Copenhagen interpretation Entanglement qubits 2 quantum coins 2

Copenhagen interpretation

Entanglement - qubits 2 quantum coins 2 spins ( spin “up” or spin “down”) Entangled state: Entangled state many qubits:

Entanglement – collective variables 2 ensembles of quantum coins 3 heads, 3 tails 4 heads, 2 tails

Now, imagine 1012 spins in each ensemble… 2 gas samples • Only interactions/measurements of the collective spin of each ensemble are necessary • Atoms are indistinguishable - high symmetry of the system – - robustness against losses of spins • No free lunch: limited capabilities compared to ideal maximal entanglement

Outline Continuous quantum variables Atoms: Collective spin of the sample Light: Stokes parameters of the pulse Hald, Sorensen, Schori, Polzik PRL 83, 1319 (1999) Entangling atoms via interaction with light Theory Kuzmich, Polzik PRL 85, 5639 (2000) Duan, Cirac, Zoller, Polzik PRL 85, 5643 (2000) Experiment: entangled state of two Cs gas samples – two macroscopic entangled objects Julsgaard, Kozhekin, Polzik Nature 413, 400 (2001). Quantum communication protocols with entangled atomic samples Proposals Kuzmich, Polzik PRL 85, 5639 (2000) Duan, Cirac, Zoller, Polzik PRL 85, 5643 (2000)

Quantum limits on the communication rate n photons, frequency n, duration dt t 1 2 3 4 frequency multiplexing Quantum memory

Quantum State (information) Processing Teleportation of atoms: Entangled ensembles Ensemble to be teleported Spin rotations target Light pulse Quantum memory for light: Write in the memory: map polarization state of light onto atomic spin Entangled ensembles memory Rotations of spin Unknown quantum state of light Memory read-out: map atomic spin state onto polarization of light Output beam memory Entangled light Polarization rotations Teleportation of light Innsbruck Rome Caltech-Aarhus

Why use ensembles of atoms? • Quantum information processing often requires efficient interaction between light and atoms • Entangled (squeezed) states of atomic ensembles are required in applications such as frequency standards Basic light - matter interaction: 1 Must be large Increase E with cavity: A photon gets many chances to interact with the atom. Caltech (H. J. Kimble) Munich (G. Rempe), . . . 2 And increase dipole moment d: Atoms in Rydberg states n=50, 51 are large and easy to hit with a photon Paris ( S. Haroche et al) 3 Use ensemble of atoms: Aarhus, since 1997

Spin memory with Coherent Spin States Quasi-continuous encoding Indistinguishable coherent states 90 0 180 270 • Densely coded states are impossible to read but possible to transfer via teleportation

Entangled or inseparable continuous variable systems • EPR example 1935 2 particles entangled in position/momentum • EPR state of light Ou, Pereira, Kimble 1992 Perfect EPR state Simon PRL (2000) Duan, Giedke, Cirac, Zoller PRL (2000) Necessary and sufficient condition for entanglement

Uncorrelated atoms Coherent state of spin-½ atoms x j=1/2 + y z j=1/2 + Along x: all tails d. Jz, y 2=Jx/2=N/4 Jx = y = N/2 z Along y, z: random misbalance between heads and tails

EPR state of two macro-spin systems [Jz, Jy] = i. Jx N and S condition for entanglement: Along x: all tails x z z y J 1 J 2 y Along y, z: ideally no misbalance between heads and tails of the two ensembles, or, at least, less than random misbalance

Two samples oppositely polarized x y Two entangled samples z y z -x

Total z and y components of two ensembles with equal and opposite macroscopic spins can be determined simulteneously with arbitrary accuracy x z x y z Therefore entangled state with Can be created by measurement

How to measure the total spin projections? • Send off-resonant light through two atomic samples • Measure polarization of light (Faraday effect)

6 P D 3/2 z x Cesium y 6 S 1/2 , F=4 Light / Atom - Interaction Faraday effect: Atomic spins rotate polarization of light z Continuous variables Bell measurement Back action: Light rotates spins of atoms A. Kuzmich and E. S. Polzik, Phys. Rev. Lett. , 85, 5639 (2000) x y Lu-Ming Duan, J. I. Cirac, P. Zoller, E. S. Polzik, Phys. Rev. Lett. , 85, 5643 (2000)

Entangled state of 2 macroscopic objects Entangling beam J 1 s+ pump Y X Z Y spump Z J 2 Polarization detection

