8 Incoherent fields and the Hanbury Brown Twiss effect

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1 8 Incoherent fields and the Hanbury Brown Twiss effect References: Loudon, Quantum Theory of Light Ch. 3, Ch. 6 A. Aspect, D. Boiron, C. Westbrook, "L'optique quantique atomique", Reflets de la physique, No. 4, p , mai 2007, disponible au english version: "Quantum atom optics with bosons and fermions", Europhysics News, 39 vol. 1, p. 22 (2008), arxiv: G. Baym, The physics of Hanbury Brown-Twiss intensity interferometry: from stars to nuclear collisions, arxiv:nucl-th/ A. Review of results for coherent and 1 photon states We have used the model of photodetection via the photoelectric effect to discuss the processes of measuring photon count rates and pairwise coincidence rates for various states of a quantum electric field. Detection of 1 photon corresponds to the quantity w 1, and of 2 photons to w 2 : w 1 E r, t Ψ i 2, w 2 E r b, t E r a, t Ψ i 2 Using these formulae, we have analyzed detection after a beam splitter. For a single photon state, Ψ 1 we find: w c t 2, w d r 2, w 2 c, d 0. For a coherent state, Ψ Α we find: w c t 2 Α 2, w c r 2 Α 2, w 2 c, d w 1 1 c w d The coherent state result is indentical to what you would find for a Poissonian flux of particles which are sent randomly towards detectors c and d by the beam splitter. Would you get the same result for a lamp? The answer is NO, as shown by Hanbury Brown and Twiss in A diagram of their experimental apparatus is shown below. For a thermal source HB and T found: w 2 c, d 2 w c w d. Show Import " Users Chris Documents chris working enseign calculs figures movies figures from papers hbt lab expt.png" This result is a bit puzzling when viewed in terms of photons. If anything, a thermal source seems more random that a coherent one. But totally random photon counts ought to obey the relation w 2 c, d w 1 c w 1 d. There seems to be something not so random about a thermal source. We will give an explanation of this result using the quantum theory, but the HBT effect is in fact easier to understand using classical electromagnetism. In addition, the classical explanation is very instructive and therefore we will begin with a discussion of classical sources. We will see that the HBT effect is the result of fluctuating electromagnetic fields, and the study of these fluctuations is 2 useful both in a quantum and a classical context. The key quantity determining w c, d is the expectation value of E c Ed Ed Ec. To understand its properties we are thus led to a study of correlation functions in both classical and quantum electrodynamics.

2 2 13 OQ 8 hbt.nb B. Theory of a fluctuating classical light source Imagine that we have a source of negligible spatial extent which emits a fluctuating electric field. In a later section we will discuss a possible physical realization of such a source. The idea is that the complex electric field due to this source at some detector is given by: Ω t Φ t E t A t e where A and Φ fluctuate in time but are stationary. We neglect any spatial dependence or propagation effects for the moment. Stationary means that they fluctuate in such a way that their averages and higher moments A, A 2,... Φ, Φ 2,... are independent of time if the averaging time T is much longer than a correlation time a quantity which we will make more precise later. That is we can write A 1 T 2 T t A t T With these assumptions, the Wiener-Khintchine theorem applies and relates the correlation function of the field to its power spectral density (PSD). Wiener-Khintchine Theorem This useful theorem relates the correlation function of a fluctuating quantity to its power spectral density. Suppose E t is a fluctuating, stationary, quantity. Its Fourier transform is: E Ω 1 T 2 Π t e i Ω t E t assuming that T Τ c the correlation time of E t. T The power spectral density (PSD) of E is E Ω 2. E Ω 2 1 T T 2 Π 2 t t e i Ω t t E t E t T T 1 T T 2 Π 2 Τ t E t E t Τ e i Ω Τ T T T 1 2 Π 2 2 T T Τ E t E t Τ e i Ω Τ assuming T Τ c 1 2 Π 2 2 T Τ E t E t Τ e i Ω Τ The PSD is the Fourier transform of the correlation function and vice versa. A useful, possibly obvious corrollary: Ω E Ω 2 E t E t, which is proportional to the optical intensity. The PSD is the quantity one measures by analyzing the light from the source by a spectrometer. Correlations and interference contrast In addition to giving a way of describing the results of spectroscopy measurements. The correlation function described above also describes what happens in interferometry. Consider a simple Mach-Zehnder interferometer as in the diagram. Import " Users Chris Documents chris working enseign calculs figures movies 06 coh mach.pdf" A source emits a field E s t which is divided at the 1st beam splitter and recombined at the second. The field at the output arm c is the sum of the fields at the second beam splitter with an appropriate phase: E c t t 2 E s t r 2 E s t Τ where t 2 and r 2 are the transmission and reflection coefficients of the beam splitters, which here we will assume to be 1/2, and Τ is the time delay between the two arms and is related to the path difference by Τ d. A typical detector detects the intensity of the light, proportional c to the modulus of the electric field and averaged over many optical cycles:

