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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates Michikazu Kobayashi (Kyoto Univ.) Leticia F. Cugliandolo (Paris VI) Bose-Einstein condensates at finite temperatures Stochastic Gross-Pitaevskii equation

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Michikazu Kobayashi (Kyoto Univ.) Leticia F. Cugliandolo (Paris VI)

  • Jan. 18, 2016 “量子渦と非線形波動2016”
  • Bose-Einstein condensates at finite temperatures
  • Stochastic Gross-Pitaevskii equation and thermodynamic

phase transition

  • Geometric transition of quantized vortices
  • Phase ordering and quantized vortices in quench dynamics
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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Ultracold atomic Bose gas

87Rb, 23Na, 7Li, 1H, 85Rb, 41K, 4He, 133Cs, 174Yb, 52Cr, 40Ca, 84Sr, 164Dy, 168Er

Trapping atoms Laser cooling Evaporative cooling

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Ultracold atomic Bose gas

4£10-7 K 2£10-7 K 1£10-7 K JILA, 1995

Condensation Condensation (not dynamics) (not dynamics)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Ultracold atomic Bose gas

4£10-7 K 2£10-7 K

Condensation Condensation (not dynamics) (not dynamics) Phase transition of noninteracting Bose gas

Uniform system Harmonically trapped system

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Bose-Einstein condensates at finite temperatures

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Thermodynamic phase transition

Total energy and specific heat Total energy and specific heat Correlation length Correlation length

Critical behaviors of specific heat and correlation length Critical behaviors of specific heat and correlation length near the critical temperature near the critical temperature → → 2nd ordered phase transition 2nd ordered phase transition

PRL 77, 4984 (1996) Science 315, 1556 (2007)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Efgects of interparticle interaction

a a : s-wave scattering length : s-wave scattering length

Infinitesimal interaction a changes the universality class for uniform system (a =0 is singular) → It is difgicult to determine ¢Tc

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Theories for BEC at finite T

Boltzmann & Gross-Pitaevskii（ZNG theory） Stochastic Gross-Pitaevskii eq. Complex Ginzburg-Landau eq. Classical-field Monte Carlo Bogoliubov theory Projected Gross-Pitaevskii eq. Path-integral Monte Carlo Trancated Wigner method Complex Stochastic Gross-Pitaevskii eq.

  • Simple
  • Not widely used
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SGP equation and thermodynamic phase transition

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

SGP equation in uniform system

JPhysB 38, 4259 (2005)

Unapplicable near the zero temperature due to neglecting the commuation relation [Ã,Ãy]=i± (complexification of à is needed)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

SGP equation in uniform system

JPhysB 38, 4259 (2005)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

SGP equation in uniform system

Ito's lemmna

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

SGP equation in uniform system

SGP equation gives (at least) equilibrium property with GP energy functional

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Numerical Simulation of SGP eq.

Space : 3-dimensional space with periodic boundary condition

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Thermodynamic transition

Order parameter Specific heat Non-analytic behavior emerges at T ≈ 2 . 2 6

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Critical exponents and universality class

Nature of thermodynamic transition for interacting Bose gas → Symmetry breaking of global U(1) phase shifu : ψ → ψ e

i φ

→ Universality class : XY model

Critical exponents and comparison with XY model

SGP equation can describe the BEC transition as spontaneous U(1) symmetry breaking

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Geometric transition of quantized vortices

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Thermodynamic and geometric transitions

Question : Are there geometric transition corresponding to the BEC transition (thermodynamic transition)?

For free bosons : percolation transition of particle worldlines

PRE 63, 026115 (2001)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Thermodynamic and geometric transitions

β ℏ Example of particle worldlines w = 1 w = 3 β ℏ β ℏ β ℏ For ¹ = 0 at T = Tc, large worldline loops emerge and worldline percolation occurs (critical exponents are same)

Probability weight for loop w

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Geometric transition of vortex loops

Interacting bosons : discussion of particle worldlines is difgicult (It cannot be discussed within SGP equation) Can we expect the percolation of vortex loops instead at the thermodynamic transition point?

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Vortex line density

Small loops are generated through the tunneling process

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Vortex snapshots (longest loop is highlighted)

T = 0.6 Tc T = 0.8 Tc T = Tc

Long loops are generated close to the critical point

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Loop length distribution

Power-law structure emerges near Tc → Vortex percolation

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Loop length distribution

Power-law fitting P(l)_l{¿ is best at T =0.98Tc → Vortex percolation occurs at Tpº0.98Tc Discrepancy between Tp (geometric transition) and Tc (thermodynamic transition)

Bump structure at large l for T=0.99Tc and T=Tc → Percolating loops (finite-size efgect)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Discrepancy between geometric and thermodynamic transitions

Discrepancy between geometric and thermodynamic transitions are observed in several interacting models

  • SU(2) local gauge field model : Tp t 0.994 Tc
  • RP2 model (nematic liquid crystal) : Tp t 0.996 Tc
  • Nonlinear O(2) sigma model : Tp t 0.992 Tc
  • Phys. Lett. B 482, 114 (2000)

PRB 72, 094511 (2005)

  • Geometric transition of line defects occurs as a precursory

phenomenon of thermodynamic transition (both are independent).

  • Thermally excited long vortices may be detectable between

Tc and Tp

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Critical exponents and order parameter

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Critical exponents and order parameters

Number of percolating loops

Critical exponents of order parameters are also consistent with universality of self-seeking random work

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Phase ordering and quantized vortices in quench dynamics

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Melting dynamics

Critical slowing down near the (thermodynamic) critical temperature

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Temperature quench dynamics

Line length density

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Temperature quench dynamics

The same critical exponent as that in the equilibrium at Tp (not Tc) emerges → Spontaneous formation of critical percolating state (not critical thermodynamic state) : dynamics is dominated by vortices! Line length density Loop length distribution

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Summary

  • We consider the statistical properties such as transition of interacting

BEC in equilibrium.

  • There are two kinds of transitions: well-known thermodynamic

transition and geometric transition of quantized vortices and both are independent.

  • Universality class:

XY-model for thermodynamic transition Self-forcusing random walk for geometric transition

  • Geometric critical state emerges in the quench dynamics.
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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Uniform system vs trapped system

PRL 77, 4984 (1996)

Trapped system Trapped system Toward uniform system Toward uniform system

PRL 110, 200406 (2013)

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Zaremba-Nikuni-Griffin theory

JLTP 116, 277 (1999)

Noncondensate particle : Boltzmann's eq. Condensate particle : GP eq. Exchange between two components

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Zaremba-Nikuni-Griffin theory

JLTP 116, 277 (1999)

Exchange process : Markov process SGP eq. (discrepancy from Gaussian noise is renormalized into γ ) Condensate particle : GP eq. Exchange between two components

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Estimation of °

Damping of Scissors mode

PRL 86, 3938 (2001)

°/(na3)1/3 from ZNG theory

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Thermodynamic phase transition and quantized vortices in Bose-Einstein condensates

Loop length distribution