Detuned mechanical parametric amplification as a quantum non-demolition measurement

Here, we will instead take a value of the detuning $\Delta$ close to the instability threshold $\left| {{\Delta }_{\text{th}}} \right|=\sqrt{{{\gamma }^{2}}+{{\chi }^{2}}}$. In particular, one obtains extremely simple dynamics in the case where $\Delta =-\chi$, as the Hamiltonian takes the form

$$\begin{eqnarray}&&\tilde{H}\,=-\frac{\hbar \chi }{2}\left( {{\hat{X}\,}^{2}}+1 \right).\end{eqnarray} \tag{ 6 }$$

In this case, the squeezing and rotation operations conspire to produce simple coherent dynamics analogous to that of a free particle: similar to momentum, the $\hat{X}\,$ quadrature is a constant of the motion, while similar to position, $\hat{Y}\,$ grows at a rate determined by $\hat{X}\,$. That is, in the absence of external influences, $\left( \text{d}/\text{d}t \right)\hat{Y}\,=\chi \hat{X}\,$. It follows trivially that while the $\hat{X}\,$ quadrature is unaffected by the parametric driving, at long times (or low frequencies) the $\hat{Y}\,$ quadrature becomes an amplified version of $\hat{X}\,$. To make this more precise, we include mechanical dissipation in the standard way. With mechanical amplitude damping at rate γ, the quantum Langevin equations describing the mechanical resonator take the form

$$\begin{eqnarray}&&\left[ \begin{matrix} \text{d}\hat{X}\, \\ &&\text{d}\hat{Y}\, \\ \end{matrix} \right]=\left[ \begin{matrix} -\gamma & 0 \\ \chi & -\gamma \\ &&\end{matrix} \right]\left[ \begin{matrix} \hat{X}\, \\ &&\hat{Y}\, \\ &&\end{matrix} \right]\;\text{d}t+\sqrt{2\gamma }\left[ \begin{matrix} \text{d}{{\hat{X}\,}_{\text{in}}} \\ &&\text{d}{{\hat{Y}\,}_{\text{in}}} \\ &&\end{matrix} \right],\end{eqnarray} \tag{ 7 }$$

where ${{X}_{\text{in}}},{{Y}_{\text{in}}}$ describe the input noise from the mechanical bath. The above is easily solved in the frequency domain as

$$\begin{eqnarray}&&\hat{X}\,\left( \omega \right)={{\hat{X}\,}_{0}}\left( \omega \right),\quad \hat{Y}\,\left( \omega \right)={{\hat{Y}\,}_{0}}\left( \omega \right)+\frac{2\chi }{\gamma -i\omega }{{\hat{X}\,}_{0}}\left( \omega \right),\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{{\hat{X}\,}_{0}}\left( \omega \right)=\frac{{{\hat{X}\,}_{\text{in}}}}{\gamma -i\omega },\quad {{\hat{Y}\,}_{0}}\left( \omega \right)=\frac{{{\hat{Y}\,}_{\text{in}}}}{\gamma -i\omega },\end{eqnarray} \tag{ 9 }$$

are the mechanical quadratures when $\chi =0$; they simply correspond to the quadratures of a mechanical resonator in thermal equilibrium.

Equation (8) express the key idea underlying our DMPA-based backaction evasion scheme: for low frequencies and large $\chi /\gamma$, the detuned parametric drive causes $\hat{Y}\,$ to become an amplified version of $\hat{X}\,$, whereas $\hat{X}\,$ is completely unaffected by the parametric driving. The situation is reminiscent of a QND measurement: the mechanical $\hat{Y}\,$ quadrature 'measures' the $\hat{X}\,$ quadrature, without any backaction disturbance. The amplification induced by the detuned parametric driving also means that a standard continuous position measurement made using the cavity output effectively becomes a single-quadrature measurement of $\hat{X}\,$. We make this precise in what follows.