3 13 OQ 8 hbt.nb 3 I detector 1 2 Ε 0 c E c t Ε 0 c 1 4 E s t 2 E s t Τ 2 2 Re E t E t Τ 1 4 Ε 0 c E s t 2 Re E t E t Τ We have assumed in the second line that the field is stationary. If we suppose that E s is of the form E s A e i Ω t Φ t and that the phase Φ is constant, we have ordinary (Mach-Zehnder) interference fringes. That is, the detected intensity varies with the path delay as I 1 cos Ω Τ. If on the other hand Φ is a fluctuating variable, we see that the E t E t Τ term is not so simple. The fringe contrast will be reduced for time delays which are too long. Thus interference contrast is determined by a correlation function and because of the WK theorem, by the spectral width of the source. A model of a fluctuating source (Collisional broadening) To discuss the role of the correlation function a little more concretely and to illustrate the Wiener-Khintchine theorem, it is instructive to consider a specific model of a radiation source. This discussion is adapted from Loudon Ch. 3. We will suppose the source consists of many atoms which, if they were isolated, would radiate light at a fixed frequency Ω 0. Let us assume however, that these atoms occasionally collide and that such a collision shifts the phase of the radiation of that atom by a random amount. E j t A e i Ω 0 t i Φ j t The label j denotes a particular atom, and we assume that the amplitude A radiated by each atom is fixed and the same for all atoms. The quantity Φ j t of a given atom is constant between collisions but undergoes large instantaneous changes at random times at a rate Γ Let us now assume that our source consists of a large number N of such atoms and that we can observe the electric field resulting from the sum over all the atoms. E tot t A e i Ω 0 t N j 1 e i Φ j t (1) The sum of the phase terms is a fluctuating quantity which occasionally can sum to a complex number with modulus N, occasionally to zero, and to anything in between. The phase of the sum is random and therefore the average value of the field is zero. Correlation time of the electric field (collisional broadening) What is the correlation time of this field? That is, what is the typical time over which the electric field remains unchanged in amplitude and phase? To find this time, we must compute the correlation function of the field. E t E t Τ A 2 e Ω 0 Τ j N k N e Φ j t Φ k t Τ A 2 e Ω 0 Τ j e Φ j t Φ j t Τ j k e Φ j t Φ k t Τ The second term in the (...) is zero because the phases of different atoms are uncorrelated and so each term e i Φ j t Φ k t Τ in the sum is zero after a sufficient averaging time. The first term can be written as N times the correlation function of a single atom: E t E t Τ A 2 N e Ω 0 Τ e i Φ j t Φ j t Τ For Τ 0 the quantity in brackets is unity, for other times it is equal to the probability p Τ that no collision has happened in the interval t, t Τ, because in our model, any collision results in a completely different phase. If collisions take place at a rate Γ, then p Τ e Γ Τ, and we have E t E t Τ A 2 N e Ω 0 Τ Γ Τ. This formula tells us how long a delay is possible in an interferometer like the one above while still observing interference fringes. This time, 1 Γ can be converted into a length, which is often referred to as the coherence length l coherence c Γ. We can also use the Wiener- Khintchine theorem to find the power spectral density of the source: E Ω 2 1 Ω Ω 0 2 Γ 2 We find we have a Lorentzian line shape, and we see that line width, the collision rate, and correlation time are closely related quantities. It is clear from the definition of the correlation function used in the Wiener-Khintchine theorem that the correlation function depends only on the relative time at which two field are evaluated: E t E t Τ E t Τ E t E t E t Τ This property allows us to give a correlation function for negative relative times: E t E t Τ A 2 N e Ω 0 Τ Γ Τ. A model of a fluctuating source (Doppler broadening) To discuss the role of the correlation function a little more concretely and to illustrate the Wiener-Khintchine theorem, it is instructive to consider a specific model of a radiation source. We will suppose the source consists of many atoms which, in their rest frame, radiate light at a fixed frequency Ω 0. Let us assume however, that these atoms are moving and that along a certain direction their frequencies in the fram of the observer are Doppler shifted: E j t A e i Ω 0 j t where each atom is assumed to have a different Doppler shift. We will assume that these Doppler shifts have a Gaussian distribution which one would expect from a Maxwell-Boltzmann velocity distribution. The total electric field is E tot t A e i Ω 0 t N j 1 e j t (1)

