Prospects of charged-oscillator quantum-state generation with Rydberg atoms

We explore the possibility of engineering quantum states of a charged mechanical oscillator by coupling it to a stream of atoms in superpositions of high-lying Rydberg states. Our scheme relies on the driving of a two-phonon resonance within the oscillator by coupling it to an atomic two-photon transition. This approach effectuates a controllable open system dynamics on the oscillator that in principle permits versatile dissipative creation of squeezed and other nonclassical states which are central to sensing applications or for studies of fundamental questions concerning the boundary between classical and quantum-mechanical descriptions of macroscopic objects. We show that these features survive thermal coupling of the oscillator with the environment. We perform a detailed feasibility study finding that current state-of-the-art parameters result in atom-oscillator couplings which are too weak to efficiently implement the proposed oscillator state preparation protocol. Finally, we comment on ways to circumvent the present limitations.

Figure 1

(a) Setup of the system. Atoms pass one at a time above a micromechanical oscillator. An arm with charge $+Q$ oscillates vertically, while another arm with charge $-Q$ is fixed at position $z_{-Q}$. Atoms pass the oscillator at a rate $r$ (see text for details). (b) In the single-phonon process one deexcitation of the atom excites a single-phonon transition in the oscillator. (c) In the two-phonon process a two-photon transition in the atom via an intermediate manifold excites a two-phonon transition in the oscillator.

Figure 2

(a) Minimum variance $\mathrm{\Delta }{\chi }_{{\varphi }_{\min }}^2$ and (b) negative volume of the Wigner function $V_{\text{neg}}$ after $k=30$ atoms have passed, as a function of atomic excited state population ${\left|\beta \right|}^2$ and two-phonon integrated coupling strength $G_2$. (c)–(f) Wigner function $W$ of the oscillator state for different parameter choices $(G_2,|\beta {|}^2)$: (c) $(0.06,0.1)$, (d) $(0.2,0.2)$, (e) $(1,0.1)$, and (f) $(1,0.4)$.

Figure 3

(a) Negative volume $V_{\mathrm{neg}}$ of the Wigner function of the charged oscillator state and (b) minimum variance $\mathrm{\Delta }{\chi }_{{\varphi }_{\min }}^2$ after 30 atoms have passed as a function of relative thermal coupling strength ${\mathrm{\Gamma }}_{\text{m}}/r$ and thermal bath occupation ${\overline{n}}_{\mathrm{th}}$. Parameters $(G_2,|\beta {|}^2)$ used: (a) $(0.2,0.2)$, (b) $(0.06,0.1)$.

Figure 4

(a) Level scheme and transitions for a four-level manifold. (b) Coupling strength ${\gamma }_i$ for $i=x, y$, and $z$ in short-dashed, long-dashed, and solid lines, respectively, for an atom at position $\mathbf{R}=[A{Z}_0,0,Z_0]$. The coupling strength has been scaled to the maximum of ${\gamma }_z$.