Linear optical quantum information processing in silicon nanophotonics

Transcription

1 Linear optical quantum information processing in silicon nanophotonics Edwin E. Hach, III School of Physics and Astronomy Rochester Institute of Technology 1

2 I am grateful to pursue the work described here in collaboration with Dr. Stefan F. Preble, Microsystems Engineering, RIT Dr. Ali W. Elshaari, KTH Royal Inst. of Tech., Stockholm Dr. Paul M. Alsing, Air Force Research Lab, Rome, NY Michael L. Fanto, Air Force Research Lab, Rome, NY Dr. A. Matthew Smith, Air Force Research Lab, Rome, NY Dr. Christopher C. Tison, Air Force Research Lab, Rome, NY RIT Students: Jeffrey Steidle* Ryan Scott** Thomas Kilmer** David Spiecker** ** * 2

3 Prologue: Photonic Control of Photons (Deterministic) We wish this would happen V H 0 0 V H and V H 1 H V but it requires an intensity on the order of W/cm 2 So we search for nonlinear crystals to mediate the interaction H 0 1 χ 2, χ 3 H V but media having nonlinear susceptibilities large enough for practical applications are not readily available 3

4 Prologue: Photonic Control of Photons (Probabilistic) Linear Optical Quantum Information Processing Linear Optical Transformations Measurements/Post selection Feedback/Feedforward Control Target In Target Out Success 1 16 of the time* Figure: pieter-kok.staff.shef.ac.uk dˆ Linear Optical Elements Desired Characteristics: Scalable Dynamically Tunable η â bˆ ĉ cˆ d ˆ * r r aˆ * b ˆ 4

5 Prologue: Photonic Control of Photons (sort of) 50/50 Beam Splitter The Hong-Ou-Mandel Effect Post Selection P = 50% out ~ P coinc = 0% 1 η 1 2 in = P = 50% 14 Sept

6 Outline The Goal Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 14 Sept

7 Goal: Scalable, tunable, on-chip quantum circuits in silicon nanophotonics 14 Sept

8 DEVICE FABRICATION I R. I. T Nanophotonics Group Spin coat XR-1541 SOI piece (This slide is included to emphasize the power of multidisciplinary collaboration personally, I have no business trying to explain the fabrication process) Soft baking 2 min E-beam exposure SiO 2 cladding Si etch (RIE) nanophotonics.rit.edu Resist developing 8

9 DEVICE FABRICATION II R. I. T Nanophotonics Group ˆ ˆ RIT acornell grad students and post docs; no device a ˆ ˆ RIT acornell grad students and post docs; device a about 24 hours later nanophotonics.rit.edu 9

10 Outline The Goal Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 10

11 Linear Quantum Optics on-a-slide The usual program U(2) b, b a, a c d = τ κ κ τ a b d, d c, c Invert Hermitian Adjoints ψ in = A mn a m b n m=0 n=0 m! n! a = τc κ d b = κc + τ d vac a, a = 1 c, c = 1 b, b = 1 d, d = 1 κ 2 + τ 2 =1 m m ψ out = A mn j n m=0 n=0 j=0 l=0 n l τ j κ m j κ l τ n l m! n! c j+l d m+n j l vac 11

12 Linear Optical Quantum Information Processing Knill-Laflamme-Milburn (KLM) Proposal U(2) Figure: pieter-kok.staff.shef.ac.uk DiVincenzo Criteria Scalable Initialization of Qubits Faster than Decoherence Time Universal Gates Easy Readout Quantum Portability (Qcomm) 14 Sept

13 To address the problem of scalability for use in an integrated quantum circuit, we consider U(2) Coupled Micro-Ring Resonator (MRR): A Fundamental Circuit Element for Quantum Optics on-a-chip l, l γ,η f, f φ 2 θ φ 1 a, a κ,τ c, c Simple Model: Continuous Wave (CW) Operation Internal Modes not Pictured Lossless (Unitary) Operation Practical Advantages: Scalable Can be tuned dynamically Stefan et al can make them (they do it all the time) a, a = 1 f, f = 1 c, c = 1 l, l = 1 14 Sept

