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# Bose-Einstein Condensates - Science topic

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1. Bose-Einstein condensation: How do we rigorously prove the existence of Bose-Einstein condensates for general interacting systems? (Schlein, Benjamin. "Graduate Seminar on Partial Differential Equations in the Sciences – Energy, and Dynamics of Boson Systems". Hausdorff Center for Mathematics. Retrieved 23 April 2012.)
2. Scharnhorst effect: Can light signals travel slightly faster than c between two closely spaced conducting plates, exploiting the Casimir effect?(Barton, G.; Scharnhorst, K. (1993). "QED between parallel mirrors: light signals faster than c, or amplified by the vacuum". Journal of Physics A. 26 (8): 2037.)
Regarding the first problem, there are many examples. For instance, the paper by O. Penrose, Bose-Einstein condensation in an exactly soluble system of interacting particles'', esearchportal.hw.ac.uk/en/publications/bose-einstein-condensation-in-an-exactly-soluble-system-of-intera
Cf. also, the paper by E. Lieb and R. Seiringer, Proof of Bose-Einstein Condensation for Dilute Trapped Gases'',
Regarding the second problem, the boundary conditions break Lorentz invariance. That's why the question isn't well-posed, whether in the classical limit or when quantum effects must be taken into account. In a finite volume it requires care to define the propagation velocity properly, since the equilibrium field configurations describe standing waves.
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Hello, someone could provide me your code to analyze it, investigating I have seen that the Gross-Pitaevskii equation is solved for simulation, is there any other simpler way or in any case what would be the simplest case to perform the simulation for the Gross-Pitaevskii equation?
Dear Dr Diego Espinoza . Kindly see the following useful RG link:
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Dear colleagues, what do you think of a possible mathematical analogy between a one-sided surface and a Bose-Einstein condensate?
Yes, dear Narasim Ramesh, but not only this. For example, one can consider the behavior of a superfluid liquid on the surface of a Mobius strip or a Klein bottle.
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in trying to undestad conformal field theory with holography,probably i need help bacause ¿how ads/cft can have implications in experimental areas?
Hi Jaun,
Thanks!
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Could quasi-periodic tilings play a part in High temperature Superconductivity ? Even in a heated disordered state, could the quasi-crystalline nature of the material enforce some kind of order wherein the global and local properties are interconnected ? Breaking this could possibly require very large amount of energy which could lead to a cooling of the system which could make the system amenable to the influence of electromagnetic fields which could lead to a disorder - order transition (associated with many phenomenon such as Bose Einstein Condensation, Superfluidity, Superconductivity etc) with associated dissipation of energy and further cooling.
Could the above play a part in diffractive rsdiative processes dominating over conductive diffusive processes ?? (Possibly through smooth but non-differentiable solutions of the associated transport equations due to singularities at multiple scales which could lead to localization). Could this be a mechanism for inducing superconductivity ? Could such fields / modes induce waveguides for photons / electrons / phonons and their bound states , especially through solitons etc ? And could this be related to superconductivity ??
Dear Sundaram,
The phase of wave function plays the vital role for arising Superconductivity.
Now it is a very common practice to develop single crystals.
Superconductivity arises when the phase symmetry is broken.
When the phase of each site of atoms add in an ordered manner , Superconductivity arises.
Then forming a single crystal of that type of Materials for which vibrational energy is very small and can't break the phase coherency at ambient temperature , that should, I think, show Superconductivity at ambient temperature .
If not, please discuss what parameters I am not including in my Discussion and what is wrong in my Discussion .
Best Wishes
Thanks
N Das
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The occupation number of bosons can be any number from zero to infinity, guiding us to the Bose-Einstein statistics. On the other hand, for example, a classical wave can be considered a superposition of any number of sine or cosine waves. Isn't it similar to say the occupation number of a classical wave can be any number from zero to infinity and utilizing Bose-Einstein statistics for classical waves in particular and classical fields in general?
Dear Rasoul Kheiri, your question is interesting and two-fold, I explain:
As you state, the occupation number of bosons (and the ground boson state) can allow infinite numbers of particles (infinite modes for classical waves), but the integer-spin of bosons is an entire number. This is why I mean by two-fold.
I elaborate, for solid-state physics, phonons have a zero spin, but for electromagnetic waves seen as photons, the spin is equal to one. Phonons are longitudinal in nature, photons are transversal in nature.
In EMW, we have the coherent states for photons, which are quantum in nature but that shows some features as the Poisson distribution and not the Bose-Einstein distribution which reflects the entire spin. Those fields are all within the harmonic oscillator - HO approximation.
In addition, the second quantization in terms of creation and annihilation operators with commuting algebraic properties is needed.
Furthermore, to show the spin nature (structure) of the photon, further, the QED should be used, their relativistic origin, or at least the Klein Gordon equation instead of the Schroedinger equation.
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Rogue waves are an important topic in water waves, plasma physics, nonlinear optics, Bose Einstein condensate and so on
How can we predict a rogue wave?
Are you interested in theoretical aspects? If yes, decouple your system in the form of a dinNPDE. Apply preferably numerical simulations to see the wave amplitude behaviours, and so forth. Have a needful comparison with the latest reports on it.
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Many thinkers reject the idea that large scale persistent coherence can exist in the brain because it is too warm, wet, noisy and constantly interacts, and consequently, is 'measured' by the environment via the senses.
The problem of decoherence is, I suggest, in part at least, a problem of perception - the cognitive stance that we adopt toward the problem. If we examine the problem of interaction with the environment, common sense suggests that we perceive the primary utility of this interaction as being the survival of the organism within its environment. It seems to follow that if coherence is involved in the senses then evolution must have found a way of preserving this quantum state in order to preserve its functional utility - a difficult problem to solve!
I believe that this is wrong! I believe that the primary 'utility' of cognition is that it enables large scale coherent states to emerge and to persist. In other words, I believe that we are perceiving the problem in the wrong way. Instead of asking 'How do large scale coherent states exist and persist given the constant interaction with the environment?', we should ask instead - 'How is cognition instrumental in promoting large scale robust quantum states?'
I think the key to this question lies in appreciating that cognition is NOT a reactive process - it is a pre-emptive process!
Let us take an extreme position and see if we can make progress
If we assume, that instead of quantum coherence being a subsequent add-on to the living process, that it is, in fact, intrinsic to the living process. And if we further assume that quantum coherence in living systems is intrinsically robust, and necessarily so, in order to perform its biological function. Then we may be able to address the problem a different way: by paring the issue down to its very basics we may simplify it enough to see the way forward:-
If it is true that consciousness correlates with a macroscopic quantum coherent state.
