Probing Many Body Quantum Systems by Interference

Transcription

1 Probing Many Body Quantum Systems by Interference Jörg Schmiedmayer Vienna Center for Quantum Science and Technology, Atominstitut, TU-Wien J. Schmiedmayer: Probing Many-Body Quantum Systems 1 Outline One-dimensional systems Coherently split 1d quantum gas Tools to probe many body physics Full distribution functions of interference Probing the Dynamics Dephasing dynamics: Two regimes Pre-Thermalization Generalized Gibbs ensemble Emergence of a new length scale Local emergence of the relaxed state Light cone like spreading AtomChip Other Non-equilibrum systems Decay of excited state -> twin beams Fast cooling in 1d Outlook J. Schmiedmayer: Probing Many-Body Quantum Systems 2

2 1-dimensional Systems J. Schmiedmayer: Probing Many-Body Quantum Systems 3 1-dimensional Systems Weakly interacting 1d Bose gas quasi-condensate ω All energies µ, k B T << ħω uniform density fluctuating phase true 1D condensate longe range phase coh. quantum fluctuations T=0 T ϕ n 1d ξ h ω ξ h = ~5 nk T φ : l ϕ =L 1D quasi condensate macroscopic wave function fluctuating phase: l φ <L quasi condensate regime experiments: nk thermal gas T / d = Nω k B ~1000 nk With interaction energy: I = n g and 1d interaction strength: m n 1d g 1d 1d 1d g1 d = 2ω as T J. Schmiedmayer: Probing Many-Body Quantum Systems 4

3 1-dimensional Systems Weakly interacting 1d Bose gas quasi-condensate ω All energies µ, k B T << ħω uniform density fluctuating phase Excitations play an enhanced role in 1d Homogeneous 1d Bose gas integrable thermalization expected to be slow thermally populated Kinoshita et al. Nature 440, 900, 2006 The longitudinal phase fluctuations are key for our experiments J. Schmiedmayer: Probing Many-Body Quantum Systems 5 Coherent splitting of a 1d BEC RF traps on a chip manipulation using near-field radio-frequency fields: Schumm et al, Nature Phys. 1, 57 (2005) Coupling of atomic Zeeman levels by RF fields smoothly deforms the trap into a (fully tunable) double-well: A coherent beamsplitter for matterwaves! J. Schmiedmayer: Probing Many-Body Quantum Systems 6

4 Probing the quantum state full distribution function of interference J. Schmiedmayer: Probing Many-Body Quantum Systems 7 FDF of phase and contrast Matter-wave interferometry: repeat many times phase, contrast i>100 This is only possible as we prepare a single 1d gas! Plot as circular statistics contrast i full distribution function of phase & contrast phase i Polkovnikov et al. PNAS 103, 6125 (2006) Gritsev et al. Nature Phys. 2, 705(2006) FDF contains information about all order correlation functions in solid state: Full Counting Statistics J. Schmiedmayer: Probing Many-Body Quantum Systems 8

5 interference of phase fluctuating 1D condensates How can we study the dynamics and noise of phase fluctuations ϕ 1 create a copy ϕ 2. ϕ n create two independet samples ϕ 1 ϕ 2. ϕ n ϕ 1 ϕ 2. ϕ n ϕ 1 ϕ 2. ϕ n J. Schmiedmayer: Probing Many-Body Quantum Systems 9 x 1 x 2 d Interfering independent 1d Quantum Liquids A is a quantum operator. Its measured value will fluctuate from shot to shot. A = 1 n 1d L/2 L/2 a 1 + (z)a 2 (z)dz A. Polkovnikov, et al., PNAS 103, 6125 (2006). V. Gritsev, et al., Nature Phys. 2, 705 (2006). For independent BEC s: expectation value of contrast: A = 0 due to random rel. Phases Look at A 2 A 2 = 1 n 1d 2 nd moment of fringe ( average contrast ) L/2 dz 2 a + 1 (z 1 )a 1 (z 2 ) a + 2 (z 2 )a 2 (z 1 ) dz 2 1 L/2 2 nd order correlation function E. Demmler, A. Polkovnikov, V. Gritsev, A. Imambekov E. Altman J. Schmiedmayer: Probing Many-Body Quantum Systems 10

