Tutorial:Tutorial: quantumquantum informationinformation && quantumquantum opticsoptics // atomicatomic physicsphysics

Peter Zoller University of Innsbruck, Institute for Theoretical Physics SFB Coherent Control €U TMR 1 Road Map

• our goal is to implement quantum networks

• nodes: local quantum computing – quantum processors with atoms channel • channels: communication – transmission of photons

node

• quantum optics toolbox – photons –atoms

2 Quantum information processing

• quantum computing quantum weirdness: ouput |out 9 superposition 9 entanglement quantum |out  Û|in 9 interference processor 9 nonclonability and uncertainty |in input 9 no decoherence! • quantum communications

| 9 teleportation | | 9 crytography | |

transmission of a quantum state 3 Qubits, Quantum Gates etc.

• quantum bits or qubits

example: two qubits n spin1/2 1 | i |  c00|00  c01|01  c10 |10  c11|11 0 | i

• single qubit gate

 |   Û1| U1

• two qubit gate |a |a

|b |a  b control target Controlled-NOT 4 Examples from quantum optics

• ion traps '95 • cavity QED cavity

fiber

• neutral atoms: laser

µ1 µ2 laser

Vdipr interacting Rydberg dipoles optical lattice

These systems realize manipulation on the single quantum level. 5 Quantum computing: check list

• quantum memory • issues • single qubit gate – zero vs. finite temperature • two-qubit gate – single system vs. ensemble – quantum control – scalability • preparation – robustness • read out – speed – "controlled decoherence" – simplicity – ... • decoherence

6 Atoms as quantum Memory

single trapped atom:

|0 |1 qubit in longlived trap internal states

or: atomic ensembles:

+

7 • trapping and cooling of atoms and ions

linear ion trap atom "chips" 9 traps: conservative potential 9 load the trap 9 laser cooling: prepare motional ground state Remarks: long coherence times Blatt demonstrated.

Schmiedmayer

laser traps arrays of microtraps: optical lattice

FORT laser laser

8 Single qubit gates

Raman process Requirements:

Ω1 Ω2 laser addressing single qubit

laser |0 |1

1 Ω1Ω2 Rabi frequency Ω = eff 4 ∆ 2 spontaneous 1 Ω1,2 Γeff Γ emission ∼ 4 ∆2

Remarks: 9 state preparation by optical pumping 9 decoherence due to spontaneous emission (and 9 collisions) Two-qubit gates

• implement entanglement of two qubits

example: |00 |00 phase gate |01 |01 U 2 |10 |10 |11ei|11

• How?

9 auxiliary collective mode as data bus 9 controllable two body interactions

9 dynamical phases 9 geometric phases (holonomic quantum computing) 10 Two-qubit gates via quantum databus

- Entanglement via collective auxiliary quantum degree of freedom

quantum Collective mode data bus Examples: Ion traps (Cirac and Zoller PRL '95) switch Cavity QED (Pellizzari et at PRL '96) 1 2 qubits state vector: collective mode |  x cx|xN1xN2 x0  |  quantum register data bus gate: swap

requirement: cooling of the collective mode (= prepare a pure state)

11 Two-qubit gates via two-body interactions

- Controlled two-body interaction We must design a Hamiltonian

H Et|1 11|  |121| V(R) so that 12 i qubits |11  |12  e |11  |12

Examples: Cold collisions (Jaksch et al PRL ' 99) Ion trap 2000 (Cirac and Zoller, Nature 2000) Rydberg gate (Jaksch et al. PRL 2000)

12 Ion Trap Quantum Computer '95

J. I. Cirac, P. Zoller PRL '95

• Cold ions in a linear trap

Qubits: internal atomic states Quantum gates: entanglement via exchange of phonons of quantized center-of-mass mode laser pulses entangle ion pairs

• State vector

Ψ = cx xN 1,... ,x0 at om 0 phonon | i | − i | i X quantum register databus

• QC as a time sequence of laser pulses • Read out by quantum jumps (100 % efficiency) 13 Physics behind the quantum gate

