Optical Implementations of QIP

1. Optical Implementations of QIP Kevin ReschIQC, Department of PhysicsUniversity of Waterloo

2. Quantum optics and Quantum Info. group Optical Imaging Quantum computing Entanglement sources Tests of nonlocality Tomography Interferometry

3. Goal of the talk • To understand from basic principles how a quantum information protocol works in theory and in practice using optics • I chose quantum teleportation where we can understand the discrete (polarization) and continuous variable versions of this protocol

4. Quantum teleportation • A process for transmitting quantum information using a classical channel and shared entanglement Figure credit: Bouwmeester et al. Nature 390, 575 (1997).

5. Outline • Introduction to quantum optics • photons, encodings, entanglement • QIP with polarization • waveplates, CNOT, teleportation • QIP with continuous variables • Wigner function, measurements, teleportation

6. Photons

7. Field quantization • A procedure for finding a quantum description of light • Starting point is Maxwell’s Equations in vacuum • Introducing the potentials and gauge

8. Field quantization • From these derive wave equation for the vector potential • Spatial mode expansion (exact form depends on boundary conditions) Plane wave solutions Periodic BC, cubic volume

9. Field quantization • Also from classical physics, the energy stored in an EM field • Energy for a single mode • Rewriting complex A in terms of real quant • Gets us onto familiar territory Classical Harmonic Oscillator (mass = 1)

10. Field quantization

11. Field quantization • Promote the classical parameters to operators • Which defines field operators

12. Field quantization • And find the energy for each mode • Which simplifies to

13. Field quantization • Harmonic oscillator The excitations of the EM modes are “photons” – particles of light q “number” operator

14. Experimental evidence for photons • Particles can only be detected in one place Ca Atomic Cascade Grangier, Roger, Aspect Europhysics Lett 1, 173 (1986)

15. Properties of photons • A single photon has just three properties: • Colour/energy, • Polarization, • Direction/momentum, • Its quantum state can be described as a superposition of these properties

16. Single photon QI encodings • Polarization • Spatial modes i = + |V> |H> • Time-bin • Freq. encoding

17. QIP with optics • Pros: • Low decoherence* • High speed • Flexible encodings • Cons: • Negligible photon-photon interactions • Loss • Hard to keep in one place • Some encodings unsuitable for some situations, ex., polarization/modes in fibre Ideal for quantum communication *can be susceptible to coupling internal DOF

18. But an optical mode is more complicated… • Photons are bosons, so we can have many per mode • Important multi-photon states of a single mode: • Fock or number state • Coherent state • Squeezed state • Thermal state • (Things can get very complicated with a large number of modes and all the DOF)

19. “Mode” observables: Quadratures • We can write quadrature operators analogous to x and p (but do not correspond to pos/mom of the photon!) • Since , there must be an uncertainty relation

20. Useful operator identities • Baker-Campbell-Hausdorff lemma • Glauber’s identity valid when [A,[A,B]]= [A,[A,B]]=0.

21. Phase shift operators • Phase shift operator (exp free-field) • Using BCH • Or • Free-field evolution converts one quadrature into the other in the form of a rotation

22. Quadratures • These observables correspond to components of the electric field • There is an uncertainty relation between the E field ‘now’ and the E field a quarter cycle ‘later’

23. Coherent states • Defined as eigenstates of lowering operator a is not Hermitian so α can be complex • Uncertainties in mode variables: • Min uncertainty, equal between q and p

24. Displacement operator • Coherent states can be generated using the displacement operator: • This can be seen by rewriting the operator using Glauber’s identity and comparing

25. Displacement operator • Useful identities and properties:

26. Coherent states in quantum optics • Coherent states play an important role as a basis in quantum optics • But coherent states with different amplitudes are orthogonal • And the basis is “overcomplete” (projectors do not sum to identity)

