It is defined that if more than one eigenvalue of the one-body density matrix are macroscopically occupied the condensate is said to be fragmented.

$$ n^{(1)},n^{(2)},...=\mathcal{O}(\mathcal{N}) $$ If two of the eigenvalues are of the order of the number of particles the condensate is two-fold fragmented.


The ground state of repulsively interacting bosons in a 1D double well potential (symmetric) are found to be two-fold fragmented, if the strength of the interactions are large and the height of the barrier is high enough.

Important quantities:-

One-body density matrix, \begin{split} \hat{\rho}^{(1)}(r_{1}|r_{1}';t)&=\mathcal{N}.{Tr}_{2,..,{\mathcal{N}}}[|\psi(t)\rangle\langle\psi(t)|]\\&=\mathcal{N}\int dr_{2}...dr_{N}\psi(r_{1},..,r_{N};t)\psi^{*}(r'_{1},..,r'_{N};t) \end{split} The one-body density matrix can be expanded in its eigenfunctions as $$ \hat{\rho}^{(1)}(r_{1}|r_{1}';t)=\sum_{i}n^{(1)}_{i}(t)\alpha^{(1)}_{i}(r_{1},..,r_{p};t)\alpha^{(1)\ast}_{i}(r'_{1},..,r'_{p};t) $$

where $n^{(1)}_{i}$ and $\alpha^{(1)}_{i}$ are the eigenvalues and eigenfunctions of the one-body density matrix $\hat{\rho}^{(1)}$ and $\psi$ is the wavefunction for $\mathcal{N}$ identical, spinless bosons.

My understanding:-

The eigenvalue of a density matrix $\hat{\rho}=\sum_{i}P_{i}|\psi_{i}\rangle\langle\psi_{i}|$ is the probability $P_{i}$ of the system to be in the state $|\psi_{i}\rangle$ and the eigenfunctions are the corresponding state vectors $\psi_{i}$.

For a composite system of $\mathcal{N}$ particles (indistinguishable) with density matrix $\hat{\rho}$, the reduced density matrix of one particle is $\hat{\rho}^{(1)}=Tr_{2,..,N}(\hat{\rho})$ which describes the state of one particle.


What exactly are fragmented condensates and how do I get a physical picture of it based on the definition of density matrices ?

  • $\begingroup$ Sec.II in this article cat.phys.s.u-tokyo.ac.jp/~ueda/EJMuellerPRA74.pdf explains the concept in terms defined in my answer to your previous question, physics.stackexchange.com/questions/110371/…. I think you are not yet familiar with the theory of indistinguishable particle systems (a.k.a many-particle systems) and second quantization, but are trying to find your way using general concepts, like the reduced density matrix. These are indeed fundamental, but the way you attempt to use them is not quite correct. $\endgroup$ – udrv Jan 31 '16 at 18:31
  • $\begingroup$ Try reading the intro chapter(s) in a good Many-particle book or course. Personally I recommend this one: fulviofrisone.com/attachments/article/448/Fetter%20Walecka.pdf (old but good, notation and concepts as current as ever), but any other that suits your learning style would do. $\endgroup$ – udrv Jan 31 '16 at 18:31

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