This "computes first/think later" attitude of mine with the one-atom lasing might not be the wisest. I've opted to tolerate the black lines as a margin within computer limits, that can be refined later—and there is some grounds to that, I still think a little bit—but I might be lulling myself into structures that don't exist physically or will vanish on being-observable constraint. This becomes important when odd-looking features need to be brushed off. The Pac-Man look is genuine indeed, but takes a different form, smoother, just as predicted, and I haven't implemented weighting of the eigenvalues yet, as I was doing in Ref. [1]. Getting rid of the black (too small to have a sign) lines, the transition from condensation to boiling is cleaner and much deeper. A needle from the classical field into the quantum ladder:
Browsable
There's also beautiful fractal-looking structure into the continuum, at high $\omega_p$, in the form of those weaving patterns of red and blue making zebra-looking shapes on a continuum. This is in a classical counterpart to the quantum fractals at small $\omega_p$. So this is nice. What is less nice is that most of the interesting quantum fractals (when the bubbles collide) are black, so would be brushed off if blown over.
The morning update of what computers spent their time doing.
In the boiling window, first, the job completed as planned. Surprisingly, but reassuringly, the pac-man shape survives finder density and higher precision.
This is still probably some artifact of the representation but a second-order one, by what I mean that the transition from lasing to boiling is certainly genuine and capture something, but not that dramatic:
Weighting the eigenvalues with their $L_p$ weight would produce something similar but smoother, not without such hard black boundaries (which mean that we faded into zero).
80-digit precision is not needed in the boiling windows, baring singular exceptions, it matches the 40-digit calculations. There is some structure missed at the end of the condensation window:
but none in the boiling one, so I can probably safely interpolate in this window, and ask a nursing process to provide more points in regions of eigenvalue color-change or abrupt variations of slopes. Those would be the orange markers below, 140 additional pumps:
Note that the nurse from the Meiji era, carries cherry blossoms, not lillies... at some point, the universe wants you to move on, even if you don't want to. It has gravity, and you only have memories, not a fair fight.
In the condensing window, then, calculation is still some time to complete, especially as I keep chunking in new ones in between. In van der Corput order, that however refines the picture, and our nurse mending could be used there too (at least for other $\gamma_a/\gamma_\sigma$ cases):
Two photons that appear correlated on a visible donut are in fact sampled independently from a hidden dipole, whose orientation changes from shot to shot: the donut—and the correlations—arise from averaging over dipoles.
The condensing to boiling transition doesn't get clarified by patching the smooth condensation to the crude boiling part. I've started a $19.002\le P_\sigma/g\le 19.998$ calculation at 80 digits (by steps of 0.004, so 250 pumps, result for tomorrow morning). The question is whether this gap opening is genuine or a collapse of precision.
Ten-Go's progress on uijin. The long continuous stretch at the end is the boiling window, that was computed separately at a different grid resolution, whence the jump in pumps on the left vertical. I put the finer-grid end of the condensation window to higher priority (top red bars, all taken up by azag) so as to have a closer look at the condensation-boiling transition, and get a hinch at where closer attention should be given, as the pacman shape of Fig. (0PlMJ) looks weird and could be numerical artifacts.
My plans are then to acquire full-span (by the end of this week), plus high-regular grid (non-adaptative) pictures for other $\gamma_a/\gamma_\sigma$ to get another idea of how the skeleton morphs for different parameters, and when having a bit of time on my hands, investigate closer the numerical stability of it all and the nature of the isolated points in Fig. (0PgZn): fractal or noise.
Ten-Go's progress on uijin, through this BitTorrent looking interface:
I might leave the filler complete in the background with little less intervention until then (couple of days). With such small ($\approx 15\%$) progress, there is almost no visible gain to the eye. The order is van der Corput's, so it overall looks better, meaning locally the same everywhere:
Browsable
This is the full $\max{\omega_p}$ this time, as promised to always show something different. You need to take a deep dive to see the quantum bubbles. Now it becomes urgent to develop the dynamic-tile viewing of such a picture, but we should survive at least through what remains of October. The fractal nature is not resolved yet. But I'll have to settle the numerical stability seriously first, which is not trivial. For now, I increased to 80-digit precisions. I should also investigate the nature of the black indeterminacy of the weights and also explore the $\gamma_p$ structure (the real part of the eigenvalues, or broadening, we're looking here at the imaginary part $\omega_p$, of frequency). I did that 17 years ago and still have in mind the chilling vision of those extremely messy forking trees. It'll be fun to look at it again, but like going into the jungle, so not now, probably not even these coming weeks. And no more about this from me for today (only for the computers, to compute).
