Coherent population trapping and dynamical instability in coupled atom-molecule condensates

Coherent population trapping (CPT) is an important concept in many subfields of physics and chemistry. Here we analyze the collective excitation spectrum of the CPT states in a coupled atom-molecule condensate system. We find that collisions between particles can cause the CPT state to be dynamically unstable, which is a unique feature of the nonlinear system. We obtain a set of analytical criteria for determining the stability properties of the CPT state in the long-wavelength limit. We construct stability diagrams and provide a systematic classification of various instabilities according to collisional interaction strengths.

Figure 1

The energy diagrams of three-level atom-molecule system involving free-bound-bound transitions. Conversion of atoms in $\mid a⟩$ to quasibound molecules in $\mid m⟩$ is accomplished by Feshbach resonance, while the coupling between $\mid m⟩$ and the ground molecular state $\mid g⟩$ is provided by laser light.

Figure 2

The quasiparticle dispersion spectra for $n=1$, $\Omega =1$, and $ϵ=-2$. Solid lines represent the collisionless case while dotted lines are for a system with ${\lambda }_a=0.625$, ${\lambda }_m={\lambda }_g={\lambda }_{am}={\lambda }_{ag}={\lambda }_{mg}=0.3{\lambda }_a$. Units are defined in Sec. 4.

Figure 3

Division of the positive quarter of $({\lambda }_g,{\lambda }_a)$ space into three regions by a straight line ${\lambda }_g=-4{\lambda }_a+4{\lambda }_{ag}$ and a hyperbola ${\lambda }_a{\lambda }_g={\lambda }_{ag}^2$.

Figure 4

Classification of the instabilities in the limit of low momentum. The shaded areas of the diagrams in columns 1 and 2 correspond to the instabilities due to the imaginary sound speed and the complex gaps, respectively. The sets of diagrams in rows 1, 2, and 3 are the instability maps for systems operating in regions I, II, and III as specified in Fig. 3, respectively.

Figure 5

$\mathrm{Im}\left[\omega \left(p\right)\right]$ as a function of $\Omega$ for a fixed momentum $p=0.005$. The curves from left to right correspond to $ϵ=1.3$, 1.5, 2, and 3, respectively. Other parameters are $n=1$, ${\lambda }_a=0.625$, and ${\lambda }_m={\lambda }_g={\lambda }_{am}={\lambda }_{mg}={\lambda }_{ag}=0.3{\lambda }_a$. Units are defined in Sec. 4.

Figure 6

(Color online) A three-dimensional view of $\mathrm{Im}\left[\omega \left(p\right)\right]$ as a function of $\Omega$ and $ϵ$ for two branches of roots to Eq. (23). Parameters are the same as in Fig. 5. Units are defined in Sec. 4.

Figure 7

The time evolution of the particle number densities in the condensate mode probed by a zero-momentum perturbation with $\delta {\Omega }_0=0.001$ for ${\omega }_0=$ (a) 1.1 and (b) 0.9, while the Feshbach detuning is fixed at $ϵ=3$. Other parameters are the same as in Fig. 5. Units are defined in Sec. 4.

Figure 8

The time evolution of ${\mid {\varphi }_g(l,t)\mid }^2$ ($l=0$, 1, and 2) after a finite-momentum perturbation with $\delta {\Omega }_1=0.001$ at $p=0.01$ $(\delta {\Omega }_0=0)$ is applied to a system initially prepared in a CPT state of $ϵ=-4$ and ${\Omega }_0=1$. We include modes ranging from $l=-10$ to 10 in the simulation. Other parameters are ${\lambda }_a=0.625$, ${\lambda }_m={\lambda }_g={\lambda }_{mg}=0.5{\lambda }_a$, ${\lambda }_{am}={\lambda }_a$, ${\lambda }_{ag}=1.4{\lambda }_a$, and $n=1$. Units are defined in Sec. 4.