The next step of the analysis is to understand how a standard conditional position measurement [18] is modified by the effective amplification described above. As described in [19], the transformation into a rotating frame allows for simple equations of motion for the conditional state of the oscillator. As the mechanical Hamiltonian is quadratic and we are making a linear measurement, an initially Gaussian mechanical state will also remain Gaussian at all times. Hence, the conditional evolution equations reduce to equations for the means and covariance matrix (see appendix A). Letting ${{V}_{X}}\equiv \left\langle \left\langle {{\hat{X}\,}^{2}}\right\rangle \right\rangle$, ${{V}_{Y}}\equiv \left\langle \left\langle {{\hat{Y}\,}^{2}}\right\rangle \right\rangle$ and $C\equiv \left\langle \left\langle \left\{ \hat{X}\,,\hat{Y}\, \right\}/2\right\rangle \right\rangle$ denote the elements of the conditional covariance matrix, one finds (again taking $\Delta =-\chi$)

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}{{V}_{X}}=-2\gamma {{V}_{X}}+2\gamma \left( N+1/2+{{N}_{BA}} \right)-4\eta \mu \left( V_{X}^{2}+{{C}^{2}} \right)\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}{{V}_{Y}}=-2\gamma {{V}_{Y}}+4\chi C+2\gamma \left( N+1/2+{{N}_{BA}} \right)-4\eta \mu \left( V_{Y}^{2}+{{C}^{2}} \right)\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}C=-2\gamma C+2\chi {{V}_{X}}-4\eta \mu C\left( {{V}_{X}}+{{V}_{Y}} \right),\end{eqnarray} \tag{ 12 }$$

where N is the mean bath phonon number, μ is the measurement rate, and η is the efficiency of the measurement ($\eta =1$ corresponds to a quantum-limited continuous position measurement), while ${{N}_{BA}}=\mu /2\gamma$ describes the backaction heating of both quadratures from the continuous position measurement, parameterized as an additional mean phonon occupation. These equations have a simple interpretation: the χ terms correspond to the coherent dynamics due to the detuned parametric drive, whereas the nonlinear μ-dependent terms correspond to the conditioning terms due to the measurement (i.e. the measurement leads to an effective nonlinear damping of the variances). Throughout this paper, we assume a situation where the mechanical susceptibility is not modified by the measurement. In cavity optomechanics, this is achieved by driving the cavity on resonance rather than on the red or blue sidebands.

For comparison, the corresponding conditional evolution equations for a near-QND measurement of the mechanical X quadrature (via, for example, dual sideband driving) take the general form [13, 18]

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}{{V}_{X}}=-2\gamma {{V}_{X}}+2\gamma \left( N+1/2+{{N}_{\text{bad}}} \right)-4\eta \mu V_{X}^{2}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}{{V}_{Y}}=-2\gamma {{V}_{Y}}+2\gamma \left( N+1/2+{{N}_{\text{BA}}} \right)-4\eta \mu {{C}^{2}}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\frac{\text{d}}{\text{d}t}C=-2\gamma C-4\eta \mu {{V}_{X}}C,\end{eqnarray} \tag{ 15 }$$

where ${{N}_{\text{BA}}}$ is defined as above, and ${{N}_{\text{bad}}}<{{N}_{\text{BA}}}$ is the spurious backaction heating of the $\hat{X}\,$ quadrature due to imperfect QND measurement. Here, the measurement conditioning terms now reflect the fact that only the $\hat{X}\,$ quadrature is being measured. Further, in the ideal QND limit, the parameter ${{N}_{\text{bad}}}=0$, and there is no backaction heating of the measured $\hat{X}\,$ quadrature. Small deviations from the ideal QND limit result in a small amount of backaction heating of the X quadrature; we parameterize this (as in [13]) by ${{N}_{\text{bad}}}$. Note that one can easily verify from equation (14) that the stationary conditional state has C = 0.

