Jonathan P. Dowling Methods of Entangling Large Numbers of Photons for Enhanced Phase Resolution quantum.phys.lsu.edu Hearne Institute for Theoretical Physics Quantum Science and Technologies Group Louisiana State University Baton Rouge, Louisiana USA Entanglement Beyond the Optical Regime ONR Workshop, Santa Ana, CA 10 FEB 2010

H.Cable, C.Wildfeuer, H.Lee, S.D.Huver, W.N.Plick, G.Deng, R.Glasser, S.Vinjanampathy, K.Jacobs, D.Uskov, J.P.Dowling, P.Lougovski, N.M.VanMeter, M.Wilde, G.Selvaraj, A.DaSilva Not Shown: P.M.Anisimov, B.R.Bardhan, A.Chiruvelli, G.A.Durkin, M.Florescu, Y.Gao, B.Gard, K.Jiang, K.T.Kapale, T.W.Lee, D.J.Lum, S.B.McCracken, S.J.Olsen, G.M.Raterman, C.Sabottke, K.P.Seshadreesan, S.Thanvanthri, G.Veronis Hearne Institute for Theoretical Physics Quantum Science & Technologies Group

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Optical C-NOT with Nonlinearity The Controlled-NOT can be implemented using a Kerr medium: Unfortunately, the interaction  (3) is extremely weak*: at the single photon level — This is not practical! *R.W. Boyd, J. Mod. Opt. 46, 367 (1999). R is a  /2 polarization rotation, followed by a polarization dependent phase shift . R is a  /2 polarization rotation, followed by a polarization dependent phase shift .  (3) R pol PBS zz |0  = |H  Polarization |1  = |V  Qubits |0  = |H  Polarization |1  = |V  Qubits

Two Roads to Optical Quantum Computing Cavity QED I. Enhance Nonlinear Interaction with a Cavity or EIT — Kimble, Walther, Lukin, et al. II. Exploit Nonlinearity of Measurement — Knill, LaFlamme, Milburn, Franson, et al.

Photon-Photon XOR Gate Photon-Photon Nonlinearity Kerr Material Cavity QED EIT Cavity QED EIT Projective Measurement LOQC KLM LOQC KLM WHY IS A KERR NONLINEARITY LIKE A PROJECTIVE MEASUREMENT?

G. G. Lapaire, P. Kok, JPD, J. E. Sipe, PRA 68 (2003) KLM CSIGN Hamiltonian Franson CNOT Hamiltonian NON-Unitary Gates   Effective Unitary Gates A Revolution in Nonlinear Optics at the Few Photon Level: No Longer Limited by the Nonlinearities We Find in Nature! A Revolution in Nonlinear Optics at the Few Photon Level: No Longer Limited by the Nonlinearities We Find in Nature! Projective Measurement Yields Effective “Kerr”!

Single-Photon Quantum Non-Demolition You want to know if there is a single photon in mode b, without destroying it. *N. Imoto, H.A. Haus, and Y. Yamamoto, Phys. Rev. A. 32, 2287 (1985). Cross-Kerr Hamiltonian: H Kerr =  a † a b † b Again, with  = 10 –22, this is impossible. Kerr medium “1”“1” a b |  in  |1|1 |1  D1D1 D2D2

Linear Single-Photon Quantum Non-Demolition The success probability is less than 1 (namely 1/8). The input state is constrained to be a superposition of 0, 1, and 2 photons only. Conditioned on a detector coincidence in D 1 and D 2. |1|1 |1|1 |1  D1D1 D2D2 D0D0  /2 |  in  = c n |n   n = 0 2 |0  Effective  = 1/8  22 Orders of Magnitude Improvement! Effective  = 1/8  22 Orders of Magnitude Improvement! P. Kok, H. Lee, and JPD, PRA 66 (2003)

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Quantum Metrology H.Lee, P.Kok, JPD, J Mod Opt 49, (2002) 2325 Shot noise Heisenberg

Sub-Shot-Noise Interferometric Measurements With Two-Photon N00N States A Kuzmich and L Mandel; Quantum Semiclass. Opt. 10 (1998) 493–500. SNL HL

a † N a N AN Boto, DS Abrams, CP Williams, JPD, PRL 85 (2000) 2733 Super-Resolution Sub-Rayleigh

