My previous blog discussed some historical papers related to the intensity interferometer and its connection to quantum optics. Here, I explain the basic physics of an intensity interferometer.
In the context of spatial coherence, the coherence theory expresses the degree of spatial coherence as,
with \( U_i(t) \) representing the fields of sources \( i = 1 \) and \( 2 \).
An intensity interferometer measures the intensity correlation function between such sources. If \( I_1(t) \) is the intensity of source 1 and \( I_2(t) \) is the intensity of source 2, then the intensity correlation function is given by:
If one ignores the background (the first term in the sum of the above equation) and considers only the fluctuations in the signal (the second term), then the term of relevance will be:
The signal in the intensity interferometer is thus proportional to \( \left| \gamma_{12} \right|^2 \).
A conventional interferometer measures a signal that is proportional to \( \left| \gamma_{12} \right| \), which includes the amplitude and phase, whereas an intensity interferometer measures a signal proportional to \( \left| \gamma_{12} \right|^2 \), which is not sensitive to the phase.
Intensity interferometers have certain advantages compared to conventional interferometers (such as the Michelson interferometer). Below is a partial list:
Intensity measurements (unlike amplitude or phase) can be done directly using optoelectronic instruments.
They do not require precise, sub-wavelength optical alignment, unlike amplitude- or wavefront-dividing interferometers.
They can be used with two detectors that are placed far apart, thereby improving the spatial resolution of the measurement (relevant in astronomy).
A constraint of an intensity interferometer is that the intensity of the participating source should be bright.
There is an important connection between quantum optics and radio astronomy. Hanbury Brown and Twiss in the 1950s devised the intensity interferometer.
Particularly, they were interested in measuring the ‘diameter of discrete radio sources’. The title of their seminal paper reads “A new type of interferometer for use in radio astronomy”. As the authors claimed in their paper: “The principle of the instrument is based upon the correlation between the rectified outputs of two independent receivers at each end of a baseline, and it is shown that the cross-correlation coefficient between these outputs is proportional to the square of the amplitude of the Fourier transform of the intensity distribution across the source.”(Brown and Twiss, 1954)
First, they tested their technique in a laboratory situation and followed it up with a measurement of the diameter of Sirius. Their technique was a game-changer in measuring the diameter of bright stars.
As the intensity interferometers were being developed, the laser was realized in the early 1960s. Unlike conventional light sources, laser light is coherent, and this brings in unique features that can be used to understand the nature of light. In the context of laser optics, intensity interferometers had immediate utility in studying coherence through correlation measurement. It was logical to combine lasers with intensity interferometers and study the correlation. This combination is what led to the discovery of some fascinating aspects of quantum properties of light, including anti-bunching.
If the book by Born and Wolf is considered a classic on the electromagnetic theory of light, the quantum extrapolation is the book by Leonard Mandel and Emil Wolf titled Optical Coherence and Quantum Optics.
This book discusses the interface of statistical optics, optical coherence, and quantum optics. The core argument of the book starts with probability theory and its connection to fluctuations of light and builds optical coherence, polarization, and eventually quantum optical effects of light. It is a well-written treatise on light with a flavor of experiments (Mandel did some pioneering experiments in quantum optics) and theoretical explanation (a hallmark of Wolf).
In the preface of the book, they bring together the importance of intensity interferometers and the discovery of lasers and explain how and why it led to a deeper understanding of quantum optics:
“Prior to the development of the first lasers in the 1960s, optical coherence was not a subject with which many scientists had much acquaintance, even though early contributions to the field were made by several distinguished physicists, including Max von Laue, Erwin Schrodinger and Frits Zernike. However, the situation changed once it was realized that the remarkable properties of laser light depended on its coherence. An earlier development that also triggered interest in optical coherence was a series of important experiments by Hanbury Brown and Twiss in the 1950s, showing that correlations between the fluctuations of mutually coherent beams of thermal light could be measured by photoelectric correlation and two-photon coincidence counting experiments. The interpretation of these experiments was, however, surrounded by controversy, which emphasized the need for understanding the coherence properties of light and their effect on the interaction between light and matter.” (Mandel and Wolf, 1995, p. 1)
This further led to a series of studies on light-matter interaction from a coherence perspective, and included analysis of the fluctuation of light by understanding the randomness and the associated statistics of the fluctuations. Mandel, Wolf, Glauber, E.C.G. Surdarshan and many others across the world laid the foundation and connection between optical coherence and quantum optics. What started as a technical development in radio astronomy turned out to be a vital tool in quantum optics.
