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Grangier2017 - Cours4.Pdf 1 University Paris-Saclay - IQUPS Content of the Lecture Optical Quantum Engineering: From fundamentals to applications Philippe Grangier, Part 1 - Quantum optics with discrete and continuous variables Institut d’Optique, CNRS, Ecole Polytechnique. 1.1 Homodyne detection and quantum tomography 1.2 Generating non-Gaussian Wigner functions : kittens, cats and beyond Lecture 1 (7 March, 9:15-10:45) : • Qubits, entanglement and Bell’s inequalities. Part 2 - Towards optical quantum networks Lecture 2 (14 March,11:00-12:30) : 2.1 Entanglement, teleportation, and quantum repeaters • From QND measurements to quantum gates and quantum information. 2.2 Some experimental achievements Lecture 3 (21 March, 9:15-10:45) : Part 3 - A close look to a nice single photon • Quantum cryptography with discrete and continuous variables. 3.1 Single photons: from old times to recent ones 3.2 Experimental perspectives Lecture 4 (28 March, 11:10-12:30) : • Non-Gaussian quantum optics and optical quantum networks. Quantum description of light Quantum description of light Discrete Photons Continuous Wave Discrete Photons Continuous Wave Parameters : Number of photons n Amplitude & Phase (polar) Quadratures X & P (cartesian) Destruction operators a A single "mode" of the α Creation operators a+ quantized electromagnetic field Number operator N = a+a φ (a plane wave, or a "Fourier transform limited" pulse) is described as a quantized harmonic oscillator : X = (a + a+)/√2 P = (a - a+)/i√2 operators a, a+, N = a+ a, etc... [ X, P ] = i Quantum description of light Quantum description of light Discrete Photons Continuous Wave Discrete Photons Continuous Wave Parameters : Number & Coherence Amplitude & Phase (polar) Parameters : Number & Coherence Amplitude & Phase (polar) Quadratures X & P (cartesian) Quadratures X & P (cartesian) Representation: Density matrix Wigner function W(X,P) Representation: Density matrix Wigner function W(X,P) ρ0,0 ρ0,1 ρ0,2 … Measurement : Counting : APD, VLPC, TES... Demodulation : Homodyne detection ρ ρ ρ … Local ρ = 1,0 1,1 1,2 Oscillator ρ2,0 ρ2,1 ρ2,2 … + Quantum state … … … … X = (a + a )/√2 P = (a - a+)/i√2 [ X, P ] = i Coherences < n | ρ | m > Heisenberg : ∆X. ∆P ≥ 1/2 Interference, then subtraction of measurement of both X and P photocurrents : measurement of Xθ = X cosθ + P sinθ V1 - V2 ∝ EOLEEQ(θ) ∝ Xθ = Xcosθ + Psinθ Homodyne detection Homodyne detection, Wigner Function and Quantum Tomography 2 2 2 - i θ LO * i θLO I1 = |ELO + ES| /2 = { |ELO| + |ES| + |ELO| (ES e + ES e ) }/2 2 2 2 – i θLO * i θLO • Quasiprobability density : I2 = |ELO - ES| /2 = { |ELO| + |ES| - |ELO| (ES e + ES e ) }/2 Wigner function W(X,P) – i θLO * i θLO I1 - I2 = |ELO| (ES e + ES e ) • Marginals of W(X, P) = |E | (E + E *) + => Probability distributions (Xθ) LO S S => √nLO ( a + a ) X meas. (θLO = 0) P • Probability distributions (Xθ) + P = |ELO| (ES - ES*)/i => √n ( a - a )/i P meas. (θ = π/2) LO LO => W(X, P) (quantum tomography) X and P do not commute : P ^ Heisenberg relation XLO W(X,P) 50/50 + Low-noise V(X) V(P) ≥ N 2 0 BS - amplifier θ 2.