6 higher order correlations: full contrast statistics A Q 2m A Q 2 m α m = W (α)α m dα 0 with α = A A Q Q 2 2 normalized moment of order m full distribution of fringe contrast to reconstruct the full distribution, one has to calculate ALL moments of interference contrast How to predict the full distribution function: Quantum impurity problem: interacting one dimensional electrons scattered on an impurity Conformal field theories with negative central charges mapping to: - generalized Coulomb gas model - related problem of fluctuating random surfaces Ornstein-Uhlenbeck stochastic process (for thermal occupation of excitations) V. Gritsev, et al., Nature Phys. 2, 705 (2006) A. Polkovnikov, et al., PNAS 103, 6125 (2006) A. Imambekov et al. PRA 77, (2008) HP. Stimming et al. et al. PRL 105, (2010) J. Schmiedmayer: Probing Many-Body Quantum Systems 11 Analysis of interference patterns: contrast analysis Hofferberth et al Nature Phys. 4, 489 (2008) wave vector of fringe separation: contrast of integrated profile: Q = md / expectation value of contrast: A = 0 A / 2 1 L + a1 1d L / 2 J. Schmiedmayer: Probing Many-Body Quantum Systems 12 Q Q = n t TOF ( z) a 2 ( z) dz due to random rel. phases

7 full contrast statistics theory predictions A. Polkovnikov, et al., PNAS 103, 6125 (2006). V. Gritsev, et al., Nature Phys. 2, 705 (2006). Semi-classical approach: Stimming et al. et al. PRL (2010) theoretically expected distribution functions for the average contrast: quantum coherence: asymetric Gumbel distribution (low temp. T or short length L) α = A A 2 Q 2 Q thermal fluctuations: broad Poissonians distribution (high temp T or long length L) intermediate regime: double-peak strukture J. Schmiedmayer: Probing Many-Body Quantum Systems 13 full contrast statistics experiment experimentally measured distribution functions for the average contrast: Hofferberth et al Nature Phys. 4, 489 (2008) ω T = 2π 3 khz n 1d = 60 µm -1 K = 46 T = 30 nk ξ T = 0.9 µm quantum coherence: asymetric Gumbel distribution (low temp. T or short length L) ω T = 2π 3 khz n 1d = 60 µm -1 K = 46 T = 60 nk ξ T = 0.45 µm thermal fluctuations: broad Poissonians distribution (high temp T or long length L) intermediate regime: double-peak strukture No free parameters! experiment records entire distribution function of interference contrast high order correlations can be derived J. Schmiedmayer: Probing Many-Body Quantum Systems 14

8 Bogoliubov Approach calculate tunnel coupled systems Luttinger liquid treatment is based on statistical independence of fluctuations in the two quasi condensates and can not easily be extended to 1D tunnel-coupled systems: Stimming et al. et al. PRL 105, (2010) Symbols: Luttinger L. Line : Bogoliubov Mora/Castin showed equivalence of Luttinger Liquid and Bogoliubov approach (PRA 2003) Applying it to the approach of Whitlock and Bouchoule allows to describe also tunnel-coupled systems. The relative phase can be modelled by: with: random numbers [0, 1] include quantum fluctuations (zero-point oscillations of the atomic field): J. Schmiedmayer: Probing Many-Body Quantum Systems 15 Semi-Classical calculation Gaussian fluctuations thermal fluctuations lead to a simple statistical model: Semi-classical expression for A(L) Stimming et al. et al. PRL 105, (2010) Compare full quantum and semi-classical Symbols: Bogoliubov Line : Gaussian mod. Gaussian fluctuations for θ(z) The relative phase evolution along z is described by an Ornstein-Uhlenbeck stochastic process f(z) is the random force with the properties <f(z)>= 0, <f(z 1 )f(z 2 )> = 2κ T δ (z 1 z 2 ), and 1/l J plays the role of the friction coefficient. Where can we see ground state quantum fluctuations? J. Schmiedmayer: Probing Many-Body Quantum Systems 16