The most difficult however is the condensation buildup, that goes fractal, so I keep accumulating fractal dust, i.e., new unconnected points, and my connecting grid goes beserk. We're working at finite (albeit very high) truncation so in principle it should stop at some point. But so far I feel like Mandelbrot when he was looking at this:
As far as my bubbling is concerned, this is what I have as of now 0Pecs(af):
I'll need to change the way I plot this as you don't look at a fractal by dropping points on paper. I'll need to tile-zoom into it. Not important for now, this is a background project, after all. The most interesting are all those scattered points. So I'm now running a zoom on
$$5.20\le P_\sigma/g\le 5.22$$
Let's see how many digits down the decades we'll need to go.
I've deployed a basic queue-managing process to complete such things dynamically in the background. I called it ten-go, after a famous military operation and with keywords of the bushido as instructions to the pooling process. It's very aggressive! Appropriating myself things that most Japanese have turned their back to.
It's running since this morning on the above "campaign" namecoded uijin.
I also need to:
cover the full range of the condensation window at the step of 0.004 (that of yzao's 9‒9.5 window), to get a nice overall visual, estimated 1,760 kernel-hours.
Produce same window for another $\gamma_a/\gamma_\sigma$ (possibly more than one) at this resolution.
Plot finely the higher-res, adaptive grid structures, extending {{pan}} to do so.
Depending on how fractal the thing is, some of the above might be done this week. I'd like it to be very fractal, of course, although that'll become a serious numerical project. I'm already into 40-digits bignum precision. Tomorrow might bring clarity, or more confusion.
v. Enk and Fuchs[2] argue that what matters for a laser is phase coherence, not a fixed absolute phase, and that coherent states are privileged in describing laser light, so spot-on with us. The quantum de Finetti representation also is directly related (although we're still mainly with semi-classical multiphoton states instead).
A smooth classical-to-quantum transition will be costly. Even putting both azag and yzao on the job almost full power, I could only improve over the week-end on Fig. (0OG5J), which is in the range
$$0.15 \le P_\sigma/g\le 12.51\quad\text{(condensing)}$$
as shown in this update, that also—as promised—now shows the extended outer region up to $|\omega_p/g|\le 0.3$:
Browsable
The panorama zooming make this larger window not hinder the fine structure and next time I could also show the whole structure (not now as this would be distracting and I'll probably be doing several such updates before this figure completes so I'll have something qualitatively new to report).
I had $5\le P_\sigma/g\le 6$ (azag) and then $0.5\le P_\sigma/g\le 1.5$ (yzao) and later $9.002\le P_\sigma/g\le9.598$ (yzao) run at finer density, which is however not enough to close the gaps, and then put azag to close the $5.2\le P_\sigma/g\le 5.6$ sub-window but this is more costly than expected as this works on a dynamic grid and this one gets refined and refined as computation progress. It also fails to show the new structure when plotted in this way, so I'll have to develop some dynamical zooming, working out (pre-computed) tiles as one zooms in. At the beginning of the week, we are anyway left with the above numerical effort completed.
I must also look at the boiling window in Fig. (0OaVM), i.e., just right past the one we're now computing, in the range
$$12.51 \le P_\sigma/g\le 43\quad\text{(boiling)}$$
This should be looked, first, on a linear grid, then on a refined one and then on a continuous one. Probably other windows besides the condensing and boiling ones are safe to extrapolate from what we already have.
I should also compute such graphs for other $\gamma_a/\gamma_\sigma$, as those change nature too and it'd be interesting to see that in action, maybe even as an animation. This could be enough at the finest non-continuous grid level, maybe with continuous windows over small regions of interest (bubbling, boiling, ...)
As I said, it looks like a home-alone Christmas project.
I implemented my own browsable viewer of big images, of which I expect to produce a few regarding my one-atom lasing quantum-to-classical transition. But it's nice on conventional panoramas too.
Here's how it shows us the deep-lasing case I chose to run in the background while I do other things. You can browse this around, pan and zoom. As resolution is still low, there is not much to see, but you can imagine how it'll look like:
Browsable
I'll see if it makes sense to display the whole vertical axis: remember, we're looking very close to the cavity here, to see its boiling, its burbling. There's much more structure elsewhere but it's moot. However since this allows you to decide where to go, size allowing, I'll display the whole thing next.
You don't see this anywhere else, do you? What we published previously[[3],[1]] was short of the lasing regime, so whoever's looking at that is one of the very few to have this inside view of what quantum condensation of photons really looks like.
