The more we are ”ensemble” Two lectures and a talk . Klaus Mølmer University of Aarhus.

1. The more we are ”ensemble”Two lectures and a talk.Klaus MølmerUniversity of Aarhus. Lectures: • ”The simple, and not so simple, oscillator description of atomic ensembles” • ”Quantum optical states of interacting atoms and light” Talk (this evening): ”Quantum computing with collective encoding”

2. Lecture I Lecture II Research talk • Collaboration over the past decade with: • Lars Bojer Madsen, Vivi Pedersen, Jacob Friis Sherson, • Uffe V. Poulsen, Anne E. B. Nielsen, Zoltan Kurucs, • Etienne Brion, Line Hjortshøj Petersen, Aarhus • Eugene Polzik, Anders Sørensen, Copenhagen • Mark Saffman, Thad Walker, Madison • Rob Schoelkopf and Dave Schuster, Yale • Andrew Briggs, Hua Wu, Arzhang Ardavan, John Morton, Oxford • Janus Wesenberg, Singapore

3. Lecture I Lecture II Research talk • Collaboration over the past decade with: • Lars Bojer Madsen, Vivi Pedersen, Jacob Friis Sherson, • Uffe V. Poulsen,Anne E. B. Nielsen, Zoltan Kurucs, • Etienne Brion, Line Hjortshøj Petersen, Aarhus • Eugene Polzik, Anders Sørensen, Copenhagen • Mark Saffman, Thad Walker, Madison • Rob Schoelkopf and Dave Schuster, Yale • Andrew Briggs, Hua Wu, Arzhang Ardavan, John Morton, Oxford • Janus Wesenberg, Singapore

4. Lecture I: The simple (and not so simple) quantum description of a large atomic ensemble.Outline The ”cold” needle in the hot atomic hay stack: definition & addressing of a simple collective quantum system ”Doing by learning” the Keppler orbits and the ”hääyöaie” of K. F. Gauss. quantum measurements and state control A few complications, and how experimentalists can be right even when they do not deserve it.

5. Internal state: |b> |a> or |b> |a> |a> |b> or The quantum needle in the atomic hay stack Collective internal state: State with all atoms in |a>: …………….. |0> Symmetric state with one atom in |b>: ... |1> (|b,a, .. a>+|a,b,a .. a> + )/√N Symmetric state with k atoms in |b>: … |k> k= 0,1, … … N. If k << N ~103-1012,  harmonic oscillator If k ~ N ~ 1-100,  collective spin (N spin ½) Next: how useful is this assigment ? Rb vapour cell

6. = √N < |H1| > + + + + + + uper atom Σ Hn • Only needs Hamiltonian, invariant under permutations among particles

7. |↑> |↓> Collective state of N spin ½ particles • All atoms in (|↑>+|↓>)/√2  <Jx>=N/2, • Jz = (N↑-N↓)/2, <Jy>=<Jz>=0 • Binomial distribution: Var(Jz)=N/4 • Var(Jy)Var(Jz) = |<Jx>|2/4 is minimum allowed by Heisenberg uncertainty relation. • The polarized ensemble is in a collective quantum state ! • J+|J,M>=√(J-M)(J+M+1) |J,M+1> • J+|J,-J>=√2J √1 |J,-J+1> = √N |j,-J+1> Normally, ρ1 atom is quantum. Here, collective spins are our Quantum Variables !

8. |n> Complete equivalence with oscillator/collective spin • Eigenstates • Effective couplings • ”Standard coupling matrix elements”: J-,J+ a, a+ • And the Hermitian observables: Jy, Jz x, p • Imperfections and loss  density matrix theory. |J,Mx>

9. uper atom = quantum needle in a hot hay stack √N times stronger coupling to uniform mode (than to random (thermal) spin excitations). But … Assume 1012 spins: 99 % spin down, 1 % spin up That is 1010 spin up >> √1012 = 106. Can we still find and address the ”needle” collective excitation in this hay stack ?