And if it is also true that this coherent state can effect change in the world of classical physics
Then, given the evidence of our own ontology, the beginning of life on this planet would have coincided with the moment that quantum coherence found a way of breaking through the de-coherence barrier and maintaining coherence employing Occam's razor] as a direct consequence of the way in which
that change is effected.
If this argument holds, and if the soliton instrumental in the
process of catalysis maintains coherence through the process then, we should discover that cognition is not an aspect of life, -but definitive of it.
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As you know I had proposed
Achimowicz formulae stating that Information can be transformed to energy by the relation: (1) E = I x c2 in analogy to reasoning that E= m x c2 = h x omega as proposed by Planck
The next step is quantomize information so I shoud write:
E= I x c2 = h x omega
where h - Planck constant and omega is the information frequency.
Next question is : Does DNA have the own frequencies i.e. stable information frequencies at which it resonates?
. So what the quanta of information means ????
What is its interpretation ???
Any one wishing to answer this question ???
The Epistemological Crisis in Modern Physics
Amrit Sorli*, Steven Kaufman*
ABSTRACT
In physics today it often happens that experimental data is interpreted as proof of a phenomenon that has not been
directly observed, but for which phenomenon there is a theoretical model. With the obtained data acting thereby as
proof, the model then becomes recognized as “real,” after which the theoretical phenomenon that the model
describes also then becomes recognized as “real” – that is, the heretofore purely theoretical phenomenon is
acknowledged as a physical reality, even though it has never been observed, by either instruments or human
senses. This relatively new situation, in which unobserved phenomenon come to be treated as if they had been
directly observed, has lead modern physics into deep epistemological crisis of which it is not yet aware. The
solution to overcoming this crisis.
Key Words: Epistemology, Higgs field, gravitational waves
DOI Number: 10.14704/nq.2018.16.2. NeuroQuantology 2018; 16, 2:
Introduction
Recently, two Nobel prizes were given for the
discovery of phenomena that have not yet been
observed by either instruments or human senses,
namely: the Higgs field and gravitational waves.
In the Higgs field research, it was found that
extremely rarely (one in millions of collisions of
protons) a characteristic flux of energy can be
measured that has been named the “Higgs
boson.” For modern physics, the discovery of the
Higgs boson stands as proof of the existence of
the Higgs field, even though such a field has been
neither measured by instrument nor observed by
human senses. Similarly, in gravitational wave
research, it has been found that the laser light
motion in the LIGO interferometer sometimes
takes a bit longer or shorter time when passing
the beams, and this has been interpreted as
occurring when a gravitational wave is
theoretically passing through the interferometer.
For modern physics, the minimal time variability
of the laser light stands as proof of the existence
of gravitational waves, even though such waves
have been neither measured by instrument nor
observed by human senses. In both of these cases,
there is an “epistemological gap” between
obtained data and the interpretation of that data
that represents a serious problem from the
standpoint of the epistemology of physics.
The weak point of the methodology of modern
physics
The Special Theory of Relativity (STR), published
in 1905, deeply changed the methodology of
physics. As a result of STR, it became and remains
an accepted truth that time is the 4th dimension of
space, and as such has an actual physical
existence. The formalism X 4 = ict has
convinced the majority of physicists that time is
the 4th dimension in the space-time model. And
so, as a consequence of the acceptance of the
space-time model, physicists are also convinced
that time, as the 4th dimension of space, has a real
physical existence, although there is no direct
experimental evidence whatsoever for this, nor
Corresponding author: Amrit Sorli, Steven Kaufman
Address: Foundations of Physics Institute – FOPI, Slovenia
Relevant conflicts of interest/financial disclosures: The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential conflict of interest.
Received: 12 July 2017; Accepted: 7 September 2017
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Collective effects are evident in billions and billions of particles or entities in physics, such as In lasers, electromagnetism [1], superconductivity, critical mass in nuclear physics, physics of fluids, thixotropic and other non-newtonian effects, fusion and fission, binding energy, gravity, and quantum mechanics.
There are applications also in maths. We discussed its application in social movements, where statistics is not used, nor psychology, but a causal model is introduced, based on physics of fluids and collective effects.
The problem is that a system made of billions of billions of particles or entities, as usual in physics of natural systems, is much harder to study, for example, in quantum behaviour or even classical.
In network theory, comes the example of 6 degrees of separation. Now, in physics [2,3], comes the example of 10 photons. Studying quantum behaviour of particles is much easier with fewer particles, so the fact that phase transitions occur in these small systems means we can better study quantum properties such as coherence.
Could we start to see behaviour of collective effects with 10 electrons or less? Can we use them to better study coherence also in non-quantum behaviour? What is the lower limit?
[1] Carver Mead, Collective Electrodynamics: Quantum Foundations of Electromagnetism,
[3] Driven-dissipative non-equilibrium Bose–Einstein condensation of less than ten photons, https://www.nature.com/articles/s41567-018-0270-1
I am aware of at least one highly nontrivial collective effect with 3 entities: Efimov states. Much simplified, it's existence of trimer where there is no dimer.
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All the research papers I found so far, are just showing measurement of the squeezing parameter or quantum Fisher Information (QFI). Of course authors mention that, due to large QFI or strong squeezing this setup can be used for metrological purposes beyond standard quantum limit (SQL). I could not find any papers, which actually perform estimation of the unknown phase and show that the precision is beyond SQL. I am curious from the point of view of estimation in the presence of decoherence (which is always present). Theoretical papers indicate that entangled states are basically useless if frequency is estimated (e.q. Ramsey spectroscopy).
I am not actually expert in this field and I am not sure whether the following paper is of your help, I just referred this if I could learn something from you and others.
Entanglement-free Heisenberg-limited phase estimation
BL Higgins, DW Berry, SD Bartlett, HM Wiseman, GJ Pryde, Nature 450 (7168), 393
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Many nonlinear equations are solved for soliton solutions using Hirota bilinearization method but those equations are proven to be integrable either in inverse scattering sense or Lax pair method. Is it correct to use the Hirota bilinearization for nonintegrable systems?
A short answer is: yes, it is, because the existence of HIrota bilinear representation per se does not guarantee integrability. It more or less guarantees existence of a solution of a special form that in many cases can be interpreted as a two-soliton solution. Integrability, however, requires existence of four-soliton solution, see e.g.
Hietarinta, J. Hirota's bilinear method and its connection with integrability. Integrability, 279–314, Lecture Notes in Phys., 767, Springer, Berlin, 2009.
