University aris-saclay - IQUS Optical Quantum Engineering: From fundamentals to applications hilippe Grangier, Institut d Optique, CNRS, Ecole olytechnique. Lecture (7 March, 9:5-0:45) : Qubits, entanglement and Bell s inequalities. Lecture 2 (4 March,:00-2:0) : From QND measurements to quantum gates and quantum information. Lecture (2 March, 9:5-0:45) : Quantum cryptography with discrete and continuous variables. Lecture 4 (28 March, :0-2:0) : Non-Gaussian quantum optics and optical quantum networks. Content of the Lecture art - Quantum optics with discrete and continuous variables. Homodyne detection and quantum tomography.2 Generating non-gaussian Wigner functions : kittens, cats and beyond art 2 - Towards optical quantum networks 2. Entanglement, teleportation, and quantum repeaters 2.2 Some experimental achievements art - A close look to a nice single photon. Single photons: from old times to recent ones.2 Experimental perspectives Quantum description of light Quantum description of light Discrete hotons Continuous Wave Discrete hotons Continuous Wave arameters : Number of photons n Amplitude & hase (polar) Quadratures & (cartesian) A single "mode" of the quantized electromagnetic field (a plane wave, or a "Fourier transform limited" pulse) is described as a Destruction operators a Creation operators a + Number operator N = a + a φ α quantized harmonic oscillator : operators a, a +, N = a + a, etc... = (a + a + )/ 2 = (a - a + )/i 2 [, ] = i
Quantum description of light Quantum description of light Discrete hotons Continuous Wave Discrete hotons Continuous Wave arameters : Number & Coherence Amplitude & hase (polar) Quadratures & (cartesian) arameters : Number & Coherence Amplitude & hase (polar) Quadratures & (cartesian) Representation: Density matrix Wigner function W(,) Representation: Density matrix Wigner function W(,) ρ 0,0 ρ 0, ρ 0,2 ρ,0 ρ, ρ,2 ρ = ρ 2,0 ρ 2, ρ 2,2 Coherences < n ρ m > = (a + a + )/ 2 = (a - a + )/i 2 [, ] = i Measurement : Counting : AD, VLC, TES... Demodulation : Homodyne detection Local Oscillator Quantum state Heisenberg :. /2 measurement of both and measurement of θ = cosθ + sinθ Interference, then subtraction of photocurrents : V - V 2 E OL E EQ (θ) θ = cosθ + sinθ Homodyne detection I = E LO + E S 2 /2 = { E LO 2 + E S 2 + E LO (E S e - i θ LO + E S * e i θlo ) }/2 I 2 = E LO - E S 2 /2 = { E LO 2 + E S 2 - E LO (E S e i θlo + E S * e i θlo ) }/2 I - I 2 = E LO (E S e i θlo + E S * e i θlo ) = E LO (E S + E S *) = E LO (E S - E S *)/i and do not commute : Heisenberg relation V() V() N 0 2 Signal Field E S Local Oscillator Field E LO (classical) => n LO ( a + a + ) meas. (θ LO = 0) => n LO ( a - a + )/i meas. (θ LO = π/2) 50/50 BS hotodiode + Low-noise - amplifier hase control : Measurement of or Homodyne detection, Wigner Function and Quantum Tomography ^ LO ^ 2. θ θ Quasiprobability density : Wigner function W(,) Marginals of W(, ) => robability distributions (θ) robability distributions (θ) => W(, ) (quantum tomography) W(,)
Homodyne detection, Wigner Function and Quantum Tomography Quantum description of light Discrete hotons Continuous Wave arameters : Representation: Number & Coherence Density matrix Amplitude & hase (polar) Quadratures & (cartesian) Wigner function W(,) Measurement : Counting : AD, VLC, TES... Demodulation : Homodyne detection ^ LO ^ 2. θ θ Squeezed State : Gaussian Wigner Functions «Simple» states Fock states (number states) Sources : - Single atoms or molecules - NV centers in diamond - Quantum dots - arametric fluorescence... Sources : Lasers : coherent states α Gaussian states φ. /2 Non-linear media : squeezed states α φ Basic question : Non-Gaussian States amplitude & phase? Consider a single photon : can we speak about its quadratures &? Single mode light field hotons n photon state Harmonic oscillator Quanta of excitation n th eigenstate robability n () robability Ψ n (x) 2 Basic question : Can the Wigner function of a Fock state n = ( with all projections have zero value at origin ) be positive everywhere? Non-Gaussian States Consider a single photon : can we measure its amplitude & phase? quadratures &? Ψ n (x) 2 n=0 photons n= photon n=2 photons NO! The Wigner function must be negative It is not a classical statistical distribution! Hudson-iquet theorem : for a pure state W is non-positive iff it is non-gaussian Many interesting properties for quantum information processing
