/sci/ - Science & Math

Physics is a pseudo science anyway.

>>5408947

>>5408947
uhhhhh.
no.

>>5408951

All your truths are based on another truth, which is based on another truth, which is based on the truth of another truth, till you turn full circle.

Check mate

What? A gas? At near absolute zero? What fucking particle is that?

>>5408961
Not just near. BELOW.

>>5408924
Perhaps it doesn't go negative, and the scale simply goes lower than we thought. Negative energy would be a bit weird.

>>5408964
I know but now, I'm wondering how the fuck a particle could be a gaz at that point. (Sorry, chemistry and physics pleb. I'm a mathematician).

>>5408969
No, see, it's quantum.
>spooky

>>5408975
That's what I don't get.
What the fuck is a Quantum gas and why the fuck do physicists add "Quantum" in the beginning of anything nowadays? Not every thing related to classical mechanics has "Classical" as a prefix.

>>5408977
Because quantum is
>spooky
and makes things sound more interesting

The gas has negative temperature. Negative temperature is not colder than absolute zero.

>>5408975
lol'd

http://www.scientificamerican.com/article.cfm?id=quantum-gas-goes-below-absolute-zero

>>5408924

Did you even read the article? It explained it perfectly for fucks sake

>>5408983

What about "billionths of a degree Kelvin BELOW absolute zero" don't you understand?

>>5408998
What about "Colder" don't you understand?

T = dE/dS

all negative temperature means is that the entropy of the system decreases when you add energy. It doesn't feel colder than absolute zero if you touch it.

>hotter/colder
>subjective measures of temperature

>>>/b/

>>5409010
> entropy decreases

>>5409051
>entropy cannot decrease in open systems

Entropy can only decrease if everyone adopts free (as in freedom) software. Stay blind, sheeple.

>>5409051
It's not a closed system, hence why you can add energy.

>>5409051
You cannot into thermodynamics.
Unisolated systems decrease in entropy all the time.

>If built, such systems would behave in strange ways, says Achim Rosch, a theoretical physicist at the University of Cologne in Germany, who proposed the technique used by Schneider and his team3. For instance, Rosch and his colleagues have calculated that whereas clouds of atoms would normally be pulled downwards by gravity, if part of the cloud is at a negative absolute temperature, some atoms will move upwards, apparently defying gravity4.

>Another peculiarity of the sub-absolute-zero gas is that it mimics 'dark energy', the mysterious force that pushes the Universe to expand at an ever-faster rate against the inward pull of gravity. Schneider notes that the attractive atoms in the gas produced by the team also want to collapse inwards, but do not because the negative absolute temperature stabilises them. “It’s interesting that this weird feature pops up in the Universe and also in the lab,” he says. “This may be something that cosmologists should look at more closely.”

So we antigravity and Alcubierre drive-able mateirals now?

>>5409109
MY DIIIIICK

>>5409109
> if part of the cloud is at a negative absolute temperature, some atoms will move upwards, apparently defying gravity.
Holy fuck.

>Another peculiarity of the sub-absolute-zero gas is that it mimics 'dark energy'
HOLY FUCK.
This could be a glorious time indeed.

>>5408951
Whatt he means it is a fractal science with dimension of 2.9

Don't forget that things can also have negative temperatures, as in -1000 Kelvin.

Negative Kelvin is routinely achieved with powerful lasers, which means that they are hotter than an infinite positive temperature. In other words, an object that is -1K is hotter than the +100000000K, but something that is -1000 K is colder than something that -1K.

Negative Kelvin is routinely achieved with powerful lasers, which yield temperatures hotter than an infinite positive temperature. In other words, -1K is hotter than +100000000K, however, -1000 K is colder than -1K.

>>5409180
Please explain how that made any sense before i google it and my brain leaks out of my ear.

>>5409180
wut

>tfw relative absolute 0
>0 is relative

>>5409189
Just look up "negative temperature" on Wikipedia. It's a statistically based concept of temperature, I believe.

>>5408965
It pretty much IS negative energy
>For instance, Rosch and his colleagues have calculated that whereas clouds of atoms would normally be pulled downwards by gravity, if part of the cloud is at a negative absolute temperature, some atoms will move upwards, apparently defying gravity

>>5409010
>T = dE/dS

Fuck no, it's 1/T=∂S/∂E
negative and infinite temps are fine, 0K is the only thing not allowed

>>5409241
1/(dS/dE)=dE/dS

problem math autists?

