Momentum correlations of a few ultracold bosons escaping from an open well

The dynamical properties of a one-dimensional system of two and three bosons escaping from an open potential well are studied in terms of the momentum distributions of particles. In the case of a two-boson system, it is shown that the single- and two-particle momentum distributions undergo a specific transition as the interaction strength is tuned through the point where tunneling switches from the pair tunneling to the sequential one. Characteristic features in the momentum spectra can be used to quantitatively determine the participation of specific decay processes. A corresponding analysis is also performed for the three-boson system, showing a scheme for generalizations to higher particle numbers. For completeness, the time-dependent Tan contact of the system is also examined and its dynamics is found to undergo a similar transition. The results provide insight into the dynamics of decaying few-body systems and offer potential interest for experimental research.

Figure 1

The shape of the trapping potential at $t<0$ [$V_0\left(x\right)$, black dashed line] and after a sudden change at $t\ge 0$ ($V\left(x\right)$, solid red line). Energy and length are given in units of $\hslash {\mathrm{\Omega }}_0$ and $\sqrt{\hslash /m{\mathrm{\Omega }}_0}$, respectively.

Figure 2

The two-particle probability density ${\rho }_2(x_1,x_2)$ of the initial state for the two-particle system with zero interactions $g=0.0$ (top left), attractive interactions $g=-1.0$ (top right), and repulsive interactions $g=1.0$ (bottom left). Bottom right: The single-particle density profile ${\rho }_1\left(x\right)$ of the initial state for different interactions. It can be seen that repulsive interactions broaden the single-particle density profile and reduce the two-particle density along the $x_1=x_2$ diagonal, while attractive interactions narrow the single-particle density profile and concentrate two-particle density along the diagonal. Positions are in units of $\sqrt{\hslash /m{\mathrm{\Omega }}_0}$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$ two-particle probability density in units of $m{\mathrm{\Omega }}_0/\hslash$, single-particle density in units of $\sqrt{m{\mathrm{\Omega }}_0/\hslash }$.

Figure 3

The eigenenergies of the many-body Hamiltonian (1) after the well is opened at $t=0$, for $N=2$ bosons and different interaction strength $g$. The eigenstates can be divided into two groups: states which describe a pair of almost independent particles for all $g$ (red), and states which for $g>0$ describe independent particles but for $g<0$ describe bound boson pairs with energy strongly dependent on $g$ (blue). Energies are in units of $\hslash {\mathrm{\Omega }}_0$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$.

Figure 4

Time evolution of the two-particle probability density ${\rho }_2(x_1,x_2;t)$ of an initially trapped two-boson system for various interaction strengths $g$. The dashed lines demarcate the well boundary $x_B=3x_0$. For the noninteracting and repulsive systems ($g=0.0,g=1.0$), essentially the entire decay process takes place via sequential tunneling, while for sufficiently strong attractions ($g=-1.0$) the system decays almost solely via pair tunneling. Positions are in units of $\sqrt{\hslash /m{\mathrm{\Omega }}_0}$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$, time in units of $1/{\mathrm{\Omega }}_0$ two-particle probability density in units of $m{\mathrm{\Omega }}_0/\hslash$. Compare also to Fig. 2 in [30].

Figure 5

(a) The single-particle momentum distribution ${\pi }_1(k;t)$ and the two-particle momentum distribution ${\pi }_2(k_1,k_2;t)$ of the two-boson system for various interaction strengths $g$, at a specific moment $t=120$. Black dashed lines indicate predicted values of characteristic momenta calculated from the system energy (see text). In the noninteracting system ($g=0.0$) two bosons are emitted sequentially with identical momenta. In the repulsive system ($g=1.0$) two bosons are emitted sequentially with two different momenta. In a system with sufficiently strongly attractions ($g=-1.0$) the bosons are emitted as a bound pair with a well-defined center-of-mass momentum. For $g=-1.0$, the thin black line shows ${\pi }_1$ when Eqs. (4) are redefined to exclude the part of the wave function corresponding to the trapped particles. (b) The noise correlation $\mathcal{G}(k_1,k_2;t)$ for the two-boson system at $g=\pm 1.0$ and $t=120$. The correlations and anticorrelations that cannot be captured by a single-particle description become clearly visible. Momenta are in units of $\sqrt{\hslash m{\mathrm{\Omega }}_0}$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$, time in units of $1/{\mathrm{\Omega }}_0$ two-particle momentum distribution in units of $1/\left(m{\mathrm{\Omega }}_0\hslash \right)$, single-particle momentum distribution in units of $\sqrt{1/\left(m{\mathrm{\Omega }}_0\hslash \right)}$.