14 Outline The Goal Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 14

15 Transition Amplitudes, Quantum Interference, and Path Integrals The transition amplitude for a system to evolve from some initial state to some final state along a particular path connecting the initial and finals states is a product of transition amplitudes for taking each individual step along the path φ f,i =φ f,f 1 φ f 1,f 2 φ i+2,i+1 φ i+1,i The transition amplitude for a system to evolve from some initial state to some final state is determined by the quantum interference between transition amplitudes along all possible paths connecting the initial and final states Φ i f = all paths φ f,i (Propagator) The transition amplitude for a system to evolve from some initial state to some final state is determined by the quantum interference between transition amplitudes along all possible paths connecting the initial and final states f = i Φ i f i 15

16 Discrete Path Integral* Approach to Linear Quantum Optics To adapt the Feynman prescription for use in linear quantum optics 1) Imagine that the Boson operators represent** photons and that photons are little, classical, localized balls of light*** 2) Enumerate the classical paths of the Boson operators through the optical system 3) Construct the transition element along each path by multiplying the appropriate transition element for each step (phase shift, reflection, transmission, etc.) along the path 4) Propagator: sum over all paths connecting particular pairs of input and output modes 5) Use the propagator to express each output Boson operator as a linear superposition of the inputs 6) Forget Step 1 and treat the Boson operators like Boson operators * Skaar et al, and some others, thought of this, too ** They don t *** They aren t 16

17 Phase Shifter U(1) (rectilinear propagation through linear optical medium) a in ϴ a out θ = n ωl c = ωt int a in a out = e iθa in a in a in e iθa in a in = a in e iθ Path a in a out Transition Amplitude e iθ Linear Quantum Optics: discrete stepwise transition amplitudes are generally easy to determine using the Heisenberg Picture Feynman s Spacetime Formulation: infinitesimal transition amplitudes are determined using the classical action 17

18 Directional Coupler (or any SU(2)) b, b τ d, d a, a κ τ κ c, c κ 2 + τ 2 =1 Path (input mode output mode) a c a d b c b d Transition Amplitude Along Path τ κ κ τ Heisenberg Picture: Rotations generated by J 1 = 1 2 a b + ab and J 2 = 1 2i a b ab 18

19 Discrete Path Integral (DPI) Formulation of the Basic Ring Resonator Quantum Circuit Element: CW Operation l, l γ,η f, f φ 2 θ φ 1 a, a κ,τ c, c Path #Round Trips Transition Amplitude Along Path ac 0 τ 1 κ e iφ 1 η e iφ 2 κ 2 κ e iφ 1 η e iφ 2 τ e iφ 1 η e iφ 2 κ n κ e iφ 1 η τ e iφ 1 η e iφ 2 fc 1/2 γ e iφ 2 κ 3/2 γ e iφ 2 τ e iφ 1 η e iφ 2 κ n + 1/2 γ e iφ 2 τ e iφ 1 η e iφ 2 al 1/2 κ e iφ 1 γ 3/2 κ e iφ 1 η e iφ 2 τ e iφ 1 γ n + 1/2 κ e iφ 1 η e iφ 2 τ e iφ 1 fl 0 η 1 γ e iφ 2 τ e iφ 1 γ n 1 κ 2 γ e iφ 2 τ e iφ 1 η e iφ 2 τ e iφ 1 γ n γ e iφ 2 τ e iφ 1 η e iφ 2 τ e iφ 1 c = sum over paths in class a c a + sum over paths in class f c f n κ n γ n 1 γ Similar idea for (a,f)l 19

