1. 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)

2. NUCLEAR SPINS HAVE LONG RELAXATION TIMES Spin 1/2  No electric quadrupole moment coupling 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 In practice Ground state He3 has a purely nuclear spin ½ Nuclear magnetic moments are small  Weak magnetic couplings  dipole-dipole interaction contributes for T 1 ≃ 10 9 s (1mbar)

3. Examples of Relaxation times in 3 He T 1 =344 h T 2 =1.33 h 0.Moreau et al. J.Phys III (1997) Magnetometer with polarized 3 He Ming F. Hsu et al. Appl. Phys. Lett. (2000) Sol-gel coated glass cells

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

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

6. How to access the ground state of 3 He ? 11S011S0 23S123S1 23P23P 20 eV metastable ground Metastability exchange collisions During a collision the metastable and the ground state atoms exchange their electronic variables He* + He -----> He + He* Colegrove, Schearer, Walters (1963) 1.08  m Laser transition R.F. discharge

7. Typical parameters for 3He optical pumping Metastables : n = 10 10 -10 11 at/cm 3 n / N =10 -6 Metastable / ground : Laser Power for optical pumping P L = 5 W An optical pumping cell filled with ≈ mbar pure 3 He The ground state gas can be polarized to 85 % in good conditions

8. A SIMPLIFIED MODEL : SPIN ½ METASTABLE STATE S = s i I = i n N Rates Partridge and Series (1966) d /dt = - + Heisenberg Langevin equations Langevin forces = D ab (t-t’) dS /dt = - S + I + f s dI /dt = - I + S + f i

9. 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 3 Spin quadratures S x =(S 21 +S 12 )/2 ; S y =i (S 12 - S 21 )/2 S x = S y = n/4 I x = I y = N/4

10. SQUEEZING TRANSFER TO FROM FIELD TO ATOMS Linearization of optical Bloch equations for quantum fluctuations around the fully polarized initial state Control classical field of Rabi freqency Raman configuration >, A, A † cavity mode squeezed vaccum H= g (S 32 A + h.c.) 12 3 ↓ ↑ Exchange collisions

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

12. SQUEEZING TRANSFER TO GROUND STATE ATOMS The metastable coherence S 12 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 2 ↑ 1 3 A ↓ Exchange collisions give a linear coupling beween S 12 and I ↑↓ Resonant coupling condition : E ↑ –E ↓ =

13. SQUEEZING TRANSFER TO GROUND STATE ATOMS Resonance conditions in a magnetic field ↓↑ Zeeman effect : E 2 -E 1 = 1. 8 MHz/G E ↑ –E ↓ = 3.2 kHz/G ↓↑ A

14. GROUND AND METASTABLE SPIN VARIANCES Pumping parameter Strong pumping the squeezing goes to metastables Slow pumping the squeezing goes to ground state Cooperativity ≃ 100 When polarization and cavity field are adiabatically eliminated

15. GROUND AND METASTABLE SPIN VARIANCES

16. EXCHANGE COLLISIONS AND CORRELATIONS When exchange is dominant, and for A weak squeezing in metastable maintains strong squeezing in the ground state Exchange collisions tend to equalize the correlation function

17. WRITING TIME OF THE MEMORY Build-up of the spin squeezing in the metastable state 10 0.1 Build-up of the spin squeezing in the ground state

18. TIME EVOLUTION OF SPIN VARIANCES css 0.0 1 2 0.0 0.5 1.0 1.5 2.0 34 5t(s) The writing time is limited by (density of metstables)

19. READ OUT OF THE NUCLEAR SPIN MEMORY 1 2 ↓↑ 3 Pumping 10 s 1 ↓ 2 ↑ 3 Storage hours 1 2 ↓↑ 3 Writing 1 s 1 ↓ 2 ↑ 3 Reading 1 s 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 A in A out P(t 0 ) = E LO (t) E LO (t’) Fourier-Limited Spectrum Analyzer Temporally matched local oscillator Read-out function for

21. REALISTIC MODEL FOR HELIUM 3 Real atomic structure of 3He Exchange Collisions 11S011S0 23S123S1 23P023P0 AA F=3/2 F=1/2

22. REALISTIC MODEL FOR HELIUM 3 Real atomic structure of 3He Lifetime of metastable state coherence: =10 3 s -1 (1torr) 11S011S0 23S123S1 23P023P0 Exchange Collisions AA F=3/2 F=1/2

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 Effect of Adiabatic elimination not justified 1 mbarr T=300K

24. EFFECT OF A FREQUENCY MISMATCH IN GROUND STATE We have assumed resonance conditions in a magnetic field E 4 - E 3 + = E ↑ – E ↓ = E 4 - E 3 ≃ 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 e 2r 1

25. HOMOGENEITY REQUIREMENTS ON THE MAGNETIC FIELD Example of working point : B B = 5x10 - 4 10 - 1 3 10 -2 10 -1 10 0 10 1 10 2 10 3 0.5 0.6 0.7 0.8 0.9 1.0 E-71E-61E-51E-41E-30.01 B=0

26. LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES Correlated macroscopic spins (10 18 atoms) ? OPO correlated beams Julsgaard et al., Nature (2001) : 10 12 atoms =0.5ms

27. INSEPARABILITY CRITERION Two modes of the EM field Quadratures Two collective spins Duan et al. PRL 84 (2000), Simon PRL 84(2000)

28. LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES OPO 12 3 ↓ ↑ 12 3 ↓ ↑ correlated beams coupling

29. LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES In terms of inseparability criterion of Duan and Simon noise from Exchange collisions coupling Spontaneous Emission noise