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Introduction to Quantum Optics for Cavity QED Quantum correlations of the intensity and the field. Luis A. Orozco Joint Quantum Institute Department of Physics at UMD and NIST State Equation of the dimensionless fields. ⎡ 2C ⎤ y = x ⎢1+ 2 ⎥, ⎣ 1+ x ⎦ 2C x=y− 2 1+ x Output=Input-Atomic polarization that can saturate € Atomic Polarization -2Cx 1+x2 y x Drive Transmission Steady State: Exchange of Excitation: Intensity response to step down of the atoms-cavity system. Steady State: Exchange of Excitation: Jaynes-Cummings Model 1 + − + + H = ω aσ z + ω c a a + g(σ a + σ a) 2 No interaction term: ground state With interaction term: ground state degenerate excited states excited states This interaction splits the degenerate excite states. This is the so-called vacuum Rabi splitting. Same Physics as two coupled oscillators (field mode and atomic polarization). We only have one excitation in the system, there is no difference between boson and fermion. The study of noisy signals is done with correlation functions. We have learned a lot about characteristic times, sizes and some dynamics with such correlation functions in statistical mechanics. They have the following form: Example of a Noisy photocurrent: <F(t) F(t+τ) > <F(t) G(t+τ)> For Optical signals the variables we want to correlate to themselves or to each other can be: Field and Intensity How do we measure these functions? G(1)(t+τ) = <E(t) E*(t+τ)> field-field G(2)(t+τ) = <I(t) I(t+τ)> intensity-intensity Η(t+τ) = <I(t) Ε(t+τ)> intensity-field Wave-Wave Correlation Michelson Interferomenter g (1) (τ ) = E * (t ) E (t + τ ) I (t ) Spectrum of the source F (ω ) = 1 (1) e xp ( i ωτ ) g (τ )dτ ∫ 2π Basis of Fourier Transform Spectroscopy Classical Intensity Correlation Functions: Hanbury Brown and Twiss. (HTB) two persons only! At equal time: With the variance: The equal time correlation function is related to the variance, and will be greater than 1. Discretize the time series: Ii Multiply by the delayed time series Ii+n Average the result and see the “correlation time” Another way to calculate the correlation function is with the waiting time distribution. Measure the separation between two consecutive pulses (start and stop) Histogram the distribution of separations. This is the same as g(2)(τ) if the fluctuations are very rare. Make sure the intensities are low, so that seldom you get coincidences. Intensity (photons) time Comparison of g (2) (τ) from photon counting (a), and from a time series of intensities measured with a single PMT (b). Correlation functions in quantum optics are conditional measurements. The detection of the the first photon prepares a state that then evolves in time. The correlation functions can be Field-Field: Mach-Zehnder, Interferogram. Intensity-Intensity: g(2)(t) Hanbury-Brown and Twiss. Use a digital Storage oscilloscope Start = trigger Stop = conditional average Very brief and incomplete history of the intensity correlation functions. 1955 Forrester measures “Photoelectric mixing of Incoherent light” 1956 Hanbury-Brown and Twiss; astronomy to measure the size of a star looking at the intensity with two different detectors, not interfering the fields as Michelson had proposed and doneo (bunching). 1963 Glauber and others (Mandel, Sudarshan, Wolf) formalize the quantum correlation funcions “Quantum theory of Coherence”. 1976 Experiment and Theory of single atom resonance fluorescence Kimble, Dagenais, Mandel; Carmichael and Walls. Shows the non-classical effect of antibunching. Earlier experiments for Bell Inequalities by Clauser with a cascaded atomic source can be interpreted as well as measurements of nonclassical properties of light. Quantum Mechanically (time and normal order): The variance of the number of photons is related to the probability of coincident photons given by g(2)(0). Light with Poissonian statistics has g(2)(0) = 1. Light with a super-Poissonian statistical distribution has g(2)(0)> 1. A sub-Poissonian distribution has g(2)(0) < 1, a clear signature of a nonclassical field. Intensity correlation function measurements: g ( 2 ) (τ ) = Iˆ(t ) Iˆ(t + τ ) Iˆ(t ) 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 7 663 536 starts 1 838 544 stops Regression of the field to steady state after the detection of a photon. Steady State: Exchange of Excitation: The state of the cavity QED system for N atoms is: The values of the coefficients are: The probability density of two simultaneous transmission of photons is then: 2 00 aˆ ψ 2 This can be zero if p is zero 2 = α pq 2 The conditional field prepared by the click is: A(t)|0> + B(t)|1> with A(t) ≈ 1 and B(t) << 1 Mostly one prepares the vacuum! The main object of interest is the FIELD Can we measure it? One photon over the coherence time of the cavity has a field of about 5 mV/m. HOMODYNE and record as a function of the phase of the local oscillator to obtain the field. We need interference. However, the random arrivals of the photons do not permit us to obtain it CONDITIONAL detection. Homodyne Measurement There is an interference between the Local Oscillator and the signal that is proportional to the amplitude of the signal |L.O. × cos(φ) + S.|2 = |L.O.|2 + 2 L.O.× S. × cos(φ) + |S.|2 Perfect detector I(t) = |α + Δα|2 I(t) = |α| 2 + 2 α Δα + |Δα|2 ; <α*α>=n DC~n Shot noise ~n1/2 neglect. Intensity-Field correlations H(t) = <I(t)E(t)> = <E(t)>conditioned Detection: Homodyne in a Mach Zehnder interferometer Conditional Measurement: Only measure when we know there is a photon. Source: Cavity QED Condition on a Click Measure the correlation function of the Intensity and the Field: <I(t) E(t+τ)> Normalized form: hθ(τ) = <E(τ)>c /<E> From Cauchy Schwartz inequalities: 0 ≤ h0 (0) − 1 ≤ 2 h0 (τ ) − 1 ≤ h0 (0) − 1 Photocurrent average into 50 Ω after one average, note the size of the fluctuations (about 100 mV) After 6,000 averages After 18,000 averages A signal has appeared with a size of about 3 mV peak to peak out of an original noise of 100 mV After 59,000 averages Photocurrent average with random conditioning Flip the phase of the Mach-Zehnder by 146o The fluctuations of the electromagnetic field are measured by the spectrum of squeezing. Look at the noise spectrum of the photocurrent out of the Mach Zehnder interferometer. ∞ S (ν ,0 ) = 4 F ∫ cos(2πντ )[h0 (τ ) − 1]dτ , 0 F is the photon flux into the correlator. The spectrum of squeezing is the Fourier transform of the normalized correlation function h(t) when third order moments are negligible. Regression of the field to steady state after the detection of a photon. Monte Carlo simulations of the wave-particle correlation and the spectrum of squeezing in the low intensity limit for an atomic beam. Spectrum of Squeezing from the Fourier Transform of h0(t) This is the conditional evolution of the field of a fraction of a photon [B(t)] from the correlation function. hθ(τ) = <E(τ)>c /<E> The conditional field prepared by the click is: A(t)|0> + B(t)|1> with A(t) ≈ 1 and B(t) << 1 We measure the field of a fraction of a photon! Fluctuations are very important.