20 DPI Solution for CW Quantum Transfer Function for Basic Ring Resonator Quantum Circuit Element c = sum over paths in class a c a + sum over paths in class f c f = τ + κ η κ e iθ η τ iθ m e m=0 a + γ e iφ 2 κ η τ iθ m e m=0 f = τ η κ 2 e iθ 1 η τ e iθ a γ κe iφ2 1 η τ f e iθ l = γκ e iφ 1 1 η τ e iθ a + η τ γ 2 e iθ 1 η τ e iθ f a = tc + sl f = s c + t l t η τe iθ η τ e iθ s γκ e iφ 2 η τ e iθ s κγ e iφ 1 η τ e iθ t τ ηe iθ η τ e iθ 20

21 Ring Resonator/Fabry-Perot Etalon Equivalence a in t 1 t 2 a out r 1 r 1 ϕ r 2 r 2 b out t 1 t 2 b in A schematic representation of an externally driven Fabry-Perot etalon. Each distinct input/ouput optical mode is labeled with its Bosonic annihilation operator. The phase conventions for the transmission and reflection amplitudes adopted here are valid parameterizations for the SU(2) transformations that occur at the lossless, partially reflective mirrors forming the cavity; to facilitate our comparison this is in agreement with the notation in Ref. [8]. The linear phase shift, ϕ, is the phase shift for a single crossing of the cavity; that is, a round trip through the cavity is accompanied by a phase shift of θ = 2ϕ. 21

22 Simple Model for a Lossy Ring Coupled to a Single Waveguide Drop Port l, l? l γ,η 0 f f, f φ 2 θ φ 1 a, a κ,τ c, c Γ θ a, a κ,τ c, c 1) Solve the deterministic problem: ψ out = tc + sl = t 1 c 0 l + s 0 c 1 l t 1,0 + s 0,1 2) Express the deterministic solution using (pure state) global density operator: (G) ρ out = ψout ψ out = t 2 1,0 1,0 + ts 1,0 0,1 +st 0,1 1,0 + s 2 0,1 0,1 3) Trace over loss channel to obtain (mixed state) reduced density operator for transmission mode: (c) ρ out = Trmode l (G) ρ out = t s γ 2 κ 2 P 0,1 1,0 = 1 + η 2 τ 2 2Re ητe iθ 14 Sept

23 Comparison with Classical Electrodynamics (Sanity Check) Parametric Limit Highly excited ordinary coherent state input along mode a: a α = α α ψ in = α, 0 = D a α I f vac =exp αa α a vac The output (pure state) is a direct product of ordinary coherent states: ψ out = D c tα D l sα vac = tα c sα l (c) Specifically, in mode c ρ out = tα tα where Quantity Yariv Treatment Circulation Method Classical Factor Fully QM Treatment w/ Coherent State Physical Loss Model (Drop Port Method) Input Field Amplitude 1 α in Output Field Amplitude (waveguide) b 1 α out CirculationFactor /Internal Transmission Amplitude at Drop Port α η Round Trip Phase θ θ ac transition Amplitude t τ α out = tα = η τe iθ η τ e iθ α in 14 Sept 2018 A. Yariv, Electronics Letters 36, 321 (2000) 23

24 Outline The Goal Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 24

25 Hong-Ou-Mandel Effect (Beam Splitter/Bulk Optics) At the 50/50 operating point: 1,1 2 0 = 1 2 2,0 0,2 HOM 2-photon interference r, t 1 F in = The 50/50 point is a zero dimensional HOMM H-O-M Output State Fidelity: F 2 0 out 2 0 F t 25

26 Two Photon Driving: Hong-Ou-Mandel Manifolds (HOMM) l, l γ,η f, f t η τe iθ η τ e iθ s γκ e iφ 2 η τ e iθ a, a κ,τ c, c t τ ηe iθ η τ e iθ Input state: ψ in = 1,1 = a f s κγ e iφ 1 η τ e iθ Output state: ψ out = 2ts 2,0 + tt + ss 1,1 + 2st 0,2 HOMM: tt + ss = 0 κ 2 + γ 2 + κ 2 γ 2 + 2Re ητe iθ = 2 26

