Engineering nonlinear boson-boson interactions using mediating spin systems

TL;DR

This paper introduces two deterministic protocols to synthesize nonlinear boson-boson interactions by mediating spins. By leveraging linear spin-boson couplings, spin drivings, a spin-spin coupling, and carefully chosen initial states and intermediate spin operations, it derives effective cross-Kerr and nonlinear beam-splitter Hamiltonians that enable entangled coherent states (ECS) and N00N/N00M states between two bosonic modes. Numerical simulations validate the approach, showing ECS with coherent amplitudes $|\alpha|$ up to around 3 and high-fidelity N00N/M states for modest photon numbers, while elucidating the trade-offs imposed by the Lamb-Dicke condition and higher-order terms. The results offer a flexible, platform-agnostic route to strong nonlinear bosonic interactions, with potential impact on quantum communication, metrology, and computation across superconducting, trapped-ion, and related quantum technologies.

Abstract

We present a protocol to create entangled coherent states by engineering cross-Kerr interactions between bosonic systems endowed with (externally driven) internal spin-like degrees of freedom. With slight modifications, the protocol is also able to produce N00N states through nonlinear beam splitter interactions. Each bosonic system interacts locally with its spin and by suitably tuning the model parameters, various classes of effective bosonic interaction Hamiltonians, mediated by the coupled spins, can be engineered. Our approach is benchmarked by numerical simulations aimed at studying the entanglement within a bosonic register and comparing it with the expected one resulting from the target Hamiltonians.

Figures (4)

• Figure 1: Schematic diagram showing the steps needed to derive the Hamiltonian in Eq. \ref{['eq:HamAfterApprox']}.
• Figure 2: (a)- (b): Comparison of the state evolved through $H_\mathrm{linear}$ and the ideal state resulting from the effective Hamiltonian $H_\mathrm{ideal}$ in Eq. (\ref{['eq:noVarphi']}). In these simulations we have used the parameters $\alpha_{1,2}= 0.5$, $\epsilon_{1,2}/\omega=0.2, \eta_{1,2}/\omega=0.1$ with $\omega$ the value taken by the bosonic frequency (identical for both subsystems). In panel (a) we report the logarithmic negativity, while the state fidelity is plotted in panel (b). (c)- (e): Entanglement dynamics quantified with logarithmic negativity for increasing values of the coherent-state amplitudes. We have taken $\alpha_{1,2}=2$ in panel (c), $2.5$ in (d) and, finally, $3$ in (e). The other parameter values are $\epsilon_{1,2}/\omega=0.5, \eta_{1,2}/\omega=0.05$, and $\lambda/\omega=10$.
• Figure 3: (a): Difference between the logarithmic negativity of the target N00M state and the time evolved state. For $n=m=1$, we take $\eta/\omega=0.25$ and $\epsilon = \lambda/6$. For $n=m=2$, we take $\eta/\omega=0.31$ and $\epsilon = 2\lambda/5$. For $n=1, m=2$, we take $\eta/\omega=0.35$ and $\epsilon = 5\lambda/34$. (b): Infidelity between the same states as in (a). (c): Density matrix elements for $n=m=2$ taking $\epsilon/\omega=1/750$. (d): Density matrix elements for $n=1, m=2$ taking $\epsilon/\omega=1/750$.
• Figure 4: Fidelity between the state evolved with $H_\mathrm{linear} (t)$ and the N00N state $\ket{n,0}+e^{i\theta}\ket{0,m}$. For each case, the driving strength is taken to be $\epsilon=1/750$.