Next will probably be what the computer is computing now, which is the region $5\le P_\sigma/g\le 6$ at higher resolution, where some strange things happen: the sign of the weight of the eigenvalue (whether it's emitting or absorbing) is unknown to the extended machine precision, which is 40 digits; eigenvalues are well-defined to this precision and are certified to higher truncation, but something happens here with the weight, in which case I plot it black; also the bubbling looks fractal, bubbles seem to collide, to interfere, it looks like some other type of higher-rungs falling into the photon soup takes place, with sploshing that causes ripples throughout the bubbles, maybe breaking or merging them. At this very discretized level—which still took 340 kernel-hours (24h and 43min on the 24 kernels I allocated)—one doesn't see much, of what remains completely unclear phenomenology anyway. To the best of my knowledge, nobody understand what is going on even in very simple cases.[4]
Computer computed, and this is what if gets for Fig. (0OcN9) in linear scale:
which will need the same treatment as Fig. (0K719b) to become smooth and neatly resolved. I'll also implement some viewer here on laussywiki so that one can explore such structures interactively. I had done something similar back in the time, in Southampton, when doing this for our 2010s papers, but this was local and of course completely lost in space and time. Everything that doesn't get shared publicly eventually gets lost.
The computer computed the lasing skeleton for the smallest $\gamma_\sigma=0.004\gamma_a$ of the good-cavity series, over the whole pumping range, and it's what I got (overlaied with the cavity population $n_a$ in green):
Its interesting pump range for lasing in between the two vertical lines and looks like this:
The interesting part of this is around the cavity frequency, where the photon condensation takes places to build the lasing mode. Its zoom looks like this:
It's in log-scale because I wanted to span initially from much lower pumps to much highers. The same in linear scaling looks like that:
So now, I've asked the computer to produce a high-res version of this in linear scale. It's a much enlarged region of the previous bubbling window considered (in Fig. (0NxFT)), to which I might come back. As a fractal object, it's begging for zooms inside zooms inside zooms.
There is also an interesting revival of seemingly much bigger bubbling in Fig. (0OaVM), like some boiling point, at the top of the condensation, when the population saturates, before quenching. This of course will require a separate analysis. So far I am into the condensation buildup phase.
What is clear deep in the lasing regime is that bubbles
are associated to photons: each new photon diving in the cavity condensate opens a bubble. This is not unexpected as those are due to rungs of the JC ladder interfering but it's nice to see the correspondence:
Here we see the superposition of the bubble counts $k$ vs the population (dotted line). Each bubble sits at the center of the integer number of photons in the mean population.
The case where $n_\sigma$ also exhibits a clear plateau is not accessible numerically (at least not easily for eigenvalues) since $n_\sigma\approx{1\over2}\left(1+{\Gamma_\sigma\over\kappa_\sigma}\right)$ requires $P_\sigma\ll\kappa_\sigma$ where $\kappa_\sigma\equiv 4g^2/\gamma_a$. So I'll go with these parameters:
$$g=1,\quad \gamma_a=0.1\,g,\quad \Delta=P_a=\gamma_\varphi=0,$$
and eleven $\gamma_\sigma$ chosen so that
$g^{(2)}(P_\sigma\to0)$ runs evenly from $\tfrac{2}{3}$ (smallest possible $g^{(2)}$) to 2:
$$\gamma_\sigma/\gamma_a = 0.004,\ 0.34,\ 0.76,\ 1.30,\ 2.02,\ 3.03,\ 4.55,\ 7.10,\ 12.2,\ 28.0,\ \text{and } 509.$$
which gives rise to this family of one-atom lasing curves:
I'll first check what happens with the skeleton when changing the $\gamma_\sigma/\gamma_a$ ratio. This changes $g^{(2)}$ a lot but possibly doesn't affect the skeleton itself. The lasing bubbles might be more sensible to this parameter...
The one-atom laser $g^{(2)}$(0) in the good cavity limit (needed to lase well anyway) is not 1, as we implied in Ref. [1] and assumed to derive approximate (still excellent) expressions. Instead, one can find it by pushing the moment recurrence we used at small pumping to higher orders and to this case, as more accurately given by:
where $x\equiv{P_\sigma\gamma_a}/({4g^2})$. This shows that lasing is actually sub-Poissonian for $P_\sigma$ below $4g^2/(3\gamma_a)$—less the small shift $(\gamma_a+39\gamma_\sigma)/3$ that comes from the next order, and super-Poissonian from there to the quenching.
Before I commit to one particular case deep in the classical (lasing) regime at the full quantum level, I should check more carefully what happens with the spectrum structure from the $\gamma_\sigma/\gamma_a$ imbalance. This affects $g^{(2)}$ as we know since at least Ref. [1] but the effect on the eigenvalue skeleton, I don't think (or don't remember if) we ever looked at it. I'm doing it now.
The one-atom lasing case for which I mapped its exceptional-point bubbling, was not entering deep lasing, although it still reached max population of $n_a\approx 4.92$:
I want to see this quantum to classical transition, deep in the lasing regime, so I'll be running its eigenvalue spectrum in the background, which probably will take much more time. Maybe ready for Christmas. The four dashed lines are where the eigenvalues from higher rungs close onto their exceptional points. I'm prospecting which range of parameters would work.
