Quantum Optics with Surface Plasmons

5. 18 at CYCU

國家理論科學中心（南區） 成大物理系

陳光胤

Outline

• Introduction• SE of excitons into surface plasmons• Coherent single surface plasmon

transport• Experiment proposal• Summary• Outlooks

I.Introduction

What is surface plasmon?Classical :

Quantum :Surface plasmon modes on the surface of metals

Nature 424, 824 (2003)

Lycurgus Cup and Stained Glass

normal (reflected light) held up to the lightTrustees of The British Museum

SP excitation

Gothic Window in Notre-Dame de Paris Himmelsfahrskirche, Dresden

Spontaneous Emission (SE) of Quantum Dot (QD)

Pulse laser inject

QD

he

Vacuum

life time of QD~ ns

II. SE of excitons into nanowire surface plasmons

nanowires

sp

dots

Recent experiment

Model

zmetal nanowire

e

hstrong interaction

Why strong?

20 | |spkE dV

h

SP fields

2 2

The dielectric function is

assumed as ( ) (1 / ),

9.6 (for Ag), and 5.3 (for GaN).

p

C. A. Pfeiffer et al., Phys. Rev. B 10, 3038 (1974)

Tanscendental Equation

0 10 20 300.0

0.2

0.4

0.6

0.8

K

R=0.1 R=0.2 R=0.3 R=0.5

,

,

( 1 53.8 ).

p

z

p

p

k cK

aR

cR nm

APL, 87, 111104 (2005)

Indep. of φ

wire, n=0

Dispersion relations of SP

thin film

, , ; one unit of 53.8 .pz

p p

ak cK R R nm

c

Dispersion relations of SP

0 10 20 300.4

0.5

0.6

0.7

0.8

n=1

Re[

]

K

, , .pz

p p

ak cK R

c

0 10 20 30

0.74

0.76

0.78

0.80

n=2 R=0.1 R=0.2 R=0.3 R=0.5

Re

[]

K

Key feature: nonlinear dispersion with local minimum

approach : Fermi’s golden rules (with dipole approx.)

2

0

20

nn=0 n=0 ,

2 30

0 3

2( ),

| ( ) |2

= .( )

| |

4 free space .

3

i

zi

zi

sp sp eg k

sp zk

eg n kk

z

eg

dk e d E g

d E k

d

dk

d

c

SE rate calculations

e

helectric dipole moment

+

-

SE rates into SP

0.74 0.76 0.78 0.800

50

100

0.74 0.76 0.780

1000

2000

3000

0.74 0.76 0.78 0.800.0

0.1

0.2

0.3

0.74 0.76 0.78 0.800

100

200

0/

p

n

0.72 0.76 0.800.0

0.5

1.0

n=3

n=2

n=1

n=0

n

0/

p

n=3

n

(b) n

n=1

(a)

n n

n=0

0.76 0.78 0.800.0

0.1

0.2

n0.5 0.6 0.7 0.80

2

4

0.5 0.6 0.7 0.80

10

20

30

n

n=2

R=0.1 R=0.5

strongly enhancedSE rates

OBTAINED!

G.Y Chen, Y. N Chen and D. S. Chuu, Opt. Lett. 33, 2212 (2008).

0

is normalized to the

free-space decay rate n

/eg p /eg p

Band-edge effect

2

0

20

nn=0 n=0 ,

2( ),

| ( ) |2

= .( )

| |

i

zi

zi

sp sp eg k

sp zk

eg n kk

z

dk e d E g

d E k

d

dk

0

∞

perturbation treatment is inappropriate !

Markovian : weak interactionsNon-Markovian : strong interactions

0 10 20 300.74

0.76

0.78

0.80

K

n=1, R=0.1

Non-Markovian treatment

ˆ( ) ( )ex sp

di t H tdt

Schrödinger’ s Equation

Laplace Transf.

with ;ij i j

Decay dynamics

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

|be(t

)|2

0t

=0 =-0.4

0

=0.40

=0.80

(b)

|be(t

)|2

0t

d=0.2 d=0.3 d=0.35

(a)

Y. N. Chen, G. Y. Chen*, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).

1, zeg n k 0

0 10 20 300.74

0.76

0.78

0.80

K

n=1, R=0.1

1, 0.74647zn k p

0.34d

III. Coherent single surface plasmon transport

† †, , ,

'[ ] [( ]

2eg e e g k k e g k g e kH i dk v k a a g dk a a

D. E. Chang, A. S. Sǿrensen, E. A. Demler, and M. D. Lukin, Nature Physics 3, 807 (2007).

,with a b a b

Single QD

coupling strengthbetween QDsand SP

Scattering of nanowire SPReal space H :

tr

'

'

:decay into surface plasmon modes.

