Quantum Optics and Quantum Information with Continuous Variables

Transcription

1 Quantum Optics and Quantum Information with Continuous Variables Philippe Grangier Laboratoire Charles Fabry de l'institut d'optique, UMR 8501 du CNRS, Palaiseau, France

2 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography (Gaussian!) 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks (non-gaussian!) 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

3 «!Discrete!» vs «!continuous!» Light Light is : Discrete Photons Continuous Wave We want to know : We describe it with : Density matrix n,m their Number & Coherence its Amplitude & Phase (polar) its Quadratures X & P (cartesian) Wigner function W(X,P) We measure it by : Counting: APD, VLPC, TES... Demodulating : Homodyne Detection Local Oscillator Quantum State V 1 -V 2! X _ =Xcos +Psin «Simple» States Fock States Gaussian States

4 Coherent (Homodyne) Detection, Wigner Function and Quantum Tomography QIPC Quasiprobability density : Wigner function W(X,P) Marginals of W(X, P) => Probability distributions P(X") Probability distributions P(X") => W(X, P) (quantum tomography) P ^ X LO W(X,P) " ^ 2."X " X P X

5 Non-Gaussian States Basic question : amplitude & phase? Consider a single photon : can we speak about its quadratures X & P? Single mode light field Harmonic oscillator Photons Quanta of excitation n photon state n th eigenstate Probability P n (X) Probability # n (x) 2 n=0 photons n=1 photon n=2 photons # n (x) 2

6 Non-Gaussian States Basic question : amplitude & phase? Consider a single photon : can we speak about its quadratures X & P? Can the Wigner function of a Fock state n = 1 ( with all projections have zero value at origin ) be positive everywhere?

7 Non-Gaussian States Basic question : amplitude & phase? Consider a single photon : can we speak about its quadratures X & P? NO! The Wigner function must be negative Classical statistical distribution Hudson-Piquet theorem : for a pure state W is non-positive iff it is non-gaussian Many interesting properties for quantum information processing

8 Wigner function of a single photon state? (Fock state n = 1) where ˆ! = 1 1 and N 0 is the variance of the vacuum noise : One may have N 0 =! / 2, N 0 = 1/2 (theorists), N 0 = 1 (experimentalists) Using the wave function of the n = 1 state : one gets finally :

9 Coherent state $ Squeezed state Fock state n=1 Cat state $ + %$ P. Grangier, "Make It Quantum and Continuous", Science (Perspective) 332, 313 (2011)

10 !"#$$%&#"'$()%"#*+%",-(%'#'(-&%.

11 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

12 «!!Schrödinger s Cat» state Classical object in a quantum superposition of distinguishable states Quasi - classical state in quantum optics : coherent state $ & Q I P C P "X."P = 1 P Coherent state X X Schrödinger cat state Resource for quantum information processing Model system to study decoherence Wigner function of a Schrödinger cat state

13 How to create a Schrödinger s cat? Suggestion by Hyunseok Jeong, calculations by Alexei Ourjoumtsev : Fock state n& Vacuum 50/50 0& Homodyne Detection squeezed cat&! p < ' : OK p > ' : reject Squeezed Cat state For n! 3 the fidelity of the conditional state with a Squeezed Cat state is F! 99% Size : Same Parity as n : Squeezed by : $ 2 = n " = n*( 3 db

14 Experimental Set-up Femtosecond Ti-S laser pulses 180 fs, 40 nj, rep rate 800 khz Q I P C SHG OPA Homodyne APD V 2 -V 1 Time

15 Resource : Two-Photon Fock States Femtosecond laser n=0 OPA APD1 n=1 APD2 Homodyne detection n=2 Experimental Wigner function (corrected for homodyne losses) Phys. Rev. Lett 96, (2006)

16 Squeezed Cat State Generation Homodyne Conditional Preparation : Success Probability = 7.5% Two-Photon State Preparation Tomographic analysis of the produced state