To define the effective measurement strength, we return to equations (10)–(12) and solve for the stationary value of the covariance C. We obtain

$$\begin{eqnarray}&&C=\frac{\chi }{\gamma +2\eta \mu \left( {{V}_{X}}+{{V}_{Y}} \right)}{{V}_{X}}\equiv g{{V}_{X}}.\end{eqnarray} \tag{ 16 }$$

Unlike the BAE case, here the stationary value of C is non-zero. We now use this result to eliminate C from the steady-state equation of motion for ${{V}_{X}}$, obtaining

$$\begin{eqnarray}&&0=-2\gamma {{V}_{X}}+2\gamma \left( N+1/2+{{N}_{\text{bad},\text{eff}}} \right)-4\eta {{\mu }_{\text{eff}}}\left( {{V}_{X}},{{V}_{Y}} \right)V_{X}^{2},\end{eqnarray} \tag{ 17 }$$

where we have described the effective measurement strength:

$$\begin{eqnarray}&&{{\mu }_{\text{eff}}}\left( {{V}_{X}},{{V}_{Y}} \right)=\mu \left( 1+\frac{{{C}^{2}}}{V_{X}^{2}} \right)=\mu \left( 1+{{g}^{2}} \right),\end{eqnarray} \tag{ 18 }$$

and introduced an effective bad-cavity parameter

$$\begin{eqnarray}&&{{N}_{\text{bad},\text{eff}}}=\mu /\left( 2\gamma \right).\end{eqnarray} \tag{ 19 }$$

Comparing against equation (13) describing backaction evasion, we see that equation (17) is now of the same form. For a large measurement enhancement ($g\gg 1$), there is a strong similarity to a near-ideal QND measurement of the X quadrature in that the measurement conditioning parameter ${{\mu }_{\text{eff}}}$ is enhanced far above μ without a coinciding increase in the backaction heating ${{N}_{\text{bad},\text{eff}}}$, which is independent of g.

In the complete absence of measurement ($\mu =0$), the coherent amplification alone determines the covariance so that $g=C/{{V}_{X}}={{\chi }^{\prime }}$ (where from here onward ${{\chi }^{\prime }}$ denotes the dimensionless ratio $\chi /\gamma$, equalling unity at the self-oscillation threshold of a non-detuned parametric amplifier). As the measurement strength is increased, the ratio g is attenuated by the conditioning of the quadratures. This can be seen in equation (16), where even though increasing μ reduces ${{V}_{X}}$ and ${{V}_{Y}}$, the product $\mu \left( {{V}_{X}}+{{V}_{Y}} \right)$ is a monotonically increasing function of μ. This attenuation of g reflects the fact that the damping effect of the position measurement on the covariance counteracts the coherent amplification due to the parametric drive. In the limit of a perfect measurement ($\mu /\gamma \to \infty$), g approaches zero and the parametric drive becomes irrelevant.

A bandwidth picture provides a useful heuristic explanation for the form of equation (16), as follows. The measurement conditioning terms $\eta \mu {{V}_{X}}$ and $\eta \mu {{V}_{Y}}$ in this equation also appear in equations (10) and (11), where they may be understood as damping rates in addition to the intrinsic rate γ. Accordingly, the conditioning associated with a position measurement makes use of information gathered over time-scales $1/\left( \eta \mu {{V}_{X}} \right)$ and $1/\left( \eta \mu {{V}_{Y}} \right)$. However, the effective amplification dynamics are only significant on timescales longer than the mechanical decay time, given by $1/\gamma$. This is shown by equation (8), where the additional term in $\hat{Y}\,\left( \omega \right)$ decays for $|\omega |\gg \gamma$. Therefore, for sufficiently short measurement timescales the amplification is effectively frozen out and plays no role in the conditioning ⁴ . This explains the appearance of the rates $\eta \mu {{V}_{X}}$ and $\eta \mu {{V}_{Y}}$ as attenuating terms in the measurement gain given by equation (16).

