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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 One may have N = ! / 2 , N = 1/2 (theorists), N = 1 (experimentalists) Non-Gaussian 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 Schrödinger s 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 ( ) For n ≥ 3 the fidelity of the | odd cat⇥ o | ⇥ | ⇥ conditional state with a Squeezed Cat state is F ≥ 99% ⇥even cat = ce( + ) | ⇥ | ⇥ | ⇥ 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 〉 rep rate pulses Femtosecond Ti-S laser Tomographic analysis of the produced state analysis Tomographic 180 fs 800 kHz , Success ProbabilitySuccess = 7.5% Preparation : Homodyne Conditional 40 nJ Squeezed Cat State Generation
Homodyne OPA ,
SHG
APD
Experimental Set-up
V2-V1 Time
Preparation Preparation Two-Photon State
Homodyne detection Bigger cats : NIST (Gerrits, 3-photon subtraction), (Haroche, ENS microwave cavity UCSB… QED), laser Femtosecond Experimental Wigner function Corrected for the losses of the final homodyne detection. detection. homodyne of for the final the losses Corrected A. OurjoumtsevA. et al, Nature Resource : Two-PhotonFockStates Reconstructed with a Maximal-Likelihood algorithm algorithm a Maximal-Likelihood with Reconstructed OPA APD2 APD2 Wigner function of the prepared state of the prepared function Wigner APD1 APD1 n=2 n=1 n=0 448 , 784,16august 2007 Phys. Rev. Lett (corrected for homodyne (corrected losses) Experimental WignerExperimental function
96,
213601 (2006) 213601 ? ?
JL Basdevant 12 Leçonsde Mécanique Quantique
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: e + 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 | ⌅ | ⌅ + 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. Exchange 1. Exchange |ψ 〉 |ψ 〉 of entangled B of entangled B states states
2. Entanglement a|+〉 + b|-〉 2. Entanglement a|+〉 + b|-〉 M M distillation distillation
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 1. Exchange |ψ 〉 |ψ 〉 of entangled B of entangled B states states
2. Entanglement a|+〉 + b|-〉 M M 2. Entanglement a|+〉 + b|-〉 a|+〉 + b|-〉 distillation distillation
3. Entanglement 3. Entanglement swapping M M swapping M M
4. Quantum teleportation
One needs to : * distribute (many) entangled states * store them (quantum memories) * process them (distillation)
Quantum Teleportation Quantum Teleportation C. H. Bennett et al, 1993 C. H. Bennett et al, 1993 QIPC / S4P QIPC / S4P state Ψ Step Pair of entangled 1 1 1 2 3 qubits
Qubit in an unknown state Ψ * one cannot « read » it Step Joint Bell * one cannot duplicate it 1 2 3 2 measurement (« non-cloning » theorem) * but it is possible to « teleport » it ? result ( = to make a remote copy, 3 Step destroying the original) 3 Operation on qubit 3 The answer is yes ! (4 possible operations) Qubit 3 is now in state Ψ Quantum Teleportation Quantum Teleportation with Photons
+ | Ψ〉123 = | Φ〉1⊗ | Φ 〉 23 Classical transmission + of information = | Φ 〉12 ⊗ (α | 0〉3 + β |1〉3 )+ Photon to be − teleported | Φ 〉12 ⊗ (α | 0〉3 − β |1〉3 )+ + | Ψ 〉12 ⊗ (α |1〉3 + β | 0〉3 )+ Pair of entangled − | Ψ 〉12 ⊗ (α |1〉3 − β |01 〉3 ), photons where Initial state ± 1 | Φ 〉12 = (| 0〉1 | 0〉2 ± |1〉1 |1〉2 ) | Φ〉 = α | 0〉 + β |1〉 1 1 1 2 Experiment realized in 1997 by the ± 1 group of Anton Zeilinger, Austria The shared entangled pair | Ψ 〉12 = (| 0〉1 |1〉2 ± |1〉1 | 0〉2 ) 2 + 1 | Φ 〉23 = (| 0〉2 | 0〉3 + |1〉2 |1〉3 ) Drawback of this scheme: only one Bell state out of 4 can be identified 2 Bennett, Brassard, Crepeau, Josza, Peres, Wooters 1993
Entanglement swapping Innsbruck University + VICTOR - Two entangled pairs 12 and 34 NIST Ψ− Φ− Boulder Ψ+ - Bell measurement by Alice on Ψ− 2004 photons 2 and 3 Φ+ Ψ− - Photon 1 and 4 become entangled ALICE without having ever met ! Bell-State Analyzer - This can be checked using a Bell test between 1 and 4.
