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30 September (2024)
29 September (2024)
28 September (2024)
Replying to @JeanBricmont
Avoir écrit un ouvrage sur la bibliothèque de Napoléon en Anglais, qu'il crève lui aussi.
Area for a simple polygon with vertices in polar coordinates rₖ exp(iθₖ) (angles must be ordered in [0,2π[). There's a famous formula (shoelace) in Cartesians, but I don't know of any source providing the following. Even the triangle N=3 is not always well covered. Here you go:
Another way to write it, using the modulo (%) and now labelling the N vertices as rₖ exp(iθₖ) for 0≤k≤N-1:
which appears to look like the Shoelace formula (determinants when going to the cartesian form of z). Isn't this result more general, compact and simple than what we find everywhere? Am I or the Wikipedia and references therein missing something obvious?
23 September (2024)
Replying to @SchneiderQMat
Is it Christopher without badge?
22 September (2024)
Fabrice Laussy reposted
The Phase as a Key to Unlock Quantum Correlations
Read more in HEISINGBERG blog post at heisingberg.eu/2024/09/20/the… https:/…
Read more in HEISINGBERG blog post at heisingberg.eu/2024/09/20/the… https:/…
«Purity is possible if you turn your life into a line of poetry written with a splash of blood.»
21 September (2024)
Give someone with enough talent a mirror and he can raise himself, for the moment of an insight, above Kubrick and Tarkovsky. That's the real equality among men: that of the mind.
20 September (2024)
For people interested in the quantum/classical correspondence wp.icmm.csic.es/seminars/2024/…
19 September (2024)
Replying to @frederic_RTfr
ça ne sera pas nucléaire tout de suite... mais quand même, des frappes Russes en Europe, même si sur des sites militaires, quelle fracture du monde, quel plongeon dans l'abîme!
10 September (2024)
Replying to @QNFoton
dear Alexis; yes it is allowed (Göppert Mayer) and yes it is continuous. If there is any reason why faster processes are suppressed, which is what happens in planetary nebula, you may even see it directly, i.e., in PL. If not, it's always there through correlations.
9 September (2024)
𝗧𝘄𝗼 𝗣𝗵𝗼𝘁𝗼𝗻𝘀 𝗘𝘃𝗲𝗿𝘆𝘄𝗵𝗲𝗿𝗲
Part 4: The crème de la crème.
As promised earlier this year (part 1 here ), we collect the best snippets of our paper that the Royal Society published today. A fast way to read the paper: royalsocietypublishing.org/doi/10.1098/rs…
Part 4: The crème de la crème.
As promised earlier this year (part 1 here ), we collect the best snippets of our paper that the Royal Society published today. A fast way to read the paper: royalsocietypublishing.org/doi/10.1098/rs…
Fabrice Laussy @fplaussy · 22 Feb 2024
𝗧𝘄𝗼 𝗣𝗵𝗼𝘁𝗼𝗻𝘀 𝗘𝘃𝗲𝗿𝘆𝘄𝗵𝗲𝗿𝗲
Part 1: Kindergarten.
Niels Bohr founded 𝑄𝑢𝑎𝑛𝑡𝑢𝑚 Mechanics by 𝑄𝑢𝑎𝑛𝑡𝑖𝑧𝑖𝑛𝑔 angular momentum of the electron going round the nucleus. He became to Rydberg's empirical spectral lines what Newton was to Kepler's orbits.
Part 1: Kindergarten.
Niels Bohr founded 𝑄𝑢𝑎𝑛𝑡𝑢𝑚 Mechanics by 𝑄𝑢𝑎𝑛𝑡𝑖𝑧𝑖𝑛𝑔 angular momentum of the electron going round the nucleus. He became to Rydberg's empirical spectral lines what Newton was to Kepler's orbits.
I don't know how many people are interested in the two-frequency ℑₖ (fluctuations, anomaly & squeezing) decomposition of resonance fluorescence—our Figs. 2(g-o)—let alone understand what this is, but someone has to do it, and if not us (& now), then who? (& when?)
Here I focus only on the salient points as expressed in the paper, not even spending time on funny things like, say, Ref. 12's dreadful typo. (It's the typesetter's fault, no doubt; nowadays one should proofread v°2.0 as they manage to hide mistakes even in the hyperlinks)
Here we go! We approach two-photon physics from dissipative (Lindblad) quantum optics, as opposed to typical QED. This allows us to solve the problem exactly. No Fermi golden rule.
We provide the exact shape for the photoluminescence (one-photon) spectrum:
ℒ is a Lorentzian (broadening of the δ peak) while ℳ is more messy but this covers everything:
At resonance, this becomes Eq. (2.7) in presence of a detector of bandwidth Γ. Assuming an ideal detector, i.e., Γ→0, gives a strong δ peak sitting on top of a weak fluorescent (squared) Lorentzian:
It's difficult to see that the previous seemingly trivial statements, actually become subtle and important when going to two-photon (quantum) physics.