Since equation (16) is an implicit equation, the net effect of the parametric drive and measurement in the regime where the measurement is significant $\left( 2\eta \mu \left( {{V}_{Y}}+{{V}_{X}} \right)\gg \gamma \right)$ is not immediately clear. For instance, increasing ${{\chi }^{\prime }}$ will further increase the amplified variance ${{V}_{Y}}$, while increasing the measurement strength μ will condition ${{V}_{Y}}$ to a smaller value. The situation simplifies in the case of a strong parametric drive ($\chi \prime \gg 1$), such that the squeezing is strong and ${{V}_{Y}}\gg {{V}_{X}}$. The net heating of ${{V}_{Y}}$ is then found by keeping only the $\mu {{V}_{Y}}$ term in the denominator of equation (16) so that

$$\begin{eqnarray}&&C\approx \chi {{V}_{X}}/2\eta \mu {{V}_{Y}},\end{eqnarray} \tag{ 20 }$$

and substituting this into equation (11). A cubic equation is then obtained for ${{V}_{Y}}$ in the steady-state, with the solution

$$\begin{eqnarray}&&{{V}_{Y}}\approx {{\left( \frac{\chi }{\eta \mu } \right)}^{2/3}}{{\left( \frac{{{V}_{X}}}{2} \right)}^{1/3}}.\end{eqnarray} \tag{ 21 }$$

Inserting this back into equation (20), and then into equation (10) using the steady-state condition gives an equation for ${{V}_{X}}$ that can be solved

$$\begin{eqnarray}&&0=-2\gamma {{V}_{X}}+2\gamma \left( N+1/2+{{N}_{BA}} \right)-4\eta \mu V_{X}^{2}-{{\left( 4{{\chi }^{2}}\eta \mu \right)}^{1/3}}V_{X}^{4/3}.\end{eqnarray} \tag{ 22 }$$

We can see that the extra conditioning term due to the covariance is now proportional to $V_{X}^{4/3}$. That is, in the regime where the measurement and parametric drive are both significant, the overall effect of the conditioning via the Y quadrature lies between that of damping (linear in ${{V}_{X}}$) and that of direct conditioning (quadratic in ${{V}_{X}}$).

In the strong driving limit, ${{V}_{X}}$ becomes small enough that the terms proportional to ${{V}_{X}}$ and $V_{X}^{2}$ in equation (22) can be neglected, yielding the simple solution

$$\begin{eqnarray}&&{{V}_{X}}\approx {{\left[ \frac{{{\left( 2N+2{{N}_{\text{BA}}}+1 \right)}^{3}}}{4\chi {{\prime }^{2}}\eta \mu /\gamma } \right]}^{1/4}}.\end{eqnarray} \tag{ 23 }$$

Since ${{N}_{\text{BA}}}$ is proportional to the measurement strength μ, there is clearly an optimum value of μ that minimizes ${{V}_{X}}$, located around where this backaction term becomes important. Differentiating to find the optimal measurement strength in this limit yields

$$\begin{eqnarray}&&{{\mu }_{\text{opt}}}\left( \chi \prime \to \infty \right)=\gamma \left( N+1/2 \right),\end{eqnarray} \tag{ 24 }$$

which corresponds to the backaction noise equalling half of the original noise in the oscillator. This trade-off between conditioning and backaction is in contrast to backaction evasion, where the conditional variance of the measured quadrature decreases monotonically with μ, even with spurious heating.

Substituting this optimal measurement strength back into equation (23) leaves

$$\begin{eqnarray}&&{{V}_{X}}\approx \frac{{{3}^{3/4}}}{2{{\eta }^{1/4}}}\sqrt{\frac{2N+1}{\chi \prime }},\end{eqnarray} \tag{ 25 }$$

Therefore, arbitrarily strong quantum squeezing is possible if $\chi \prime \gg 2N+1$. This can be compared with the variance obtained for backaction evasion in the strong measurement regime (where $\mu /\gamma \gg 1$)

$$\begin{eqnarray}&&{{V}_{X}}\approx \frac{1}{{{\eta }^{1/2}}}\sqrt{\frac{2N+1}{\mu /\gamma }}.\end{eqnarray} \tag{ 26 }$$

Notably, the DMPA scheme is clearly more suited to a sub-optimal efficiency η, consistent with previous numerical analysis [10]. This is especially relevant to nanomechanical systems, where even the best state-of-the-art optomechanical devices have loss factors of the order of 10% [20].

