Two-component Bose gases with one-body and two-body couplings

We study the competition between one-body and two-body couplings in weakly interacting two-component Bose gases, in particular as regards field correlations. We derive the mean-field theory for both ground-state and low-energy pair excitations in the general case where both one-body and two-body couplings are position dependent and the fluid is subjected to a state-dependent trapping potential. General formulas for phase and density correlations are also derived. Focusing on the case of homogeneous systems, we discuss the pair-excitation spectrum and the corresponding excitation modes, and use them to calculate correlation functions, including both quantum and thermal fluctuation terms. We show that the relative phase of the two components is imposed by that of the one-body coupling, while its fluctuations are determined by the modulus of the one-body coupling and by the two-body coupling. One-body coupling and repulsive two-body coupling cooperate to suppress relative-phase fluctuations, while attractive two-body coupling tends to enhance them. Further applications of the formalism presented here and extensions of our work are also discussed.

Figure 1

Coupled two-component Bose gas. The gas is made of bosonic particles of a single atomic species, which can be in two different internal states (labeled 1 and 2). It is described by the two field operators ${\widehat{\psi }}_1\left(\mathbf{r}\right)$ and ${\widehat{\psi }}_2\left(\mathbf{r}\right)$, corresponding to each component. In this work, we assume that the two components are coupled by one-body and/or two-body interactions of coupling constants $\Omega$ and $g_{12}$, respectively. In the most general case, the two coupling constants can be position dependent.

Figure 2

Bogoliubov spectrum of the coupled excitations in a homogeneous two-component Bose gas with $g_{12}\ne 0$ and $\Omega \ne 0$. Plotted are the two energy branches $E_{\mathbf{k}}^{\mathrm{in}/\mathrm{off}}$ [Eqs. (59) and (60)] in the case $g_1=g_2$, for $g_{12}=0.7g_1$ and $\hslash {\Omega }_0=0.4g_{1}n$. This corresponds to a situation where $g_{12}n>\hslash {\Omega }_0$ and the two branches cross at a certain momentum $k^c$ (see text). For $g_{12}n<\hslash {\Omega }_0$, there is no crossing point and the “off” branch is always above the “in” branch. Here, ${\mu }_0=g_{1}N/2\mathcal{V}$ is the chemical potential in the absence of any coupling, and ${\xi }_0=\hslash /\sqrt{2m{\mu }_0}$ is the corresponding healing length.

Figure 3

Amplitudes of the wave functions $f_{\sigma \mathbf{k}}^{\mathrm{p}/\mathrm{m}}$ of the coupled Bogoliubov excitations for a homogeneous two-component Bose gas with $g_{12}\ne 0$ and $\Omega \ne 0$. Plotted are the absolute values, $|f_{\sigma \mathbf{k}}^{\mathrm{p}/\mathrm{m}}|$ [see Eqs. (61) to (64)] for the same parameters as in Fig. 2. Since $g_1=g_2$, the absolute values are independent of the component $\sigma$. The excitations are in phase in the “in” branch ($E_{\mathbf{k}}^{\mathrm{in}}$) and off phase for the “off” branch ($E_{\mathbf{k}}^{\mathrm{off}}$).

Figure 4

Correlation function of the relative phase for a 1D two-component Bose gas with one-body (${\Omega }_0\ne 0$) and two-body ($g_{12}\ne 0$) couplings, plotted for various temperatures ($k_{\mathrm{B}}T/{\mu }_0=0$, 1, $1.5$, 2) in the case where $g_1=g_2\equiv g$. The parameters here correspond to $N={10}^4$ atoms of ${}^{87}$Rb ($m\simeq 144\times {10}^{-27}$ kg) in a 1D box of size $2L={10}^{-4}$ m and interacting via the scattering length $a_1=a_2=5.95$ nm. It corresponds in the absence of any coupling to the chemical potential ${\mu }_0=gn=7.88\times {10}^{-31}$ J, which we choose as the energy unit. In these units, we use the parameters $\hslash {\Omega }_0=1{\mu }_0$ and $g_{12}n=0.75{\mu }_0$.

Figure 5

Relative-phase fluctuations as a function of temperature for a 1D two-component Bose gas with one-body (${\Omega }_0\ne 0$) and two-body ($g_{12}\ne 0$) couplings, plotted for the same parameters as in Fig. 4. The solid blue line with dots is the exact calculation, corresponding to Eq. (71) in $\mathbf{r}=0$. The solid red line is the expansion (75) and the dotted red line corresponds to the first left-hand-side term in Eq. (75). While the quantum fluctuations are small, the thermal contribution increases with temperature. At high temperature, the exact calculation is accurately reproduced by the high-temperature expansion (75), whereas the linear dominant term proves insufficient to do so.