1. Quantum Information with Continuous Variables Klaus Mølmer University of Aarhus, Denmark Supported by the European Union and The US Office of Naval Reseach

2. Continuous variables: Collective variables for macroscopic atomic samples. Optical field variables - continuous wave fields ! Interaction: Dispersive: Phase shifts, Faraday polarization rotation Homodyne measurement on the fields Goals: Precision probing, atomic clocks Squeezing, entanglement, cats and Fock states Teleportation, quantum memory, quantum computing

3. Outline Continuous variable physical systems Gaussian state formalism Three applications: –Squeezing and entanglement –Magnetometry –Photons from fields Outlook

4. Continuous variables for two-level atoms Many atoms in (|↑>+|↓>)/√2  =N at /2, = =0 Var(J y )Var(J z ) = | | 2 /4  binomial noise (M.U.S.). let p at = J z /√, x at = J y / /√, [x at,p at ]=i harmonic oscillator degrees of freedom. ”Ground state” is Gaussian in x at,p a Also without 100 % optical pumping! |↓>|↑> Normally, ρ 1 atom is quantum. Here, collective p at, x at, are our Quantum Variables !

5. Continuous variables for polarized light x-polarized light has = N ph /2, = = 0. let p ph = S z /√, x ph = S y / /√, [x ph,p ph ]=i harmonic oscillator ground state, Gaussian in x ph,p ph Interaction Dispersive atom-light interaction: σ + (σ - ) light is phase shifted by |↑> ( |↓> ) atoms  Faraday polarization rotation, proportional to H int = g S z J z = к p at p ph |↓>|↑>

6. Update of atomic state due to interaction with a light pulse H int = g S z J z = к p ph p at p ph unchanged x ph  x ph + к t p at x ph is measured: we learn about p at, we ”unlearn” about x at p at unchanged x at  x at + к t p ph

7. Gaussian states State characterized by vector of (x’s and p’s) with mean values m γ = matrix of covariances, γ ij = 2 P(y) = N exp(-(y-m) T γ -1 (y-m)) Gaussian states (m and γ) transform under xx, xp and pp interactions (linear optics, squeezing), decay and losses. Gaussian states (m and γ) transform under measurements of x’s and p’s (Stern-Gerlach and homodyne detection). Quantum case: Heisenberg uncertainty limit on γ

8. Update of Gaussian atomic state due to interaction with a light pulse Loss of light: I γ ph  (1-ε)γ ph +ε I Atomic decay: I γ at  (1-ηΔt)γ at +2(ηΔt) I

9. Interaction with a continuous beam Continuous  frequent probing (weak pulses/short segments of cw beam): Before interaction: optical state is trivial * After interaction: state is probed or discarded Differential equation for atomic covariance matrix This is a non-linear matrix Ricatti equation. *: not true for finite bandwidth sources

10. Application 1: Squeezing and Entanglement

11. Atomic spin squeezing due to optical probing L. B. Madsen and K.M., Phys. Rev. A 70, 052324 (2004) For the simple atom-light example (binomial distribution): Entanglement of two gases by optical probing : Duan, Cirac, Zoller, Polzik Measure (x 1 +x 2 ) and (p 1 -p 2 )

12. Entanglement and The Swineherd (Hans Christian Andersen 1805-1875) ” … when one put a finger into the steam rising from the pot, one could at once smell what meals they were preparing on every fire in the whole town”

13. Entanglement costs: ”Ten kisses from the princess” “Ask him,” said the princess, “if he will be satisfied with ten kisses from one of my ladies.” “No, thank you,” said the swineherd: “ten kisses from the princess, or I keep my pot.”

14. How much entanglement can be generated over a lossy optical transmission lines ? Ans.: ”Any amount, with use of repeaters and distillation.” What is the best possible entanglement, obtained with Gaussian operations ? (Distillation forbidden). Does the no-distillation theorem put an ultimate limit to the entanglement that we can squeeze into a single pair of distant oscillator-like systems ?

15. Direct transmission I = I 0 exp(-L/L 0 ) = I 0 (1-ε), loss ε Two-mode entanged state: (x 1 -x 2 ), (p 1 +p 2 ) EPR uncertainty: Δ EPR = (Var(x 1 -x 2 )+Var(p 1 +p 2 ))/2 < 1. Δ EPR  (1-ε) Δ EPR + ε (  1, for large loss) x 1 p 1 x 2 p 2

16. Entanglement of two gases, GEoF Polarisation rotation Polarisation rotation loss Does the read-out teach us more about gas 1 or 2 ? Faraday rotation with x-polarized beam H = κ 1 p p 1 + κ 2 p’ p 2 = κ√(1-ε) p’ (p 1 +p 2 ) - κ√ε p vac p 1