>tfw you didn't take statistical mechanics and learned babied down engineering thermo instead

>>5409241
do you not understand reciprocals

>>5409252
>>5409258
For any value not 0

>>5408960
Haha! I don't care if troll, physics is too fun to talk about. See, it all starts with observation of unknown phenomena. Once patterns are recognized, underlying laws are guessed at (this can take an incredibly long about of time), and experiments are carried out (this can take incredibly expensive toys) to gather more data and refine the guesswork until prediction with a high degree of accuracy is possible! Whoo! Physics forever!

>>5409253
feels bad man

chemical engineer here.

the article posted by OP is mathematical trickery.

as the article states, most particles are at overall average energy states, but if you make more of the particles have higher energy states, then you play some thermodynamic trickery, and mathematically because of some derivatives involving entropy, temperature becomes negative. (because entropy is concerned with the probability of certain energy states microscopically), it should also be noted that the entropy of a system does not have to increase, it can decrease, it's based on probability, and most of the time the probability is 99.99999999999999% sure that the entropy will increase and 0.0000000000001% sure that it will decrease. overall, entropy is related to probabilities of energy states microscopically, and anyone who taught you that entropy has to increase misled you (but didnt really mislead you, it's touchy).

>>5409308
How much energy would it take to create large scale systems whose entropy is more likely to decrease than increase?
Is this a solution to the problem of heat dissipation in heat engines?

>>5409341
think of it this way. imagine a ton of tiny boxes, each with a ping pong ball inside.

at energy level 1 imagine you have 1,000,000 possible boxes, and at energy level 50 you have 1 possible box. therefore if you threw a ping pong ball in a random box the chances are that it would end up in a box with a lower energy level.

so when you put... say... 2,000 ping pong balls in the boxes, there's a really small chance that they all ended up in the 1 box with energy level 50.

THIS is entropy. entropy (in the microscopic sense) describes the chances for certain arrangements of ping pong balls.

when you take a normal course (not a statistical course) in thermodynamics, everything assumes your pingpong ball distribution came out with a typical outcome, and nothing crazy happened in the distribution.

i'm not sure if that's clear or even if it helps.

this is stuff you don't really learn about (unless you have a rigorous teacher) in a normal thermo class, it's more reserved for graduate level statistical thermodynamics classes.

I think explaining how entropy works has become an official /sci/ sport.

i'd also like to say, imagine a similar case where there are 3 energy levels:

and energy level 1 = 100 boxes
energy level 2 = 50 boxes
energy level 3 = 20 boxes

if you have 2 pingpong balls already placed, and 1 ball happens to be in level1, and the other ball is in level2. if both balls happen to go into... say... level1 at the same time, then guess what, the entropy of the system decreased.

>>5409359

>this is stuff you don't really learn about (unless you have a rigorous teacher) in a normal thermo class, it's more reserved for graduate level statistical thermodynamics classes.

Thank god I did. I always pondered how you could design a system that has a passive heat pump to where the temperature could increase naturally (or decrease entropy in a large localized area to be technically correct).

>>5409361
understanding something deeply takes a lot of time, even if it seems like a concept that should be simple.

it's all about how far down the rabbit hole you want to go (and engineers stop going down the rabbit hole when it's not economical to go further)

>>5408924

Why does that make you mad OP? I find that incredibly interesting.

>Phonons

Wasn't this the thing needed for the Alqubierre/ warp drive that was mentioned some months ago?

Well fuck, we antigravity now.

>>5409435
yup...

but what I wonder is

as mass increases time slows down.. but if these have negative mass, becasue they are repelled by gravity.. does that mean they go faster.. or does the 0 point flip and they go backwards in time, explaining all the reversed features they exhibit

>mfw all the ultra hard sci-fi fags claiming that you can't have negative mass or exotic matter for Alcubierre warp drive now have to contend with negative temperatures and still tell me that exotic matter is impossible

>>5409615
>negative temperatures

I don't think you know what this means.