27 P(1,1) τ = η (Real) Hong-Ou-Mandel Manifolds: Robust Two-Photon Bunching HOMM: tt + ss = 0 κ 2 + γ 2 + κ 2 γ 2 + 2Re ητe iθ = d Case τ θ = 2 cos θ 2 cos θ 2 1 τ = η (Real) θ θ Parameter space of ring resonator circuit element features a differentiable manifold of operating points on which the HOM Effect occurs this will translate into enhanced robustness in the operation of devices that rely on this effect. EEHIII, S.F. Preble, A.W. Elshaari, P.M. Alsing and M.L. Fanto, Phys. Rev. A89, (2014) 27

28 Hong-Ou-Mandel Manifolds Allow for Operational Optimization of a Scalable U(2) Device Dynamic Tuning F A < 1 F B = 1 B (On HOMM) Experimentally adjustable τ = η A (Off of HOMM) θ 14 Sept

29 Hong-Ou-Mandel Manifolds: Robust Two-Photon Bunching HOMM: tt + ss = 0 κ 2 + γ 2 + κ 2 γ 2 + 2Re ητe iθ = 2 A Two-Dimensional HOMM: Constraint: P 1,1 = 0 τ θ This figure was featured in the Physical Review A online Kaleidoscope c. April 2014 η 29

30 Outline The Goal Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 30

31 The Ring Resonator Circuit Theory b out l 21 γ 1,η 1 γ 2,η 2 b in θ 1 θ 2 Notation: cout d out = We have: a in A B C D cin d in b out l 12 = κ 1,τ 1 l 12 κ 2,τ 2 Block 1 Block 2 A B C D l 21 a in l 21 a out = F G H J a out b in l 12 Overall Network We want: b out a out = T 11 T 12 T 21 T 22 b in a in We Introduce a set of Mode Swap Transformations 31

32 Mode Swap Transformation b out l 21 γ 1,η 1 γ 2,η 2 b in θ 1 θ 2 a in κ 1,τ 1 κ 2,τ 2 a out l 12 Upper Rail cout d out = A B C D cin d in c in d out = A B C D cout d in Lower Rail cout d out = A B C D cin d in c out d in = A B C D cin d out A = 1, A B = B, A C = C, A D = 1 A det B A C D Upper Rail A = 1 A det B D C D, B = B D, C = C D,D = 1 D Lower Rail 14 Sept

33 Quantum Transfer Function for Circuit b out l 21 γ 1,η 1 γ 2,η 2 b in θ 1 θ 2 a in κ 1,τ 1 κ 2,τ 2 a out local upper swap on block 1 l 12 b out l 12 = A B C D local upper swap on block 2 l 21 a out = F G H J l 21 a in l 21 l 12 = b in l 12 b in a out = A B C D F G H J b out a in l 21 l 12 result of local swaps b in a out = global upper swap F G H J A C B D b out a in = S 11 S 12 S 21 S 22 b out a in b out a out = T 11 T 12 T 21 T 22 b in a in q.e.d T 11 = S 11, T 12 = S 12, T 21 = S 21, T 22 = S Sept

34 A Ring Resonator Based Nonlinear Phase Shifter Many adjustable, controllable parameters to search ψ in cin Block 1 Block 2 Block 3 cout ψ shifted SUCCEED! 1 out b out θ 2 b in 1 in Local Swaps 0 in a in θ 1 θ 3 a out 1 out FAIL cin b out l 13 = cout l 21 l 13 = cout l 32 = a out A B 0 C D P Q 0 R S F G 0 H J cin l 21 a in cin l 32 l 13 cout b in l 13 cin l 21 l 13 = cout l 32 l 13 = cout b in = a out A B 0 C D P Q 0 R S F G 0 H J cin b out a in cin l 21 l 13 cout l 32 l 13 cout b in = a out = F G 0 H J α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 P Q 0 R S cin b out a in A B 0 C D cin b out a in Global Middle Rail Swap cout b in = a out α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 cin b out a in cout b out = a out 1 M 33 α 12 M 31 α 21 1 α 23 α 22 M 13 α 32 M 11 cin b in a in 34

