controlling one- and two photon transports in one-dimension

ChangChang--PuPu

SunSun

Institute of Theoretical Physics Chinese Academy of Sciences

Controlling oneControlling one--

and two photon transports and two photon transports in onein one--dimensiondimension

Sept.,2010Sept.,2010

http://www.itp.ac.cn/~suncp

Outline

1. L. Zhou, Z. R. Gong, Y.X., Liu,

CPS

, F. Nori，

Phys. Rev. Lett

101, 100501 (2008)

2. T. Shi, CPS, Phys. Rev. B 79, 205111 (2009) 3. T. Shi, S.H. Fan, CPS, arXiv:1009.2828

•

Background and motivations•

Single photon transport with a controller

•

Two photon transport in waveguide •

Towards active manipulation for photons

Relevant papers

1. Controlling Quasibound States in 1D Continuum Through Electromagnetic Induced Transparency Mechanism Z. R. Gong, H. Ian, Lan Zhou, CPS, Phys. Rev. A 78, 053806 (2008)

2. Intrinsic Cavity QED and Emergent Quasi-Normal Modes for Single Photon H. Dong, Z. R. Gong, H. Ian, L. Zhou, CPS, Phys. Rev. A 79, 063847(2009)

3. Quantum super-cavity with atomic mirrors Lan Zhou, H. Dong, Yu-xi Liu, CPS, F.Nori. Phys. Rev. A 78, 063827 (2008)

4. Lehmann-Symanzik-Zimmermann Reduction Approach to Multi-Photon Scattering in Coupled-Resonator Arrays T. Shi, CPS, Phys. Rev. B 79, 205111 (2009)

5.Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array J.Q. Liao, J.F. Huang, Y. Liu, L.M. Kuang, CPS, Phys. Rev. A 80, 014301(2009)

6.Observable Topological Effects of Mobius Molecular Devices Nan Zhao, H. Dong, Shuo Yang, CPS, Phys. Rev. B 79, 125440 (2009)

7. Möbius graphene strip as a topological insulator Z. L. Guo, Z. R. Gong, H. Dong, CPS, Phys. Rev. B 80, 195310 (2009)

1. Controlling Quasibound States in 1D Continuum Through Electromagnetic Induced Transparency Mechanism Z. R. Gong, H. Ian, Lan Zhou, CPS, Phys. Rev. A 78, 053806 (2008)

2. Intrinsic Cavity QED and Emergent Quasi-Normal Modes for Single Photon H. Dong, Z. R. Gong, H. Ian, L. Zhou, CPS, Phys. Rev. A 79, 063847(2009)

3. Quantum super-cavity with atomic mirrors Lan Zhou, H. Dong, Yu-xi Liu, CPS, F.Nori. Phys. Rev. A 78, 063827 (2008)

4. Lehmann-Symanzik-Zimmermann Reduction Approach to Multi-Photon Scattering in Coupled-Resonator Arrays T. Shi, CPS, Phys. Rev. B 79, 205111 (2009)

5.Quantum switch for single-photon transport in a coupled superconducting transmission-line-resonator array J.Q. Liao, J.F. Huang, Y. Liu, L.M. Kuang, CPS, Phys. Rev. A 80, 014301(2009)

6.Observable Topological Effects of Mobius Molecular Devices Nan Zhao, H. Dong, Shuo Yang, CPS, Phys. Rev. B 79, 125440 (2009)

7. Möbius graphene strip as a topological insulator Z. L. Guo, Z. R. Gong, H. Dong, CPS, Phys. Rev. B 80, 195310 (2009)

Quantum information and future quantum devices

Emergent quantum phenomenain artificial structures and meta-materials

Quantum information Quantum coherent devices

Based on whole wave function rather than state density only:

Phase effect dominated

From electronic to single electron transistor (SET)

based on current and voltage from the density of electrons rather than phases of the states

Controlling quantum state at the level of single electron

All optical device in quantum level: Controlling one photon by one photon

Optical switch to single photon transistor (SPT)

http://www.gizmag.com/optical-transistor-made-from-single-molecule/12157/

Why controlling photon by photon is difficult ?

