LONG-LIVED QUANTUM MEMORY
USING NUCLEAR SPINS
Laboratoire Kastler Brossel
A. Sinatra, G. Reinaudi, F. Laloë (ENS, Paris)
A. Dantan, E. Giacobino, M. Pinard (UPMC, Paris)
NUCLEAR SPINS HAVE LONG RELAXATION
TIMES
Ground state He3 has a purely nuclear spin ½
Nuclear magnetic moments are small
 Weak magnetic couplings
 dipole-dipole interaction contributes for T1≃ 109 s (1mbar)
Spin 1/2  No electric quadrupole moment coupling
In practice
T1 relaxation can reach 5 days in cesium-cotated cells
and 14 days in sol-gel coated cells
T2 is usually limited by magnetic field inhomogeneity
Examples of Relaxation times in 3He
T1=344 h
T2=1.33 h
Ming F. Hsu et al.
Appl. Phys. Lett. (2000)
Sol-gel coated glass cells
0.Moreau et al. J.Phys III (1997)
Magnetometer with polarized 3He
3He
NUCLEAR SPINS FOR Q-INFORMATION ?
Quantum information with continuous variables
Squeezed light
Squeezed light
Spin Squeezed atoms
Quantum memory
SQUEEZING OF LIGHT AND SQUEEZING OF SPINS
One mode of the EM field
Y=i(a†-a)
N spins 1/2
Coherent spin state
CSS
(uncorrelated spins)
Sx= Sy= |<Sz>|/2
X=(a+a†)
X = Y=1
 
X >1
Squeezed state
Y <1
 
Coherent state
Projection noise in
atomic clocks
Squeezed state
 Sx2 < N/4
 Sy2 > N/4
R.F. discharge
How to access the ground state of 3He ?
Laser transition
1.08 m
23P
23S1
Metastability
exchange collisions
metastable
20 eV
11S0
ground
He* + He -----> He + He*
During a collision the metastable and the ground state atoms
exchange their electronic variables
Colegrove, Schearer, Walters (1963)
Typical parameters for 3He optical pumping
An optical pumping cell filled with ≈ mbar pure 3He
Metastables :
n = 1010-1011 at/cm3
Metastable / ground :
n / N =10-6
Laser Power for optical
pumping
PL= 5 W
The ground state gas can be polarized to 85 % in good conditions
A SIMPLIFIED MODEL : SPIN ½ METASTABLE STATE
n
n
S=
s
N
I=
i
i1
N
d< I >/dt = -
m< S >
g < I >
+
+
i
i1
Partridge and Series (1966)
d< S >/dt = -
i
 
g<I>
 m< S >
Rates
m N

g n
Heisenberg Langevin equations
/dt = dS
dI
/dt = -
S +  g I + fs
 m 
 g 
I +  m S + fi
Langevin forces
<fa
(t) fb(t’)> = Dab (t-t’)
PREPARATION OF A COHERENT SPIN STATE
Atoms prepared in the fully polarized CSS by optical pumping
Spin quadratures
3
Sx=(S21+S12)/2 ; Sy=i (S12- S21)/2
↓

Sx=
= n/4
2
1

↑

 N/4
Ix
=
Sy
Iy =
This state is stationary for metastability exchange collisions


SQUEEZING TRANSFER TO FROM FIELD TO ATOMS
Raman configuration
Control classical field
of Rabi freqency 
>  , 
3


1
A , A† cavity mode
squeezed vaccum


2
H= g (S32 A + h.c.)
Exchange collisions

 
↓
↑
Linearization of optical Bloch equations for quantum
fluctuations around the fully polarized initial state
SQUEEZING TRANSFER TO METASTABLE ATOMS
Linear coupling for quantum fluctuations
between the cavity field and metastable
atoms coherence S21
3

1
A
2
2 


d
gn
S21  i  
A
S21 
dt

  
shift
coupling
Adiabatic elimination of S23
Resonant coupling condition :

  (E 2  E1 )  ( 2  1 )

+shift = 0
Two photon detuning
Dantan et al. Phys. Rev. A (2004)
SQUEEZING TRANSFER TO GROUND STATE ATOMS
3

2

The metastable coherence S12
evolves at the frequency (1   2 )
A
1
2
Exchange collisions give a linear

coupling beween S12 and I↑↓
Resonant coupling condition :
↓
↑
E↑ –E↓ = (1   2 )
Resonant coupling in both metastable and ground state is
Possible in a magnetic fieldsuch that the Zeemann effect in
the metastable compensates the light shift of level 1
SQUEEZING TRANSFER TO GROUND STATE ATOMS

A
Zeeman effect :
2

E2-E1 = 1. 8 MHz/G

↓
E↑ –E↓= 3.2 kHz/G
↑
Resonance conditions in a magnetic field

2
E 2  E1 

 E↑  E↓  (1   2 )
GROUND AND METASTABLE SPIN VARIANCES
When polarization and cavity field are adiabatically eliminated
Pumping parameter

N 
m
2
in 2
Iy  1
(1 Ax )
4     m


n 

in 2
S  1
(1 Ax )
4     m

2
y


 2

2
(1 C)
Cooperativity
2
C
gn

≃ 100
Strong pumping    m the squeezing goes to metastables
Slow pumping    m
the squeezing goes to ground state

