Squeezing and superposing many-body states of Bose gases in confining potentials

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1 Squeezing and superposing many-body states of Bose gases in confining potentials K. B. Whaley Department of Chemistry, Kenneth S. Pitzer Center for Theoretical Chemistry, Berkeley Quantum Information and Computation Center UC Berkeley

2 Collaborators Jonathan DuBois (currently at Quantum Simulations Group, LLNL) Jan Korsbakken Ignacio Cirac (Munich)

3 Outline 1 from cold atoms to helium; quantum Monte Carlo methods 2 squeezing trapped cold atoms - repulsive bosons in a double well 3 mesocopic superposition states - attractive bosons and helium

4 Range of interactions a > a < 7 Li, 85 Rb + Feshbach, Cs + Feshbach...

5 BEC in a double well split condensate meso(macro)scopic coherences interference squeezing superpositions many-body tunneling a > : V ext (r) = x 2 + y 2 + z 2 + V b ((z/l) 2 1) 2 a < : V ext (r) = x 2 + y 2 + z 2 + V b 2πl e (z/2l)2

6 quantum Monte-Carlo Methods given a general many-body bosonic problem H = N i 2 2m i 2 i + N V ext (r i ) + i N V int (r ij ) (1) i<j QMC solves for full many-body ground state Ψ (r 1, r 2,..., r N ) realistic interaction potentials V ( r i r j ) exact : good for finding unexpected emergent behavior VPI: ground state energy, densities, ρ(r, r ) (the OBDM) fixed phase VPI, POITSE: excited states PIMC, PICF: finite temperature properties, excited states

7 the one body density matrix (OBDM) For a general time dependent N particle many body state Ψ N = i c i ψ (i) (r 1,, r N ; t) the single particle density matrix is ρ(r, r ; t) = ˆΨ (r, t) ˆΨ(r, t) = N c i d Rψ (i) (r, R; t)ψ (i) (r, R; t) i where R = {r 2,, r N }

8 natural orbitals The OBDM is Hermitian and so can be diagonalized ρ(r, r ) = i n i φ i (r)φ i (r ) where n i are real and i n i = N. φ i (r) are the natural orbitals. simple condensate :: lim N n /N 1, n {i>} /N fragmented condensate :: n { i k} /N 1, n {i>k} /N

9 fragmentation vs. depletion In strongly interacting quantum fluids (e.g. bulk He 4 ), even when there is only one single particle state with large occupation, the fraction of particles occupying φ can be very small n 1% while the rest of the eigenvalues of the OBDM still have essentially zero occupation in the thermodynamic limit. large depletion implies that the ground state is a highly entangled many body state (bulk He 9% depletion) in contrast, a fragmented state has more than one single particle state with large occupation. Each of these states is a real condensate and is internally phase coherent.

10 VPI and the one-body density matrix Variational path integral Monte Carlo (VPI): T =, non-periodic paths in imaginary time. Probability of a path is { 2M 1 } P Ψ T (R )Ψ T (R 2M ) G (R m, R m+1 ; τ) m= At center of the path, distribution is independent of Ψ T for sufficiently long paths with length = 2Mτ The one-body reduced density matrix (OBDM) is defined as: ρ(r, r ) = ψ (r, r 2,..., r N )ψ(r, r 2,..., r N )dr 2... dr N Diagonalization of the OBDM yields the single-particle eigenfunctions (natural orbitals) and corresponding eigenvalues (occupation numbers)

11 Interaction potential models for cold atoms characterized by s wave scattering length a > a < semiclassical approximation { rij > a V int (r ij ) = r ij a ) V int (r ij ) = ɛ (( σrij ) 12 ( σrij ) 6 (VV Flambaum, GF Gribakin, C Harabati PRA 59, 1998 (1999)) a cos( π 4 )(ɛ1/2 σ 3 4 choose ɛ, σ to eliminate bound states ) 1/2 [1 tan( π 4 ) tan(γ σ 2ɛ π 8 )]Γ(3/2) Γ(5/4)

12 Propagators for VPI N H = i + V (R) + i i<j spatial configuration V int (r ij ) R = r 1, r 2,..., r N external potential V (R) = i 1 2 (x 2 i + y 2 i + V b [ (zi /l) 2 ] 2 ) a < : 4th order propagator a > : 4th order propagator for T + V (R), modify for hard sphere V int (r ij ) with image construction

13 Modified 4th order propagator for hard sphere interactions G(R, R, τ) = dr e τ/6v (R) e τ/2t (R,R ) e 2τ/3Ṽ (R ) e τ 2T (R,R ) e τ/6v (R ) Ṽ = V + τ 2 [V, [T, V ]] 48 - modify to restricted path Green s function G r (R, R, τ), with G r (R, R, τ) = G(R, R, τ) G(I (R), R, τ) - I (R) is the mirror reflection of R in the plane of the hard wall - for hard sphere interactions, this gives image correction factor to first kinetic term G(r, r, τ) [1 ] e (r ij a)(r ij a)/λτ i<j

14 a > : Ground state energetics energy grows more slowly with higher barriers, suggests increasing fragmentation deviations from mean field with increasing N and larger a values Vb =5 Vb = 1 Vb = 2 Vb = 4 E(Vb,N)/E(Vb) N a a a DMC, a=.1 a VPI

15 a > : Natural orbitals z 1 n φ (z) n 1φ 1(z) n 2φ 2(z) 2 3 i n i /N.75(4) 1.2() 2.2(5) 3.1(2) 4.(2) N = 4, a/a ho.24, Na.99, na σ N = ( N) 2 N ρ(z, z ) = i n iφ i (z)φ i(z )