10. Yes we can !!! Janus Wesenberg and Klaus Mølmer, Mixed collective states of many spins; PRA, 65, 062304 (2002). State of single spin: Collective tensor product state: • expansion in total collective spin basis: (spin, projection, symmetry) 2 spin ½ particles couple to singlet state and triplet manifold. n spin ½ particles couple to spin (j=n/2) manifold, with m = -n/2, -n/2+1 , … n/2 n-1 different spin (j=n/2-1) manifolds, … .  State is a direct sum of components in each of these manifolds

11. m m j + + + + ”Locking” in incoherent sum of very pure statesJanus Wesenberg and Klaus Mølmer, PRA, 65, 062304 (2002). Population of |j,m> ~ pj+m (1-p)j-m, w(m=-j+k) ~w(m=-j)∙pk m=-j cannot be further de-excited, by collective coupling. ”Locked entropy”. j ~n/2, narrow distribution for small p: same evolution in each j manifold,

12. How to address the collective quantum state Interaction Hamiltonian invariant under permutations (everybody interacts with everybody) Interact with a uniform probe field • Use a Bose-Einstein condensate • Use a hot gas: • Time averaged interaction • Doppler free, fx colinear Raman • Phase shift of far-detuned probes • Polarization rotation probe • Avoid decay and loss

13. Applications √N enhancement, collective adressing is crucial in • Slow and stopped light Harris, Hau, Lukin, Fleischhauer, … • Quantum state storage and teleportation Kuzmich, Polzik, Fiuracek, Cirac, Sørensen, Lukin … • Quantum spin squeezing and entanglement Kuzmich, Bigelow, Duan, Cirac, Zoller, Polzik, Mabuchi, K.M, Wiseman … • Quantum repeaters Lukin, Duan, Cirac, Zoller, Kimble, Kuzmich, Sørensen … • Precision probing, magnetometry Budker, Geremiah, Mabuchi,Polzik, … • Quantum computing on ensembles, blockade gates Jaksch , Lukin, Cirac, Saffman, Grangier, Pillet …

14. Charles Babbage (1791-1871)Inventor of the computer ”… . 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.” (.. and of the atomic ensemble memory:)

15. Atom light interaction. Atoms get excited Light field is damped, phase shifted or rotated (pol.) Maxwell: … as two coupled oscillators

16. |↑> |↓> Atom light interaction, as two coupled oscillators. Light oscillator: Nph x-polarized photons have Stokes vector components <Sx> = Nph/2, <Sy> = <Sz> = 0. let pph = Sz/√<Sx>, xph = Sy/ /√<Sx>, [xph,pph]=i (oscillator ”n” is number of σ-x photons) Interaction: Dispersive atom-light interaction: σ+ (σ-) light is phase shifted by |↑> (|↓>) atoms • Faraday polarization rotation, proportional to <Jz> Hint = g SzJz = к pat pph

17. Update of atomic state due to interaction with a light pulse Hint = кpphpat pphunchanged xph xph+ к tpat patunchanged xatxat+ к tpph xph is measured: we learn about pat, we ”unlearn” about xat • Probing of atoms: clocks, magnetometers, .. • Transfer of quantum states: from atoms to light, and from light to atoms • Measurement induced back action: squeezing, entanglement, … .

18. Hääyöaie = wedding night attempt (in Finnish) 1795, K. F. Gauss (cf. Daniel Kehlmann ”Measuring the World”, 2005) Estimate Keppler orbit parameters from noisy position measurements Law: x(t)x(t+dt); Observations: xobs(ti) Least squares fit (used in 1801 to locate ”lost” planet Ceres, without revealing his method) Problem; New matrix problem when data set accumulates. Solution: Kalman filter theory (Kalman, 1960). Probabilistic description of system parameters, continuosuly updated with each new piece of information.

19. Quantum measurements and classical parameter estimation. Kalman: Update with each new piece of information the current probabilistic description of system parameters. Quantum mechanics: Quantum state is ”state-of-knowledge”, quantum state reduction is ”enhanced knowledge” (Copenhagen). We can use ideas and techniques from the Kalman filter theory in quantum dynamics. Particularly simple for Gaussian distributions and Gaussian states.

20. Measurement back action on atomic state: New Gaussian . y2: Field y1: Atoms Linear transformation y Sy. Measurement of one component y2* of y = (y1,y2): New Gaussian iny1 (new mean (depends on y2*) and variance (determinstic)

21. Probing 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 variance This is a non-linear so-called Riccati equation. Can include effects of loss and decay. *: not true for finite bandwidth sources

22. Atomic spin squeezing due to optical probing For the simple atom-light example (binomial distribution): Review on Gaussian states, updates, sqeezing etc.: L.B.Madsen and K. Mølmer: Continuous measurements on continuous variable quantum systems: The Gaussian description, in "Quantum Information with Continuous Variables of Atoms and Light", Eds. N. Cerf, G. Leuchs, and E. S. Polzik. Imperial College Press, 2007; quant-ph/0511154.

