Questions tagged [bose-einstein-condensate]

A Bose–Einstein condensate is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. In this state, a large fraction of the bosons occupy the *lowest quantum state* so that macroscopic quantum phenomena are in evidence. Use for all related BEC processes.

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What new insights are gained in quantum information by doing tasks (e.g. creating BECs) in space?

Recently, I have seen a lot of discussion surrounding NASA and quantum technologies. Specifically, there is a paper that came out in Nature titled "Quantum gas mixtures and dual-species atom ...
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$s$-wave scattering approximation of the two-body interaction

In quantum gases, one usually approximates the two-body interaction $V(\boldsymbol{r})$ with a s-wave delta potential $$V_{\text{pseudo}}(\boldsymbol{r}) = \frac{4\pi \hbar^2 a}{m} \delta (\boldsymbol{...
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Quantum pressure of a Bose gas in a harmonic trap: Why is the result divergent?

The quantum pressure $E_{kin}$ of a Bose gas in the Thomas Fermi limit with contact interactions in a symmetric harmonic trap is determined by (Pitaevskij & Stringari, 2016): $E_{kin}=\int d\...
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Non-Saturation in Interacting Bose Gas Integral

I am independently working through some problems on Bose-Einstein condensation. In particular, I am trying to show that—in the Hartree-Fock mean-field approximation—for a Bose gas with contact ...
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Critical Temperature of a Bose-Einstein Condensate

I found that most sources and derivations of a relationship between the fraction of bosons in the ground state and normalised temperature are given as $$\frac{N_0}{N} = 1-\left(\frac{T}{T_C}\right)^{\...
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How does one produce a condensate?

In physics textbooks, one learns about Bose-Einstein condensate and it is all about taking thermodynamic limits. Of course, in real life, infinite systems do not exist. So, picture the following ...
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Tricky Integral: Evaluating Renormalized Ultraviolet "Divergent" Integral

I am trying to rederive the results presented in the paper, in particular equation (30). That is, I am trying to compute the correction to the ground-state energy of a dipolar condensate due to beyond-...
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Trying to derive Bose-Einstein Condensation using the canonical ensemble

My approach: Write down the partition function $Z_1$ of a single particle, approximate the summation with integral so $Z_1 = \int e^{-\beta E} g(E) dE$ where $g(E)$ is the density of states. Mark the ...
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Experimental evolution of condensates

I was talking to a colleague professor the other day and he said something that got me curious. The way I remember it, he said basically that in experiments a Bose-Einstein condensation is usually ...
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Under which condition we can use semi-classical approach in non-interacating bosons system?

I am studying BEC and interacting Bosons system. I take Bose-Einstein condensation in dilute gases (C. J. Pethick, H. Smith) as my reference. In section 2.3.1, they mentioned: $$ n(\vec{r})=\sum_{\nu}|...
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Bose-Einstein condensation explanation

I am reading the introduction of The mathematics of the Bose gas and its condensation by E. Lieb, R. Seiringer, J. Solovej and J. Yngavason. The authors explain how, in the free case, the density in ...
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A small question about off-diagonal long ranged order

I am studying the off-diagonal long ranged order from the book Superconductivity, Superfluids, and Condensates by James F. Annett. Thereof, I got stuck in a small step from (5.63) to (5.64)(Section 5....
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Bose-Einstein condensate and one-particle state

I am a little confused about the definition of a Bose-Einstein condensate. It is said that, in such a condensate, a huge number of particles are in the same state of lower energy. The term state of ...
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Free expansion of a harmonic trapped atoms

I read the book by Bose-Eistein condensation in dilute gas by C. J. Pethick, H. Smith. I met a question in page 31 (chapter 2,3,1). Thereof, they said that the distribution function of harmonic ...
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Derivative of a integral $|\Psi(\mathbf{r})|^2\int d^3r'\frac{1}{|\mathbf{r}-\mathbf{r}'|^{3}}|\Psi(\mathbf{r}')|^2$

I've been deriving equation of motion for Bose-Einstein condensation (BEC), and I've run into a slight problem while trying to do the derivative of $|\Psi(\mathbf{r})|^2\int d^3r'\frac{1}{|\mathbf{r}-\...
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How can Cooper pairs be stable?

In the BCS theory, a Cooper pair is formed by two electrons of nearly opposite momentum $(k \sim -k')$. What I don't understand is how can this structure remain stable in the metal: Assume that the ...
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How does the used wavefunction look like for the mean-field approximation in BEC?

