Mesoscopic field state superpositions in Cavity QED: present status and perspectives Serge Haroche, Ein Bokek, February 21 st 2005 Entangling single atoms with larger and larger fields: an exploration of the quantum classical boundary
1. A reminder about Cavity QED: a story about springs and spins 2. Microscopic atom-field entanglement in Cavity QED 3. Mesoscopic atom field entanglement in dispersive regime 4. Mesoscopic atom field entanglement in resonant regime 5. Probing decoherence of mesoscopic superpositions 6. Perspectives: non-local mesoscopic entanglement
1. A story about springs and spins
Cavity QED: a story about springs and spins Towards Classical e g The spin: 2-level atom Mesoscopic field (n~10-100) 2 Microscopic 1 field n=0 The spring: Cavity mode Standard Hamiltonian H = hω at σ z 2 + hω ca a + hω σ + a + σ a Strong coupling 1 1 Ω >>, T c T a 50kHz : 1kHz : 30Hz
Coherent field state n Produced by coupling for a short time cavity to a classical source Superposition of photon number state with Poissonian distribution: α = C n n, C n = e α 2 /2 α n, n n! n = α 2, P(n) = C n 2 = e n n n n! Phase space representation Im(α) n=0 Amplitude α ϕ=ϕ 0 - ω c t Re(α)
Cavity QED set up Rev. Mod. Phys. 73, 565 (2001) Oven Source for photon injection in cavity High Q Cavity Circular Rydberg atom preparation (principal quantum number 51 or 50) Ramsey pulses source Field ionization detector
2. Microscopic atom-field entanglement in Cavity QED (atom and field behave as qubits)
Non resonant atom-field coupling: atomic light shift Non-resonant atom-field coupling e, n Ω 2 (n + 1) 4δ n? Vacuum Rabi frequency Ω Atom-field detuning δ g, n Ω2 n 4δ Measuring the atomic frequency inside box yields photon number Detecting these shifts without «looking» inside the box: Measure atomic coherence phase shift during the atom box crossing time by Ramsey interferometry! Amounts to non-destructive (QND) detection of single photon
How to realize a π atomic phase shift per photon? Dispersive and resonant methods e, n g, n Ramsey e g i Ω 2 (n + 1) Φ = Ω2 (2n + 1) 4δ t Φ = π per photon if Ω2 t δ Dispersive coupling = 2π Resonant atom-cavity interaction on e-g transition g,1 i,1 4δ Ω2 n 4δ ( i + g ) 1 i + e iπ g Ωt =2π e iπ g,1 Ωt =2π i,1 ( ) 1 ( i + g ) 0 i + g ( ) 0 2π Rabi rotation in 1 photon induces π phase shift of atom state (like spin 1/2) Works only in the 0-1 photon subspace
Quantum Non Demolition measurement of a photon G.Nogues et al, Nature, 400, 239 (1999) Single photon in C dephases fringes 0 photon 0 photon R 1 R 2 1 photon 1 photon R 1 R 2 Phases adjusted so that atom exits in one state if there is 0 photon in C, in the other if there is 1 photon (π phase shift in a one photon field) QND method with atom acting as a meter measuring the field: quantum gate operation If field is in a superposition of photon states, entanglement created: e c 0 0 + c 1 1 c 0 e,0 + c 1 g,1
3. Mesoscopic atom field entanglement in dispersive regime
Exchanging the roles of matter and field: Use mesoscopic coherent field for a QND measurement of atoms with 100% efficiency Im(α) An atom in level e or g dephases coherent field by Φ -Φ Φ = ± Ω2 4δ t g e g e Re(α) (single atom index) Measuring the phase of the field after the atom has crossed the cavity reveals the state of the atom (or the absence of atom) in a non-destructive way. In order to measure the field phase distribution, use an homodyne method: mix field with a reference with variable phase and measure resulting field intensity
