Title. Quantum communication and other quantum information technologies

Transcription

1 Title Quantum communication and other quantum information technologies F Stef Roux CSIR National Laser Centre Presented at the University of Pretoria 20 February 2014 p. 1/41

2 Contents Over view of quantum mechanics Quantum communication Quantum state preparation Quantum teleportation Decay of entanglement in turbulence Theory Numerical simulations Experimental results p. 2/41

3 Quantum mechanics Paul Dirac Lasers Werner Heisenberg Computers Transistors Neils Bohr Microelectronics Erwin Schroedinger p. 3/41

4 Einstein-Podolsky-Rosen Albert Einstein Boris Podolsky Nathan Rosen x x Quantum mechanics: measurements on one particle dictate the state of the other particle. Unstable particle Spontaneous decay p p p. 4/41

5 Parametric down conversion One incoming photon Two outgoing photons Energy conservation: ω pump = ω signal +ω idler Momentum conservation: k pump = k signal +k idler Laser Spontaneous parametric down conversion (SPDC) Nonlinear crystal A Type II phase matching B Type II phase matching photons have perpendicular polarization: θ B = θ A π/2 However, each beam on its own is unpolarized contains all states of polarization. p. 5/41

6 Multiple realities Ψ H V V H A B 2 A B Reality #1 Reality #2 A A B B p. 6/41

7 Separability Ψ = 1 2 H A V B 1 2 H A H B V A V B 1 2 V A H B... can be factored (separated) Ψ = 1 2 ( H A + V A )( H B V B ) A B Ψ 1 2 Separability Not entangled p. 7/41

8 Quantum communication What does quantum communication have that classical communication doesn t? Fundamental security! One cannot copy a quantum state one cannot eavesdrop without sender/receiver knowing Eavesdropper Alice Optical fibre Bob Quantum protokol (Quantum Key Distribution QKD) to produce an encryption key that is fundamentally secure p. 8/41

9 Quantum Key Distribution BB84 Basis 0 1 Bob Alice Measure Prepare Alice Bob Key p. 9/41

10 Polarization vs modes Can always specify polarization with two polarization states: Ψ = C H H +C V V = C L L +C R R Polarization 2-dimensional Hilbert space each photon can carry one qubit of information For more information per photon (larger channel capacity) need larger Hilbert space Transverse spatial modes have an infinite dimensional Hilbert space p. 10/41

11 Laguerre-Gaussian modes General solutions of the paraxial wave equation in normalized polar coordinates: Mpl LG (r,φ,t) = exp(ilφ)(1+it) p ( ) Nr l (1 it) p+ l +1 L l 2r 2 p 1+t 2 exp ( r 2 1 it ) x = rw 0 cosφ, y = rw 0 sinφ, z = z R t (z R = πw0 2/λ) L l p associate Laguerre polynomials p radial mode index (non-negative integer) l azimuthal index (signed integer) N normalization constant p. 11/41

12 Bessel-Gaussian modes In normalized polar coordinates: 2 ( σr ) Ml BG (r,φ,t;χ) = π J l exp 1 it ( iσ 2 t 4r 2 ) 4(1 it) exp(ilφ izk z ) x = rw 0 cosφ, y = rw 0 sinφ, z = z R t (z R = πw 2 0 /λ) J l Bessel function σ = w 0 k r normalized radial scale parameter k r,k z radial and longitudinal wavenumber p. 12/41

13 QKD in higher dimensions Need mutually unbiased bases in higer dimensions φ a,n φ b,m 2 = 1 d for a b Bases Basis functions 2 3 p. 13/41

14 Quantum state preparation Experimental setup to prepare and measure entangled photon states: Signal SMF Laser BBO A B 50/50 beam splitter Idler SLM APD APD CC SLM SMF p. 14/41

15 Spatial light modulators Pure phase modulation or complex amplitude modulation p. 15/41

16 Quantum state tomography To reconstruct density matrix ρ mn = φ m ρ φ n for density operator ρ = n P n ψ n ψ n p. 16/41