>>5409627

I don't think you've been reading the thread

>
Anand Ramanathan said:

As a former quantum physicist, I find this News and Views as well as the Science article quite misleading. Although the work itself is very impressive, the authors misuse the definition of temperature to claim to achieve negative temperatures. Using their definition, anyone who creates a 2-level (or 3-level) quantum energy system with higher particle occupation at the higher energy states can claim negative temperature. This can easily be done in any cold atom laboratory. I would like someone to correct me if I am wrong.

>>5408924
>That's it.
>I quit physics.
Then you probably never did enough physics for your "quitting" to mean anything, this isn't surprising to anyone who knows what temperature is or how logarithms work. It's also been mathematically known for over a century and questions around it are a staple of undergraduate thermodynamics courses.

>>5408977
>>5408981
Quantum states and classical states are treated differently in statistical mechanics, classical is the default so it's good to clarify when that's not what's being talked about.

As for what it means in general, it's any property that is quantized. You see it very often because much of physics now deals with phenomenon small enough where quantum effects are relevant.

>>5408998
You should probably progress beyond a highschool physics course before you start condescending to other people.

Not the guy you're responding to by the way.
In general, this whole thread has made me more depressed about /sci/ than any other in a very, very long time.

>>5409706
Tell us about your physics background.
If it's strong, I'll want to suck your dick

>>5409359
>this is stuff you don't really learn about (unless you have a rigorous teacher) in a normal thermo class, it's more reserved for graduate level statistical thermodynamics classes.
Really? I've TAd at two universities and done undergraduate work at a third and they all strongly stressed the rigorous math behind statistical mechanics.

>>5409711
I have an undergraduate degree in mathematics and physics from the University of Maryland, but I am now studying Logic for my doctorate.

You shouldn't be impressed by me, you shouldn't really hero-worship scientists in general; that's the kind of "fuck yeah science!" mindset that's made so many highschoolers make /sci/ the insufferable mess it is these days.

Just work on your education and read mathematically rigorous, sober-minded educational materials if you're actually interested in physics. The enjoyment probably won't come from flashy "omg inverse the polarity and reverse gravity, negative force," especially at first, but more from interesting problems and puzzles. If that's not your thing, I'd suggest you rethink majors (I mean that in a nice way, a competent physics program won't spend any time on teaching you neat empirical stuff about the world).

>>5409726
How is one supposed to find such materials? I found this
http://www.ocf.berkeley.edu/~abhishek/chicmath.htm
for math, but I don't have anything nearly as good for physics or the other sciences.
C-could we p-p-perhaps become f-friends?

>>5409729
The best way is to go to college. If you want to start learning physics, the usual progression would be something like (assuming a normal unambitious highschool path that included trig)
Calc 1, intro classical physics 1
Calc 2, intro classical physics 2
Multivar, classical mechanics
DiffEQ, Quantum physics (I THINK these two can be taken at the same time, but it's been a while)
Statistical mechanics
Electrodynamics

Where the ones on the same line can be taken concurrently. That's a kind of bare-bones undergraduate load to let you be competent in whatever you then want to study.

There's a site called "hyperphysics" I know we used to recommend to freshmen, and I've heard good thing about the Khan Academy and Pauls Online Notes for calc.

>>5409740
If you just meant statmech&therm, you'd only really need differential equations (which carries with it calc1&2) and some slight differential equations and multivariate calculus, though not a ton of either.

>>5409740
What about learning from textbooks on your own?

>>5409744
The outline I gave was based on the kind of background knowledge you'd probably need, it'd apply the same if you were self-teaching.

You could probably skip most of quantum physics and classical mechanics and some of multivar and a little of diffeq if you were just aiming for statmech&thermo (which you shouldn't, eletro is the hardest undergraduate physics course bu statmech is by far the worst), but figuring out what you can and cannot skip would probably be more work than just learning the material properly.

>>5409755
I don't intend to skip anything.
I want to learn as much as possible, though since I'm at a fairly early stage, I don't think it's possible for me to understand how deep the rabbit hole goes.
Do you have a skype, so we can continue this conversation later.
Obviously not on voice/video chat.

>>5409758
I probably don't have the time, enthusiasm, or patience; sorry. I don't go on this board very often anymore so I can't say for sure, but I think if you ask around next time someone sensible is talking you might have a better shot- look at more boring threads or ones based on mathematics.