35 Outline The Goal Essential Ideas About Quantum Optics/Information Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 35

36 (a) (Bulk Optics) cin φ cout b in 2 b out 1 3 a in Block 1 Block 2 Block 3 a out (b) (Scalable Silicon Nanophotonics) ψ in cin Block 1 Block 2 Block 3 cout ψ shifted SUCCEED! 1 out b out θ 2 b in 1 in 0 in a in θ 1 θ 3 a out 0 out SUCCEED Figure 2 A schematic diagram of a NonLinear Sign Gate (a) in bulk optics and (b) as implemented via our proposal using directionally coupled silicon nanophotonics waveguides and microring resonators (mrr). The nonlinear sign flip is effected on the state in mode c, as given in Eq. (1); modes a and b are auxiliary modes required for the probabilistic action of the gate. The black arrow connecting parts (a) and (b) of the figure effectively summarizes the advancement we discuss in detail in this paper. 36

37 Our Proposal for the Essential Piece of a Scalable KLM CNOT Gate A Ring Resonator Based NonLinear Phase Shifter (NLPS) ψ a 1,in Block 1 Block 2 Block 3 3 δ 1 c12 κ 2 τ 2 c23 δ 3 a 1,out ψ a 2,out 1 out SUCCEED! γ 1 η 1 b 21 θ 2 b 32 γ 2 η 2 γ 3 η 3 a 2,in 1 in 0 in a 3,in θ 1 κ 1 τ 1 a 12 δ 2 a 23 θ 3 κ 3 τ 3 a 3,out 0 out SUCCEEDL T (1) T (2) T (3) 37

38 The NonLinear Phase Shifter (we ll have two, please) What it does: How it works (when it works): S 3 Unitary How often it works: Unitary Part: DPI Formulation: Mode Swap Algebra: 1 g12 g13 1 G g m m ,3 3,2 g11 g31 m2,3 m2, NLPS ˆ NLPS U 1 ˆ NLPS p P NLPS success aˆ aˆ 1,in 1,out T aˆ S aˆ 2,in 2,out aˆ aˆ 3,in 3,out aˆ1,out aˆ1,in aˆ ˆ 2,out S2 S2 T S2 T S2 T a2,in aˆ aˆ 3,out 3,in S 3 m g m 1 G g 1 g 3,3 12 3, g22 m1,3 g32 m1,1 2 S Projection 3 NLPS m m g 1 G m m g 2,2 2, ,2 1,1 23 g33 g31 g32 1 Pˆ NLPS Pˆ NLPS ψ in cin Block 1 Block 2 Block 3 cout ψ shifted SUCCEED! 1 out b out θ 2 b in 1 in 0 in a in θ 1 θ 3 a out 0 out SUCCEED 38

39 Case 1: On-resonance, no phase delays: δ i = 0, θ i = 2π, t i = t i * θ i = 2π, mrrs on resonance 0 1 δ i = 0, no phase delays t 2 * = 2( 2 1), t 1 * = t 3 * = 1/ Figure 3: The one dimensional manifolds η 2 2 i τ i ; T i vs τ i (i = 1,3, black solid; i = 2, black dashed) from Eq. 57 on which optimal operation of the scalable NLPSG occurs under conditions of resonant (θ i =0mod2π), balanced mrrs and δ i =0mod2π phase shifts in the waveguides. In contrast with bulk optical realizations, these curves provide theoretical evidence for vastly enhanced flexibility in implementation and integration of the NLPSG based on directionally coupled mrrs in silicon nanophotonics. 39