• Swap the qubit (internal state) to phonon data bus

laser |0A  |1A |0phonon |0A |0phonon  |1phonon 

• Two qubit gate 1 2

step 1: laser swap qubit 1 to phononic bus

CNOT qubit 2 + phonon mode |0A|0phonon  |0A|0phonon

step 2: |0A|1phonon  |0A|1phonon laser |1A|0phonon  |1A|1phonon

|1A|1phonon  |1A|0phonon step 3: laser 14 The Innsbruck Linear Ion Trap

R. Blatt et al. 1997 - 2001 Achievements & expectations: • storing ~ 10 qubits • ground state cooling • addressing single ions • two qubit gate ... soon!? • few qubits within the next years? • decoherence times: ~ 40 gates

15 Boulder Linear Ion Trap

C. Monroe, D. Wineland et al., Nature 2000

• Lithographic trap • Maximally entangled state of N=4 ions

|0000  |1111

Mølmer - Sørensen protocol (no indivdual addressing) ions

200 µm • Bell state measurements • Two atom entangled state interferometry

16 A fast 2-qubit gate with Rydberg atoms

D. Jaksch, J.I. Cirac, P.Z., S. Rolston, M. Lukin and R. Cote, PRL 2000

• Rydberg atom in constant electric field • setup • linear Stark effect • permanent dipole moment

energy |r

n ~ 15 ...60 atom E   n2 huge!

laser

|g E

• Large dipole-dipole interaction E  1kV/cm µ1 µ2 R  opt/2  300 nm E  60 GHz for n  15 Vdipr Vdip  4 GHz 17 Two-qubit gate: internal dynamics

• atomic configuration • two qubit gate Atom 1 Atom 2

Vdip  u 11 11 | i → | i |r1 |r2 10 10 laser1 laser 2 | i → | i

|11 |01 |12 |02 01 01 qubit qubit | i → | i

00 eiφ 00 | i → | i

• dipole – dipole interaction = phase shift • force !? / 18 Scheme: 1,2  u gate time t  2/1,2  1/trap

• Energy levels of two atom 1 + atom 2

atom 1 atom 2 | rr〉 Vdip  u R

 2 | r1〉 | 1r〉 | r0〉 | 0r〉 Large dipole-dipole interaction shifts level  1  2  1 off resonance: No double excitation - | 11〉 | 01〉 | 10〉 | 00〉 no force!

• Laser pulse sequence "dipole blockade" 1 2 1

π 2π π time 19 • Laser pulse 1

atom 1 atom 2 | rr〉 Vdip  u R

| r1〉 | 1r〉 | r0〉 | 0r〉

 1  1

| 11〉 | 01〉 | 10〉 | 00〉

1 2 1

π 2π π time 20 • Laser pulse 2

atom 1 atom 2 | rr〉 Vdip  u R

 2 No excitation! | r1〉 | 1r〉 | r0〉 | 0r〉 "dipole blockade"  2

| 11〉 | 01〉 | 10〉 | 00〉

minus sign!

1 2 1

π 2π π time 21 • Laser pulse 3

atom 1 atom 2 | rr〉 Vdip  u R

9 Fast phase gate | r1〉 | 1r〉 | r0〉 | 0r〉 |00 |00 |01 |01  1  1 |10 |10 |11 |11 | 11〉 | 01〉 | 10〉 | 00〉 9 no force

1 2 1

π 2π π time 22 Atomic ensembles

M Lukin, L.M. Duan et al., PRL 2001 • mesoscopic atomic ensembles (instead of microscopic quantum objects)

- coherent manipulation of collective excitations of atomic ensembles

|r

laser laser N  100 atoms |q |g  10m ground state

-underlying physics:

dipole blockade

23 Manipulating collective excitations |r

|q • ground state |g

laser N |g   |g1|g2  |gN

• one excitation (Fock state)

|gN1 q |g |q |g blockade   i 1 i N laser

• two excitations

|gN2 q |g |q |q |g   i,j 1 i j N etc. 24 We can store and manipulate qubits. cont.

