Limit to Spin Squeezing in BEC : from two-mode to multimode A. - PowerPoint PPT Presentation

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Limit to Spin Squeezing in BEC : from two-mode to multimode A. Sinatra, Y. Castin, E. Witkowska ∗ , Li Yun, J.-C. Dornsetter Laboratoire Kastler Brossel, Ecole Normale Sup´ erieure, Paris ∗ Institute of Physics, Polish Academy of Sciences, Warsaw Warsaw, September 10 th 2012

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Plan 1 INTRODUCTION 2 DEPHASING MODEL 3 LOSSES 4 TEMPERATURE

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Spin squeezing and atomic clocks N two-level atoms : Collective spin : S x = � j ( | a �� b | + | b �� a | ) j / 2, S z = � j ( | a �� a | − | b �� b | ) j / 2 Uncorrelated atoms 1 ∆ ω unc √ = ab NT Squeezed state ξ ∆ ω sq ab = ξ ∆ ω unc = √ ab NT Spin squeezing parameter ξ 2 = N ∆ S 2 ⊥ Kitagawa, Ueda, (1993) ; Wineland (1994) � S x � 2

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Spin squeezing schemes in atomic ensembles Light-Atoms interaction Quantum Non Demolition measurement of S z ξ 2 = − 3 . 0 dB = 0 . 5 Vuleti´ c PRL (2010) ξ 2 = − 3 . 4 dB = 0 . 46 Polzik J. Mod. Opt (2009) Cavity feedback ξ 2 = − 10 dB = 0 . 1 Vuleti´ c PRL (2010) Interactions in BEC Stationary method for BEC in two external states In a double well ξ 2 = − 3 . 8 dB = 0 . 42 Oberthaler, Nature (2008) In a double well on a chip Reichel PRL (2010) Dynamical method for BEC Feshbach ξ 2 = − 8 . 2 dB = 0 . 15 Oberthaler, Nature (2010) State-dependent pot. ξ 2 = − 2 . 5 dB = 0 . 56 Treutlein, Nature (2010)

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Dynamical generation of spin squeezing in a BEC At t < 0 all the atoms are in condensate a . At t = 0 , π/ 2 -pulse Factorized state just after the pulse � a † + b † � N 1 � | x � = √ √ | 0 � = C N a , N b | N a , N b � N ! 2 Expansion of the Hamiltonian Castin, Dalibard PRA (1997) H (ˆ ˆ N a , ˆ E (¯ N ǫ ) + µ a (ˆ N a − ¯ N a ) + µ b (ˆ N b − ¯ N b ) = N b ) 1 N a ) 2 + . . . 2 ∂ N a µ a (ˆ N a − ¯ + Non linear Hamiltonian H NL = � χ S 2 z

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Dynamical generation of spin squeezing in a BEC Predictions at T = 0 without decoherence : Best squeezing time H NL = � χ S 2 z 1 1 ξ 2 best ∼ χ t best ∼ N 2 / 3 N 2 / 3 No limit to the squeezing ? Kitagawa, Ueda, PRA (1993) ; Sørensen et al. Nature (2001) What limits spin squeezing for N → ∞ ? Particle losses : Li Yun, Y. Castin, A. Sinatra, PRL (2008) 1 / 3 √  � 2 � 7 � � 5 3 m t ,ω, N ξ 2 = min 2 K 1 K 3   28 π � a Non-zero temperature : A. Sinatra et al. PRL (2011) ; Frontiers of Phys. (Springer) (2011) ; Eur. Phys. Journ. D (2012)

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Spin squeezing scaling for N → ∞ Squeezed state Uncorrelated atoms Heisenberg limit 1 ab ∝ ξ ( N ) ab ∝ 1 ∆ ω unc √ ∆ ω sq ∝ ∆ ω H √ ab N N N Two mode model H NL = � χ S 2 Kitagawa Ueda z 1 1 ∆ ω sq N → ∞ , ξ ∼ ⇒ ab ∼ N 1 / 3 N 5 / 6 Two mode model with dephasing Two mode model with decoherence Multimode description at finite temperature or zero temperature ab ∼ ξ min ∆ ω sq N → ∞ , ξ ∼ ξ min � = 0 ⇒ √ N Explicit calculations to obtain ξ min ( dephasing ), ξ min ( losses ), ξ min ( temperature ), ...

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Two-mode dephasing model hamiltonian with a dephasing term S 2 � � H = � ω ab S z + � χ z + DS z G. Ferrini et al. PRA 2011, Sinatra et al. Frontiers of Physics 2012 D is a time-independent Gaussian random variable, � D � = 0 � D 2 � → ǫ noise ; N → ∞ N Although the analytical solution holds ∀ ǫ noise , typically ǫ noise ≪ 1 ǫ noise ⇔ Fraction of lost particles ǫ noise ⇔ Non-condensed fraction in the thermodynamic limit.