Speaking about journeys in time, $I_{\alpha\beta}$ is the weak value introduced by Aharonovet al.[5], in our case of the intensity operator $\hat I(\mathbf r)\equiv\hat\Psi^\dagger(\mathbf r)\hat\Psi(\mathbf r)$ for a field prepared in the coherent state $|\boldsymbol\alpha\rangle$ (or in the past) but later found in the coherent state $|\boldsymbol\beta^*\rangle$ (or in the future), so that makes Eq. \eqref{eq:0LiYJb} a positively weighted average of weak values. Progress! It is something...
It is my feeling that when past and future agree, the weak-value becomes a classical probability.
When they don't, then we enter this weak-value business of contextuality and realizations as two-time objects. Now what it all means, I don't know, but it seems it has some place to go.
Not a long night but at least a long day. And I have to leave early, so I'll go to bed but taking this with me below the sheets: the thermal-aligned donut is still a donut but with a pulsating profile, making it look like a pinched donut, a wannabe dipole
$$\rho_{\rm oracle}(r,\theta)=\frac{r^2e^{-r^2}}{\pi}\left[1+\frac{\pi}{4}\cos2\theta\right].$$
Along any ray the normalized radial law is the donut's $r^2e^{-r^2}$: the cuts along and across the axis differ only by the weights $1\pm\pi/4$ and coincide once normalised to unit area.
Around the rim the alignment is a cosine of contrast $\pi/4\simeq0.785$, between the coherent donut ($0$) and the critical dipole ($1$): $1+\pi/4\simeq1.785$ on the axis, $1-\pi/4\simeq0.215$ across it:
While I'm still in China thinking about the Simpson's paradox, let me share the animation I did for the slide opening the 3rd (三) part of my talk: Laozi contemplating birds.
The flock as a whole goes upward. Each bird in its own group feels it's getting downward, as birds appear out of the fog upwards from it. Each birds independently of anything else flies in a straight horizontal line. Correlations are thus dependent on your reference frame and to what you compare yorurself to.
I found the way out, it's smoking hot: we cut everything in two: left, ensemble averages; right, single-collapse averages. Because they are the same, nobody will notice, which is basically the point. But we'll have a cornerstone "Ensemble average" vs "Single-collapse average" on both sides of an almost invisible dotted yellowish line, and $C_\alpha=1/6$ and $1/4$ on two columns, also introducing RCPS in a very natural way. It will be beautiful!
We promised with Pavlos to write to the editor of Nature within a couple of week, with something big. We were
9 d 16 h 26 min 14 s overdue
when we submitted the manuscript on 1November (2026)(af). What we had got increasingly bigger with time, but passed the above, we were not people of our words, and at some point one must stop and submit something...
I'm thinking in putting the one-collapse observables for $g^{(2)}$ and $g^{(3)}$ too:
but we don't quite have the room. Logic demands them, though... The above is Monte Carlo for 1000 points.
The steps are:
Take the collapse's pairs for $g^{(2)}$ (triples for $g^{(3)}$) in relative angles, and normalize relative to chance: histogram of Δθ over all its pairs, divided by the number of pairs and by the uniform density.
Multiply by the pair and triple rates $R \equiv \langle N(N−1)\rangle/\langle N\rangle^2$ (and $R^{(3)} \equiv \langle N(N−1)(N−2)\rangle/\langle N\rangle^3$. That is 3/2 and 3 for thermal light, 1 and 1 for Poissonian numbers, and $1−1/N$ and $(1−1/N)(1−2/N)$ for fixed $N$. Here this is put by us, with knowledge of what we have or want, this is not provided by the single collapse. The ensemble or knowledge of the state must provide them.
Note that if we normalized by the singles, we get no correlations by construction, which is a proof of Simpsons' paradox, and thus also a nice point!
We can reproduce thermal light exactly with $C=1/6$ up to fourth order. We need two atoms otherwise! I don't think we make this point in the manuscript but probably should... It'll be a long night.
Last animation for my keynote, the removal of the phase confounding factor for thermal light:
I want to discuss three points, 一 二 三, and because I'll prepare and rehearse their delivery, I should have time to cover them all. I won't have enough time to go into the details, like fully removing the Simpsons paradox with random-phase coherent states, but the main message should be conveyed.
My new animation of the azimuthal HBT effect:
It's faster than distances, less things to introduce, and I can carry on right away on the confounding factors and Pavlos' experimental data.
I must also start to look into our Simpson paradox in parallel spaces.
We now have two oscillators (or two universes in which they live), in the doubled phase space
$(\boldsymbol\alpha,\boldsymbol\beta)\in\mathbb C^M\times\mathbb C^M$, and the state is now
This is my hope for the alternative-universe description. It becomes standard, classical one-universe when $\boldsymbol{\alpha}=\boldsymbol{\beta}^*$. The de Finetti representation then reads:
\begin{equation}
\label{eq:0LiYJb}
\rho^{(N)}(\mathbf r_1,\dots,\mathbf r_N)=\int
P(\boldsymbol\alpha,\boldsymbol\beta)\prod_{j=1}^N
I_{\boldsymbol\alpha\boldsymbol\beta}(\mathbf r_j)\,
d^{2M}\!\alpha\,d^{2M}\!\beta\,,
\end{equation}
where, and this is where the key to the problem lies:
\begin{equation}
\label{eq:0LdNRb}
I_{\boldsymbol\alpha\boldsymbol\beta}(\mathbf r)\equiv
\Psi^*_{\boldsymbol\beta^*}(\mathbf r)\,\Psi_{\boldsymbol\alpha}(\mathbf r)\,.