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If its possible to create a Bose-Einstein condensate for which no possible Fermi interactions can prevent further collapse and the mass was large enough, could the system collapse into a black hole? For integer spin particles might this be not only for baryons, but also for photons, since they have effective mass and are acted on by gravity? This might be if intensity of photon beam, ie energy density were large enough, perhaps even if the beam were slowed down in a condensate, to increase the energy density.
Dear Neil,
In principle this could be a good idea if a Bose-Einstein would guarantee that you can get a very high density. But that is not the case because the Bose-Einstein condensate is the occupation of the fundamental state by many bosons and the state dosen't need to be localized in one point. For instance, the Cooper pairs of a superconductor are in the same state but they are not localized in one point.
It is true that the condensate of the bosons gives the idea that at low temperatures they are very localized and therefore with a great density, at least much higher if they were in a gass phase. But in any case the highest density is far of being contained in one sphere of their corresponding Schwarzschild radius. For instance, to introduce all the Moon mass in a sphere of 0.1 mm of radius. Could you imagine Moon condensate in such small particle? If you could get a condensate in such conditions thus you would obtain a black hole.
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I am interested in coupling between collective modes of a BEC. Is there any experiment observing several collective modes simultaneously, and, l=0 mode with other modes?
Dear Sofia,
Good idea! I think I need to contact these authors.
Best wish
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I would like to know if the energy gap between ground state and the first excited state in a Bose-Hubbard model have been determined? In particular I would like to know how does it scale with the system size? So far I couldn't find any numerical data for 1D, 2D or 3D.
Thank you very much! I wonder if you know how does the situation looks like in the Mott insulator phase (n = 1)? If there is no hoping (t = 0) the gap is \Delta E = U, so it does not depend on the lattice size. Do you know how does the gap scale with lattice size when we use perturbation theory in hoping term of the Hamiltonian (strong-coupling expansion)? Do you know any paper where I can find this information?
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https://en.wikipedia.org/wiki/Superfluidity: Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without loss of kinetic energy. When stirred, a superfluid forms cellular vortices that continue to rotate indefinitely. Superfluidity occurs in two isotopes of helium (helium-3 and helium-4) when they are liquified by cooling to cryogenic temperatures. It is also a property of various other exotic states of matter theorized to exist in astrophysics, high-energy physics, and theories of quantum gravity The phenomenon is related to Bose–Einstein condensation, but neither is a specific type of the other: not all Bose-Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates
Bose Einstein condensates held in an optical lattice have a superfluid regime, wherein the atoms are not localised to any particular well of the lattice. This can happen for compound bosons, that is particles for which the spins sum up to an integer. An example of these is lithium 7. Lithium 6 is a fermion and cannot do some of the same things because of the Fermi exclusion principle. Nevertheless, there are some aspects of degenerate fermion behaviour that can lead to them being considered as superfluids. Fermions are not my specialty, so I won't comment further.
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Can any one suggest me best articles for quantum simulations of ultracold quantum gases
Bloch I, Dalibard J, Nascimbene S. Quantum simulations with ultracold quantum gases[J]. Nature Physics, 2012, 8(4): 267-276.
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Hello all,
I have started reading about diffractive optics. The papers and books I have read till now, all claim that
"It is extremely difficult (impossible?) to find closed-form diffraction solutions
using the Rayleigh–Sommerfeld expression for most apertures. The Fresnel
expression is more tractable, but solutions are still complicated even for simple cases such as a rectangular aperture illuminated by a plane wave."
I want to know why does this happen?
Look at the Sommerfeld problem, which is arguably the simplest conceivable diffraction problem: you consider a two-dimensional system, with a plane wave obliquely incident on a half line. There are 3 different regions in which the behaviour is qualitatively different
1) the reflection region, where the wave is roughly given by a superposition of two waves, one incident, one reflected.
2) the transmission region, in which one wave goes through
The exact solution must account for these 3 different behaviours, and display a continuous transition between all of them. Sommerfeld's solution does exactly that, but it should not be a surprise that, in order to be able to do that, it is fairly complicated.
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Dear Colleagues,
In automatized quantum chemical calculations (such as Chem Bio 3D Ultra) sometimes the IR/Raman spectra are not generated due to the fact that out of the first 6 frequencies (which obviously belong to the translation and rotation degrees of freedom) one is negative. This happened in relatively simple, symmetric structures, including ring compounds, after performing geometry optimization. I am fully aware of the fact that in saddle point geometries there may happen negative frequency, but I would not expect it in stable ring structures. I would like to know what is the reason (I am sure it is  a kind of artifact) and, most importantly, how can it be avoided. Apparently the problem is only that the simulated spectrum is not plotted, otherwise the rest of the frequencies seem to be reasonable.
Imaginary frequencies are usually a sign of structures that aren't in a fully relaxed ground state, usually example transition states in reactions paths.
If you get unwanted imaginary frequencies in ground state geometry optimisations, it often can help to tighten the convergence criteria, for example by adding the keywords SCF=tight. (Unless you are already using the most recent version of Gaussian, G16, were this is already the default.)
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Dear Friends,
I want to get the article which clearly illustrate how the atomic s-wave scattering length in Bose-Einstein Condensates (BEC) can be tuned spatially by means of external magnetic and/or optical trapping fields. Please let me know the references. Thanks in advance.
With warm regards
jbSudhan
Dear Prof. Boris,
Sincerely Yours
jbSudhan
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Fröhlich proposed that when proteins absorb a terahertz photon the added energy forces the oscillating molecules into a single, lowest-frequency mode which supports the Orch-OR theory of consciousness by Hameroff and Penrose. In contrast, other models predict that the protein will quickly dissipate the energy from the photon in the form of heat. This was challenged by Reimers and group in 2009, where the group demonstrated that no amount of mechanical energy can produce a coherent Fröhlich condensate. But in a recent 2015 study, Katona and his colleagues concluded that the long-lasting structural changes that were observed in the helical structure of the lysosome crystal could only be explained by Fröhlich condensation, a quantum-like collective state in which the molecules in a protein behave as one.
My take would be that if an area of study is reductionistic, or radically reductionistic, it will not be able to explain or 'find' consciousness because it is an emergent field where the sum of the parts do not equal the whole and cannot be reduced back to their parts. This is why it is hard to imagine classical science as able to find things like ghosts, ESP NDE etc, including consciousness. Therefore, Frohlich is on the right track because he allows for an emergent field. Where is gets complicated, though in that presupposes a giant, universal collective field of fields.