4 4 13 OQ 8 hbt.nb The sum of the phase terms fluctuates in time. Occasionally it can sum to a complex number with modulus N, occasionally to zero, and to anything in between. The phase of the sum is random and therefore the average value of the field is zero. Correlation time of the electric field (Doppler broadening) What is the correlation time of this field? That is, what is the typical time over which the electric field remains unchanged in amplitude and phase? To find this time, we must compute the correlation function of the field. E t E t Τ A 2 e Ω 0 Τ j N k N e i j t k t Τ A 2 e Ω 0 Τ j e i j Τ j k e i j k t k Τ The second term in the (...) is zero because the phases of different atoms are uncorrelated and so each term e i j k t k Τ in the sum gives zero after sufficient time averaging. The first term can be written the integral over a distribution of Doppler shifts D : and we have j e i Ω 0 j ) Τ e Τ 1 D e Τ e 2 Σ 2 e Σ2 Τ 2 2 Σ 2 Π 2 E t E t Τ A 2 N e Ω 0 Τ Σ 2 Τ 2 2. This formula tells us how long a delay is possible in an interferometer like the one above while still observing interference fringes. This time, 1 Σ can be converted into a length, which is often referred to as the coherence length l coherence c Σ. We can also use the Wiener- Khintchine theorem to find the power spectral density of the source: Ω Ω E Ω e 2 Σ 2 A Gaussian distribution, as we had constructed initially. Visualizing the field It is interesting to try to visualize the field E tot. The sum in Eq. 1 is the sum of random complex numbers of unit modulus. In the complex plane the sum can be represented by adding up randomly oriented unit length vectors in 2 dimensions, that is it corresponds to a random walk in 2D. Thus, if I ask what is the value of the electric field E tot at several different, widely separated times, I will find the same distribution as that of the distance traveled by a random walker in 2 dimensions. By widely separated I mean separated by more than the correlation time. By thinking in terms of random walks, we immediately see that the distribution of electric field values will be Gaussian, centered at the origin with an RMS deviation of A N. Below is a simulation of such a field. It s a little easier to simulate a Doppler broadened source so I begin with a Gaussian distribution of 100 frequencies, with an rms width of unity. The spectral width is thus unity Then I calculate j e j t for many, widely separated values of the time t. The rms width of the frequency distribution is unity, therefore the correlation time is also unity. Note that I begin running the time at t 100. At t 0, the fields would necessarily all be in phase.

5 13 OQ 8 hbt.nb = RandomVariate@NormalDistribution@D, 100D; etot = Ht + 100LD, 8j, 1, 100<D; ListPlotATableA8Re@etot D, Im@etot D<, 9t, 0, 105, 10=E, AspectRatio 1E I can also plot the modulus of the summed electric field etot. Below it is shown for 3 correlation times and for 50 correlation times. 5

6 6 13 OQ 8 hbt.nb Plot Abs e tot, t, 0, 3, PlotRange 0, 20, Plot Abs e tot, t, 0, 50, PlotRange 0, , We see that indeed, on time scales short compared to unity, the field is approximately constant, whereas on longer time scales, the field modulus fluctuates strongly, sometimes going close to zero. You can see that the RMS value of the field is about 100. Below is the value of the phase, whose correlation time is the same.