• qubits

N N1 |  |g   |g q +

superposition

• entanglement of ensembles

25 Atomic ensembles: quantum memory for light

• purpose

[ Atomic ensemble ] incoming light pulse storage medium outgoing light pulse

9unknown (arbitrary) state 9same state 9known shape of wave 9reshaping packet

• how? example ... cavity write a1 read

atoms

cavity laser theory: Lu Ming Duan; M. Lukin and M. Fleischhauer PRL '00 exp: R. Walsworth et al, PRL '01 26 L. Hau et al., Nature '01 Teleportation with coherent light + atomic ensembles

• theory: Lu Ming Duan, J.I. Cirac, P.Z. and E. Polzik, PRL Dec 2000 • experiment: E. Polzik et al., Nature 2001

• features – so far: quantum computing and communications requires 9 single atoms and single photons cavity 9 high-Q cavities atom

laser laser – now: can we get away with ... 9 atomic ensembles? 9 free space? ensemble simpler! of atoms

– continuous variable quantum communications – we use measurements to generate EPR states 27 Quantum Communication

• the goal of quantum communication is to transmit a quantum state reliably Alice φ Bob | i φ φ φ | i | i φ | i | i

transmission of a quantum state

• The quantum state can be a qubit |  |0  |1 or a continuous variable quantum state

 x  position |   dx|x x x,p  i  p  momentum 28 Teleportation of Continuous Quantum Variables

• continuous variable quantum states

x  position |   dx|x x x,p  i p  momentum

• continuous variable protocol 9 prepare an EPR state as a resource

TA B Vaidman Braunstein Kimble (exp) |EPR   dP|P A|  PB   dX |XA|XB

eigenstate of commuting operators:

2 X A  X B |EPR  X1|EPR  X A  X B  0 2 P A  P B |EPR  P1|EPR  P A  P B  0 29 cv teleportation cont. 9 the initial state is TA B

|in  |T  |EPR AB 9 we make a Bell measurement

X T  X A X2   projective measurement on |X2, P2TA P T  P A P2

result of measurement: |out  |X2, P2TA TA X2,P2|in

iX P iP X  |X2,P2TA  e 2 B e 2 B |B apart from a displacement B is in |

9 classical communication and displacement

X2 P2

30 |out  |X2, P2TA  |B we have teleported the state Physical realization with squeezed light

Braunstein and Kimble '96, Caltech exp '98

• squeezed vacuum as a cv EPR state

  a ra b ab pump |EPR  e |0A  |0B

1 2  n|n |n tanhr b    n0  A  B   

1  with quadrature components a  X A  iP A with a,a   1 2

– position representation: Gaussian

2 2r 2 2r XA,XB|E  expXA  XB e  XA  XB e  – variances 2 2r XA  XB  e 2 2r PA  PB  e

31 Implementation of cv teleportation

Caltech present

1. EPR: two mode squeezed light 1. atomic / spin squeezing

1 2 coherent parametric light downconversion EPR atoms 1 atoms 2 source measure EPR

9 quantum variables - quadrature components - atomic collective "spin" 9 EPR correlations - spin squeezing - light squeezing

• Questions: interactions? noise? 32 teleportation cont. Caltech present

2. Bell measurement

Bell classical communications

out

atoms 1 atoms 2 Bell EPR 1 2 in

EPR source atoms 3 teleport 3 to 2

coherent light

• Questions: - efficiency of Bell measurement - can we do better than with light?33 Scheme

ex atoms coherent light

• atomic level scheme • incident light: - coherent 3 4 - linear light polarization

+ ex = σ + σ− σ + σ - 1 2

1 1 M = M = + −2 2 • atoms initially: - prepared in superposition

1 2 34 Scheme

ex atoms coherent light

• atomic level scheme • incident light: - coherent 3 4 - linear light polarization