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Recently I was interested in the Efimov effect. I read some reference of related experimental research. I found all these experiments used ultra_cold atoms but not Bose-Einstein condensates(BECs) although all these group are able to creat BECs. So I want to know why people did not use BECs in this kind experiment?
Dose it due to the high density or the many-body correlation of BECs?
Note also that in the first detection of an Efimov state it was seen as a resonance in the three-body loss at a scattering length -850a0 (i.e. where the Efimov state joins the continuum). If you want to create a BEC you need to avoid this resonance because of atom loss + BEC:s with negative scatteirng length are generally unstable.
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In dealing with Bose-Einstein statistics, assuming that the ground state energy is zero, it is concluded that the chemical potential of Bose gas system must be negative. How can we explain the zero energy ground state and its meaning?
The ground state energy of any stable system, that's not coupled to gravity, can be set equal to zero, as a direct calculation of the partition function of the free scalar field-that describes a free Bose gas-shows. This doesn't have anything to do with the chemical potential (that controls the density, not the energy).  The reason is  that the ground state energy of any stable system is bounded from below and global time translation invariance (that's where the absence of coupling to gravity is used)  implies that only energy differences are  observable. Therefore the value of the ground state energy isn't observable and can be set to zero.
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Hi all,
This is a pure curiosity question. Obviously, Superfluid helium has to do with BECs (in particular, it is implicitly suggested by Feynman in his famous lectures). However, my idea of a BEC is that it is like a Maser, or a matter Laser, right ? So in principle, all the bosons in the BEC have the same QM state vector, don't they ? But in superfluid helium, we can see a thin film climbing up and down the walls of the beaker, which shows that at least some particles don't have the same momentum as everyone else in the beaker. So is superfluid helium really a BEC ? And if not, is it because it is in a quite dense state, whereas most BECs are obtained in a low density gaseous state ?
I think the connection between BEC and superfluidity is not that obvious. The theory of superfluidity by Landau makes no mention of the statistics of the particles (Bose or Einstein), as was often noticed. It does not mean though that BEC are not superfluid: they are, as a consequence of Bogoliubov theory and of the finite speed of zero momentum phonons (because of the weak interaction between particles)..  Incidentally, what you call BEC is not superfluid without interactions because the low momentum excitation have zero speed, as is well known too. Superfluidity requires that zero energy fluctuations have finite speed (or a gap) and that the temperature is less than a critical value. This is correct for interacting bosons, like Helium 4 but also for fermions like Helium 3, but there things are more complex than what is described by the original Landau theory because of a complex relation between the order parameter and the spin of the particles, but it remains within the general framework of this theory.  About the Maser, I believe it is a kind of non equilibrium system that can hardly be understood with equilibrium theories.  The frictionless flow (all particles not having the same momentum) you refer to is well explained by Landau theory.
Yves Pomeau
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To obtain some nonlinear equation from physical models we use the stretched variables in a very limited approximation.
For example, to obtain KdV equation which is a nonlinear equation with weak dispesrions and weak nonlinearities. We use the transformation X and T with n=1/2.
Please, note that to obtain a KdV or mKdV (modified KdV) equation, we introduce stretched coordinates and our dependent variables are then expanded in power series (around their unperturbed uniform states) of the small parameter "a" characterizing the strength of the nonlinearity or dispersion [see H. Washimi and T. Taniuti, Phys. Rev. Lett. 17, 996 (1966), for more details]. Your question should nt be about the validity of the stretched variables but about the validity of your perturbative expansion: your analysis should be of small but finite amplitude.
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Hello Dr. Schaff,  Thank you.  I look forward to reading the references that you suggested. Detecting gravitational wave involves a large amount of physical space. If GW can be detected using cold atom physics techniques, this may result in more applications of GW.
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What are the experimental methods to find quasiprobability distributions/density matrix of a two-mode Bose-Einstein condensate.
Thank you
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All that I have about the Bose-Einstein distribution is the statistics of the occupation number n of the j-th energy level,
(1) <nj> = 1/{exp[(μj - ϵj)/kBT] - 1}
where μj is the chemical potential on the j-th level, and ϵj the level energy. However, I would like to calculate the average kinetic energy in the gas, i.e.
j nj Ekinetic,j /∑j nj , also the most probable kinetic energy in the gas, and the similar quantities for velocity.
Does somebody know a simple way to do it?
Don't you mean
< ∑j nj Ekinetic,j /∑j nj >?
Or perhaps
< ∑j nj Ekinetic, j> /< ∑j nj >?
For a macroscopic gas one would think the last expression is sufficient (with some qualifications in the Bose condensation regime).
If you really want to calculate the probability distribution of the quantity you wrote down, it might still be possible by use of the central limit theorem (provided you have sufficiently many j-values). The numerator and denominator should both be approximately gaussian distributed (but not quite independent as probability distributions).
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It has been a popular belief that super fluidity of He-II and of trapped dilute Bose gases arises due to macroscopically large fraction of particles in a single particle state of momentum p=0.   It appears that the number of He atoms in He-II in such state is as low as 10% as concluded presumably by theory and experiment.   However, if 10% He-atoms in p=0 state can produce super fluidity, then 1% can also do and if 1% can do so, 0.1% can also do and so on ...., unless there is a minimum bound by some law of nature.   It is natural therefore to know the law which decides this lower bound and what is the value of the lower bound number of He-atoms in p=0 state which can produce super fluidity ?.
Dear Yatendra,
Superfluidity in Bose condensates is characterised by a macroscopically large eigenvalue of the first order reduced density matrix.
As you say the fraction condensed is always much less than 1 and goes to zero at the critical point. The condensate density must be big enough to supress the fluctuations which destroy the ODLRO ( off-diagonal long range order).
Best wishes- Lawrence Dunne
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It is known that bosons have a bunching tendency. Assume that the bosons are atoms of a gas in a magnetic trap. At high temperatures this effect is not obvious. But, if we decrease the temperature (though keeping it above the temperature of transition to BEC), what I see in reports is that the bunching tendency increases.
Why does this happen? It seems to me that the bunching effect should become less and less obvious as we get closer to the transition temperature. Due to the increase in density the atoms get anyway closer to one another, and on the other hand, there is electric repulsion between the electron clouds of the atoms. So, the bunching should become less visible.
Whatever I can say about lowering the temperature, is that the average linear momentum of the atoms decreases and the inter-atomic distance decreases, s.t. the average distance between the atoms in the position-momentum phase-space decreases.
But bunching is a two-particle interferometry effect. I don't see the connection with the distance in the phase-space. To the contrary, as the temperature decreases and the density in the ordinary space increases, it seems to me that it would be more difficult to distinguish between a pair of atoms close to one another because of the increased density, and a pair of atoms close because of the bunching.