23. Probing of a classical magnetic field Treat atoms AND light AND B field by a joint Gaussian probability distribution in variables (B,xat,pat,xph,pph). Analytical covariance matrix solution (no noise, constant field) • Independent of ΔB0 • not as 1/Nat ,1/t2 Long times: Var(B)~ 1/(N2at t3)

24. Real atoms Fine structure: Coupling between electron spin and orbital motion. Well separated excitation frequencies for different excited states. Ground state is degenerate (spin up and spin down, but there is more: hyperfine structure !)

25. Hyperfine structure: nuclear spin I + electron J Real Atoms not to scale Find the closed two-level transition Clock- frequency • Cs: I=7/2 • Fg= 3 or 4

26. Simple oscillator description of multi-level ensembles. Simple version: Assume that one sub-level |0> is strongly populated, e.g., by optical pumping. Treat occupation ni in all other states |i> as oscillator number operators. Define corresponding raising and lowering (and hence x and p) op’s. (A ”real” theorist would call this second quantization of the matter field). Harder (better) version: Use physical operators (Hamiltonians, Measured quantities) to define only the relevant oscillators. Collective O = ∑ Oi , withO |0> =α|0> + β|ψO> Then we can assign a standard collective X-observable (with Var(X)=1/2)), by the equation: O = N α + √2N βX + smaller terms. Time evolution under a single collective Hamiltonian, or measurements of a single operator, introduces only one effective oscillator. (Phys. Rev. A 81, 032314 (2010)):

27. O1 O2 Interactions and measurements combined Spin Squeezing of Atomic Ensembles via Nuclear-Electronic Spin Entanglement T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sørensen, and E. S. Polzik Phys. Rev. Lett. 101, 073601 (2008) Background: Collection of N spin-F atoms (Cs, Rb, … ), Squeezing of total angular momentum by internal squeezing of each spin-F atom, or by measurement of collective spin of all atoms. Two different observables: • Same X quadrature of a single oscillator • X and P quadratures of a single oscillator • X1 and X2 quadratures of two independent oscillators • X and P qudratures of partially correlated oscillators. Internal Fz squeezed state = ∑ ck |F,Mx = F-2k >, O1 = F+2 + F-2 (wrt. x). Measured observable (wrt. x-direction) O2 = Fz = F+ + F- Manipulation of either oscillator cause collective spin to be squeezed, but does it improve the system as storage device for a single field oscillator?

28. Answer is YES ! And it improves the memory by exactly the ”naive” expectation (as if the internal state squeezing, precisely squeezes the collective atomic mode of interest, before we store the light field.) But the explanation is non trivial: Recall: Collective O = ∑ Oi , defines a quadrature X operator O = N α + √2N βX via O |0> =α|0> + β|ψO>. If all atoms are internally squeezed, the macroscopically populated single atom state |0> in this formula is not the optically pumped state |F,Mx=F>, but the new reference state |0’> = ∑ ck |F,Mx = F-2k >,. Fz acting on that state has a larger β|ψO> component. This implies a larger probing strength of the collective Fz.

29. Conclusions part I Atomic ensembles are simple quantum systems They couple strongly to light They have lots of uses They can be managed theoretically Polzik group has pioneered collective state manipulation of ensembles, and with their demonstration of internal state control opened a great field to play with simultaneous external/internal state quantum control, and we seem to have the formalism to understand their results! So far, my atoms never interacted with each other. Squeezed and entangled states were caused by their interaction with light. Gaussian analysis is fine, but restricts states to be Gaussian. Stay tuned forLecture II: Ensemble performance with (suitably) interacting atoms.

30. Lecture II: Ensemble performance with (Rydberg) interacting atoms.

31. Internal state: |b> |a> or |b> |a> |a> |b> or Ensemble quantum physics:”a quantum needle in an atomic hay stack” Collective internal state: A single qubit: |0> = State with all atoms in |a> |1> = Symmetric state with one atom in |b>: (|b,a, .. a> + |a,b,a .. a> + … )/√N Rb vapour cell

32. = √N < |H1| > uper atom + + +

33. uper Qubit |0> |1> + + +

34. How do we restrict the collective population to zero and unity (a qubit) ? • How do we scale to more than a single qubit ?

