EXPLORING NATURE WITH A QUANTUM SIMULATOR

CollaboratorsHuo Mingxia,

Dimitris G. Angelakis, Elica Kyoseva

Kwek Leong Chuan,Center for Quantum Technologies, National

University of SingaporeInstitute of Advanced Studies, Nanyang

Technological University

Simons Conference on New Trends in Quantum Computation, 18 Nov 2010

12 May 2009

There is a story about two friends, who were classmates in high school, talking about their jobs.

One of them became a statistician and was working on population trends…..

He (the statistician) showed a reprint to his former classmate…

Disclaimer:This story is meant for educational purposes only. Any resemblance to real persons, living or dead, is

purely coincidental.

Also…it is not politically correct!!!

12 May 2009

Pictures stolen from various “dubious” sources : http://images.huffingtonpost.com/gen/48069/thumbs/r-OBAMA-AND-BUSH-large.jpg; http://powerltd.com/wp-content/uploads/2009/01/obama-bush.jpg; http://euroross.blogspot.com/Confused%20Bush.jpg

Let me show you a reprint on the

Gaussian distribution…

The Gaul distribution?

Asterix?

12 May 2009

So what are those symbols…..

12 May 2009

Oh…this is pi…

What is that?

The ratio of the circumference of the circle

to its diameter

Well, now you are pushing your joke too far…surely

the population has nothing to do with the

circumference of the circle

Pie???Is it edible?

19 April 2010 Talk Presented at KIAS

Pictures stolen from various “dubious” sources : http://images.huffingtonpost.com/gen/48069/thumbs/r-OBAMA-AND-BUSH-large.jpg; http://powerltd.com/wp-content/uploads/2009/01/obama-bush.jpg; http://euroross.blogspot.com/Confused%20Bush.jpg

Oh, I am trying to mimic physical

processes in Nature with a quantum

simulator!

What are you working on this days now?

What is that?

There are many processes in Nature that are

described by equations that cannot be solved by

ordinary computers…

So by building a quantum simulator experimentally, one

could in principle solve computationally difficult

physical processes through observation of laboratory processes that mimic the

physical processes….

19 April 2010 Talk Presented at KIAS

So what can you do with quantum

simulator? Can you do experiment with them like what I did

in Iraq?

Basic Idea behind simulation….of any kind

Physical System Simulating system

Common EquationsCommon Equations

“Nature isn't classical, dammit, and if you want to make a simulation of Nature, you'd

better make it quantum mechanical, and by golly it's a wonderful problem, because it

doesn't look so easy.”

The rule of simulation that I would like to have is that the number of computer elements required to simulate a large physical system is only to be proportional to the space-time volume of the physical system. I don’t

want to have an explosion!!

I would like to have an exact simulation, in other words, that the computer will do “exactly” the same as nature

R. P. Feynman

Universal Quantum Simulator

R. P. Feynman

Dimitris G. Angelakis et al, Phys. Rev. A 2007

See also New Scientist, 2007; Innovation 2010

M. J. Hartmann et al Nat. Phys. 2006; Andrew D. Greentree et al, Nat. Phys.2006

http://www.physorg.com/news199512137.html

Elmar Haller, et al. Nature, 2010

One dimensional electronic gas

Luttinger liquid theory

Collective excitations:Spin excitationsCharge excitations

Different velocities for spin and charge: spin-charge separation

We will simulate spin-charge separation in quantum optical systems

http://online.itp.ucsb.edu/online/exotic04/yacoby/oh/02.html

Recent efforts for measurement are inconclusive

PK EE

PK EE

The physics of Lieb-Liniger model (bosonic atoms):

zzzzUzz

mdzH zz2

1

U

Realized in 1D optical lattices (experiments by Bloch et al.)

,)}()(2

)](2

)()(2

1[{ 21

2,1

2

zzV

zU

zzm

dzHi

iiziz

where the density . )()()( zzz iii

Charge density: )()()( 21 zzzC

Spin density: )()()( 21 zzzS

U

V

Two-component Lieb-Liniger model:

PRL 95, 150402 (2005)

PRA 77, 013607 (2008)

Strongly correlated effects with photons and polaritons• DGA, M. Santos, S. Bose, “Photon blockade induced Mott transitions and XY spin models in

coupled cavity arrays”, (arXiv: June 06) Phys. Rev. A (Rap. Com.) vol. 76, 031805 (2007).• M. Hartmann, F. Brandao, M. Plenio, “Strongly interacting polaritons” (arXiv: June 06 ), Nat.

Phys. 2, 849 (06); • Greentree et al, arXiv:cond-mat/0609050 Quantum Phase transition of light, (arXiv: Sept 06,

Nat. Phys. 2006)

PRL 102, 203902 (2009)

Nature 424, 847 (2003)

PRL 94, 093902 (2005)

2

2

1ng

cg

tzE ,ˆ t

1

2

3

Experiments have shown that the pulse can be slowed down and eventually stored in atomic excitations.

dark state polariton:

Group velocity:

tzgntzEtz ,ˆ,ˆ,ˆ13

N

n

kkn aa

nD 10ˆ

!