GROUND AND METASTABLE SPIN VARIANCES
in 2
x
(1 A )  0.5
C  500
1.0
0.9
0.8
0.7
0.6
0.5
10-3
2
y
2
y
I
10-2
S
10-1
100
 / m
101
102
103
EXCHANGE COLLISIONS AND CORRELATIONS
n
n
n
 S     s    si s j    n(n  1)  si s j 
4
i1
i j
2
2
i
N
n
N
 I     i    ii i j    N(N  1)  ii i j 
4
i1
i j
2
2
i
Exchange collisions tend to equalize the correlation function
When exchange is dominant ( m  ) , and for
n,N 1
 I 2
 N S 2  A weak squeezing in metastable
1 
1 maintains strong squeezing

/4  in the ground state
N /4  n n

WRITING TIME OF THE MEMORY
Build-up of the spin squeezing in the metastable state
 Sy2  (t)  Sy2 s  0.55 exp(2t)
 10  m  5 107 s1
1.0
0.9
0.8
0.7
0.6
0.5
2
2
Sy
Iy
10
/m
Build-up of the spin squeezing in the ground state
 2
 Iy  (t)  Iy2 s  0.55 exp(2g t)
2 g
g 
 1 s1
m  
1.0
0.9
0.8
0.7
0.6
0.5
2
2
Sy
Iy
0.1
/m
TIME EVOLUTION OF SPIN VARIANCES
Ix2
2.0
1.5

css
1.0
Iy2
0.5
0.0
0.0
Ax2
1
2
3

The writing time is limited by
4
5
g  n
t(s)
(density of metstables)
READ OUT OF THE NUCLEAR SPIN MEMORY
Pumping
Writing

3
Reading
Storage

3
3
3
1
2
1
2
1
2
1
2
↓
↑
↓
↑
↓
↑
↓
↑

10 s
1s

hours
1s
By lighting up only the coherent control field  ,
in the same conditions as for writing, one retrieves
transient squeezing in the output field from the cavity

READ OUT OF THE NUCLEAR SPIN MEMORY
Ain

Aout
E LO (t)  exp(g t)
T
Fourier-Limited
Spectrum Analyzer
Temporally matched
local oscillator


d
P(t0) = 

 T
toT
i (tt' )
ELO(t) ELO(t’) <Aout(t) Aout(t’)>
dt
e

to
Read-out function for T 1/g

Rout (t 0 )  1

P(t 0 ) /
P(t ) / 
0
CCS
 (1 Iy2 )exp(2g t 0 )
REALISTIC MODEL FOR HELIUM 3
Real atomic structure of 3He
23P0

A

A
F=1/2
F=3/2
23S1


11S0
Ýg   g (g  Tre m )

Ým   m (m  g  Trn m )

Exchange
Collisions
REALISTIC MODEL FOR HELIUM 3
Real atomic structure of 3He
23P0

23S1

0
A
0
0

11S0





A
0
F=1/2
F=3/2
Exchange
Collisions
Lifetime of metastable state coherence:
 0 =103s-1 (1torr)
NUMERICAL AND ANALYTICAL RESULTS
We find the same analytic expressions as for the simple model
if the field and optical coherences are adiabatically eliminated
1.0
0.9
Adiabatic elimination
not justified
0.8
Effect of
0  
0.7
 
2
 Iy
0.6
2
Sy
0.5
10
-3
10
-2
10
-1
10
0
10
1
10 10
2
3
1 mbarr
/  m
T=300K
 m  5 106 ,   2 107,  0 103,  100 , C  500,   2 103
EFFECT OF A FREQUENCY MISMATCH IN GROUND STATE
We have assumed resonance conditions in a magnetic field
E4 - E3 +
2

= E↑ – E↓ =
(1  2 )
E4 - E3 ≃ 1. 8 MHz/G
E↑ –E↓= 3.2 kHz/G
What is the effect of a magnetic field inhomogeneity ?
 the squeezed field and the ground state
De-phasing between
coherence during the squeezing build-up time
e2r
e-2r
1
HOMOGENEITY REQUIREMENTS ON THE MAGNETIC FIELD
1E-7 1E-6 1E-5 1E-4 1E-3 0.01
1.0
0.9
0.8
B Tesla 
B = 5x10- 4
B
0.7
B
4

10
0.6 B

B=0
0.5
10-3 10-2 10-1 100 101 102 103
 / m

Example of working point :
  0.1 m , B  57 mG, Larmor  184 Hz

LONG-LIVED NON LOCAL CORRELATIONS
BETWEEN TWO MACROSCOPIC SAMPLES
Julsgaard et al., Nature (2001) : 1012 atoms
OPO
correlated beams

Correlated macroscopic
spins (1018 atoms) ?
=0.5ms
INSEPARABILITY CRITERION
Duan et al. PRL 84 (2000), Simon PRL 84(2000)
Two modes of the EM field

Quadratures Ax1(2)  (A1(2)  A1(2) )

1 2
 F   (Ax1  Ax2 )  2 (Ay1  Ay 2 ) 2
2
Two collective spins


Ay1(2)  i(A1(2)  A1(2) )
Iz1  Iz2
N

2
2 2
 A   (Iy1  Iy2 )  2 (Ix1  Ix 2 ) 2

N

LONG-LIVED NON LOCAL CORRELATIONS
BETWEEN TWO MACROSCOPIC SAMPLES
in
1
A



3
A
OPO
in
2
correlated beams
1
2
↓
↑




 
coupling
in


i

I


A
 g  y1(2)
x1(2)  noise
in


i

I



A
 g  x1(2)
y1(2)  noise


3
1
2
↓
↑
LONG-LIVED NON LOCAL CORRELATIONS
BETWEEN TWO MACROSCOPIC SAMPLES
In terms of inseparability criterion of Duan and Simon
coupling
 
  m 1 
A 
 F  2 
 
 
m  
 m    m   C 
m
noise from
Exchange collisions

Spontaneous
Emission noise
C
   m
 A  F
g2n

1