Correcting noise in optical fibers via dynamic decoupling

Transcription

1 Introduction CPMG Our results Conclusions Correcting noise in optical fibers via dynamic decoupling Bhaskar Bardhan 1, Petr Anisimov 1, Manish Gupta 1, Katherine Brown 1, N. Cody Jones 2, Hwang Lee 1, Jonathan Dowling 1 1 Department of Physics & Astronomy, Louisiana State University, Baton Rouge 2 Edward L Ginzton Laboratory, Stanford University, Stanford December 2, 2011

2 Introduction CPMG Our results Conclusions Outline 1 Introduction 2 CPMG dynamic decoupling 3 Our results 4 Conclusions

3 Introduction CPMG Our results Conclusions What is dynamic decoupling good for? Logical qubits Error correction techniques Error mitigation and quantum control Physical qubits

4 Introduction CPMG Our results Conclusions Previous work on using dynamic decoupling for photons Massar & Popescu [NJP 9 158] polarization mode dispersion 2007 Optical fibers Wu and Lidar [PRA ] analytical proof 2004 Dephasing Lucamarini et al. [PRA ] [PRL ] photons in a ring cavity 2009/2011

5 Introduction CPMG Our results Conclusions Sending a photon through an optical fiber We use a polarization maintaining fiber ~ 10m changes in birefringence caused by changes in environment Corrects for dephasing in two orthogonal directions Only need to correct for dephasing in one direction

6 Introduction CPMG Our results Conclusions How our noise effects our states Not all of the states of our qubits will be effected by the noise. We can see this by looking at the states on the Bloch sphere. Our noise is modelled as being along the Z axis X Y The 0 and 1 states are not effected by the noise. However all other states will be effected, and this is something we need to correct for. The worst case scenario is the + and states.

7 Introduction CPMG Our results Conclusions A simple example of dynamic decoupling z z z z Y Y Y Y X X X X (a) (b) (c) (d) A simple example of dynamic decoupling using two π pulses. At the end of our sequence the state is back in its original position on the Bloch Sphere.

8 Introduction CPMG Our results Conclusions CPMG dynamic decoupling We want our dynamic decoupling sequence to be resistant to errors in the waveplates. Carr-Purcell-Meiboom-Gill dynamic decoupling is resistant to errors in the waveplates for states + and [Morton et al. PRA ]

9 Introduction CPMG Our results Conclusions Modelling our fiber The overall accumulated phase in one noise region is given by φ = 2π [ ] L λ n L L (1)

10 Introduction CPMG Our results Conclusions The + state gives the worst fidelity The fidelity of the state is given by F = cos 2 θ + ( α 2 β 2 ) 2 sin 2 θ (2) This is smallest when α 2 = β 2, and therefore this is the state with the largest error. When α 2 = 0 or β 2 = 0 then the fidelity is always 1.

11 Introduction CPMG Our results Conclusions How the number of waveplates effects fidelity Here we assume L = 10m, and σ L = 3m with a total fiber length of 10km.

12 Introduction CPMG Our results Conclusions How σ L and σ φ effect fidelity Here we assume 500 waveplates on a fiber of length 10km with L = 10m.

13 Introduction CPMG Our results Conclusions Conclusions We have shown that using a CPMG sequence of dynamic decoupling we can correct for dephasing errors in an optical fiber. When L = 10m, σ L = 3m and σ φ = ±100 radians for a 10km fiber, 400 waveplates are required to achieve a fidelity suitable for BB84. In future work we will go on and explore what happens if we introduce errors onto the waveplates.

14 Quantum Control For Generation of High-N00N States Cable, H; Dowling, JP, Physical Review Letters, 99 (16): Art. No OCT

15 N00N states - background µ N0i+ 0Ni 2 Path entangled state. Maximal path number entanglement. Applications: - beating the classical diffraction limit (lithography, microscopy ), - Heisenberg limited interferometry (gyroscopy, accelerometers, magnetometers ), - spectroscopy. Experimentally N00N states experiments with up to 6 photons: P. Walther et al Nature (429) (2004), M. W. Mitchell et al Nature (429) (2004), K. J. Resch et al quant-ph/ Problems of efficient generation and engineering with many photons. Conceptual issues.

16 Quantum Interferometric Lithography source of two-mode correlated light NOON Generator mirror a b N-photon absorbing substrate phase difference along substrate Deposition rate: Classical input :! N (" ) = ( â + e #i" ) ˆb N ( â + e +i" ˆb ) N! N (" ) = cos 2 N (" / 2) N00N input :! N (" ) = cos 2 ( N" / 2) Super-resolution, beating the single-photon diffraction limit. Boto, Kok, Abrams, Braunstein, Williams, and Dowling PRL 85, 2733 (2000)

17 Super-Resolution λ/ν λ

18 Phase Estimation Optical N00N states in modes a and b, Unknown phase shift on mode b so. Cramer-Rao bound mode b phase shift Heisenberg Limit!. parity measurement mode a

19 !" =! ˆP d ˆP / d" Super-Sensitivity Δϕ = 1/Ν = Heisenberg Limit Δ ϕ = 1/ = shot-noise limit

20 Many photon N00N state generator using a Kerr non-linearity Generate large N00N states using a cross-kerr interaction Ĥ = hka ab b and a source of photon number states. e.g. modify Christopher Gerry PRA (59) page 4095 (1999) macroscopic (single-mode) cat states. BUT required non-linearity very large.