We can now easily see that setting μ to near the backaction-dominated regime allows the equivalent QND measurement strength ${{\mu }_{\text{eff}}}$ to be deeply within it. Measurement of the proxy $\hat{Y}\,$ quadrature can therefore be used to condition the $\hat{X}\,$ quadrature to below the level of the zero-point motion. This can be shown in the general case by using numerical solutions to equations (10)–(12). Figure 1 shows the minimum parametric drive strength required to achieve a fixed level of quantum squeezing using the optimum measurement rate ${{\mu }_{\text{opt}}}$, as well as the purity of the final conditional state. The required measurement strength and achievable purity for backaction evasion are shown for comparison. In the limit of strong squeezing, the parametric drive takes over the measurement's role in backaction evasion, while the optimal measurement strength approaches the constant given by equation (24) as expected. For low temperatures, this is currently an experimentally feasible parameter, with recent electromechanical [21] and optomechanical [14, 15] experiments demonstrating backaction noise exceeding zero-point and thermal fluctuations ($\mu >2{{\mu }_{\text{opt}}}$).

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With the measurement strength optimized, the squeezing is limited only by the normalized parametric drive strength $\chi \prime$. In this analysis, the rotating wave approximation forces the restriction $\chi \ll \prime Q$, where $Q={{\omega }_{m}}/\gamma$. Experimental limits on $\chi \prime$ are also set by the linear response range of the resonator, since the antisqueezed quadrature has variance exceeding the thermal variance by a factor of $1+\chi {{\prime }^{2}}$. Finally, the condition $\Delta =\pm \chi$ requires precise frequency control of both the resonator and parametric modulation to avoid the instability threshold $\left| {{\Delta }_{\text{th}}} \right|=\sqrt{{{\chi }^{2}}+{{\gamma }^{2}}}$, which becomes closer with increasing $\chi \prime$. Therefore, environmental influences on the oscillator frequency such as temperature fluctuations are detrimental in the strong driving regime, as is also the case for BAE protocols [22].

So far in this analysis, the parallels between DMPA and backaction evasion have been demonstrated for the dynamics and statistics of the $\hat{X}\,$ quadrature. It is interesting to note that these parallels do not extend to the orthogonal $\hat{Y}\,$ quadrature, which is amplified and conditioned in the DMPA scheme rather than heated. The variance of the $\hat{Y}\,$ quadrature is relevant to future quantum applications, many of which rely on a pure or almost-pure squeezed Gaussian state as a building block. These include the production of exotic non-classical states [23], entanglement between multiple oscillators [24] and continuous variable quantum computing [25]. To illustrate the difference between the two schemes considered, we compare the quantity $P=V_{g}^{2}/\left( {{V}_{X}}{{V}_{Y}}-{{C}^{2}} \right)$, which reaches a maximum value of one for a pure state.

For a BAE measurement, the purity can be obtained from the solutions of (13)–(15)

$$\begin{eqnarray}&&{{P}_{\text{BAE}}}=\frac{\eta }{1+\gamma \left( 2N+1 \right)/\mu }\frac{2}{\sqrt{1+4\eta \mu \left( 2N+2{{N}_{\text{bad}}}+1 \right)/\gamma }-1}.\end{eqnarray} \tag{ 27 }$$

In the ideal good cavity limit ${{N}_{\text{bad}}}=0$ and for a strong measurement $\eta \mu \prime \gg 2N+1$, the backaction causes a decrease in purity towards zero as μ is increased

$$\begin{eqnarray}&&{{P}_{\text{BAE}}}\left( \eta \mu /\gamma \gg 2N+1 \right)\approx \sqrt{\frac{\eta \gamma }{\mu \left( 2N+1 \right)}}.\end{eqnarray} \tag{ 28 }$$