Wigner function of a single photon state? (Fock state n = ) Gaussian Wigner Function where ˆρ = and N 0 is the variance of the vacuum noise : Coherent state α Squeezed state One may have N 0 =! / 2, N 0 = /2 (theorists), N 0 = (experimentalists) Non-Gaussian Wigner Function Using the wave function of the n = state : one gets finally : Fock state n= Cat state α + α. Grangier, "Make It Quantum and Continuous", Science (erspective) 2, (20) Content of the Lecture art - Quantum optics with discrete and continuous variables. Homodyne detection and quantum tomography.2 Generating non-gaussian Wigner functions : kittens, cats and beyond art 2 - Towards optical quantum networks 2. Entanglement, teleportation, and quantum repeaters 2.2 Some experimental achievements art - A close look to a nice single photon. Single photons: from old times to recent ones.2 Experimental perspectives Small sample, many more papers!
. = «Schrödinger s Cat» state Classical object in a quantum superposition of distinguishable states Quasi - classical state in quantum optics : coherent state α Coherent state Resource for quantum information processing Model system to study decoherence Schrödinger cat state odd cat = c o ( ) even cat = c e ( + ) Wigner function of a Schrödinger cat state Fock state n How to create a Schrödinger s cat? Suggestion by Hyunseok Jeong, proofs by Alexei Ourjoumtsev : Vacuum 50/50 0 Homodyne Detection squeezed cat! p < ε : OK p > ε : reject For n the fidelity of the conditional state with a Squeezed Cat state is F 99% Squeezed Cat state Size : Same arity as n : Squeezed by : α 2 = n θ = n*π db Another hint The rebirth of the cat H n (p) e p2 /2 H 0 (p 0 ) e p2 0 /2! Make a n-photon Fock state Fock state n Vacuum 0 Homodyne Detection repared state ψ p=0 : OK H n ((p p 0 )/ p 2) e (p p 0) 2 /4 e (p+p 0) 2 /4 = For a large n (n ) H n ((p p 0 )/ p 2) e (p2 +p 2 0 )/2 H n (p/ p 2) e p2 /2! x n e x2 /2 Fourier transform Cat state squeezed by db 50/50 BS : n/2 photons transmitted R=50% 0 << OK Homodyne measurement - hase dependence - arity measurement : 0 =0 2k+ = 0 Reflected : even number of photons Transmitted : same parity as n Squeezed cat state (from n=2) = 2/ 2 - / 0
Femtosecond Ti-S laser pulses 80 fs, 40 nj, rep rate 800 khz Experimental Set-up Resource : Two-hoton Fock States SHG Femtosecond laser OA n=0 Homodyne AD OA AD n= AD2 V 2 -V Time Homodyne detection n=2 Experimental Wigner function (corrected for homodyne losses) hys. Rev. Lett 96, 260 (2006) Squeezed Cat State Generation Experimental Wigner function A. Ourjoumtsev et al, Nature 448, 784, 6 august 2007 Two-hoton State reparation Homodyne Conditional reparation : Success robability = 7.5% Tomographic analysis of the produced state Wigner function of the prepared state Reconstructed with a Maximal-Likelihood algorithm Corrected for the losses of the final homodyne detection.?? JL Basdevant 2 Leçons de Mécanique Quantique Bigger cats : NIST (Gerrits, -photon subtraction), ENS (Haroche, microwave cavity QED), UCSB
