g 0 g ground excited H JC = ~ ! a a + 2 z + g + a + a - - PowerPoint PPT Presentation

Waveguide QED: Quantum Transport

• f Strongly Correlated Photons

Harold Baranger, Duke University, with Huaixiu Zheng and Dan Gauthier Topic: 1D bosons [photons, plasmons, cold atoms, ...] interacting with two-level systems (2LS) [atom, qubit, quantum dot, ...] Outline:

• motivation:

quantum optics, quantum information, phases of “interacting” light

• experimental systems: optical, superconducting qubits, nanowires, ...
• simple model → surprising result, 2 photon correlated state
• our results: photon blockade using a 4 level system; 2 qubits
• conclusions and speculations...

Motivation: Quantum Optics

Two level atom in a cavity: cavity QED

Quantum Optics:

k g g

HJC = ~!a†a + ✏ 2z + g

• +a + a†−

Jaynes-Cummings model:

ω

1 2 3 ground 2 1

✏

excited

strong coupling shifts the modes of the cavity → no longer an evenly spaced harmonic osc. ladder → nonlinear spacing produces photon blockade Does coupling to a continuum of modes bring in something new? ~ effective interaction between photons

√ 2g

g

c

• ut(t)

Node A Node B k

in(t)

Y

A

W Y

B

W g

a b

Quantum

quantum node quantum channel

from H. J. Kimble, “Quantum Internet”, Nature 2008

Motivation: Flying Qubits for Quantum Networks

Is there a cavity-free way to do this? Scheme being developed based

• n cavity QED:

“rotating wave approximation”: neglect 2 terms in the dipole coupling; valid when

Γ

(1D)%Waveguide.QED%

• |gihe| + |eihg|

a + a†

H = X ~!ka†

kak + ✏|eihe| +

X V ⇣ a†

k|gihe| + ak|eihg|

⌘

1D Waveguide QED: Strong Coupling

Dipole'coupling'

two level system

Γ/✏ ⌧ 1

Waveguide QED: Variety of Potential Physical Systems

Emitter–surface-plasmon coupling Surface-plasmon–waveguide coupling

Losses

Tapered nanowire Dielectric waveguide

nanofiber guided modes atoms [from LeKien & Hakuta, PRA 2008] [from Shen & Fan, PRL 2007]

nanofiber + atom photonic crystal + defect plasmonic wire + quantum dot superconducting qubit + stripline

Transmission- line cavity 10 GHz in C

• p

e r

• p

a i r b

• x

a t

• m

Out 10 µm

a

[Chang, Sorensen, Demler & Lukin, Nat.Phys. 2007] Cooper-pair box transmission line [Schoelkopf & Girvin, Nature 2008]

Experimental Systems: Example 1: Atom + Dielectric

z r(t) Pin PR PT φ ρ

g/ 0

γ 80 40 [from Alton, et al. (Kimble group) N.Phys. 2010] Observed strong interaction of single atom and single photon.

Experimental Systems: Example 2: Quantum Dot + Nanowire

InAs quantum dot embedded in GaAs nanowire:

β/2 β/2 rm HE11 α 2θ 200 nm 2.5 µm

b

Au SiO2

[from Claudon, et al. (Grenoble CEA group), N.Photonics 2010 & PRL 2011]

P=9

Experimental Systems: Example 3: Superconducting Qubit Rabi Splitting!

Cooper pair box [from Wallraff, et al. (Yale group) Nature 2004]

A)

VR

Vin

VT

10um

100nm 320 um

1

C)

Data Theory P [dBm] T

T R

1.0 0.8 0.6 0.4 0.2 0.0

• 140
• 130
• 120
• 110

Experimental Systems: Example 3: Superconducting Qubit

• 30dB
• 30dB
• 10dB
• 20dB
• 10dB
• 20dB

DC Block BPF LPF RT 4.2K 1K 50mK “Atom” Cc

B)

Amplifier

ωp 1

2 ∼

g

2(τ)

2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0

• 128
• 124
• 120

2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0

• 150
• 100
• 50

50 100 150

P

• 129dBm
• 128dBm
• 127dBm
• 125dBm

g

2(0)

[dBm]

B)

g

2(τ)

P

τ [ ]

ns

g

2(τ)