7 13 OQ 8 hbt.nb 7 Plot Arg e tot, t, 0, Intensity distribution We can also discuss the distribution of the values of the intensity. The intensity is related to the field modulus by I 1 2 Ε 0 c E tot Ε 0 c E r 2 E i Ε 0 c E 2 where E r and E i are the real and imaginary parts of E tot. We can find the distribution of intensities for a Gaussian electric field. First, we will define the function D E to be the distribution for the electric field in the complex plane as in the diagram above: D E E 1 exp E r 2 2 E i 2 2 Π Σ E 2 Σ E 2 The value of Σ E in our model source is A N. (The factor of 1/ 2 comes from the fact that Σ E is the RMS of the real and imaginary parts, whereas the mean of E is D E E E r E i D E E E E Φ D I I I times bigger.) The intensity distribution is defined by: We see that D I I Π D E E I 1 I exp I I. The average intensity is given by I 1 2 Ε 0 c 2 Σ E 2. On the other hand, the most probable intensity is zero. This is reminiscent of the photon number distribution in a thermal state. Recall that P n the probability to have n photons in a give thermal state is given by P n e n ħω k B T. You can verify that: I D I I I I I 2 I 2 D I I I 2 I 2 We conclude that for a Gaussian electric field, 2 I 2 I 2. In the HBT experiment we a classical analysis consists in assuming that the photocurrent in each detector is proportional to the incident intensity, and that the correlator takes the product of the two intensities before squaring. If we think of the the beam splitter as simply making 2, copies of the classical field, the HBT experiment consists in comparing I 2 and I 2. The observation simply corresponds to observing a fluctuating intensity and verifying that the mean squared is larger than the square of the mean, and there is nothing mysterious about it. Intensity correlation We are now ready to compute the intensity correlation function, a function that strongly resembles the quantity w c, d 2 we have discussed in earlier lectures: I t I t Τ E t E t Τ E t Τ E t j,k,l,m E j t E k t Τ E l t Τ E m t j E j t E j t Τ E j t Τ E j t j k E j t E k t Τ E k t Τ E j t j l E j t E j t Τ E l t Τ E l t... In the last line we have included only terms in which the field of each atom is multiplied by its complex conjugate. The contributions of different atoms are uncorrelated therefore the mean of the product is the product of the mean and so we can write: j k E j t E k t Τ E k t Τ E j t j l E j t E j t Τ E l t Τ E l t j k E j t E j t E k t E k t j l E j t E j t Τ E l t Τ E l t And since the contribution of each atom to the sum is the same field of each atom has the same mean we can write:

8 8 13 OQ 8 hbt.nb E t E t Τ E t Τ E t N E j t E j t Τ E j t Τ E j t N N 1 E j t E j t 2 E j t E j t Τ 2 If the number of sources is very large, the 1st term can be neglected and we can say N N 1 N 2. Then we have E t E t Τ E t Τ E t E t E t 2 E t E t Τ 2 If we now normalize the correlation function to E t E t 2 we have g 2 Τ 1 g 1 Τ 2 Where g 2 Τ E t E t Τ E t Τ E t and g 1 Τ E t E t Τ, are referred to as normalized 2nd and 1st order correlation functions. For a E t E t 2 E t E t collisionally broadened source we find that the correlation function varies as g 2 Τ 1 e 2 Γ Τ. Plot 1 Exp 2 Τ, Τ, 0, 3, PlotRange 0, Thus correlation measurements can, for thermal sources, obtain the same information as spectroscopic or interferometric measurements. The decay time in this case is 2 times shorter, but if we know the source has an exponential correlation function, we determine Γ. C. Spatial coherence effects, measurements of source sizes We will now make a small aside to briefly discuss spatial coherence, i.e. a situation in which several spatial modes and propagation effects of the radiation field are present. We do this partly because these spatial effects have had some practial importance in fields outside of quantum optics, especially in astronomy. Michelson' s stellar interferometer How can one measure the size of a star? To get an idea of the task, consider that the diameter of the sun is roughly 10 6 km and that the nearest star is 4 light years km away. We deduce that a typical angular radius for a star is of order 10 8 radians ( seconds of arc). The most effective method is interferometric, and was first succesfully carried out by A. Michelson in the 1930s. The method was for a long time only a curiosity, but since the advent of speckle interferometry and adaptive optics astronomers have been able to make many such measurements. The principle of stellar interferometry is illustrated in the figure below. It is just a double slit experiment. We consider a source of size s with an angular size Θ s, a pair of slits with separation d, and a viewing plane at distance l from the slits. L