35. How do we restrict the collective population to zero and unity (a qubit)? [ Excite the system with single quanta. (Cooper-Pair-Box + stripline cavity field ) Tonight’s talk ] This lecture: Rydberg excitation blockade. 1. Rydberg physics 2. Quantum computing 3. Quantum communication Rev. Mod. Phys to appear (arXiv:0909.4777) M. Saffman, T. G. Walker, K. Mølmer, Quantum information with Rydberg atoms,

36. Rydberg atoms Very large orbitals, large dipoles ~n2, and long lifetimes (msec) (Johannes Robert Rydberg 1854-1919) Interacting Rydberg atoms No mean dipole in atomic product states, but huge transition dipoles between states ! δ C/R3 C/R3 0 On/Off switch 1012

37. Rydberg excitation blockade Rydberg blockade: a single atom is excited at the resonant atomic transition frequency, but a second atom is not resonant at the same frequency, due to the interaction energy shift • A resonant laser will only excite one atom, further excitation is blocked • Partial suppression of excitation in large clouds • Controlled dynamics of two trapped atoms (Grangier, Saffman, 2009) Rydberg blockade gate: Jaksch et al PRL 85, 2208 (2000)

38. Rydberg blockade in ensembles If all atoms are within ~ 10 microns of each other, the isotropic Rydberg-Rydberg resonant interaction implements the ”super qubit”.

39. Interface to light Distributed computing on different clouds with flying qubits Long distance quantum communication Multi-atom collective effects in light emission (phase matching and ”superradiance”). Directional single photon source (collective em.).

40. Collective qubits in atom light interfacesL. H. Pedersen and K.M., Phys. Rev. A 79, 012320 (2009); arXiv:0807:3610 Directed absorption/emission by small sample (7 x 7 x 20 atoms) within 10 micron Rydberg blockade radius. Solution of coupled atomic equations  spatio-temporal field mode, Deterministic single photon source, mapping (< 0.2 rad) better than 95 %

41. Scalable quantum computing and long distance quantum communication • Few bit quantum repeater: • Interfaces well with light: Light emission in ”nice mode” • Light absorption (”single mode EIT”) Few bit distillation/purification Swap of qubit to four auxiliary states, e.g., distillation of entanglement Fidelity 0.95 0.999. (c.f. Jiang et al, PRA 76, 062323 (2007) (Line Pedersen and K.M, PRA 79, 012320 (2009);

42. Use multi-level atoms Atoms with N+1 internal levels Ensemble with all atoms in state |0> … |1> |N> |2> … |0>

43. Quantum computing with ensemble encoding Atoms with N+1 internal levels Ensemble with all atoms in state |0> Note: Analogy with optical computing. … |1> |N> |2> … |0> Symmetric collective excitation of a single atom into different internal states: encoding bit value 1 Number of bits ~ number of states (linear scaling)!!!

44. |r> i,iii) ii) … |i> … |0> One- and two- qubit quantum gates One-bit gates Two-bit gates |ri> |rj> Note: internal state Addressing only! … |i> |j> … |0> i: swap |i> - |ri> ii: drive |j>-|rj>-|j>  sign change Iii: swap |ri> - |i>  conditioned j-phase (if |0i>) i: swap |i> - |r> ii: drive |0>-|r> Iii: Swap |r> - |i>  rotation of |0i>,|1i>

45. We achieve 14 bits in a small cloud of cesium!E. Brion, K. M., M. Saffman, Phys. Rev. Lett. 99, 260501 (2007) Blockade works over 5 μm scale, e.g., an optical dipole trap or small lattice with 100-1000 atoms Cs hyperfine ground states coupled to n ~ 70 Rydberg states (Zeeman addressing)

46. We achieve 127 bits in a cloud of holmium!Mark Saffman and K.M., Phys. Rev. A 78, 012336 (2008) Ho 4I15/2 ground state contains 128 hyperfine states!!! And a thousand bits in a few coupled clouds.

47. Prepare and emit quantum states on demandA. E. B. Nielsen and KM, Phys. Rev. A 81, 043822 (2010); arxiv:1001.1429. Atomic ensemble is universal few-bit quantum computer  1. prepare any state of collective atomic register 2. release as photon/no photon or polarization qubit 3. Entangled wave packets

48. EPR polarization entangled states Atomic state:  Field Bell-state: Implement reference frame-free quantum communication (G. Tabia and B.-G. Englert, arXiv:0910.5375)

49. Arbitrary-N photon number GHZ states. Arbitrary-N polarization cluster* state (*: Eigenstate of [σxi Πjσzj ] for all vertices i, and n.n. j) N. H. Lindner, T. Rudolph, Phys. Rev. Lett. 103, 113602 (2009)

50. Polarization cluster state double barrel