Engineering multi-mode bosonic squeezed states using Monte-Carlo optimization

Abstract

Bosonic systems, such as photons and ultracold atoms, have played a central role in demonstrating quantum-enhanced sensing. Quantum entanglement, through squeezed and GHZ states, enables sensing beyond classical limits. However, such a quantum advantage has so far been confined to two-mode bosonic systems, as analogous multi-mode squeezed states are non-trivial to prepare. Here, we develop a Monte-Carlo based optimization technique which can be used to efficiently engineer a Hamiltonian control-sequence for multi-mode bosonic systems to prepare multi-mode squeezed states. Specifically, we consider a Bose-Einstein condensate in an optical lattice, relevant for applications in gravimetry and gradiometry, and demonstrate that metrologically useful squeezed states can be generated using the Bose-Hubbard Hamiltonian which includes on-site atomic interactions, tunable via Feshbach resonances. By analyzing the distribution (density) of the quantum Fisher information (QFI) over the Hilbert space, we identify a characteristic \textit{intermediate scaling} of the QFI: $\mathcal{O}(N^2 L+L^2 N)$, which lies between the standard quantum limit (SQL) and the Heisenberg limit (HL) for $N$ atoms in $L$ modes. We show that in general, within the Hilbert space there is a finite, $\mathcal{O}(1)$ measure subset of Hilbert space with an intermediate QFI scaling. Therefore, one can find a Hamiltonian control sequence using a Monte Carlo optimization over random control sequences, that produces a state with intermediate scaling of the QFI. We assume an experimentally accessible range of the control parameters in the Hamiltonian resources and show that the intermediate scaling can be readily achieved. Our results indicate that the HL can be approached in quantum gravimetry using realistic experimental parameters for systems with $L=\mathcal{O}(1)$ and $N\gg L$.

Figures (5)

• Figure 1: Squeezing in multi-mode bosons:a Atoms in a double well can be used in quantum sensing and their states are represented by the standard Bloch sphere. b Bosons in a lattice with $L$ sites can be considered as a spin-$\frac{L-1}{2}$ system. This system can be used to measure gravity along the axis of the lattice (see text). The corresponding Heisenberg limit in quantum sensing with this system is $\frac{1}{N(L-1)}$. c. Intermediate scaling (red line) between the SQL and the HL (see text). The left (right) panel shows the scaling for $L=2$ ($L=100$).
• Figure 2: Monte-Carlo optimization:a. QFI optimized using Monte-Carlo for $L= 3, 5$ and $N$ up to $200$. The markers correspond to the numerically optimized QFI using $\nu=10$ samples. The solid lines are the intermediate scaling, corresponding to $\mu$ in Eq. (\ref{['mean_std']}). b. Similar plot for various $L$. c. The expected error $\Delta \phi \sim 1/\sqrt{F}$ as a function of $N$ (data from part (a)). d. The optimal QFI obtained using Monte-Carlo, with $N=20$, $\nu=10$ for various evolution times $T$. The dashed lines show the intermediate scaling $\mu$ for the same cases. The states $\psi(T)$ are not sufficiently entangled for small $T$ resulting in this asymptotic behavior. The code used to produce the above data is described in sec. \ref{['sec:code_description']} of the supplementary information supplements.
• Figure 3: Possible experimental design: In contrast to a standard optical lattice formed out of standing waves, we propose a lattice formed out of repulsive barriers on an attractive optical dipole trap. The walls can be produced by painting a tweezer array, where the intensity of each wall and the separations can be tuned independently.
• Figure S1: Left: The pulse sequence that steers the system from a Fock state $\ket{4000}$ to a GHZ-like state. Right Top: The QFI of the state $\ket{\psi(t)}$ evolves in time. The initial QFI is zero since all bosons are in site 1 at $t=0$, and the final QFI is very close to the HL since the final state is a GHZ-like state. Right Bottom: The infidelity between the quantum state $\ket{\psi(t)}$ and the GHZ-like state. Note that here we optimize the parameters $J(t)$, $U_i(t)$ and $\Delta(t)$ are explicitly smooth in contrast to Eq. (10) in the main text, resulting in the smooth behavior of the QFI.
• Figure S2: Top: The initial and final population of two Fock states, $\ket{4000}$ and $\ket{0004}$. Since GHZ-like state is a superposition between the two Fock states, half of the population is in $\ket{4000}$ while the other half in $\ket{0004}$. Bottom: Amplitude of the initial and final density matrices $\rho_0 = \ket{\psi(0)}\bra{\psi(0)}, \rho_f = \ket{\psi(T)}\bra{\psi(T)}$. The Fock basis index ranges from 1 to 35, starting with Fock state $\ket{4000}$ and ending with $\ket{0004}$.