Rabi oscillation at the quantum-classical boundary Generation of Schrödinger cat states

Rabi oscillation at the quantum-classical boundaryGeneration of Schrödinger cat states Alexia Auffèves-Garnier Soutenance de thèse 29 juin 2004

About coherent superpositions Any superposition of states is a possible state Physical meaning of a "coherent superposition"? Young's hole experiment Interference fringes Addition of the probability amplitudes of each path Signature of a coherent superposition=interferencefringes Quantum phase of the superposition=phase of the fringes

Entanglement Coherent superposition for a bipartite system "Entangled state" = non factorisable state No system is in a definite state Quantum correlations Violation of Bell's inequalities Correlations in all the basis V V H The two photons form an EPR-pair H S Anticorrelation

The Schrödinger-cat paradox Entanglement of a microscopic system with a macroscopic one A two-level atom and acatin a box Total correlation The cat "measures" the atomic state Linear evolution The system form an EPR pair Quantum correlations Atom projected on |e>+|g> Cat projected on |dead>+|alive> Macroscopic state superposition

What enforces classicality: concept of complementarity No information about the state A superposition of states remains coherent A slit acts as a Which-Path detector Two possible states for the slit correlated with the two particle's states The particle-slit system gets entangled Contrast of the fringes No interference if there exist a Which-Path information

Decoherence A macroscopic object interacts with its environment and gets entangled with it No interference between macroscopic states Coherent superposition (dead> and |alive>) Decoherence Statistical mixture (|dead> or |alive>) Quantum correlations in all the basis Classical correlations in the « natural » basis Time of decoherence all the faster as the two cat states are more different

The quantum-classical boundary Microscopic object Environment -2 time scales -3 parts Mesoscopic object -Continuous monitoring of the environment -No entanglement -Classical behavior -Entanglement -« Schrödinger cat » states -Quantum behavior Classical world Quantum world Continuous parameter to explore the quantum-classical boundary?

Tools of CQED 51 (level e) 51.1 GHz 50 (level g) Vacuum Rabi period A two level system 1 mode of the electromagnetic field Rydberg states |e> and |g> High Q superconducting cavity Long life time Field relaxation Strong coupling regime

From quantum to classical Rabi oscillation Microscopic object Environment Rydberg atom Infinity of external modes Mesoscopic object Coherent field in the cavity mode Resonant interaction between the atom and the field Rabi oscillation Quantum Rabioscillation Classical Rabi oscillation -Atom-field entanglement -Preparation of mesoscopic coherences of the field -Field unsensitive to the atom -Factorized atom-field state

Outline • Elements of theory • Results on Rabi oscillation • Rabi oscillation at the quantum-classical boundary: the atomic point of view • Effect of the Rabi oscillation on the field • Experimental study • The atom-field system • Preparation of mesoscopic states superpositions of the field: experimental results • Evidence of correlation between atomic and field state • Coherence of the prepared superposition: where are the cats?

Basics on two-level systems |e> |e> Two-level system {|e>;|g>} |g> |g> General state A vector in the Bloch sphere |e> Z Dynamics of the system Precession around the eigenstates of the hamiltonian Y X |g>

Classical Rabi oscillation Z | - > Y |+> X |g> Time (a.u.) |e> Two-level system {|e>;|g>} interacting with a resonant field |g> Rotating frame Rotating wave approximation |e> Eigenstates of thehamiltonian Rabi oscillation at frequency

Rabi oscillation as an interference Time (a.u.) Evolution Detection in {|e>,|g>} basis Classical Rabi oscillation: a quantum beat between two indistinguishable paths |e> |+> |e> Evolution Change of basis Detection Change of basis | - >

Rabi oscillation in a quantized field |n+1> |e> |n> |g> |n-1> |e,n> |g,n+1> Rabi oscillation between |e,n> and |g,n+1> at frequency Two-level system {|e>;|g>} interacting with a resonant quantized field |n> Jaynes-Cummings hamiltonian Exchange of a quantum of energy |e,n>; |g,n+1> : two levels coupled and degenerate Eigenstates of the hamiltonian: "Dressed states"

The vacuum Rabi oscillation Initial state |e,0> Rabi oscillation at (t) e P 0.8 0.6 0.4 0.2 m time ( s) 0.0 0 30 60 90 Vacuum Rabi frequency Atom-field entangled state Maximal entanglement at Formation of an EPR-pair Rabi oscillation in a quantum field entanglement Rabi oscillation in a classical field No entanglement Continuous evolution?

Coherent states of the field Field radiated by a classical source in the mode Poissonian distribution of the photon number Representation in the complex plane "Quantum" field Big fluctuations "Classical" field Small fluctuations : a continuous parameter to explore the quantum classical boundary

Rabi oscillation in a mesoscopic coherent field with Collapses and revivals Collapse when the side components are phase-shifted by Spectrum of the frequencies for a mesoscopic field Revival when two successive components recover the same phase Distribution width Spectrum width

Classical limit Spectrum of the Rabi frequencies In a coherent field In the classical limit Atomic signal only depends on the energy of the field « Classical limit » Effect on the phase of the field?