9 Fluctuating interference for coupled 1d systems Experiment: Betz et al. PRL 106, (2011) Theory: Stimming et al. et al. PRL 105, (2010) J. Schmiedmayer: Probing Many-Body Quantum Systems 17 Phase correlation comparison to semi-classical model T. Betz et al. PRL 106, (2011) Phase locking for l J < λ Τ J. Schmiedmayer: Probing Many-Body Quantum Systems 18

10 Quantum <-> Thermal full contrast statistics Where can we see quantum fluctuations? Low temperature Short length scale Quantum fluctuations broaden the distribution function W(α) Stimming et al. et al. PRL 105, (2010) L=10μm n 1D = 60μm -1 ω=2π 3 khz T=10 nk ω=2π 60 khz T=10nK Look at the ratios of the higher moments of W(α) R s < 1 Quantum fluctuations dominate J. Schmiedmayer: Probing Many-Body Quantum Systems 19 phase correlation function Has the form phase correlation function quantum <-> thermal Stimming et al. et al. PRL 105, (2010) With asymptotic behaviour: Thermal fluctuations become dominant at: J. Schmiedmayer: Probing Many-Body Quantum Systems 20

11 Probing the Dynamics of Experiment: M. Gring, M. Kuhnert, T. Langen et al. (VCQ, Vienna) Theory: T. Kitagawa, E. Demler (Harvard) I. Mazets (VCQ, Vienna) J. Schmiedmayer: Probing Many-Body Quantum Systems 21 Relaxation in an Isolated Quantum System after a Quench Does an isolated many body quantum system relax? Quench populates many body states Evolution is unitary -> de-phasing of the quantum states populated in the quench How long does a quantum system retain memory of its initial state? How does the time averaged state look like Eigenstate thermalization hypothesis Srednicki Phys.Rev.E, 50, 888 (1994) Rigol et al. Nature 452, 854 (2008) Systems with dynamical constraints (close to integrable point) Pre-thermalization Berges et al., PRL, 93, (2004) Generalized Gibbs ensembles Rigol et al., PRL, 98, (2007) rich dynamics expected J. Schmiedmayer: Probing Many-Body Quantum Systems 22

12 Split 1d-Bose-gas as non-equilibrium system Initial state coherently split quantum gas + ˆn 1 (x), φ ˆ 1 (x) ˆn 2 (x), φ ˆ 2 (x) symmetric anti symm. ˆn s = ˆn 1 + ˆn 2, ˆn a = ˆn 1 ˆn 2, φs ˆ = φ ˆ 1 + φ ˆ 2 thermally populated φa ˆ = φ ˆ 1 φ ˆ 2 populated by quantum fluctuations hold time Thermal equilibrium state two independent quantum gases + equal (thermal) populated symmetric anti symm. ˆn s = ˆn 1 + ˆn 2, ˆn a = ˆn 1 ˆn 2, φs ˆ = φ ˆ 1 + φ ˆ 2 equal thermal population φa ˆ = φ ˆ 1 φ ˆ 2 J. Schmiedmayer: Probing Many-Body Quantum Systems 23 Experimental Procedure Gring et al., Science 337, 1318 (2012) J. Schmiedmayer: Probing Many-Body Quantum Systems 24

13 Theoretical Description of Non- Equilibrium Dynamics Interacting many body system is described by an effective field theory with quasi particle excitations Luttinger-liquid (integrable) [ ρ H s = 8m ( ˆφ z s ) 2 + gˆn 2 s [ ρ H c = 8m ( ˆφ z c ) 2 + gˆn 2 c φ s (z) = relative phase φ c (z) = sum phase ] dz ] dz n s (z) = relative density n c (z) = sum density Kitagawa et al. PRL (2010) NJP (2011) Excitations are soundwaves that scramble the local relative phase on different length scales Change of relative phase is directly related to the interference contrast Model describes the dynamics through the dephasing of the momentum modes (k-modes) does not describe a thermalisation process Just after splitting thermally populated ˆφ c (z) = ˆφ A (z)+ ˆφ B (z) ˆφ s (z) = ˆφ A (z) ˆφ B (z) populated only by quantum shot noise from the splitting process Equilibrium thermally populated ˆφ c (z) = ˆφ A (z)+ ˆφ B (z) ˆφs (z) = ˆφ A (z) ˆφ B (z) thermally populated Two-observable FDFs J. Schmiedmayer: Probing Many-Body Quantum Systems 25 FDF of Phase and Contrast comparison to theory theory: Kitagawa et al. Kuhnert et al., PRL 110, (2013) J. Schmiedmayer: Probing Many-Body Quantum Systems 26

