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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Physical Interpretation of BEC Formation

I have recently worked through a theoretical argument for why Bose-Einstein Condensation occurs. If we are comparing an ensemble of distinguishable versus indistinguishable particles, we can compute ...
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Proof that Bose-Einstein condensation (BEC) discontinuity happens at the thermodynamic limit

From http://www.damtp.cam.ac.uk/user/tong/statphys/sp.pdf page 89 (95/191 in the pdf file), We can see that by looking again at the expressions (3.30) and (3.31), which tell us $$ z = \left( 1+ \...
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Connection between superconductivity and breaking of $U(1)$ symmetry in superconductors

$\newcommand{\Ket}[1]{\left|#1\right>}$Suppose I have a total Hamiltonian $H = H_0 + V$ given by the usual kinetic term $$H_0 = \frac{\hbar^2}{2m} \sum_{\mathbf{k}, \sigma = \uparrow, \downarrow} \;...
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Gas or liquid? a Bose-Einstein condensate

It is often said that a Bose-Einstein condensate of cold atoms is a gas. But because of Andrews' discovery of the critical temperature, we know that a gas and a liquid is not fundamentally different. ...
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Density of states-phase uncertainty relation

I came across this uncertainty relation for Density of states $N$ and phase $\theta$ in "Introduction to Many-Body Physics" by P Coleman on Page 15, equation (2.20). $$\Delta N\Delta\theta &...
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When a BECs mixture Hamiltonian posses a Galileian invariance?

I'm currently studying BECs mixtures (with and without the presence of a coupling term between them). While studying the symmetries of the system I stuck on the following question: does this system ...
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Scattering with attractive potential

In the context of low-energy scattering, as the potential becomes more and more attractive, the scattering length varies from negative, diverging then to positive values. It corresponds to ...
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Why is a bosonic condensate not easy to be scattered in a superconductor? [duplicate]

Why does an individual fermionic electron scatter off lattice defects easily while the Cooper pairs are insusceptible? I know they condensate to a same macroscopic quantum state, but still, why can ...
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Why are Alkali atoms used in many Cold Atom experiments?

It seems that alkali atoms are often used in cold atom experiments. The first BEC was formed with alkali atoms, and many modern experiments used Alkalis. What is special about having a single electron ...
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Depth of dipole trap

I am wondering about the relation between a harmonic potential (e.g. generated by a dipole trap) and temperature. I have seen couple of times that you can express the depth of the trap as a ...
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Lindhard Function for Boson

We know the density-density response for a non-interacting system with electrons is given by \begin{equation} \chi(q,\omega)=\sum_{k} \dfrac{f_{k}-f_{k+q}}{\omega+\epsilon_{k}-\epsilon_{k+q}+i\eta} \...
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Determine the entropy of an ideal bosonic gas

I am struggling with this problem. So we're told that any macroscopic state of a gas can be characterized by microscopic stated of a single particle. We divide the microscopic states into groups. So $...
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Can real-space-eigenstates of conduction electrons in crystal cause formation of electronic singlet pairs?

Crystals may contain electronic real-space-eigenstates as ground states, which are spatially much larger than one unit cell, such as impurity states, standing waves at Brillouin zone edges, Anderson ...
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Condensate vs superfluid fraction in $\rm{}^4He$

In the superfluid state of $\rm{}^4He$, it is known from neutron scattering measurements and Monte Carlo simulations that about 10% of the $\rm{}^4He$ atoms condense into the lowest energy state. Said ...
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Sign of momentum doesn't affect Bogoliubov coefficients in Bogoliubov transformation for BEC

I'm running into some issues with the deriving Bogoliubov transformation. Specifically, in order to diagonalize the Hamiltonian $$\begin{align}H = \sum_p \frac{p^2}{2m} \hat a^\dagger_p \hat a_p + \...
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Derivation of density of bosons below Bose-Einstein condensation temperature

I am trying to understand the explanation of Bose-Einstein condensation for non-interacting bosons given in Piers Coleman's "Introduction to Many-Body Physics", pg. 85-86. Coleman first ...
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What can cause viscosity in a BEC gas?

I know that Bose-Einstein condensate is not always a superfluid. So what can cause viscosity in a BEC gas?
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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 ...
pratik misra's user avatar
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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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