Ground-State Cooling of Levitated Magnets in Low-Frequency Traps

We present a ground-state cooling scheme for the mechanical degrees of freedom of mesoscopic magnetic particles levitated in low-frequency traps. Our method makes use of a binary sensor and suitably shaped pulses to perform weak, adaptive measurements on the position of the magnet. This allows us to precisely determine the position and momentum of the particle, transforming the initial high-entropy thermal state into a pure coherent state. The energy is then extracted by shifting the trap center. By delegating the task of energy extraction to a coherent displacement operation, we overcome the limitations associated with cooling schemes that rely on the dissipation of a two-level system coupled to the oscillator. We numerically benchmark our protocol in realistic experimental conditions, including heating rates and imperfect readout fidelities, showing that it is well suited for magnetogravitational traps operating at cryogenic temperatures. Our results pave the way for ground-state cooling of micron-scale particles.

Figure 1

Setup sketch. A levitated magnetic particle of radius $R$ and magnetization $M$ follows a harmonic motion in the $Z$ direction with frequency ${\omega }_M$. A magnetically sensitive TLS (e.g., a NV center in diamond) of bare energy splitting ${\omega }_{\mathrm{TLS}}$ is placed in the axis of oscillation at a distance $r_0$ from the center of the harmonic potential. The energy splitting of the TLS is a function of the position of the magnet $r_z$. The position of the particle is initially described by a thermal distribution (indicated as the purple blurred area). The equilibrium position of the trap can be shifted.

Figure 2

Cooling scheme. The upper panels show the Wigner function of the particle from its initial state till the end of the second part of the protocol. The middle section shows the amplitude profile of the pulse in each step. The lower section shows the evolution of the Wigner function during a single step of the protocol and highlights the displacement that is induced by the pulse.

Figure 3

Numerical simulations. (a) GS fidelity $\mathcal{F}$ at the end of the protocol versus readout fidelity for different initial phonon numbers. (b) The same data as in (a) plotted as a function of the initial phonon number. These results were obtained by simulating the von Neumann equation of the closed system. (c) Time needed for part one of the protocol in units of inverse coupling $1/g$. (d) GS fidelity $\mathcal{F}$ as a function of the heating rate for different readout fidelities and an initial state with $\overline{n}=100$ that is in thermal equilibrium with the environment. The data were generated by simulating the Lindblad equation with ${\omega }_M=2\pi g/300$. The shaded areas mark the region with GS fidelity below 0.5, which for a thermal state corresponds to a mean phonon number larger than one. The GS fidelity of state $\rho$ is $\mathcal{F}=⟨0|\rho |0⟩$ [44].