+ ex = σ + σ− σ + σ - 1 2

1 1 M = M = + −2 2 • atoms initially: • Remarks: - prepared in superposition 9 light propagation problem 9 spontaneous emission noise 1 2 35 1. Atoms as a continuous variable quantum system

• continuous atomic operators number of atoms 2Na  AL  1

1 zzi zz atoms  z, t  lim | |   z0 Az i i z

• collective spin operators for the atomic ground states

L a ρA S = σ + σ † dz x 2 12 12 Z 0 L ³ ´ a ρA S = σ12 σ † dz angular momentum: y 2i − 12 Z 0 Bloch vector L ³ ´ a ρA S = (σ11 σ22) dz z 2 − Z 0 36 atoms cont.

• superposition of the two ground states: coherent spin state

N 1 |1  |2 1 2 2 Bloch picture Bloch vector z Sa  Sa, Sa, Sa N a x y z   2 ,0,0

• quantum fluctuations P a Sa, Sa  iSa a a 1 a y z x Sy Sz  2 |Sx | y

a we treat Sx classically and rescale a x X Xa, Pa   i XaPa  1 2 9coherent spin state =vacuum state 9there are many cv quantum states canonical communation relations around it: a a a a |   dX |X X 37 2. Light (polarization) as cv quantum system

• one dimensional light propagation problem: polarization

 0 ik z t E z, t   aiz,te 0 0 ex 40A i, z    aiz,t, aj z , t  ijz  z 

• input • Stokes parameter strong coherent input with linear T p c † † polarization Sx = a1a2 + a2a1 dτ 2 0 ai (0,t) = α t Z ³ ´ h i T photon number p c Sy = a1† a2 a2†a1 dτ T 2 2i 0 − 2Np =2c α t dt 1 Z ³ ´ 0 | | À T p c R Sz = a1†a1 a2†a2 dτ 2 0 − Z ³ ´ 38 light cont.

• input and output

ex Rem.: free field aiz, t  ait  z/c atoms z commutation relations for a free field:

p p p p p Sx,y,z Sx,y,z Sy , Sz   iSx input output

• for linear light polarization the expectation value of the Stokes vector is

Stokes picture p p p p Np S   Sx, Sy , Sz    ,0,0 2 z p p we treat Sx  Sx   Np classically

P p Xp, Pp  i y |p   dXp |XpXp  Xp x 39 3. Dynamics: quantum noise propagation problem

3 4 coherent light atoms 1 atoms 2 σ + σ - in out 1 2

• Maxwell-Bloch equations

2 2 ∂ i g ρAσii g ρAγσii ai (z, τ)= | | ai (z, τ) | | ai (z, τ)+noise ∂z ∆c − 2∆2c 2 2 ∂ i g a2† a2 a1† a1 g γ 0 a2†a2 + a1†a1 σ12 = | | − σ12 | | σ12 +noise ∂τ ³ ∆ ´ − ³ 2∆2 ´

a p Kuzmich approx. by a spatial H Sz Sz PpPa mean field ∼ ∼ Polzik effective Hamiltonian: 40 dispersive interaction Dynamics: effective Hamiltonian

• we model the atom-light interaction as effective time evolution

Kuzmich Ueff  expiPp Pa atoms Polzik input output 9phase shift noise !? 9quadratic   Xp  Xp  UeffXpU eff   Pp  Pp  UeffPpU eff

• whole system + process

coherent light atoms 1 atoms 2 input output measurement

• we can describe dynamics in Heisenberg or Schrödinger picture 41 3.1 Dynamics in the Heisenberg picture

• effective dynamics • Heisenberg equation:

light: atoms  X Xp Pa input output p       Pp  Pp Xp  Xp  UeffXpU eff P P U P U atoms: p  p  eff p eff  Xa  Xa  Pp  Pa  Pa Ueff  expiPp Pa

42 Teleportation Step 1: atomic EPR correlations

• first round coherent measure light atoms 1 atoms 2 in out

 Xp1 Xp1

 Xp1  Xp1  Pa1  Pa2

measure out in: vacuum noise to obtain

Measurement of Xp1 projects the two atomic ensembles into an (approximate) eigenstate of Pa1  Pa2 2 1 2r Pa1  Pa2   e 1 2 2   43 cont.

