Multimode NOON-state generation with ultracold atoms via geodesic counterdiabatic driving

TL;DR

This paper tackles the challenge of rapidly generating multimode NOON states with ultracold atoms by combining geodesic driving with counterdiabatic corrections in a star-shaped Bose-Hubbard model. The authors derive a reduced two-level Hamiltonian in the self-trapping regime and obtain perturbative parameters that capture the essential dynamics, then construct a geodesic path in parameter space and a time-independent counterdiabatic term that saturates the quantum speed limit. They show exponential improvements in creation times for NOON states as the particle number grows and demonstrate metrological advantages via quantum Fisher information, including Heisenberg-limited phase sensitivity for 3-NOON states. The approach is made experimentally viable by mapping the CD contribution onto effective constant parameters using Floquet engineering or gauge fields, enabling scalable preparation of large entangled states with potential impact in quantum metrology and information processing.

Abstract

We present a protocol for the generation of NOON states with ultracold atoms, leveraging the Bose-Hubbard model in the self-trapping regime. By the means of an optimized adiabatic protocol, we achieve a significant reduction in the time required for the preparation of highly entangled NOON states, involving two or more modes. Our method saturates the quantum speed limit, ensuring both efficiency and high fidelity in state preparation. A detailed analysis of the geodesic counterdiabatic driving protocol and its application to the Bose-Hubbard system highlights its ability to expand the energy gap, facilitating faster adiabatic evolution. Through perturbation theory, we derive effective parameters that emulate the counterdiabatic Hamiltonian, enabling experimentally viable implementations with constant physical parameters. This approach is demonstrated to yield exponential time savings compared to standard geodesic driving, making it a powerful tool for creating complex entangled states for applications in quantum metrology and quantum information. Our findings pave the way for scalable and precise quantum state control in ultracold atomic systems.

Figures (10)

• Figure 1: Schematic representation of the star shaped Bose-Hubbard model for $L=6$ and a central initially populated well. The bosons can tunnel from the central well to any of the outer wells with a hopping rate $J$. Particles localized in the same well present an interaction $U$. Initially, all bosons are localized in the central well, whose energy $\varepsilon$ is isolated far from the rest of the spectrum. Over time, the energy of the central well is varied by a function $\varepsilon (t)$ to induce an adiabatic transfer of the system’s state to the desired coherent superposition $\vert \text{L-NOON}\rangle$. At the end of the protocol, the central well is again isolated from the rest of the spectrum, leaving only the entangled state across all $L$ wells.
• Figure 2: (a) Spectrum as a function of $NU/J$ for $N=10$, $L=3$ and $\varepsilon=0$. It is clearly possible to define a self-trapping regime, within which several effectively separable states in the energy spectrum can be identified. In these subsystems, the size of the Hilbert space is significantly reduced, allowing for the determination of a simplified Hamiltonian. The validity of the approximation can be understood as the overlap $F_{i} = \vert \langle \psi(t=0) \vert N;0,0,0\rangle\vert^{2}$, depicted in a red fade, between the initial state $\vert \psi(0)\rangle$ and the Fock state having all the bosons located in the central well. (b) Spectrum as a function of driving parameter $\varepsilon/J$ , with $N=10$, $L=3$ and $U=20J$. Once the self-trapping regime is set, the top energy levels are isolated and a reduced system can be defined. The validity of the approximation is verified in (c), where the eigenenergies of the reduced Hamiltonian are compared with the top energies of the full Bose-Hubbard Hamiltonian for several numbers of particles $N$, $L=3$ and $U=20J$ .
• Figure 3: Perturbative diagram of the energy corrections to the state $\vert N;0,0,0\rangle$, for $L=3$, at fourth order. Each path contributes to the state's energy in the manner of a Feynman path integral. Each arrow represents a possible transition path and carries a probabilistic amplitude proportional to $J/U$. The correction to the energy of the state $\vert N;0,0,0\rangle$ is given by the sum over all paths of length $l$—where $l$ corresponds to the desired perturbative order—that start at and return to this state. The number of such paths (24 in this example) is determined by the $l^{\text{th}}$ power of the adjacency matrix associated with the diagram.
• Figure 4: (a) Spectrum of the Bose-Hubbard Hamiltonian under geodesic driving (G, red curves) and geodesic counterdiabatic driving (GCD, blue curves) as a function of the parameter $\varepsilon /J$, for $U=20J$ and $N=4$. The states of interest, namely those where all particles are in the same level, are protected by a gap of value $U(N-1)$ from the rest of the spectrum, enabling the definition of a reduced system. A black rectangle highlights the avoided crossing region. (b) Zoom into the avoided crossing region. The gap is significantly widened due to the GCD protocol, allowing much faster adiabatic creation of triple-NOON states. (c) Population of the states $\vert N;0,0,0\rangle$ (thin lines) and $\vert \text{3-NOON} \rangle$ (thick lines) over time, for a total protocol time $T=10^{3}\hbar/J$. Population inversion is complete with the GCD protocol, whereas it does not even reach 10% with simple geodesic driving (G). (d) Various values of the time required to create a triple-NOON state with G and GCD driving for different numbers of particles $N$, with a fixed $U=20J$ and a fixed fidelity $F=0.99$. An exponential growth in creation time is observed with the G protocol, while it is effectively mitigated with the GCD protocol, achieving experimentally feasible creation times.
• Figure 5: Evolution of the total phase variance for the state $\vert \psi(\theta_{1},\theta_{2},\theta_{3})\rangle$ as a function of the population root of the $\vert \text{3-NOON}\rangle$ state for different particle numbers $N$. This graph highlights the minimum of $\vert \Delta \bm{\theta}\vert ^{2}$ predicted by Humphrey et al.PRLNOON at $\alpha = 1/\sqrt{d + \sqrt{d}}$. In the inset, minimal values of $\text{Tr}[F_{\mu\nu}^{-1}]$ are plotted as a function of the number of particles to confirms the Heisenberg scaling of $\sim 1/N^{2}$ for pure NOON states, which occurs when the GCD protocol is fully implemented