Quantum Magnetomechanics: Ultrahigh-$Q$-Levitated Mechanical Oscillators

Engineering nanomechanical quantum systems possessing ultralong motional coherence times allows for applications in precision quantum sensing and quantum interfaces, but to achieve ultrahigh motional $Q$ one must work hard to remove all forms of motional noise and heating. We examine a magneto-meso-mechanical quantum system that consists of a 3D arrangement of miniature superconducting loops which is stably levitated in a static inhomogeneous magnetic field. The motional decoherence is predominantly due to loss from induced eddy currents in the magnetized sphere which provides the trapping field ultimately yielding $Q\sim {10}^9$ with motional oscillation frequencies of several hundreds of kilohertz. By inductively coupling this levitating object to a nearby driven flux qubit one can cool its motion very close to the ground state and this may permit the generation of macroscopic entangled motional states of multiple clusters.

Figure 1

Cluster of three insulated superconducting loops levitating in a magnetic field generated by a magnetized sphere with magnetization vector along ${\widehat{x}}_1$. In the laboratory frame the axes are labeled $\{{\widehat{x}}_j{\}}_{j=1}^3$ with angular coordinates ${\alpha }_j$ around each. The reference system has the center of the sphere as origin. We depict the magnetic vector field generated by the spherical magnet and the nearby flux qubit (which we take to be a 3-junction phase qubit), sitting under the cluster on a yellow substrate.

Figure 2

Cooling performance. Final resonator phonon occupation number $n_f$ as a function of the initial occupation number $N_{\mathrm{th}}$ for cooling via dissipation. The phonon number $N_{\mathrm{th}}$ corresponds to the fixed temperature $T_r$ of the resonator bath. The dot-dashed lines represent the low temperature limit $n_{\mathrm{LD}}$. When $n_f$ is below the identity dotted line, some cooling is achieved. The system parameters for the light gray (red) curves are $({\omega }_r,\lambda ,{\Gamma }_{\perp })/2\pi =(6\times {10}^5,1.6\times {10}^3,{10}^5)\mathrm{Hz}$ and $(\Omega ,\delta ,T_q)=(0.5{\omega }_r,-\sqrt{{\omega }_r^2-{\Omega }^2},10\mathrm{mK})$. The dark gray (blue) curves represents a case where the bath is at higher temperature but the decay rate is smaller: $({\omega }_r,\lambda ,{\Gamma }_{\perp })/2\pi =(6\times {10}^5,1.6\times {10}^3,{10}^4)\mathrm{Hz}$ and $(\Omega ,\delta ,T_q)=(0.5{\omega }_r,-\sqrt{{\omega }_r^2-{\Omega }^2},100\mathrm{mK})$. The excess dephasing has been set to satisfy ${\Gamma }_{||}={\Gamma }_{\perp }$. Inset: Resonator-qubit coupling constant $\lambda$ as a function of the distance $d$ between the qubit and the center of the cluster of loops. For large distances the coupling scales as $d^{-3}$ following its linear dependency on the mutual inductance between the horizontal resonator loop and the qubit loop.