• second round: we first rotate the atomic spins

Xa1  Pa1 Xa2  Pa2

Pa1  Xa1 Pa2  Xa2 and then we measure  Xp2  Xp2  Xa1  Xa2

measure out in: vacuum noise to obtain

• result: we have generated a cv EPR state

2 1 2r Pa1  Pa2  2  e 12 for   5 we have r  2 2 1 2r Xa1  Xa2   e 122

44 Teleporation Step 2: Bell measurement

Bell X0 = X κ(X X ) p1 p1 − a1 − a3

X0 = Xp2 κ(Pa1 + Pa3) p2 −

atoms 1 atoms 2 EPR measure out in: vacuum noise to obtain

1 atoms 3 efficiency   1  12 2

• combining 1 and 2: teleporation

F  1 fidelity 1 1 1 for   5 we have F  96%  2  2 12 2 45 Noise & Imperfections

• noise, and modelling of the noise as "beam splitters"

vacuum vacuum vacuum

coherent atoms 1 atoms 2 light t

spontaneous propagation loss as a spontaneous emission "beam splitter" emission

• example: propagation losses

 teleporation fidelity F  1/ 1  t

Even with a notable transmission loss rate t  0. 2, quantum teleportation with a remarkable high fidelity F  0.7 is still achievable. 46 light 3.2 Wave function atoms 1 atoms 2

• initial state

ψ = ψ p ψ a1 ψ a2 | i | i | i | i

∞ 1 p2 ∞ 1 p2 ∞ 1 p2 dp p e− 2 dpa1 pa1 e− 2 a1 dpa2 pa2 e− 2 a2 ∼ | i | i | i Z−∞ Z−∞ Z−∞

vacuum noise light atoms 1 and 2 z z

Xp Xa y y a P p P x x 47 light atoms 1 atoms 2

• interaction – time evolution

U =exp(iκpˆpˆa2)exp(iκpˆpˆa1)

– wave function

∞ ∞ ∞ 1 2 2 2 iκp(pa1 +pa2) (p +p +p ) ψ dp dpa1 dpa2 p pa1 pa2 e e− 2 a1 a2 | i ∼ | i| i| i Z−∞ Z−∞ Z−∞

light + atom initial vacuum noise entangled by interaction

48 light atoms 1 atoms 2

measure x • measurement – the atomic wave function after the measurement is

ψ a = x¯ ψ | i h | i ∼

∞ ∞ ∞ 1 2 2 2 ip[κ(pa1 +pa2)+¯x] (p +p +p ) dp dpa1 dpa2 pa1 pa2 e e− 2 a1 a2 ∼ | i| i Z−∞ Z−∞ Z−∞

49 light atoms 1 atoms 2

measure x • measurement – the atomic wave function after the measurement is

ψ a = x¯ ψ | i h | i ∼

1 2 1 2 2 ∞ ∞ (κ(pa1 +pa2)+¯x) (p +p ) dpa1 dpa2 pa1 pa2 e− 2 e− 2 a1 a2 ∼ | i| i Z−∞ Z−∞ initial: uncorrelated

pa2 interaction + measurement

pa1

1 ∆p = √2 50 light atoms 1 atoms 2

measure x • measurement – the atomic wave function after the measurement is

ψ a = x¯ ψ | i h | i ∼

1 2 1 2 2 ∞ ∞ (κ(pa1 +pa2)+¯x) (p +p ) dpa1 dpa2 pa1 pa2 e− 2 e− 2 a1 a2 ∼ | i| i Z−∞ Z−∞ initial: uncorrelated EPR pa2 pa2 interaction + measurement

pa1 pa1

1 1 ∆p = ∆(p + p )2 = e 2r a1 a2 1+2κ2 − √2 ≡51