Correlation functions in optics; classical and quantum 2. TUW, Vienna, Austria, April 2018 Luis A. Orozco

Transcription

1 Correlation functions in optics; classical and quantum 2. TUW, Vienna, Austria, April 2018 Luis A. Orozco

2 Correlations in optics

3 Reference that includes pulsed sources: Zheyu Jeff Ou Quantum Optics for Experimentalists World Scientific, Singapore 2017

4 The study of noisy signals uses correlation functions. <F(t) F(t+τ) > <F(t) G(t+τ)> For optical signals the variables usually are: Field and Intensity. Photocurrent with noise:

5 How do we measure these functions? G (1) (τ) = <E(t)* E(t+τ)> field-field G (2) (τ) = <I(t) I(t+τ)> intensity-intensity H(τ) = <I(t) E(t+τ)> intensity-field

6 Correlation functions tell us something about fluctuations. The correlation functions have classical limits. They are related with conditional measurements. They give the probability of an event given that something has happened.

7 Mach Zehnder or Michelson Interferometer Field Field Correlation ) ( ) ( ) ( ) ( * (1) t I t E t E g τ τ + = τ τ ωτ π ω d g i exp F ) ( ) ( 2 1 ) ( (1) = Spectrum: This is the basis of Fourier Spectroscopy

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9 Handbury Brown and Twiss Can we use fluctuations, noise, to measre the size of a star? They were radio astronomers. R. Hanbury Brown and R.Q. Twiss, A New Type of Interferometer for Use in Radio Astronomy, Phil. Mag. 46, 663 (1954).

10 Flux collectors at Narrabri R.Hanbury Brown: The Stellar Interferometer at Narrabri Observatory Sky and Telescope 28, No.2, 64, August 1964

11 Narrabri intensity interferometer with its circular railway track R.Hanbury Brown: BOFFIN. A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics (1991)

12 The HBT controversy

13 Source a and b are within a Star. Can we measure the angular distance R/L~θ so that we could know the diameter? Source a: Source b:

14 The amplitude at detector 1 from sources a and b is: And the intensity I 1 The average over the random phases φ a and φ b gives zero And the product of the intensity of each of the detectors <I 1 ><I 2 > is independent of the separation of the detectors.

15 Multiply the two intensities and then average. g (2) This function changes as a function of the separation between the detectors. with

16 Relation to the Michelson Interferometer The term in parenthesis is the associated to the fringe Visibility (first order coherence) if we now take the square of the fringe visibility and average it:

17 The solution of E. M. Purcell, Nature 178, 1449 (1956). Mentions the work of Forrester first real optical intensity correlation. A. T. Forrester, R. A. Gudmundsen and P. O. Johnson, Photoelectric Mixing of Incoherent Light, Phys. Rev. 99, 1691 (1955). Mentions that bosons tend to appear together Does the calculation and relates it to the first order coherence.

18 Hanbury Brown and Twiss; Intensity Intensity correlation g (2) ( τ ) = I( t) I( t + τ ) I( t) 2 R. Hanbury Brown and R.Q. Twiss, Correlation between Photons in Two Coherent Beams of Light, Nature 177, 27 (1956).

19 Correlations of the intensity at τ=0 It is proportional to the variance

20 Intensity correlations (bounds) 2I ( t) I( t Cauchy-Schwarz 2 + τ ) I ( t) + I ( t +τ ) 2 The correlation is maximal at equal times (τ=0) and it can not increase.

21 How do we measure them? Build a Periodogram. The photocurrent is proportional to the intensity I(t) n i M i N n i j i I I t I t I I t I I t I + = = ) ( ) ( ) ( ) ( τ τ Discretize the time series. Apply the algorithm on the vector. Careful with the normalization.

22 Discretize: I i Multiply with displacement : I i+n Add and average:

23 Another form to measure the correlation with with the waiting time distribution of the photons. The minimum size of the variance of the electromagnetic field. Store the time separation between two consecutive pulses (start and stop). Histogram the separations If the fluctuations are few you get after normalization g (2) (τ). Work at low intensities (low counting rates). Intensity (photons) time

24 Use a 4me stamp card. Later process the data, then you can calculate all sorts of correla4ons. Digi4ze the full signal (important to iden4fy the nature of the event e.g. par4cle physics).

25 Simultaneous measurement of g (2) (τ). Digital storage oscilloscope (DO) captures the photocurrent from the photomultiplier tube (PMT). The correlation is calculated later Waiting time distribution with avalanche photodiodes (APD). The time to digital converter (TDC) stores the intervals for later histogramming.