BOB Polarization Polarization - With photons + beam-splitter + Analysis Analysis counters the Bell state analysis remains incomplete 1 2 3 4 Entangled Entangled Photon I II Photon Source Source How to produce CV entangled beams ? Quantum teleportation of coherent states Signal 1. Combine two orthogonally squeezed beams P input The state received by Bob P is « re-created » in the XC , PC X X squeezed X , P initial coherent state X C A A XC = ( XA + XB )/√2 EPR PC = ( PA + PB )/√2 Modulators source entangled ! XD = ( XA - XB )/√2 50-50 BS P P = ( P - P )/√2 Bob D A B LOs Alice X XD , PD X , P B B (π/2) PD squeezed OK !
Alice measures XA and PA on 2. Use a Non-degenerate Optical Parametric Amplifier (NOPA) half an EPR beam, mixed with Measured XA and PA an unknown coherent state P XA , PA Vacuum state NOPA X Pump Experiments : (khi-2 entangled ! beam A. Furusawa et al, Science 282, 706 (1998) P crystal) W. Bowen et al, Phys. Rev. A 67, 032302 (2003) Vacuum state X , P B B T.C. Zhang et al, Phys. Rev. A 67, 033802 (2003) X
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 332, 330-333 (2011). Remark : the initial squeezed state is CW, not pulsed !
Alice measures XA and PA on half an EPR beam, mixed with a cat state (obtained from photon subtraction) Content of the Lecture Single Photon from a single polariton L.M. Duan, M.D. Lukin, (DLCZ protocol) J.I. Cirac, and P. Zoller, Nature 414, 413 (2001) Part 1 - Quantum optics with discrete and continuous variables 1.1 Homodyne detection and quantum tomography 87Rb MOT: 1.2 Generating non-Gaussian Wigner functions : kittens, cats and beyond Density ρ ≈ 4 1010 cm-3 Part 2 - Towards optical quantum networks Cooperativity
2.1 Entanglement, teleportation, and quantum repeaters C ≈ 200 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
E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014)
Single Photon from a single polariton Single Photon from a single polariton (DLCZ protocol) (DLCZ protocol)
E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014) E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014) Single Photon from a Single Photon from a single polariton single polariton (DLCZ protocol) (DLCZ protocol)
E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014) E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014)
Single Photon from a Single Photon from a single polariton single polariton (DLCZ protocol) (DLCZ protocol)
Corrected for detection Overall efficiency detection Overall extraction efficiency : efficiency : 57% 82%
E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014) E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014) Looking at the Blinking Photon Single Photon from a single polariton (DLCZ protocol)
Time Quantum memory effect : the memory time (1 µs) is limited by 0 X motional decoherence due to finite temperature (50 µK) (ns) Density plot of homodyne data, time averaged with the expected 0 "photon wave packet" duration
200
400 blink !
η = PDoppler (t) × PCoop × PRead × PPumping × PMode × PCav 600 0.94 × 0.97 × 0.96 × 0.965 × 0.97 = 0.82 : ok ! E. Bimbard et al, arXiv:1310.1228 (2013), PRL 112, 033601 (2014)
Photonic Controlled-Phase Gates (Temporary) Conclusion Through Rydberg Blockade in Optical Cavities
S. Das, A. Grankin, I. Iakoupov, E. Brion, Many potential uses for Quantum Continuous Variables… J. Borregaard, R. Boddeda, I. Usmani, A. Ourjoumtsev, P. Grangier, A. S. Sørensen, arxiv:1506.04300 * Conditional preparation of « squeezed » non-gaussian pulses / cats
Dual-rail photonic gate (used here for a Bell measurement) * Big family of phase-dependant states with negative Wigner functions ! * Many new experimental results...
First * Quantum cryptography photon { * Coherent states protocols using reverse reconciliation, secure against any (gaussian or non-gaussian) collective attack Second photon { * Working fine in optical fibers @1550 nm (SECOQC, SeQureNet...) * Quantum repeaters Significant increase in the swap efficiency in a quantum repeater scheme, * Basic elements do work in proof-of-principle experiments... compared to linear quantum gates (with efficiency bounded to 50%) * … but not well enough yet to get a whole system with acceptable efficiency. => secret bit rate increased by about one order of magnitude. * So more work is needed, both on theory and implementations... See also : J. Borregaard et al, arXiv:1504.03703 (2015).