17 Spiral bandwidth Orbital angular momentum (OAM) ( azimuthal index) is conserved in SPDC OAM entanglement High dimensional entanglement broad OAM spectrum p. 17/41

18 Quantum teleportation (2-dim) Output quantum state (qbit) Bell (joint) measurement Classical communication Unitary transformation Input quantum state (qbit) Entangled (Bell) state p. 18/41

19 Quantum teleportation (n-dim) Output quantum state (n-dim) Joint measurement Classical communication Unitary transformation Input quantum state (n-dim) Entangled (m-dim) state p. 19/41

20 Turbulence vs Scintillation Turbulence: velocity distribution in fluid Refractive index: n(r) = 1+δn(r) Scintillation: what happens to light in turbulence Random phase modulations + diffraction. Input plane Turbulent atmosphere Output plane z Input beam Scintillated beam p. 20/41

21 Kolmogorov model Refractive index structure function: a D n = [δn(r 1 ) δn(r 2 )] 2 = C 2 n( r 1 r 2 ) 2/3 C 2 n Refractive index structure constant Power spectral density: Φ n (k) = 0.033C 2 n k 11/3 x 1 x Turbulent atmosphere θ Phase structure function: D θ (d) = [θ(x 1,y 1 ) θ(x 2,y 2 )] 2 = 6.88 x 2 ( d r 0 ) 5/3 where d = (x 2 x 1 ) 2 +(y 2 y 1 ) 2 ( ) λ 2 3/5 and Fried parameter (distance): r 0 = Cnz 2 a LC Andrews and RL Phillips, Laser beam propagation through random media, 2nd ed. SPIE Press (2005) p. 21/41

22 Single phase screen Assuming weak scintillation (only affects the phase) a (a) L (b) L Free space Turbulent atmosphere Phase screen Use single phase screen with single parameter (r 0 Fried parameter): ρ mn (z) = Em(r 1 )E n (r 2 )ψ(r 1 )ψ (r 2 ) [ ] exp 1 2 D θ( r 1 r 2 ) d 2 r 1 d 2 r 2 Function of only w 0 /r 0 (contains all parameters including z) Evaluated at z = 0 a C. Paterson, Phys. Rev. Lett., 94, (2005) p. 22/41

23 Entanglement decay Decay of qubit OAM entanglement (concurrence C) in turbulence a use quadratic structure function approximation: D ( x r 0 ) 5/3 ( x r 0 ) 2 Observations: Concurrence decays as function of w 0 /r 0 only Decays to zero (sudden death) at w 0 /r 0 1 Last longer for larger azimuthal indices a B.J. Smith and M.G. Raymer, Phys. Rev. A, 74, (2006) p. 23/41

24 Infinitesimal propagation Propagate over infinitesimal distance Instead of going from 0 to z in 1 step, we proceed in many small steps of dz dz Turbulent atmosphere ρ(z) ρ(z+dz) p. 24/41

25 Infinitesimal propagation equation Infinitesimal propagator equation (IPE): a z ρ mnpq = i(p mx ρ xnpq ρ mxpq P xn +P px ρ mnxq ρ mnpx P xq ) +Λ mnxy ρ xypq +Λ pqxy ρ mnxy 2Λ T ρ mnpq ρ = m,n m p ρ mnpq n q P mp (z) = 1 a 2 G 2k m(a,z)g p (a,z) d 2 a Λ mnpq = k 2 Wmp(a,z)W nq (a,z)φ 0 (a,0) d 2 a W mn (a,z) = G m (a +a,z)g n(a,z) d 2 a Λ T = k 2 Φ 0 (a,0) d 2 a p. 25/41

26 Properties of the IPE Derived in Fourier domain Based on power spectral density: Φ n (k) The resulting density matrix is hermitian Follows from identity: Λ mnpq = Λ nmqp Expressible as Master equation in Lindblad form (However z-derivative and not time-derivative) valid density matrix Transverse spatial modes infinite dimensional Hilbert space IPE is an infinite set of coupled differential equations To solve them one needs to truncate the set truncated IPE is not trace preserving: tr{ρ} 1 p. 26/41