17 Experimental Wigner function A. Ourjoumtsev et al, Nature 448, 784 (2007) Wigner function of the prepared state Reconstructed with a Maximal-Likelihood algorithm Corrected for the losses of the final homodyne detection.?? JL Basdevant 12 Leçons de Mécanique Quantique Bigger cats : NIST (Gerrits, 3-photon subtraction), ENS (Haroche, microwave cavity QED), UCSB

18 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

19 Coherent States Quantum Key Distribution P X * Essential feature : quantum channel with non-commuting quantum observables -> not restricted to single photon polarization or phase! -> Design of Continuous-Variable QKD protocols where : * The non-commuting observables are the quadrature operators X and P * The transmitted light contains weak coherent pulses (about 10 photons) with a gaussian modulation of amplitude and phase * The detection is made using shot-noise limited homodyne detection

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21 QKD protocol using coherent states with gaussian amplitude and phase modulation Efficient transmission of information using continuous variables? -> Shannon's formula (1948) : the mutual information I AB (unit : bit / symbol) for a gaussian channel with additive noise is given by (a) Alice chooses X A and P A within two random gaussian distributions. (b) Alice sends to Bob the coherent state X A + i P A > I AB = 1/2 log 2 [ 1 + V(signal) / V(noise) ] (c) Bob measures either X B or P B (d) Bob and Alice agree on the basis choice (X or P), and keep the relevant values. V A P P A P B Reminder : I(X; Y) = H(X) - H(X Y) = H(Y) - H(Y X) = H(X) + H(Y) - H(X; Y) X B X A X N 0 V A

22 Security of coherent state CV-QKD : collective attacks Alice-Bob mutual information : I AB Eve-Bob mutual information : I BE (Shannon : individual attacks) * BE (Holevo : collective attacks) Bounded from channel evaluation! Secret Key Rate :!I = I AB - I BE (Shannon)!I = I AB - * BE (Holevo)! For both individual and collective attacks Gaussian attacks are optimal ) Alice and Bob consider Eve s attacks Gaussian and estimate her information using the Shannon quantity I BE or the Holevo quantity * BE Fig : V A = 21 (shot noise units) ' = (shot noise units), + = 0.5 M. Navasqués et al, Phys. Rev. Lett. 97, (2006) R. García-Patrón et al, Phys. Rev. Lett. 97, (2006)

23 Error correcting codes efficiency - Errorless bit strings must be extracted from noisy continuous data - Binning + error correction with very efficient LDPC codes, efficiency, < 1 Alice-Bob mutual information : I AB Available after error correction :, I AB Eve-Bob mutual information : * BE (Holevo : collective attacks) Imperfect correction efficiency induces a limit to the secure distance

24 R. Renner and J.I. Cirac, Phys. Rev. Lett. 102, (2009) Also : finite size effects, A. Leverrier, F. Grosshans and P. Grangier, Phys. Rev. A 81, (2010)

25 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

26 All-fibered 1550 nm Field test of a continuous-variable quantum key distribution prototype S Fossier, E Diamanti, T Debuisschert, A Villing, R Tualle-Brouri and P Grangier New J. Phys. 11 No 4, (April 2009)

27 The SECOQC Quantum Back Bone Real-size demonstration of a secure quantum cryptography network by the European Integrated Project SECOQC, Vienna, 8 october 2008 * CV link - 9 km - 8 kb/s realized by CNRS / Institut d'optique and Thales 8 kb/s CV secret bit rate during the demo (8h)

28 Symmetric Encryption with QUantum key REnewal SEQURE Symmetric Cryptography Symmetric Cryptography QKD Infrastructure QKD Infrastructure secret key rate 1 kbit/sec at 25 km secret key rate 1 kbit/sec at 25 km Alice Alice Secret key Secret Interface key Interface Raw key QKD Raw key QKD message Ciphered message Bob Thales 1 Gbit/sec Bob 1 Gbit/sec Secret key Classical link for QKD Interface Secret key ENST Classical link for QKD Raw Interface key Quantum link for QKD QKD IO/ Thales Raw key Quantum link for QKD QKD Thales ENST IO/ Thales! Thales : Mistral Gbit