21 N00N state generation by measurementinduced non-linearity Several schemes using: - linear optics, - single photon sources or OPO, - conditioned on photodetection measurements. H. Lee, Pieter Kok, N. J. Cerf and J. P. Dowling, PRA (65) (R); J. Fiurášek PRA (65) (2002); X. Zou, K. Pahlke and W. Mathis quant-ph/ But exponential resource scaling! Analogy to quantum computing - KLM scheme. Efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. E. Knill, R. Laflamme and G. J. Milburn, Nature (409) pg. 46 (2001).

22 N00N states and ±π/2 relational Schrödinger cats (a) A 50:50 beam splitter followed by a π/2 phase shift converts an input N00N state to a relational Schrödinger cat with components with relative phases ±π/2. (b) achieves the reverse. U ps1 (π/2) U bs (50: 50) { Ni 0i F + 0i Ni F } Z 2π 0 dφ 2π e inφ + ( 1) NZ 2π 0 s s + N + N 2 eiφ 2 ei(φ π/2) cs s s + N + N 2 eiφ 2 ei(φ+π/2) cs dφ 2π e inφ (a) (b)

23 Generating relational Schrödinger cats by a simple interference procedure OPO Ni Ni 0i 0i beam splitters couple principle modes to ancillae 50:50 beam splitter A simple experiment can demonstrate interference phenomena between two modes initially in a state with no prior phase correlations. ψ r.s.c i l, r counts at left, right photodetectors Localisation of the relative phase plays a central role.

24 Localising relative optical phase Representation in terms of coherent states, ne iφ ic.s. = e n/2 P r=0 q n r r! eirφ ri F The detection process acts on the input state as follows, Ni Ni F R dθdφe in(θ+φ) Ne iθ E Ne iφ E c.s.. R dθdφe i 2 (2N r l)(θ+φ) c l,r ( ) Ne iθ E Ne iφ E c.s. c l,r ( ) for r=l=0,1,2,5,15 counts multi-valued: ± 0 =±2arccos ³q r/(r+l) Gaussian approximation: exp h ³ l+r 4 ( ± 0 ) 2i φ θ.

25 Fidelities for converting relational Schrödinger cats to N00N states Generation of relational Schrödinger cat states is probabilistic. No range for picked out. Only ±π/2 cats can be immediately converted to N00N states. Define a fidelity where Λ N00N maximises F. ± 0 F = ½ ¾ h0 hn +e iλ NOON hn h0 2 ψ converted i 2 µ F cos 2N 0 π/2 2 whenever 0 π/2. Poor scaling whenever different from π 0 2.

26 A beam splitter as a two mode rotation A beam splitter implements a rotation on two modes, angle ζ determined by transmittivity. Acting on a state with perfectly defined relative phase: (1) RELATIVE PHASE STRETCH 1 =arctan[tan ( 0 )/cos(2ζ)] ± 0 ±π/2!!!! Desired ±π/2 components are fixed points! Breaks down by (not a cat!) 0 =0 (2) CHANGE OF MODE INTENSITIES

27 Stretching relational Schrödinger cat states Stretching relational Schrödinger cat reduces overlap of cat components. A unitary transformation cannot be sufficient! Simple measurement yields ψ r.s.c. ( 0 )i localised at ± 0. Additional beam splitter corrects relative phases but distorts the cat U (50:50) ψ r.s.c ( 0 ) : We want ψ r.s.c. (π/2)i: Need to correct intensity difference!

28 ψ rsc (± 0 )i Correction by feed-forward delay lines ³ ψ rsc ± π 2 i 50: 50 0ibeam splitter beam splitter reflectivity η(l,r) photodetection RELATIVE PHASE CORRECTION INTENSITY CORRECTION Variable beam splitter reflectivity η =2 2 [1+cos( 0 )] implemented by Mach-Zehnder setup

29 Summary Scalable N00N state generation Analytic results for all aspects of scheme. Feedforward correction circuit is efficient! Photon loss probabilities ~ binomial in. Compatible with a twin beam source, ρ source = P p(n) ni nihn hn. total cos( 0 ) photons Relational schrödinger cat states produced by simple measurement procedure are done so efficiently. Efficiency of scheme independent of size of required N00N state. Scaling to many photons, spending two thirds of the input photons on the localisation process suffices for F=0.99. Lose further fixed fraction to correction process.