40 Case 2: Outer MRR not on resonance δ i = 0, θ 2 = 2π, θ 1,3 2π, t i = t i * 0 θ 2 = 2π, on resonance θ 1,3 = 2π, not on resonance 1 δ i = 0, no phase delays t 2 * = 2( 2 1), t 1 * = t 3 * = 1/ Sept

41 Outline The Goal Essential Ideas About Quantum Optics/Information Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 41

42 Reality Check: Losses, etc. Increasing Losses (decreasing effective α) Theoretical analysis of losses ( whistles and spectral effects (bells ): P.M. Alsing, EEHIII, C. C. Tison, and A. M. Smith, Phys. Rev. A95, (2017) Theoretical analysis of photon pair generation (all the whistles and bells): P.M. Alsing and EEHIII, Phys. Rev. A96, (2017) P.M. Alsing and EEHIII, Phys. Rev. A96, (2017) 42

43 Outline The Goal Essential Ideas About Quantum Optics/Information Input/Output Theory: A General Solution Under Ideal Conditions Hong-Ou-Mandel Manifolds: A Promising Result Formulation of a Quantum Optical Circuit Theory A Scalable KLM CNOT Gate!? Engineering and Design: The Fine Print Summary and Outlook 14 Sept

44 Summary Ring resonators in silicon nanophotonics offer a promising architecture for quantum information processing devices (scalable, tunable, easy to fabricate) Passive Quantum Optical Feedback leads to a topological enhancement of the parameter space of this class of device first example: Hong-Ou-Mandel Manifolds (HOMM) The existence of Hong-Ou-Mandel Manifolds (HOMM) suggest that there are robust regimes for operating quantum information processors based on ring resonator technology We now have a systematic way to formulate and search the parameter spaces of these systems under realistic conditions with respect to finite pulse widths and environmental losses Outlook There are many quantum gates and networks that could be pivotal to quantum information processing, communications, metrology, and imaging we are starting with the KLM CNOT We are working to develop numerical device design software to inform the planning and assembly of experimental tests of our theoretical results, and, from there, to better inform device design and integration Do you have any ideas for interesting quantum optical systems to investigate? We love to collaborate! 44

45 Thank You for Listening! 45

46 Back-Up Slides 46

47 M.S. and Professional Master s Programs Coming Fall

48 Single Photon Transport: Quantum Dynamics (Scattering Theory Approach) ˆ Heff dxc ( x) 0 ivg c( x) c i a a dx x c 1 ˆ i 1t Hˆ eff 1 t ˆ t ~ a 1 t dx x, tcˆ x e t x ~ ˆ 0, 0 cav Vc ( x) aˆ V * aˆ c( x) 14 Sept 2018 EEHIII, A.W. Elshaari, & S.F. Preble, Phys. Rev. A82, (2010) 48

49 Single Photon Transport: Quantum Dynamics Finite Difference Time Domain Results: Wave Packet α 1 direct + α 1 delay 14 Sept

50 Adiabatic Invariant: Single Photon Transport: Quantum Dynamics Finite Difference Time Domain Results: Adiabatic Wavelength Change pdq = U ω = Nħ λ λ = U U 14 Sept