Schrödinger cats with with continuous light beams Groups of A. Furusawa (Tokyo), M. Sasaki (Tokyo), E. olzik (Copenhagen), U. Andersen (Copenhagen), J. Laurat (aris),... Other methods for bigger / better cats... J. Etesse, M. Bouillard, B. Kanseri, and R. Tualle-Brouri, hys. Rev. Lett. 4, 9602 (205) Conditionnally prepared state : N. Lee et al, Science 2, 0 (20) Fidelity 99% with an even cat state with α =.6, squeezed by s =.52 along x : Reconstructed Wigner function, corrected for losses. Schrödinger cats with microwaves in superconducting cavities Content of the Lecture Some examples... Serge Haroche group (cacity QED, aris) Nature 455, 50 (2008) art - Quantum optics with discrete and continuous variables. Homodyne detection and quantum tomography.2 Generating non-gaussian Wigner functions : kittens, cats and beyond Rob Schoelkopf group (circuit QED, Yale) Atom field interaction: e + g Nature 495, 205 (20) e + g Cats with 2, or 4 "legs"... Measure atom in state + 2 John Martinis group (circuit QED, Santa Barbara) Nature 459, 546 (2009) Quantum state synthetizer art 2 - Towards optical quantum networks 2. Entanglement, teleportation, and quantum repeaters 2.2 Some experimental achievements art - A close look to a nice single photon. Single photons: from old times to recent ones.2 Experimental perspectives
Long distance quantum communications How to fight against line losses? Long distance quantum communications How to fight against line losses? Amplification a + + b - G G G a + + b - Amplification a + + b - G G G a + + b -. Exchange of entangled states ψ B Quantum relays ψ B ψ B ψ B a + + b - Long distance quantum communications How to fight against line losses? Amplification a + + b - G G G a + + b - Long distance quantum communications How to fight against line losses? Amplification a + + b - G G G a + + b -. Exchange of entangled states ψ B. Exchange of entangled states ψ B 2. Entanglement distillation a + + b - 2. Entanglement distillation a + + b - M M
Long distance quantum communications How to fight against line losses? Amplification a + + b - G G G a + + b - Long distance quantum communications How to fight against line losses? Amplification a + + b - G G G a + + b -. Exchange of entangled states ψ B. Exchange of entangled states ψ B 2. Entanglement distillation a + + b - M M 2. Entanglement distillation a + + b - a + + b -. Entanglement swapping M M. Entanglement swapping M M 4. Quantum teleportation One needs to : * distribute (many) entangled states * store them (quantum memories) * process them (distillation) Quantum Teleportation C. H. Bennett et al, 99 QIC / S4 Quantum Teleportation C. H. Bennett et al, 99 QIC / S4 Step state Ψ 2 air of entangled qubits Qubit in an unknown state Ψ * one cannot «read» it * one cannot duplicate it («non-cloning» theorem) * but it is possible to «teleport» it? ( = to make a remote copy, destroying the original) The answer is yes! Step 2 Step Joint Bell measurement 2 result Operation on qubit (4 possible operations) Qubit is now in state Ψ
Quantum Teleportation Quantum Teleportation with hotons Ψ 2 = = + Φ Φ Φ Φ Ψ Ψ + 2 2 + 2 2 2 ( α 0 + β ) ( α 0 β ) ( α + β 0 ) + ( α β 0 ), + + hoton to be teleported Classical transmission of information air of entangled photons Initial state Φ = α 0 + β The shared entangled pair Φ + 2 = ( 2 0 2 0 + 2 ) where Φ ± 2 = ( 2 0 0 2 ± 2 ) Ψ ± 2 = ( 2 0 2 ± 0 2 ) Bennett, Brassard, Crepeau, Josza, eres, Wooters 99 Experiment realized in 997 by the group of Anton Zeilinger, Austria Drawback of this scheme: only one Bell state out of 4 can be identified Entanglement swapping VICTOR Ψ Φ Ψ+ Ψ Φ+ Ψ ALICE Bell-State Analyzer - Two entangled pairs 2 and 4 - Bell measurement by Alice on photons 2 and - hoton and 4 become entangled without having ever met! - This can be checked using a Bell test between and 4. Innsbruck University + NIST Boulder 2004 BOB olarization Analysis Entangled hoton Source I olarization Analysis 2 4 Entangled II hoton Source - With photons + beam-splitter + counters the Bell state analysis remains incomplete
