Hugenholtz–Pines relations and the critical temperature of a Rabi coupled binary Bose system

Abstract

Using a theoretical field Gaussian approximation, we have studied Rabi coupled binary Bose system at low temperatures. We have derived extended Hugenholtz–Pines relations taking into account one body interaction (e.g. Rabi coupling) and studied the critical temperature $T_c$ of Bose–Einstein condensate transition. We have shown that, the shift of $T_c$ due to this interaction cannot exceed $\sim 60 \%$ and goes to a plateau with increasing the parameter $\Omega _R/T_{c}^{0}$, where $\Omega _R$ is the intensity of the coupling and $T_{c}^{0}$ is the critical temperature of the system with $\Omega _R=0$. Moreover, the shift is always positive and does not depend on the sign of the one body interaction.

Graphic abstract

Appendices

Appendix

Relation between the phases and phase invariance

In present “Appendix” we show that, the stability condition with respect to the relative phase $\xi =\exp (i\theta )$ leads to the relation $\xi \omega _R=-1$. In fact, the relative phase angle $\theta$, should correspond to the minimum of $\Omega _0$:

$$\begin{aligned} \frac{\partial \Omega _0}{\partial \theta }=0, \quad \frac{\partial ^2\Omega _0}{\partial \theta ^2}>0 \end{aligned}$$

(A.1)

In general, $\Omega _0$ is given by

$$\begin{aligned} \Omega _0= & {} V[-\mu \rho _{0a}-\mu \rho _{0b}+\displaystyle \frac{g_a\rho _{0a}^2}{2}+\displaystyle \frac{g_b\rho _{0b}^2}{2} \nonumber \\{} & {} +g_{ab}\rho _{0a}\rho _{0b}+\displaystyle \frac{\Omega _R\sqrt{\rho _{0a}\rho _{0b}}\xi (\omega _R+\omega _R^\star )}{2}]\nonumber \\ \end{aligned}$$

(A.2)

Besides, in stable equilibrium, the variational parameters $\rho _{0a}$ and $\rho _{0b}$ should satisfy the saddle-point equations [12, 27, 28]

$$\begin{aligned}{} & {} \frac{\partial \Omega _0}{\partial \rho _{0a}}{=}0, \quad \frac{\partial \Omega _0}{\partial \rho _{0b}}{=}0, \quad \nonumber \\{} & {} \left( \frac{\partial ^2\Omega _0}{\partial \rho _{0a}^2}\right) \left( \frac{\partial ^2\Omega _0}{\partial \rho _{0b}^2}\right) {-}\left( \frac{\partial ^2\Omega _0}{\partial \rho _{0a}\partial \rho _{0b}}\right) >0 \end{aligned}$$

(A.3)

Now bearing in mind $\omega _R=e^{-i\theta _R}$, and using Eqs. (A.1), (A.2), we obtain

$$\begin{aligned} \begin{aligned}&\frac{\partial \Omega _0}{\partial \theta }=-V\Omega _R\sqrt{\rho _{0a}\rho _{0b}}\sin (\theta -\theta _R){=}0, \\&\frac{\partial ^2\Omega _0}{\partial \theta ^2}{=}{-}V\Omega _R\sqrt{\rho _{0a}\rho _{0b}}\cos (\theta {-}\theta _R)>0 \end{aligned} \end{aligned}$$

(A.4)

which lead to the well-known [1, 9, 10] result: $\theta -\theta _R=\pi (2n+1)$ i.e. $\omega _R\xi =-1$. This physically means that, when the system goes into its equilibrium state, it will choose the relative phase between the two condensates by itself, preferring the case with $\xi =-\omega _R$, where $\omega _R$ is in fact the sign of the one body interaction. For example, if the latter is negative, $\omega _R=-1$, then the condensates will coexist with the same phase $\xi =+1$, and vise versa. Moreover, particularly, in both cases, with $\xi =\pm 1$ there always exist in phase and out of phase excitations. The former correspond to the density branch $\omega _d$, while the latter to the spin branch $\omega _s$, as it has been clarified in Refs. [39, 42]

Below we show that physical observables do not depend on the relative phase $\xi$. Actually, in the symmetric case for the self energies we have

$$\begin{aligned} \begin{aligned}&X_1=X_3=-\mu +3\rho _{0a}g+\rho _{0a}g_{ab}=\Sigma _n+\Sigma _{an}-\mu \\&X_2=X_4=-\mu +\rho _{0a}g+\rho _{0a}g_{ab}=\Sigma _n-\Sigma _{an}-\mu \\&X_5=2\xi g_{ab}\rho _{0a}+\frac{\Omega _R\omega _R}{2}, \quad X_6=\frac{\Omega _R\omega _R}{2} \end{aligned}\nonumber \\ \end{aligned}$$

(A.5)

and for the dispersions:

$$\begin{aligned} \begin{array}{l} \omega _1=\sqrt{(\varepsilon _k+X_2+X_6)(\varepsilon _k+X_1+X_5)}, \\ \omega _2= \sqrt{(\varepsilon _k+X_2-X_6)(\varepsilon _k+X_1-X_5)} \end{array} \end{aligned}$$

(A.6)

Now from Eqs. (33), (A.5) and (A.6) it is seen that, e.g. $\rho _{1a}$ is phase invariant, that is $\rho _{1a}(\xi ,\omega _{R})=\rho _{1a}(-\xi ,-\omega _{R})$, since the transformation $(\xi ,\omega _{R})\leftrightarrow (-\xi ,-\omega _{R})$ is equivalent to the following replacements: $X_5\leftrightarrow -X_5$, $X_6\leftrightarrow -X_6$, $\omega _1\leftrightarrow \omega _2$. Note that, this statement holds true both for the condensed as well as normal states. Phase invariance of other observables can be proven in the same way.