9 13 OQ 8 hbt.nb 9 If the source is point like Θ 0) and is temporally coherent, then one will observe fringes as shown with a spacing Σ l Λ. If we think of d displacing the source by an angle Θ, the fringes shift by the same angle, or in the viewing plane, by the distance Σ l Θ. If instead of displacing the source, we think of it as having an angular size, the observed fringes will consist of the sum of many fringe patterns displaced by various values of Σ. If the source is too big the fringes will be washed out. This happens if the angular source size is of order Θ Λ. Thus, in Michelson s stellar inteferometer, if one observes the variation of the fringe contrast as the distance d is varied, one can d deduce Θ the angular size. In close analogy with the reasoning we used in the Mach-Zehnder interferometer, you can show that the complex electric field at a point r in the viewing plane is E r, t K 1 E r 1, t 1 K 2 E r 2, t 2 where r 1 and r 2 are the positions of the slits. The times t 1 and t 2 are defined such that t t i l i c, where l i is the propagation distance from the slit i to the point r in the viewing plane. K i is a complex quantity that takes into account the transmission of the slit and any attenuation of the field due to propagation (varying as 1 R). The time averaged intensity at the screen is given by: I r 1 8 Ε 0 c K 1 E r 1, t 1 2 K 2 E r 2, t Re K 1 K 2 E r 1, t 1 E r 2, t 2 We see that, as in the Mach-Zehnder example, the contrast is given by a 1st order correlation function, except that in this case the spatial dependence of the electric field also enters. See slides for more images. The Hanbury Brown and Twiss intensity interferometer. The Michelson method is extremely difficult to implement. Atmospheric and mechanical fluctuations in the telescope and wash out fringes too and this can prevent observation of the contrast due to the source size. Recent technology, especially adaptive optics, allow one to circumvent these problems. In the 1950 s however, there was another proposal to measure stellar sizes. The point of departure is the idea of speckle. For a source of radiation consisting of many independent, and possible fluctuating emitters, much like in section B, but having a finite size, you would expect to see a speckle pattern at some observation plane.

10 10 13 OQ 8 hbt.nb The typical size of a speckle is d speckle Λ L. The speckle size is the diffraction limited size of a source of angular size L. If you could s s observe this speckle, you could deduce the size of the source. In the light from an ordinary star, it is very difficult to see the speckle because as we have seen the time scale of the fluctuations is on the order of the inverse spectral width of the source. For nearly white light, this corresponds to femtoseconds. Our eyes cannot see these fluctuations, but this might be possible electronically. If you had two very rapid detectors and placed them on the same speckle spot, they would observe the same temporal fluctuations. This would work to some extent even if the detectors were not as fast as the fluctuations. (Say we had 100 ps resolution. 10 fs fluctuations would be attenuated by 10 4 but not wiped out entirely.) If on the other hand the two detectors were observing different speckle spots the intensity fluctuations would be uncorrelated. The product of the correlated intensities would necessarily be bigger. I r 1 2 I r 1 I r 2 A measurement of I r 1 I r 2 as a function of the separation d of the detectors would look like this: Plot 1 Exp d 2 2, d, 0, 4, PlotRange 0, I have assumed that the correlation is Gaussian. In fact you can show that, as in the temporal case, the shape of the correlation function is related to the Fourier transform of the spatial distribution of the source, and as in the temporal case the 1st and 2nd order correlation functions have a simple relationship for chaotic sources. Michelson s method uses interference contrast (the 1st order correlation) to determine the speckle size of a source, the Hanbury Brown Twiss method uses the 2nd order correlation. The HBT method has a big advantage: it is much less sensitive to atmospheric fluctuations or to mechanical stability than the Michelson method. HBT managed to study detector separations of ~100 m. But there is also a big disadvantage: because the fastest dectors today are still at least 10 4 times slower than the fluctuations of starlight, the signal is very weak (g g 1 2 ). HBT could only make measurements on a small number (36) of very bright stars. D. Thermal sources: statistical mixtures