How to produce CV entangled beams?. Combine two orthogonally squeezed beams C, C C squeezed D, D D squeezed A, A entangled! B, B 2. Use a Non-degenerate Optical arametric Amplifier (NOA) Vacuum state Vacuum state ump beam NOA (khi-2 crystal) C = ( A + B )/ 2 C = ( A + B )/ 2 D = ( A - B )/ 2 D = ( A - B )/ 2 OK! A, A entangled! B, B Signal input LOs (π/2) Alice measures A and A on half an ER beam, mixed with an unknown coherent state Quantum teleportation of coherent states 50-50 BS Alice ER source Measured A and A Experiments : A. Furusawa et al, Science 282, 706 (998) W. Bowen et al, hys. Rev. A 67, 0202 (200) T.C. Zhang et al, hys. Rev. A 67, 0802 (200) The state received by Bob is «re-created» in the initial coherent state Modulators Bob Quantum teleportation of cat states The state received by Bob is «re-created» in the initial cat state (with some loss in fidelity : 0.75 => 0.45 ) Experiment : N. Lee et al, Science 2, 0- (20). Remark : the initial squeezed state is CW, not pulsed! Alice measures A and A on half an ER beam, mixed with a cat state (obtained from photon subtraction)
Content of the Lecture art - Quantum optics with discrete and continuous variables. Homodyne detection and quantum tomography.2 Generating non-gaussian Wigner functions : kittens, cats and beyond art 2 - Towards optical quantum networks 2. Entanglement, teleportation, and quantum repeaters 2.2 Some experimental achievements L.M. Duan, M.D. Lukin, J.I. Cirac, and. Zoller, Nature 44, 4 (200) Single hoton from a single polariton 87 Rb MOT: Density ρ 4 0 0 cm - Cooperativity C 200 art - A close look to a nice single photon. Single photons: from old times to recent ones.2 Experimental perspectives E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) Single hoton from a single polariton Single hoton from a single polariton E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204)
Single hoton from a single polariton Single hoton from a single polariton E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) Single hoton from a single polariton Single hoton from a single polariton Overall detection efficiency : 57% Corrected for detection efficiency Overall extraction efficiency : 82% E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204)
Looking at the Blinking hoton Single hoton from a single polariton Time (ns) 0 0 Density plot of homodyne data, time averaged with the expected "photon wave packet" duration Quantum memory effect : the memory time ( µs) is limited by motional decoherence due to finite temperature (50 µk) 200 400 blink! 600 η = Doppler (t) Coop Read umping Mode Cav 0.94 0.97 0.96 0.965 0.97 = 0.82 : ok! E. Bimbard et al, ariv:0.228 (20), RL 2, 060 (204) hotonic Controlled-hase Gates Through Rydberg Blockade in Optical Cavities S. Das, A. Grankin, I. Iakoupov, E. Brion, J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev,. Grangier, A. S. Sørensen, arxiv:506.0400 Dual-rail photonic gate (used here for a Bell measurement) First photon Second photon { { Significant increase in the swap efficiency in a quantum repeater scheme, compared to linear quantum gates (with efficiency bounded to 50%) => secret bit rate increased by about one order of magnitude. See also : J. Borregaard et al, ariv:504.070 (205). (Temporary) Conclusion Many potential uses for Quantum Continuous Variables * Conditional preparation of «squeezed» non-gaussian pulses / cats * Big family of phase-dependant states with negative Wigner functions! * Many new experimental results... * Quantum cryptography * Coherent states protocols using reverse reconciliation, secure against any (gaussian or non-gaussian) collective attack * Working fine in optical fibers @550 nm (SECOQC, SeQureNet...) * Quantum repeaters * Basic elements do work in proof-of-principle experiments... * but not well enough yet to get a whole system with acceptable efficiency. * So more work is needed, both on theory and implementations...