New York Times Discovery Could Mean Faster Computer Chips

Quantum Lithography Experiment |20>+|02 > |10>+|01 >

Canonical Metrology note the square-root P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811 Suppose we have an ensemble of N states |  = (|0  + e i  |1  )/  2, and we measure the following observable: The expectation value is given by: and the variance (  A) 2 is given by: N(1  cos 2  ) A = |0  1| + |1  0|    |A|  = N cos   The unknown phase can be estimated with accuracy: This is the standard shot-noise limit.  = = AA | d A  /d  |  NN 1

Quantum Lithography & Metrology Now we consider the state and we measure High-Frequency Lithography Effect Heisenberg Limit: No Square Root! P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, (2002). Quantum Lithography*: Quantum Metrology:  N |A N |  N  = cos N    H = = ANAN | d A N  /d  |  N 1

Recap: Super-Resolution  N=1 (classical) N=5 (N00N)

Recap: Super-Sensitivity dP 1 /d  dP N /d  N=1 (classical) N=5 (N00N)

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Showdown at High-N00N! |N,0  + |0,N  How do we make High-N00N!? *C Gerry, and RA Campos, Phys. Rev. A 64, (2001). With a large cross-Kerr nonlinearity!* H =  a † a b † b This is not practical! — need  =  but  = 10 –22 ! |1  |N|N |0  |N,0  + |0,N  N00N States In Chapter 11

N00N & Linear Optical Quantum Computing For proposals* to exploit a non-linear photon-photon interaction e.g. cross-Kerr interaction, the required optical non-linearity not readily accessible. *C. Gerry, and R.A. Campos, Phys. Rev. A 64, (2001). Nature 409, page 46, (2001).

Measurement-Induced Nonlinearities G. G. Lapaire, Pieter Kok, JPD, J. E. Sipe, PRA 68 (2003) First linear-optics based High-N00N generator proposal: Success probability approximately 5% for 4-photon output. e.g. component of light from an optical parametric oscillator Scheme conditions on the detection of one photon at each detector mode a mode b H. Lee, P. Kok, N. J. Cerf and J. P. Dowling, PRA 65, (2002).

Towards High-N00N! Kok, Lee, & Dowling, Phys. Rev. A 65 (2002) the consecutive phases are given by:  k = 2  k N/2 a’a b’b c d |N,N   |N-2,N  + |N,N-2  PS cascade 123 N 2 |N,N   |N,0  + |0,N  p 1 = N (N-1) T 2N-2 R 2  N  e22e2 with T = (N–1)/N and R = 1–T Not Efficient!

Implemented in Experiments!

|10::01 > |20::02 > |40::04 > |10::01 > |20::02 > |30::03 >

N00N State Experiments Rarity, (1990) Ou, et al. (1990) Shih, Alley (1990) …. 6-photon Super-resolution Only! Resch,…,White PRL (2007) Queensland photon Nagata,…,Takeuchi, Science (04 MAY) Hokkaido & Bristol photon Super-sensitivity & Super-resolution Mitchell,…,Steinberg Nature (13 MAY) Toronto , 4-photon Super- resolution only Walther,…,Zeilinger Nature (13 MAY) Vienna

N00N

Physical Review 76, (2007) N00N LHV 

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Efficient Schemes for Generating N00N States! Question: Do there exist operators “U” that produce “N00N” States Efficiently? Answer: YES! Question: Do there exist operators “U” that produce “N00N” States Efficiently? Answer: YES! Constrained Desired |N > |0 > |N0::0N > |1,1,1 > Number Resolving Detectors

Phys. Rev. Lett. 99, (2007)

M-port photocounter Linear optical device (Unitary action on modes) Terms & Conditions Only disentangled inputs are allowed ( ) Modes transformation is unitary (U is a set of beam splitters) Number-resolving photodetection (single photon detectors) VanMeter NM, Lougovski P, Uskov DB, et al., General linear-optical quantum state generation scheme: Applications to maximally path-entangled states, Physical Review A, 76 (6): Art. No DEC Linear Optical N00N Generator II

