long-lived quantum memory using nuclear spins a. sinatra, g. reinaudi, f. laloë (ens, paris)...
TRANSCRIPT
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LONG-LIVED QUANTUM MEMORY USING NUCLEAR SPINS
A. Sinatra, G. Reinaudi, F. Laloë (ENS, Paris)
Laboratoire Kastler Brossel
A. Dantan, E. Giacobino, M. Pinard (UPMC, Paris)
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NUCLEAR SPINS HAVE LONG RELAXATION TIMES
Spin 1/2 No electric quadrupole moment coupling
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
In practice
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)
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Examples of Relaxation times in 3He
T1=344 h
T2=1.33 h
0.Moreau et al. J.Phys III (1997) Magnetometer with polarized 3He
Ming F. Hsu et al. Appl. Phys. Lett. (2000) Sol-gel coated glass cells
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3He NUCLEAR SPINS FOR Q-INFORMATION ?
Quantum information with continuous variables
Squeezed light Squeezed light
Spin Squeezed atomsQuantum memory
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SQUEEZING OF LIGHT AND SQUEEZING OF SPINS
N spins 1/2
Coherent spin state CSS
(uncorrelated spins)
Projection noise in atomic clocks
Sx= Sy= |<Sz>|/2
€
Δ
€
Δ
X >1 Y <1
Squeezed state
€
Δ
€
ΔSqueezed state
Sx2
< N/4 Sy2
> N/4
€
Δ
€
Δ
One mode of the EM field
X=(a+a†)
Y=i(a†-a)
Coherent state X = Y=1
€
Δ
€
Δ
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How to access the ground state of 3He ?
11S0
23S1
23P
20 eV
metastable
ground
Metastability exchange collisions
During a collision the metastable and the ground state atomsexchange their electronic variables
He* + He -----> He + He*
Colegrove, Schearer, Walters (1963)
1.08 m
Laser transition
R.F
. d
isch
arg
e
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Typical parameters for 3He optical pumping
Metastables : n = 1010-1011 at/cm3
n / N =10-6
Metastable / ground :
Laser Power for opticalpumping PL= 5 W
An optical pumping cell filled with ≈ mbar pure 3He
The ground state gas can be polarized to 85 % in good conditions
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A SIMPLIFIED MODEL : SPIN ½ METASTABLE STATE
S =
€
i=1
n
∑ s i I =
€
i=1
N
∑ i i
n
N
€
γm
γ g
=N
n
RatesPartridge and Series (1966)
d< S >/dt = - < S > + < I >
d< I >/dt = - < I > + < S >
€
γm
€
γg
€
γm
€
γg
Heisenberg Langevin equations
Langevin forces
<fa(t) fb(t’)> = Dab (t-t’)
€
δdS /dt = - S + I + fs
€
γm
€
γg
dI /dt = - I + S + fi
€
γg
€
γm
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PREPARATION OF A COHERENT SPIN STATE
This state is stationary for metastability exchange collisions
↓ ↑
Atoms prepared in the fully polarized CSS by optical pumping
1 2
3Spin quadratures
Sx=(S21+S12)/2 ; Sy=i (S12- S21)/2
€
Δ
€
ΔSx = Sy = n/4
Ix = Iy = N/4
€
Δ
€
Δ
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SQUEEZING TRANSFER TO FROM FIELD TO ATOMS
Linearization of optical Bloch equations for quantum fluctuations around the fully polarized initial state
€
ΩControl classical field
of Rabi freqency
Raman configuration > ,
A , A† cavity mode
squeezed vaccum
H= g (S32 A + h.c.)€