Where am I wrong?
Dear friends,
Here are some facts that shed some light on the problem, although I can't be sure that they provide a full answer.
1. As the temperature decreases, the Maxwell-Boltzmann distribution of velocity moves its peak toward low velocities, but also becomes narrower. The probability of finding two (almost) equal velocity increases. At low temperatures, the Maxwell-Boltzmann distribution will become, for bosons Bose-Einstein, and for fermions Fermi distributions - see below.
2. Assuming also that the considered gas is provided by a reservoir of particles, s.t. with decreasing the temperature the density increases, the atoms become (on average) closer to one another, as said in the question. That has in fact a negative consequence, the bunching is blurred.
3. In consequence of 1 and 2, the average number of particles per phase-space cell increases. (For the usual 3D space, the volume of the unit cell in the phase-space is ħ3.)
4. Bunching of bosons and anti-bunching of fermions are two-particle (or multi-particle) interference phenomena. The anti-symmetry of the wave-function of two fermions identically polarized, has the effect that the closer the two fermions are in linear momentum, the amplitude of probability for them to get close to one another is destroyed. In one cell of the phase-space there can be at most 1 fermion. For identically polarized bosons the situation is opposite due to the symmetry of the wave-function. The closer they are in linear momentum and position, the amplitude of probability is enhanced. Thus in one phase-space cell tend to gather more than 1 boson.
As shown at the points 1 to 3, lowering the temperature increases the density in the phase-space i.e. the above quantum effects become more observable. The Maxwell-Boltzmann distribution splits into a Bose-Einstein one and a Fermi one.
5. The repulsion between clouds of electrons may be a more perturbing or less perturbing effect, depending on the type of atoms and other experimental details.
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In the recent Australian National University experiment by Dr. Truscott, Wheeler's Delayed Choice thought experiment was carried out with ultra-cooled helium atoms. The Nature Physics publication is below, with other explanations available online. What I'm interested in is a plausible explanation that occurred to me.
So, the state of the atom is indeterminate between the first light grating and the second possible light grating. This second event, only determined to actually happen (or not) after the first event occurs, seems to influence the behavior (either particle or wave behavior) of the atom between the first and second grating. There are two plausible explanations. Either the atom doesn't actually exist as a wave or particle between these two events, and this behavior is only determined at measurement, or (obviously controversially) the future event effects the past.
I'm interested in this second explanation, because in my mind it resembles the local-realism violations of certain entangled pairs. There is support to the idea that a particle can be entangled with itself (though the usefulness of this until now was moot), so my question, as stated above, is this: could this single helium atom be entangled with itself between the first and second gratings? As a measurement of one entangled particle causes the other to take on the measured state, this behavior seems at the very least parallel. When this atom is measured at the second grating (let's call this t=1), not only is the behavior determined for that point, but as that collapses the superposition of having traveled as either a wave or particle, it also collapses it at its state immediately after the first grating (for now, t=0).
This does require the atom as it exists at t=0 and at t=1 to be considered an entangled pair to have a consistent explanation, hence the question.
Dear Devin,
I believe that your problem "...to be entangled with itself across time, between measurements?" is directly related to temporal correlations referred to as temporal steering. I am sending you my recent paper on this phenomenon. However, it might be a good idea to read the original work of YN Chen et al.  [Phys. Rev. A 89, 032112 (2014)].
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We need to wind a pair of anti-Helmholtz coils and we would like to use wire that has a rectangular cross section to maximize the filling ratio.
Can anyone recommend a company that sells something usable (not to thick) and reasonably priced?
You can browse the web site of phywe company, here you are
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Any suggestion or discussion about the question
Dear Dr. Sanjeet,
I have just come across the following paper where Bessel functions (of the first kind) pop up in the context of a 2D photonic BEC:
A.-W. de Leeuw, H. T. C. Stoof and R. A. Duine, Phys. Rev. A 89, 053627 (2014).
I hope this will inspire you to move forward.
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Does anyone have experience with this system, and, if so, what is the minimum achievable linewidth?
Furthermore, do you know some problems that would lead to linewidth broadening and how to solve them?
In my experience, the linewidth of DL100 after stabilization by using LIR 100 plus saturation absorption or polarization spectroscopy is about 300KHz-500KHz. You can check the linewidth by heterodyne detecting two DL 100 lasers to get the beating signal.
Numerous factors can broaden the linewidth. Besides mechanical vibration or acoustic noise, I believe that there are at least three probable reasons can induce the situation worse. 1) Misalignment of the grating in DL 100. You can check the threshold current of DL 100. If it is distinctly bigger than the default value, you need to adjust the grating. 2) Reflected laser into DL 100. It is likewise a common situation especially when you couple laser into a fiber. If this is the cause of linewidth broadening, you need to use a better optical isolator. 3) Electronic noise, which can couple into the laser or a frequency stabilization circuit from ground or space, especially when you use some kind of RF apparatus, for example, RF　sources for AOMs or function generators. It is better to use a clean ground for the laser system.
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In BEC no. of atoms should remain constant. That's why we can not use same statistics for photon. but,
1)what are the minimum no of atoms?, which will show BEC.
2)What about one atom(BEC) and why?
3) Can we somehow use such condensation for Photons?
Dear Dr. Chauhan,
1) As usual in statistical physics, many of its notions have exact meaning in the so-called thermodynamic limit only.  This refers also to the phase transition of Bose-Einstein condensation (BEC). The well-known estimate for the temperature of this phase transition (in TeX notation and in units with Boltzmann constant equal to unity)
T_c \sim (\hbar^2/m)(N/V)^{2/3}
has exact meaning in the limit N\to\infty and V\to\infty but N/V=const. Only in this limit the thermodynamic quantities (as, for example, heat capacity) have singularities as functions of the temperature T. In practice, for finite N and V, we have a fast change of the behavior of these thermodynamic quantities in vicinity of T_c, and the greater are N and V, the smaller is the width of the transition region from one dependence to another one. For small N this width can become of the same order of magnitude as T_c and then it does not make sense to speak about "the phase transition of BEC".
2) If you take formally N=1 and V=a^3 ("a" is a linear size of the volume V) then you get T_c \sim \hbar^2/ma^2 \sim E_0
where E_0 is the ground state energy of a particle confined in the volume V. According to the laws of statistical physics, the probability to find the particle at this state is w_0 \sim \exp(-E_0/T) \sim \exp(-T_c/T).
This means that for T<<T_c this probability is close to unity. If you wish, you can call this as "condensation" of a single particle at its ground state. IMHO, however, this wording for such an obvious behavior does not have much physical sense.