Figure 6

The center-of-mass momentum distributions ${\pi }_{\mathrm{CM}}(K;t)$ of the two-boson system for various values of $g$, at a specific moment $t=120$. For the repulsive ($g=1.0$) and sufficiently strongly attractive ($g=-1.0$) systems, only one decay process is available (sequential and pair tunneling, respectively) and it is reflected in the distribution as a single peak. For a system with weaker attractions ($g=-0.5$) both sequential and pair tunnelings are possible, and two peaks appear in the spectrum, each corresponding to a different process. Momenta are in units of $\sqrt{\hslash m{\mathrm{\Omega }}_0}$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$, time in units of $1/{\mathrm{\Omega }}_0$ momentum distribution in units of $\sqrt{1/\left(m{\mathrm{\Omega }}_0\hslash \right)}$.

Figure 7

The relative participation of pair and sequential tunneling in the overall dynamics of the two-boson system, for various interaction strengths $g$. Green and red symbols show the participation of pair tunneling and sequential tunneling, respectively, calculated from the areas of the corresponding peaks in the center-of-mass momentum distribution ${\pi }_{\mathrm{CM}}$ at $t=180$. For comparison, the corresponding results from Ref. [30] are shown as green dashed and red solid lines, respectively. It can be seen that sequential tunneling dominates in a wide range of interaction strengths, but its participation falls abruptly to zero as $g$ approaches the critical value $g=-0.9$. Interaction strength is given in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$.

Figure 8

Time evolution of Tan's contact relative to its initial value, $\mathcal{C}\left(t\right)/\mathcal{C}\left(0\right)$, for the two-boson system for various values of $g$. The magnitude of the contact is seen to decay exponentially within the shown time frame. Inset: The fitted decay rate of the exponential decay of $\mathcal{C}\left(t\right)/\mathcal{C}\left(0\right)$, as a function of $g$. Time is given in units of $1/{\mathrm{\Omega }}_0$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$, decay rate in units of ${\mathrm{\Omega }}_0$.

Figure 9

The single-particle momentum distribution ${\pi }_1(k;t)$ and the center-of-mass momentum distribution ${\pi }_{\mathrm{CM}}(K;t)$ of the three-boson system for two different interaction strengths, at specific moments $t$. In the repulsive case (top row) three bosons are emitted sequentially with well-defined momenta $k^{\prime \prime },k^{\prime },k_0$. In the attractive case (bottom row) the bosons can additionally tunnel as bound pairs with well-defined center-of-mass momenta $K$ or $K^{\prime }$, or as a trimer with center-of-mass momentum $P$. Each peak in the distributions can be associated with specific characteristic momenta, as indicated by arrows. Momenta are in units of $\sqrt{\hslash m{\mathrm{\Omega }}_0}$, interaction strength in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$, time in units of $1/{\mathrm{\Omega }}_0$ momentum distributions in units of $\sqrt{1/\left(m{\mathrm{\Omega }}_0\hslash \right)}$.

Figure 10

The relative participation of trimer tunneling in the overall dynamics of the three-boson system, for various interaction strengths $g$. Blue symbols show the participation of trimer tunneling calculated from the areas of the corresponding peaks in the center-of-mass momentum distribution ${\pi }_{\mathrm{CM}}$ at $t=140$. For comparison, the corresponding result from Ref. [30] is shown as a blue solid line. It can be seen that the participation of trimer tunneling remains near zero for approximately $g>-0.46$, increases throughout the region $-0.65<g<-0.46$, and becomes close to 1 for $g<-0.65$. Interaction strength is given in units of $\sqrt{{\hslash }^3{\mathrm{\Omega }}_0/m}$.