U This example disproves the N00N Conjecture: “That it Takes At Least N Modes to Make N00N.” The upper bound on the resources scales quadratically! Upper bound theorem: The maximal size of a N00N state generated in m modes via single photon detection in m-2 modes is O(m 2 ). Upper bound theorem: The maximal size of a N00N state generated in m modes via single photon detection in m-2 modes is O(m 2 ). Linear Optical N00N Generator II

Entanglement Seeded Dual OPA Idea is to look at sources which access a higher dimensional Hilbert space (qudits) than canonical LOQC and then exploit number resolving detectors, while we wait for single photon sources to come online. R.T.Glasser, H.Cable, JPD, in preparation.

Quantum States of Light From a High-Gain OPA (Theory) G.S.Agarwal, et al., J. Opt. Soc. Am. B 24, 270 (2007). We present a theoretical analysis of the properties of an unseeded optical parametric amplifier (OPA) used as the source of entangled photons. The idea is to take known bright sources of entangled photons coupled to number resolving detectors and see if this can be used in LOQC, while we wait for the single photon sources. OPA Scheme

Quantum States of Light From a High-Gain OPA (Experiment) In the present paper we experimentally demonstrate that the output of a high-gain optical parametric amplifier can be intense yet exhibits quantum features, namely, a bright source of Bell-type polarization entanglement. F.Sciarrino, et al., Phys. Rev. A 77, (2008) This is an Experiment! State Before Projection Visibility Saturates at 20% with 10 5 Counts Per Second! Lovely, Fresh, Data!

Optimizing the Multi-Photon Absorption Properties of N00N States (submitted to Phys. Rev. A) William N. Plick, Christoph F. Wildfeuer, Petr M. Anisimov, Jonathan P. Dowling In this paper we examine the N-photon absorption properties of "N00N" states, a subclass of path entangled number states. We consider two cases. The first involves the N-photon absorption properties of the ideal N00N state, one that does not include spectral information. We study how the N-photon absorption probability of this state scales with N. We compare this to the absorption probability of various other states. The second case is that of two-photon absorption for an N = 2 N00N state generated from a type II spontaneous down conversion event. In this situation we find that the absorption probability is both better than analogous coherent light (due to frequency entanglement) and highly dependent on the optical setup. We show that the poor production rates of quantum states of light may be partially mitigated by adjusting the spectral parameters to improve their two-photon absorption rates.

Return of the Kerr Nonlinearity! |N,0  + |0,N  How do we make High-N00N!? *C Gerry, and RA Campos, Phys. Rev. A 64, (2001). With a large cross-Kerr nonlinearity!* H =  a † a b † b This is not practical! — need  =  but  = 10 –22 ! |1  |N|N |0  |N,0  + |0,N 

Implementation of QFG via Cavity QED Kapale, KT; Dowling, JP, PRL, 99 (5): Art. No AUG Ramsey Interferometry for atom initially in state b. Dispersive coupling between the atom and cavity gives required conditional phase shift

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Computational Optimization of Quantum LIDAR forward problem solver INPUT “find min( )“ FEEDBACK LOOP: Genetic Algorithm inverse problem solver OUTPUT N: photon number loss A loss B Lee, TW; Huver, SD; Lee, H; et al. PHYSICAL REVIEW A, 80 (6): Art. No DEC 2009 Lee, TW; Huver, SD; Lee, H; et al. PHYSICAL REVIEW A, 80 (6): Art. No DEC 2009 Nonclassical Light Source Delay Line Detection Target Noise

10/20/ Loss in Quantum Sensors SD Huver, CF Wildfeuer, JP Dowling, Phys. Rev. A 78 # DEC 2008 N00N Generator Detector Lost photons LaLa LbLb Visibility: Sensitivity: SNL--- HL— N00N No Loss — N00N 3dB Loss ---

Super-Lossitivity Gilbert, G; Hamrick, M; Weinstein, YS; JOSA B 25 (8): AUG dB Loss, Visibility & Slope — Super Beer’s Law! N=1 (classical) N=5 (N00N)

Loss in Quantum Sensors S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # DEC 2008 N00N Generator Detector Lost photons LaLa LbLb Q: Why do N00N States Do Poorly in the Presence of Loss? A: Single Photon Loss = Complete “Which Path” Information! A B Gremlin