[Hoi, et al. (Delsing/Wilson group) arXiv 2012)]

P > 20

2 Photon States: Correlation (handwaving...)

Scattering by 2LS depends on whether it is in excited state. Strong coupling ➔ 1 photon near 2LS strongly influences others ➔ generates a strong correlation between the photons Example result (Shen&Fan, PRL-PRA 2007): ... part of outgoing wave: the 2 photons are exponentially correlated * connected to stimulated emission (bosons!) inject 2 photons in uncorrelated resonant plane wave states

+(✏ iΓ0)| "ih" | Waveguide QED: Waveguide + Two-Level System (2LS)

H = −i~c Z dx  a†

R(x) d

dxaR(x) − a†

L(x) d

dxaR(x)

• +V

h σ+aR(0) + σ−a†

R(0) + σ+aL(0) + σ−a† L(0)

i

1D continuum (bosons)

+ +

coupling 2 level system (described by Pauli matrices)

σ+ a†

R(0)

caveat: strong coupling but not “ultra-strong” → ¡rotating wave approximation, so neglect (⇒ makes exact solution straight forward)

• 1. number of “excitations” conserved (no “Kondo effects”...)
• 2. L-R symmetry ⇒ make even and odd combination

⇒ only even mode couples to 2LS (and it is chiral...)

• 3. bosonic statistics of is crucial!

a†

R(

|ψ1i = Z dx g1(x) a†

e(x)|0, gi + e1|0, ei

|ψ2i = Z dx1dx2 g2(x1, x2) a†

e(x1)a† e(x2)|0, gi +

Z dx e2(x)a†

e(x)|0, ei

g2(x1, x2) e2(x) g2(x1, x2) ∼ Sym n ei(k1x1+k2x2) + Bei(k1+k2)x2e−Γ|x2−x1|o

• 3. Scattering of wave-packet with definite photon number: Fock state

use a Gaussian wave-packet-- spectral width σ

Waveguide QED (2LS): 2-Photon Scattering State

• 1. Scattering wave states for incoming even plane waves
• 2. Transform back to Left-Right basis
• 4. Scattering of coherent state: mean photon number

n=1 can satisfy with plane waves n=2 no plane wave solution: can’t make and consistent correlated state need a continuum for this

Γ ≡ 2V 2/c Waveguide QED (2LS):

n=4 variety of more complicated correlations:

1-photon transmission:

incoming photon at resonance with the 2LS; Gaussian wave-packet

2 4 6 8 10 0.2 0.4 0.6 0.8 1 / P(1) P(1)

R

P(1)

L

decay rate from 2LS into waveguide:

Waveguide QED (2LS): 2-Photon Transmission

PW = “plane wave”

• corr. = correlated state + interference part

RR = right-right RL = right-left (1 photon transmitted, 1 reflected) 5 10 0.5 1 P(2)

RR

PW BS

5 10 0.5 1 P(2)

RL

PW BS

5 10 −0.5 0.5 1 / P(2)

LL

PW BS

5 10 0.5 1 / P(2)

P(2)

RR

P(2)

RL

P(2)

LL

[coupling strength] large correlated state effects!!

corr. corr. corr.

− 10 (x2−x1)

• g(2)

R (x2 x1) = houtα|a† R(x2)aR(x2)a† R(x1)aR(x1)|outαi

houtα|a†

R(x1)aR(x1)|outαi2

Waveguide QED (2LS): Two-point Correlation

10 − 5 10 10 (x −x ) 5 10 5 10

• (f) =0.4

0.5 gR

(2)(x2−x1)

• −
• bunching!

anti-bunching! uncorrelated ⇒

[Huaixiu Zheng]

Spectral Entanglement: Two-Photon State

Spectrally entangled photon pair ¡ ¡ ¡ ¡(possible large-alphabet quantum communication) [Huaixiu Zheng]

− − − − − −

−

k1−k0 k2−k0 −1 −0.5 0.5 1 −1 −0.5 0.5 1 fRR 0.5 1 1.5 2 2.5 x 10

−3

− − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

−

− − − − − −

−

− − − − − − − − − − − − − − − − − − k1−k0 k2−k0 −1 −0.5 0.5 1 −1 −0.5 0.5 1 fLL 2 4 6 8 10

FLL FRR

System'parameters:' Γ=9,%Γ’=1,'δ=0,%%σ=0.01'

0.1

Waveguide QED: Four-Level System (4LS)

Consider a more complicated local quantum system: Why??