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Spin dynamics and relative phase dynamics a = e i θ a √ N a [ N a , θ a ] = i √ b = e i θ b √ N b Initially : N a − N b ∼ N [ N b , θ b ] = i 1 and θ a − θ b ∼ N ≪ 1 √ a † b = � N a ( N b + 1) e − i ( θ a − θ b ) Spin components S x ≃ N S y ≃ − N S z = N a − N b 2 ; 2 ( θ a − θ b ) ; ; 2 Heisenberg equation of motion for the phase difference ( θ a − θ b )( t ) = ( θ a − θ b )(0 + ) − χ t (2 S z + D ) χ t ≫ 1 ρ gt S y becomes a copy of S z : squeezing as ↔ � ≫ 1 N √ ρ gt 1 Phase spreading ( θ a − θ b ) ∼ 1 as χ t ≃ ↔ � ≫ N √ N

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Best spin squeezing and spin-squeezing time min = minimum of ξ 2 over time ξ 2 Best squeezing Close to best squeezing time � D 2 � N →∞ ξ 2 ( t η ) = (1 + η ) ξ 2 ξ 2 → = ǫ noise min min N 1 t η t min t’ η ρ gt η 1 = � � ηξ 2 min 2 (t) 0.1 ρ gt min ξ ∼ N 1 / 4 � 2 (1+ η ) ξ min ρ gt ′ η ∼ N 1 / 2 2 ξ min � -1 0 1 2 3 4 10 10 10 10 10 10 ρ gt/ / h

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE A different conclusion in the weak-dephasing limit S 2 � � H = � χ z + D S z � D 2 � → constant ; N → ∞ (e.g. N → ∞ at fixed non-condensed particles or lost particles) cf. A. Sørensen PRA 2001 � 1 3 2 + � D 2 � min = 3 2 / 3 1 � ξ 2 Best squeezing N 2 / 3 + + o 2 N N √ ρ gt min 3 = 3 1 / 6 N 1 / 3 − Best time 4 + o (1) � We recover in this case the scaling of H = � χ S 2 z plus corrections .

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Particle losses: Monte-Carlo wave functions Interaction picture with respect to H nl = � χ S 2 z H nl t H nl t H nl t H nl t c a = e i � a e − i c b = e i � b e − i � � Effective Hamiltonian and Jump operators for m-body losses i � � � 2 γ ( m ) c † m ǫ c m γ ( m ) c m H eff = − S ǫ = ǫ ǫ ǫ = a , b Evolution of one wave function with k jumps | ψ ( t ) � = e − iH eff ( t − t k ) / � S ǫ k e − iH eff τ k / � S ǫ k − 1 . . . S ǫ 1 e − iH eff τ 1 / � | ψ (0) � Quantum averages � � � � ˆ � ψ ( t ) | ˆ O� = dt 1 dt 2 · · · dt k O| ψ ( t ) � 0 < t 1 < t 2 < ··· t k < t k { ǫ j }

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Jumps randomly kick the relative phase Relative phase distribution at t = 0 and χ t = 2 π in single Monte Carlo realizations with 3, 1 and 0 quantum jumps Sinatra, Castin EPJD 1998 c a ( t ) | φ � N ∝ | φ − χ t / 2 � N − 1 c b ( t ) | φ � N ∝ | φ + χ t / 2 � N − 1 After k jumps | ψ ( t ) � ∝ | φ + χ t 2 D� N − k with D = 1 � k l =1 t l ( δ ǫ l , b − δ ǫ l , a ) t N.B. : e − i � χ DS z t | φ � = | φ − χ t 2 D �

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Best squeezing and best time for N → ∞ We use the exact solution for one-body losses : γ t = fraction of lost particles at time t N → ∞ γ t ≡ ǫ loss = const ≪ 1 For long times ρ gt � ≫ 1 � � � 2 ξ 2 ( t ) ≃ �D 2 � �D 2 � ≃ γ t + [1 + O ( γ t )] ρ gt N 3 N 0 10 � 2 / 3 min = 3 � 4 � γ ξ 2 4 3 ρ g 2 -1 10 ξ ρ gt min 1 = � � 3 ξ 2 4 min -2 10 10 20 30 40 ρ gt/ h /

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Unified view between dephasing noise and losses Particle Losses Dephasing model | ψ ( t ) � ∝ | φ + χ t ( θ a − θ b )( t ) = ( θ a − θ b )(0 + ) − χ t [2 S z + D ] 2 D� D from quantum jumps D from a dephasing H �D 2 � � D 2 � ξ 2 ( t ) ξ 2 ( t ) ≃ ≃ N N ρ gt / � > 1 ρ gt / � > 1 �D 2 � � D 2 � = γ t 3 = ǫ loss = ǫ noise N 3 N

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Multimode description Hamiltonian for component a (idem for b ) a ( r ) h 0 ψ a ( r ) + g � ψ † 2 ψ † a ( r ) ψ † H = dV a ( r ) ψ a ( r ) ψ a ( r ) . r Before the pulse, the system is in thermal equilibrium in a with T ≪ T c . the pulse mixes the field a with the field b that is in vacuum : ψ a ( r )(0 + ) = ψ a ( r )(0 − ) − ψ b ( r )(0 − ) √ 2 After the pulse the two fields evolve independently

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Bogoliubov description Bogoliubov expansion : weakly interacting quasi-particles � ǫ k c † H a = E 0 + a k c a k + cubic terms + quartic terms k � = 0 Spin components S z = N a − N b � ψ † S + ≡ S x + iS y = dV a ( r ) ψ b ( r ) 2 r In the Bogoliubov description � N � S + = e i ( θ a − θ b ) 2 + F ( θ a − θ b )( t ) = ( θ a − θ b )(0 + ) − gt � V [( N a − N b ) + D ] D and F depend on Bogoliubov functions and occupation numbers of quasi particles c † a k c a k after the pulse

Plan INTRODUCTION DEPHASING MODEL LOSSES TEMPERATURE Squeezing parameter evolution Double expansion in ǫ size = 1 / N → 0 and ǫ Bog = � N nc � / N → 0. Spin squeezing saturates to a finite value Spin squeezing as a function of a renormalized time ( τ ≃ ρ gt / (2 � )) 0 10 -1 10 2 > / N <D 2 ξ Two-modes result -2 10 -3 10 -2 -1 0 1 2 3 10 10 10 10 10 10 τ The limit � D 2 � / N depends on temperature and interaction strength