: decay into all other possible channels.

P (Purcell factor)

sp

sp

/ sp

Scattering of nanowire SP

P=20

J. T. Shen, and S. Fan, Opt. Lett. 30, 2001-2003 (2005); D. E. Chang, A. S. Sǿrensen, E. A. Demler, and M. D. Lukin, Nature Physics 3, 807 (2007).

/ sp

Our model

d

Transmitted

Transmitted

Reflected

Incident

Reflected

｜ e1 〉 ｜ e2 〉

Method The model Hamiltonian :

,with a b a b 1 1 2 2

†,

1,2

, ,

'[ ( )] ,

2

[( ) . .]

j jeg e e g k kj

ikde g e g k

H i dk v k a a

g dk e a h c

The stationary state :

1 2

† † † †, , 1 2

1 2 1 2

[ ( ) ( ) ( ) ( )] , ,0

, ,0 , ,0

k k R R k L L

k k

E dx x c x x c x g g

e e g e g e

The probability amplitude of each QD in excited state

tr

-5 0 50.0

0.5

1.0

-5 0 50.0

0.5

1.0

-5 0 50.0

0.5

1.0

-5 0 50.0

0.5

1.0

(d)(c)

(b) kd= or 2, '=0.05sp

kd=/3, '=0.05sp

Reflection

Transmission

(a)

Reflection

Transmission

Sc

att

eri

ng

Pro

ba

bilit

y

kd= or 2, '=0.25spkd=/3, '=0.25

sp

Reflection

Transmission

Sc

att

eri

ng

Pro

ba

bilit

y

/pl

Reflection

Transmission

/pl

Results

2 2 2min 'tan ( ) 4( ) ( )

sp sp

kd

1 2

1 2

2 ,

(2 1)

k k

k k

kd n e e

kd n e e

I.

'tan( )

2sp kd

II.

maximalentanglement

reflected trapped transmittedkE

1 21 2 1 2, ,0 , ,0k ke e g e g e

entangled state !

1 21 2 1 2 1 2 3 41 1 2 2, ,0 , ,0 ( ) ( )k ke e g e g e c e c g c e c g

concurrenceA property to quantify entanglement

For two qubits state ρ:

spin-flip state

C=1 maximal entanglement

iwhere, is eigenvalues of '

W. Wootters, PRL 80, 2245 (1998)

0kd 2kd kd

'tan( )

2sp kd

G. Y. Chen, Y. N. Chen, F. Mintert, N. Lambert, D. S. Chuu, and A. Buchleitner, in preparation.

Experimental realizations

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, H. J. Kimble, Science 319, 1062 (2008).

D. E. Chang, A. S. Sǿrensen, E. A. Demler, and M. D. Lukin, Nature Physics 3, 807 (2007).

Experiment proposal

d

dielectric waveguide

|g1 > |g2 >

|e1 > |e2 >

minimize the dissipations

Summary

• The SE rate of QD exciton can be strongly enhanced by coupling to SP.

• The decay dynamics around the band edge should be treated with non-Markovian way.

• Through the scattering of SP, the maximal entanglement between two QDs can be achieved.

Outlook I: surface-plasmonic switch

z

CdSe QD

GaN

Ag

10.76 nm

e

h

An external constant magnetic field Bz

Bz

The dispersion relations would be variated.

Surface-Plasmonic Switch

Outlook II: Quantum Phase Transition of Surface Plasmons

Simulation of QPT from a Superfluid to Mott insulator by using utralcold atoms :

hopping On-site interaction

U > t : Insulator

t > U : SuperfluidL. Buluta and F. Nori, Science 326, 108 (2009).

|g1 > |g2 >

|g3 >

J J

|e1 > |e2 > |e3 >

Simulation by using surface plasmons :

Detection of Mott Transition

|g1 > |g2 >

|g3 >

J J

waveguide

†k ka a

|e1 > |e2 > |e3 >

insulator or superfluid ?

Collaborators :

Prof. Dr. Andreas Buchleitner(Uni. Freiburg, Germany)

Dr. Florian Mintert(Uni. Freiburg, Germany)

Prof. Franco Nori(The Uni. of Michigan, Ann Arbor, USA and RIKEN, Japan )

Dr. Neil Lambert (RIKEN, Japan)

Prof. Dr. Tobias Brandes (TU Berlin, Germany)

Prof. 陳岳男 成大物理系＆ NCTS （ south ）

Prof. 褚德三 交大電子物理系 退休教授