Brown, R. Hanbury, and R. Q. Twiss. ‘LXXIV. A New Type of Interferometer for Use in Radio Astronomy’. Philosophical Magazine 45, no. 366 (1954): 663–82. https://doi.org/10.1080/14786440708520475.
Brown, R. Hanbury, and R. Q. Twiss. ‘Correlation between Photons in Two Coherent Beams of Light’. Nature 177, no. 4497 (1956): 27–29. https://doi.org/10.1038/177027a0.
Hanbury Brown, R., and R. Q. Twiss. ‘A Test of a New Type of Stellar Interferometer on Sirius’. Nature 178, no. 4541 (1956): 1046–48. https://doi.org/10.1038/1781046a0.
I have been studying quantum mechanics for almost 30 years now. Every time I go back to study and understand something, it reminds me of a quote by Nelson Mandela: “There is nothing like returning to a place that remains unchanged to find the ways in which you yourself have altered” ~ Long Walk to Freedom (1994)
On further contemplation, I find the same with other branches of physics and certain aspects of mathematics, too.
14th of March is Einstein’s birthday. There is so much written about Einstein, and every time you read about him or a text written by him, there is always something interesting to learn. Recently, I came across a wonderful paper by Shankland, who compiled his conversation with Einstein over a period of ten years and published it in 1962 in the American Journal of Physics. Below are three excerpts from the paper to give you a taste of the conversation. I would urge you to read the conversation in full, and it is a delight.
(Shankland 1963, 1)
“When I asked him how he had learned of the Michelson-Morley experiment, he told me that he had become aware of it through the writings of H. A. Lorentz, but only after 1905 had it come to his attention! “Otherwise,” he said, “I would have mentioned it in my paper.” He continued to say the experimental results which had influenced him most were the observations on stellar aberration and Fizeau’s measurements on the speed of light in moving water. “They were enough,” he said. I reminded him that Michelson and Morley had made a very accurate determination at Case in 1886 of the Fresnel dragging coefficient with greatly improved techniques and showed him their values as given in my paper. To this he nodded agreement, but when I added that it seemed to me that Fizeau’s original result was only qualitative, he shook his pipe and smiled, “Oh it was better than that!” He thought Zeeman’s later precise repetition of this experiment was very beautiful. He seemed really delighted when I mentioned to him how elegant I had found (as a student) his method of obtaining the Fresnel dragging coefficient from his composition of velocities law of special relativity.” (Shankland 1963, 2)
“I asked Professor Einstein how long he had worked on the Special Theory of Relativity before 1905. He told me that he had started at age 16 and worked for ten years; first as a student when, of course, he could only spend part-time on it, but the problem was always with him. He abandoned many fruitless attempts, “until at last it came to me that time was suspect!” Only then, after all his earlier efforts to obtain a theory consistent with the experimental facts had failed, was the development of the Special Theory of Relativity possible. This led him to comment at some length on the nature of mental processes in that they do not seem at all to move step by step to a solution, and he emphasized how devious a route our minds take through a problem. “It is only at the last that order seems at all possible in a problem.”” (Shankland 1963, 2)
“Our conversation then returned to the Michelson-Morley experiment and the Special Theory of Relativity. I could not help feeling that this elegant special theory, the product of his youthful efforts, held the place nearest to his heart. I asked him if he felt that writing out the history of the ;v[ichelson-Morley experiment would be worthwhile. He said, “Yes, by all means, but you must write it as Mach wrote his Science of Mechanics.” Then he gave me his ideas on historical writing of science. “Nearly all historians of science are philologists and do not comprehend what physicists were aiming at, how they thought and wrestled with their problems. Even most of the work on Galileo is poorly done.” A means of writing must be found which conveys the thought processes that lead to discoveries. Physicists have been of little help in this because most of them have no “historical sense.” Mach’s Science of Mechanics, however, he considered one of the truly great books and a model for scientific historical writing. He said, “Mach did not know the real facts of how the early workers considered their problems,” but Einstein felt that Mach had sufficient insight so that what he says is very likely correct anyway.” (Shankland 1963, 4)
There is a lot more to explore in the wonderful conversation paper. Link below.
Shankland, R. S. 1963. ‘Conversations with Albert Einstein’. American Journal of Physics 31 (1): 47–57. https://doi.org/10.1119/1.1969236.