\end{equation}
This is complex. Even if it'd be real, I'm not sure what it'd mean so I'm not entirely concerned about it being complex. We're about at the stage of Cardano finding $i$ such that $i^2=-1$ something so "complex" that it's even "imaginary"! It's not if you look in the extended space (2D), then it's not a point in the line, it's a point in the plane. You bring rotation into that, but little else (formally). I expect something similar here. Something has to be weird, or complex, or imaginary somewhere... otherwise it'd be boring classical physics all the way down.
A quick proof of Marinoet al.[6]'s Eq. (3) (which is Eq. \eqref{eq:0LoKmb} below for violating CSI for bright beams). If one decomposes $$g^{(2)}_{ij}\equiv 1+\epsilon_{ij}$$ as we like to do,[7] then the CSI reads
$(1+\epsilon_{ab})^2\le(1+\epsilon_{aa})(1+\epsilon_{bb})$ which, expanded, yields $2\epsilon_{ab}-(\epsilon_{aa}+\epsilon_{bb})\le\epsilon_{aa}\epsilon_{bb}-\epsilon_{ab}^2$. The lhs is of first order, so setting the rhs, of second order, to zero gives their Eq. (3):
\begin{equation}\label{eq:0LoKmb}\epsilon_{ab}\le\frac{\epsilon_{aa}+\epsilon_{bb}}{2}\,.\end{equation}
This is not equivalent, though. If CSI is obeyed, then Eq. \eqref{eq:0LoKmb} is obeyed. By contraposition, therefore, violating Eq. \eqref{eq:0LoKmb} ⟹ violating CSI. But one can violate CSI while satisfying Eq. \eqref{eq:0LoKmb}.
One gets the above logical implication by adding one to both sides: Eq. \eqref{eq:0LoKmb} is the same as
\begin{equation}
\label{eq:0LSc7b}
g^{(2)}_{ab}\le\frac{g^{(2)}_{aa}+g^{(2)}_{bb}}{2}\,,
\end{equation}
i.e., a weaker bound on the cross-correlation, by the arithmetic mean of the autocorrelations instead of their geometric mean. But since $g^{(2)}\ge0$, the CSI reads $g^{(2)}_{ab}\le\sqrt{g^{(2)}_{aa}g^{(2)}_{bb}}$, and the geometric mean is bounded by the arithmetic one, $\sqrt{g^{(2)}_{aa}g^{(2)}_{bb}}\le(g^{(2)}_{aa}+g^{(2)}_{bb})/2$.
Therefore, every classical field satisfies both inequalities. The converse is not true (it is for symmetric fields, $g^{(2)}_{aa}=g^{(2)}_{bb}$; otherwise the CSI violations it misses lie in a window of width $(\sqrt{g^{(2)}_{aa}}-\sqrt{g^{(2)}_{bb}})^2/2$).
The high-res version of the quantum to classical transition, which I computed here in the background, with an adaptative grid (each time there was a gap, Claude instructed the computer to try harder; since it's never bored, he could do that efficiently):
As it's better to show pictures than equations, here they are, at the threshold $k=2\pi/L$ where the violation becomes total:
$\phi_a$ is real and simply decays, $\phi_b$ grows exactly as much, so that their product—and thus $G^{(2)}(\mathbf r,\mathbf r)$—is the same everywhere, while its phase winds once over the box, which is also what makes the two modes orthogonal. The intensity itself, $G^{(1)}=2c\cosh k(x-L/2)$, is a bathtub: anything but uniform. You can't have both at once, that's a theorem of that time.
The condition for total violation, $\kappa\equiv kL/2\pi\ge1$, reads better as $|\nabla\Lambda|\ge|\nabla\varphi|$: the log-ratio of the amplitudes must ramp up at least as steeply as the relative phase. That's the collinear case of the local analysis, holding over the whole box.
There's also a one-liner for the "iff" that I missed then: $\cosh\kappa s+\cos s-2=2\big[\sinh^2(\kappa s/2)-\sin^2(s/2)\big]$, and $\sinh\kappa t\ge\sinh t>t>\sin t$ when $\kappa\ge1$, while $\sinh\kappa t\approx\kappa t<\sin t$ near $t=0$ when $\kappa<1$.
