Integrable Atomtronic Interferometry

High sensitivity quantum interferometry requires more than just access to entangled states. It is achieved through the deep understanding of quantum correlations in a system. Integrable models offer the framework to develop this understanding. We communicate the design of interferometric protocols for an integrable model that describes the interaction of bosons in a four-site configuration. Analytic formulas for the quantum dynamics of certain observables are computed. These expose the system’s functionality as both an interferometric identifier, and producer, of NOON states. Being equivalent to a controlled-phase gate acting on 2 hybrid qudits, this system also highlights an equivalence between Heisenberg-limited interferometry and quantum information. These results are expected to open new avenues for integrability-enhanced atomtronic technologies.

Figure 1

Schematic representation of the interferometric circuit with tunneling between nearest neighbors. An initial state is prepared with $M$ particles in site 1, and $P$ particles in a (generally entangled) state across sites 2 and 4. After Hamiltonian time-evolution, measurement of particle number at site 3 is used to deduce information about the initial, or postmeasurement, state across sites 2 and 4.

Figure 2

Time evolution of expected fractional imbalance $⟨N_1-N_3⟩/M$ (dot points) for the Hamiltonian [Eq. (1)] as a function of time $t$ in units of seconds, with $U/J\simeq 1.2$, $U/\hslash \simeq 2\pi \times 19.5\mathrm{Hz}$, $J/\hslash \simeq 2\pi \times 16.2\mathrm{Hz}$, and different initial states: (a) $|4,0,0,0⟩$, (b) $|4,11,0,0⟩$, (c)–(d) $(|4,11,0,0⟩+\exp (i\varphi )|4,0,0,11⟩)/\sqrt{2}$ with $\varphi =0$ (c) and $\varphi =\pi$ (d). The top panels display agreement with the formula (3) (solid lines), while the bottom panels are in agreement with Eq. (5) (solid lines).

Figure 3

Normalized linear entropy and NOON correlation function. The red (green) line depicts $\tilde{\mathcal{E}}[{\rho }_3(t)]$ [$C_{1,3}(t)$] calculated with the effective Hamiltonian $H_{\mathrm{eff}}$ of Eq. (2), while the dots illustrate the numerical values obtained with the Hamiltonian [Eq. (1)]. The initial state is $|{\mathrm{\Psi }}_0⟩=|4,11,0,0⟩$, and the Hamiltonian parameters are $U/\hslash =2\pi \times 19.5\mathrm{Hz}$ and $J/\hslash =2\pi \times 16.2\mathrm{Hz}$.

Figure 4

Dependence of $⟨N_1-N_3⟩/M$ as a function of time $t$ (in units of seconds) and phase $\phi$, for initial state [Eq. (4)] with $M=4$, $P=11$, $\varphi =P\phi$, and $U/J\simeq 1.2$. Upper surface: the colors range from light to dark blue, indicating lower and higher values for the imbalance fraction. The green color represents the region where $⟨N_1⟩\approx ⟨N_3⟩$. Lower plane: the effect on the system’s dynamics is highlighted, specifically for the limiting cases $\phi =0$ and $\phi =\pi /P$, where it is seen that there is a minimum-maximum inversion at $\phi =\pi /(2P)$.