14 Experimental observation two regimes Initially reduced phase spread shows coherence of the splitting Over time, two regimes emerge: long length scale: significant occupation of phonon modes with λ<l leads to random phaseswith in L and to loss of contrast on the same timescale as the phase diffuses contrast decay regime (spin decay) short length scale: only significant occupation of phonon modes with λ>l -> only phase diffusion phase diffusion regime (spin diffusion) theory: Kitagawa et al., PRL 104, (2010); NJP (2011) experiment: Kuhnert et al., PRL 110, (2013) J. Schmiedmayer: Probing Many-Body Quantum Systems 27 evolution time te integration length L FDF of (Contrast) 2 Gring et al., Science 337, 1318 (2012) 110µm 22µm Long length scale Short length scale 27.5ms 17.5ms 12.5ms 7.5ms 4.5ms 2.5ms Initial coherence decays over time distribution becomes indistiguishable from a random distribution. Correlations between the two gases appear to be lost. The interferometer seems to have lost it s coherence Distribution stays peaked for all times Correlations still present in the system for long times J. Schmiedmayer: Probing Many-Body Quantum Systems 28

15 FDF of (Contrast) 2 FULL CLOUD 60 μm 38 μm 27 μm 22 μm Gring et al., Science 337, 1318 (2012) (red) theory with experimental parameters Probability Density ms ms ms ms ms ms Contrast Squared, C 2 Experiment: D. Smith et al. (Vienna) Theory: Kitagawa et al. (Harvard) J. Schmiedmayer: Probing Many-Body Quantum Systems 29 full contrast statistics experiment experimentally measured distribution functions for the average contrast: Theory: A. Polkovnikov, et al., PNAS 103, 6125 (2006). V. Gritsev, et al., Nature Phys. 2, 705 (2006). Experiment: S. Hofferberth et al Nature Phys. 4, 489 (2008) ω T = 2π 3 khz n 1d = 60 µm -1 K = 46 T = 30 nk ξ T = 0.9 µm quantum coherence: asymetric Gumbel distribution (low temp. T or short length L) ω T = 2π 3 khz n 1d = 60 µm -1 K = 46 T = 60 nk ξ T = 0.45 µm thermal fluctuations: broad Poissonians distribution (high temp T or long length L) intermediate regime: double-peak strukture No free parameters! experiment records entire distribution function of interference contrast high order correlations can be derived J. Schmiedmayer: Probing Many-Body Quantum Systems 30

16 Thermalization Gring et al., Science 337, 1318 (2012) Measure effective temperature from the distributions: exp T kin ~ 100nK T eff =12 ± 3 nk L th (T eff ) = 12 ± 4 µm ~ L 0 Effective temperature is ~ 8 times colder than the initial kinetic temperature! Experiment: D. Smith et al. (Vienna) Theory: Kitagawa et al. (Harvard) J. Schmiedmayer: Probing Many-Body Quantum Systems 31 Thermalization? Gring et al., Science 337, 1318 (2012) Measure effective temperature from the distributions: y FDF for independent condensates at 60 nk T eff =12 ± 3 nk Effective temperature is ~ 8 times colder than the initial kinetic Normalised temperature! Squared Contrast, C 2 /<C 2 > J. Schmiedmayer: Probing Many-Body Quantum Systems 32

