Dilute Bose gas

The Hamiltonian of a system of spinless bosons, interacting through the pair potential $U$ and immersed in the external field $V({\bf r})$ is given, in second quantization, by

$$\hat H = \int \hat \Psi^{\dagger}({\bf r}) \left(-\frac{\hbar^2}{... ... r_1 - r_2}\vert)\, \hat \Psi({\bf r_1}) \hat \Psi({\bf r_2})\,{\bf dr_1 dr_2},$$ (1.1)

here $m$ is the mass of a particle, $\hat\Psi({\bf r})$ and $\hat\Psi^\dagger({\bf r})$ are the boson field operators that annihilate and create a particle at the position ${\bf r}$. If the gas is dilute and cold, then the two-body potential can be replaced by the pseudopotential $U({\bf r'-r}) = g \delta({\bf r'-r})$ which is fixed by a single parameter, the $s$-wave scattering length $a$, through the coupling constant

$$g = \frac{4\pi\hbar^2a}{m}$$ (1.2)

The Hamiltonian (1.1) takes the form

$$\hat H = \int \hat \Psi^{\dagger}({\bf r}) \left(-\frac{\hbar^2}{... ...) \hat \Psi^{\dagger}({\bf r}) \hat \Psi({\bf r}) \hat \Psi({\bf r})\,{\bf dr},$$ (1.3)

The field operator can decomposed as $\hat\Psi({\bf r}) = \sum_{\bf k} \hat a_{\bf k} \phi_{\bf k}({\bf r})$, where $\phi_{\bf k}({\bf r})$ are single-particle wavefunctions with quantum number ${\bf k}$. The bosonic creation and annihilation operators $\hat a_k$ and $\hat a^\dagger_k$ are defined in Fock space through the relations

$$\hat a^\dagger_{\bf k}~\vert n_0,\,n_1,\,...,\,n_k,\,...\rangle = \sqrt{n_k+1} \vert n_0,\,n_1,\,...,\,n_k+1,\,...\rangle,$$ (1.4)

$$\hat a_{\bf k}~\vert n_0,\,n_1,\,...,\,n_k,\,...\rangle = ~~~\sqrt{n_k}~~\vert n_0,\,n_1,\,...,\,n_k-1,\,...\rangle,$$ (1.5)

where $n_k$ are the eigenvalues of the operator $\hat n_{\bf k} = \hat a^\dagger_{\bf k} \hat a_{\bf k}$ giving the number of atoms in the single-particle state ${\bf k}$. The operators $\hat a_{\bf k}$ and $\hat a^\dagger_{\bf k}$ obey the usual bosonic commutation rules

$$[\hat a_{\bf k}, \hat a_{\bf k'}^\dagger] = \delta_{\bf k k'},\qu... ...at a_{\bf k'}] = 0,\quad [\hat a_{\bf k}^\dagger, \hat a_{\bf k'}^\dagger] = 0.$$ (1.6)

Bose-Einstein condensation occurs when the number of particles in one particular single-particle state becomes very large $N_{\bf0} \gg 1$. In this limit the states with $N_{\bf0}$ and $N_{\bf0} \pm 1 \approx N_{\bf0}$ correspond to the same physical configuration and, consequently, the operators $\hat a^\dagger_{\bf 0}$ and $\hat a_{\bf0}$ can be treated as complex numbers

$$\hat a^\dagger_{\bf0} = \hat a_{\bf0} = \sqrt{N_{\bf0}}$$ (1.7)

For a uniform gas in a volume $V$ the good single-particle states correspond to momentum states and BEC occurs in the single-particle state $\psi_0 = 1 / \sqrt{V}$ having zero momentum. Thus, the field operator $\hat\Psi_{\bf k}({\bf r})$ can be decomposed in the form $\hat\Psi_{\bf k}({\bf r}) = \sqrt{N_0/V} + \hat\Psi'({\bf r})$. The generalization for the case of nonuniform and time-dependent configurations is given by

$$\hat\Psi_{\bf k}({\bf r},t) = \Phi({\bf r},t) + \hat\Psi'({\bf r},t),$$ (1.8)

where the Heisenberg representation for the field operators is used. Here $\Phi({\bf r},t)$ is a complex function defined as the expectation value of the field operator $\Phi({\bf r},t)~=~\langle\hat\Psi({\bf r},t)\rangle$. The function $\Phi({\bf r},t)$ is a classical field having the meaning of an order parameter and is often called the wave-function of the condensate. The mean-field theory, which describes the behavior of the classical field $\Phi({\bf r},t)$ and ignores the fluctuations $\hat\Psi'({\bf r},t)$ is contained in the Gross-Pitaevskii theory. A more refined approach, which takes into account the fluctuations of the field operator was proposed by Bogoliubov.