LONG-LIVED QUANTUM MEMORY USING NUCLEAR SPINS

Transcription

1 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)

2 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 T s (1mbar) Spin 1/2 No electric quadrupole moment coupling In practice T 1 relaxation can reach 5 days in cesium-cotated cells and 14 days in sol-gel coated cells T 2 is usually limited by magnetic field inhomogeneity

3 Examples of Relaxation times in 3 He T 1 =344 h T 2 =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 3 He

4 3 He NUCLEAR SPINS FOR Q-INFORMATION? Quantum information with continuous variables Squeezed light Squeezed light Spin Squeezed atoms Quantum memory

5 SQUEEZING OF LIGHT AND SQUEEZING OF SPINS One mode of the EM field Y=i(a -a) X=(a+a ) Coherent state X = Y=1 X >1 Squeezed state Y <1 N spins 1/2 Coherent spin state CSS (uncorrelated spins) S x = S y = <S z > /2 Projection noise in atomic clocks Squeezed state S x2 < N/4 S y2 > N/4

6 How to access the ground state of 3 He? R.F. discharge Laser transition 1.08 µm Metastability exchange collisions 2 3 P 2 3 S 1 metastable 20 ev 1 1 S 0 ground He* + He -----> He + He* During a collision the metastable and the ground state atoms exchange their electronic variables Colegrove, Schearer, Walters (1963)

7 Typical parameters for 3He optical pumping An optical pumping cell filled with mbar pure 3 He Metastables : n= at/cm 3 Metastable / ground : n / N =10-6 Laser Power for optical pumping P L = 5 W The ground state gas can be polarized to 85 % in good conditions

8 A SIMPLIFIED MODEL : SPIN ½ METASTABLE STATE n N n S = s i I = i=1 N i i i=1 Partridge and Series (1966) γ m γ g γ g γ m d< S >/dt = - < S > + < I > d< I >/dt = - < I > + < S > Heisenberg Langevin equations γ m γ g Rates = N n ds /dt = - γ S + γ I + fs m g di /dt = - γ I + γ g m S + fi Langevin forces δ <f a (t) f b (t )> = D ab (t-t )

9 PREPARATION OF A COHERENT SPIN STATE Atoms prepared in the fully polarized CSS by optical pumping 3 Spin quadratures S x =(S 21 +S 12 )/2 ; S y =i (S 12 -S 21 )/2 1 2 S x = S y = n/4 I x = I y = N/4 This state is stationary for metastability exchange collisions

10 SQUEEZING TRANSFER TO FROM FIELD TO ATOMS γ Raman configuration >, δ Control classical field of Rabi freqency Ω Exchange collisions γ γ A, A cavity mode squeezed vaccum H= g (S 32 A + h.c.) Linearization of optical Bloch equations for quantum fluctuations around the fully polarized initial state

11 SQUEEZING TRANSFER TO METASTABLE ATOMS Ω 3 A Linear coupling for quantum fluctuations between the cavity field and metastable atoms coherence S Adiabatic elimination of S 23 d dt S = i δ + Ω2 21 shift S 21 + Ωgn A coupling Resonant coupling condition : +shift δ = 0 δ = (E 2 E 1 ) (ω 2 ω 1 ) Two photon detuning Dantan et al. Phys. Rev. A (2004)

12 SQUEEZING TRANSFER TO GROUND STATE ATOMS Ω 3 A The metastable coherence S 12 evolves at the frequency (ω 1 ω 2 ) Ω Exchange collisions give a linear coupling beween S 12 and I Resonant coupling condition : E E = (ω 1 ω 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

13 SQUEEZING TRANSFER TO GROUND STATE ATOMS Ω 2 Ω A Zeeman effect : E 2 -E 1 = 1. 8 MHz/G E E =3.2 khz/g Resonance conditions in a magnetic field E 2 E 1 + Ω2 = E E = (ω ω ) 1 2

14 GROUND AND METASTABLE SPIN VARIANCES When polarization and cavity field are adiabatically eliminated I 2 y = N 4 1 γ m (1 A x Γ+γ m in 2 ) Pumping parameter Γ= γ Ω2 2 (1 + C) S 2 y = n 4 1 Γ (1 A x Γ+γ m in 2 ) Cooperativity C = g2 n κγ 100 Strong pumping Γ >> γ m the squeezing goes to metastables Slow pumping Γ << γ m the squeezing goes to ground state