Anthony James Leggett (26 March 1938 – 8 March 2026)
pic credit: Britannica
I was informed that Anthony Leggett passed away…he made physics wonderful by asking profound questions, as below:
“……there is no good reason to accept thisdivision of the world into a microscopic regime where QM reigns and a macroscopic one governed by classical physics; QM is a very ‘totalitarian’ theory, and if it applies to an individual and electrons, then it should prima facie equally apply to the macroscopic objects made up of them, including any devices which we have set up as measuring apparatus….” [1]
The above snapshot is from OpenAI, which has claimed to have derived a new result in theoretical physics. What is it about, and how good are the claims? Below, I discuss them.
Let me start with some background. Except for the hydrogen atom, the nucleus of all elements in the periodic table consists of neutrons and protons. Neutrons and protons are made of quarks. Quarks interact through gluons. How do these gluons interact? This is a contemporary question.
In this particular case, the authors of the study say: “We’ve published a new preprint showing that a type of particle interaction many physicists expected would not occur can, in fact, arise under specific conditions. The work focuses on gluons, the particles that carry the strong nuclear force.”[1]
The interaction can be computed in terms of probabilities[2], and these probabilities depend on quantum mechanical amplitudes (also called scattering amplitudes). Finding these amplitudes requires a deeper knowledge of strong nuclear forces. Computing such amplitudes is expensive and requires a lot of effort. Physicists, under physical constraints, take a guess on which interaction is more probable and which is not. This study shows that one of the interactions that physicists thought was not probable turns out to be probable, but under specific conditions. “The preprint studies a central concept in particle physics called a scattering amplitude. A scattering amplitude is the quantity physicists use to compute the probability that particles interact in a particular way. …….One case, however, has generally been treated as absent (having zero amplitude)……..As a result, this configuration has largely been set aside. The preprint shows that this conclusion is too strong.”[1]
Of course, this has been possible using the brute force computational capability of the GPT 5.2 model, and it has come up with a particular formula that shows the amplitude to be probable and has further validated it with a formal proof. It is a methodological breakthrough, and the authors claim, “An internal scaffolded version of GPT‑5.2 then spent roughly 12 hours reasoning through the problem, coming up with the same formula and producing a formal proof of its validity.” [1]
I think it is a good development in computational physics and helps in calculating parameters that have relevance in finding probabilities of interaction in particle physics. Overall, my hunch is that it is an important step in computational physics.
Notes:
[1] OpenAI has put out an excellent summary of this problem (without jargon), and it needs basic physics, and the flow of text is good.
A gentle reminder: Digital infrastructure is not equal to physical infrastructure. The former is a smaller set of the four-dimensional space-time world we live in. AI-based tech is fantastic for an upgrade in digital infrastructure and has already made tremendous progress. But the real deal is in the physical domain. This also indicates where the future action is, and will be influenced by our understanding of physical sciences, including engineering domains beyond computer science. What we are witnessing in AI is probably the peak of Gartner’s hype cycle.
If you need to admire complex analysis for its elegance and visual utility, try quantum optics. Specifically, the description of quantum states. Thanks to creation and annihilation operators, the position and momentum states of a quantum optical field can be represented as quadratures. These entities can now be represented on the orthogonal axes of a complex plane. The representation of Argand diagrams starting with a classical electromagnetic field and then extrapolating them to quantum theory is a tribute to its geometrical representation. The fact that two axes can be utilized to represent real and imaginary parts of the defined state is itself an interesting thing. By certain operations within the plane, one can realize the vacuum state, the coherent state, and the squeezed state of quantum optics.
The Vacuum Spread – One of the major consequences of quantum theory, and especially the second quantization, is the realization of the vacuum states. Even when there are zero photons, there is a residual energy in the system that manifests as vacuum states. How to define the presence or absence of a photon is a different proposition because vacuum states are also associated with something called virtual photons. That needs a separate discussion. Anyway, in a complex plane of quadrature, a vacuum state is represented by a circular blob and not a point (see fig. 1). It is the spread of the blob that indicates the uncertainty. In a way, it is an elegant representation of the uncertainty principle itself because the spread in the plane indicates the error in its measurement. Importantly, it emphasizes the point that no matter how low the energy of the system is, there is an inherent uncertainty in the quadrature of the field. This also forms the fundamental difference between a classical and a quantum state. The measurement of the vacuum fluctuation is a challenging task, but one of the most prevalent consequences of vacuum fluctuation is the oblivious spontaneous emission. If one looks at the emission process in terms of stimulated and spontaneous pathways, then the logical consequence of the vacuum state becomes evident in some literature on quantum optics. Spontaneous emission is also defined as stimulated emission triggered by vacuum state fluctuations. It is an interesting viewpoint and helps us to create a picture of the emission process vis-à-vis the stimulated emission.