11 13 OQ 8 hbt.nb 11 Now we will analyse the same situation using quantized fields. We will model a source as a black body, filtered spectrally by a filter of width Ω. We will show that for delays smaller than Τ 1 Ω, we find that w 2 a, b w 1 a w 1 b. Reminder from lecture 1: A thermal field is characterized by an incoherent sum over number states n with occpations P n : P n 1 e ħω k T e n ħω k T with n n 0 n P n N.B. this is NOT the equivalent of the state: Ψ n 0 a n n, avec a n P n. 1 e ħω k T 1. Such a state would have Ψ E t Ψ ħω 2 Ε 0 V i Ψ a e i Ω t a e i Ω t Ψ 0. By contrast, the expectation value for a quadrature of the field for a thermal state is zero (no definite phase). In general the expectation value of an observable O in a thermal mixture is: O s 0 P n n O n You can see that we are doing two different averages: a quantum expectation value expressed by... and a weighted statistical average over the occupations of different quantum states. Single mode examples Let us consider a single mode and calculate some mean values for thermal states. E s 0 E E n 0 P n n E E n ħω s 2 Ε 0 V n 0 P n n n E E E E s ħω ħω 2 Ε 0 V 2 n 0 2 Ε 0 V 2 n 0 P n n a a a a n P n n a a a a a a n ħω 2 Ε 0 V 2 n 0 P n n 2 n... ħω 2 Ε 0 V 2 2 n 2 n ħω 2 Ε 0 V n The expectation value E E E E for a single mode thermal state is 2 times larger than that for a single mode coherent state s Α E E E E Α with the same number of photons. Thermal mean values A computational aside. Recall the thermal populations of the harmonic oscillator states: We have: P n 1 e ħω k B T e n ħω k B T n P n 1. Now, let a ħω k B T, then n n n P n 1 e a a n e n a This is the Bose distribution. To find the 2nd moment: e a 1 1 e a e ħω k B T 1 n 2 n n 2 P n 1 e a 2 a 2 n e n a 1 e a e a 1 2 n 2 n 2 Count rates at a beam splitter The count rate after beam splitter is easy to calculate (using, E c t E a, E d r E a ): ħω w c n 0 P n n E c E c n 2 Ε 0 V t2 n w d ħω 2 Ε 0 V r2 n w 2 c, d n 0 P n n Ec Ed From which we find, for a thermal state: w 2 c, d 2 w c w d. Recall that for a coherent state: w c, d 2 w c w d. Multi-frequency calculation Ed Ec n ħω 2 Ε 0 V 2 t 2 r 2 2 n 2 As for coherent and single photon states, a single frequency thermal state is a little artificial. A multi-frequency approach allows us to discuss the temporal behavior of the count rate. As before we define a multi-frequency state as a sum over modes.