16 Finite Size Effects on Fragmentation occupation of lowest symmetric natural orbital as function of a/a a is perpendicular trap length,.1 a/a ho.5, N = 64 expect more fragmentation as a increases see non-monotonic dependence because a approaches length scale of trapping potential N V b = 1 V b = 2 V b = a/a

17 a > : squeezed states P ( N) P( N) = 2 δ( i Θ(z i) N R ) N N N = 64, a/a ho =.1, na , V b = 5

18 a > : squeezed states P ( N) N N = 64, a/a ho =.1, na , V b = 1

19 a > : squeezed states P ( N) N N = 64, a/a ho =.1, na , V b = 2

20 a > : squeezed states.6 P ( N) N Ψ N/2, N/2 N = 64, a/a ho =.1, na , V b = 4

21 Scaling of number fluctuations with interaction strength large N analytic two mode invalid N = 128 is too few particles - compare exact diagonalization or two mode Hamiltonian is not valid in this regime δ = 5x1 4, 1 2 < a/a ho <.3 Analytic two mode (based on harmonic analysis in large N limit): for κ > Nδ/2 8/3.1 n = N/4(δ/(δ + 4Nκ)) 1/4 n = δn/(8 2κ)

22 a > : Tunneling energetics excited state energy gives effective tunneling which decreases for higher barrier tunneling depends on N and a strong finite size effects for high barrier.1.1 E1 E.1 1e-4 1e N

23 a < : cat states possible ideal cat N, +, N : CN(z) C(z) z z non-ideal cat 1 1 k c k k, N k : C1(z) C N 1 (z) z z 1 2 3

24 Distinguishability Measure of Cat State Size consider 2-branch cat state A + B define cat size as the largest number of partitions, such that branches can be distinguished with probability 1 δ by measuring any one of the partitions or, if all particles are equivalent: N/n min, where n min is the smallest number of particles that must be measured branch distinguishability problem: determine which of two possible states, A or B, a given unknown state is 1 motivated by recognition that 2 ( ) requires only 1 measurement if 1 = but more if 1 [Korsbakken, Whaley, DuBois, Cirac PRA (27)]

25 How to distinguish branches How do we describe a measurement to distinguish branches of cat-like state A + B? Outcomes of measurement associated with POVM elements E A and E B Positive Operator Valued Measurements Generalization of projective measurements. Set {E i } of non-negative Hermitian operators satisfying i E i = I. Probability of outcome i for density matrix ρ is P i = tr (ρe i ). Success probability given by P = 1 2 tr ( A A E A) tr ( B B E B)

26 For an n-particle measurement, this reduces to P = 1 ( 2 tr ρ (n) A E (n) A )+ 1 ( ) 2 tr ρ (n) B E (n) B, E A,B E (n) A,B I(N n) where ρ (n) A,B are n-particle reduced density matrices Optimal measurement is projective measurement in the basis where ρ (n) A ρ(n) B is diagonal Corresponding success probability is (Helstrom 1976): P = tr ρ (n) A ρ(n) B If each branch is a separable state, same success probability is obtained with an adaptive scheme using only one-particle non-entangled measurements (PRA (27))

27 fit QMC fluctuation number distribution P( N) to form obtained from 2-state model [ ( ) N ( ) ] N dθ f (θ) â cos θ + ˆb sin θ + â sin θ + ˆb cos θ f (θ) = C N exp( (θ θ ) 2 2σ 2 ) orthogonal branches for θ = completely overlapping branches for θ = ±π/4 σ measures spread of branches (also causes overlap) n-rdm calculations possible with this form, yields cat size C δ Korsbakken, DuBois, Cirac, Whaley, PRA 27

28 a < : Number distribution.16. P ( N) P( N) = 2 δ( i Θ(z i) N R ) N 1 N N= 4, N a /a ho =.11, V b = 1 ω

29 a < : Number distribution.16 size C δ = δ = 1 2 P ( N) N N= 4, N a /a ho =.11, V b = 1 ω

30 a < : Number distribution.16. P ( N) N N= 4, N a /a ho =.11, V b = 15 ω

31 a < : Number distribution.16 size C δ = 1 δ = 1 2 P ( N) N N= 4, N a /a ho =.11, V b = 15 ω

32 a < : Number distribution P ( N) N N= 4, N a /a ho =.11, V b = 2 ω

33 a < : Number distribution.24 size C δ = 2 δ = 1 2 P ( N) N N= 4, N a /a ho =.11, V b = 2 ω

34 a < : Number distribution MEOW!! P ( N) N N= 4, N a /a ho =.22, V b = 12 ω

35 a < : Number distribution MEOW!!.5.4 size C δ = 4 δ = P ( N).2.1 Ψ N, +, N N N= 4, N a /a ho =.22, V b = 12 ω

36 liquid 4 He in a double well potential 4 He in a double well : parameters Aziz-1992 potential N = 2 barrier width a b =.15 Å barrier height 1 K trap frequency ω =.1 Hz

37 Cats in liquid 4 He: N=2.35 P ( N) T=2.5 K T=1.25 K T=.625 K T=.3125 K N cattiness at T=.31 K (cold cat) for δ =.1, C δ = 1; for δ = 1 5, C δ = 5

38 Summary VPI for strongly interacting bosons in double well potential repulsive bosons - finite size effects on squeezing attractive bosons - cat states, also for helium validity of two mode approximations excitations...