Hybrid Systems for the Generation of Nonclassical Mechanical States via Quadratic Interactions

We present a method to implement two-phonon interactions between mechanical resonators and spin qubits in hybrid setups, and show that these systems can be applied for the generation of nonclassical mechanical states even in the presence of dissipation. In particular, we demonstrate that the implementation of a two-phonon Jaynes-Cummings Hamiltonian under coherent driving of the qubit yields a dissipative phase transition with similarities to the one predicted in the model of the degenerate parametric oscillator: beyond a certain threshold in the driving amplitude, the driven-dissipative system sustains a mixed steady state consisting of a “jumping cat,” i.e., a cat state undergoing random jumps between two phases. We consider realistic setups and show that, in samples within reach of current technology, the system features nonclassical transient states, characterized by a negative Wigner function, that persist during timescales of fractions of a second.

Figure 1

(a) Proposed configuration: an NV center in a diamond film (red) is placed on top of a mechanical oscillator (e.g., a thin membrane) and placed between two magnets with aligned magnetization. Dimensions have been altered for visual clarity. (b) Magnetic field lines, computed for two Dy magnetized cylinders of 30 nm diameter separated by a gap of 30 nm. Red (blue)-shaded areas mark the regions where $\partial B_{z(x)}/\partial z\approx 0$. Both first derivatives are zero at the position of the NV center—represented by a circle—when the oscillator is at rest. (c) Magnetic field along the $z$ axis for $x=0$.

Figure 2

(a) Two-photon coupling rate versus zero-point fluctuation amplitude $z_{\mathrm{zpf}}$ of the oscillator, for two Dy magnets separated by 30 nm. (b) Two-photon coupling rate versus the magnet separation for three different magnetic materials, for a resonator with $z_{\mathrm{zpf}}=200\mathrm{fm}$. (c) Cooperativity versus the oscillator quality factor $Q$ and the zero-point fluctuations, for a magnet separation of 30 nm, oscillator frequency ${\omega }_m/(2\pi )=1.8\mathrm{MHz}$, temperature $T=10\mathrm{mK}$, and pure-dephasing rate ${\gamma }_z/(2\pi )=10\mathrm{Hz}$. The white line $C=1$ marks the onset of quantum effects. Point A corresponds to a feasible point for state-of-the-art technology at mK temperatures [88], with $z_{\mathrm{zpf}}=43\mathrm{fm}$ and $Q=4.2\times {10}^9$; B is the point taken in most part of the text for clarity of results: $z_{\mathrm{zpf}}=200\mathrm{fm}$ and $Q=4.2\times {10}^8$

Figure 3

Signatures of a dissipative phase transition in the steady state. (a) Phonon number and fluctuations of the position operator versus the driving amplitude $\mathrm{\Omega }$. (b) Wigner function for three values of the driving amplitude. Here, $g_2/(2\pi )=5\mathrm{Hz}$, $Q=4.2\times {10}^8$, ${\omega }_m/(2\pi )=1.8\mathrm{MHz}$, $T=10\mathrm{mK}$, ${\gamma }_z/(2\pi )=10\mathrm{Hz}$.

Figure 4

Dynamics of the density matrix and of a single quantum trajectory. (a) Phonon population versus time. Vertical lines depict times when a phonon emission process takes place. (b) Wigner function of the mechanical mode given by the density matrix (upper) and the wave function of a single trajectory (lower), at the times marked with arrows on the top of panel (a). The last two columns of the bottom row depict a jumping cat, at times before and after a phonon emission process. Parameters are those of Fig. 3.

Figure 5

Cattiness $\mathcal{C}$ versus time for an initial vacuum state. (a) Parameters are those of Fig. 3, with ${\gamma }_z/(2\pi )=10$ (solid, blue), 20 (dashed, yellow), and 50 Hz (dotted, red). Inset: Wigner function at the time of maximum $\mathcal{C}$, $t\approx 51\mathrm{ms}$. (b) $\mathrm{\Omega }/(2\pi )=3.18\mathrm{Hz}$ and parameters of point A in Fig. 2, consistent with Ref. [88] at $T=10\mathrm{mK}$: $z_{\mathrm{zpf}}=43\mathrm{fm}$ ($g_2\approx 0.23\mathrm{Hz}$), $Q=4.2\times {10}^9$ (estimated), and ${\gamma }_z/(2\pi )=1\mathrm{Hz}$, giving $C\approx 4$.