If you have any question now while I stay up waiting for my girlfriend to get home I'll answer them though.

dumping...

Negative Absolute Temperature for Motional Degrees of Freedom

Absolute temperature T is one of the central concepts of statistical mechanics and is a measure of, for example, the amount of disordered motion in a classical ideal gas. Therefore, nothing can be colder than T = 0, where classical particles would be at rest. In a thermal state of such an ideal gas, the probability Pi for a particle to occupy a state i with kinetic energy Ei is proportional to the Boltzmann factor
Pi \propto e^(Ei/kBT)(1)
where kB is Boltzmann’s constant. An ensemble at positive temperature is described by an occupation distribution that decreases exponentially with energy. If we were to extend this formula to negative absolute temperatures, exponentially increasing distributions would result. Because the distribution needs to be normalizable, at positive temperatures a lower bound in energy is required, as the probabilities Pi would diverge for Ei → –∞. Negative temperatures, on the other hand, demand an upper bound in energy (1, 2). In daily life, negative temperatures are absent, because kinetic energy in most systems, including particles in free space, only provides a lower energy bound. Even in lattice systems, where kinetic energy is split into distinct bands, implementing an upper energy bound for motional degrees of freedom is challenging, because potential and interaction energy need to be limited as well (3, 4). So far, negative temperatures have been realized in localized spin systems (5–7), where the finite, discrete spectrum naturally provides both lower and upper energy bounds. Here, we were able to realize a negative temperature state for motional degrees of freedom.

>>5409765
Alright.
Have you ever used Sears and Zemansky's University Physics?
Do you know of any resources that can be used alongside Euclid's Elements as exercises of a sort?

>>5409771
>Have you ever used Sears and Zemansky's University Physics?
No idea, doubt it.
>Do you know of any resources that can be used alongside Euclid's Elements as exercises of a sort?
Not off the top of my head, but something like that would be more useful if you're more interested in pure math than physics (not that there'd be anything wrong with that!)

//

Fig. 1 [see previous post image]

Negative absolute temperature in optical lattices. (A) Sketch of entropy as a function of energy in a canonical ensemble possessing both lower (Emin) and upper (Emax) energy bounds. (Insets) Sample occupation distributions of single-particle states for positive, infinite, and negative temperature, assuming a weakly interacting ensemble. (B) Energy bounds of the three terms of the 2D Bose-Hubbard Hamiltonian: kinetic (Ekin), interaction (Eint), and potential (Epot) energy. (C) Measured momentum distributions (TOF images) for positive (left) and negative (right) temperature states. Both images are averages of about 20 shots; both optical densities (OD) are individually scaled. The contour plots below show the tight-binding dispersion relation; momenta with large occupation are highlighted. The white square in the center indicates the first Brillouin zone.

//

In Fig. 1A, we schematically show the relation between entropy S and energy E for a thermal system possessing both lower and upper energy bounds. Starting at minimum energy, where only the ground state is populated, an increase in energy leads to an occupation of a larger number of states and therefore an increase in entropy. As the temperature approaches infinity, all states become equally populated and the entropy reaches its maximum possible value Smax. However, the energy can be increased even further if high-energy states are more populated than low-energy ones. In this regime, the entropy decreases with energy, which, according to the thermodynamic definition of temperature (8) (1/T = ∂S/∂E), results in negative temperatures. The temperature is discontinuous at maximum entropy, jumping from positive to negative infinity. This is a consequence of the historic definition of temperature. A continuous and monotonically increasing temperature scale would be given by −β = −1/kBT, also emphasizing that negative temperature states are hotter than positive...

...temperature states, i.e., in thermal contact, heat would flow from a negative to a positive temperature system.