Here's how much CSI is violated (green) or satisfied (black) for each pair of points, $(x_1,y_1)$ and $(x_2,y_2)$ where only the $x$ coordinates matter, below and at threshold:
Homogeneous—I remind ourselves—means that the violation is the same everywhere for a given separation, not that it's the same everywhere. So the maps are striped along the diagonal, and a pair of points violates CSI by the same amount, wherever you put it in the box. The violation starts from nothing on the diagonal (as $\delta^4$) and grows with the separation, remaining the same anywhere for a given separation.
This is the homogeneity of $G^{(2)}$. The normalized $g^{(2)}=G^{(2)}/G^{(1)}G^{(1)}$ is not homogeneous, since the intensity is not. Whether CSI is violated doesn't care (the $G^{(1)}$ cancel), but "antibunching" does: with $g^{(2)}$, it changes from place to place.
Alexandros suggested to really go into the multiphoton regime with photon-number resolving camera. Since then (this was the doggy-park discussion), I've looked at what the theory says about this. Instead of the continuous plane, we pixelate a 3000$^2$ grid covering $[-4,4]^2$ (in units of the vortex's radius, putting its rim at 375 pixels from the center).
With $N=1.2\times 10^7$ photons, we have about 10 photons averages in a pixel on the rim for the donut. That makes for neat, classical pictures:
The pair distance has $\binom N2=7.2\times10^{13}$ sample points. Now, that's large numbers!
Within granularity of the grid, we get the expected results:
Something which has been on my mind since 2009 is the the nature of the quantum to classical transition in the Jaynes-Cummings model. I described it with this type of structure in our paper with del Valleet al.[3].
This has everything. For the life of me, I don't remember now who at the HPM (2026) was doing something similar but, of course, in Tavis-Cummings or much more complicated systems. People always jump to the difficulty and invoke big machinery and hopeless cases when everything is, always, in two coupled oscillators (one truncated). The discussion also popped up with Najmeh on the nature of nonlinearity in quantum mechanics. I'm referring a paper now who goes again to great circumlocutions to miss the point very badly. When Miguel Bastarrachea-Magnani visited us in Madrid (and when I met him again at the PLMCN26), I tried to interest him in solving this problem, him being an expert on quantum chaos and integrability. I had spent then much time digging up my old codes that were running in the Southampton clusters so that we could reproduce the results then. I was concerned about convergence and numerical stability, because the computer capacities had demultiplied. Now in times of AI, you can have Claude do these things for you, find the code, cleanse them, run them and check convergence by itself:
It's working indeed as a very efficient, very motivated PhD student or obedient Post. Doc. I'll settle this question once and for good, then. Expect to see beautiful fractals in the coming weeks. I'll just have it use the color code of our later (del Valle and Laussy[1]) paper.
Violation of CSI from two $\ell_a\neq\ell_b$ vortices. Never everywhere, always somewhere:
In the plane of radial contrast and angular separation, the CSI is satisfied in a band around equal radii, $r_1=r_2$, whose width oscillates with the angle between zero and a factor $(3+2\sqrt2)^{1/||\ell_b|-|\ell_a||}$ in $r_2/r_1$, and are violated everywhere outside this band. No choice of charges removes the band: $|\ell_b|\neq|\ell_a|$ implies $\ell_b\neq\ell_a$, so the relative phase always reaches $\pi$ somewhere, which fights violation. The band narrows with increasing $\big||\ell_b|-|\ell_a|\big|$.
Working on Violation of Cauchy-Schwarz Inequalities in Space and Time, I realize that the infamous $g^{(2)}$<1/2 criterion might have some sound meaning in violations of CSI inequalities with a thermal field as a reference, as opposed to a coherent one, which gives the antibunching scenario. If using another reference $b$, then CSI violations read:
$$g^{(2)}_{aa}<\frac1{g^{(2)}_{bb}}\,.$$
Have $b$ thermal, you get $g^{(2)}_{aa}<1/2$. It doesn't make the criterion any better, except that such fields—still not being single-photon source ones—are able to violate CSI in an environment where standard fluctuations are the thermal ones.
I like to find the connection I have to every Nobel prize awarded. This year, this is to be found in our Referee reply to the Two photons everywhere[7] paper, where the Referee was complaining that something rare, even if big, is not significant. He wanted to rob us of my title and have it read "Two-photon correlations everywhere" instead. I told him "how about IceCube?" I didn't tell him "this'll get the Nobel prize in a couple of years" but could have:
Of course, the authors are the first to be aware that a correlated photon pair consists of two individual photons and that strong bunching at vanishing intensity does not imply that a relevant rate of photon pairs is available. Instead, it follows from strong bunching that two-photon correlations occur much more frequently than predicted by the Poisson statistics. Thus, for a sufficiently small intensity, one correlated photon pair per year can correspond to a value of g^(2) of 10^5. The authors will agree with me that from a practical point of view, i.e. in the presence of finite detection noise and with applications in mind, it is not justified to list such a region of the two-photon spectrum under the title "Two photons everywhere". I therefore suggest that the authors adapt their presentation in such a way that even an uninformed reader will not be misled.
