Time crystal behaviour - th- · Mean- eld approximation - SSB (CM localization) Mean- eld...

Time crystal behavior of excitedeigenstates

Jagiellonian University in Krakow

A. Syrwid, J. Zakrzewski, and K. Sacha

A. Syrwid, J. Zakrzewski, K. Sacha Jagiellonian University in Krakow 16.06.2017 Zakopane 1 / 11

Space Crystal vs Time Crystal

Space Crystal

H =∑j

~p 2j

2mj+∑i<j

V (~ri −~rj)

invariant under translation of all particlesby the same vector in space and so arethe eigenstates.

Spontaneous symmetry breaking=⇒ Space crystal formation

ground state t = const

Time Crystal

The time-independent hamiltonianis also invariant under time transla-tions.

Analogy to the case of space crystalformation:

Spontaneous breaking of continuoustime translation symmetry=⇒ Time crystal formation

ground state ~r = const

.....A. Syrwid, J. Zakrzewski, K. Sacha Jagiellonian University in Krakow 16.06.2017 Zakopane 2 / 11

Discrete Time Crystals

Theoretical prediction:

K. Sacha, PRA 91, 033617 (2015).

V. Khemani et al., PRL 116, 250401 (2016).

D. V. Else et al., PRL 117, 090402 (2016).

Experiments:

J. Zhang et al., Nature 543, 217 (2017).

S. Choi et al., Nature 543, 221 (2017).


Mean-field approximation - SSB (CM localization)

Mean-field approximation =⇒ the Gross-Pitaevski equation (all particles occupy the same single-particle state) (

−1

2∂2x + g0(N − 1)|φ0|2

)φ0 = µφ0

GPE solution does depend only ong0(N − 1).

In thermodynamic limit we considerN →∞, g0 → 0g0(N − 1) = const.

The weaker interactions we have thelonger lifetime we expect.

Phase transition:

When g0(N − 1) < −π2

particles want to keep each other

GS: uniform solution =⇒non-uniformsolution

..

..

Fano factor: F =

⟨n2(θ)

⟩− 〈n(θ)〉2

n(θ)

R. Kanamoto, H. Saito, M. Ueda, Phys. Rev. A 73, 033611 (2016).


Wilczek’s idea of Time Crystal formation

N bosons on a 1D Aharonov-Bohm ring (α↔ magnetic flux, g0 < 0 and m = ~ = 1).

H =N∑j=1

(pj − α)2

2+

g0

2

∑i 6=j

δ(xi − xj ),

Switching to the CM coordinates (P - CM momentum, H ↔ relative d.o.f.)

H =(P − Nα)2

2N+ H(xi , pi ), P = Pj = 2πj , GS :

∂H

∂Pj= 2π

j

N− α ≈ 0.

SSB =⇒ CM localization

The CM probability current corresponds to (PN = 2πN)

XCM ↔∂H

∂PN= 2π − α 6= 0

In the limit N →∞ for all α exists j for which

∂H

∂Pj= 2π

j

N− α = 0

The eigenstate corresponding to PN = 2πN is not theground state!

CM would not move in ground state!!!

E

j

α ≠ 0

α = 0

j = N

GS




H =N∑j=1

(pj − α)2

2+

g0

2

∑i 6=j

δ(xi − xj ),


H =(P − Nα)2

2N+ H(xi , pi ), P = Pj = 2πj , GS :

∂H

∂Pj= 2π

j

N− α ≈ 0.



XCM ↔∂H

∂PN= 2π − α 6= 0


∂H

∂Pj= 2π

j

N− α = 0



E

j

α ≠ 0

α = 0

j = N

GS


Spontaneous breaking of continous t-translation symmetry

Idea of experiment:

α = 2π α = 0

SSB| |2ψ2πN

v = 2π

L = 1T = 1/2π

2|ψ |2πN

2|ψ |2πN

excited stateground state

TIME CRYSTAL BEHAVIOR OF EXCITED EIGENSTATE!


Numerical simulations: exact diagonalization + evolution

We consider time evolution of density-density correlation function asssuming that the firstparticle was measured in time moment t = 0 at the position x1 = 0.5 (P = PN , α = 0)

ρ2(x , t) ∝⟨ψ

PN0

∣∣ψ†(x , t)ψ(x , t)ψ†(x1, 0)ψ(x1, 0)∣∣ψPN

0

⟩, T =

1

2π, g0(N − 1) = −15.

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

pro

bab

ilit

y d

ensi

ty N=20 N=35 N=50

t/T = 0.4π

0 0.2 0.4 0.6 0.8

t/T=0.8π

0 0.2 0.4 0.6 0.8position

t/T = 1.2π

0 0.2 0.4 0.6 0.8

t/T = 1.6π

0 0.2 0.4 0.6 0.8 1

t/T = 2π

0 5 10 15 t/T0

0.2

0.4

0.6

0.8

con

tras

t

20 30 40 50number of particles

3

6

9

12

tc/T

(a)

(b)

contrast = max(ρ2)−min(ρ2)max(ρ2)+min(ρ2)

The time evolution reveals smearing of ρ2 in time.

The lifetime grows linearly with N.

Gaussian-like spreadσ(t) ∝

√σ4 + σ2t2

σ4N2→ t

N.


Numerical simulations: time evolution of CM distribution

Measurement of 20% fraction of particles =⇒ CM is well localized.

Time evolution of many-body CM distribution.

tD - time after the width of the CM distribution becomes significantly wider thanthe initial width (σCM

M−B ≈ σGPE/2)

20 25 30 35 40 45 50initial number of particles

0.1

0.2

0.3

0.4

0.5

0.6

tD/T

0 0.1 0.2 0.3 0.4 0.5fraction of particles measured

0.85

0.90

0.95

1.00

con

den

sate

fra

ctio

n

Using Central Limit Theorem we may esti-mate the evolution of CM distribution

σ(t) ∝

√(σ/√N)4 + (σ/

√N)2t2

(σ/√N)4N2

→ t√N

The many-body symmetry broken state evo-lution converges to the CLT predictions


Recapitulation

Spontaneous breaking of a continuous time translation symmetry to the discretesymmetry in the time crystal model introduced by Frank Wilczek

Spontaneous rotation of a non-uniform density can not be observed for largenumber of particles if the system is prepared in the ground state.

Excited eigenstate, although the initial single particle density is uniform and doesnot display any motion, measurement of the position of a single particle reveals arotation of the remaining particle cloud.

The spontaneous rotation can be observed in ultra-cold atomic gases!

AS, J. Zakrzewski, K. Sacha, Time crystal behavior of excited eigenstates,arXiv:1702.05006 (2017)


Time crystal behaviour - th- · Mean- eld approximation - SSB (CM localization) Mean- eld...

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