Quantum description of light. Quantum description of light. Content of the Lecture

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1 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

2 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(,)

3 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

4 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!

5 . = «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

6 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

7 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

8 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

9 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

10 Quantum Teleportation Quantum Teleportation with hotons Ψ 2 = = + Φ Φ Φ Φ Ψ Ψ ( 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 = ( ) where Φ ± 2 = ( ± 2 ) Ψ ± 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

11 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 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)

12 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 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)

13 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)

14 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) blink! 600 η = Doppler (t) Coop Read umping Mode Cav = 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: 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: (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 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...