One of the points we wish to highlight is precisely that two-photon physics (multiphoton physics in general) is not related to the amount of signal, which is one-photon physics (or classical physics). Whether two-photon phenomena are, or are not, useful from a practical point of view is a secondary concern at our present level of description. We want to attract instead attention on the two-photon spectrum structure, with its two-photon Mollow triplet (of two-photon leapfrog transitions), circles of antibunching, unconventional bunching in the detuned Heitler regime, etc. All this is happening in the abstract two-photon space, which is not directly visible, but exists nonetheless, with no (theoretical) concern for signal. In this context, does one correlated-photon pair per year, to take the extreme case of the Referee, make a phenomenon unphysical? The IceCube collaboration recently reported the observation of seven (7) events in a 1 km$^3$ detector measured over the course of 10 years [Phys. Rev. Lett. 132:151001 (2024)]. The present text is, conceptually, at this level of description. In the two-photon spectrum, one does not need to—indeed, should not—refer to intensity: the structure exists regardless of the one-photon brightness.
What I couldn't tell him, of course, is that there were 124 people before the one guy who would get the prize (and he was in the top list):
For most people not from the neutrino community, that is the most obvious thing to comment upon: Halzen got the prize alone. The last one was Charpak in 1992. Before him, De Gennes (also French) also got it alone. For many physicists I know, a single awardee is a first time in their life. Since Cecil Powell in 1950, and Yukawa before him (the one after whom an institute is named in Kyoto), it is indeed more common to award various physicists. Before that, couples were the exception, even when such couples were Schrödinger and Dirac. The only triplet for the first 55 years of the prize was Becquerel and the two Curie, who were really one. Times change.
Wilhelm Conrad RöntgenHendrik A. Lorentz Pieter ZeemanHenri Becquerel Pierre Curie Marie CurieLord RayleighPhilipp LenardJ.J. ThomsonAlbert A. MichelsonGabriel LippmannGuglielmo Marconi Ferdinand BraunJohannes Diderik van der WaalsWilhelm WienGustaf DalénHeike Kamerlingh OnnesMax von LaueWilliam Bragg Lawrence Braggnot awardedCharles Glover BarklaMax PlanckJohannes StarkCharles Edouard GuillaumeAlbert EinsteinNiels BohrRobert A. MillikanManne SiegbahnJames Franck Gustav HertzJean Baptiste PerrinArthur H. Compton C.T.R. WilsonOwen Willans RichardsonLouis de BroglieSir Chandrasekhara Venkata Ramannot awardedWerner HeisenbergErwin Schrödinger Paul A.M. Diracnot awardedJames ChadwickVictor F. Hess Carl D. AndersonClinton Davisson George Paget ThomsonEnrico FermiErnest Lawrencenot awardednot awardednot awardedOtto SternIsidor Isaac RabiWolfgang PauliPercy W. BridgmanEdward V. AppletonPatrick M.S. BlackettHideki YukawaCecil PowellJohn Cockcroft Ernest T.S. WaltonFelix Bloch E. M. PurcellFrits ZernikeMax Born Walther BotheWillis E. Lamb Polykarp KuschWilliam B. Shockley John Bardeen Walter H. BrattainChen Ning Yang Tsung-Dao LeePavel A. Cherenkov Il´ja M. Frank Igor Y. TammEmilio Segrè Owen ChamberlainDonald A. GlaserRobert Hofstadter Rudolf MössbauerLev LandauEugene Wigner Maria Goeppert Mayer J. Hans D. JensenCharles H. Townes Nicolay G. Basov Aleksandr M. ProkhorovSin-Itiro Tomonaga Julian Schwinger Richard P. FeynmanAlfred KastlerHans BetheLuis AlvarezMurray Gell-MannHannes Alfvén Louis NéelDennis GaborJohn Bardeen Leon N. Cooper Robert SchriefferLeo Esaki Ivar Giaever Brian D. JosephsonMartin Ryle Antony HewishAage N. Bohr Ben R. Mottelson James RainwaterBurton Richter Samuel C.C. TingPhilip W. Anderson Sir Nevill F. Mott John H. Van VleckPyotr Kapitsa Arno Penzias Robert Woodrow WilsonSheldon Glashow Abdus Salam Steven WeinbergJames Cronin Val FitchNicolaas Bloembergen Arthur L. Schawlow Kai M. SiegbahnKenneth G. WilsonSubrahmanyan Chandrasekhar William A. FowlerCarlo Rubbia Simon van der MeerKlaus von KlitzingErnst Ruska Gerd Binnig Heinrich RohrerJ. Georg Bednorz K. Alex MüllerLeon M. Lederman Melvin Schwartz Jack SteinbergerNorman F. Ramsey Hans G. Dehmelt Wolfgang PaulJerome I. Friedman Henry W. Kendall Richard E. TaylorPierre-Gilles