# Eigenvalue of the first-order reduced density matrix (1-RDM) and condensation of bosons

It is defined that a system of identical bosons is said to be condensed, if the largest eigenvalue of the first-order reduced density matrix (1-RDM) is of the order of the number of particles in the system. Check Phys. Rev. 104, 576 (1956) and chapter 2 of Many-Body Schrödinger Dynamics of Bose-Einstein Condensates.

For a given wavefunction $\psi(r_{1},...,r_{N};t)$ of $\mathcal{N}$ identical spinless bosons with spatial coordinates $r_{i}$, the first-order reduced density matrix (1-RDM) is defined as $$\rho^{(1)}(r_{1},r^{'}_{1};t)=N\int\psi(r_{1},r_{2},...,r_{N};t)\psi^{*}(r^{'}_{1},r_{2},...,r_{N};t)dr_{2}dr_{3}...dr_{N}=\langle\psi(t)|\hat{\Psi}^{\dagger}(r{'}_{1})\hat{\Psi}(r_{1})|\psi(t)\rangle$$

where $\hat{\Psi}(r)$ is the bosonic field operator and the wavefunctions are properly normalized $\langle\psi(t)|\psi(t)\rangle=1$. The first-order reduced density matrix can be regarded as the kernal of the operator $$\hat{\rho}^{(1)}=\mathcal{N}.{Tr}_{\mathcal{N}-1}[|\psi(t)\rangle\langle\psi(t)|]$$

What exactly are the 1-RDM and its eigenvalues and how do I get a physical picture of it?

What is the physical meaning of the above statement based on the definition of the 1-RDM?

• Note that for indistinguishable particles, the "first-order reduced density matrix" typically refers to the matrix $M$ with elements $M_{ij}=\langle a^\dagger_i a_j\rangle$. – Norbert Schuch Jan 26 '16 at 9:09
• BTW, would help if you could focus your question more. As it stands, it is very broad (and not very clear). – Norbert Schuch Jan 26 '16 at 9:10
• Closely related (?) question (though with no answer): physics.stackexchange.com/questions/110371 – Norbert Schuch Jan 26 '16 at 9:17
• @NorbertSchuch I have edited the actual question.please check. – ss1729 Jan 27 '16 at 10:00
• I'm afraid that while you have added definitions, the question as such is still quite unclear. – Norbert Schuch Jan 27 '16 at 10:11

The 1st order density matrix, $$\rho({\bf x}, {\bf x'}) = \langle \hat\psi^\dagger({\bf x'})\hat\psi({\bf x}) \rangle$$ is a Hermitian operator since $\rho({\bf x}, {\bf x'}) = \rho^*({\bf x'}, {\bf x})$ and its diagonal entries, $$\rho({\bf x}, {\bf x}) = \langle \hat\psi^\dagger({\bf x})\hat\psi({\bf x}) \rangle = n({\bf x})$$ give the number density of particles at location ${\bf x}$, such that $$\int{d{\bf x} \;\rho({\bf x}, {\bf x})} \equiv \int{d{\bf x} \;n({\bf x})} = N$$ is the (average) total number of particles in the system. Its orthonormal eigenfunctions, defined by $$\int{d{\bf x'} \rho({\bf x}, {\bf x'}) \phi_j({\bf x'})} = n_j \;\phi_j({\bf x})\\ \int{d{\bf x} \;\phi^*_j({\bf x})\phi_k({\bf x})} = \delta_{jk}\\ \sum_j{\phi^*_j({\bf x})\phi_j({\bf x'})} = \delta({\bf x} - {\bf x'})$$ represent natural orbitals. The eigenvalues $$n_j = \int{\int{d{\bf x}d{\bf x'} \;\phi^*_j({\bf x})\; \rho({\bf x}, {\bf x'})\; \phi_j({\bf x'})}}$$ satisfy $$\sum_k{n_k} = \int{d{\bf x} \;n({\bf x})} = N$$ and represent the occupation numbers of the natural orbitals.
In fact, if $a^\dagger_j$, $a_j$ are creation and annihilation operators of these orbitals, $[a_j, a^\dagger_k]=\delta_{jk}$, the field operators are expanded as $$\hat\psi({\bf x}) = \sum_j{\phi_j({\bf x})\;a_j}$$ and we have $$\rho({\bf x}, {\bf x'}) = \sum_{jk}{\phi^*_j({\bf x'})\phi_k({\bf x}) \langle a^\dagger_j a_k \rangle}$$ and $$\rho_{jk} \equiv \langle a^\dagger_j a_k \rangle = \int{\int{d{\bf x}d{\bf x'} \phi^*_k({\bf x}) \rho({\bf x}, {\bf x'}) \phi_j({\bf x'})}} = \rho_{jj}\delta_{jk}\\ \rho_{jj} \equiv \langle a^\dagger_j a_j \rangle = n_j$$ In the condensate state one orbital achieves a high occupation number $n_0 \sim N$, such that $$N = n_0 + \sum_{j\neq 0}{n_j}$$ and the largest eigenvalue of $\rho({\bf x}, {\bf x'})$ becomes $$\rho_{00} = n_0$$
• Good answer. It might be worth adding (w.r.t. to BEC) that if the largest eigenvalue $\eta_0$ is close (or proportional) to $N$, this means that the mode $a_0$ is occupied by $\eta_0$ particles, which corresponds to Bose-Einstein condensation. – Norbert Schuch Jan 27 '16 at 11:23