Dynamics of phase transitions in ion traps

A. Retzker, A. Del Campo, M. Plenio, G. Morigi and G. De Chiara

Quantum Engineering of States and Devices: Theory

and Experiments

Contents

Nucleation of defects in phase transition(the Kibble-Zurek mechanism)

Trapped ions: structural phases &kinks

Inhomogeneous vs. Homogeneous setup

Conclusion & Outlook

1) The

The Kibble Zurek mechanism

• Cosmology : symmetry breaking during

expansion and cooling of the early universe

• Condensed matter:

• Vortices in Helium

• Liquid crystals

• Superconductors

• Superfluids

T. W. B. Kibble, JPA 9, 1387 (1976); Phys. Rep. 67, 183 (1980)W. H. Zurek, Nature (London) 317, 505 (1985); Acta Phys. Pol. B. 1301 (1993)

Cosmology in the lab

Similar free-energy landscape near a critical point

Second order phase transition

Landau theory: Free energy landscape, changes across a 2PT from single to double well potential, parametrized by a relative temperature

Second order phase transition

Universal behaviour of the order parameters: divergence of correlation/healing length dynamical

relaxation time

€

τ =τ 0

εμ

€

ξ =ξ0

ευ

€

ε =TC − T

TC

KZM in the Ginzburg-Landau potential

What is the size of the domain? The density of defects?

Defect = boundary between domains

Laguna & Zurek, PRL 78, 2519(1997)

Real field in the GL potential

Langevin dynamics

Linear quench

adiabatic adiabatic

Non-adiabatic

freezing

Freeze-out time

The Kibble Zurek mechanism

€

τ =τ 0

εμ

€

ε =t

τQ

adiabatic adiabatic

Non-adiabatic

freezing

Linear quench

The Kibble Zurek mechanism

€

τ =τ 0

εμ

€

ε =t

τQ

€

ξ =ξ(t) =ξ0

ε ˆ t ( )υ

€

ξ =ξ0

τ Q

η

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 4

The average size of a domain is given by the correlation length at the freeze out time

€

ξ ∝ ξ0

τ Q

η

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 4

2) A test-bed: ion chains

arXiv:1002.2524

Cold ion crystals

Boulder, USA: Hg+ (mercury)

Aarhus, Denmark: 40Ca+ (red) and 24Mg+ (blue)

Innsbruck, Austria: 40Ca+

Oxford, England: 40Ca+

Birkl et al. Nature 357, 310(1992)

Structural phases in trapped ions

N ions on a ring with harmonic transverse confinement

Fishman et al. PRB 77, 064111 (2008)Morigi et al, PRL 93, 170602 (2004); Phys. Rev. E 70, 066141 (2004).

Structural phases in ion traps

€

υ t(c )2 = 4

Q2

ma(0)3

Zig

Zag

N ions on a ring with a harmonic transverse confinement

€

υ t(c ) = 3Nν /4 logN

Zig Zag movie

Wolfgang Lange, Sussex

Zig Zag movie

Wolfgang Lange, Sussex

20 0,

21 sinh / coshx x x

2 23tanh 1 3 tanhikxk x e x k ik x 2 21

2 ,2k k

21

3,

2

20 1/ coshx x

Quantum kink properties

Normal modes of the double well model

Spectrum structure

0.1 {

Dispersion relation for the ion trap and profile of the two most localized states.

High localized mode

Low localized mode

Landa etal, Phys. Rev. Lett. 104, 043004 (2010)

Kinks

T. Schaetz MPQ

Left kink

Right kink

Kinks

T. Schaetz MPQ

Two kinks

Extended kink

KZM in the Ginzburg-Landau potentialChain of trapped ions on a ring: transverse normal modes, Re/Im

parts of

€

Ψkσ =

1

Ne iknσ n

n

∑

€

σn = yn,zn

€

k ∈ −π /a,π /a[ ]

Effective Ginzburg-Landau potential for a small amplitude of transverse osc .

€

V = V (2) +V (4 )

Continuous description of the field for the transverse modes

€

−1( )nσ n →ψ σ (x)

€

xn → x

Long wavelength expansion wrt the zig-zag mode

Zig Zag

Control parameter

Linear

€

δ =υt2 −υ t

(c )2

€

δ > 0

€

δ < 0

Thermodynamic limitEq. of motion of the slow modes in the presence of Doppler

cooling

Get the input of the Kibble-Zurek theory

Estimate the correlation length and relaxation time

€

τ 0 ∝ η /δ

€

τu ∝ 1/ δ

€

ξ ∝ h /δ1/ 2

Relaxation time: Correlation length:

anti-kink kink

Dynamics through the structural phase PT

Consider a linear in time quench in the square of the transverse frequency

Coupled Langevin equations:

€

m˙ ̇ r i +∂riV ri{ }, t( ) + mη ˙ r i +ε (t) = 0

After the transition:

Dynamics through the structural phase PT

0.256

On the ring

Homogeneous system:

Inhomogeneous KZMAxial and transverse harmonic potential (instead of a ring

trap):

€

vF

€

vF

€

vF

€

vF

Inhomogeneous ground state Dubin, Phys. Rev. D, 55 4017 (1997)

Thomas-Fermi

Inhomogeneous KZM

Inhomogeneous transition!

€

υ t(c )2 = 4

Q2

ma(0)3

€

υ t(c )2 = 4

Q2

ma(x)3

Spatially dependant critical frequency (within LDA)

€

n(x) =3

4

N

L1−

x 2

L2

⎛

⎝ ⎜

⎞

⎠ ⎟

?

On the trap KZM scaling

Density of defects as a function of the rate

50 ions

Kink interaction

0.995

On the trap KZM scaling

Density of defects as a function of the rate

Kink interaction

On the trap: dynamics

Slow quench: adiabatic limit

Fast quench: nucleation of kinks

Inhomogeneous KZM

€

δ(x, t) = ν t (t)2 −ν tc (x)2 = ν t (0)2 −ν t

c (x)2 −δ 0

t

τ Q

Inhomogeneous density, spatially dependent critical frequency .

Linear quench:

Adiabatic dynamics is possible even in the thermodynamic limit when

In contrast with the homogeneous KZM

Solitons: Zurek, PRL 102 (2009)

€

vF < vx

Causality restricts the effective size of the chain

Front satisfying: Versus sound velocity:

€

vx = ξx /τ x

€

δ(xF , tF ) = 0

Inhomogeneous KZM

Defects appear if :

€

vF > vx

Effective size of the chain for KZM :

€

2 X*

€

X* = (vF − vx )t

0.995

Theory vs. “experiment”

Nice agreement

Model

Soft mode

The soft mode renormalizes the

double wells

Fishman et al, PRB 77, 064111 (2008)

Adiabatic limit

Model

Soft mode

The soft mode stops the double

wells

Simulations

model

€

δ

€

d

Outlook

3D case

Kinks dynamics

With optical lattice: transverse Kontorova model

Full counting statistics

Other systems

Thanks

PhD position available