Linear and quadratic reservoir engineering of non-Gaussian states

We study the dissipative preparation of pure non-Gaussian states of a target mode which is coupled both linearly and quadratically to an auxiliary damped mode. We show that any pure state achieved independently of the initial condition is either (i) a cubic phase state, namely, a state given by the action of a non-Gaussian (cubic) unitary on a squeezed vacuum or (ii) a (squeezed and displaced) finite superposition of Fock states. Which of the two states is realized depends on whether the transformation induced by the engineered reservoir on the target mode is canonical (i) or not (ii). We discuss how to prepare these states in an optomechanical cavity driven with multiple control lasers, by tuning the relative strengths and phases of the drives. Relevant examples in (ii) include the stabilization of mechanical Schrödinger-cat-like states or Fock-type states of any order. Our analysis is entirely analytical: it extends reservoir engineering to the non-Gaussian regime and enables the preparation of novel mechanical states with negative Wigner function.

Figure 1

Synoptic scheme of the paper. As shown on the left panel, we study the open dynamics of two interacting bosonic modes: the target ($\widehat{b}$) and the auxiliary ($\widehat{a}$) mode. The coupling has both a linear and a quadratic part in $\widehat{b}$ and the coefficients $c_j,j=1,...,5$, can be independently tuned. Depending on the choice of the initial state and the coefficients, either pure or mixed steady states can be stabilized. Among them, we focus on unconditionally pure states of mode $\widehat{b}$, namely, pure states that are reached asymptotically regardless of the initial condition. These states must be annihilated by the operator $\widehat{f}=\widehat{f}(b,b^{†})$ and there must be no conserved quantities. For the case of a linear transformation $\widehat{f}={\widehat{f}}_{\mathrm{lin}}$, unconditionally pure states coincide with (rotated) squeezed states, while there are no such states for a purely quadratic transformation $\widehat{f}={\widehat{f}}_{\mathrm{quad}}$; these cases are displayed in the dashed boxes. On the other hand, for the transformation $\widehat{f}={\widehat{f}}_{\mathrm{lin}}+{\widehat{f}}_{\mathrm{quad}}$ any unconditionally pure state belongs to either one of two distinct classes, depending on the choice of $c_j$. If the coefficients are such that $\widehat{f}$ is bosonic, then the steady state is a cubic phase state, namely, the state obtained by the action of a non-Gaussian (cubic) unitary on a squeezed vacuum; otherwise, it is written as a Gaussian unitary acting on a finite superposition of Fock states. All these states are non-Gaussian, and in particular nonclassical. Furthermore, two distinct families of unconditionally pure states can be found in (ii). Relevant examples are macroscopic quantum superposition states and states that can approximate with arbitrary precision any (displaced) Fock state.

Figure 2

Wigner function $W_{\gamma ,r}(x,p)$ of the state cubic phase state $\left|\gamma ,r\right.〉$ [Eq. (19)] for (a) cubicity $\gamma =0.1$ and squeezing $r=0.6$ ($\approx 5\mathrm{dB}$) and (b) $\gamma =0.4,r=0.8$ ($\approx 7\mathrm{dB}$).

Figure 3

Relevant examples of finite superpositions of Fock states. (a)–(c) Wigner function of the state $\left|{\mathcal{C}}_n^{\mathrm{\Phi }}\right.〉$ [Eq. (40)] for $n=1,3,6$; (d)–(f) Wigner function of the state $\left|{\mathcal{C}}_n^{\mathrm{\Psi }}\right.〉$ [Eq. (42)] for $n=1,3,6$; (g)–(i) Wigner function of the state $\left|{\mathcal{F}}_n^{\mathrm{\Psi }}\right.〉$ [Eq. (44)] with $n=5$ for $c_2/c_1=0.7,0.9,0.99$ ($\approx 7,13,23\mathrm{dB}$).

Figure 4

(a) Maximum fidelity between the macroscopic quantum superpositions $\left|{\mathcal{C}}_n^{\mathrm{\Phi }}\right.〉$ [Eq. (40)], $\left|{\mathcal{C}}_n^{\mathrm{\Psi }}\right.〉$ [Eq. (42)], and the Schrödinger-cat state $\left|{\mathcal{C}}_{\alpha }^{\pm }\right.〉\propto \left|\alpha \right.〉\pm \left|-\alpha \right.〉$ for different values of $n$; the fidelity is optimized over $\alpha$. (b) Fidelity between $\left|{\mathcal{F}}_n^{\mathrm{\Psi }}\right.〉$ [Eq. (44)] and the Fock state $\left|n\right.〉$ as a function of the amount of squeezing, from $n=1$ (blue) to $n=10$ (red).

Figure 5

Fidelity between the mechanical steady state in the presence of thermal noise and the ideal target state [${\gamma }_m=0$ in Eq. (47)] as a function of $\overline{n}$ and ${\gamma }_m$ (parametrized by the cooperativity $\mathcal{C}$). (a) Cubic phase state [Eq. (19)] with $\gamma =0.1,r=0.6$. (b) Displaced Fock-type state [see Eq. (44) and following paragraph] for $n=5,r=0.69$. (c), (d) Displaced and squeezed finite superposition of Fock states [see Eq. (42) and following paragraph] for $n=3$ (c) and $n=5$ (d).

Figure 6

Fidelity between the target state annihilated by Eq. (43) and the steady state obtained with perturbed couplings. The fidelity is plotted against relative errors ${\delta }_1$ (${\delta }_2$) in the coupling with the term ${\widehat{b}}^{†2}$ ($\{\widehat{b},{\widehat{b}}^{†}\}$) for $n=1$ (a) and $n=3$ (b). In the above plots, the shown contours correspond to fidelities of 0.95 (red, solid), 0.97 (yellow, dashed), and 0.99 (white, dotted dashed).

Figure 7

(a) Linear and quadratic optomechanical interaction between a mechanical resonator ($\widehat{b}$) and a cavity mode ($\widehat{a}$) [Eq. (48)], with the cavity coherently driven on multiple frequencies. (b) In a displaced frame and after discarding fast oscillating terms, the mechanical mode $\widehat{b}$ gets “dressed” by the linear and quadratic interaction (nonlinear operator $\widehat{f}$) and couples via a beam-splitter interaction to the cavity fluctuation $\widehat{d}$ [Eq. (53)], thus implementing the model of Eq. (1). The different contributions to $\widehat{f}$ are determined by the relative strengths and phases among the drives, as symbolized by the inner circle. (c) An optomechanical crystal implementation of a tunable linear and quadratic coupling. Photons of the confined photonic mode ${\widehat{a}}_{L,R}$ hop through a mechanical resonator ($\widehat{b}$) and hybridize into supermodes ${\omega }_{\pm }$ delocalized over the two slots. For a central beam equidistant from the two slots ($X_0=0$), a purely quadratic optomechanical coupling is realized, while with a tunable offset of the beam displacement $X_0$, arbitrary linear and quadratic admixtures can be achieved [Eqs. (67) and (68)]. (d) Hybridization of the right and left modes into supermodes and avoided crossing.