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

Waveguide QED: Quantum Transport of 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 Quantum Optics: Two level atom in a cavity: cavity QED 2 3 √ 2 g 2 1 k g 0 1 g ω ✏ 0 g ground excited H JC = ~ ! a † a + ✏ 2 � z + g � + a + a † � − � � Jaynes-Cummings model: strong coupling shifts the modes of the cavity → no longer an evenly spaced harmonic osc. ladder → nonlinear spacing produces photon blockade ~ effective interaction between photons Does coupling to a continuum of modes bring in something new?

Motivation: Flying Qubits for Quantum Networks a b from H. J. Kimble, Quantum quantum “Quantum Internet”, node Nature 2008 Scheme being developed based on cavity QED: c quantum Node A g channel in ( t ) W Y Node B B k Y out ( t ) W A Is there a cavity-free way to do this?

1D Waveguide QED: Strong Coupling (1D)%Waveguide.QED% two level system Γ a + a † � � � � Dipole'coupling' | g ih e | + | e ih g | ⇣ ⌘ X X ~ ! k a † a † k a k + ✏ | e ih e | + k | g ih e | + a k | e ih g | H = V “rotating wave approximation”: neglect 2 terms in the dipole coupling; Γ / ✏ ⌧ 1 valid when

Waveguide QED: Variety of Potential Physical Systems photonic crystal + defect nanofiber + atom guided modes nanofiber atoms [from LeKien & Hakuta, PRA 2008] [from Shen & Fan, PRL 2007] superconducting qubit + stripline plasmonic wire + quantum dot Out transmission a Emitter–surface-plasmon coupling line Transmission- line cavity Surface-plasmon–waveguide coupling Losses Tapered nanowire C o o Cooper-pair p e r - p a i r Dielectric waveguide o m b o x a t box 10 µ m 10 GHz in [Schoelkopf & Girvin, Nature 2008] [Chang, Sorensen, Demler & Lukin, Nat.Phys. 2007]

Experimental Systems: Example 1: Atom + Dielectric P T P in z r ( t ) ρ P R φ g / 0 γ 0 40 80 Observed strong interaction of single atom and single photon. [from Alton, et al. (Kimble group) N.Phys. 2010]

Experimental Systems: Example 2: Quantum Dot + Nanowire InAs quantum dot embedded in GaAs nanowire: P=9 b 2 θ α 2.5 µ m HE 11 β /2 β /2 SiO 2 r m 200 nm Au [from Claudon, et al. (Grenoble CEA group), N.Photonics 2010 & PRL 2011]

Experimental Systems: Example 3: Superconducting Qubit Rabi Splitting! Cooper pair box [from Wallraff, et al. (Yale group) Nature 2004]

Experimental Systems: Example 3: Superconducting Qubit A) V R V T 2 ( τ ) g V in B) 2 (0) 2.4 g P 2.4 320 um 2.2 -129dBm 2.2 -128dBm 2.0 2.0 10um -127dBm 1.8 -125dBm 1.8 1.6 C) 2 ( τ ) 1 g 1.4 1.0 B) ω p R 1.6 T 1.2 1 2 1.0 1.4 0.8 -128 -124 -120 Data 0 2 ( τ ) g P [dBm] 1.2 RT 0.6 Theory T 1.0 Amplifier 0.4 -20dB 4.2K LPF -20dB -150 -100 -50 0 50 100 150 0.2 DC Block -10dB 1K -10dB τ [ ] ns 0.0 -30dB BPF -30dB -140 -130 100nm -120 -110 50mK ∼ P [dBm] P > 20 Cc “Atom” [Hoi, et al. (Delsing/Wilson group) arXiv 2012)]

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

Waveguide QED: Waveguide + Two-Level System (2LS)  � Z R ( x ) d L ( x ) d 1D continuum a † dxa R ( x ) − a † H = − i ~ c dxa R ( x ) dx (bosons) + +( ✏ � i Γ 0 ) | "ih" | 2 level system (described by Pauli matrices) h i + σ + a R (0) + σ − a † R (0) + σ + a L (0) + σ − a † + V L (0) coupling caveat: strong coupling but not “ultra-strong” σ + a † → ¡ rotating wave approximation, so neglect R (0) ( ⇒ 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...) a † R ( 3. bosonic statistics of is crucial!

