Negative-Mass Effects in Spin-Orbit Coupled Bose-Einstein Condensates

Negative effective masses can be realized by engineering the dispersion relation of a variety of quantum systems. A recent experiment with spin-orbit coupled Bose-Einstein condensates has shown that a negative effective mass can halt the free expansion of the condensate and lead to fringes in the density [M. A. Khamehchi et al., Phys. Rev. Lett. 118, 155301 (2017)]. Here, we show that the underlying cause of these observations is the self-interference of the wave packet that arises when only one of the two effective mass parameters that characterize the dispersion of the system is negative. We show that spin-orbit coupled Bose-Einstein condensates may access regimes where both mass parameters controlling the propagation and diffusion of the condensate are negative, which leads to the novel phenomenon of counterpropagating self-interfering packets.

Figure 1

(a) Schematic of the experimental configuration. The BEC initially resides at the bottom of the lower branch shaped by the spin-orbit coupling (see energy level diagram). The trap is then released in the $x$ dimension to let the BEC expand. (b) Parameter space for $m_1$, defined by the momentum range $|k_3–k_4|$ for which $m_1$ is negative. Blue dots: configurations considered in [18]. Green dots: configurations explored here. (c) SOCBEC dispersion properties for $\mathrm{\Omega }=2.5$, $\delta =1.36$, $k_R=1$ (from Ref. [18]). Blue line: lower branch $E_{-}$. Green line: effective mass $m_1$. Purple line: effective mass $m_2$. Red dashed line: group velocity $v$. The lower branch presents a region with $m_2<0$, where $v$ decreases with increasing $k$. (d) As for (c), but $\mathrm{\Omega }=1$, $\delta =0.7$, $k_R=1$, $E_{-}$ has regions with both $m_1<0$ and $m_2<0$, and $v$ has an opposite sign to the momentum. (i–iii) Examples of wave packet propagation, the colored stars indicate where $E_{-}$ is excited.

Figure 2

BEC expansion with the SOC dispersion of Fig. 1. The first column shows the linear case with $g=0$ and the second column the interacting case. (a,b) Spacetime evolution of $|\psi (x,t){|}^2$, with a condensate initial size of (a) $0.25\mu \mathrm{m}$ and (b) $5\mu \mathrm{m}$ Thomas-Fermi radius, obtained from the ground state of a $10^4$ particles BEC released from a 100 Hz trap. Insets: density at $t=35\mathrm{ms}$. (c,d) Wavelet decomposition of $\psi (x)$ at $t=35\mathrm{ms}$. Vertical blue lines: limits of the diffusion cone in $x$. Purple horizontal lines: position of the inflection points in $k$. Red dashed curve: classical displacement $d(k)$ of each $k$-wave vector. An animation (video S1) of this figure is provided in the Supplemental Material [19].

Figure 3

BEC expansion with the SOC dispersion of Fig. 1 with both $m_1<0$ and $m_2<0$. The first column shows the noninteracting case with $g=0$ and the second column the interacting case. (a,b) Spacetime evolution of $|\psi (x,t){|}^2$, with a condensate initial size of (a) $0.25\mu \mathrm{m}$ and (b) $5\mu \mathrm{m}$ Thomas-Fermi radius, obtained from the ground state of a $\approx 2.5\times {10}^4$ particles BEC released from a 100 Hz trap. Insets: density at $t=35\mathrm{ms}$. (c,d) Wavelet decomposition of $\psi (x)$ at $t=35\mathrm{ms}$. The color code is the same as Figs. 2 and 2 with additional horizontal green lines delimiting the $m_1<0$ region. An animation (video S2) of this figure is provided the Supplemental Material [19].