Because negative temperature systems can absorb entropy while releasing energy, they give rise to several counterintuitive effects, such as Carnot engines with an efficiency greater than unity (4). Through a stability analysis for thermodynamic equilibrium, we showed that negative temperature states of motional degrees of freedom necessarily possess negative pressure (9) and are thus of fundamental interest to the description of dark energy in cosmology, where negative pressure is required to account for the accelerating expansion of the universe (10).
Cold atoms in optical lattices are an ideal system to create negative temperature states because of the isolation from the environment and independent control of all relevant parameters (11). Bosonic atoms in the lowest band of a sufficiently deep optical lattice are described by the Bose-Hubbard Hamiltonian (12)
(2)
Here, J is the tunneling matrix element between neighboring lattice sites 〈i,j〉, and and are the annihilation and creation operator, respectively, for a boson on site i, U is the on-site interaction energy, is the local number operator, and V ∝ ω2 describes the external harmonic confinement, with ri denoting the position of site i with respect to the trap center and ω the trap frequency.
In Fig. 1B, we show how lower and upper bounds can be realized for the three terms in the Hubbard Hamiltonian. The restriction to a single band naturally provides lower and upper bounds for the kinetic energy Ekin, but the interaction term Eint presents a challenge: Because in principle all bosons could occupy the same lattice site, the interaction energy can diverge in the thermodynamic limit. For repulsive interactions (U > 0), the interaction energy is only bounded from below but not from above, thereby limiting the system to positive temperatures; ...

>>5409778
http://thepiratebay.se/torrent/7466949/University_Physics_with_Modern_Physics_%2813th%29_-_Sears__Zemansky..
I was told that calculus doesn't really form the basis of the material, but I've also heard that the mathematical demands increase as you go through the book. It's for undergrads, as you can see.

>>5409782
You can take the freshmen intro courses without (much) calculus, even *technically* without any if you're willing to just stake some secretly non-sensical explanations at face-value and memorize a lot of needless stuff. But it's far better to take it concurrently with your calc courses, or even to take calc 1 first if you can help it.

>>5409788
Calculus not forming the basis of the material is bad for me.
And part of the reason why I wanted to discuss this in private is because I'm not in university at this point.

...in contrast, for attractive interactions (U < 0), only an upper bound for the interaction energy is established, rendering positive temperature ensembles unstable. The situation is different for the Fermi-Hubbard model, where the Pauli principle enforces an upper limit on the interaction energy per atom of U/2 and thereby allows negative temperatures even in the repulsive case (13, 14). Similarly, a trapping potential V > 0 only provides a lower bound for the potential energy Epot, whereas an anti-trapping potential V < 0 creates an upper bound. Therefore, stable negative temperature states with bosons can exist only for attractive interactions and an anti-trapping potential.

To bridge the transition between positive and negative temperatures, we used the n = 1 Mott insulator (15) close to the atomic limit (|U|/J → ∞), which can be approximated by a product of Fock states . Because this state is a many-body eigenstate in both the repulsive and the attractive case, it allows us to switch between these regimes, ideally without producing entropy. The employed sequence (Fig. 2A) is based on a proposal by Rapp et al. (4), building on previous ideas by Mosk (3). It essentially consists of loading a repulsively interacting Bose-Einstein condensate (BEC) into the deep Mott insulating regime (I in Fig. 2A), switching U and V to negative values (II), and finally melting the Mott insulator again by reducing |U|/J (III). For comparison, we also created a final positive temperature state with an analog sequence.

The experiment started with a BEC of 1.1(2) × 105 39K atoms in a pure dipole trap with horizontal trap frequency ωdip (V > 0) at positive temperature (T > 0) and a scattering length of a = 309(5) a0, with a0 the Bohr radius. We ramped up a three-dimensional (3D) optical lattice (I) with simple cubic symmetry to a depth of Vlat = 22(1) Er.

>>5409782
"It's for undergrads" is generous with S&Z - it's a solid intro level text but you'll never see it past freshman, MAYBE sophomore level. It's quickly supplanted by superior, calc-based treatments, but it's fantastic for the ultra-basic fundamentals.

>>5409793
Where could I go from there?
A friend wanted me to go through some of these after that
>classical mechanics taylor
>electrodynamics griffiths
>quantum mechanics zettili
>reif statistical/thermal
>carroll spacetime and geometry
>sakurai quantum mechanics
He also had a corresponding list of math books to go through.