3) There is no such a phenomenon as BEC for photons since number of photons in a cavity is not preserved and it changes with temperature: if T is decreased, then photons are absorbed by the walls of the cavity and their number N decreases according to the Stefan-Boltzmann law
N \sim (T/\hbar c)^3 V.
Formal substitution of this formula into the above expression for T_c gives
T_c \sim T^2/mc^2,
that is T_c\to 0 with T\to 0 and condensation of photons cannot be reached by decrease of T on the contrary to situation with atoms confined in a trap whose number is preserved when T\to 0. Sometimes one can speak about "light condensates" in a loose sense, but this is not a standard BEC.
Best regards,
Anatoly.
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BEC represents order in momentum space, which is fictitious.  It occurs at k = 0 (or, more precisely, at the lowest momentum state, depending on the observer's reference frame).  This corresponds to the infinite- (or large-) r limit in configuration space.  How exactly?  How does no (the condenate fraction) appear in this space (which is related to  momentum space through, of course, a Fourier transform)?
BEC in experiments with ultracold dilute gases of atoms are done inside atom traps. You can think of these as a harmonic trapping potential in real space. In this case, condensation occurs into the lowest energy eigenstate (neglecting atom-atom interaction for arguments sake), or in other words into the ground state of the harmonic trap. This state has a finite width both in real space and in momentum space.
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There exists a vast literature on superfluidity.  It is growing daily, now that we are drawn more and more to low-dimensional systems, both theoretically and experimentally.  One thing is clear:  superfluidity and Bose-Einstein condensation are two distinct phenomena, the one not necessarly implying the other.  I sympathize with beginners in the field; there is so much confusion, although the phenomenon is now a few years older than a century!
It depends what you mean by superfluidity theory. If you mean the reason for which the superfluid can move without any viscosity then I develop a kinetic theory that explains this phenomenon. It extends the Boltzmann equation to the case of a dilute Bose condensed gas and tell why there is no friction. See P. Navez, Physica A 356, 241 (2005). I have more references if you wish. Let me tell also that even in this simplest case, a lot of questions remain unanswered. That makes the field still open. Contact me if you want to know more.
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Solitons are the 1D topological defect with phase singularity of \pi, whereas vortices have \2pi phase jump around its core. Recently the obsevation of solitonic vortex in BEC has been confirmed. Can anybody elaborate solitonic vortex clearly?
For solitary waves, there are two types. Bright solitons can exist in BECs with attractive interactions, while dark solitons are topological excitation in repulsive BECs.  Bright solitons are self-focusing, nondispersive, particlelike solitary waves (see PRA.85.013627 for detailed description of one dimensional bright solitons in condensates with and without a trap).  They have uniform phase distribution. A dark soliton is an envelope soliton that has the form of a density dip with a phase jump across its density minimum (see review in J. Phys. A: Math. Theor. 43 (2010) 213001).  Vortices are also topological excitations in condensates. The quantization of circulation of the fluid leads to quantized vortex states which are characterized by the phase circulation about the vortex core being $2s\pi$ (Phys. Rev. Lett. 81, 5477, Phys. Rev. A 64, 031601(R) ), where $s$, the winding number or vortex charge, is a nonzero integer. This phase singularity is independent of atomic and trap parameters. Vortices with opposite charge (opposite phase circulation, i.e. clockwise and anticlockwise) are called vortex and antivortex. If the winding number satisfies the relation $|s|>1$, vortices are not stable when perturbed. They will dissociate into singly charged vortics ($s=\pm 1$).
Svortices are vortices conﬁned to essentially one-dimensional dynamics, which obey a similar phase-oﬀset–velocity relationship as solitons. Marking the transition between solitons and vortices, svortices are a distinct class of symmetry-breaking stationary and uniformly rotating excited solutions of the 2D and 3D Gross-Pitaevskii equation, , which have properties of both vortices and solitons (J. Phys. B: At. Mol. Opt. Phys. 34 L113, New J. Phys. 15 113028).
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Klein-Gordon relativistic equation (although not a fancy of a majority) with addition of magnetic vector potential will also give rise to GPE equation. So, whatever be the route, the BEC formation should be judged by number density and temperature as well as effective mass that should be large for condensation. This is again related to temperature and scattering-length parameter.
Could you perhaps provide links or references to the work which you describe so that we may read about it?
Cheers, Gordon.
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When we are dealing with a system of small or large number of interacting bosons confined in a potential, what are the parameters which provides information about the Bose-Einstein condensation and condensate fractions?
Dear Kuldeep, defining BEC stricly in the thermodynamic limit ( N->infinity and V-> infinity with N/V = constant ) is problematic because a) N/V is never constant in experiments and b) the number of particles is always finite. Current BEC experiments work with as few as 40-200 atoms: see e. g.
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The dynamics of BEC trapped in optical lattice is governed by the discrete nonlinear Schrodinger equation (DNLSE). In tight binding limit, one can expand the condensate order parameter in terms of Wannier basis function. This basis function is further used in the calculation of the tunneling energy (J), offset energy (V_n) and interatomic interaction strength (U). Is there a way to calculate the Wannier basis numerically?
Since your equation is nonlinear, there may be a problem with the standard methodology of summing up the eigenstates to obtain the Wanniers since a linear combination of two eigenstates is not an eigenstate.
We recently showed how to do obtain Wanniers directly by minimizing the energy functional with an extra term to ensure localization in real space (so-called L1 norm). There is no reason why this should not work for a nonlinear Hamiltonian, although we have not yet tried. Link to the paper is attached. If you think this approach fits your problem, I will be happy to help.
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I am trying to understand the nature of Bose-Einstein Condensates and the possibility that they may be associated with living processes. My intuition tells me that they must be scale -free processes. Can anyone confirm this intuition with some references?
Leaving the question of whether BEC's play a role in biological processes aside, is my intuition that BEC's may be more easily 'created' if some sort of fractal scaffold was employed correct?
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I am no physicist, but I am a writer and a fan of SF. The Bose-Einstein Condensate is surely one of the most fascinating 'artifacts' to be produced in recent years and may have significance in as wide-ranging fields of enquiry as consciousness to safe energy.
I want to write (or rather complete) a novel that explores these possibilities, but I want to stray as little as possible from current theoretical models, whilst indulging myself and the reader with some of the fantastical explorations true to the spirit of SF.
Can anyone help me? A sufficiently great contribution will entitle my collaborator to acknowledgement, if not the title of co-author.