No direct inter-photon interaction and direct coupling to external E.M field

according to QED

Photon self interaction must be mediated by some massive particles in higher order processes

http://upload.wikimedia.org/wikipedia/commons/b/ba/Feynman_EP_Annihilation.svg

Single photon based devices

• Single-photon sourceAn ideal triggered source of single photon emits one and only one photon in each pulse

distributed-Bragg- reflector (DBR) cavity

Photonic-crystal cavity

Our proposal based on superconducting artificial atoms ( PRB 75, 104516 2007)

• Single-photon detection Toshiba setup single photon detector

Signature of single photon by its statistics

A regulated sequence of optical pulses that contain one-and-only-one photon

2

2

2 2

1. , g (0)>1, superPoissonian, classical2. , g ( )=1, Poissonian, classical3. , g (0)<g ( )<1, subPoissonian, quantum

n nn nn n

τ

τ

Δ >

Δ =

Δ <

2)2(

)(:)()(:

)(tItItI

gτ

τ+

=

1

0 τ

1.

2.

3.

Single photon transistor (SPT) proposal

D. E. Chang et al , Nature Physics, 3,807(2007)

The setup was based on the theory by a series papers in S.H. Fan, et. al (Stanford), e.g., J. T. Shen and S. Fan, Phys. Rev. Lett. 95, 213001 (2005); 98, 153003 (2007); ibid. 98, 153003 (2007);Opt. Lett. 30, 2001 (2005)

Model : Linear waveguide coupled to a local two-level system

With electromagnetically induced transparency (EIT) mechanism

Our questions about this SPT Setup

One –shot control : one photon by one photon?

No, nly

strong light controls the EIT

Wide band or narrow band ?

Narrow one due to the single resonate point

Localize photon for quantum memory ?

No, this localization need bound state of Photon !

The linear dispersion that could not trap photon

Dirac Type particle ,Klein paradox

Our questions about this SPT Setup

Evanesce wave coupling for Photonic crystal defect cavity

g

e

Controlling photons with local atoms

Quantum Devices

Photon transistor\switch

Quantum storage

Photonic logic device Physics:

Lee-Fano-Aderson model Quasi-Normal Mode Quasi-Bound State Feshbach Resonance

Physical ImplementationCircuit QED with Superconducting qubitPhotonic Crystal Defect cavityCoupled Nanomechanical resonators

Bethe AnsatzDiscrete Coordinate Scattering Equation Quantum Field Theory

kk cos2ξω −=Ω kk cos2ξω −=Ω

kk ≈sin

2/1cos 2kk −≈

Higher E

Low E

Simulating waveguide in high energy limit

Tight-binding boson model

Non-Linear dispersion

0 π2/π− 2/π

( )1.... . .c j jj

H a aξ ++= − +∑ h c

CRA Based single photon transistor (SPT)

L. Zhou, Z. R. Gong, Y.X. Liu, C. P. Sun, F. Nori, Phys. Rev. Lett

101, 100501 (2008)

g

e

Hc ∑j

ajaj − ∑

jaj

aj1 h. c.

HI |e⟨e| Ja0† |g⟨e| |e⟨g|a0,

Фx

Local controllerCircuit QED setup

Discrete coordinate scattering equation

( ) 0 0k k j kej

E u j a g u e+= +∑

Stationary eigen-state

| |k k kH EΩ ⟩ = Ω ⟩

Vacuum state of the cavity field

Single-photon amplitude

Excited state amplitude

)0()(

)]1()1([)()( 0

kkek

jkekkkk

JuuE

JujujujuE

=Ω−

+−++−=− δξω

Two channel scattering equation

( ) ( ) ( ) ( )( ) 1 1k k k k kV E u j u j u jω ξΩ − − = − + + −⎡ ⎤⎣ ⎦

Resonate potential in effective scattering equation

20( ) j

kk

JV E

Eδ

=−Ω

Resonance Potential

Energy dependent

Working mechanism of SPT

Ω<kE Ω=kEΩ>kE

g

e

Solution 1: 2 bound photon states

, 0( )

, 0

ikx

ikx

Ae xu j

Ae x

−⎡ ⎤>= ⎢ ⎥<⎣ ⎦

E 2J

E g2

E − 2 − 4J2

ω

E − g2

E − 2 − 4J2

2Jω +

1BE

2Jω −

2BE

2J

2J

2

2 0ik gE JeE

ω −− + − =−Ω

2 cosE J kω= −

2E Jω< −

For j<0

For j>0

( ) ikjikjLk reeju −+=

( ) ikjRk seju =

( ) 2

2

cos2sin2 JkkiJr

−−Ω−=

ξωξ

The boundary condition at j=0

Solution 2: single photon scattering

( )[ ] 4222

4

4)(

JJR

+ΔΔ−Ω−−=Δ

ωξ

kcos2ξω −Ω−=Δ

Breit-Wigner and Fano

line shape

Phase Diagram of reflectionhigh energy limit

Low energy lim

it

Super-cavity: analog of super-lattice

Super-cavity:

g

e

g

e

Zhou, Dong, Liu, Sun, Nori

Phys. Rev. A 78, 063827

(2008)

