Localization of phonons in chains Localization of phonons in chains of trapped ionsof trapped ions

Alejandro Bermúdez, Miguel Ángel Martín-Delgado Alejandro Bermúdez, Miguel Ángel Martín-Delgado and Diego Porrasand Diego Porras

Department of Theoretical Physics Department of Theoretical Physics Universidad Complutense de MadridUniversidad Complutense de Madrid

• Quantum effective spin models

Introduction Quantum many body physics with trapped ions

• Interacting phonons: Phonon - Hubbard model

• D. Porras, J.I. CiracD. Porras, J.I. Cirac, Effective quantum spin systems with trapped ions, Phys. Rev. , Effective quantum spin systems with trapped ions, Phys. Rev. Lett. (2004)Lett. (2004)

• A. Friedenauer, H. Schmitz, D. Porras, T. Schätz, A. Friedenauer, H. Schmitz, D. Porras, T. Schätz, Simulating a quantum magnet with Simulating a quantum magnet with trapped ionstrapped ions, , Nature Physics (2008)Nature Physics (2008)..

• D. Porras, J.I. CiracD. Porras, J.I. Cirac, BEC and strong correlation behavior of phonons in ion traps. , BEC and strong correlation behavior of phonons in ion traps. Phys. Rev. Lett. (2004)Phys. Rev. Lett. (2004)

Anderson Localization

• A single particle moves in a disordered potential

1jj j

H U j j t j j 1U2U

3U

4U

• Observed in electronic transport, light/sound waves. Recently in the field of ultracold bosons (recent experiments at Inguscio‘s group and A. Aspect‘s group)

• Anderson (1958) A particle initially localized at a given site may be localized by quantum effects (Anderson localization)

log( )jn0 L| |/j j

jn e

0j

log( )jn

0j

• Vibrations around the equilibrium positions → axial or transverse to the chain

Vibrational modes of an ion chain

2 2, 2 2 2

mot , , ,0 0 3x,z ,

1( )

2 2 | |j

j j lj j j l j l

p eH m q c q q

m z z

jzjx

2 30

2

/e d

m

• Ratio: Coulomb coupling / trapping energy

• determines the phonon properties

1

1 Soft modes (gapless) → ions strongly coupled

Stiff modes (gaped) → vibrations almost independent

Vibrational modes of an ion chain

jx

• Radial vibrations can be controlled by tightening the radial trapping potential...

2

3 20 x

1x

e

d m

x,

x

n

(mode number)n

x x

→ stiff (gaped) modes

... can be controlled by tightening the radial trapping potential, such that

x x, x, x,n n nn

H a a

fast rotating terms

2† †

Coul ,0 0 3, ,

( )( )| | i j i j i i j j

i j i ji j

eV x x t a a a a

z z

† †

i j i ja a a a

resonant terms - tunneling

4 4 † 4 † 2 2anh 0 ( ) ( ) ( )j j j j j

j j j

V Vx V x a a U a a

2i te

,, phonon conservationi jU t

effective Hubbard interaction

Tight-binding Hamiltonian for phonons

† †i j i ja a a a

• In the phonon-number conserving limit, we get a Bose-Hubbard model• This limit is realized by the radial vibrations of a chain of trapped ions

† † † † 2,

,

( ) ( )j j i j i j i j j jj i j j

H a a t a a a a U a a

Numerical calculations Density Matrix Renormalization Group)

Mott phase (U >> t)

Phonon superfluid (U << t)

The phonon Bose-Hubbard model

X.-L. Deng, D. Porras, J.I. Cirac, X.-L. Deng, D. Porras, J.I. Cirac, Phys. Rev. A (2008)Phys. Rev. A (2008)

Two paths towards disorder

(a) Compositional disorder

1jj j

H U j j t j j

• Large samples + self-averaging

describe the potential as a stochastic variable

•B. Paredes, F. Verstraete, J.I. Cirac, Exploiting quantum paralelism to simulate quantum B. Paredes, F. Verstraete, J.I. Cirac, Exploiting quantum paralelism to simulate quantum random many-body systems, random many-body systems, Phys. Rev. Lett. (2004)Phys. Rev. Lett. (2004)

jU W

W

( ) 1/ 2

( ) 1/ 2

p U W

p U W

(b) Disorder induced by coupling to a system of spins

jU U

U

• Consider the case in which (random binary alloy)jU W

1z

j j

H U j j t j j • Potential is a true stochastic variable !!

Statistics quantum statistics of spin

Localization of phonons - introduction

• Phonons in a chain of trapped ions may be described by a tight-binding model

• We will show that by using lasers, the local trapping energy depends on the internal state (effective spin) of the ions

† † †,

,

( )j j j i j i j i jj i j

H a a t a a a a

jx

zj jU

• Thus, the local trapping energy becomes a stochastical variable with the same statistical properties of internal (effective spin) operators

•A. Bermúdez, M.A. Martín-A. Bermúdez, M.A. Martín-Delgado, D. Porras, arXiv:1002.3748Delgado, D. Porras, arXiv:1002.3748

Inducing a disordered potential for phonons

L L L L( ) ( )†L 2

j ji k x t i k x t

j jj

H e e

• Start with a laser that shines the chain in the radial direction.

