Figure 1

Artistic sketch of the phonon maser model, where one spin is approaching the magnetic tip ($M_{\mathrm{tip}}$) at the time interval $\mathrm{\Delta }t$ and interacts longitudinally with a mechanical oscillator during the time $\tau$. The damping mechanisms to thermal environments are present for the oscillator at the rate $\kappa$ in units of the oscillator's frequency, ${\omega }_m$. By this scheme we want to evidence the two possible outputs that depend crucially on the type of spin measurement, used at an intermediate step of the micromaser model.

Figure 2

(Left panel) Time evolution of the average phonon number calculated: (i) using analytical expression Eq. (5), which converges to the steady-state ${\overline{n}}_{SS}=9.7$ (red solid line) and (ii) using numerical simulation of the ME (4) which converges to the steady-state ${\overline{n}}_{SS}\approx 9.3$ (dot-dashed). (Inset) Steady state phonon number vs pump $\propto r^2$, as in Eq. (5).

(Right panel) Probability distribution function evidencing the steady-state solution (green bars) and compared to the standard Poisson distribution for the same mean value (red line). (Inset) Wigner function plots and the thermal initial distribution. The model parameters are: ${\overline{n}}_0=0.1$, $\lambda =0.001$, $\kappa =0.01*\lambda$ and $\mathrm{\Delta }t=41*\tau$, for $\tau =\pi$.

Figure 3

Second-order coherence degree, evidencing the formation of coherent state when $g^{\left(2\right)}\left(0\right)\to 1$. Here we compare: (i) analytical expression Eq. (7) (red dashed) and (ii) numerical solution resulted from the ME (4) (black dot-dashed). Both curves fit perfectly. (Inset) Linewidth vs pump $\propto r^2$, as in Eq. (8).

Figure 4

(Left panel) Time evolution of the average phonon number calculated: (i) using heralded postselection in the state $\left|{\psi }_{\text{f}}\right.〉$ results in lasing with the steady-state ${\overline{n}}_{SS}=9.2$ (magenta line); (ii) using postselection in the same $\left|{\psi }_{\text{f}}\right.〉$ state but with failures, results in lasing with the steady-state ${\overline{n}}_{SS}=2.5$ (blue dashed line); (iii) using spin tracing, which gives no lasing (red dot-dashed line); (iv) using postselection or spin tracing with the initial state $|\downarrow \rangle$ or $|\uparrow \rangle$ result in lasing with the steady-state ${\overline{n}}_{SS}=4.9$ (green line-diamonds). For the first three cases the initial unnormalized state is $\left|{\psi }_{\text{i}}\right.〉=0.4|\uparrow \rangle +0.6|\downarrow \rangle$. (Right panel) Second-order coherence degree, evidencing the formation of coherent state when $g^{\left(2\right)}\left(0\right)\to 1$. The result for spin tracing operation, i.e., case (iii), is far from a coherent state. The model parameters here are: ${\overline{n}}_0=0.1$, $\lambda =0.001$, $\kappa =0.014*\lambda$ and $\mathrm{\Delta }t=41*\tau$, for $\tau =\pi$.

Figure 5

(a)–(c) Phonon number as function of the number of spins calculated for different postselection success probabilities ($P_S$ in legend) and several values of the steady-state phonon number, $n_{SS}^{\mathrm{eigen}}$, obtained by spin tracing operation for the initial eigenstate of ${\widehat{\sigma }}_z$. (d) The steady-state phonon number vs post-selection probability. The parameters here are: ${\overline{n}}_0=0.1$, $\lambda =0.06$, $\mathrm{\Delta }t=35*\tau$, $\tau =\pi$ and the preselected spin state for the postselection is $\left|{\psi }_{\text{i}}\right.〉=\left(\right|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}$.