Download Introduction: effective spin

Localization of phonons in chains
of trapped ions
Alejandro Bermúdez, Miguel Ángel Martín-Delgado
and Diego Porras
Department of Theoretical Physics
Universidad Complutense de Madrid
Introduction
Quantum many body physics with trapped ions
• Quantum effective spin models
• D. Porras, J.I. Cirac, Effective quantum spin systems with trapped ions, Phys. Rev.
Lett. (2004)
•
A. Friedenauer, H. Schmitz, D. Porras, T. Schätz, Simulating a quantum magnet with
trapped ions, Nature Physics (2008).
• Interacting phonons: Phonon - Hubbard model
• D. Porras, J.I. Cirac, BEC and strong correlation behavior of phonons in ion traps.
Phys. Rev. Lett. (2004)
Anderson Localization
• A single particle moves in a disordered potential
U3
U1
U2
U4
H  U j j
j
j  t j
j 1
j
• Anderson (1958)  A particle initially localized at a given site may be localized
by quantum effects (Anderson localization)
log( n j )
n j  e| j  j0 |/ L
log( n j )
j0
• 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)
j0
Vibrational modes of an ion chain
• Vibrations around the equilibrium positions → axial or transverse to the chain
zj
H mot
xj
2
 p2 , j

1
e
2 2
2
  
  m q , j  c  0 0 3 (q , j  q ,l ) 


  x,z  j 2 m
j 2
j ,l | z j  zl |

e2 / d03
 

2
m
  1
  1
• Ratio: Coulomb coupling / trapping energy
• determines the phonon properties
Stiff modes (gaped) → vibrations almost
independent
Soft modes (gapless) → ions strongly coupled
Vibrational modes of an ion chain
• Radial vibrations can be controlled by tightening the radial trapping potential...
xj
... can be controlled by tightening the radial trapping potential, such that
e2
x  3 2
d0 mx
1
→ stiff (gaped) modes
H x   x,n ax, n ax,n
n
 xx
 x,n
x
n (mode number)
Tight-binding Hamiltonian for phonons

U , ti , j  phonon conservation
e
VCoul  
i, j
2
e
x x   ti , j (ai  ai† )(a j  a †j )
0
0 3 i j
| zi  z j |
i, j
Vanh  V x 4j  V  x04 (a j  a †j )4  U  (a †j )2 (a j ) 2
j
j
j
2i t fast rotating
terms
ai a j  ai†a†j
ai†a j  ai a†j
resonant terms tunneling
effective Hubbard
interaction
The phonon Bose-Hubbard model
• 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
H    a †j a j   ti , j (ai†a j  ai a †j )  U  (a †j a j ) 2
j
i, j
j
Phonon superfluid
(U << t)
Numerical calculations 
Density Matrix
Renormalization Group)
Mott phase (U >> t)
X.-L. Deng, D. Porras, J.I. Cirac, Phys. Rev. A (2008)
Two paths towards disorder
• Consider the case in which U j  W (random binary alloy)
(a) Compositional disorder
Uj
W
W
• Large samples + self-averaging
 describe the potential as a
stochastic variable
H  U j j
j
j  t j
j 1
j
p (U  W )  1/ 2
p (U  W )  1/ 2
(b) Disorder induced by coupling to a system of spins
Uj
U
U
H  U  z j
j
j  t j
j 1
j
• Potential is a true stochastic variable !!
Statistics  quantum statistics of spin
•B. Paredes, F. Verstraete, J.I. Cirac, Exploiting quantum paralelism to simulate quantum
random many-body systems, Phys. Rev. Lett. (2004)
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
xj





H    j a †j a j   ti , j (ai† a j  ai a †j )
j
i, j
 j  U zj
•A. Bermúdez, M.A. MartínDelgado, D. Porras, arXiv:1002.3748
• Thus, the local trapping energy becomes a stochastical variable with the same
statistical properties of internal (effective spin) operators
Inducing a disordered potential for phonons
• Start with a laser that shines the chain in the radial direction.
HL 
•

 i ( kL x j L t )
† i ( kL x j L t )

e


e
 j
j
2 j
L  x   , Lamb Dicke limit
HL 
 L
†
 i t

a
e
 H.c 


j j
2 j
H eff   t j ,l a †j al  U   zj a †j a j
j ,l
j


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
Inducing a disordered potential for phonons
• The local trapping potential depends on the effective spin of the ions
H eff   t j ,l a †j al  U  zj a †j a j
j ,l
j
Uˆ a a
j
†
j
j
j
• Assume the following separable spin state

S
 
1
 2 ... 
,
N

j


1
 
j
2
j

• Local potentials are uncorrelated and show the following mean-value/variance
Uˆ j  U  zj  0
Uˆ jUˆ l  U 2  zj  lz  U 2 j ,l
 zj  1
random binary alloy
model
Inducing a disordered potential for phonons
• General case: evolution of a phonon state
 (t )  ei H t ph (0) S S ei H t
S 
s j 1
ph (t )  TrS ei H t ph (0) S S ei H t    | cs ,s ,... |2 e
 
1

i H
cs1 , s2 ,...,sN s1 ,...sN
s j  t
2
ph (0)e
i H
s j t
sj
Hs   U  s j a †j a j   ti , j ai† a j
j
j
i, j
• Time evolution of the reduced phonon density matrix is a statistical mixture
p,,, ...
 ph (0)
p,,, ...
p,,, ...
ph,,, ... (t )
ph,,, ... (t )
ph,,, ... (t )
 
ph
(t )
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.
N  50
U  0.5t
 L  10
Typical values
U  10 kHz
t 100 kHz
time e xp. 100 / t 1 ms
• Difficult experiment  cool ions to the ground state, create a single phonon
at one ion, measure the vibrational state
Phonon localization: Outlook
• Including anarmonicities (standing-wave)  study Anderson localization with
interactions, bose glass models
H    a †j a j   ti , j (ai†a j  ai a †j )  U j (a †j a j ) 2
j
i, j
j
• By controlling the spin internal state  realizations of 1D systems with
correlated disorder (random dimer models)
 S   j  odd  j ,  j 

1

2
j

j 1

j

j 1

products of Bell-pairs - may be created
with quantum gates
Perfect correlation between
U j , U j 1
Thanks for your attention
Phonon 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)
Fock state
(1 phonon)
cooled ion
(0 phonons)
n=0
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=1
0  a1† ... aN† 0
We prepare a Mott
insulator Phase
Adiabatic
evolution
t
U
t
U
 1
0  
 N
N
†
a
j j  0

Superfluid
The trapped ion toolbox
Summary
input
Trapped Ion
Quantum Simulator
of many-body
physics ???
(0)
output
U  ei H t
(T )
• Effective spins
• Stiff and soft phonons
• Schemes to prepare and measure
quantum states of spins/phonons
Some work for theorists to
be done…
• Spin-phonon couplings
Quantum magnetism in trapped ions
iz zj  iz  zj
Remark: How big must a quantum simulator be?
critical phase
Critical
exponent is a
bulk property
• 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).
• “Intractable problems“  non-equilibrium properties, decoherence...
Quantum magnetism in trapped ions
Spin-spin interactions: Scaling of errors
• A more detailed analysis allows us to understand the limitations of our quantum spin
simulator.
• Consider the case of coupling by transverse (stiff) modes
J i, j 
J    x
2
J
z z
0
i
0 3
j
Error
J=
x
2n  1
Error   2 (2n  1)
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:
x  20 MHz, J = 10 kHz, Error = 10-2
• Same is true for any scheme that relies on adiabatic elimination of phonons
(walking wave, couplings by magnetic field gradients ...)