12 12 13 OQ 8 hbt.nb E t k E k Ek The expectation value of an operator O in a multimode thermal state is given by: O s n 1... n k O n 1... n k P n1... P nk n 1... n k with: P ni 1 e ħω i k B T e n i ħω i k B T. The values of n i define the spectrum of the source. First we calculate w 1 : w 1 1 j ħω j n j 2 Ε 0 Ω E Ω 2 V The last expression is the total intensity (integrated over the spectrum). We can also calculate a 1st order correlation function: G 1 Τ E t Τ E t 1 j ħω j n j e i Ω j Τ 2 Ε 0 d Ω E Ω 2 i ΩΤ e V G 1 Τ is the unnormalized correlation funtion and is proportional to the Fourier transform of the PSD. It is therefore a function whose modulus has a width 1/ Ω. Two photon rate For the two photon rate, we calculate (t 1 t, t 2 t Τ): ħ w 2 t Τ, t j,k,l,m 2 Ε 0 V 2 Ω j Ω k Ω l Ω m e i Ω j t Ω k t Τ Ω l t Τ Ω m t a j a k a l a m s In the average a j a k a l a m s, only terms with an a a in the same mode contribute (n j a j a j ): a j a k a l a m s j,l k,m n j n k s j,k n j s j,m k,l n j n k s j,k n j s In the sum, taking into account the s, we find: w 2 ħ t Τ, t j,k 2 Ε 0 V 2 Ω j Ω k e i Ω j Ω k Τ 1 n j n k s j,k n j s To evaluate n j n k s : n j n k s n j s n k s si j k, n j n j s 2 n j s 2 nj s therefore: n j n k s n j s n k s j,k n j s 2 nj s. But in the sum j,k, terms proportional to j,k, only contribute N times while the terms n j s n k s contributes N 2 times. If N 1, one can neglect all terms except n j s n k s n j n k. ħ w 2 t Τ, t j,k 2 Ε 0 V 2 Ω j Ω k e i Ω j Ω k Τ 1 n j n k ħ 2 Ε 0 V 2 j Ω j n j e i Ω j Τ k Ω k n k e i Ω k Τ j Ω j n j k n k 1 2 Ε 0 V 2 j ħω j n j e i Ω j Τ 2 j ħω j n j 2 Connection to the correlation function Finally we find that the coincidence detection rate as a function of the delay Τ is: w 2 t Τ, t G 1 Τ 2 G With G 1 Τ 0 when t 1 Ω In the jargon of quantum optics, g 2 is called the second order correlation function. Its definition is: g 2 Τ w 2 t Τ,t w c t w d t E t E t Τ E t Τ E t E t E t 2 G Τ 2 G 0 2 G 0 2 g 1 Τ 2 1 A classical source corresponds to g 2 1. A "non-classical" one corresponds to g 2 1. E. A problem that the classical approach cannot treat: fermions If the particles used in the experiment are identical fermions, instead of commutation relations a j, a k j,k, one must use anticommutation relations a j, a k a j a k a k a j j,k. In other words: a j a k a k a j et a j a j 1 a j a j Therefore: a j a k a j a k a j a j a k j,k a k a j a j a k a k a j a j i,k and a j a k a k a j a j a k a j a k a j a j a k a k a j a j i,k In both cases the expectation value for j k vanishes. The correlation function is: G 2 j,k Ω j Ω k a j a j a k a k 1 e i Ω j Ω k Τ The sum can include k j because its contribution is zéro. In terms of normalized correlation functions g 2 Τ 1 g 1 Τ 2 and a correlation experiment would observe anti-correlations.

13 Huntingdon and Broad Top Mountain RR The$Hanbury$Brown$&$Twiss$effect:$ from$stars$to$cold$atoms Chris Westbrook Institute Optique, Palaiseau! Michelson stellar interferometer principle Fringe contrast indicates the spatial coherence of the source. When d is too big, fringes disappear: θ ~ λ/d

14 ! Michelson stellar interferometer realization d θ ~ λ/d d ~ 6m θ ~ 10-7 Michelson measured the angular diameters of 6 (big) stars.! Fringes from a real star from the European Southern Observatory

15 Intensity interferometry reflecting telescope d Robert Hanbury Brown I 1 I 2 correlator (C ~ I 1 I 2 ) The noise in two optical (or radio) telescopes should be correlated for sufficiently small separations d. Reminiscent of Michelson's interferometer to measure stellar diameters, but less sensitive to vibrations or atmospheric fluctuation.! HBT stellar interferometer principle Starlight produces rapidly fluctuating speckle on the earth. The size of the speckle is: d speckle ~ λl/d ~ λ/θ Intensity fluctuations within one speckle are correlated so that I1 I1 I1I2

16 Visualisation de speckle 7 "The" Hanbury Brown and Twiss experiment 1956 How does the product of the photo currents (or the joint probability of detecting a photon at each detector) vary as one detector is moved with respect to the image of the other?

17 Answer C ~ I 1 I 2 R. Hanbury Brown and R. Q. Twiss, Nature 177, 27 (1956) The experiment shows beyond question that the photons in the two coherent beams of light are correlated and that this correlation is preserved in the process of photoelectric emission. HBT Telescope 10

18 First measurement of the diameter of Sirius At a telescope (actually, a search light mirror) θ = radians R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956) HBT avec des atomes (Orsay) Cold cloud of He* ( µk) 20 ev énergie interne Détection d atomes individuels à 3D (x, y, t z) Résolution 300 µm (x,y), 3 nm (z) 12

19 Example 1: Orsay Detection of metastable atoms by µ-channel plate. (He* has ~ 20 ev) Excellent time (vertical) resolution. Delay-line anode gives in plane resolution. Long time of flight increases correlation length. Fonctions de corrélation normalisées Nature 445, 402 (2007) Bunching for bosons (He-4) as for photons. Antibunching for fermions (He-3). Pauli principe - no wave interpretation. Difference is due to quantum statistics, not to interactions.