27 Example: symmetric qubits For initial state: ( l, l l,l )/ 2, density matrix: 1 R ρ mnpq = T 0 1+R 2 2R R 1+R R 2 [ where T = exp 127 ] 36 Z(t) R = exp [ 572 ] Z(t) t-dependence: Z(t) σ t where σ = π3/2 Cnw 2 11/3 0 6Γ(2/3)λ 3 0 (1+τ 2 ) 5/6 dτ σt Concurrence: C = 1 2 (R2 +2R 1) p. 27/41

28 Comparison of results Single phase screen IPE p. 28/41

29 Do we need the IPE? Scintillation strength Strong scintillation w /r =1 0 0 Weak scintillation Normalized propagation distance p. 29/41

30 Numerical simulations General procedure: Prepare input state Split-step method: Multiply mode by random phase function Propagate through free-space (without turbulent) Extract density matrix Compute concurrence p. 30/41

31 Input state Bell state: Ψ = 1 2 ( l A l B + l A l B ) 4 modes separately propagated through turbulence l, l LG modes at the waist (z = 0) with p = 0 For l = 1: U (LG) 01 = 2 πrexp(iφ)l 1 0(2r 2 )exp( r 2 ) r normalized radial coordinate φ azimuthal angle L 1 0 ( ) associated Laguerre polynomial p. 31/41

32 Split-step method BBO Turbulent atmosphere Quantum state tomography Phase screens BBO Simulated turbulent atmosphere Quantum state tomography p. 32/41

33 Random phase screens Phase function: θ(x,y) = k 0 z k 2π F 1{ χ n (K)[Φ 0 (K,0)] 1/2} χ n (K) random complex spectral function: delta correlated: χ(k 1 ) χ (K 2 ) = (2π k ) 2 δ 2 (K 1 K 2 ) normally distributed zero mean k sample spacing in frequency domain ( 1/outer scale) p. 33/41

34 Numerical of results S = log 10 ( π 3 C 2 nw 11/3 0 λ 3 ) (a) (b) Concurrence Concurrence w 0 /r 0 w 0 /r 0 p. 34/41

35 Experimental setup Signal SMF Laser BBO A B 50/50 beam splitter Idler SLM APD APD CC SLM SMF p. 35/41

36 Comparison of results Qubit (Bell state) both photons through turbulence: (a) (b) Concurrence l = 1 l = 3 w 0 /r 0 (c) w 0 /r 0 (d) Concurrence l = 5 l = 7 w 0 /r 0 w 0 /r 0 p. 36/41

37 Decay distance Distance scale for entanglement decay: L dec (l) = 0.06λ2 l 5/6 w 5/3 0 C 2 n For w 0 = 10 cm, λ = 1550 nm and C 2 n = m 2/3 : L dec l km km km km p. 37/41

38 Higher dimensional states l = 1 l = 3 l = 5 l = Tangle (Lower bound for entanglement): τ{ρ} = 2tr{ρ 2 } 2tr{ρ 2 R } 2(d 1) max(τ) = d p. 38/41

39 Experimental results (a) (b) w 0 /r 0 (c) (d) w 0 /r 0 w 0 /r 0 w 0 /r 0 p. 39/41

40 Comparison of results Numerical and experimental data: (a) (b) w 0 /r 0 w 0 /r 0 Theoretical single phase screen calculations: Tangle w0/r0 l = 1 l = 2 l = 3 l = 4 l = 5 l = 6 l = 7 p. 40/41

41 Conclusions Quantum communication, which enables fundamentally secure communication, is a new technology that is actively being develop by international research groups Quantum communication requires various other technologies: Quantum state preparation Quantum teleportation etc. Free-space quantum communication suffers decay of entanglement due to turbulence in the atmosphere By studying the effect of scintillation on entanglement we can determine design constraint for free-space quantum communication p. 41/41