∆^X Signal Field E θ S Photodiode Phase control : P Measurement of X or P X Local Oscillator Field ELO (classical) X Homodyne detection, Quantum description of light Wigner Function and Quantum Tomography Discrete Photons Continuous Wave Parameters : Number & Coherence Amplitude & Phase (polar) Quadratures X & P (cartesian) Representation: Density matrix Wigner function W(X,P) Measurement : Counting : APD, VLPC, TES... Demodulation : Homodyne detection « Simple » Fock states Gaussian states states (number states) Sources : Sources : P ^ - Single atoms or molecules X Lasers : Non-linear media : LO - NV centers in diamond coherent states squeezed states - Quantum dots θ - Parametric fluorescence P P ^ .... 2.∆Xθ α α φ φ Squeezed State : X X X Gaussian Wigner Functions ∆X.∆P ≥ 1/2 Non-Gaussian States Non-Gaussian States Basic question : Basic question : amplitude & phase? Consider a single photon : can we measure its quadratures X & P ? amplitude & phase? Consider a single photon : can we speak about its quadratures X & P ? Can the Wigner function of Single mode light field Harmonic oscillator a Fock state n = 1 ( with all projections have Photons Quanta of excitation zero value at origin ) n photon state nth eigenstate be positive everywhere ? 2 Probability Pn(X) Probability |Ψn(x)| n=0 photons n=1 photon n=2 photons NO ! The Wigner function must be negative It is not a classical statistical distribution ! |Ψ (x)|2 n Hudson-Piquet 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 = 1) Gaussian Wigner Function where ρ ˆ = 1 1 and N0 is the variance of the vacuum noise : Coherent Squeezed state |α〉 state Non-Gaussian One may have N = ! / 2 , N = 1/2 (theorists), N = 1 (experimentalists) 0 0 0 Wigner Function Using the wave function of the n = 1 state : one gets finally : Fock Cat state state |n=1〉 |α〉 + |α〉 P. Grangier, "Make It Quantum and Continuous", Science (Perspective) 332, 313 (2011) Content of the Lecture Part 1 - Quantum optics with discrete and continuous variables 1.1 Homodyne detection and quantum tomography 1.2 Generating non-Gaussian Wigner functions : kittens, cats and beyond Part 2 - Towards optical quantum networks 2.1 Entanglement, teleportation, and quantum repeaters 2.2 Some experimental achievements Part 3 - A close look to a nice single photon 3.1 Single photons: from old times to recent ones 3.2 Experimental perspectives Small sample, many more papers ! « Schrödinger’s Cat » state How to create a Schrödingers cat ? Suggestion by Hyunseok Jeong, proofs by Alexei Ourjoumtsev : • Classical object in a quantum superposition of distinguishable states • “Quasi - classical” state in quantum optics : coherent state | α 〉 Vacuum Squeezed Cat state |0〉 50/50 |squeezed cat〉 ! Fock |n〉 P ∆X.∆P = 1 P state |p|< ε : OK Homodyne |p|> ε : reject X Detection X Coherent state Schrödinger cat state ⇥ = c ( α α ) For n ≥ 3 the fidelity of the | odd cat⇥ o | ⇥| − ⇥ conditional state with a Squeezed Cat state is F ≥ 99% ⇥even cat = ce( α + α ) | ⇥ | ⇥ | − ⇥ 2 • Resource for quantum information processing Wigner function of a Size : α = n Schrödinger cat state Same Parity as n : θ = n*π • Model system to study decoherence Squeezed by : 3 dB Another hint… The rebirth of the cat p2/2 p2/2 Hn(p) e− H0(p0) e− 0 • Make a n-photon Fock state ⌦ 2 2 ! (p p0) /4 (p+p0) /4 Vacuum Hn((p p0)/p2) e− − e− = • 50/50 BS : ≈ n/2 photons transmitted |0〉 − (p2+p2)/2 