• 1. would like “classic” effect of interacting photons,

such as photon blockade

• 2. want to change number statistics-- non-classical light

classical controllable field couples levels 2 and 3: made in optics and supercond. qubits

[Majer et al., PRL 2005]

[Huaixiu Zheng]

Waveguide QED: Four-Level System (4LS)

1D continuum

+ +

coupling 4 level system

H = Z dx (i)~c h a†

R(x) d

dxaR(x) a†

L(x) d

dxaL(x) i + Z dx ~V (x)

• [a†

R(x) + a† L(x)]

• |1⇤⇥2| + |3⇤⇥4|
• + h.c.

+

4

X

j=2

~ ⇣ ⇥j iΓj 2 ⌘ |j⇤⇥j| + ~Ω 2 ⇣ |2⇤⇥3| + h.c. ⌘

Multi-Photon: Cavity-free Photon Blockade

true 2-photon transmission compared to 2 independent photons:

?"

δ = 0

photon blockade

[coupling strength]

[Huaixiu Zheng]

detuning from 4LS:

Γ0 = 1

loss in 4LS:

σ = 0.2

wavepacket spectral width:

• log10[P0/P0,Poisson]

10 20 2 4 −0.04 −0.02

log10[P1/P1,Poisson]

10 20 2 4 −0.1 0.1

• log10[P2/P2,Poisson]

10 20 2 4 −1 1

• log10[P3/P3,Poisson]

10 20 2 4 −1 1 2 (a) (b) (c) (d)

n=0 n=1 n=2 n=3

Toward a Single Photon Source

Inject coherent state pulse with 1 photon on avg. Number statistics

• f transmitted

pulse?? sub-Poissonian statistics: multi-photon states are suppressed single photon source important for secure quantum communication

• Minimal'system'for'scalable'quantum'networks'
• Rela6vely'unexplored'for'1D'waveguide'case'
• So'far,'no'analy6c'solu6on'found@@@in'sharp'contrast'to'

single@qubit'case'

Two@Qubit'Problem'

'

• Original(Hamiltonian(
• Hamiltonian(with(bosonic(sites(

Numerical(Green(Func7on(Method(I(

• Non$interac+ng-sca/ering-eigenstates-
• Full-interac+ng-solu+on-
• Photon$photon-correla+on-(g2)-

Numerical-Green-Func+on-Method-II-

† † 2 † †

| ( ) ( ) ( ) ( ) | ( ) | ( ) ( ) | | ( ) ( ) | a x a x ct a x ct a x g t a x a x a x ct a x ct

α α α α α α α α

ψ ψ ψ ψ ψ ψ 〈 + + 〉 = 〈 〉〈 + + 〉

Note: g2 = 1 for product state or coherent state (classical)

Photon&Photon'Correla-on:'Quantum'Beats'

k0# ??" ??"

[Huaixiu Zheng]

Markovian)vs.)Non-Markovian)Regime)

)

Dynamics)on)a)very)long)6me)scale!)

Qubit-Qubit Entanglement: Concurrence

Long%distance- entanglement!-

a b

Quantum

quantum node quantum channel

Future Directions

• 1. best way to observe

correlated photon effects?

• 3. effect of many 2LS?
• - correlations of many photons, QPT?
• 2. quantum gates: transfer q.information from qubit to photon
• - key building block of a quantum network...
• - photon-qubit pi-phase gate possible

z r(t) Pin PR PT φ ρ

g/ 0

γ 80 40 Isc Isc tI0 I0

B

1 µm

Æ 1D waveguide + strongly coupled 2LS or 4LS

• effective photon-photon interaction

Æ photon-photon correlated state & photon blockade

• g2, number statistics, spectral entanglement, ...

Æ two qubit system

• persistent quantum beats,

long-distance entanglement Waveguide QED ➔ interactions between photons ➔ interest for q.optics, q.information, and QPT

CONCLUSIONS

Fitzpatrick Institute for Photonics

Huaixiu Zheng, Dan Gauthier, and HUB

PRA 2010; PRL 2011; PRA 2012; arXiv 2012

1/ Γ