de GennesGeorges CharpakRussell A. Hulse Joseph H. Taylor Jr.Bertram N. Brockhouse Clifford G. ShullMartin L. Perl Frederick ReinesDavid M. Lee Douglas D. Osheroff Robert C. RichardsonSteven Chu Claude Cohen-Tannoudji William D. PhillipsRobert B. Laughlin Horst L. Störmer Daniel C. TsuiGerardus 't Hooft Martinus J.G. VeltmanZhores Alferov Herbert Kroemer Jack KilbyEric Cornell Wolfgang Ketterle Carl WiemanRaymond Davis Jr. Masatoshi Koshiba Riccardo GiacconiAlexei Abrikosov Vitaly L. Ginzburg Anthony J. LeggettDavid J. Gross H. David Politzer Frank WilczekRoy J. Glauber John L. Hall Theodor W. HänschJohn C. Mather George F. SmootAlbert Fert Peter GrünbergYoichiro Nambu Makoto Kobayashi Toshihide MaskawaCharles K. Kao Willard S. Boyle George E. SmithAndre Geim Konstantin NovoselovSaul Perlmutter Brian P. Schmidt Adam G. RiessSerge Haroche David J. WinelandFrançois Englert Peter HiggsIsamu Akasaki Hiroshi Amano Shuji NakamuraTakaaki Kajita Arthur B. McDonaldDavid J. Thouless F. Duncan M. Haldane J. Michael KosterlitzRainer Weiss Barry C. Barish Kip S. ThorneArthur Ashkin Gérard Mourou Donna StricklandJames Peebles Michel Mayor Didier QuelozRoger Penrose Reinhard Genzel Andrea GhezSyukuro Manabe Klaus Hasselmann Giorgio ParisiAlain Aspect John Clauser Anton ZeilingerPierre Agostini Ferenc Krausz Anne L’HuillierJohn J. Hopfield Geoffrey HintonJohn Clarke Michel H. Devoret John M. MartinisFrancis Halzen
I have successfully simulated the entangling Gerace gate (H+CNOT) with coherent states. It fails in a different way than I'd have thought. If fails gracefully. It's not immediately useful for anything I can see (no obvious quantum supremacy) but it's not trivially useless either. There might be scheme to push it into a computational resource. And at a fundamental level, it's a beautiful thing to characterize.
The most interesting is that the block we're interested in remains shielded all down the circuit, so it remains good for whatever comes next, that's what the deterministic scheme brings us. Next is to implement a real application, Deutsch, Simon, GHZ, teleportation, whatever... and show that with probability $e^{-2}$ or $e^{-3}$ depending on what we do, we get our result. It'll certainly won't beat a direct classical approach, but it's a first foot into quantum computation with only resources already available to our experimentalists (interacting waveguides). It'd be interesting to also consider hybrid approach, e.g., a Fock state + a coherent one. That might bring us over the edge.
The sea is not calming down (I've spent the afternoon at the seafront in a restaurant, with Baklava and more Greek coffees) but the sun is coming down. I'll have the chance today to see the sunset from the castle. I'll see if the other prisoners would be interested in that.
The Gerace gates could be useful as deterministic even on Gaussian inputs, and thus semi-scalable, shifting the indeterminacy to the number of qubits as opposed to the number of gates. This seems to be $e^{-n}$ for the $|\alpha|^2=1$ (optimum) case.
I'm not sure how well known this is. See for instance Descampset al.[8] It would be interesting to devise a practical problem that could be implemented and actually solved demonstrating a quantum advantage. Maybe using Grover's algorithm, for instance.
Truncated coherent states to $\nu$ photons per rail quickly degrade the LOQC CNOT probability of success from $P_{\rm succ}=1/9$ for $\nu=1$ (Fock states) to, already with $\nu=2$, $P_{\rm succ}\simeq|\alpha|^2/27$ (ratios $0.111$ and $0.037$ at $|\alpha|^2=0.01$): already $3.7\times10^{-4}$ at $|\alpha|^2=0.01$.
Besides, which should also be looked at closer, it's not only that the gate fire less, but it fires wrong, in contrast to Gerace gates, which fail gracefully. Anyway, it's now established that while Fock states are an essential resource for LOQC (as they should), it's not the case for Gerace scheme, surprisingly.
Milburn[9] implemented a Fredkin gate with cross-Kerr. Chuang and Yamamoto[10] implemented the dual-rail logical encoding of Scala et al. (before them). As for Adhikariet al.[11], they rely on superconducting qubits to implement those ideas.
Hanbury Brown-Twiss effect and classical entanglement with OAM-carrying light. J. Laatikainen, S. Pokharel and O. Korotkova in arXiv:2604.14452 (2026).