Fock | n 〉 |ψ 〉 Hn((p p0)/p2) e− 0 R=50% state − p=0 : p2/2 n x2/2 Homodyne OK Hn(p/p2) e− x e− Detection Fourier transform! |P0|<<1 ⇒ OK Prepared state Cat state squeezed by 3 dB Homodyne measurement - Phase dependence For a large - Parity measurement : 〈P =0|2k+1〉 = 0 n (n ≥ 3) 0 Reflected : even number of photons Transmitted : same parity as n Squeezed cat state (from n=2) = √2/3 | 2 〉 - √1/3 | 0 〉 Experimental Set-up Resource : Two-Photon Fock States Femtosecond Ti-S laser pulses 180 fs , 40 nJ, rep rate 800 kHz Femtosecond laser SHG OPA n=0 APD OPA APD1 n=1 Homodyne APD2 1 -V 2 n=2 Experimental Wigner function V Homodyne detection (corrected for homodyne losses) Phys. Rev. Lett 96, 213601 (2006) Time Squeezed Cat State Generation Experimental Wigner function A. Ourjoumtsev et al, Nature 448, 784, 16 august 2007 Two-Photon State Preparation Homodyne Conditional Preparation : Success Probability = 7.5% ? ? JL Basdevant 12 Leçons de Mécanique Tomographic analysis Wigner function of the prepared state Quantique of the produced state Reconstructed with a Maximal-Likelihood algorithm Corrected for the losses of the final homodyne detection. Bigger cats : NIST (Gerrits, 3-photon subtraction), ENS (Haroche, microwave cavity QED), UCSB… Schrödinger cats with with continuous light beams Other methods for bigger / better cats... J. Etesse, M. Bouillard, B. Kanseri, and R. Tualle-Brouri, Groups of A. Furusawa (Tokyo), M. Sasaki (Tokyo), E. Polzik (Copenhagen), Phys. Rev. Lett. 114, 193602 (2015) U. Andersen (Copenhagen), J. Laurat (Paris), .... Conditionnally prepared state : Fidelity 99% with an even cat state with α = 1.63, squeezed by s = 1.52 along x : N. Lee et al, Science 332, 330 (2011) Reconstructed Wigner function, corrected for losses. Schrödinger cats with microwaves in superconducting cavities Content of the Lecture Some examples... Serge Haroche group Part 1 - Quantum optics with discrete and continuous variables (cacity QED, Paris) Nature 455, 510 (2008) 1.1 Homodyne detection and quantum tomography 1.2 Generating non-Gaussian Wigner functions : kittens, cats and beyond Part 2 - Towards optical quantum networks Rob Schoelkopf group Atom field interaction: α e + α g 2.1 Entanglement, teleportation, and quantum repeaters (circuit QED, Yale) ⇤ | ⌅⇥| ⌅ | − ⌅⇥| ⌅ 2.2 Some experimental achievements Nature 495, 205 (2013) e + g Measure atom in state | ⌅ | ⌅ α + α Cats with 2, 3 or 4 "legs"... ⇧ 2 ⇤ | ⌅ | − ⌅ Part 3 - A close look to a nice single photon 3.1 Single photons: from old times to recent ones John Martinis group 3.2 Experimental perspectives (circuit QED, Santa Barbara) Nature 459, 546 (2009) Quantum state synthetizer Long distance quantum communications Long distance quantum communications How to fight against line losses ? How to fight against line losses ? Amplification a|+〉 + b|-〉 G G G a|+〉 + b|-〉 Amplification a|+〉 + b|-〉 G G G a|+〉 + b|-〉 1. Exchange |ψ 〉 of entangled B Quantum relays states |ψB〉 |ψB〉 |ψB〉 a|+〉 + b|-〉 Long distance quantum communications Long distance quantum communications How to fight against line losses ? How to fight against line losses ? Amplification a|+〉 + b|-〉 G G G a|+〉 + b|-〉 Amplification a|+〉 + b|-〉 G G G a|+〉 + b|-〉 1.
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