Wide-Band Scattering of Single Photon

Yue Chang, Z. R. Gong, C. P. Sun

. arXiv:1005.2274

Two photon transport in CRA waveguide

Two photon effect:

The very quantum nature of light

T. Shi and C. P. Sun, Phys. Rev. B 79, 205111 (2009); arXiv:0907.2776.

Tow photons in one dimension

Anti bunching single photon case two photon case

Photon blockadePhoton blockade

T. Shi, CPS, arXiv:0907.2776(2009)

Signature of photon blockade via statistics

A two photon interference effect, tends to enhance the single photon effect for single photon counting or source

2

2

2 2

1.g (0)>1, No Blockade2.g ( )=1, No Blocade3.g ( )<g (0)<1, Blockade

τ

τ

(2) ( )g τ1

0 τ

1.

2.

3.

Photon bunching

Photon antibunchingPhoton antibunching

Photon BunchingPhoton Bunching

Photon AntiPhoton Anti--BunchingBunching

1

0 τ

2)2(

)(:)()(:

)(tItItI

gτ

τ+

=

Coulomb (electron) blockade

Coulomb interaction prevents electron from tunneling to Island

1.

Non-linear potential

2.

For certain gate voltage

2

2QCH =

2( )2

Q eCH −=

2 2( ) ( /2)2 2

0 ( )0 ( )

Q e Q e Q eC C CE

E tunnelingE no tunneling

− −Δ = − =

Δ <Δ >

Photonic analog of Coulomb blockade effect

Strong repulsive interaction of photons is induced by nonlinear medium effectively the excitation of medium by a first photon can block the transport of a second photon.

nonlinear medium

† † 2( )H aa k aaξ=− +

Imamoglu, A.,et

al . Rev. Lett. 79, 1467 (1997).

Mechanism of photon blockade

λ +(0)2

ω c

g0c

λ −(0)2

λ −(0)1

ω c

λ +(0)1

2g

g

Spectrum of JC model

( ) ( ) | , ( | 1,n n n e n n gλ α β± ± ±= ⟩ + + ⟩

2 21( ) ( ) ( ) 42 c cn n ngλ ω ω± = Ω+ − ± Ω− +

2 2 2 2

(2) (1)

( ) 16 ( ) 4

2 ( resonance)c

c

E

g g

g

λ λ

ω ω ω

ω

− +Δ = −

= − Ω− + + Ω− +

= −

K. M. Birnbaum

et al., Nature (London) 436, 87 (2005))

c gω −

ω c

ω c

Anti-bouncing means photon blockade?

K. M. Birnbaum


λ +(0)2

ω c

g0c

λ −(0)2

λ −(0)1

ω c

λ +(0)1

2g

g

c gω −

EΔ

Mechanism and Experiment of photon blockade

K. M. Birnbaum


U

P B S

1D

e

g 2D

B S

Photon blockade due to anharmonicity

of energy levels

Transmission line coupled to nonlinear Nano-mechanical resonator via quantum transducer setup [CPS, L. F. Wei, Y Liu, F. NoriPhys. Rev. A 73, 022318 (2006)]

Y.D. Wang, CPS C. Bruder, in preparation, 2010

2. Numeircal

Master equation approache.g., R. J. Brecha et al., Phys. Rev. A 59, 2392 (1999).

1.Quantum trajectory approach:L. Tian

and H. J. Carmichael, Phys. Rev. A 46, 6801 (1992).

3.Mean field approach: K. Srinivasan

and O. Painter, Phys. Rev. A 75, 023814 (2007).

4. Exact solution with Bethe Ansatz

and QFT

J. T. Shen

, S. Fan, Phys. Rev. Lett. 98, 153003 (2007);

L. Zhou et al.,Phys. Rev. Lett. 101, 100501 (2008); H. Dong et al., Phys. Rev. A 76, 063847 (2009); T. Shi and C. P. Sun, Phys. Rev. B 79, 205111 (2009); arXiv:0907.2776.

