Achieving Sub-Zeptonewton Force Sensitivity and Spin-Motion Entanglement in Levitated Diamond via Pulsed Backaction Evasion

Abstract

We propose a system to achieve sub-zeptonewton force sensing and robust spin-mechanical entanglement in a levitated diamond system. By coupling a Nitrogen-Vacancy (NV) center spin to the motion of its host diamond within a magnetic trap, we develop a platform designed to surpass the standard quantum limit. We develop and compare three distinct pulse sequences--Ramsey, Hahn echo, and Carr-Purcell-Meiboom-Gill (CPMG)--to create increasing amounts of backaction evasion while mitigating the effects of shot noise and thermal decoherence. Our results show that the CPMG sequences yield the most significant performance gains, reaching a force sensitivity of better than $10^{-23} \text{ N}/\sqrt{\text{Hz}}$ for broadband sensing around $10^4 \text{ Hz}$. Furthermore, we derive an entanglement witness protocol that accounts for pulsed dynamical decoupling, proving that spin-motion entanglement remains detectable even when occurring much faster than the mechanical period. These findings provide a more practical path for using levitated nanodiamonds both as high-precision sensors and as non-classical mechanical systems for fundamental tests of quantum mechanics.

Figures (7)

• Figure 1: (a) A diamond (yellow) with an NV center spin is levitated in a magnetic trap (magnets shown in gray). A green laser is used for initialization and readout. The photoluminescence emitted by the NV center is shown in red. Purple depicts the infrared (IR) laser; one of these is the feedback laser which is used to push the diamond and scatter light off the diamond, while a second is used to detect the motion of the diamond. The inset shows the three levels of the spin triplet state; the $m_{s}=-1$ and $m_{s}=+1$ state separate due to Zeeman splitting ("Z.S"). In our setup, we make use of only the $m_{s}=0$ and $m_{s}=+1$ states. (b) The operators under Heisenberg evolution. Here, $\zeta = \sqrt{1/2 m\omega}~\lambda$, where $\lambda = 2g/\omega$. $\phi_{\rm{ent}}$ is as given in Equation \ref{['eq:PhiEnt']} and the $p(t)$ and $q(t)$ plots describe Equations \ref{['eq:qHeis']} and \ref{['eq:pHeis']}.
• Figure 2: To implement backaction evasion, three pulse sequences, Ramsey, Echo, and Carr-Purcell are investigated. The $v,x$ plots in red and black show the differing levels of success; we see that Ramsey, Echo, and Carr-Purcell have increasingly smaller spin-dependent final mechanical states, leading to reduced backaction.
• Figure 3: (a) EW violation when the mass is cooled to its ground state at the start of the experiment and remains in contact with a continuous thermal bath. (b) EW violation when the mass starts in an initial thermal state and remains in contact with a continuous thermal bath. The plots get truncated when the violation becomes negative.
• Figure 4: Plots of sensitivity vs various parameters. Different colors indicate the various types of pulse sequences, while solid lines are for $\bar{n}/Q = 1$ and dashed lines are for $\bar{n}/Q = 10^{4}$ (as also shown in the plot legend). (a) 1 $\mu$m radius diamond, 100 Hz trap, $Q=10^{6}$, cooling time $t_c$ = 100 $\mu$s, cooling rate $\gamma_c =$ 1 kHz, and an expected signal frequency of 2 kHz. (b) Optimal g, fixed input of 2 kHz (c) Optimal g, fixed time of 100 $\mu$s.
• Figure 5: A visual representation of the resulting states that occur with pulse sequences, with and without a thermal bath. $\gamma$ refers to the parameters of an arbitrary coherent state $\ket{\gamma}$. Diagram is not to scale. (a) Depiction of Eq. \ref{['eq:pulselessState']}, where $\ket{-\alpha}$ (in grey) is translated by $\pm 2g/\omega$ (to pink). (b) Depiction of Eq. \ref{['eq:pulseState']}, where $\ket{\alpha}$ (in grey) is translated by $\pm (1/4)\omega g \tau^{2}$ (to purple).