Here, Er = h^2/(2m λlat^2) is the recoil energy with Planck’s constant h, the atomic mass m, and the lattice wavelength λlat = 736.65 nm. The blue-detuned optical lattice provides an overall anti-trapping potential with a formally imaginary horizontal trap frequency ωlat that reduces the confinement of the dipole trap, giving an effective horizontal trap frequency ωhor=(ωdip^2+ωlat^2)^(1/2). Once the atoms are in the deep Mott insulating regime where tunneling can essentially be neglected [tunneling time τ = h/(2πJ) = 10(2) ms], we set the desired attractive (repulsive) interactions (II) to prepare a final negative (positive) temperature state using a Feshbach resonance (16). Simultaneously, we decreased the horizontal confinement to an overall anti-trapping (trapping) potential by reducing ωdip.
Subsequently, we lowered the horizontal lattice depths (III), yielding a final value of U/J = −2.1(1) [U/J = + 1.9(1)], and probed the resulting momentum distribution by absorption imaging after 7 ms time-of-flight (TOF). The whole sequence was experimentally optimized to maximize the visibility of the final negative temperature state. We chose a 2D geometry for the final state to enable strong anti-trapping potentials and to avoid detrimental effects due to gravity (9).

//


Fig. 2
Experimental sequence and TOF images. (A) Top to bottom: lattice depth, horizontal trap frequency, and scattering length as a function of time. Blue indicates the sequence for positive, red for negative temperature of the final state. (B) TOF images of the atomic cloud at various times t in the sequence. Blue borders indicate positive, red negative temperatures. The initial picture in a shallow lattice at t = 6.8 ms is taken once for a scattering length of a = 309(5) a0 (top) as in the sequence, and once for a = 33(1) a0 (bottom; OD rescaled by a factor of 0.25), comparable to the final images. All images are averages of about 20 individual shots. See also Fig. 1C.

//

In Fig. 2B, we show TOF images of the cloud for various times t in the sequence, indicated in Fig. 2A. During the initial lattice ramp [at Vlat = 6.1(1)Er], interference peaks of the superfluid in the lattice can be observed (t = 6.8 ms) (Fig. 2B, top). Because quantum depletion caused by the strong repulsive interactions already reduces the visibility of the interference peaks in this image (17), we also show the initial superfluid for identical lattice and dipole ramps, but at a scattering length of a = 33(1) a0 (t = 6.8 ms) (Fig. 2B, bottom). The interference peaks are lost as the Mott insulating regime is entered (t = 25 ms). In the deep lattice, only weak nearest-neighbor correlations are expected, resulting in similar images for both repulsive and attractive interactions (t = 28 ms). After reducing the horizontal lattice depths back into the superfluid regime, the coherence of the atomic sample emerges again. For positive temperatures, the final image at t = 30.5 ms is comparable, albeit somewhat heated, to the initial one at t = 6.8 ms, whereas for attractive interactions, sharp peaks show up in the corners of the first Brillouin zone, indicating macroscopic occupation of maximum kinetic energy. The spontaneous development of these sharp interference peaks is a striking signature of a stable negative temperature state for motional degrees of freedom. In principle, the system can enter the negative temperature regime following one of two routes: It either stays close to thermal equilibrium during the entire sequence or, alternatively, relaxes toward a thermal distribution during lattice ramp-down. Either way demonstrates the thermodynamic stability of this negative temperature state.
To examine the degree of thermalization in the final states, we used band-mapped (18) images and extracted the kinetic energy distribution, assuming a noninteracting lattice dispersion relation Ekin(qx,qy).

The result is shown in Fig. 3, displaying very good agreement with a fitted Bose-Einstein distribution. The fitted temperatures of T = −2.2J/kB and T = 2.7J/kB for the two cases only represent upper bounds for the absolute values |T| of the average temperature because the fits neglect the inhomogeneous filling of the sample (9). Both temperatures are slightly larger than the critical temperature |TBKT| ≈ 1.8J/kB (19) for the superfluid transition in an infinite 2D system but lie below the condensation temperature |TC| = 3.4(2)J/kB of noninteracting bosons in a 2D harmonic trap for the given average density (9).

//


Fig. 3
Occupation distributions. The occupation of the kinetic energies within the first Brillouin zone is plotted for the final positive (blue) and negative (red) temperature states. Points show experimental data extracted from band-mapped pictures. Solid lines are fits to a noninteracting Bose-Einstein distribution assuming a homogeneous system. (Insets) Top row: Symmetrized positive (left) and negative (right) temperature images of the quasimomentum distribution in the horizontal plane. Bottom row: Fitted distributions for the two cases. All distributions are broadened by the in situ cloud size (9).