Theoretical approaches for two photon

e

g e

g

Duality of two configurations for two photon

Side-coupling case Direct-coupling case

Reflection of photons in the side-coupling case = Transmission of photons in the direct-coupling case

J. T. Shen and S. Fan, Phys. Rev. A 79, 023837 (2009).

HW ∑k kakak

HJC caa |e⟨e| ga|g⟨e| a|e⟨g|,

HI V∑kaka H. c. / L

Lehmann-Symanzik-Zimmermann Reduction in QFT

Two -photon effect T. Shi, CPS, Phys. Rev. B 79, 205111 (2009)

1 2 1 2 1 2 1 2

1 1 2 2 2 1 1 2

; ; ,

p p k k p p k k

p k p k p k p k

S iT

S S S S

= +

+

,pk k kpS t δ=1 2

1 2 1 2

4 4 2,

2 1 11,2

( ) [( 2 )( 2 ) 4 ] .( ) ( )( )

p p Ep p k k

s i s i ss s i

E V g E E gTE k p

α δ απ λ λ λ

+

=± =± =

− −Ω − Ω − −=

− − −∏ ∏∏

QFT Calculations 2

T. Shi, Sanhui Fan, C. P. Sun, Phys. arXiv (2010).

2(2)2( ) ( , ) /g t x x Dτ τ= +

For S=L,R,

t2x 1, x 2 12 eiExc t̄ k1

t̄ k2cosΔkx − F, x,

4 41 1,2

1,21 1 2 1

( 2 )exp[ ( )]( , ) ;

4( ) [( ) ( )]

Es s s

s is i s

V g s E i xF x

E kλ λ

λλ λ λ λ

=± −

=± =+ −

− −∑=

− − −∏ ∏

(2) ( ) ( ) ( ) ( ) ( ) |out S S S S outG S a x a x a x a x Sτ τ τ+ += + +

-10 0 100

4

8

τ

gH2LHτL−

1

-20 0 200

0.5

1

τ

gH2LHτL

5 10 15-1

258

11

Eê2

gH2LH0L

2T

( )a

( )c

( )b

( )d

kΔpΔ 110−

210

510

810

1110

anti-bunching=blockade

large bunching

Strong coupling regime :

2g V

10aω ω= =

12E λ −= 12

E λ +=

(2) ( ) 1g τ −

R=Reflection T=Transmission

RT

R T R T

2g V

Weak coupling regime :

9.5 10 10.50

4

8

12

Eê2

gH2LH0L

2T

kΔpΔ 010

410

810

1210

kΔpΔ

-20 0 200

0.5

1

τ

gH2LHτL

-10 -5 0 5 100.00.51.01.52.02.5

τ

gH2LHτL−

1

( )a

( )c

( )b

( )d

photon blockade effect vanishes

Reflected anti-Bouncing photons

T. Shi, CPS, Phys. Rev. B 79, 205111 (2009)；2010，in Arxive

reflection

2nd order coherence

Summary for two photon

The two photon transports in waveguide coupled to a cavity embedded a TLS : Exact solution by LSZ reduction.

Photon blockade effect in strong coupling regime.

Vanishing of Photon blockade effect in weak coupling regime.

Analytic results agree with observations in recent experiment

Towards active manipulation for photon

via

•Quantum Zeno dynamics•Photonic Feshbach

Resonance

•Induced gauge field with Mobius

topology

Active control via quantum Zeno dynamics

High frequency modulation

Band structure and bound states in frequency Domain

a → At a cost

PHYSICAL REVIEW A 80, 062109 2009

L.Zhou ,S. Yang ,Y-x Liu ,C. P. Sun, F. Nori

0[ cos( )] | | | | h.c. ....aH t e e G J e gω ννΩ⎛ ⎞= +Ω ⟩⟨ + ⟩⟨ + +⎜ ⎟

⎝ ⎠I

Dynamic Quantum Zeno Effect

( )exp sin ( )exp( )n nix J x inγ γ= ∑

( )

exp[ ( sin ] | | h.c.,

| | h.c.,i n tn

n

H G i t e g

G J e e gν

νν

ν

+∞−Δ

=−∞

Ω= Δ − ⟩⟨ +

Ω⎛ ⎞= ⟩⟨ +⎜ ⎟⎝ ⎠

∑

I

0 | | h.c.,H G J e gνΩ⎛ ⎞≈ ⟩⟨ +⎜ ⎟

⎝ ⎠I

/ 2.4048, 5.5201,...νΩ =

Decoupling at the zeros of some Bessel function

!!

