A guide to numerical experiments

Transcription

1 BECs and and quantum quantum chaos: chaos: A guide to numerical experiments Martin Holthaus Institut für Physik Carl von Ossietzky Universität Oldenburg Quo vadis BEC? at MPIPKS Dresden August 13, 2010

2 What this is about 0. What this is about Technical: Validity of mean-field ansatz (mfa) under chaotic conditions Operational: Suggestion of new type of experiments with Bose-condensed atoms Conceptual: Link between ultracold-atom physics and knowledge previously gained from quantum chaos Based on work done with Stephan Arlinghaus, André Eckardt, Bettina Gertjerenken, Maria Tschikin, and Christoph Weiss

3 Basic system 1. Basic system Bosonic Josephson junction: H 0 = Ω 2 with ( a 1 a 2 + a 2 a ) ( 1 + κ a 1 a 1 a 1a 1 + a 2 a 2 a ) 2a 2 [a j, a k ] = δ j,k (j, k = 1, 2) See, e.g., Milburn, Corney, Wright, & Walls, PRA 55, 4318 (1997), Smerzi, Fantoni, Giovannazi, & Shenoy, PRL 79, 4950 (1997), Parkins & Walls, Phys. Rep. 303, 1 (1998), and many pre-bec authors

4 Within mean-field approximation, this leads to (τ = Ωt ) i d dτ c 1(τ) = 1 2 c 2(τ) + 2 Nκ Ω c 1(τ) 2 c 1 (τ) i d dτ c 2(τ) = 1 2 c 1(τ) + 2 Nκ Ω c 2(τ) 2 c 2 (τ) Consider absolute values and phases, c j (τ) = c j (τ) e iϑ j(τ) (j = 1, 2) and introduce population imbalance and relative phase: p(τ) = c 1 (τ) 2 c 2 (τ) 2 ϕ(τ) = ϑ 2 (τ) ϑ 1 (τ)

5 Their equations of motion ṗ = 1 p 2 sin ϕ p Nκ ϕ = cos ϕ p2 Ω p constitute Hamiltonian equations for a nonrigid pendulum: ϕ = H nrp p where H nrp (p, ϕ) = α p 2, ṗ = H nrp ϕ 1 p 2 cos ϕ with α Nκ Ω. Solutions in terms of elliptic functions ( self-trapping ): p(τ) = cn(τ, α ) for α < 1 sech(τ) for α = 1 dn(ατ, 1/ α ) for α > 1 for p(0) = 1, say. (Kenkre & Campbell, PRB 34, 4959 (1986))

6 Realization with small BECs: Heidelberg group Typical data: 2π/Ω some 10 ms Nκ/Ω = O(1) for N = O(10 3 ) Albiez et al., PRL 95, (2005), Gati & Oberthaler, J. Phys. B 40, R61 (2007).

7 2. Extension Add time-periodic forcing: H(t) = H 0 + (µ 0 + µ 1 sin ωt) ( a 2 a 2 a 1 a 1). Within mfa, this corresponds to a driven nonlinear pendulum, ( µ0 H mf (τ) = H nrp 2p Ω + µ 1 Ω sin ω ) Ω τ, which, of course, admits chaotic solutions. M.H., PRA 64, (R) (2001), M.H. and Stenholm, EPJ B 20, 451 (2001).

8 Phenomena studied so far (1): Coherent control of self-trapping 1.0 Utilizes ( 2µ1 Ω eff = ΩJ n ω for ) p Np 0.0 n ω =! 2 µ τ Same rescaling governs control of SF-MI-transition: Eckardt, Weiss & M.H., PRL 95, (2005), Zenesini et al. (Pisa group), PRL 102, (2009).

9 Pisa experiments prove feasibility! Lignier et al., PRL 99, (2007), Sias et al., PRL 100, (2008), Eckardt et al., PRA 79, (2009).

10 Phenomena studied so far (2): Photon -assisted tunneling and fractional Shapiro steps 0.5 <J z > t / N ω/ω Eckardt, Jinasundera, Weiss & M.H., PRL 95, (2005)

11 Phenomena studied so far (3): Generation of entanglement Weiss & Teichmann, PRL 100, (2008)

12 Mfa for N-particle sys 3. Mfa for N-particle systems Folk wisdom: As everybody knows, the order parameter Ψ(t) is the expectation value ψ(t) of the field operator... wait a minute!

