and Three-body Interactions

quantum.ustc.edu.cn

and Three-body Interactions

Using spins to simulate

quantum mechanical systems:

a simple system with

competing two- and threebody

interactions

Xinhua Peng

University of Science and Technology of China

July 19, 2009


LOGO

Table of Contents

1. Introduction

2. System

3. Experiments

4. Conclusion

Your site here


LOGO

Introduction

Simulation of quantum systems

Classical computers

• Exponential growth of Hilbert space

n

|



2

2

...


i 1

1

i

n

1

c

computational

basis

i1 ... i

| i1...

i

n n


System with 50

qubits

2 50 ≈10 15 complex

amplitudes ~ 32 x 10 15

bytes of information

well beyond the capacity of existing computers

Is it possible to classically simulate faithfully a quantum system

Naïve answer: NO

Your site here


LOGO

Simulation of quantum systems

Quantum computers – Universal

quantum simulators

1982 Richard P. Feynmann

R.P. Feynman, “Simulating Physics

with Computers”, Int. J. Theor. Phys.

21, 467-488, 1982

Can we do it with a new kind of

computer – a quantum computer Now

it turns out, as far as I can tell, that you

can simulate this with a quantum

system, with quantum computer

elements. […] I therefore believe it’s true

that with a suitable class of quantum

machines you can imitate any quantum

system, including the physical world.

Your site here


LOGO

History – Quantum simulator

Feynman 1982: Quantum Computer would efficiently

simulate dynamics of quantum systems.

Lloyd 1996: Formalized conjecture, assumed tensor

product structure, showed efficient algorithm.

Recent development

Theoretical

• Physics: Energy spectrum; Many-body systems; Pairing

Models in BEC … [PRL, 79, 2586 (1997); PRL, 83, 5162 (1999);

Proc. R. Soc. Lond. A 454, 313 (1998); PRL, 89, 057904 (2002)]

• Quantum chemistry: Molecular energies; Dynamics of

chemical reactions (EPL, 80, 67008 (2007); Science, 309, 1704

(2005); PNAS 105, 18681 (2008) ]

Experimental

• NMR: Quantum harmonic and anharmonic oscillators;

Three-body interaction; Fano-Anderson model [PRL, 82,

5381-5384 (1999) ; PRA, 61, 012302 (1999) ;PRA, 71, 032344

(2005) ; PRA, 71, 012307 (2005); PRL, 100, 100501 (2008)]

• Ion trap: Quantum magnet [Nature phys, 4, 757 (2008)]

• Optics: Anyonic fractional statistics [PRL, 102, 030502 (2009)]

Your site here


LOGO

Quantum computation

Main Applications

Quantum

algorithms

Hard classical

problems

Quantum

simulations

A medium-scale

quantum simulation

with 30 to 100 qubits

can exceed the

limitations of

classical computing!

Quantum

many-body

problems

Quantum phase

transitions

Quantum

chaos

……

Quantum

chemistry

Macromolec

ular systems

Molecular

energies

Quantum

compounds

and reactions

……

Material

science

Micromecha

nism

High

temperature

supercondu

ctivity

……

……

Your site here


LOGO

System

Spin chain with competing many-body

interaction

Two-body

interactions

Three-body

interactions

Your site here


LOGO

Quantum phase transitions

Quantum phase transition (QPT) describes a qualitative

change in the ground state of a many-body system due

to its quantum fluctuations as a function of a control

parameter in its Hamiltonian.

• The nonanalyticity in the ground state could be either

the limiting case of an avoided level–crossing or an

actual level-crossing.

• QPTs occur as a result of competing ground state

phases.

• Classification of QPT:

• First-order QPT

Simple level crossing, also occur in a finite-size

system

• Continuous (or second-order) QPT

an avoided level crossing, in the thermodynamic

limit

Your site here


LOGO

Two limiting cases:

Two-spin Ising quantum model (First-order)

Ground state

J 0 2

z

0

J 2

0


… z

0

+



z

0

Paramagnet

J

/ 2

2 z

Antiferromagnet

Three-spin Ising quantum model (Second-order)

Ground state

J … 0 … 0 Paramagnet

3

0

x

0

J 3

0



x

x


J

2


+ +

x

Ferromagnet


+

Your site here


LOGO

A simple case with 3 spins

Magnetic field

B z

Two-body interaction

Three-body interaction

B x

Ground state

Qualitative change of QM ground state

Your site here


LOGO

2D phase diagram

x

0

IA

IB

IC

Your site here


LOGO

Experimental verification

Main challenges:

to identify experimentally accessible systems,

where the three-body interaction Hamiltonian

is present;

to experimentally control the variation of the

system Hamiltonian in a sufficiently precise

manner;

to characterize the resulting ground state.