Towards A Realistic Quantum Sensor S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # DEC 2008 Try other detection scheme and states! M&M Visibility M&M Generator Detector Lost photons LaLa LbLb M&M state: N00N Visibility M&M’ Adds Decoy Photons

Try other detection scheme and states! M&M Generator Detector Lost photons LaLa LbLb M&M state: M&M State — N00N State --- M&M HL — M&M SNL --- N00N SNL --- A Few Photons Lost Does Not Give Complete “Which Path” Towards A Realistic Quantum Sensor S. Huver, C. F. Wildfeuer, J.P. Dowling, Phys. Rev. A 78 # DEC 2008

Optimization of Quantum Interferometric Metrological Sensors In the Presence of Photon Loss PHYSICAL REVIEW A, 80 (6): Art. No DEC 2009 Tae-Woo Lee, Sean D. Huver, Hwang Lee, Lev Kaplan, Steven B. McCracken, Changjun Min, Dmitry B. Uskov, Christoph F. Wildfeuer, Georgios Veronis, Jonathan P. Dowling We optimize two-mode, entangled, number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number.

Lossy State Comparison PHYSICAL REVIEW A, 80 (6): Art. No DEC 2009 Here we take the optimal state, outputted by the code, at each loss level and project it on to one of three know states, NOON, M&M, and Generalized Coherent. The conclusion from this plot is that The optimal states found by the computer code are N00N states for very low loss, M&M states for intermediate loss, and generalized coherent states for high loss. This graph supports the assertion that a Type-II sensor with coherent light but a non-classical number resolving detection scheme is optimal for very high loss.

Super-Resolution at the Shot-Noise Limit with Coherent States and Photon-Number-Resolving Detectors (submitted to JOSA B) Yang Gao, Christoph F. Wildfeuer, Petr M. Anisimov, Hwang Lee, Jonathan P. Dowling There has been much recent interest in quantum optical interferometry for applications to metrology, sub-wavelength imaging, and remote sensing, such as in quantum laser radar (LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum state of light, so that for high-photon loss the optimal strategy is to transmit coherent states of light, which suffer no worse loss than the Beer law for classical optical attenuation, and which provides sensitivity at the shot-noise limit. This approach leaves open the question -- what is the optimal detection scheme for such states in order to provide the best possible resolution? We show that coherent light coupled with photon number resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot-noise in terms of the detected photon power. Classical Quantum  Waves are Coherent! Magic Detector!

Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit (submitted to Phys. Rev. Lett.) Petr M. Anisimov, Gretchen M. Raterman, Aravind Chiruvelli, William N. Plick, Sean D. Huver, Hwang Lee, Jonathan P. Dowling We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum ( photons on average). We show that super-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/. SNL HL TMSV & Parity

Resolution and Sensitivity of a Fabry-perot Interferometer With a Photon-number-resolving Detector Phys. Rev. A (2009) Christoph F. Wildfeuer, Aaron J. Pearlman, Jun Chen, Jingyun Fan, Alan Migdall, Jonathan P. Dowling With photon-number resolving detectors, we show compression of interference fringes with increasing photon numbers for a Fabry-Perot interferometer. This feature provides a higher precision in determining the position of the interference maxima compared to a classical detection strategy. We also theoretically show supersensitivity if N-photon states are sent into the interferometer and a photon-number resolving measurement is performed.

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Heisenberg Limited Magnetometer: N00N in Microwave Photons Transferred into N00N in SQUID Qubits and Reverse! Entangled “SQUID” Qubit Magnetometer A Guillaume & JPD, Physical Review A Rapid 73 (4): Art. No APR 2006.

Quantum Magnetometer Arrays Can Detect Submarines From Orbit

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer

Microwave to Optical Transducer KT Kapale and JPD, in preparation

Outline 1.Nonlinear Optics vs. Projective Measurements 2.Quantum Imaging vs. Precision Measurements 3.Showdown at High N00N! 4.Efficient N00N Generators 5.Mitigating Photon Loss 6. Microwave-Entangled SQUID Qubits 7. Microwave to Optical Photon Transducer