Let's have a two-photon look. The two-photon observable of choice, g⁽²⁾, has the corresponding expressions with detection (2.9) or for an ideal detector (2.10):
Do you see it? With textbook, ideal (no detector) cases, you'd think you have an antibunched δ(ω) line. But even notations show this is mistaken: one needs Γ→0, the other Γ→∞. Finite Γ gives you one 𝒐𝒓 the other: Γ is not a mere technical concern but a central consideration!
Still, there is a way to have both together (antibunched narrow line), but one first needs to understand the nature of antibunching when one admixes coherence (driving laser) and quantum fluctuations (the two-level system)
@QuantSciTech can give you Joint subnatural-linewidth and single-photon emission doi.org/10.1088/2058-9… by showing you that antibunching is the interference between a coherent and a squeezed state, and if you don't have the right balance you can correct it yourself with a laser.
Eq. (2.9) above in the thread [you're reading a paper] hides that it measures photons at a given frequency ω (in a Γ window). It's a particular case of something bigger. The bigger thing being: all photons at all the frequencies. You could correlate those too. Stupid? Read on.
Now is the time for quantum optics to do something different.
The new way to look at things: the two-photon spectrum.
First lesson from the two-photon spectrum: emission from the spectral peak is classical! The quantum hides in the dark.
This is again the manifestation that two-photon space is independent of the other-photon spaces, so when doing two-photon physics, try to forget about one-photon physics. Looks obvious, but say that to an experimentalist! Signal for them is one-photon signal, intensity, counts...
It's why many refuse to believe (most refuse to hear about) leapfrog photons, which exist as multiphotons but don't show up in one-photon observables. People are oblivious to basically all of the big picture. Grabbing the quantum beast by its classical tail.
To understand two-photon emission, we must first do a bit of easy two-photon quantum states physics.
It's a student exercise to compute correlators of two-mode squeezed states. For small squeezing, it's all very simple and one gets two-photon bunching all over the place. But, we can repeat the @QuantSciTech trick and admix coherent states. Then we get antibunching—for two modes.
So the quantum state that describes everything is this one: the coherent-displaced two-mode squeezed state:
Next exercise, compute correlators for those. Indeed, now they can exhibit antibunching. I let your gaze wander on the equations to satisfy yourself that much tuneability is now in our hands, just mixing fields:
Before, when looking for antibunching when mixing a coherent state with a squeezed state, we had one condition. With two modes, we now have an infinite set of conditions which define patterns in the parameter space.
This was happening with energies for two-photon emission (bunching), something we knew since 2013. It also happens with phases for two-photon interferences (including antibuching, and unconventional bunching), something we report in this paper for the first time.
Interferences of what? Of four terms, one is 1, three are correlations of the fluctuations ℑᵢ for i=0, 1, 2.
These four terms are those we previously considered degenerate for one mode, explaining antibunching as 0=1+1+0-2 (squeezed/coherent admixture in the Heitler regime) or 0=1-1+0+0 (saturation of a two-level system in the Mollow regime).
And now the leap to greatness, turning the exercises into a new picture of the world: the two modes become the two frequencies at which we put our detectors. The patterns in parameter space become visible in the two-photon spectrum, imprinting the multiphoton interferences there.
And if it's true, then before the real two-photon resonance fluorescence—which we say produces squeezing that it interferes with the driving laser—then let's look directly at squeezing + lasing together:
Is it true? It is. A squeezed cavity produces photons correlated in frequencies exactly as the Heitler regime of a two-level system does (there you need to remember things like ):
Fabrice Laussy @fplaussy · 18 Apr 2023
Why there is a circle of antibunching in two-photon spectra...
And the agreement becomes exact in the limit of vanishing driving. You were thinking of doing big non-Gaussian two-level-system physics; you were mainly admixing the two most basic Gaussian states.
The two-photon spectrum for both cases is this beautiful expression, which is to two-photon physics what the Lorentzian is to the one-photon one.
I discussed it in detail already, , but I can't help marvelling again at the apparition of an exact circle in the photon correlations.
Fabrice Laussy @fplaussy · 11 Mar 2024
𝗧𝘄𝗼 𝗣𝗵𝗼𝘁𝗼𝗻𝘀 𝗘𝘃𝗲𝗿𝘆𝘄𝗵𝗲𝗿𝗲
Part 3: the circle.
Part 1 here:
Hello again, conspiracy physicists...
Part 3: the circle.
Part 1 here:
Hello again, conspiracy physicists...
There are many more beautiful things to look at, e.g., the redundancy of our ℑᵢ terms in the Gaussian wrapping of this regime
but we can leave it there for now. We have not only revealed the two-photon physics of resonance fluorescence, we have also explained it.
That's only the first step towards multiphoton physics. We only, after all, tackled the case N=2. Even there you can see some evidence of N=3 physics, and at some point, someone should go and look at the three-photon space in the eyes. People who are not scared of hypervolumes.
If you go there and get lost in the Hilbert space, we agreed to be held accountable.
4 September (2024)
Replying to @sylvaingigan
When I said to J.-J. Greffet that Bell asked Aspect if he had a permanent position before discussing whether Bell inequalities were a good PhD topic, he told me this was common practice in France and that he, himself, also had a permanent position ahead of his PhD.