Figure 1. Vacuum state representation. Note that their centre is at the origin and has a finite spread across all the quadrants. Figure adapted from ref. 2.
Another manifestation of the vacuum state is the Casimir effect, where an attractive force is induced as you bring two parallel plates close to each other. The distance being of the order of the wavelength or below this triggers a fascinating phenomenon which has deep implications not only in understanding the fundamentals of quantum optics and electrodynamics, but also in the design and development of quantum nanomechanical devices.
A shift in the plane – Coherent states are also described as displaced vacuum states, and this displacement is evident in the Argand diagrams. The quadrature can now help us visualize the uncertainty in the phase and the number of photons in the optical field. One of the logical consequences of the coherent state is the number-phase uncertainty. This gets clear if one observes the spread in the angle of the vector and the radius of the blob represented (see Fig. 2). Notice that the blob still exists. The only difference is that the location of the blob has shifted. The consequence of this spread has a deeper connection to the uncertainty in the average number of photons and the phase of the optical field. The connection to the number of photons is through the mod alpha, which essentially represents the square root of the average number of photons. Taken together, the blob in the Argand diagram represents the number-phase uncertainty.
Figure 2. Coherent state representation. Note that their centre is displaced. Figure adapted from ref. 2.
Lasers are the prototypical examples of coherent states. The fact that they obey Poissonian statistics is the direct consequence of the variance in the photon number, which is equivalent to the square root of the average number of photons. This means one can use photon statistics to discriminate between sources that are sub-Poissonian, Poissonian, or super-Poissonian in nature. The super-Poissonian case is the thermal light, and the sub-Poissonian state represents photon states whose number can reach up to 1 or 0. The coherent states sit in the middle, obeying the Poissonian statistics.
Everything has a cost – Once you have a circle with a defined area, it will be interesting to ask: Can you ‘squeeze’ this circle without changing its area? The answer is yes, and that is what manifests as a squeezed state. In this special state, one can squeeze the blob along one of the axes at the cost of a spread in the orthogonal direction. This converts the circle into an ellipse (see Fig. 3).
Figure 3. Squeezed State. Note the circle has been squeezed into an ellipse. Figure adapted from ref. 2.
Note that the area must be conserved, which means that the uncertainty principle still holds good; just that the reduction in the uncertainty along one axis is compensated by the increment in another. This geometrical trick has a deep connection to the behaviour of an optical field. If one squeezes the axis along the average number of photons, it means that you are able to create an amplitude-squeezed state. This means the uncertainty in the counting of photons in that state has reduced, albeit at the cost of the uncertainty in the measurement of phase. Similarly, if one squeezes the blob along the axis of the phase, then we end up with a lowering of the uncertainty for the optical phase. Of course, this comes at the cost of counting of number of photons. I should mention that the concept of optical phase itself is not clearly defined in quantum optics. This is because an ill-defined phase can have a value of 2π, which creates the problem. An interesting application of the phase-squeezed quantum states is in interferometric measurements. By reducing the uncertainty in the phase, one can create highly accurate measurements of phase shifts. So much so that this can have direct implications on high-precision measurements, including gravitational wave detection. The anticipation is also that such tiny shifts can be helpful in observing feeble fluctuations in macroscopic quantum systems.
Pictures can lead to more than 1,000 words. And if you add them to a quantum optical description, as in the case of the states that I have defined, they create a quantum tapestry. Perhaps this is the beauty of physics, where there is a coherence between mathematical language, geometrical representation, and physical reality. Feynman semi-jokingly may have said, “Nobody understands quantum mechanics,” but he forgot to add that there is great joy in the process of understanding through mathematical pictures. After all, he knew the power of diagrams.
References:
Ficek, Zbigniew, and Mohamed Ridza Wahiddin. Quantum Optics for Beginners. 1st edition. Jenny Stanford Publishing, 2014.
Fox, Mark. Quantum Optics: An Introduction. Oxford Master Series in Physics 15. Oxford University Press, 2006.
Gerry, Christopher C., and Peter L. Knight. Introductory Quantum Optics. Cambridge, United Kingdom ; New York, NY, 2024.
Saleh, B. E. A., and M. C. Teich. Fundamentals of Photonics. 2nd edition. Wiley India Pvt Ltd, 2012.