Your site here


LOGO

Experiments

Nuclear Magnetic Resonance

B 0

Nuclear spins

I = 1/2

RF pulses

Sample: Diethyl-fluoromalonate

Your site here


LOGO

Experimental procedure

Step 1 Step 2 Step 3


Initial state

preparation

Pseudo-pure

state

pp

^

c1

d

g


g

Hamiltonian

Simulation and

adiabatic

passage

Average

Hamiltonian

Theory

Optimized

discrete scan

Measurement

Quantum state

tomography

Detection of

entanglement

Your site here


LOGO

Quantum simulation

General scheme

U


Simulated (S) Physical

(P) s p

e

iH I T

sT


1


pT


Pulse sequence for Hamiltonian simulation

V

T

e

i

H

p

T



k

e

iH

p

t

k

V

k

iH

iH

x

z

e e e e

/2 iH

iH

/2

x





3

O , H

i

, H H H

x x x z x

Your site here


LOGO

Case A: Two-spin Ising model

First-order quantum phase transition

x

0

x

0

Level crossing

Lifting degeneracy

+ +

Your site here


LOGO

Case B: Three-spin Ising model

Second-order QPT in the thermodynamic limit

An avoided level

crossing

In the finite-size system, the sharp

quantum phase transition is

smoothed into a graduate change

in the properties of the ground

state from paramagnetic to

ferromagnetic order.

J 3

Your site here


LOGO

Optimized discrete scans

Experiments: M = 8

Your site here


LOGO

Experimental procedure

Step 1 Step 2 Step 3


Initial state

preparation

Pseudo-pure

state

pp

^

c1

d

g


g

Hamiltonian

Simulation and

adiabatic

passage

Average

Hamiltonian

Theory

Optimized

discrete scan

Measurement

(Detecting Critical

points)

Standard twospin

correlation

Quantum state

tomography

Detection of

entanglement

Your site here


LOGO

Detection of Critical points

Standard two-spin correlation


i j

C

xx i


j x

x

/ 3

Critical behavior

Nothing

New critical phenomena induced by three-body interaction

cannot be detected by the standard two-spin correlation.

Your site here


LOGO

Detection of Critical points

Entanglement witnesses (Detecting entangled

states)

Three qubits

Three-qubit entanglement

All GHZ-type sates

Rules out full separability

Your site here


LOGO

Experimental results

Case A: Two-spin Ising model

x

0

Phase I

Phase II

Your site here


LOGO

Experimental results

Case B: Three-spin Ising model

Phase I

Phase III

IA

IB

IC

Your site here


LOGO

Experimental results

Quantum state tomography

• Attenuated correlation

(with decoherence)

• Fidelity (without decoherence)

Your site here


LOGO

Conclusion

Our work provides a first experimental evidence to the

interesting properties of the ground state (entangling

phase transitions) by adiabatically simulating a threespin

system involving three-body interaction.

A novel type of quantum critical behavior induced by

three-body interactions, which cannot be observed via

the standard two-spin correlations but can be made

visible by quantifying different types of entanglement,

or by using suitable entanglement witnesses.

The evolution in the experiment is coherent and

quantum mechanical. The generation of the

entanglement confirms that the transitions are not

driven by thermal fluctuations but are of quantummechanical

origin (analogous to quantum fluctuations

driving quantum phase transitions).

Experiments on larger quantum systems ……

Your site here


LOGO

Thanks to

Prof. Dieter Suter

Prof. Jiangfeng Du

Dr. Jingfu Zhang

Prof. Y. J. Deng

Prof. D. Bruss

Dr. H. Kampermann

Thanks for your attention!

E-mail: xhpeng@ustc.edu.cn

Hefei National Laboratory for Physical Science at

Micro-scale &

Department of Modern Physics,

University of Science and Technology of China,

Hefei, Anhui, P.R.China

Your site here

More magazines by this user
Similar magazines