15 GROUND AND METASTABLE SPIN VARIANCES (1 A x in 2 ) = 0.5 C = I y 2 S y Γ /γ m

16 EXCHANGE COLLISIONS AND CORRELATIONS n i=1 n i j < S 2 >= <s i 2 >+ <s i s j >= N i=1 n i j < I 2 >= <i i 2 >+ <i i i j >= n 4 + n(n +1) < s i s j > N 4 + N(N +1) < i i i j > Exchange collisions tend to equalize the correlation function (γ m >> Γ) n,n >>1 When exchange is dominant, and for I 2 N /4 1 = N n S 2 n /4 1 A weak squeezing in metastable maintains strong squeezing in the ground state

17 WRITING TIME OF THE MEMORY Build-up of the spin squeezing in the metastable state < S y 2 > (t) <S y 2 > s = 0.55 exp( 2Γt) Γ=10 γ m = s I y 2 S y 2 10 Γ/γ m Build-up of the spin squeezing in the ground state < I y 2 > (t) <I y 2 > s = 0.55 exp( 2Γ g t) Γ g = 2Γγ g γ m +Γ 1 s I y S y 2 Γ/γ m

18 TIME EVOLUTION OF SPIN VARIANCES 2.0 I x A x 2 I y 2 css t(s) The writing time is limited by γ g n (density of metstables)

19 READ OUT OF THE NUCLEAR SPIN MEMORY Pumping Writing Storage Reading 3 Ω 3 3 Ω s 1 s hours 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

20 READ OUT OF THE NUCLEAR SPIN MEMORY ω dω T A in A out E LO (t) = exp( Γ g t) ω T Temporally matched local oscillator to+ T to Fourier-Limited Spectrum Analyzer P(t 0 ) = E LO (t) E LO (t ) <A out (t) A out (t )> dt e iω(t t') Read-out function for T >1/Γ g R out (t 0 ) =1 P(t 0 )/ ω = (1 I [ P(t 0 )/ ω] y CCS 2 )exp( 2Γ g t 0 )

21 REALISTIC MODEL FOR HELIUM 3 Real atomic structure of 3He 2 3 P 0 A A 2 3 S S 0 Ω Ω F=1/2 F=3/2 Exchange Collisions Ý ρ g = γ g ( ρ g + Tr e ρ m ) Ý ρ m = γ m ( ρ m + ρ g Tr n ρ m )

22 REALISTIC MODEL FOR HELIUM 3 Real atomic structure of 3He 2 3 P S S 0 Ω A Ω A γ 0 γ 0 γ 0 γ 0 F=1/2 F=3/2 Exchange Collisions Lifetime of metastable state coherence: γ 0 =10 3 s -1 (1torr)

23 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 Effect of γ 0 Γ I y 2 S y 2 Adiabatic elimination not justified Γ γ mbarr T=300K Γ/γ m γ m = ,γ = , γ 0 =10 3, κ =100γ, C = 500, =

24 EFFECT OF A FREQUENCY MISMATCH IN GROUND STATE We have assumed resonance conditions in a magnetic field Ω 2 E 4 -E 3 + = E E = (ω 1 ω 2 ) E 4 -E MHz/G E E =3.2 khz/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 2r e -2r 1

25 HOMOGENEITY REQUIREMENTS ON THE MAGNETIC FIELD E-71E-61E-51E-41E B B = 5x10-4 B B =10 4 Example of working point : 3 B= B Tesla [ ] Γ /γ m Γ = 0.1γ m, B = 57 mg, Larmor =184 Hz

26 LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES Julsgaard et al., Nature (2001) : atoms τ=0.5ms OPO correlated beams Correlated macroscopic spins (10 18 atoms)?

27 INSEPARABILITY CRITERION Duan et al. PRL 84 (2000), Simon PRL 84(2000) Two modes of the EM field Quadratures A x1(2) = (A 1(2) + A 1( 2) + ) A y1(2) = i(a 1( 2) + A 1(2) ) I F = 1 [ 2 2 (A x1 A x2 ) + 2 (A y1 + A y2 )]< 2 Two collective spins I z1 = I z2 N 2 I A = 2 [ N 2 (I y1 I y2 ) + 2 (I x1 + I x2 )]< 2

28 LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES Ω γ 3 γ A 1 in OPO correlated beams A 2 in Ω γ 3 γ coupling ( in Γ g iω)i y1( 2) = β A x1( 2) + noise ( in Γ g iω)i x1( 2) = β A y1(2) + noise

29 LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES In terms of inseparability criterion of Duan and Simon coupling I A = γ m γ m +Γ I F + 2 Γ γ m +Γ + γ m 1 γ m +Γ C Γ << γ m noise from Exchange collisions I A = I F Spontaneous Emission noise C = g2 n κγ >>1