What are fragmented condensates?
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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),...=O(N)n(1),n(2),...=O(N)
If two of the eigenvalues are of the order of the number of particles the condensate is two-fold fragmented.
Example:-
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,
ρ^(1)(r1|r′1;t)=N.Tr2,..,N[|ψ(t)⟩⟨ψ(t)|]=N∫dr2...drNψ(r1,..,rN;t)ψ∗(r′1,..,r′N;t)ρ^(1)(r1|r1′;t)=N.Tr2,..,N[|ψ(t)⟩⟨ψ(t)|]=N∫dr2...drNψ(r1,..,rN;t)ψ∗(r1′,..,rN′;t)
The one-body density matrix can be expanded in its eigenfunctions as
ρ^(1)(r1|r′1;t)=∑in(1)i(t)α(1)i(r1,..,rp;t)α(1)∗i(r′1,..,r′p;t)ρ^(1)(r1|r1′;t)=∑ini(1)(t)αi(1)(r1,..,rp;t)αi(1)∗(r1′,..,rp′;t)
where n(1)ini(1) and α(1)iαi(1) are the eigenvalues and eigenfunctions of the one-body density matrix ρ^(1)ρ^(1)and ψψ is the wavefunction for NN identical, spinless bosons.
My understanding:-
The eigenvalue of a density matrix ρ^=∑iPi|ψi⟩⟨ψi|ρ^=∑iPi|ψi⟩⟨ψi| is the probability PiPi of the system to be in the state |ψi⟩|ψi⟩ and the eigenfunctions are the corresponding state vectors ψiψi.
For a composite system of NN particles (indistinguishable) with density matrix ρ^ρ^, the reduced density matrix of one particle is ρ^(1)=Tr2,..,N(ρ^)ρ^(1)=Tr2,..,N(ρ^) which describes the state of one particle.
n(1),n(2),...=O(N)n(1),n(2),...=O(N)
If two of the eigenvalues are of the order of the number of particles the condensate is two-fold fragmented.
Example:-
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,
ρ^(1)(r1|r′1;t)=N.Tr2,..,N[|ψ(t)⟩⟨ψ(t)|]=N∫dr2...drNψ(r1,..,rN;t)ψ∗(r′1,..,r′N;t)ρ^(1)(r1|r1′;t)=N.Tr2,..,N[|ψ(t)⟩⟨ψ(t)|]=N∫dr2...drNψ(r1,..,rN;t)ψ∗(r1′,..,rN′;t)
The one-body density matrix can be expanded in its eigenfunctions as
ρ^(1)(r1|r′1;t)=∑in(1)i(t)α(1)i(r1,..,rp;t)α(1)∗i(r′1,..,r′p;t)ρ^(1)(r1|r1′;t)=∑ini(1)(t)αi(1)(r1,..,rp;t)αi(1)∗(r1′,..,rp′;t)
where n(1)ini(1) and α(1)iαi(1) are the eigenvalues and eigenfunctions of the one-body density matrix ρ^(1)ρ^(1)and ψψ is the wavefunction for NN identical, spinless bosons.
My understanding:-
The eigenvalue of a density matrix ρ^=∑iPi|ψi⟩⟨ψi|ρ^=∑iPi|ψi⟩⟨ψi| is the probability PiPi of the system to be in the state |ψi⟩|ψi⟩ and the eigenfunctions are the corresponding state vectors ψiψi.
For a composite system of NN particles (indistinguishable) with density matrix ρ^ρ^, the reduced density matrix of one particle is ρ^(1)=Tr2,..,N(ρ^)ρ^(1)=Tr2,..,N(ρ^) which describes the state of one particle.
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Condensate fragmentation as a sensitive measure of the quantum many-body behavior of bosons with long-range interactions. The occupation of more than one single-particle state and hence the emergence of fragmentation is a many-body phenomenon universal to systems of spatially confined interacting bosons.
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