I have started a Project [The Bose-Einstein Condensate (A Novel] and one Bench [Drafts]
This looks like a very interesting topic . IMHO As I understand..open to correction.. ( I am an engineer) the Bose-Einstein Condensate shows that at absolute zero temperature all atoms become one
super atom.. So possible scenarios for SciFi
1. By some method ( universal meditation?) if all human thoughts (entropy) are brought to zero throughout the world then we all coalesce into one superconscious being....
perhaps we would know completely the meaning of why we are here (on earth) and where we go from here.. actually we are trapped in a time capsule( a lifetime)
with limited knowledge of the outside world.. Add to this the idea ..that only
Information is conserved not even energy.. and the time horizon is malleable around
a blackhole.. we can conjecture all sorts of scenarios where past ,present and future merge..perhaps time stands still?
2. Questions arise eg if this one superconscious is space limited?
3. will there be other galaxies where this has already been achieved?
4. will there be extra dimensions that we can move into?
5. can we become creators of universes ?
6. can we extend the ideas to emotions such as love fear anger etc..
7. from fiction point of view plots where observation changes what has
happened would lead to interesting legal interpretations of events.
..8. One can make a modern version of Sherlock Holmes .. I think..
and have a series of stories..
These were some thoughts that crossed my mind ..hope it helps
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How the von-Neumann entropy (quantum entropy) and "quantum correlations" related to each other for a system of cold atoms? Is there any other parameter to measure the quantum correlations between the atoms?
von Neumann entropy of a reduced density operator is measure for bi-partite correlations. However, in your case the best measure of particle entanglement between the atoms is the Spin Squeezing.
see,
A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, Na-
ture 409, 63 (2001).
best
ümit.
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The minimum temperature of trapped ions corresponds to the ground state of the trapping potential. Thus, in principle, the ions are cooled down to lower temperature as the frequency of the trapping potential is reduced. Where does the limitation come from?
Dear Pintu Mandal
Assuming the ions are bosonic they could be in the same ground state and form a condensate - if they would not have very strong and long range interaction. Due to this strong interaction they are very much localized and each ion interacts with all the other ions. The result is coulomb crystal rather than a condensate, where they all share the same wavefunction (gaussian for non-interacting case otherwise Thomas Fermi profile).
There is also the question whether a condensate can be a good description for a system that has only few tens of particles. This is much more similar to a nucleous and separate excitations can be detected and the notion of a temperature has little use. Sure iontrappers use temperature some times when they take statistics of trapping few ions many times and measure the occupancy of lowest or higher state. It does however make little sense to speak of the temperature for a single sample.
Last comment I would like to make is that by lowering the trap frequency you do lower the temperature if you do so slowly compared to the trapfrequency (adiabatically). However, the phase space density will not increase and so there will be no condensate even for weakly interacting atoms with short range interaction. In the case of ions the crystal will just expand and lattice vibrations (if any) will probably transform into motion of single ions eventually. I don't know what will happen to the trap depth in practice in this case, so an iontrapper should answer that more technical part. I guess it will be lowered as well
I hope this clarifies a few things.
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For imaging of nanoscale objects, optical microscopy has limited resolution since the objects are often much smaller than the wavelength of light. One can achieve a considerable improvement in resolution with instruments such as the transmission electron microscope and the scanning electron microscope that use electrons with De Broglie wavelength much smaller than that of visible light. To image picoscale and maybe attoscale objects, in principle, we may need a coherent beam of atoms because it may have a smaller De Broglie wavelength than that of electron beam!
I do not have access to the article in the link provided by Prof Joachim. Hence I could not read it. However, I doubt if the statement, " Cooling the atoms in the gas to form a Bose–Einstein condensate increases the resolution of the device." given in the abstract of this paper is correct. Decreasing the temperature increase the de Broglie wave length which means decreases the resolution.
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First principle wave mechanical formulations reveal that a quantum particle in any potential trap has non-zero energy in its ground state. This implies that no atom of trapped dilute boson gas in a harmonic potential trap used to achieve its BEC state can have zero energy/momentum. However, people in the field claim that a macroscopically large number of atoms (say about 60% just for an example) in the lowest temperature BEC state achieved by them have zero momentum. Can any body kindly help me understand the reality? This is a state of great confusion.
Dr Zaremba, hearty thanks for your response to the question. To help me further for having better understanding of the p=0 condensate, I request your valuable comments and clarification on the following points.
In what follows, all particles of an ideal Bose gas in a harmonic trap at T=0 occupy the ground state of a single particle trapped in the said trap. Evidently, they all have identically equal non-zero energy in their ground state or in what we call as BEC state at T=0. The state obviously has zero entropy.
It is well known that the ground state of a particle in a harmonic trap is not the eigen state of the momentum operator of the particle indicating that particle momentum in this state is not a good quantum number. It is also evident that the expectation value of the momentum operator in every state of a particle in a harmonic trap is zero. Nevertheless, it is interesting, that people determine the momentum distribution of particles in this state by obtaining the Fourier transform (FT) of the ground state wave function just to prove that the state has p=0 condensate. In what follows from the underlined statement, I wonder if this momentum distribution is relevant for unravelling the physical behaviour of the system. Further since the said ground state wave function is neither a plane wave of single momentum, nor a superposition of only two plane waves of equal and opposite momentum, its FT is expected to conclude that particles are distributed on states of different momenta. It sounds OK in the frame work of mathematics but not in that of physics since particles having different momenta constitute a state of non-zero entropy which can not be reconciled because the state is a zero entropy state, as concluded in the first paragraph.
The ground state wave function of a particle trapped in a harmonic trap of finite size has significant amplitude only inside the trap, particles are expected to have momentum p > h/2L (with L being the size of the trap); in other words even non-interacting bosons in the said trap are not expected to have zero momentum and hence p=0 condensate. It is a different thing that people assume that h/2L is as good as zero since it is ~10^{-2) times smaller than h/2d (where d is inter-particles separation). In summary, particles of even non-interacting boson gas in harmonic trap do not have p=0 condensate; it is a condensate in the ground state of momentum p = h/2L and this state is not an eigen state of momentum operator; rather it is a state of the superposition of two plane waves of equal and opposite momenta (p, -p).
The same is true for uniform system of non-interacting bosons, since, rigorously, speaking the ground state wave function is a standing wave (not a plane wave) where momentum ceases to be a good quantum number of the state and its lowest value is h/2L, -not ZERO. I wish to emphasize this reality because it helps in understanding the truth that superfluidity has no relation with the so called p=0 condensate as emphasized by Landau, and the BEC in a uniform system of even non-interacting bosons is not a p=0 condensate.