†LL H.c

2i t

j jj

H a e

• , Lamb Dicke limit

† z †eff ,

,j l j l j j j

j l j

H t a a U a a

EUROPHYSICS LETTERS (2004)

Quantized AC-Stark shifts and their use for multiparticle entanglement and quantum gates

F. Schmidt-Kaler, H. H¨affner, S. Gulde, M. Riebe, G. Lancaster, J. Eschner, C. Becher and R. Blatt

L x

Inducing a disordered potential for phonons

† z †eff ,

,j l j l j j j

j l j

H t a a U a a

• The local trapping potential depends on the effective spin of the ions

†ˆj j j

j

U a a

• Assume the following separable spin state

S 1 2

1... ,

2N j j j

• Local potentials are uncorrelated and show the following mean-value/variance

z

2 z z 2,

ˆ 0

ˆ ˆ

j j

j l j l j l

U U

U U U U

1z

j random binary alloy model

Inducing a disordered potential for phonons

• General case: evolution of a phonon state

ph S S( ) (0)i H t i H tt e e

1 2

2ph S ph S S , ,... ph( ) Tr (0) | | (0)s sj j

j

i H t i H ti H t i H t

s ss

t e e c e e

• Time evolution of the reduced phonon density matrix is a statistical mixture

ph (0)

, , , ...ph ( )t

, , , ...ph ( )t

, , , ...p

, , , ...ph ( )t

, , , ...p

, , , ...p

ph ( )t

1 2S , ,..., 1

1

,...N

j

s s s Ns

c s s

† †

,,

jj j j i j i js

j i j

H U s a a t a a

Observation of phonon Anderson localization

• Simplest case separable spin state, uncorrelated disorder

• Numerical result for the diffusion of a phonon initially localized at the center of chain with N = 50 ions.

L

50

0.5

10

N

U t

• Difficult experiment cool ions to the ground state, create a single phonon cool ions to the ground state, create a single phonon at one ion, measure the vibrational stateat one ion, measure the vibrational state

Typical values

10

100

time e xp. 100 / 1

U kHz

t kHz

t ms

Phonon localization: Outlook

• Including anarmonicities (standing-wave) study Anderson localization with interactions, bose glass models

† † † † 2,

,

( ) ( )j j i j i j i j j j jj i j j

H a a t a a a a U a a

• By controlling the spin internal state realizations of 1D systems with correlated disorder (random dimer modelsrandom dimer models)

S odd 1 1

1,

2j j j j j j j

products of Bell-pairs - may be created with quantum gates

Perfect correlation between Perfect correlation between 1,j jU U

Thanks for your attention

Phonon in trapped ionsPhonon in trapped ions

How to create a phonon Bose-Einstein Condensate

1. Choose the parameters of the effective Hamiltonian in such a way that you can prepare the ground state easily. (Mott insulator)

2. Change the parameters slowly (be careful with quantum phase transitions)

3. Measure the new ground state (again with the help of an internal state)

4. allows to measure: phonon-number averages and fluctuations

n=1n=0

t U

† †0 1 ... 0Na a

We prepare a Mott insulator Phase

†0

10

N

jj

aN

Superfluid

cooled ion (0 phonons)

Fock state (1 phonon)

t UAdiabatic evolution

Summary

i H tU e

input output

( )T(0)

• Effective spins

• Stiff and soft phonons

• Schemes to prepare and measure quantum states of spins/phonons

• Spin-phonon couplings

The trapped ion toolbox

Trapped Ion Quantum Simulator of many-body physics ???

Some work for theorists to be done…

Remark: How big must a quantum simulator be?

Quantum magnetism in trapped Quantum magnetism in trapped ionsions

zz

zz

ij

ij

• Enough spins to detect bulk properties: critical exponents can be obtained with 20-30 sites.

• Recall that numerical methods exists in 1D to calculate very efficiently ground states (Density Matrix Renormalization Group).

critical phase

Critical exponent is a bulk property

• “Intractable problems“ non-equilibrium properties, decoherence...

• A more detailed analysis allows us to understand the limitations of our quantum spin simulator.

Spin-spin interactions: Scaling of errors

Quantum magnetism in trapped Quantum magnetism in trapped ionsions

• Consider the case of coupling by transverse (stiff) modes

, 30 0i j

i j

JJ

z z

2xJ

2Error (2 1)n x

ErrorJ =

2 1n

Due to this scaling:

• The smallest the error, the slowest the simulation

• Ground state cooling is not necessary (one pays the price of smaller interactions)

• Typical values:

• Same is true for any scheme that relies on adiabatic elimination of phonons (walking wave, couplings by magnetic field gradients ...)

-2x 20MHz, J = 10kHz, Error = 10