17 Longer Times Time (ms) 15 ms µm 61µm 41µm 30 µm 20µm 10µm J. Schmiedmayer: Probing Many-Body Quantum Systems 33 Long Time Behaviour Gring et al., Science 337, 1318 (2012) Evaluate effective temperature of the quasi steady state from fit to equilibrium model thermalization rate : 0.14 ± 0.04 nk/ms measured heating rate: 0.11 ± 0.06 nk/ms calcualted thermalization time: 200 ms J. Schmiedmayer: Probing Many-Body Quantum Systems 34

18 Interpretation Our 1d many body quantum system is close to an integrable system (perfect 1d system) Fast evolution is the de-phasing of the phonon modes of the initial state of the split 1d system relaxation in an integrable system (quasi) steady state is the quantum state the integrable system relaxes to. It can be described by a generalized Gibbs ensemble The fast splitting process leads to equiparition of energy in the (antisymmetric) modes thermal like state Prediction: effective temperature for the quasi steady state given by the quantum shot noise introduced by the splitting process k B T eff = gρ/2, theory: Kitagawa et al. Gring et al., Science 337, 1318 (2012) Over long times the quasi steady state should slowly decay J. Schmiedmayer: Probing Many-Body Quantum Systems 35 Verification of Scaling theory: Kitagawa et al. Gring et al., Science 337, 1318 (2012) T eff scales linearely with density: T eff is independent on initial Temperature: J. Schmiedmayer: Probing Many-Body Quantum Systems 36

19 Characterization of Emerging Legthscale Kuhnert et al., PRL 110, (2013) Mean Contrast squared <C 2 > Measured thermal correlation length λ th L=18 µm L=40 µm L=60 µm L=100 µm pre-thermalized state data averaged for time t e > 10 ms λ th (T eff ) = 17±1 µm thermal equilibrium We know from the FDF that for t e > 10 ms the correlation functions are thermal. This allows us to extract the thermal correlation length λ th from < C 2 > J. Schmiedmayer: Probing Many-Body Quantum Systems 37 Pre-Thermalization Effective length scale Kuhnert et al., PRL 110, (2013) Theory introduces a new characteristic thermal-like length scale L 0 : Transition between decay and diffusion regime occurs around integration length L 0 = 8K 2 /π 2 n 1d This is much longer than the thermal coherence length: Theory: L 0 = 15.8 ± 0.9 µm Measured: λ th (T eff ) = 16.9 ± 0.9 µm ~ L 0 Initial T : λ th ~1/T = µm J. Schmiedmayer: Probing Many-Body Quantum Systems 38

20 Interpretation Fast evolution is the de-phasing of the phonon modes of the initial state of the split 1d system relaxation in an integrable system Gring et al., Science 337, 1318 (2012) Kuhnert et al., PRL 110, (2013) Smith et al., arxiv: (quasi) steady state is the quantum state the integrable system relaxes to. It can be described by a generalized Gibbs ensemble The fast splitting process leads to equiparition of energy in the (antisymmetric) modes thermal like state Prediction: effective temperature for the quasi steady state given by the quantum shot noise introduced by the splitting process Generalized Gibbs Ensemble 2 parameters to describe the statistical ensemble: kinetic temperature (related to initial temperature) Phase correlation temperature (related to the relaxation) Example of a Pre-thermalized state (Berges 2004) Over long times the quasi steady state should slowly decay J. Schmiedmayer: Probing Many-Body Quantum Systems 39 Optimal Control of Splitting fast squeezing in a multi mode system Optimal Controll applied to the problem of the fluctuation properties in splitting a BEC J. Grond et al. PRA 79, R (2009) J. Grond et al. PRA 80, (2009) T. Langen et al. Preliminary Adiabatic splitting Fancy splitting ramps inspired by OCT: t 1 +t 2 = 17ms Leads to dramatic change of statistical distribution of interference full cloud 140 µm long t 1 8ms 10 ms 20ms 30ms 40ms t 2 t Hold J. Schmiedmayer: Probing Many-Body Quantum Systems 40

21 Emergence of the relaxed state T. Langen et al arxiv: J. Schmiedmayer: Probing Many-Body Quantum Systems 41 Local observation of relaxation How does the system acquire thermal-like properties? T. Langen et al arxiv: (prethermalized) J. Schmiedmayer: Probing Many-Body Quantum Systems 42