29 Field implementation! Fibre link : Thales R&T (Palaiseau) <-> Thales Raytheon Systems (Massy)! Fiber length about 12 km, 5.6 db loss

30 Results On site, 12 km distance, 5.6 db loss Minimal direct action on hardware (feedback loops, remote control) Air conditioning failure PC dead See

31 !"#$%&'#(#)*+%,-."/*0)1%2#134&"56#)/+%7'8"01%9$$-"#8) /'01"12(%3456%7,8(&)%9&(%%:-#';17%5-,7(&&1*<%=*10&%>:5=?%-#0;(-%0;#*%65= A Last version (commercial device, 80 km) : P. Jouguet et al, Nature Phot. 7, %1&%%1"'-,C(8%D-,"%EFG%0,%FHG%D,-%#*+%!IJ%K%$'1():%;0/*"15)%<=>%?8@ %41&0#*7(%>L"? BCDEF,%<5'88):50"$%G:';#5*@%

32 Several recent exemples of quantum hacking (e.g. Vadim Makarov et al.) Exploits weaknesses in single photon detectors Will NOT work against CVQKD (PIN photodiodes, linear regime) Hackers will have to work harder and Trojan attacks will not make it (work under way, SQN + U. Erlangen)

33 Many other works on CVQKD! <= Theory and Experiments : (incomplete list!)

34 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

35 Long-Distance Quantum Communications Q I P C Need to share highly entangled states (cryptography..) Problem : losses Solution : Entanglement Distillation : Large number of weakly entangled states Small number of strongly entangled states But impossible to distill Gaussian entanglement with Gaussian means - use non-gaussian operations! (such as photon subtraction)

36 How to create entanglement at a large distance? Two main approaches (Distillation) Alice Victor Entangled states (Distillation) Bob Alice Entangled states Joint measurement Victor Entangled states Bob Exemple : Exemple : Mostly limites by losses Mostly limited by noise In the present state of technology, this is much more efficient for distances ~ 100 km

37 H#"1*#8%:)G)"*):/%I0*J% )1*"1($);%5'J):)1*%/*"*)/ 7)8'*)%)1*"1($)8)1*%'K%5"*%/*"*)/%#/01(%L;)$'5"$03);L%GJ'*'1%/#.*:"5*0'1 M#1*%#8C#*0#<(%,D%0;1&%&7;("(%K%"$8'/*%01/)1/0*0M)%*'%*:"1/80//0'1%$'//)/%A >0;(%*,*N$,7#$%7#0&%#-(%*(C(-%0-#*&"100(8%1*%0;(%$1*(? Q I P C N0;)$0*O%K':%P>%;Q%$'//)/%R% N%S%>TU %%%%N%S%>T>>V%

38 Experiment A. Ourjoumtsev et al, Nature Physics, 5, 189, 2009 Q I P C Experimental set-up Two-mode probability distributions (two phases. and "...)

39 Results A. Ourjoumtsev et al, Nature Physics, 5, 189, Full two-mode tomography : 2009 Cuts of the experimental 4D Wigner function, corrected for homodyne losses Q I P C P(x 1,", x 2,/ ) Entanglement : N = 0,25 ± 0,04 Almost insensitive to losses in the quantum channel! but still far from a quantum repeater! T filters & APD ~ 10% : ~ 100 km optical fiber

40 Quantum repeaters with entangled cats? N. Sangouard, C. Simon, N. Gisin, J. Laurat, R. Tualle-Brouri and P. Grangier, JOSA B 27, 137 (2010) Q I P C Bell measurements are deterministic for entangled cats using only BS and photon counters! BS But parity measurements (even / odd) are extremely sensitive to losses -> To avoid errors one has to use kittens rather than cats -> Increase of the «failure» probability (getting 0 0 ) -> Overall not significantly better than using entangled photons :-(( Better hardware needed! (here : deterministic parity measurement)