//

Ideally, entropy is produced during the sequence only in the superfluid/normal shell around the interim Mott insulator: While ramping to the deep lattice, the atoms in this shell localize to individual lattice sites and can subsequently be described as a |T| = ∞ system (14). Numerical calculations have shown that the total entropy produced in this process can be small (4), because most of the atoms are located in the Mott insulating core. We attribute the observed additional heating during the sequence to nonadiabaticities during lattice ramp-down and residual double occupancies in the interim Mott insulator.

>>5409797
StatMech Reif is something I remember doing. It's very, very rigorous (both physically and mathematically), I'm not saying I wouldn't recommend it but I would say that it's not ultra-accessible.

In principle, the coherence length of the atomic sample can be extracted from the interference pattern recorded after a long TOF (20). However, the experiment was limited to finite TOF, where the momentum distribution is convolved with the initial spatial distribution. By comparing the measured TOF images with theoretically expected distributions, we were able to extract a coherence length in the final negative temperature state of three to five lattice constants (9).
To demonstrate the stability of the observed negative temperature state, Fig. 4 shows the visibility of the interference pattern as a function of hold time in the final lattice. The resulting lifetime of the coherence in the final negative temperature state crucially depends on the horizontal trap frequencies (inset): Lifetimes exceed τ = 600 ms for an optimally chosen anti-trapping potential, but an increasingly fast loss of coherence is visible for less anti-trapping geometries. In the case of trapping potentials, the ensemble can even return to metastable positive temperatures, giving rise to the small negative visibilities observed after longer hold times (fig. S4). The loss of coherence probably originates from a mismatch between the attractive mean field and the external potential, which acts as an effective potential and leads to fast dephasing between lattice sites.

The high stability of the negative temperature state for the optimally chosen anti-trapping potential indicates that the final chemical potential is matched throughout the sample such that no global redistribution of atoms is necessary. The remaining slow decay of coherence is not specific to the negative temperature state because we also observe comparable heating for the corresponding positive temperature case (blue data in Fig. 4), as well as the initial superfluid in the lattice. It probably originates from three-body losses and light-assisted collisions.

>>5409804
I suppose that's why he chose it.
Do you know anything of the other books in that list?

In contrast to metastable excited states (21), this isolated negative temperature ensemble is intrinsically stable and cannot decay into states at lower kinetic energies. It represents a stable bosonic ensemble at attractive interactions for arbitrary atom numbers; the negative temperature stabilizes the system against mean-field collapse that is driven by the negative pressure.

Negative temperature states can be exploited to investigate the Mott insulator transition (22) as well as the renormalization of Hubbard parameters (23, 24) for attractive interactions. As the stability of the attractive gas relies on the bounded kinetic energy in the Hubbard model, it naturally allows a controlled study of the transition from stable to unstable by lowering the lattice depth, thereby connecting this regime with the study of collapsing BECs (25), which is also of interest for cosmology (26). Negative temperatures also considerably enhance the parameter space accessible for quantum simulations in optical lattices, because they enable the study of new many-body systems whenever the bands are not symmetric with respect to the inversion of kinetic energy. This is the case, for example, in triangular or Kagomé lattices, where in current implementations (27) the interesting flat band is the highest of three sub-bands. In fermionic systems, negative temperatures enable, for example, the study of the attractive three-component model with symmetric interactions [SU(3)] describing color superfluidity and trion (baryon) formation using repulsive 173Yb (28), where low losses and symmetric interactions are expected but magnetic Feshbach resonances are absent.

Fig. 4
Stability of the positive (blue) and negative (red) temperature states. Main figure: Visibility V = (nb − nr)/(nb + nr) extracted from the atom numbers in the black (nb) and red (nr) boxes (indicated in the TOF images) plotted versus hold time in the final state for various horizontal trap frequencies. Dark red, |ωhor|/2π = 43(1) Hz anti-trapping; medium red, 22(3) Hz anti-trapping; light red, 42(3) Hz trapping; blue, 45(3) Hz trapping. (Inset) Coherence lifetimes τ extracted from exponential fits (solid lines in main figure). The statistical error bars from the fits are smaller than the data points. The color scale of the images is identical to Fig. 2B (see also fig. S3).