22 Decay of coherence T. Langen et al arxiv: Time evolution of the phase correlation function C(z = z - z ) = i(ϕ(z) ϕ( z )) e ODLRO 1 ms 2 ms thermal prethermalized state 3 ms 4 ms 5 ms ms J. Schmiedmayer: Probing Many-Body Quantum Systems 43 Light-Cone dynamics in the decay of coherence T. Langen et al arxiv: Expansion velocity of region with final form of the phase correlation function -> sound velocity scaling with density sound velocity in infinite system LL in a trap -> Light-Cone dynamics LL theory for sound velocity in the trap: c = π/4 c 0 J. Schmiedmayer: Probing Many-Body Quantum Systems 44

23 Emergence of Light-Cone Phase correlation function: T. Langen et al arxiv: C(z,z,t) = exp( 1 2 Δφ zz (t)2 ) Initially the spitting process creates excitations (phonons with ω k =c 0 k ) in the density quadrature. (density fluctuations from the beam splitter) With time the density quadrature of the phonons oscillate into the phase quadrature (with ω k ) Equipartition created by a fast splitting results in a 1/k population of the modes Time evolution of the phase variance: Δφ zz (t) 2 = 2π2 LK 2 k 0 sin(ω k t) 2 (1 cos(k z)) Fourier decomposition of a ramp with a flat plateau starting at z=c 0 t k 2 J. Schmiedmayer: Probing Many-Body Quantum Systems 45 Emergence of Light-Cone Δφ zz (t) 2 = 2π2 LK 2 k 0 sin(ω k t) 2 (1 cos(k z)) k 2 J. Schmiedmayer: Probing Many-Body Quantum Systems 46

24 Decay of the pre-thermalized state J. Schmiedmayer: Probing Many-Body Quantum Systems 47 Quest for Decay of the Prethermalized state Improved heating in the trap < 50nK/s M. Kuhnert et al., in preparation J. Schmiedmayer: Probing Many-Body Quantum Systems 48

25 Equilibration Measured temperatures ω 70 nk M. Kuhnert T in =69±11 nk, n 3d = 4.3 x10 14 cm 3 T in =74±8 nk, n 3d = 7 x10 14 cm 3 T in =73±10 nk, n 3d = 4.3 x10 14 cm 3 T in =78±7 nk, n 3d = 3.8 x10 14 cm 3 T in =115±20 nk, n 3d = 3.8 x10 14 cm 3 Equilibrium T = (T in + T preth )/2 Equilibration T in =120±17 nk, n 3d = 4.3 x10 14 cm 3 T in =140±20 nk, n 3d = 6.2 x10 14 cm 3 T in =177±17 nk, n 3d = 2.8 x10 14 cm 3 T in =290±60 nk, n 3d = 3.6 x10 14 cm 3 T in =320±50 nk, n 3d = 6.2 x10 14 cm 3 J. Schmiedmayer: Probing Many-Body Quantum Systems 49 Thermalization due to 2-body collisions M. Kuhnert estimated time to thermalize [ms] T [nk] J. Schmiedmayer: Probing Many-Body Quantum Systems 50

26 What have we learned generalization of homodyne measurement: the full distribution functions of observables give detailed insight into (quantum) physics in ensemble averages the central limit theorem of Gaussian statistics hides the (quantum) physics thermalization in quantum systems does not proceed through a simple path. establishment of a prethermalized state generalized Gibbs ensemble Relaxed state emerges locally and spreads throughout the many body system in a light cone like fashion pre-thermalized state decays by coupling to other dimensions FDF s allow to look at the dynamics of (de-) coherence Gring et al., Science 337, 1318 (2012) Kuhnert et al., PRL 110, (2013) Smith et al., arxiv: Langen et al., arxiv: J. Schmiedmayer: Probing Many-Body Quantum Systems 51 The Team Takuya Kitagawa Eugene Demler Jörg Schmiedmayer Igor Mazets Michael Gring Max Kuhnert DAS Tim Langen Bernhard Rauer Remi Geiger J. Schmiedmayer: Probing Many-Body Quantum Systems 52