Conditional Measurements in cavity QED. Luis A. Orozco Joint Quantum Institute Department of Physics

Transcription

1 Conditional Measurements in cavity QED Luis A. Orozco Joint Quantum Institute Department of Physics

2 University of Maryland, College Park, Maryland: Matthew L. Terraciano Rebecca Olson David Norris Jietai Jing Miami University, Oxford, Ohio: Perry Rice James Clemens University of Auckland, New Zealand Howard J. Carmichael Work supported by: National Science Foundation, National Institute of Standards and Technology

3 Cavity QED: Quantum electrodynamics for pedestrians (Haroche). No need to renormalize. Only one mode of the electromagnetic field: ATOM(S) + CAVITY MODE Perturbative: dissipation >> coupling (Purcell) Non Perturbative: dissipation << coupling (Sanchez Mondragon, Eberly) 5

4 Coupling: g = d E v ħ d depends on the radial and angular parts of the wavefunctions of the electrons. For the D 2 lines in alkali (Rb) it is a few times a 0 (el radio de Bohr radius) times the electric charge. d = e 5 S r 5P = 5. 96a e 1/ 2 3/ 2 0

5 The field associated with a photon in a mode of the cavity with volume V eff is: E v = ħω 2ε 0 V eff The square of the electric field is an energy density.

6 How large is the electric field of a photon in an optical cavity as those from Maryland, CalTech or Garching? Electric field in a wall plug! ~ 100 V/cm It is possible to measure it!

7 Cavity QED System g, 2π κ, 2π γ / 2 2π ( 2, 2.6, 3)MHz

8 Jaynes-Cummings Model How does a single atom interact with a single mode of the electromagnetic field? 1 H c a ( a 2 a + + = ħωaσ z + ħω + ħg σ + σ + a) No interaction term: g,0 ground state e, 0, g,1 degenerate excited states With interaction term: g,0 ground state 1 ± = 2 excited states ( e,0 ± g,1 )

9 The interaction splits the degenerate excited state. + e, 0, g,1 2ħg ħω C This is the so-called vacuum Rabi splitting. g,0 g,0

10 Dissipative Processes κ: Loss of a photon from the cavity due to imperfect mirrors. γ: Spontaneous emission from an atom inside the cavity to modes other than the cavity mode. Be careful about impedance matching! Dissipation is not always bad. Cavity loss allows us to look inside of the cavity to study the dynamics of the system. Can we use spontaneous emission in the same way?

11 Conditional dynamics from the system wavefunction Ψ ss = 0, g + λ 1, g 2 g γ λ 0, e + 2 λ pq 2 2, g 2 2 gλ q γ 1, e λ = a ˆ, p = p( g, κ, γ ) and q = q( g, κ, γ ) A photodetection collapses the steady state into the following non-steady state from which the system evolves. aˆ Ψ ss Ψ collapse = 0, g + λ pq 1, g 2 g λ q γ 0, e Ψ [ ] ( 2 0 e O ) ( τ ) =, + λ f ( τ ) 1, g + f ( τ ) 0 g + λ 1 2, Field Atomic Polarization

12 Intensity correlation function measurements: g (2) ( τ ) = Iˆ( t) Iˆ( t Iˆ( t) + τ ) 2 Gives the probability of detecting a photon at time t + τ given that one was detected at time t. This is a conditional measurement: g (2) ( τ ) = Iˆ( τ ) Iˆ c

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27 Exchange of excitation: Coupled system, maybe entanglement.

28 Non-classical antibunched Classically g (2) (0)> g (2 )(τ) and also g (2) (0)-1 > g (2 )(τ)-1

29 Cavity cavity waist ~ 50 microns mirror separation ~ 2.2mm Transmission of mirrors = 15 ppm and 300 ppm 2κ = 6.4 MHz Finesse = 10,000 birefringence: 1:10 4, separation of modes < 1 MHz

30 Pushed MOT Apparatus Magneto-optical trap of 85 Rb atoms N ~ 10 8 atoms. Loading time ~ ms. Launched toward the cavity with a resonant push beam.

31 Transmission of Cavity Beam Transmission: X Y = ( C) 2 Y X Normalized intracavity intensity with atoms Y Normalized intracavity intensity without atoms C NC = 1 2 Ng κγ = is the cooperativity parameter. X

32 Experimental Schematic LVIS Atom beam APD Probe APD PBS B Drive the cavity with linearly (π) polarized light. Light emitted in orthogonal direction must come from a spontaneous emission. Separate the output into two polarizations to distinguish spontaneous emission from cavity drive.

33 Atomic structure of Rb atoms. Quite different from 2 levels.

34 Atom transit data, measured with APD

35 Coincidence Measurements of Atom Transits

36 with atoms no atoms The number of coincidences for a given gate time with the expected Poissonian results (black line).

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39 ΨSS = 0g + A1 g 1g + A0 e 0e + A2 g 2g + A1 e 1e + A0 ee 0ee If the steady state wave function is separable then we should be able to write the wave function as a product state: Ψ SS = ψ C ψ A ( D + D 1 + D ) ( C g C e ) = g + If we let A 1G = D 1, A 2g = D 2, and A 0e = C e, then the following condition must be satisfied for a product state: A 1e = D 1 C E = A Oe A 1g e If this is not true There is entanglement!

40 = = σ σ σ σ a a a a A A A j e g e (2) (0) = = σ σ τ σ τ σ σ σ τ σ τ σ τ c a a a a j ) ( ) ( (0) ) ( ) ( (0) ) ( (2) The function, j (2) (τ), measures correlations between transmitted and fluorescent clicks. A measurement of j (2) (0) that differs from unity, is a witness of entanglement.

41 LVIS ( Low Velocity Intense Source) Continuous source of cold atoms ~ 1 atom in cavity at all times! Similar to MOT but with a retro optic in the vacuum with a hole for extracting a beam of atoms. Can be pulsed by plugging hole in retro optic with a MOT beam and unplugging after a MOT forms.

42 Autocorrelation of Fluorescence Mode Conditional Average on Oscilloscope b b τ b τ b b b + + ( 2) (0) ( ) ( ) (0) g FF ( τ ) = + 2 Traditional Correlation Measurement

43 Quantum Trajectory Theory 15 photons in driven mode

44 Weaker driving field Measured Cross-Correlations

45 Quantum Trajectory Theory 3 photons in the driven mode, 0.1 in the undriven mode

46 Measuring the Concurrence Recall that the concurrence is given by: Con = 2 2 (2) A1 g A0 e(1 2 j (0) + j (2) (0) 2 ) j (2) (0) can be extracted from the cross-correlation histogram, if we assume (2) (2) the mapping (0) (0) j g TF is valid. 2 2 (2) Con ~ A A (1 2 g (0) + g (2) 1g 0e TF TF (0) 2 ) where A 1g (0e) = X 1g (0e) n 0 = R 1g (0e) 2κn 0 R is the flux of photons emitted from the cavity and n 0 is the saturation photon number. 2 n 0 γ = 3g 2

47 Concurrence Measurements

48 Summary We have devised a unique way of observing spontaneous emission from our cavity QED system. We have correlated spontaneous emission photons with transmitted photons to measure entanglement.