Numerical : Quantum Zeno Switch for SPT

Photon Delocalization from bound state due to Zeno effect

Photonic analog of Feshbach

Resonance

Predicted in Nuclear physics

1961

experiment with cold atoms

1998both in MIT !

http://cua.mit.edu/ketterle_group/Research_topics/Popular_papers.htm#Enc_BEC

Wave Equation of Single Photon in H-type

E − auaj −Jauaj 1 uaj − 1 gaua0 gbub0E −

gaj,0

E − bubj −Jbubj 1 ubj − 1 gaua0 gbub0E −

gbj,0

Bound state

exp( ), 0( )

exp( ) exp( ), 0a

s ik j ju j

ik j r ik j j′ >⎛ ⎞

= ⎜ ⎟′ ′+ − <⎝ ⎠

ubj B exp−ikj, j 0B expikj, j 0

s 1 − B gagb

JbJa

sin ksin k ′ .

2a aJω +

1aE

2aE

2a aJω −

aω

2b bJω +

1bE

2bE

2b bJω −

bω

Photonic Feshbach

Resonance

b

A scattering state in chain a

and a bound state chain b

.

S=0, Total Reflection

2

2 0ikb a bb b

s g g gE J eB E E

ω −= − + − =−Ω −Ω

2

2 0ikb a bb b

s g g gE J eB E E

ω −= − + − =−Ω −Ω

Numerical with FDTD

Without bound state in another chain

With bound state forming in another chain

Coupled cavity arrays with defect in photonic crystal

FDTD = Finite-

difference time-domain

How to have more controllable parameters for photon

According quantum electrodynamics (QED), no direct interaction exist between two photons , thus magnetic or electric fields could not control the photon straightforwardly.

In this sense, photon is very different from electron

Motivated by AB effect , we use the non-trivial spatial topology to induced an equivalent field for photon

ϕϕϕϕ

21=φ( ) ( )ϕϕ

ϕΨ=Ψ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

∂∂

− Ei2

21

( ) ( )πϕϕ 2+Ψ=Ψ

( ) ( )ϕψϕ ϕ 2/ie−=Ψ

( ) ( )ϕψϕψϕ

E=∂∂

− 2

2

( ) ( )πϕψϕψ 2+−=

Gauge transf.

Periodic, single-valued

anti-periodic, multi-valued

Aharanov–Bohm

effect in a mesoscopic

ring

How about a more complicated topologically non-trivial boundary ??

http://en.wikipedia.org/wiki/File:M%C3%B6bius_strip.jpg

Tight binding boson model

with Mobius

topology

Mobius

boundary condition:

,Nj

N

j jj

j j

aA

b

VVε

ε

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞

= ⎜ ⎟−⎝ ⎠M

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

0

0

0110

ba

ba

N

N

1 111 12

j j

j j

c ad b⎛ ⎞ ⎛ ⎞⎛ ⎞

=⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

Cut- off of upper band in transmission spectrum

c-ring

d-ring

Physical Realization of Mobius

systems

Chemical Reviews (2006)Tetrahedron Lett. (1964)

J. Am. Chem. Soc. (1982)

Boson:

heating a bundle of photonic crystal fibers been

Fermion: synthesizing aromatic hydrocarbons with twisted Pi-electrons

Nature (2002)

Non-Abelian

induced gauge field in continuous limit

The Mobius

boundary condition induce an effective magnetic flux in the “conduction band”.

In the pseudo-spin representation

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

00021

41zσφ

D. Loss, P. Goldbart, A. V. Balatsky, Phys. Rev. Lett. 65, 1655 (1990)

Suppression of conduction band transmission

Conclusions also valid to the fermion

system

Acknowledgements

+ some regular visitors

Students:

Hui Dong, Tao Shi, Dazi

Xu ,

Yue Chang,Jin-Feng

Huan

Post DocDr. Qing Ai

Franco Nori (Riken & Univ. Michigan

), Shan-Hui Fan (Univ. Stanford

)

Lan Zhou (HNNU), Yu-xi Liu (Tsinghua

Univ) [ my previous Post docs]

controlling one- and two photon transports in one-dimension

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