Waveguide QED (2LS): 2-Photon Scattering State 1. Scattering wave states for incoming even plane waves Z n=1 dx g 1 ( x ) a † | ψ 1 i = e ( x ) | 0 , g i + e 1 | 0 , e i can satisfy with plane waves n=2 Z Z dx 1 dx 2 g 2 ( x 1 , x 2 ) a † e ( x 1 ) a † dx e 2 ( x ) a † | ψ 2 i = e ( x 2 ) | 0 , g i + e ( x ) | 0 , e i g 2 ( x 1 , x 2 ) e 2 ( x ) no plane wave solution: can’t make and consistent n e i ( k 1 x 1 + k 2 x 2 ) + Be i ( k 1 + k 2 ) x 2 e − Γ | x 2 − x 1 | o g 2 ( x 1 , x 2 ) ∼ Sym correlated state need a continuum for this 2. Transform back to Left-Right basis 3. Scattering of wave-packet with definite photon number: Fock state use a Gaussian wave-packet-- spectral width σ 4. Scattering of coherent state: mean photon number

Waveguide QED (2LS): n=4 variety of more complicated correlations: 1-photon transmission: 1 P (1) 0.8 incoming photon at R P (1) resonance with the 2LS; 0.6 L P (1) Gaussian wave-packet 0.4 decay rate from 2LS 0.2 into waveguide: 0 Γ ≡ 2 V 2 /c 0 2 4 6 8 10 � / �

Waveguide QED (2LS): 2-Photon Transmission 1 1 PW PW corr. corr. BS BS RR RL P (2) P (2) 0.5 0.5 0 0 large correlated 0 5 10 0 5 10 state effects!! 1 1 P (2) RR PW P (2) 0.5 corr. P (2) RL P (2) BS LL 0.5 P (2) LL 0 − 0.5 0 0 5 10 0 5 10 � / � � / � [coupling strength] PW = “plane wave” corr. = correlated state + interference part RR = right-right RL = right-left (1 photon transmitted, 1 reflected)

� � � � � � − Waveguide QED (2LS): Two-point Correlation � � � � − R ( x 2 � x 1 ) = h out α | a † R ( x 2 ) a R ( x 2 ) a † R ( x 1 ) a R ( x 1 ) | out α i g (2) h out α | a † R ( x 1 ) a R ( x 1 ) | out α i 2 10 0 5 10 10 (f) � =0.4 � (2) (x 2 − x 1 ) � � � � 5 0.5 g R uncorrelated ⇒ 0 10 0 5 10 10 0 � (x 2 − x 1 ) � (x − x ) � � bunching! anti-bunching! [Huaixiu Zheng] − �

− − − − − − − − − − − − − − Spectral Entanglement: Two-Photon State − − − − − − − − − − − − − − 3 x 10 1 2.5 1 10 fRR fLL 2 F LL 8 F RR 0.5 0.5 1.5 6 k2 − k0 k2 − k0 0 − 0 − 1 4 − − 0.5 − − 0.5 0.5 2 − − 1 0 − − 1 0 − − − 1 − 0.5 0 0.5 1 − − − 1 − 0.5 0 0.5 1 − k1 − k0 − k1 − k0 Spectrally entangled photon pair ¡ ¡ ¡ ¡ (possible large-alphabet System'parameters:' quantum communication) Γ=9,%Γ ’=1,' δ=0,%%σ=0.01 ' 0.1 [Huaixiu Zheng] − − − − − − − − − − − − − − − − − − − − − − − −

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) Z R ( x ) d L ( x ) d h i a † dxa R ( x ) � a † 1D continuum H = dx ( � i ) ~ c dxa L ( x ) + Z [ a † R ( x ) + a † � � � + dx ~ V � ( x ) L ( x )] | 1 ⇤⇥ 2 | + | 3 ⇤⇥ 4 | + h . c . coupling + 4 ⇥ j � i Γ j | j ⇤⇥ j | + ~ Ω ⇣ ⌘ ⇣ ⌘ X + | 2 ⇤⇥ 3 | + h . c . ~ 2 2 j =2 4 level system

Multi-Photon: Cavity-free Photon Blockade true 2-photon transmission compared ?" to 2 independent photons: detuning from 4LS: δ = 0 loss in 4LS: Γ 0 = 1 photon blockade wavepacket spectral width: σ = 0 . 2 [coupling strength] [Huaixiu Zheng]

Toward a Single Photon Source (a) (b) log 10 [P 0 /P 0,Poisson ] log 10 [P 1 /P 1,Poisson ] n=0 n=1 Number statistics 4 4 0 0.1 of transmitted pulse?? − 0.02 0 � 2 2 − 0.1 Inject coherent − 0.04 state pulse with 0 0 0 10 20 0 10 20 1 photon on avg. (c) (d) log 10 [P 2 /P 2,Poisson ] log 10 [P 3 /P 3,Poisson ] n=2 n=3 4 2 4 1 1 0 2 � 2 0 − 1 − 1 0 0 0 10 20 0 10 20 � � sub-Poissonian statistics: multi-photon states are suppressed single photon source important for secure quantum communication

Two@Qubit'Problem ' ' • 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'

Numerical(Green(Func7on(Method(I( • Original(Hamiltonian( • Hamiltonian(with(bosonic(sites(