In order to derive out Gross-Pitaevskii equation in BEC, mean field approximation is used. $$ \hat{\psi}=\langle\hat{\psi}\rangle+\delta\hat{\psi}' $$ and $\langle\hat{\psi}\rangle$ is called as wave ...
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Perturbation theory for optical lattice

This afternoon, I wondered about the following problem, but I cannot find the continuation. Can you help me ? We consider the following unperturbed 1D optical lattice Hamiltonian $$\hat{H}_0=\frac{\...
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How does Superconductiviy change in 2D and 1D?

I am just wondering if someone can talk about what can be carried over from the case of BCS theory in 3D down to lower dimensions. It seems there are many papers which write out BdG equations and work ...
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Vacuum state of Bogoliubov quasi-particles (continued)

This question focus on another aspect of my previous question. Consider a toy bilinear Hamiltonian consisting of two bosons $\{b_i\}_{i=1}^2$: $$ \begin{align*} \mathsf{H}[b^\dagger,b] &= ...
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how to evaluate particle fluctuations in BCS [duplicate]

I'm trying to evaluate particle fluctuations in BCS theory and I've been able to explicitly calculate the average total particle number by taking the expected value of the number operator (in ...
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Bogoliubov Hamiltonian in second order of perturbation

I have a Hamitonian: $$ H = \int d^3r \Big[\hat{\psi}^{\dagger} H_0 \hat{\psi} + \frac{g_0}{2} \hat{\psi}^{\dagger} \hat{\psi}^{\dagger} \hat{\psi}\hat{\psi} - \mu \hat{\psi}^{\dagger}\hat{\psi} \Big] ...
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What is the meaning of two wavefunctions being "same"?

So I had taken a course on BEC and Cold Atoms. I have read about the properties of non-interacting Bose gas and I was a little concerned about what we mean by two wave functions (of bosons) being the ...
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Free energy of the BCS ground state in the weakly attractive limit

I have a rather involved question regarding the weakly attractive limit of the BCS ground state. We know for exampel from The book of Pitajevski and Stringari (Bose–Einstein Condensation and ...
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What is the renormalization factor of the fixed particle number BCS wave function?

The BCS wavefunction, with fixed number is written as $$\left|\Psi_{BCS}(N)\right\rangle = \frac{1}{2\pi}\int_{0}^{2\pi} \mathrm{d}\phi\, e^{-iN\phi/2} \left|\Psi_{BCS}(\phi)\right\rangle \,\, ,$$ ...
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Relation between zero resistance and diamagnetism [duplicate]

Can zero resistance and perfect diamagnetism in a type-1 superconductor be related somehow? Because we know that having infinite conductivity alone doesn't decide the material to be a superconductor
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Bose-Einstein condensation transition line

I'm having some trouble figuring out the formula for the transition line for a BEC, i.e. the function $P(v)$, where $P$ is the pressure and $v$ is the volume of the BEC. I've substituted $$k_BT_c=\...
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Why do we cool atoms with laser light opposed to normal light?

When we use laser light to cool atoms, we get into some problems, because when atom beam slows down the Doppler shift changes the frequency of light in atom's frame of reference, so they can't ...
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Resources to learn about Bose-Einstein condensation as an undergraduate

For context, I am an undergraduate student who has background in undergrad QM and Electrodynamics (I've covered everything in both Griffiths textbooks). I do not know much, however, about statistical ...
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Energy of a Bose-Einstein condensate (BEC) in a rotating ring

I was looking at a Bose-Einstein condensate rotating in a ring. Now the energy of the BEC as a function of the angular frequency $\Omega$ is parabolic depending upon the eigenvalue of the angular ...
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Can you burn a quantum fluid?

Perhaps more generally, can you have a chemical reaction between a quantum fluid (a sodium Bose-Einstein condensate for example) and another quantum fluid or a non-quantum fluid? Or as a silly ...
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Bose-Einstein condensation of Helium

So I'm recalling this derivation of Bose-Einstein condensation in helium in some Thermo-stat mech text book. I thought it was in Reif, but I couldn't find it. It calculates the condensation ...
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Hamiltonian of the BEC in 2nd quantization [closed]

If I have $N$ non-interacting particle (bosons) forming a BEC that is trapped to $x = 0 $ (assume the system to be 1D) by an applied harmonic potential $V=\frac{1}{2}m\omega^{2}x^{2}$ How can I write ...
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Are two molecules of matter in BEC phase able to occupy the same space at the same time? [duplicate]

An important property of matter taught in grade school is that it occupies space (has a volume, whether it's relatively fixed like a solid or liquid, or depends on pressure like a gas), and that ...
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Why is the degeneracy factor for Bose Einstein distribution set to 1 automatically?

