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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 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)
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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
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3 He NUCLEAR SPINS FOR Q-INFORMATION ? Quantum information with continuous variables Squeezed light Spin Squeezed atoms Quantum 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 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
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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
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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
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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
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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 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
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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 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
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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
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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 ↓ =
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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
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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
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GROUND AND METASTABLE SPIN VARIANCES
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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
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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
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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)
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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 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
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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
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REALISTIC MODEL FOR HELIUM 3 Real atomic structure of 3He Exchange Collisions 11S011S0 23S123S1 23P023P0 AA F=3/2 F=1/2
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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
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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 elimination not justified 1 mbarr T=300K
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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
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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
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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
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INSEPARABILITY CRITERION Two modes of the EM field Quadratures Two collective spins Duan et al. PRL 84 (2000), Simon PRL 84(2000)
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LONG-LIVED NON LOCAL CORRELATIONS BETWEEN TWO MACROSCOPIC SAMPLES OPO 12 3 ↓ ↑ 12 3 ↓ ↑ correlated beams coupling
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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
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