13 Mfa for N-particle sys 3. Mfa for N-particle systems Folk wisdom: As everybody knows, the order parameter Ψ(t) is the expectation value ψ(t) of the field operator... wait a minute! Consider N -particle Schrödinger equation: i d ψ(t) = H ψ(t) dt We wish to construct c i (t) such that N c i (t) 2 = ψ(t) a i a i ψ(t)

14 Let s go: At initial moment t 0, define (N 1)-particle states ψ i (t 0 ) a i ψ(t 0 ) a i ψ(t 0 ) Clearly, this implies ψ(t 0 ) a i a i ψ(t 0 ) = ψ(t 0 ) a i ψ i (t 0 ) ψ i (t 0 ) a i ψ(t 0 ) Thus, Nci (t 0 ) = ψ i (t 0 ) a i ψ(t 0 ) Define auxiliary states ψ i (t) for all t through time evolution: i d dt ψ i (t) = H ψ i (t) Then define amplitudes c i (t) : Nci (t) ψ i (t) a i ψ(t)

15 Writing ψ i (t) a i ψ(t) = ψ i (t 0 ) U a i U ψ(t 0 ) with U = U(t, t 0 ), one gets their time-evolution equation: i d dt ψ i (t) a i ψ(t) = ψ [ ] i (t) a i, H ψ(t). Taking, for example, H = Ω ( a 2 1 a 2 + a 2 a ) ( 1 + κ a 1 a 1 a 1a 1 + a 2 a 2 a ) 2a 2, one has [ ] a 1, H = Ω 2 a κ a 1 a 1a 1. Thus (with a small error) i d dt Nc1 (t) = Ω 2 Nc2 (t) + 2 κ ψ 1 (t) a 1 a 1a 1 ψ(t) (Sad: Usual hierarchy problem... )

16 Closure assumption: ψ i (t) a i a ia i ψ(t) = ψ i (t) (a i a ) i 1 a i ψ(t) ψ i (t) a i ψ(t) ψ(t) a i ψ i (t) ψ i (t) a i ψ(t) Nc i (t) [ ] = N 3/2 c i (t) c i (t) 2 1/N Record instantaneous error of closure: (3) i (t) ψ i (t) a i a i a i ψ(t) N 3/2 c i (t) 2 c i (t) Then d i Ωdt c 1(t) = 1 2 c 2(t) + 2 Nκ Ω ( c 1 (t) 2 1 )c 1 (t) + (3) 1 (t) N N 3/2

17 Now take N, keeping Nκ constant, and assuming (3) i (t)/n 3/2 to vanish sufficiently fast with N : i d dτ c 1(τ) = 1 2 c 2(τ) + 2 Nκ Ω c 1(τ) 2 c 1 (τ) i d dτ c 2(τ) = 1 2 c 1(τ) + 2 Nκ Ω c 2(τ) 2 c 2 (τ) Hooray! But what have we gained? N c i (t) 2? = ψ(t) a i a i ψ(t)

18 Now take N, keeping Nκ constant, and assuming (3) i (t)/n 3/2 to vanish sufficiently fast with N : i d dτ c 1(τ) = 1 2 c 2(τ) + 2 Nκ Ω c 1(τ) 2 c 1 (τ) i d dτ c 2(τ) = 1 2 c 1(τ) + 2 Nκ Ω c 2(τ) 2 c 2 (τ) Hooray! But what have we gained? N c i (t) 2? = ψ(t) a i a i ψ(t) For this, we actually need with ψ(t) a i a i ψ(t) = ψ(t) a i ψ i (t) ψ i (t) a i ψ(t) ψ i (t) = a i ψ(t) a i ψ(t)

19 Once again, for clarity: By construction, we have ψ(t 0 ) a i ψ i (t 0 ) U(t,t 0) ψ i (t) whereas we actually require ψ(t 0 ) U(t,t 0) ψ(t) a i ψ i (t)

