BEC has no practical use today yet. I think it is hard for cold atoms to have use in the near future. However, people often in their paper argue that their proposal with cold atoms or becs will be useful. That is totally trash or just a lie.

I think it is a very bad trend in physics is that physics is no longer physics, but engineering. In main stream journal such as prl, nature, science, there are too many such engineering papers today. No new physics at all.

We do bec just because it is funny. That is enough.

if you add one atom to a bec, the bec wave function changes a little bit, and the energy of the system changes. The energy increase is the chemical potential.

What Jiang-min Zhang said is correct. Here is just some more info to understand why it appears in the GPE.

The chemical potential is a thermodynamic internal quantity which is defined as the amount of energy needed to put a differentially small quantity of gas to it conserving the entropy.

The entropy of a BEC is by definiton minimal or zero.

The BEC wave function is a solution to the nonlinear Schroedinger equation or Gross-Pitaevskii equation (how ever you like to call it) with an energy eigenvalue.

Since the entropy of the BEC as a gas is zero this eigenvalue is the same as the chemical potential since it is determined by the interactions alone.

If you have finite temperature the chemical potential is different from the solution of wave-function of the BEC but still dominated by interactions.

The physical explanation is that the BEC is quantum matter at rest. This by itself is a wonder as quantum mechanics would say you can not have that. What this means is that it has been slowed to a point that the energy content is almost zero. The potential is that if we can manipulate this we can use it in different ways. Most people did not even understand the significance if the BEC. This could mean that we are able to manipulate quantum matter into energy use. This could be profound.

As was said before, the chemical potential gives you an idea how much energy it costs to add one boson to a condensate in the ground state. The chemical potential depends on how strong the interaction is and is therefore more suited to describe real condensates than e.g. the single-particle energies.

Note that you don't need Gross-Pitaevskii mean-field theory to define a chemical potential.

You can also compute it on the many-body level.

However, the dynamics of BECs depends on a lot more than just the chemical potential. BEC dynamics depends on (usually many) highly excited states. I don't know what you have read about dynamics and the chemical potential. Could you be more specific?

Thanks a lot to all for your kind reply. Now I understand what is chemical potential in BEC system. After reading all your clear explanation there are some more question arising in my mind.

1) As Dr Kaspar Sakmann and Mr Silvio Koller told that "The Chemical potential depends on how strong the interaction is". Okay then let me to take an ideal case - non interacting BEC. Then what happens to chemical potential? Is it comes to zero? When we make the interaction to zero the GP equation reduces to linear Schrödinger equation. For my Knowledge in some papers they studied both localization and dynamics of BEC. In both of the approach they spoke about the Chemical potential.

2) What are all the remarkable practical importance of BEC?

BEC has no practical use today yet. I think it is hard for cold atoms to have use in the near future. However, people often in their paper argue that their proposal with cold atoms or becs will be useful. That is totally trash or just a lie.

I think it is a very bad trend in physics is that physics is no longer physics, but engineering. In main stream journal such as prl, nature, science, there are too many such engineering papers today. No new physics at all.

We do bec just because it is funny. That is enough.

1) The Schroedinger equation will have an energy eigenvalue. The lowest corresponds to the energy you need to put your particle or atom into the trap. Since it does not interact with any particles in the trap, this is the chemical potential for any amount of BEC. In free space where the potential is constant, the chemical potential is zero and the particle is then not moving at all. That is sqrt()=

=0.

When you switch off the interaction the (stationary) Gross-Pitaevskii equation becomes the single-particle Schroedinger equation H psi(x) = E psi(x) and therefore the chemical potential mu becomes the single particle energy E.

I agree with jiang-min Zhang, there is no practical use for BECs at the moment. I would be very surprised if there was one within the next five years. The applications that some people are working on are mostly for further fundamental research (I'm thinking of ultra-precise interferometers here). But remember that in 1960 people said about lasers that they were "a solution looking for a problem". Today that's a funny quote, but back then people were serious. There would also be no MRT or PET scans today without once considered weird ideas like "quantum mechanics" and "antiparticles".

Regards,

Kaspar

Thanks a lot to all for your kind reply. Even-though BEC is a many body system, in the GP equation peoples considering two and/or three body interaction alone. It is worth mention that the strength of the three body interaction is very low when compared to the two body interaction (it is clearly pointed out by Gammal et.al., in J. Phys. b: At. Mol. Opt. Phys. 29 L51). That is three body interaction is depends on the two body interaction. So, it is possible to control the three body interaction also by means of Feshbach resonance technique. As s-wave scattering length is short range scattering length, the three body interaction is very negligible then two body interaction? why we consider only s-wave scattering length? why don't we consider p-wave scattering length, d-wave scattering length....?

Some peoples talking about the making of chip using BEC. I didn't saw a single research article on this work and also I don't know the full story about it. I just heard it when I attend an international conference.

I think you might be mixing two things up. When people talk about a "many-body treatment of BECs" it does not refer to the inclusion of three-body interactions or higher. Actually only few people work with more than two-body interactions.

The Gross-Pitaevskii equation is a mean-field equation, which means you treat the entire condensate as if it was one single particle that sees an effective (therefore the word 'mean-field') potential made up by all other particles. The form of the GP wave function is Psi(x), i.e. like that for a single particle.

The true many-body wave function of the many-body Schroedinger equation looks like this

Psi(x_1,x_2,....,x_N),

i.e. it has N coordinates for the N particles in the condensate. If you assume that all particles always do exactly the same thing you can make the ansatz

Psi(x_1,x_2,....,x_N)=phi(x_1)*phi(x_2)*...*phi(x_N)

for the many-body wavefunction. This ansatz reflects the assumption that all particles occupy one and the same state, phi(x). It is just a product of the same function with different coordinates. If you plug this ansatz into the energy functional and require the stationarity you will get the stationary Gross-Pitaevskii equation for the orbital phi(x).

So that's one way to see the difference between many-body and mean-field. It has nothing to do with the inclusion of three-body interactions or higher. In both cases usually two-body interactions are included and no more than that. Many things cannot be described by a GP wave function though: in real condensates not all particles always 'do exactly the same thing'. That means that the ansatz above isn't valid anymore. Maybe you have come across the term 'Mott insulator state'. That's one example, but there are many more. The GP ansatz leads to a nice and simple equation, the GPE, but it is very restrictive and therefore not always correct.

There is good answer in the course of theor. phys., L.D.Landau, E.M.Lifshic, vol 5 and in chapter 3 of E.M.Lifshic, L.P.Pitaevskij (vol.9)

Thanks lot to Dr Kaspar Sakman for clear explanation

Thanks to Dr N. Chtchelkatchev for giving good reference to understand more things.