Following the FT of the ground state of a single particle in harmonic trap and taking h/2L ~ 0, it appears that even in case of an ideal Bose gas, p=0 condensate at no temperature is 100%; this is different from the case of uniform system where it reaches 100% value at T=0.
It is obvious that these problems become more pronounced at all T in case of systems of interacting bosons
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What are the advantages of CPD over the usual single particle density?
Christian Binek has posted a reasonable clarification of CPD. However, in system like liquids one finds several correlation function which basically deals with CPD. For example two particle correlation function g(r1-r2) defines the probability of finding particle 2 at r2 while particle 1 exists at r1.
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Could we say that the reason of wavy behavior of the atoms in zero temperature in BEC is the minimum entropy (minimum distortion for creating solitons)?
@ Kordi: What you are considering as two sub-systems is based on the conventional understanding of superfluid (BEC state) state of a system of interacting bosons. In a system of interacting bosons we cannot isolate an atom (as one system) and rest of atoms (as another sub-system). The conventional microscopic theory of superfluidity consider that the system has two sub-systems : (i) particles occupying a single quantum state of momentum p=0 which define what we call as condensate state and (ii) particles occupying states of non-zero p comprise another subsystem what we call as non-condensate. Although it is understandable that this picture does not identify a particular particle as a constituent of condensate and another as a constituent of non-condensate sub-system because particles can be exchanged between the two due to their identical nature, however, this is neither consistent with highly celebrated two fluid theory of Landau (for which Landau got Nobel prize) nor with experiments. Note that in his theory Landau no where talks about p=0 condensate as a factor responsible for superfluidity. If this picture is believed, you are right in your thinking that even at T=0, the BEC state of a system of interacting bosons has non-zero entropy which you can possibly relate to von Neumann entropy and try to think in terms of minimal entanglement. But I am sorry that this picture has serious errors arising from its erroneous basic premises. In this context, I would suggest you to also read the following papers.
In fact you may find from the reference that I gave you in my above post, the T=0 state of interacting bosons has maximal entanglement, yet it has zero entropy because all particles are in a single quantum state of pair of particles moving with equal and opposite momentum (q,-q) with centre of mass momentum K=0. This state is not only consistent with Landau theory of superfluidity but also with all experiments.
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I'm using 'ListDensityPlot[]' from MATHEMATICA to do contour plot. But it seems too slow and the fiures taking large memory. Actually, I'm generating the datas using C and stored in external file in '.dat' (file size is about 10-20 MB)format. But I like to do the Contour plot using Python.
Just for having another possibility - you can use gnuplot which is quick and light. Look in http://gnuplot-tricks.blogspot.com/2009/07/maps-contour-plots-with-labels.html
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Note that theoretically the existence of this wave in BEC has been demonstrated .
Hi Axel,
In fact, we know two kinds of kink solitons in the literature: solitons for which the plot of the real part (or the amplitude) of the wave function vs. space resembles a double kink in a pipe. The first kind (kink in the real part) is referred to as Dark soliton. Many experimental observations of such a soliton have been reported (among which the Ref. your cited in your response). However, I have not yet learnt about the experimental observation of the second kind of kink soliton. That is the point in my question.
Regards,
Etienne
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When condensate is loaded in optical lattices, non-linearity (atom-atom interaction) play a vital role in the diffusion dynamics. For some critical value of non-linearity of the condensate, diffusion is suppressed and it is said that the system is in self-trapped state. This suppression of diffusion is due to steep edges developed on both side of condensate(in 1D lattice case). But after some time these steep edges dissolve and diffusion restarts, there comes another critical time where the diffusion suppressed again and so on. I am searching a review article on self-trapping of condensate in optical lattice potentials 1D, 2D and 3D.
If now we further increase the non-linearity, there comes another critical point where the diffusion restarts.
-----------------------------
really?
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When vortices in BEC annihilate then they dissipate their energy in the form of a sound wave. Is there any theoretical way to calculate that sound energy?
you forgot to take the root. Phonons allways have a linear dispersion relation for particals these are quadratic. Then the equation reads alternatively E=hbar*omega For the rest the statement is correct.
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Condensate trapped with weak optical lattice confinement in one direction and stronger one in other two form a quasi-one dimensional condensate. What is the life time observed for 1-d condensate to the date?
This is actually a problem for Cold Atoms (Molecules) trapped in Optical Lattices of any geometry. Cold atom condensates are only stable for short times. Typically ms. Also with respect to what Zhang said, you need to know the type of atoms (molecules) involved. Some people connect the Coherent Time, tau to this stability issues and so define tau = pi/Omega [arXiv:1206.5023v1]. For NaK, this yields 100 ms.
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Bec
As to how this *modifies* a magnetic trap, I am not sure to understand your question. Typically traps do have some finite height of the potential to store the desired objects, whatever the precise mechanism (electric fields, magnetic fields, electromagnetic radiation fields in optical traps etc.). Starting with an initial ensemble you could gradually reduce the depth of the trapping potential (as a function of time or along some path in space). Those objects with kinetic energies exceeding the trap potential will exit the trap, leaving behind an ensemble with reduced temperature.
So, my answer is more about what you can achieve with modifying trap parameters than how a trap is modified. So, maybe I totally missed your point.
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how can we identify Gap-Soliton in Bose-Einstein Condensates (BEC)?
Dear Sudharsan, The measure of chemical potential can identify Gap-Soliton as chemical potential corresponding to Gap-Soliton in the presence of optical lattice (OL) lie in the gap of the band spectrum of the linear Bloch-wave spectrum (energy spectrum of OL ). Another indicator can be the non-dispersive behaviour in the time evolution of matter-wave.
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Of course, the answer depends on the chosen measurement technique -- anyway I am curious especially in the case of so-called fragmented Bose-Einstein condensates or Tonks-Girardeau gases.
By one-body density, I mean the diagonal of the reduced one-body density -- that is obtained by tracing out all but one coordinate from the wavefunction multiplied with its complex conjugate.
By many-body or N-body density, I mean the diagonal of the N-body density -- that is obtained as the wavefunction multiplied with its complex conjugate.
From a physical point of view it would be very interesting to find a way/technique to distinguish the cases where the N-body is not just a product of one-body densities from the cases where it is such a product. I know, there is interference experiments, but a way relying on just imaging would be more easily managable.
What is the ultimate goal of your work? This is where we get lost in discussing things that may not play into the real goal. We need to understand what you need from this question. Is it a better understanding of the Bose Einstein condensate. Is it a possible use of the BEC? This would help in the understanding of the question.
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