Real-time dynamics of an impurity in an ideal Bose gas in a trap

We investigate the behavior of a harmonically trapped system consisting of an impurity in a dilute ideal Bose gas after the boson-impurity interaction is suddenly switched on. As a theoretical framework, we use a field-theory approach in the space-time domain within the $T$-matrix approximation. We establish the form of the corresponding $T$ matrix and address the dynamical properties of the system. As a numerical application, we consider a simple system of a weakly interacting impurity in one dimension where the interaction leads to oscillations of the impurity density. Moreover, we show that the amplitude of the oscillations can be driven by periodically switching the interaction on and off.

Figure 1

Diagrams representing the main equations of the ladder approximation discussed in the text. (a) Equation (8) for the impurity Green's function. (b) Equation (9) for the in-medium $T$ matrix. Solid thick (thin) lines stand for the impurity Green's function, $G_I$ ($G_I^0$). The filled boxes represent the in-medium $T$ matrix, $\mathrm{\Gamma }$. The wavy lines denote the interaction potential $V$. The zigzag lines depicts the condensate particles, and the dashed line is the free Green's function for a boson depleted from the condensate, $G_B^0$.

Figure 2

Density of the impurity, ${\left|I(x,t)\right|}^2$, as a function of time $t$ and position coordinate $x$. The interparticle interaction is assumed to be constant in time, $gN=0.3$. $t$ is in units of $1/\mathrm{\Omega }, x$ is in units of $\sqrt{\hslash /\left(m_I\mathrm{\Omega }\right)}$, and ${\left|I(x,t)\right|}^2$ is in units of $\sqrt{\left(m_I\mathrm{\Omega }\right)/\hslash }$.

Figure 3

Density profiles ${\left|I(x,t)\right|}^2$ as a function of the coordinate $x$ taken at different times: $t=0$, solid blue line; $t=\pi /2$, dashed orange line; $t=9\pi /2$, dot-dashed green line; $t=5\pi$, dotted red line. Inset: The interparticle interaction which is a periodic function of time with period $\pi$ and peak amplitude $gN=0.3$. $t$ is in units of $1/\mathrm{\Omega }, x$ is in units of $\sqrt{\hslash /\left(m_I\mathrm{\Omega }\right)}, {\left|I(x,t)\right|}^2$ is in units of $\sqrt{\left(m_I\mathrm{\Omega }\right)/\hslash }$, and $g$ is in units of $\hslash \mathrm{\Omega }\sqrt{\hslash /\left(m_I\mathrm{\Omega }\right)}$.

Figure 4

Density profiles ${\left|I(x,t)\right|}^2$ as a function of the coordinate $x$ for different times $t=k\pi /2$ with $k\in \{0,1,2,...,19\}$. $t$ is in units of $1/\mathrm{\Omega }, x$ is in units of $\sqrt{\hslash /\left(m_I\mathrm{\Omega }\right)}$, and ${\left|I(x,t)\right|}^2$ is in units of $\sqrt{\left(m_I\mathrm{\Omega }\right)/\hslash }$.

Figure 5

The overlap $S_i\left(t\right)$ from Eq. (41) as a function of time $t$. The blue circles correspond to $i=0$, while the yellow triangles and green squares show the cases $i=2$ and $i=4$, respectively. $t$ is in units of $1/\mathrm{\Omega }$.