26 Intensity correlations of a diode laser driven by noise. The Wiener-Khintchine-Kolmogorov gives the unnormalized correlation function G (2) (t) from the power spectral density of the noise.

27 Power spectrum of the noise source.

28 Comparison of g (2) (τ) : (a) waiting time distribution of photon counts, (b) time series of photocurrent.

29 Quantum optics The photon is the smallest fluctuation of the intensity of the intensity of the electromagnetic field, its variance. The photon is the quantum of energy of the electromagnetic field. With energy ħω at frequency ω.

30 An important point about the quantum calculation g (2) (τ)

31 Quantum Correlations (Glauber): The intensity operator I is proportional to the number of photons, but the operators have to be normal (:) and time (T) ordered. All the creation operators do the left and the annihilation operators to the right (just as a photodetector works). The operators act in temporal order. R. Glauber, The Quantum Theory of Optical Coherence, Phys. Rev. 130, 2529 (1963).

32 At equal times (normal order) : Conmutator : â + â = â â + 1 â + â + â â = â + (ââ + 1)â = â + â â + â â + â â + â + â â = ˆn 2 ˆn where ˆn = â + â The correlation requires detecting two photons, so if we detect one, we have to take that into consideration in the accounting.

33 In terms of the variance of the photon number: The classical result says:

34 The quantum correlation function can be zero, as the detection changes the number of photons in the field. This is related to the variance properties: is the variance larger or smaller than the mean (Poissonian, Super-Poissonian or Sub- Poissonian).

35 At equal times the value gives: g (2) (0)=1 Poissonian g (2) (0)>1 Super-Poissonian g (2) (0)<1 Sub-Poissonian The slope at equal times: g (2) (0)>g (2) (0 + ) Bunched g (2) (0)<g (2) (0 + ) Antibunched Classically we can not have Sub- Poissonian nor Antibunched.

36 Quantum Correlations (Glauber): g (2) (τ ) = : Î (τ ) : c : Î : If we detect a photon at time t, g (2) (τ) gives the probability of detecting a second photon after a time τ.

37 Correlation functions as conditional measurements in quantum optics. The detection of the first photon gives the initial state that is going to evolve in time. This may sound as Bayesian probabities. g (1) (t) Interferograms. g (2) (t) Hanbury-Brown and Twiss. Can be used in the process of quantum feedback.

38 Example with nanofibers

39 optical nanofibers

40 Intensity correlations from cold atoms around the nanofiber

41 g (2) (τ) Classical each atom is like a beacon while around the ONF residuals (σ) τ (µs)

42 Quantum Correlations g H2L HtL Ê Ê ÊÊ Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê ÊÊ Ê ÊÊ Ê Ê Ê ÊÊ ÊÊ Ê Ê ÊÊÊÊ Ê ÊÊ ÊÊ Ê Ê Ê Ê Ê Ê ÊÊ ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê ÊÊ Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Antibunching and Super-Poissonian Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊÊ ÊÊÊ Ê ÊÊ Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê ÊÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê ÊÊ Ê Ê Ê Ê ÊÊ Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ ÊÊ Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê t HmsL See also: S. L. Mielke, G. T. Foster, and L. A. Orozco, "Non-classical intensity correlations in cavity QED," Phys. Rev. Lett. 80, 3948 (1998).

43 Relation between temperature and correlation time

44 Are there quantum effects in optical cavity QED? Look at the intensity fluctuations

45 Optical Cavity QED Quantum electrodynamics for pedestrians. No need for renormalization. One or a finite number of modes from the cavity. ATOMS + CAVITY Regimes: Perturbative: Coupling<< Dissipation. Atomic decay suppressed or enhanced (cavity smaller than λ/2), changes in the energy levels. Non Perturbative: Coupling>>Dissipation Vacuum Rabi Splittings. Conditional dynamics.

46 Dipolar coupling between the atom and the mode of the cavity: g = d! El electric field associated with one photon on average in the cavity with volume: V eff is: E v E v =!ω 2ε 0 V eff

47 LIGHT EMPTY CAVITY PD SIGNAL

48 Spontaneous emission Coupling Cooperativity for one atom: C 1 Cooperativity for N atoms: C C = g2 1 κγ g κ γ Cavity decay C =C 1 N

49 Steady State Atomic polarization: -2Cx 1+x 2 y Excitation x Transmission x/y= 1/(1+2C)

50 Jaynes Cummings Dynamics Rabi Oscillations Exchange of excitation for N atoms: Ω g N

51 2g Vacuum Rabi Splitting Entangled Not coupled Two normal modes

52 Transmission doublet different from the Fabry Perot resonance Scaled Transmission Frequency [MHz]

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

77 Use correlations

78 79 Gracias