17. Atomic entanglement by probing is a bad idea ! Transmitted beams Optically probed atoms Finite optical squeezing NOT

18. A most surprising result: Lars B. Madsen and K.M., to be published Finite entanglement, N=1/3 for arbitrary loss: Teleportation fidelity, F = 1/(1+N) = 75 % for unknown coherent state. Probed atoms (symmetric) Infinitely squeezed light Probed atoms (one way) with squeezed light with anti-squeezed light squeezing: 0.1, 10

19. Application 2: Quantum Metrology

20. Metrology with a quantum probe Quantum system observer Classical quantity

21. A side remark For a quantum physicist, it is natural to think of the time evolution of the state as dynamical evolution of the physical state of the system, e.g., as the wave function is getting narrow, ”the particle localizes physically”. According to the Copenhagen interpretation, however, the wave function/density matrix/Wigner function is not the state of the system, but rather a representation of our knowledge about the system. When we measure, we learn something. Complete* relationship with classical theory for the estimation of a gaussian variable under noisy measurements (Riccati). (*: but recall commutators and Heisenberg’s uncertainty relation)

22. Probing of a classical magnetic field (K.M & L. B. Madsen,Phys. Rev. A 70, 052102 (2004) ) P(random signal | B)  P(B | signal) Our SIMPLE approach: Treat atoms AND light AND B field by a joint Gaussian probability Covariance matrix for (B,x at,p at,x ph,p ph ). Analytical solution (no noise): Long times: ΔB~ 1/(N at t 3/2 ) Independent of ΔB 0 not as 1/√N at,1/√t

23. Estimate a time dependent (noisy) magnetic field Vivi Petersen and K.M. to be published Noisy field Estimator (mean of Gaussian) Correct estimator for the current value of B(t) ! (See also Mabuchi et al.)

24. ”Gaussian hindsight” Noisy field Estimator (mean of Gaussian) 1. 0.02 ms delay, minimizes 2. Temporal convolution 3. Gaussian distribution for (B(t 1 ),B(t 2 ),.. B(t n ) measurements now, update the past.

25. Noisy field Estimator (mean of Gaussian) ”Get wiser in 0.02 msec” ”If I knew then, what I know now”

26. Application 3: Leaving the Gaussian states Photons from fields.

27. Leaving the Gaussian states Gaussian states do not encode qubits (two coherent states may be almost orthogonal, but their superposition is not a Gaussian) Gaussian states cannot be distilled (purified)

28. Photons from fields Field or photon description of light? Single mode: state and operator pictures equivalent. Continuous beams, multi-mode: Expansion on number states practically impossible. Field picture useful: mean values, correlation functions. Conditioned dynamics, collapses in Heisenberg picture?

30. From many to two modes: OPO output:  Chose trigger and output modes: Wigner function of four real variables: (Gaussian, prior to click)

31. Click event (photodetection theory): ρ  a 1 ρa 1 + Wigner function transformation: and then trace over mode 1 Gaussian  Gaussian times a poynomial

32. Results: (K.M., quant-pt/0602202, today) Weak OPO n=1 Fock state Strong OPO Schrödinger kitten 75 % detection

33. Experiments (Grangier group) J. Wenger et al, PRL 92, 153601 (2004):

34. Experiments (Polzik Group, 2006): (J. S. Neergaard-Nielsen et al, quant-ph/0602198) Experiments Strong OPO 65 % efficiency Experiments Weak OPO 65 % efficiency Theory Strong OPO 65 % efficiency 100 % efficiency

35. Conclusion/outlook, Many atoms and many photons are ”easy experiments” (classical fields, homodyne detection) Many atoms and many photons is ”easy theory” (readily generalized at low cost to more samples/fields) Gaussian states: squeezing, entanglement, … and also: finite bandwidth sources, finite bandwidth detection Gaussian states unify quantum and classical variables: classical B-field + atoms + light probe  other observables: interferometry, …. First step to non-Gaussian states, discrete qubits, Schrödinger Cats, ….

36. The atom-light coupling κ = -(3Γσ/TAΔ) √| | √| | | | = N phot /2 = Flux T/2 | | = N at /2 кT ~ 1 – 100, for realistic pulses and samples (10 12 atoms) (Single atom-single photon coupling: 10 -3 effect)

37. ”…. Every atom, impressed with good and with ill, retains at once the motions which philosophers and sages have imparted to it, mixed and combined in ten thousand ways with all that is worthless and base. The air is one vast library, on whose pages are for ever written all that man has ever said or woman whispered.” Charles Babbage Ninth Bridgewater Treatise, 1837