20 Once again, for clarity: By construction, we have ψ(t 0 ) a i ψ i (t 0 ) U(t,t 0) ψ i (t) whereas we actually require ψ(t 0 ) U(t,t 0) ψ(t) a i ψ i (t) Game lost? There is one way out: Suppose we have ψ i (t) = η i (t) a i ψ(t) Then 1 = η i (t) 2 ψ(t) a i a i ψ(t) together with Nci (t) = ψ i (t) a i ψ(t) = η i (t) ψ(t) a i a i ψ(t)

21 Squaring, this now yields the desired identity N c i (t) 2 = η i (t) 2 ψ(t) a i a i ψ(t) 2 = ψ(t) a i a i ψ(t) Thus, we have an integral measure of quality: R i (t, t 0 ) = ψ i (t) a i ψ(t) a i ψ(t) or, more explicitly, R i (t, t 0 ) = ψ(t 0) a i U (t, t 0 ) a i U(t, t 0 ) ψ(t 0 ) a i ψ(t 0 ) a i ψ(t) Moral of the tale: The mfa is the better, the less R i (t, t 0 ) deviates from unity.

22 experiment 4. Numerical experiments Further extension of the model: H(t) = H 0 + ( µ 0 + µ 1 (t) sin ωt ) ( a 2 a 2 a 1 a 1 ) with slowly varying amplitude: µ 1 (t) = µ 1,max exp ( t2 2σ 2 ) Rationale: Employ adiabatic principle for deliberately guiding the system to critical spots! Breuer & M.H., Z. Phys. D 11, 1 (1989), Eckardt & M.H., PRL 101, (2008), Zenesini et al. (Pisa group), PRL 102, (2009).

23 N = 20, µ 1,max Ω = 0.8 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

24 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

25 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

26 N = 100, µ 1,max Ω = 0.8 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

27 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

28 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

29 N = 100, µ 1,max Ω = 0.9 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

30 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

31 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

32 N = 1000, µ 1,max Ω = 0.9 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

33 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

34 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

35 N = 1000, µ 1,max Ω = 1.0 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

36 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

37 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

38 N = 1000, µ 1,max Ω = 1.5 (Nκ/Ω = 0.95, µ 0/Ω = 0, σ/t = 10 ) 0.5 <J z > t / T J z (t) 1 2 ψ(t) a 1 a 1 a 2 a 2 ψ(t)

39 1.2 <J z > δ 1 R t / T Defect (3) 1 (t) /N 3/2 (red), and measure R 1 (t) (black)

40 0.5 <J z > t / T N -particle solution (blue) vs. mfa (red)

41 Maximum (3) 1 /N 3/2 for increasing µ 1,max and various N 0.4 1,max / N 3/ µ 1,max /Ω N = 20, 50, 100, 500, 1000

42 Minimum R 1 for increasing µ 1,max and various N 1.0 R 1,min µ 1,max /Ω N = 20, 50, 100, 500, 1000

43 messages 5. Take-home messages Technical: Mean-field description of N interacting Bosons with closure on the lowest level of the momentum hierarchy ( Gross- Pitaevskii-equation ) is viable if the system manages to hide the essential difference between U(t, t 0 ) a i ψ(t 0 ) and a i U(t, t 0 ) ψ(t 0 ) behind the phase of η i (t). Conjecture: For sufficiently large (but still finite!) N, this can happen over long time intervals only if the nonlinear GP dynamics are regular.

44 Operational: Besides simulating condensed-matter (and other) systems, dynamics are a potentially big issue in cold-atom physics. Questions way beyond the capabilities of supercomputers (except for strongly simplified models) now are amenable to experimental investigation: Heidelberg and Pisa have paved the way.

45 Operational: Besides simulating condensed-matter (and other) systems, dynamics are a potentially big issue in cold-atom physics. Questions way beyond the capabilities of supercomputers (except for strongly simplified models) now are amenable to experimental investigation: Heidelberg and Pisa have paved the way. Conceptual: A super-question lurking in the background is under which conditions a reduced description of a complex system is possible, and when not (and if not, why so). A huge body of knowledge gained from quantum chaos is directly applicable here.