27. Entanglement

28. The characteristic trait of QM E. Schrödinger Math. Proc. Camb. Philos. Soc. 31, 555 (1935).

29. Definition of entanglement à à à ­ = A B A B ( ) ( ) ( ) à à à x x x x = 1 2 1 2 ; • Any state that can be written, is said to be separable, otherwise it is entangled • Pure states: are separable, otherwise entangled

30. Superposition and entanglement Entanglement Superposition

31. The characteristic trait of QM Enhanced Sensors Quantum Computing Phase transitions http://www.ligo.caltech.edu http://www.eng.yale.edu/rslab/ http://www.quantum-munich.de Entanglement Quantum relativistic effects Quantum Communication Foundations of QM Figure credit: Rupert Ursin Jennewein et al. PRL 84, 4729 (2000) S. Hawking Illustrated Brief History of Time

32. Nonlinear optics • Direct photon-photon interactions too weak • Instead atoms can mediate interactions between photons – Nonlinear Optics • Ex. Second-order nonlinearity Nonlinear coefficient Creates pairs of photons Destroys pairs of photons

33. Nonlinear optics • Instead of oscillating only at the frequency of the driving field, the charge can oscillate at new frequencies Example: Second Harmonic Generation ω 2ω Χ(2) material (such as BBO or KTP)

34. Second-harmonic generation

35. Entangled photons • Reverse of SHG Parametric Down-conversion “blue” photon two “red” photons c(2) wpump=ws + wi Phase matching: kpump=ks + ki ‘Conservation laws’ constrain the pair without constraining the individual  entanglement Also: QD, at. casc

36. Down-conversion movie KTP – nonlinear crystal www.quantum.at

37. PPKTP source

38. PPKTP source

39. Multiphoton sources: Pulsed SPDC y y y y h H + + g a a g a a c = H V V H 1 2 1 2 : : y y y y 2 2 j i ( ) j i H 0 0 + a a a a ! H V V H 1 2 1 2 j i j i j i H V H V H V V H 2 2 2 2 + + = 1 2 1 1 2 2 1 2 ; ; ; ; ; • Down-conversion can sometimes emit two pairs. • If a short pulse is used for an entangled photon source, the pair are properly described by a 4-photon state

40. GHZ Correlations j i j i j i H V H V H V V H 2 2 2 2 + + 1 2 1 1 2 2 1 2 ; ; ; ; ; • Measured 4-photon coincidences to post-select GHZ state • Needs at least 1H and 1V in mode 1 V H H V Bouwmeester PRL 82, 1345 (1999) Lavoie NJP 11, 073501 (2009)

41. H H V b a c V H H V

42. V V H b a c H V V

43. Three-photon GHZ states • ~4 four-fold coincidence counts per minute (3-fold coincidence + trigger) • Fidelity with target GHZ 84% from tomography

44. 2nd method: Cascaded down-conversion e 10-11 ~1 in a billion years ~1 per day Bulk crystal (BBO) ~3 per hour ~1 in a hundred thousand years 10-9 PPKTP ~1 per second ~2 per month 10-6 Waveguide PPLN *assuming 106 s-1 primary photons, no loss, perfect detectors

45. Experimental cascaded down-conversion See also Shalm Nature Physics 9, 19 (2013); Hamel arxiv: 1404.7131 4.7 ± 0.6 counts/hr

46. Two-mode squeezed vacuum • Two mode squeezing operator • Creates or destroys photons in pairs • Properties • Warning: can’t use Glauber’s theorem

47. Two-mode squeezed vacuum • The interesting properties show up in the correlations between quadrature obs.

48. Two-mode squeezed vacuum • The commutator, • And so we have the same uncertainty relation between these joint observables as the quadratures themselves:

49. Two-mode squeezed vacuum • We can calculate the uncertainty in these observables for the TMSV • Recall • To calculate this requires several applications of the squeeze operator identities, ex.,

50. Two-mode squeezed vacuum • After some algebra • Choosing • We can “squeeze” the uncertainty in one observable at the expense of the other