41 Content of the Tutorial Q I P C Part 1 : Gaussian and non-gaussian states 1. Homodyne detection and quantum tomography 2. Generating non-gaussian Wigner functions : kittens, cats and beyond Part 2 : Continuous variable quantum cryptography 1. Continuous variable quantum cryptography : principles 2. Continuous variable quantum cryptography : implementations Part 3 : Towards quantum networks 1. Entanglement for continuous variable quantum networks 2. Deterministic photon-photon interactions

42 Photon photon interactions * Basic question we want to address : is it possible to design and manipulate (dispersive and deterministic) photon-photon interactions? Non-classical state generation Control-phase gate $& + -$& 1& 1& $& Self-Kerr effect: n& (e i/ ) n(n-1)/2 n& * Present approaches : B. Yurke & D. Stoler, PRL 57, 13 (1986) We want / = ( per photon (aj!) / = ( Cross-Kerr effect n& m& (e i/ ) nm n& m& - measurement-induced non-linearities, e.g. for KLM, or for producing Fock states, Schrödinger's cat states, noiseless amplification... Nice expts but their nondeterministic character makes them hardly scalable - cavity QED : either optical domain or microwave domain (we want optical) 1& - 1&

43 Using Rydberg-Rydberg interactions Control beam * A possible approach : Rydberg atoms ( 87 Rb, n~50) r& e& g& use Rydberg-Rydberg interactions to induce photon-photon interactions - photons -> (interacting) Rydberg polaritons -> photons * Lot of ongoing work : Theory: among recent ones, A. Gorshkov, T. Pohl, F. Bariani, M. Fleischhauer, M. Lukin, G.!Kurizki, K. Moelmer & many others Experiments: groups of C. Adams, A. Kuzmich, V. Vuletic & M. Lukin...

44 Single Photon from a single polariton (DLCZ protocol) L.M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller, Nature 414, 413 (2001) 87 Rb MOT: Density cm -3 Cooperativity C E. Bimbard et al, arxiv: (2013)

45 Single Photon from a single polariton (DLCZ protocol) E. Bimbard et al, arxiv: (2013)

46 E. Bimbard et al, arxiv: (2013) Single Photon from a single polariton (DLCZ protocol)

47 E. Bimbard et al, arxiv: (2013) Single Photon from a single polariton (DLCZ protocol)

48 E. Bimbard et al, arxiv: (2013) Single Photon from a single polariton (DLCZ protocol)

49 Single Photon from a single polariton (DLCZ protocol) Overall detection efficiency : 55% Corrected for detection efficiency Overall extraction efficiency : 80% E. Bimbard et al, arxiv: (2013)

50 Single Photon from a single polariton (DLCZ protocol) Quantum memory effect : the memory time (1 µs) is limited by motional decoherence due to finite temperature (50 µk) Single photon ok! Next step : interacting photons! E. Bimbard et al, arxiv: (2013)

51 They did the hard work... CVQKD team (Vienna 2008) Jérôme Lodewyck Eleni Diamanti Simon Fossier Thierry Debuisschert Frédéric Grosshans Rosa Tualle- Brouri Non-Gaussian team (Palaiseau 2010) Q I P C Franck Ferreyrol Rosa Tualle- Brouri Marco Barbieri Rémi Blandino

52 Thank They you did for the your hard attention work...! Valentina Parigi (post-doc, exp.) Erwan Bimbard (PhD, exp.) Etienne Brion (CNRS) Alexei Ourjoumtsev (CNRS) Rydberg polariton team Jovica Stanojevic (post-doc, th.) Andrey Grankin (PhD, th.) Just arrived! : Rajiv Boddeda, Imam Usmani Technical staff: F. Fuchs, F. Moron, A. Guilbaud Collaboration : Pierre Pillet (LAC Orsay) DELPHI Deterministic Logical Photon-photon Interactions