Δ
€
δ
€
γ
1 2
3
€
γ
€
γ
↓ ↑
Exchange collisions
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SQUEEZING TRANSFER TO METASTABLE ATOMS
1
Linear coupling for quantum fluctuations between the cavity field and metastable atoms coherence S21
Dantan et al. Phys. Rev. A (2004)
2
3
A
€
Ω
Adiabatic elimination of S23
shift coupling
€
d
dtS21 = i δ +
Ω2
Δ
⎛
⎝ ⎜
⎞
⎠ ⎟S21 +
Ωgn
ΔA
Two photon detuning
Resonant coupling condition : +shift = 0
€
δ
€
δ =(E2 − E1 ) − (ω2 −ω1 )
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SQUEEZING TRANSFER TO GROUND STATE ATOMS
The metastable coherence S12
evolves at the frequency
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
€
(ω1 −ω2)
2
↑
ΔΩ2
1
3
A
↓
€
Ω
Exchange collisions give a linear coupling beween S12 and I↑↓
Resonant coupling condition :
E↑ –E↓ =
€
(ω1 −ω2)
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SQUEEZING TRANSFER TO GROUND STATE ATOMS
Resonance conditions in a magnetic field
ΔΩ2
↓ ↑
Zeeman effect :
E2-E1 = 1. 8 MHz/G
E↑ –E↓= 3.2 kHz/G
€
E2 − E1 +Ω2
Δ= E − E = (ω1 −ω2)↓↑
€
Ω A
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GROUND AND METASTABLE SPIN VARIANCES
€
Γ=γΩ2
Δ2(1+ C)
Pumping parameter
Strong pumping the squeezing goes to metastables
Slow pumping the squeezing goes to ground state
€
Γ >>γm
€
Γ <<γm
Cooperativity
≃ 100
€
C =g2n
κγ
€
ΔIy2 =
N
41−
γ m
Γ + γ m
(1− ΔAxin 2
) ⎡
⎣ ⎢
⎤
⎦ ⎥
€
ΔSy2 =
n
41−
Γ
Γ + γ m
(1− ΔAxin 2
) ⎡
⎣ ⎢
⎤
⎦ ⎥
When polarization and cavity field are adiabatically eliminated
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GROUND AND METASTABLE SPIN VARIANCES
€
(1− ΔAxin 2
) = 0.5 C = 500
10-3 10-2 10-1 100 101 102 103
0.5
0.6
0.7
0.8
0.9
1.0
Sy2Iy
2
€
Γ /γ m
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EXCHANGE COLLISIONS AND CORRELATIONS
€
< S2 > = < si2 > + < si s j > =
i≠ j
n
∑ n
4+ n(n +1) < si s j >
i=1
n
∑
€
< I2 > = < ii2 > + < ii i j > =
i≠ j
n
∑ N
4+ N(N +1) < ii i j >
i=1
N
∑
When exchange is dominant , and for
A weak squeezing in metastablemaintains strong squeezing in the ground state€
(γ m >> Γ)
€
n,N >>1
€
ΔI2
N /4−1
⎛
⎝ ⎜
⎞
⎠ ⎟=
N
n
ΔS2
n /4−1
⎛
⎝ ⎜
⎞
⎠ ⎟
Exchange collisions tend to equalize the correlation function
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WRITING TIME OF THE MEMORY
Build-up of the spin squeezing in the metastable state
0.50.60.70.80.91.0
ΔSy2ΔI
y
2
Γ/γm
10
0.50.60.70.80.91.0
ΔSy2ΔI
y
2
Γ/γm
0.1
Build-up of the spin squeezing in the ground state €
< Sy2 > (t)− < Sy
2 >s = 0.55 exp(−2Γt)
€
Γ=10 γ m = 5 ×107 s−1
€
< Iy2 > (t)− < Iy
2 >s = 0.55 exp(−2Γg t)
€
Γg =2Γγ g
γ m + Γ≈1 s−1
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TIME EVOLUTION OF SPIN VARIANCES
css
0.0 1 20.0
0.5
1.0
1.5
2.0
3 4 5 t(s)
€
ΔIx2
€
ΔIy2
€
ΔAx2
The writing time is limited by (density of metstables)
€
γg ∝ n
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READ OUT OF THE NUCLEAR SPIN MEMORY
1 2
↓ ↑
3
Pumping
10 s
1
↓
2
↑
3
Storage
hours
1 2
↓ ↑
3
Writing
1 s€
Ω1
↓
2
↑
3
Reading
1 s€
Ω
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
€
Ω
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READ OUT OF THE NUCLEAR SPIN MEMORY
Ain