Detecting quantum fluctuations of the spin Probe polarization noise spectrum 0, 0008 Noise power [arb. units] probe B Atomic density (a. u. ) Density [arb. units] -----------------1. 00 ± 0. 02 0. 56 ± 0. 01 0. 21 ± 0. 01 0, 0006 z y 0, 0004 1. 0 0, 0002 0. 2 Shot noise level 0, 0000 300000 310000 320000 330000 340000 Frequency (Hz) RF frequency Larmor frequency W=320 k. Hz Atomic Quantum Noise 2, 4 2, 2 2, 0 Atomic noise power [arb. units] , 0. 5 1, 8 1, 6 1, 4 1, 2 1, 0 0, 8 0, 6 0, 4 0, 2 0, 0 0, 2 0, 4 0, 6 0, 8 1, 0 Atomic density [arb. units] 1, 2 1, 4 1, 6 1, 8 2, 0

700 MHz 6 P 3/2 PBS Jx 2 s- Pumping beams Jx 1 s+ B-field Syout F=4 6 S 1/2 F=3 Entangling and verifying beams m=4 Optical Entangling pumping pulse W = 325 k. Hz Verifying pulse Entangling and verifying pulses 0. 5 ms Time

2) Create entangled state and measure the state variance "Probability" Distribution of CCS Distribution of the created entangled state after 0. 5 ms Uncertainty of the verifying pulse Normalized spectral variance 2. 0 ab 1. 5 CSS 1. 0 Entangled spin state Sy(1 pulse) 0. 5 0. 0 Atoms Light (1 pulse) 0 2 2 Fx Sy(1 pulse) 4 Collective spin of the atomic sample 6 Fx [1012]

Quantum communication protocols with entangled atomic samples and tunable EPR light Entangled atomic samples Entangled (EPR) light source Protocols (proposals): Quantum memory • Teleportation of atomic states • Light-to-atoms teleportation • Atom-to-light teleportation

Parametric downconversion in a resonator (OPO) w+ P=Im(E)=i( a+ - a) E- When the two fields are separated correlations – entanglement are observed: X- X+ X = Re(E)= a+ + a E+ P- P+

Frequency tunable entangled and squeezed light around 860 nm 800 MHz 107 photons per mode AOM Cavity modes Classical field LO+ AOM LO- { -

8 4 2 2 d(X + -X - ) [d. B(2 SQL)] 6 Degree of entanglement 0. 6 – observed 0. 65 – corrected for detector noise (1 - perfect ) 1 0 -2 -4 -6 -1 0 1 2 3 Phase [ p Radians] OR { 4 5 6

Quantum state of light stored in long lived spins quantum noise of light stored in atomic spin 2 Noise power RMS [V ] 0, 003 0, 002 d. S y<SQL d. S y=SQL Electronic noise 200 Hz 1. 5 msec storage 0, 001 light noise reduced by squeezing of fluctuations Freq. [Hz] 0, 000 0 200 400 600 800 1000 1200 Larmor frequency 1400 1600 Light with Controlled X and P – controlled quantum state

Quantum Teleportation of Light Furusawa et al Science, 1998 Caltech-Aarhus-Bangor + - in Classical channels Ea Eb Quantum channel Ap in Ax Actuators p EPR source in x Ap Eb Ax

Teleportation of an entangled atomic state Pulse 1 2 Pulse 1* Pulse 2 1 3 4 • Every measurement changes the single cell spin, BUT does not change the measured sum • Every pulse measures both y and z components of the sum – entanglement is created To complete teleportation of Spin 1 to cell 4: rotate spin 4 by A+B+C:

Operation: Teleportation of atoms Memory Bob Classical channel Memory Alice EPR spin Bob Coherent pulse Distance limitations: • Losses of light – fiber (3 d. B): 1 km at 850 nm 10 km at 1500 nm space: 100 km (diffraction) OR • (Life time of ERP atoms)x(speed of their transport) Currently: (0. 001 sec)x(Boeing 747) = 30 cm With 1 hour storage = 1000 km

Communication networks based on continuous spin variables Operation: Storage of light and read-out from atomic memory EPR pulses Memory Alice Memory Bob EPR spins Light Quantum channel Symbols : polarization rotation detection of light Input-Output interaction: free space off-resonant dipole interaction Continuous variables: • polarization state of light • spin state of atoms

Brian Julsgaard Christian Schori