Mach-Zehnder interferometry with interacting trapped Bose-Einstein condensates

We theoretically analyze a Mach-Zehnder interferometer with trapped condensates and find that it is surprisingly stable against the nonlinearity induced by interparticle interactions. The phase sensitivity, which we study for number-squeezed input states, can overcome the shot noise limit and be increased up to the Heisenberg limit provided that a Bayesian or maximum-likelihood phase estimation strategy is used. We finally demonstrate the robustness of the Mach-Zehnder interferometer in the presence of interactions against condensate oscillations and a realistic atom-counting error.

Figure 1

Mach-Zehnder interferometer sequence for a finite phase $\theta$ (a) in the absence and (b), (c) in the presence of interactions, visualized on the Bloch sphere ($T_t=T_e=1$). (a) A number-squeezed initial state (small width along the $z$ axis, number-squeezing factor ${\xi }_N=0.2$) is transformed into a phase-squeezed one (small width along the equator) by a BS (rotation around the $x$ axis). Next a phase is accumulated due to an external potential (rotation around the $z$ axis). A second BS transforms the state such that the phase is mapped onto a number difference. (b) Even for very small interactions ($U_{0}N=0.1$), the number squeezing in the final state is lost. (c) For larger interactions ($U_{0}N=1$), the initial state (here a Fock state) gets strongly distorted and winds around the Bloch sphere.

Figure 2

(a) Scaling of $\sqrt{m}\Delta \theta$ with $N$ for $T_t=2$ (circles), $T_t=20$ (stars), and finite detection error $\pm 2$ (crosses, for $T_t=19$) and $\pm 5$ (diamonds, $T_t=21$) ($U_{0}N=1$ and $T_e$ fixed), compared to shot noise (dashed line) and the $U_0=0$ HL $\sqrt{m}\Delta \theta =1.4\sqrt{m}/N$ (dash-dotted line). (b) The prefactor $\beta$ (obtained from fitting) is shown for optimized $T_e<40$ (dark solid line), compared to $\sqrt{m}\Delta {\theta }_{\mathit{CRLB}}N$ (dash-dotted line) and $\sqrt{m}\Delta \theta N$ for nonoptimized values of $T_e<40$ (shaded solid line). Also results for a finite detection error of $\pm 2$ are shown (dashed line).

Figure 3

Probabilities $P\left(n\right|\theta )=|\langle n|{\psi }_{\mathit{OUT}}^{\left(\theta \right)}\rangle {|}^2$ for (a), (d), (g), and (j) binomial, (b), (e), (h), and (k) moderately number squeezed (${\xi }_N=0.2$), and (c), (f), (i), and (l) Fock states for $N=100$, $U_{0}N=1$, and $T_t=1$; (d)–(f) the ideal case (no interactions), (g)–(i) the interacting case for $T_e=1$, and (j)–(l) the interacting case for $T_e=10$. The states are also shown on the Bloch sphere. (a)–(c) $P\left(n\right|\theta =0)$.

Figure 4

(a) Phase sensitivity $\sqrt{m}\Delta \theta$ for different interaction strengths vs the initial number fluctuations ${\xi }_N(t=0)$ for $U_{0}N=1$ ($N=100$). The symbols show results for a simulated Bayesian phase estimation (circles, $T_e=1$; diamonds, $T_e=10$). (b) Number fluctuations after the first BS, ${\xi }_N(t=T_t)$, which determine the CRLB, Eq. (2).

Figure 5

$\Delta \theta$ for realistic control sequences calculated with MCTDHB method for $U_{0}N=0.1$ and a binomial state with $T_t=8$, and $U_{0}N=1$ and a highly number-squeezed state with ${\xi }_N(t=0)=0.05$ and $T_t=4$. The insets show the density (top), and the optimal control and tunnel coupling (bottom) for the squeezed state and $N=100$.