In https://scholar.harvard.edu/files/schwartz/files/12-bec.pdf, the article says "With Bose-Einstein statistics, we determined that using the grand canonical ensemble the expected number of ...
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Bose-Einstein condensate interference why same phase

I understand that a quantum particle/wave free of potential has a square of the wave function 1 everywhere and a wave function $e^{i(kx-\omega t)}$ and as consequence of that in Bose-Einstein ...
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How is volume of quantum harmonic oscillator related to the trapping frequency in BEC?

For a ideal Bose gas in harmonic trap, the total particle number can be written as, , and is fugacity. Now I want to find the expression for particle density for excited states. In case of Bose gas ...
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How to compute the ground state energy?

I am trying to compute the ground state energy of the following hamiliton. The system is originally a weakly interacting bose system and I get the following hamiltonian after bogoliubov transformation ...
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What states of matter does not exist naturally on Earth?

I am already aware that each state (solid, liquid, and gas) naturally occurs on Earth. But I am not quite sure about Plasma, Time-Crystals, and Bose-Einstein condensate (BEC) And can time-crystals and ...
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How does sodium-23 manage to form a Bose Einstein condensate with 11 protons and 11 electrons?

As asked in How is a Bose-Einstein condensate produced from sodium atoms that do not have an integer spin? , Sodium 23 has been used experimentally to form a Bose Einstein condensate. Sodium 23 is the ...
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What is difference between Bose-Einstein condensation in 3D and 2D?

I read in the reference [1] below that An infinite, non-interacting two-dimensional gas of bosons has no phase transition and never develops spontaneous coherence. However, adding interactions leads ...
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In General Relativity can electric field-induced tension in a Bose-Einstein Condensate reduce the energy needed to create an anti-gravitational field? [closed]

This is my theory showing how an electric field can create an anti-gravitational field. It is based on Einstein's General Relativity (GR), and the ability of an electric field to induce tension in ...
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Traps for Bose Einstein Condensate

I was just wondering about the role and effects of the trap in formation of BE condensate. In this respect if we have two potential, one small harmonic potential of energy E/10 and length L/100 inside ...
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Can a potential box be a trap for Bose-Einstein Condensate? Without a potential box, can we quantize energy level?

I am trying to study why are traps required for Bose Einstein condensate, and what are the conditions for a trap to sustain a BEC? Equivalently, Without energy level how to define BEC? Any comments ...
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Are traps really required for a 3D Bose Einstein condensate?

I came across this article that claims to a have a B.E condensate of exciton polariton in 2D. However, the authors also seem to admit that is a different kind of phase transition different from a ...
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Single photon Rabi frequency for Exciton Photon in 2D potential well with microcavity

I was just going through an article on Exciton-polariton Condensate in 2D and in the photon-exciton interaction strength (single photon Rabi frequency) inside a cavity seems to have a term of Bohr ...
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How does the creation of Cooper pairs and their condensation explains superconductivity?

Notes I follow provide an introduction to the BCS theory where one considers a Hamiltonian consisting of a kinetic term and an attractive potential between electrons. Next by the mean-field ...
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In BCS superconductors, superfluid part = Condensed Cooper Pairs? normal part = Bogoliubov quasiparticles?

I'm studying superconductivity and BCS theory. There are two pictures of superconductivity electric transport: Two-fluid model: superconductivity electron part + normal electron part BCS theory: ...
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Under what conditions would carbon-12 form a Bose-Einstein condensate?

Helium-4 is famous for its ability to form a Bose-Einstein condensate (BEC), because the atom is, in its ground state, a boson. This happens because the number of protons equals the number of neutrons ...
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Chemical potential for BE condensate: why the normalization would (uncorrectly) give $\mu> 0$?

Consider a gas of bosons. Let $g(\epsilon)$ be the distribution of energy levels (degeneracy). Consider the following integral $$I(\beta,\mu)=\int{d\epsilon g(\epsilon}) \frac{1}{e^{\beta(\epsilon-\mu)...
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