Field phase distribution measurement How to measure a coherent field phase-shift? Cavity QED Homodyne method Injection of a coherent field i S Second injection αe φ > α > S Atom detector (field ionization) Resulting field i α(1 e φs ) > Back to the vacuum state φ S = 0 A probe atom is sent in g> -Field in the vacuum state P g 1 -Field in an excited state P g 1 / 2 φ S P ( φ ) g S = a signal to measure the field phase distribution Field phase-shifted by φ Maximum displaced by φ
Phase-shifting by single index atom Probe atom Coherent field source (used twice) Index atom Atom detector 0,70 0,65 0,60 0,55 0,50 P.Maioli et al, PRL to be published index atom in g v = 200 m/s δ = 50 khz n=29 photons φ 0 = 39 φ = φ 0 Signals conditioned to detection of index atom in e or g 0,45 0,75 0,70 0,65 0,60 0,55-150 -100-50 0 50 100 150 No index atom φ = 0 0,50 0,45 Macroscopic single atom effect Phase-shifts with opposite signs for -150 index atom in e and g -150-100 -50 0 50 100 150 0,70 0,65 0,60 0,55 0,50 0,45 Index atom in e -100-50 0 50 100 150 φ = + φ 0
Coupling the field to atom in state superposition: dispersive atom-field entanglement Classical π/2 pulse mixing e and g (R 1 ) S Classical source (initial injection of coherent field α>) Atom prepared in symmetric superposition of e and g before crossing the cavity storing α >. The two components of the atomic state rotate the phase of the field by two opposite angles: e e>+ g> χ = 1 4δ t f t i dt Ω 2 (t) The final atom-cavity state is entangled: the system evolves towards two states with different phases, correlated to the two atomic states. Ideal example of pre-masurement: the field is a meter whose state «points towards» the atomic energy: a Schrödinger cat state situation. e α R 1 1 2 e + g ( ) α C e iχ 2 e α e iχ + 1 2 g α eiχ +
4. Mesoscopic atom field entanglement in resonant regime
Resonant atom-field coupling: atomic eigenstates at classical limit Two-level system { e>; g>} interacting with a resonant field Rotating frame Rotating wave approximation Eigenstates of the hamiltonian + > = > = <σ z > P e 1 0 1 ( e > + g >) 2 1 ( e > + g >) 2 e> g> Field in rotating frame ( Ω cl ) hω hω cos( ωt) X Z g> e> cl - > +> Y 1 Rabi oscillation at frequency Ω cl Time (a.u.)
Evolution Classical Rabi oscillation: an interference effect e > ψ (t) >= 1 2 e iω cl t / 2 + > +e iω cl t / 2 > Detection in { e>, g>} basis ( ) e> Evolution Change of basis +> - > Detection Change of basis e> Classical Rabi oscillation: a quantum interference between two indistinguishable paths P 1 e P e 1 Time (a.u.) time
Coupling atomic eigenstates to mesoscopic coherent field (1 st order in n-n) e + 2 g e g 2 1 α e 2inϕ (t ) ψ at + (t) > α + (t) > 1 ( ) : αe iϕ (t ) > 2 e iϕ (t ) e > + g > α e 2inϕ (t ) ψ at (t) > α (t) > ( ) : αeiϕ (t ) > 2 eiϕ (t ) e > g > As in dispersive case, the atomic eigenstates induce opposite phase rotations on coherent fields. Field reacts back on atom and induces a correlated atomic dipole phase shift. Effect vanishes at classical limit (n ) P(n) n n ϕ(t) = Ω t 4 n Resonant phase shift
Representation in phase space Atomic state in the equatorial Representation plane in the same plane of the Bloch sphere ZY Coherent field in the Fresnel plane X +> +> Y X α > Equatorial plane Phase correlation of the Bloch sphere Atomic dipole and field «aligned»
