Bose-Einstein Condensates - Science topic

1) "According to Wikipedia, ..."

Wikipedia is not 100% reliable, because it's not peer-reviewed. The fact that it's your only reference could indicate you have never read a physics textbook. If you are interested in physics, you should start reading some serious books which could lead to better understanding physics.

2) "gravity and light are different in most ways, it might be considered an extension of this spectrum"

Not every wave is electro-magnetic wave. Not only gravity, but sound waves and waves of the sea are also exist and not part of the electro-magnetic spectrum.

3) Even if particles behaves like waves in some experiments, they do not loose their mass. It doesn't matter if you excite it or not, they will still have mass.

4) Nobody knows for sure what future holds, your goals might be achieved one day by quantum teleportation, rather than by the method you envisioned.

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.

Dear Dr Diego Espinoza . Kindly see the following useful RG link:

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.

Hi Jaun,

Maybe this work help you to know the answer to your question?

Thanks!

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

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.

Let see what other specialists have to say about this.

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.

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.

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

purpose of this article is both to identify the epistemological crisis created by this situation, as well as to present a

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

e-mail
sorli.amrit@gmail.com, skaufman@unifiedreality.com

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

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.

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

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.

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.

Dear Sofia,

Good idea! I think I need to contact these authors.

Best wish

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?

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.

as you asked

Bloch I, Dalibard J, Nascimbene S. Quantum simulations with ultracold quantum gases[J]. Nature Physics, 2012, 8(4): 267-276.

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

3) the shadow region.

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. 

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.)

Dear Prof. Boris,

           Thanks for your reference.

Sincerely Yours

jbSudhan

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. 

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.

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. 

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

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.

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. 

Thank you

Don't you mean

< ∑_j n_j E_kinetic,j /∑_j n_j >?

Or perhaps 

< ∑_j n_j E_{kinetic, j}> /< ∑_j n_j >?

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).

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

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.

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)].

You can browse the web site of phywe company, here you are

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.

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.

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.

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.

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.

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).

Could you perhaps provide links or references to the work which you describe so that we may read about it?

Cheers, Gordon.

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.

http://dx.doi.org/10.1103/PhysRevLett.112.060401

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.

Thanks Behnam, your answer is appreciated.

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?

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

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.

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.

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.

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

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.

@ 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.

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

Hi Axel,

Thanks for your answer. It talks about a different kind of kink soliton.

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

If now we further increase the non-linearity, there comes another critical point where the diffusion restarts.

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really?

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.

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.

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.

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.

Thanks a lot for updating giving reference to a nice article

You can see this article: http://iopscience.iop.org/0953-4075/38/22/014/