# reduced single particle density matrix (1-RDM) in second quantization

Is it possible to approach one body density matrix without using field operators ?

For example for a double well potential, the reduced single particle density matrix is defined as: $$\hat{\rho}^{(1)}= {\begin{bmatrix} \langle \hat{a}_1^{\dagger} \hat{a}_1\rangle & \langle\hat{a}_1^{\dagger} \hat{a}_2\rangle \\ \langle\hat{a}_2^{\dagger} \hat{a}_1\rangle & \langle\hat{a}_2^{\dagger} \hat{a}_2\rangle \\ \end{bmatrix}}$$ Similarly for a triple well system $$\hat{\rho}^{(1)}= {\begin{bmatrix} \langle \hat{a}_1^{\dagger} \hat{a}_1\rangle & \langle\hat{a}_1^{\dagger} \hat{a}_2\rangle & \langle\hat{a}_1^{\dagger} \hat{a}_3\rangle\\ \langle\hat{a}_2^{\dagger} \hat{a}_1\rangle & \langle\hat{a}_2^{\dagger} \hat{a}_2\rangle & \langle\hat{a}_2^{\dagger} \hat{a}_3\rangle\\ \langle\hat{a}_3^{\dagger} \hat{a}_1\rangle & \langle\hat{a}_3^{\dagger} \hat{a}_2\rangle & \langle\hat{a}_3^{\dagger} \hat{a}_3\rangle \end{bmatrix}}$$

Is it possible define $\hat{\rho}^{(1)}$ starting from a general state, let's say of a double well or two mode system, $\mid \psi \rangle=\sum_{l=0}^{N}C_{l}(\hat{a}_1^{\dagger})^{l}(\hat{a}_2^{\dagger})^{N-l}\mid0,0\rangle$, by using creation and annihilation operators, without making use of field operators, by taking the partial trace over its density operator with respect to particles 2...N ?

See Eq-3 in Ref.1, Ref.2, and similar questions: Condensate fraction and single-particle density matrix and Eigenvalue of the first-order reduced density matrix (1-RDM) and condensation of bosons

• Since you are reading arxiv.org/pdf/cond-mat/0605711v1.pdf, it should be pretty obvious that this is exactly what they are using. Once the origin of the matrix elements is known, there is no need to backtrack to in order to calculate them. – udrv Apr 8 '16 at 14:27
• @udrv srry...didn't get wht u r trying to say. my doubt is, can i reach the above definition of 1-RDM with creation/annihilation operators by taking the partial trace over other particles, without making use of field operators ? – ss1729 Apr 8 '16 at 14:47
• Yes, that's what I am trying to say. The expression for the 1-RDM was reached using field operators, but once its matrix elements are defined as averages of creation & annihilation ops. products, there is no need to go back to field ops. to calculate those averages. If you take a closer look at what the authors of arxiv.org/pdf/cond-mat/0605711v1.pdf do in deriving their eqs.(9-13), you'll see that this is exactly the procedure they use: take averages with $|\psi\rangle$ defined in terms of mode ops. – udrv Apr 8 '16 at 14:54