1 dark states

Dark state polaritons in Electromagnetic Induced Transparency (EIT)

tzEtz ,ˆ~,

E

switch off adiabatically

atomic excitation

switch on and adiabatically

tzgntzEztz ,ˆ,ˆ)(,ˆ13

tztz ,ˆ~,ˆ13

tztz ,ˆ~,ˆ13

E a

b

c

a

b

c

Storing light in atoms through EIT

tztzgtzm

tzi zeff

t ,,~2,2

1, 22

g

zDeff

nm

01

4

p

gDg

1~2

vgD2

1 4 zg ng 22 effAg 1 tt

zzzzgzz

mdzH zz

eff

~2

1

0 p

E

a

b

c

d

E

2

Eng z

ˆ2

1.Defining slowly varying operators2.Adiabatically eliminating fast-rotating terms 3.Solving Maxwell-Bloch equation

E

a

b

c

usual situation in EIT

converting the excitations into pure spin-wave form

pulse propagates undistorted until exiting the fiber

pulse becomes trapped and evolves under nonlinear eq.

Nature Physics 4, 884 (2008)

0 p

E

a

b

c

d

E

d

E

1

2

3

2

ˆˆˆ ',' ]ˆ,ˆ[ kkkk

Bosonic operator

Generalize to the case of and which propagate towards the left and right directions.The counter-propagating control fields will form the standing wave and trap the quantum fields.

E

zzzzgzz

mdzH zz

eff

~2

1

Tonks-Girardeau regimepheff ngm ~

Nature Physics 4, 884 (2008)

D1

v

g 2

D1

4

znz0OD

2

1

2

1

2E

1E

a4

1E 1

1

2

3

4

2E

Type a atom

1E 2

1

2

3

4

2E

Type b atom

a2

b2

b4

: the couplings of , to transitions for a and b atoms, respectively

bag ,2,1 ,1E ,2E

and : detunings of

from transitions

and

)(2

ba )2(1E

)()(21

baba

)(4

ba

)()(43

baba

arXiv:1006.1644

Our model

,ba HHH

.},.])()([

]ˆˆ)[(2

)()({

23)(

,)(

,

)(,

)(,4321

443342223333

2

1

cHetet

eEeEg

dznH

xtzkii

tzkii

tzkii

tzkii

xxxi

xiqu

xxxiqu

xxx

i

xz

x

icl

icl

icl

icl

iqu

iqu

iqu

iqu

The Hamiltonian of the system is:

bax , ,qppq 4,3,2,1, qp

iquk i

qu

iclk i

cl

iE

i

quantum field

classical field

,)}()()](2

)()(2

1[{ 2112

2,1

2

zzVzU

zzm

dzHi

ii

izizi

224

21

2

21

21

22

2

214

22

1

22

22

21

1

12

)(4

)2(1

)2(12)()2(1

)2(1

)2(12)()2(1

)2(1)2(1)(2

)2(1

)(

)()(

)(

)()(

)(2

)(22

1

ba

bbg

ab

aag

ba

gba

zba

gba

g

gg

g

ggV

gU

ngm

where )()()( zzz iii and

][ )(2)()2(1

2)2(1

)2(1)2(1 baz

bag ng Group velocity:

a4

1E 1

1

2

3

4

2E

Type a atom

1E 2

1

2

3

4

2E

Type b atom

a2

b2

b4

)8cos(])()([2 2

2022

sszs

sszsss dz

V

K

uKu

dzH

Charge: ])()([2

22cz

c

cczccc K

uKu

dzH

Spin:

,)]()][(exp[)( 2/1zziz ),(1

)( 0 zz z

0|)(1

| zz

.2

,2

21,

21,

scsc

UV

KK

U

Vuu scsc

1

,1 ,,wheremU

Km

Uu 0

20 ,

and

PRA 77, 013607 (2008)

Here,

and

and are velocities of charge and spin, respectively.The difference between two velocities results in the spin charge separation.

cu su

2

2,ph

1

1,ph21 ,

m

n

m

nUUU

UV

KK

U

Vuu scsc

12

,12

,

1

,1

and

K

Uu ,

Conditions

c

s

s

c

K

K

u

u

Fourier transformation of density-density correlation function

Parameters: 5.0,1 sc uu

10,4.0,3000OD:50 ph N

with2,ph

2

1,ph

1

n

Um

n

Um

Two peaks charge velocity and spin velocity

),(Re qS )2,(Re S

q

)2,(Im S

25.2 )2,(Re S

For smaller

The spin-charge separation is simulated in one dimensional hollow-core fiber system, where the light-matter excitations, the dark-state polaritons are shown to follow the Lieb-Liniger dynamics of a quantum liquid.

Conclusion

Thank you!

Festchriff in honor of Vladimir Korepin25-26 May 2011

in conjunction with 5th International Workshop on

Solid State Quantum Computing and

a

b

c

d

a

b

c

d

a

b

c

d

a

b

c

d

favoured