Aout
P(t0) = ELO(t) ELO(t’) <Aout(t) Aout(t’)>
€
dω
ΔTΔω
∫ dtto
to+ΔT
∫ e−iω( t−t ' )
Fourier-Limited Spectrum Analyzer
Temporally matched local oscillator€
Δω
€
ΔT
€
ELO (t) = exp(−Γg t)
Read-out function for
€
ΔT >1/Γg
€
Rout (t0 ) =1−P(t0) /Δω
P(t0 ) /Δω[ ]CCS
= (1− ΔIy2)exp(−2Γg t0 )
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REALISTIC MODEL FOR HELIUM 3
Real atomic structure of 3He
Exchange Collisions
11S0
23S1
23P0A A
F=3/2F=1/2
€
Ω
€
Ω
€
˙ ρ g = γ g (−ρ g + Tre ρ m )
€
˙ ρ m = γ m (−ρ m + ρ g ⊗Trn ρ m )
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REALISTIC MODEL FOR HELIUM 3
Real atomic structure of 3He
€
γ0
€
γ0
€
γ0
€
γ0
Lifetime of metastable state coherence: =103s-1 (1torr)
€
γ0
11S0
23S1
23P0
Exchange Collisions
A A
F=3/2F=1/2
€
Ω
€
Ω
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NUMERICAL AND ANALYTICAL RESULTS
We find the same analytic expressions as for the simple model if the field and optical coherences are adiabatically eliminated
Effect of
Adiabatic eliminationnot justified
10-3 10-2 10-1 100 101 102 103
0.5
0.6
0.7
0.8
0.9
1.0
ΔSy2ΔI
y2
Γ/γm€
γ0 ≈ Γ
€
Γ ≈γ
€
γm = 5 ×106 ,γ = 2 ×107, γ 0 =103, κ =100γ, C = 500, Δ = −2 ×103
1 mbarrT=300K
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EFFECT OF A FREQUENCY MISMATCH IN GROUND STATE
We have assumed resonance conditions in a magnetic field
E4 - E3 + = E↑ – E↓ =ΔΩ2 E4 - E3 ≃ 1. 8 MHz/G
What is the effect of a magnetic field inhomogeneity ?
De-phasing between the squeezed field and the ground state coherence during the squeezing build-up time
E↑ –E↓= 3.2 kHz/G
e-2r
e2r
1
€
(ω1 −ω2)
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HOMOGENEITY REQUIREMENTS ON THE MAGNETIC FIELD
Example of working point :
€
Γ=0.1γ m , B = 57 mG, Larmor =184 Hz
BB
= 5x10- 4
€
ΔB
B=10−4
10-
1
3 10-2 10-1 100 101 102 103
0.5
0.6
0.7
0.8
0.9
1.0
E-7 1E-6 1E-5 1E-4 1E-3 0.01
€
Γ /γ m
€
B Tesla[ ]
B=0
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LONG-LIVED NON LOCAL CORRELATIONSBETWEEN TWO MACROSCOPIC SAMPLES
Correlated macroscopic spins (1018 atoms) ?
OPO
correlated beams
Julsgaard et al., Nature (2001) : 1012 atoms =0.5ms
€
τ
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INSEPARABILITY CRITERION
€
ℑA =2
NΔ2(Iy1 − Iy2) + Δ2(Ix1 + Ix 2)[ ] < 2
€
ℑF =1
2Δ2(Ax1 − Ax2) + Δ2(Ay1 + Ay 2)[ ] < 2
Two modes of the EM field
€
Ax1(2) = (A1(2) + A1(2)+) Ay1(2) = i(A1(2)
+ − A1(2))Quadratures
Two collective spins
€
Iz1 = Iz2 ≈N
2
Duan et al. PRL 84 (2000), Simon PRL 84(2000)
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LONG-LIVED NON LOCAL CORRELATIONSBETWEEN TWO MACROSCOPIC SAMPLES
OPO
1 2
3
€
γ
€
Ω
↓ ↑€
γ
1 2
3
€
γ
€
Ω
↓ ↑€
γ correlated beams
coupling
€
Γg − iω( ) Iy1(2) = β Ax1(2)in + noise
€
Γg − iω( ) Ix1(2) = −β Ay1(2)in + noise
€
A1in
€
A2in
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LONG-LIVED NON LOCAL CORRELATIONSBETWEEN TWO MACROSCOPIC SAMPLES
In terms of inseparability criterion of Duan and Simon
€
ℑA =γ m
γ m + Γℑ F + 2
Γ
γ m + Γ+
γ m
γ m + Γ
⎛
⎝ ⎜
⎞
⎠ ⎟1
C
⎛
⎝ ⎜
⎞
⎠ ⎟
noise fromExchange collisions
coupling
Spontaneous Emission noise
€
C =g2n
κγ>>1
€
Γ <<γm
€
ℑA = ℑ F