Miri Brook.

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

• Efficiency of quantum memories depends on optical depth

• Strong nonlinearity per photon

• Collective coupling to SC circuits

• Unique model system!

Why dense atomic ensembles?

Quantum memories

2010 : - Us, Kuzmich, Porto, Rosenbusch, Bloch.…

Experimental setup

• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling

Experimental setup

• Magneto optical trapping• Sisyphus cooling• Raman sideband cooling• Evaporative cooling

KT 51

1100 scol

Hzrosc )640285(2,

100OD

5103N

05.0

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

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

• Collisional narrowing • Spectrum with discrete fluctuations

• Motional broadening • Dynamical decoupling

• Bath spectral characterization

Outline

Motional narrowing

“

”

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

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)

Experimental results

Data collapse!

2

Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)

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

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)

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)

• Telegraph noise in semiconductors

• Single molecule spectroscopy

Discrete fluctuations

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

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

How do we measure the parameters?

• 1 is measured in low density with

2

32

10 /1)(

ttR

0

• is measured by inducing oscillations in the waist of the atomic cloud and observing their decay:

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)

Comparison to Kubo’s model

Bloembergen et al, PRA 1984

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).

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).

Can fluctuations broaden the spectrum ?

Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)

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

Motional broadening: exponential decay

Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)

Effect of cutoff

Motional broadening persists until cutoff is sampled

Relation to Zeno and anti Zeno

Y. Sagi, I. Almog, R. Pugatch, M. Aizenman and N. Davidson, PRA, in press (2011)

Suppression of collisional decoherence by dynamical decoupling

Echo fails at high densities

Dynamical Decoupling

Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)

Process tomography of DD

Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010)

Process tomography of non-linear Hamiltonian“twist” of the Bloch sphere

Rubidium 87: a11+a22-2*a12 = 0.3% of a11 and a22

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)

Measured collisional bath spectrum

Trap oscillation frequency

Lorentzian

I. Almog et. al., submitted (2011)

Measured decay vs predictions from bath spectrum

I. Almog et. al., submitted (2011)

Anomalous diffusion of atoms in a 1D dissipative lattice

Motional broadening in real space

vx d

Q=1.0

Q=1.57

Measurements of 1D anomalous diffusion

Ballistic

Diffusion

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

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

• 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

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()(

Measurements of 1D anomalous diffusion

Ballistic

Diffusion

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).

Direct observation of anomalous diffusion

tFWHM 2

1D anomalous diffusion

2

2

tFWHMBallistic

Normal diffusion

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

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

Effect of cutoff

Motional broadening persists until cutoff is sampled

Optimal DD sequence for a Lorentzian bath

G. S. Uhrig, Phys. Rev. Lett. 98, 100504 (2007).

Process tomography of non-linear Hamiltonian

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

Measured collisional bath spectrum

Axial oscillation frequency

Radial oscillation frequency

Lorentzian part

• 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

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:

Measuring the bath spectrum

B

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