e,α = 1 2 Evolution of atom-field system +,α +,α ( ) Single atom-mesoscopic field ( at ( t), ( t) > ) 1 Initial state +,α > ψ α + + 2 e 2inϕ (t ) ψ at + (t) > α + (t) > +e 2inϕ (t ) ψ at (t) > α (t) > at,α > ψ ( t), α ( t) > Fast quantum phase responsible for Rabi oscillation α + ( t) > entanglement A microscopic object leaves its imprint on a mesoscopic one Schrödinger-cat situation -> ϕ ϕ D α > +> "Size" of the cat=d D = 2 n sin Ω t 0 4 n α ( t) > The field acts as a Which-Path detector Contrast of the Rabi oscillation C(t) = < α + (t) α (t) > = e D2 (t )
Rabi oscillation in mesoscopic field collapses and revives as field components separate and recombine Atom and field disentangle: Rabi oscillation revives 1 P e (t) 0.8 0.6 0.4 Atom and field entangled: Rabi oscillation collapses 0.2 Complementarity in action! _ 0 t/2_ 10 20 30 40 50 At classical limit, collapse and revival times rejected to t =
Evidence of phase splitting v=335m/s n = 36 t int = 32µs Measured phase ϕ = 23 Expected value ϕ = Ω t Experiment and theory in 0 int 4 n = 23 very good agreement
5. Probing decoherence of mesoscopic superpositions
Fast decoherence of Schrödinger cat state Entanglement with environment: complementarity again αe iχ ± αe iχ E 0 αe iχ E χ ± αe iχ E χ en tan glement with environment Very quickly state superposition is turned into statistical mixture: αe iχ ± αe iχ ρ = αe iχ αe iχ + αe iχ αe iχ tracing over environment Interference terms in photon probability distribution P ± coh (n) = 1 2 n αe iχ ± n αe iχ 2 are washed out by decoherence P ± Dec (n) = 1 2 n αe iχ 2 + n αe iχ 2 = e n n n ( ) n! 1 ± cos2nχ P coh + (n) even cat (χ = π / 2) Loss of single photon in time ~T c /n = e n n n n! P + Decoh (n) Experiment: Brune et al, P.R.L77, 4887 (1996) Mixture of even and odd cats
Demonstrating the cat state coherence with a single atom by observing Rabi oscillation revival Observing the revival of Rabi oscillation would prove that the cat state produced during the collapse time was coherent but the revival time is, at present time, too long compared to the cavity damping time Solution: induce revival at earlier time by echo technique
Test of coherence: quantum revivals induced by time reversal Initial Rabi rotation, Apply short Stark pulse equivalent Reverse to π rotation phase Collapse around rotation Oz (transforms And slow +> phase into ->) Recombination of field rotation components induces revival of Rabi oscillation A spin echo experiment Time reversal experiments in Cavity QED to study decoherence proposed by G.Morigi, E.Solano, B.G.Englert and H.Walther, Phys.Rev.A 65, 0401202(R),2002.
Observation of induced Rabi revival signals Stark pulse Revivals up to 2T= 50 µs with n=13 photons Stark pulse Evidence of «long lived» coherent mesoscopic superpositions of field states T.Meunier et al, PRL, December 2004
How big a Schrödinger cat? Competition between preparation time t int (time of flight of atom across cavity) and decoherence time T c /n: In early work (1996), T c ~150µs restricted n to ~3-5 (dispersive experiment) In more recent work (2000-2003), T c ~1ms and n~30 t int ~30µs n T c t int n = 36 Cat state prepared by resonant atom-field coupling Very recently (December 2004), we have obtained T c ~14ms, opening the possibility to realize cat states with n~200-300.
Conclusions and perspectives Larger and longer lived cats (n in the hundreds) with better cavities Non local field states in two cavities Prepare and detect α, 0> + 0, α > (similar to n,0> + 0,n >) Marry the strangeness of Schrödinger cats with non locality: an exploration of the quantum-classical boundary Similar cat experiments with matter waves (BEC) instead of fields?
Support:JST (ICORP, Japan), EC, CNRS, UMPC, IUF, CdF S.Gleyzes P.Maioli T.Meunier G.Nogues M.Brune Alexia Auffeves J.M.Raimond