Splitting bright matter-wave solitons on narrow potential barriers: Quantum to classical transition and applications to interferometry

We study bright solitons in the Gross-Pitaevskii equation as they are split and recombined in a low-energy system. We present analytic results determining the general region in which a soliton may not be split on a potential barrier and confirm these results numerically. Furthermore, we analyze the energetic regimes where quantum fluctuations in the initial center-of-mass position and momentum become influential on the outcome of soliton splitting and recombination events. We then use the results of this analysis to determine a parameter regime where soliton interferometry is practicable.

Figure 1

Numerical results of splitting a soliton traveling at velocity $v$ (a)–(c) or $v_0$ (d)–(f) at a Gaussian barrier of strength $q$ and width $\sigma =0.2$. (a),(d) Color maps of transmission as a function of $q$ and $v$ or $v_0$. The solid (red) curves are isolines of constant transmission $T_{+}$ obtained from the numerics, while the dashed (gray) curves are theoretical predictions of transmission $T_q^s$ in the linear case over the same range. (b),(e) Curves of transmission as a function of collisional velocity $v$ or $v_0$ for various barrier strengths $q$. The shaded (red) region shows energetically disallowed splitting events. (c),(f) Curves of transmission as a function of barrier strength $q$ for various values of $v$ or $v_0$. The labeled (red) curve, for which $v,v_0=0.25$ indicates the classical, untrapped lower-energy bound on the region where a continuous range of transmission is accessible.

Figure 2

Results of numerical integrations of the GPE illustrating the sensitivity of equal splitting to extreme quantum fluctuation for various particle numbers. The transmissions after extreme positive (negative) energy quantum fluctuations are displayed in panel (a) [panel (b)]. The number fluctuation measure $T_{6{\sigma }_{v_{\mathrm{b}}}}$ [Eq. (24)] is plotted in (c). For all plots we show $N=16$ ($black+$), 32 ($\times$), 64 ($\blacksquare$), and 128 ($•$).

Figure 3

Distributions of the transmission $T_{+}$ obtained from Monte Carlo simulations. Here we show results for a range of trap frequencies and particle numbers, giving a range of uncertainties in the initial c.m. position and momentum. In the range explored, we see that the effects of varying the trap frequency (and so kinetic energy) dominate the dynamics, with narrow Gaussians at high energy, but a bimodal structure arising at low energy when energetically disallowed states arise.

Figure 4

Results of Monte Carlo simulations. Here we show the dependence of transmission on $T_{+}$ on the collision velocity ($v_{\mathrm{b}}$) after quantum position and momentum fluctuations have been added to a base collision velocity ($v_0$). For each $v_0$ the barrier strength was set to ensure equal splitting in the limit of zero fluctuations. We see that in the low-energy regimes the transmission can be very sensitive to quantum fluctuations.

Figure 5

Results of Monte Carlo simulations. Shown here are the standard deviations associated with the final transmission distributions depicted in Fig. 3. We see a weak linear dependence on the sample velocity uncertainty ${\overline{s}}_{v_{\mathrm{b}}}$ for high $v_0$, which becomes stronger, but less linear, as we reduce the energy. This can be seen by the widening (shaded) $95\%$ confidence intervals of the linear fits.

Figure 6

(a) Diagram of a Mach-Zehnder interferometer utilizing a periodic confinement with two antipodal barriers. An example of the time evolution of the density for this configuration is displayed in (c). (b) Diagram of a Mach-Zehnder interferometer utilizing harmonic confinement and a single splitting barrier. Again, an example of the time evolution for such a configuration is displayed in (d).

Figure 7

Numerically calculated transmission rates after the second collision, $I_{+}$, for two Mach-Zehnder interferometry geometries. Color maps for the (b) toroidal Mach-Zehnder and (d) harmonic Mach-Zehnder cases show the full parameter space. Panels (a) and (c) show specific curves of constant $v$, $v_0$ for the same respective scenarios and highlight the transition from the high-energy sinusoidal dependence regime to the lower-energy quasilinear dependence regime.

Figure 8

Results of numerical integrations illustrating the sensitivity of interferometry to extreme quantum fluctuation for various particle numbers. The interferometry transmissions after extreme positive (negative)-energy quantum fluctuations are displayed in panel (a) [panel (b)]. The number fluctuation measure $I_{6{\sigma }_{v_{\mathrm{b}}}}$ [Eq. (33)] is plotted in (c). For all plots we show $N=16$ ($black+$), 32 ($\times$), 64 ($\blacksquare$), and 128 ($•$).

Figure 9

Distributions of the interferometry transmission $I_{+}$ obtained from Monte Carlo simulations. Here we show results for a range of trap frequencies and particle numbers, giving a range of uncertainties in the initial c.m. position and momentum. In the range explored, we see that the effects of varying the trap frequency (and so kinetic energy) dominate the dynamics, with narrow Gaussians at high energy, but a uniform structure arising at low energy when interferometry becomes impracticable.

Figure 10

Results of Monte Carlo simulations. Here we show the dependence of interferometry transmission on $I_{+}$ on the collision velocity ($v_{\mathrm{b}}$) after quantum position and momentum fluctuations have been added to a base collision velocity ($v_0$). For each $v_0$ the barrier strength was set to ensure equal splitting in the limit of zero fluctuations. We see that in the low-energy regimes the complex and velocity-sensitive structure of the transmission renders interferometry unworkable.

Figure 11

Results of Monte Carlo simulations. Here the standard deviation associated with the final interferometry distributions depicted in Fig. 9. We see a strong, weakly linear dependence on ${\overline{s}}_{v_{\mathrm{b}}}$ for high $v_0$, which becomes stronger, but less linear, as we reduce the energy. The variance saturates when the distribution becomes effectively uniform.