Role of Higher-Order Interactions on the Modulational Instability of Bose-Einstein Condensate Trapped in a Periodic Optical Lattice

Abstract

In this paper, we investigate the impact of higher-order interactions on the modulational instability (MI) of Bose-Einstein Condensates (BECs) immersed in an optical lattice potential. We derive the new variational equations for the time evolution of amplitude, phase of modulational perturbation, and effective potential for the system. Through effective potential techniques, we find that high density attractive and repulsive BECs exhibit new character with direct impact over the MI phenomenon. Results of intensive numerical investigations are presented and their convergence with the above semi analytical approach is brought out.

Appendix A: Derivation of Variational Parameters a(t) and b(t)

The MI-motivated trial wavefunction in (7) is substituted into the Lagrangian density in (6) and the effective Lagrangian is calculated by integrating the Lagrangian density as

$$\begin{aligned} L_{eff}=\int \mathcal {L} \,\,dx \end{aligned}$$

But here, we consider an annular (one-dimensional) geometry, which imposes periodic boundary conditions on the wave function $\psi (x, t)$ and integration limits $0 \leqslant x < 2\pi$. This causes the quantization of the wave numbers, i.e. $k, q = 0,\pm 1,\pm 2,\pm 3, . . .$. In this new geometry, calculating the effective Lagrangian yields

$$\begin{aligned} L_{\mathrm {eff}}= & {} \pi \Big \{-V_s \left( A_0^2+a_1^2+a_2^2\right) -g \Big [2A_0^2\left( a_1^2+a_2^2\right) -A_0^4 \\&+4a_1^2a_2^2+4A_0^2a_1a_2 \cos (b_1+b_2)+a_1^4+a_2^4 \Big ]+\eta \Big [8q^2a_1^2a_2^2 \nonumber \\&+4A_0^2q^2a_1a_2 \cos (b_1+b_2) +2A_0^2q^2\left( a_1^2+a_2^2\right) \Big ] -\chi \Big [\frac{2}{3}\left( a_1^6+a_2^6\right) \\&+4A_0^4\left( a_1^2+a_2^2\right) +6A_0^2\left( a_1^4+a_2^4\right) +6a_1^2a_2^2\left( a_1^2+a_2^2\right) +\frac{4}{3}A_0^6 \\&+4A_0^2a_1a_2 \cos (b_1+b_2)\left( 2A_0^2+3a_1^2+3a_2^2\right) +24A_0^2a_1^2a_2^2\Big ] \\&+2a_1^2\left( - 2kq-q^2-\dot{b_1}\right) +2a_2^2\left( 2kq-q^2-\dot{b_2}\right) \Big \} \end{aligned}$$

The expression of this effective Lagrangian is such that the pair $\{b_1(t), b_2(t)\}$ may be interpreted as the set of generalized coordinates of the system, while the pair $\{A_1(t), A_2(t)\}$, with $A_1(t)=2a_1^2(t)$ and $A_2(t)=2a_2^2(t)$, gives the corresponding momenta. The Hamiltonian of the system is expressed as

$$\begin{aligned} H = -L+\int _{-\infty }^{\infty }\frac{\mathrm {i}}{2}\left( \frac{\partial \psi }{\partial t}\psi ^*-\frac{\partial \psi ^*}{\partial t}\psi \right) \mathrm {d}x. \end{aligned}$$

Considering the integration limits imposed by the new geometry, we have

$$\begin{aligned} H= & {} \pi \Big \{V_s \left( A_0^2+\frac{A_1}{2}+\frac{A_2}{2}\right) +g \Big [2A_0^2(A_1+A_2)+A_0^4\\&+\frac{1}{4} \left( A_1^2+A_2^2\right) +2A_0^2\sqrt{A_1} \sqrt{A_2}\cos (b_1+b_2)+A_1A_2 \Big ] \\&-\eta \Big [2A_0^2q^2\sqrt{A_1} \sqrt{A_2} \cos (b_1+b_2) +A_0^2q^2(A_1+A_2) \\&+2q^2A_1A_2 \Big ] -\chi \Big [\frac{1}{12}\left( A_1^3+A_2^3\right) +2A_0^4(A_1+A_2) \\&+\frac{3}{2}A_0^2\left( A_1^2+A_2^2\right) +\frac{3}{4}A_1A_2(A_1+A_2) -\frac{4}{3}A_0^6 +6A_0^2A_1A_2 \\&+A_0^2\sqrt{A_1} \sqrt{A_2} \cos (b_1+b_2)\left( 4A_0^2+3A_1+3A_2\right) \Big ] \\&+A_1\left( 2kq+q^2+k^2\right) +A_2\left( -2kq+q^2+k^2\right) +2A_0^2k^2\Big \} \end{aligned}$$

In order to derive the evolution equations for the time-dependent parameters introduced in (7) , we use the corresponding Euler-Lagrange equations based on the variational effective Lagrangian $L_{\mathrm {eff}}$. In the generalized form, these equations read

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} \left( \frac{\partial L_{\mathrm {eff}}}{\partial \dot{\xi }_i}\right) -\frac{\partial L_{\mathrm {eff}}}{\partial \xi _i} = 0 , \end{aligned}$$

where $\xi _i$ and $\dot{\xi _i}$ are, respectively, the generalized coordinate and corresponding generalized momentum. Hence, the evolution equation corresponding to the variational parameter $a_1$ is

$$\begin{aligned} \frac{\partial a_1}{\partial t} = \left[ q^2\,\eta -g-\chi \left( 2A_0^2-3a_1^2-3a_2^2\right) \right] A_0^2\,a_2\,\sin (b_1+b_2). \end{aligned}$$

For the parameter $b_1$ ,the evolution equation reads

$$\begin{aligned} \frac{\partial b_1}{\partial t}= & {} -2kq-q^2-\frac{V_s}{2}-g\left( A_0^2+a_1^2+2a_2^2\right) \\&+q^2\,\eta \left( A_0^2+4a_2^2\right) -\chi \Big [2A_0^4+6A_0^2\left( a_1^2+2a_2^2\right) +a_1^4 \\&+3a_2^2\left( 2a_1^2+a_2^2\right) \Big ] +\left( q^2\,\eta -g\right) A_0^2\,\frac{a_2}{a_1}\,\cos (b_1+b_2). \end{aligned}$$

For the parameter $a_2$, we get

$$\begin{aligned} \frac{\partial a_2}{\partial t}= \left[ q^2\,\eta -g-\chi \left( 2A_0^2-3a_1^2-3a_2^2\right) \right] A_0^2\,a_1\,\sin (b_1+b_2). \end{aligned}$$

and for the parameter $b_2$, the evolution equation is

$$\begin{aligned} \frac{\partial b_2}{\partial t}= & {} -2kq-q^2-\frac{V_s}{2}-g\left( A_0^2+2a_1^2+a_2^2\right) \\&+q^2\,\eta \left( A_0^2+4a_1^2\right) -\chi \Big [2A_0^4+6A_0^2\left( 2a_1^2+a_2^2\right) +a_2^4 \\&+3a_1^2\left( a_1^2+2a_2^2\right) \Big ] +\left( q^2\,\eta -g\right) A_0^2\,\frac{a_1}{a_2}\,\cos (b_1+b_2). \end{aligned}$$

For simplicity, we may use a variant of the MI-motivated wavefunction (7) for which

$$\begin{aligned} a_1 = a_2 = a, \qquad \mathrm {and} \quad b_1+b_2=b. \end{aligned}$$

Then the coupled ordinary differential equations for a(t) and b(t) are shown (8) and (9)