Rabi oscillation in a mesoscopic field Initial state: with Mesoscopic field Atomic superposition of quantum phase Atomic superposition of quantum phase Coherent field of classical phase Coherent field of classical phase Phase correlation

Rabi oscillation in a mesoscopic field The atomic dipole and the field are phase-entangled Generation of a Schrödinger-cat state Classical limit No atom-field entanglement Field « classical » Field unchanged Classical Rabi oscillation

Geometrical representation |+> Phase correlation Atomic dipole and field « aligned » Atomic state in the equatorial plane of the Bloch sphere Coherent field in the Fresnel plane Representation in the same plane |+> Equatorial plane of the Bloch sphere

Evolution of the atom-field system Initial state A microscopic object leaves its imprint on a mesoscopic one Schrödinger-cat situation "Size" of the cat=D D |-> |+> The field acts as a Which-Path detector Contrast of the Rabi oscillation

New insights on collapse and revival Collapse as soon as the two components are well separated Factorization of the atomic state Maximal entanglement: "Schrödinger cat state" -Unconditional mesoscopic states superposition -The field has a defined parity Field states merge again Revival of the Rabi oscillation "Size" of the cat=distance D Time (a.u.)

A more realistic situation Q function evolution in 20 photons Atom initially in |g> Rabi oscillation in 20 photons ref: J. Gea-Banacloche, PRA 44, 5913-5931(1991) V. Buzek et.al., PRA 45, 8190-8203(1992)

Circular Rydberg atoms 51 (level e) 51.1 GHz 50 (level g) Atoms of High principal quantum number Maximal orbital and magnetic quantum numbers Selection rules: a two-level system Microwave transition Very long lifetime Very sensitive to small fields Stark tuning: Stable in a weak electric field Complex preparation (53 photons) on e-g transition Field ionisation detection

The superconducting cavity Mode TEM 900 Two superconducting mirrors in Fabry-Pérot configuration cooled down to 0.6K High quality factor High confinement of the field Field per photon Strong coupling regime Controlled potentiel between the mirrors Coupling with an external source Recirculation ring 2 modes: lift of degeneracy86kHz Small thermal field

General scheme of the experiment External source « Circularising box » Oven Velocity selection Detection by ionisation Lasers of preparation Atomic beam Cavity mode External source

Field phase distribution measurement Back to the vacuum state = a signal to measure the field phase distribution How to measure a coherent field phase-shift? Homodyne method Injection of a coherent field Second injection Resulting field A probe atom is sent in |g> -Field in the vacuum state -Field in an excited state Field phase-shifted by Maximum displaced by

Experimental field phase distribution Maximum<1 (thermal field) Width of the peak

Phase splitting in quantum Rabi oscillation: timing of the experiment Injection of a coherent field A first atom is sent and interacts resonantly with the field Detection of the atom in |e> or |g> Field projected on a superposition of and Vanishing of Injection of Vanishing of A probe atom is sent in |g> : two peaks corresponding to the vanishing of each component

Evidence of the phase splitting v=335m/s Measured phase Experiment and theory in very good agreement Expectedvalue

Evolution of the phase distribution Various velocities Various number of photons « Fast » atom « Slow » atom

Experiment vs theory theory (slope 1) numerical simulations -second mode -thermal field -relaxation experimental points Experiment and simulations in very good agreement Measured phase vs theoretical phase

Atom-field correlations in the { , } basis Entangled state Correlations in all the basis Mesoscopic superposition states of the field Atomic state with defined energy Evidence of the superposition Estimation of coherence (Wigner function) Correlation between and or between and and Selective preparation of

Selective preparation of Z Fast Stark pulse rotation around Z Preparation of |-> |g> Slow rotation in the equatorial plane End of Rabi oscillation Atom in its initial state Beginning of Rabi oscillation Field in state

Phase distribution analysis v=335m/s

Coherence of the superposition Preparation of a superposition of and Coherent superposition or statistical mixture? Experimental test based on induced revivals of the Rabi oscillation cf. Tristan Meunier’s thesis A quasi-probability distribution: the Wigner function -Acts on the phase space of a harmonic oscillator -Positive for a quasi-classical field Signature of coherence: Interference fringes Statistical mixture Coherent superposition

Coherence of the superposition: where are the cats? "Fast" atom "Slow" atom Statistical mixture Coherent superposition

Requirements for a cat living Size of the cat D independant of the photon number -The two components must be separated -The superposition must remain coherent with Condition to generate a Schrödinger cat:

Requirements for a cat living No cat Classical field Quantum behavior of the field No cat Classical field Observation of mesoscopic coherences With our setup Preparation and decoherence at the same time v=150m/s (reasonable hypothesis) Efficient and promising method to generate cats

Conclusions-perspectives Experimental study of the Rabi oscillation at the quantum-classical boundary Field classical No entanglement with the atom Generation of mesoscopic state superpositions Estimation of their coherence Experimental test of coherence (cf. Tristan Meunier’s thesis) Continuous monitoring of the decoherence process -2 atoms experiment -Direct measurement of the Wigner function

Rabi oscillation at the quantum-classical boundary Generation of Schrödinger cat states