51 Single Photon Transport: Quantum Dynamics Single Photon Storage: Toward Quantum Memory 14 Sept

52 Logical {1} mode a mode b Logical {0} mode a mode b 14 Sept

53 Outline KLM Design for Nonlinear Sign Gate with beam splitters Essential element of KLM CNOT gate NLSG design with mrr Review of single-bus and double-bus mrr, Extension of dimension of parameter space by using; HOM Manifolds Results Papers Raymer, M. and McKinstrie, C., Quantum input-output theory for optical cavities with arbitrary coupling strength: application to two-photon wave-packet shaping," Phys. Rev. A 88, (2013) E. E. Hach III, S. F. Preble, A. W. Elshaari, P. M. Alsing, and M. L. Fanto, Scalable Hong-Ou-Mandel manifolds in quantum-optical ring resonators, Phys. Rev. A 89, (2014). Alsing, P. M., Hach III, E. E., Tison, C. C., and Smith, A. M., A quantum optical description of losses in ring resonators based on field operator transformations," Phys. Rev. A 95, (2017) Alsing, P. M. and Hach III, E. E., Photon-pair generation in a lossy-microring resonator. I. Theory," Phys. Rev. A 96 (3), (2017); ( v2) Alsing, P. M. and Hach III, E. E., Photon-pair generation in a lossy-microring resonator. II. Entanglement in the output mixed Gaussian squeezed state," Phys. Rev. A 96 (3), ( ) (2017)

54 KLM Nonlinear Sign Gate Ralph et al, PRA 65, (2002); Skaar and Landro, Am J Phys 72, 1385 (2004); Last Year (SPIE: Warsaw 2017): Nonlinear Sign Gate mrr Proposed KLM Nonlinear Sign Gate with mrrs Wider parameter range of operation (akin to HOM manifolds in mrr, in 2017 talk) ψ 1 t 1 t 2 t 3 ψ KLM CNOT gate

55 KLM Nonlinear Sign Gate Ralph et al, PRA 65, (2002); Skaar and Landro, Am J Phys 72, 1385 (2004); ψ 1 t 1 t 2 t 3 ψ 1 0 0

56 Quantum Input-Output Theory for MRR Standard Langevin Quantum I/O Theory Wall & Milburn, Quantum Optics Springer (1994) Raymer & McKinstrie, PRA 88, (2013) Alsing, Hach, Tison, Smith, PRA 95, (2017)

57 High Cavity Q Limit Raymer & McKinstrie, PRA 88, (2013) Alsing, Hach, PRA 96, & (2017) (see also) Schneeloch, et al., through coupling loss rate ( ) - internal propagation loss rate - cavity round trip time The high cavity Q limit recovers the usual Langevin results valid near cavity resonances

58 Dual Rail MRR: HOM Manifolds Alsing, Hach, PRA 96, (2017); PRA 96, (2017); HOM in MRR

59 Dual Bus MRR: HOM Manifolds E. E. Hach III, et al., Phys. Rev. A 89, (2014); Alsing, P. M., et al. Phys. Rev. A 95, (2017) no loss 5% loss 10% loss 15% loss 20% loss 25% loss no loss

60 S 0 S S S S 1 S S S 2S S S ˆ ˆ ˆ ˆ ˆ ˆ j2 a j,out 1 S j1sk 2 a j,out ak,out S j1sk1sl 2 a j,out ak,out al,out 0, 0, 0 j2 j, k 1,2, 2,1 2 j, k, lperm1,1,2 S S S S S S S S 2S S S NLPS success p S S S 14 Sept

61 Conceptual Picture of Evanescent Coupling ~e z δ Affected by carrier injection Affected by thermal strain 61

62 S 0 S S S S 1 S S S 2S S S ˆ ˆ ˆ ˆ ˆ ˆ j2 a j,out 1 S j1sk 2 a j,out ak,out S j1sk1sl 2 a j,out ak,out al,out 0, 0, 0 j2 j, k 1,2, 2,1 2 j, k, lperm1,1,2 S S S S S S S S 2S S S NLPS success p S S S 62

63 θ i = 2π, mrrs on resonance Case 3: vary middle interblock phase delay: δ 1 = δ 3, δ 2 0, θ i = 2π, t i = t i * δ 2 0, vary middle phase delay t 2 * = 2( 2 1), t 1 * = t 3 * = 1/ t 2 *=t 2 *(η 2, τ 2, θ 2 ) = 2( 2 1) 1D intersection of the two surfaces for δ 2 =π/30 1D manifold of solutions δ 2 = θ A2 (η 2, τ 2, θ 2 ) = π/30 14 Sept