In contrast to the above, the purity of the steady-state conditional state after applying a detuned parametric drive with the QND condition $|\Delta |=\chi$ and $\mu \ne 0$ can be derived from the general solutions of the variances in [11], and written as

$$\begin{eqnarray}&&{{P}_{\text{DMPA}}}=\frac{\eta }{1+\gamma \left( 2N+1 \right)/\mu }\left( 1+\frac{2}{\chi \prime /g-1} \right).\end{eqnarray} \tag{ 29 }$$

In the limit of weak measurement, this purity approaches a very small value due to the amplification of the Y quadrature. However, with an intermediate measurement strength, the conditioning of the Y quadrature allows for a higher purity than the equivalent BAE measurement. Since $\chi \prime /g-1$ is always positive, it is possible to assign a lower bound from the above that is independent of the parametric drive

$$\begin{eqnarray}&&{{P}_{\text{DMPA}}}>\frac{\eta }{1+\gamma \left( 2N+1 \right)/\mu }.\end{eqnarray} \tag{ 30 }$$

In the strong measurement limit this lower bound on the purity approaches η, in contrast to equation (27) where the purity approaches zero for BAE measurement. This difference is attributed to the fact that in the DMPA scheme, both quadratures are conditioned by the measurement. Therefore, even though the $\hat{Y}\,$ variance is amplified, this quadrature is still kept confined by a nonlinear conditioning term. In contrast, backaction evasion heats the unmeasured $\hat{Y}\,$ quadrature, causing ${{V}_{Y}}$ to increase linearly with $\mu \prime$.

If we consider the optimal measurement strength ${{\mu }_{\text{opt}}}$ that minimizes ${{V}_{X}}$ with a fixed parametric drive χ, the purity is reduced from the maximum of η. This purity is plotted in figure 1 for a squeezed $\hat{X}\,$ variance (i.e. ${{V}_{X}}<1/2$), where it is compared with the BAE case. It can be clearly seen that while the purity deteriorates as squeezing improves for backaction evasion, the DMPA purity approaches the lower bound of $\eta /3$ (since in this limit ${{\mu }_{\text{opt}}}/\gamma \to N+1/2$ and $\chi \prime \gg g$). Furthermore, a compromise can be made by increasing the measurement strength beyond the optimal level, reducing the strength of QND squeezing of the X quadrature in return for higher state purity. This preservation of purity in the strong squeezing limit is in stark contrast to conventional QND quadrature measurement of an oscillator and other methods for steady-state mechanical squeezing. One notable recent proposal using dissipative optomechanics results in purity scaling more favourably than for backaction evasion [26], however in this case the purity also degrades in the strong squeezing limit.

Some additional light can be shed on the parallel between DMPA and backaction evasion by quantifying the effective measurement enhancement ${{\mu }_{\text{eff}}}/\mu$. This was found to be equal to $\left( 1+{{\chi }^{\prime 2}} \right)$ in the limit of no measurement, and reduced to unity in the strong measurement limit. It is between these two limits, where weak measurement and strong parametric driving work in concert, that our scheme finds utility in QND measurement. This intermediate regime—described above for the limit of strong driving—will now be examined in detail. Making use of already derived exact solutions to equations (10)–(12) [11], we can explicitly find ${{\mu }_{\text{eff}}}$ in terms of experimental parameters. This also allows direct comparisons to be made with state-of-the art backaction evasion experiments.

The ratio ${{\mu }_{\text{eff}}}/\mu$, quantifying the ratio of conditioning measurement to backaction-inducing measurement, is given by (see appendix B)

$$\begin{eqnarray}&&\frac{{{\mu }_{\text{eff}}}}{\mu }=\frac{2\left( 1+\chi {{\prime }^{2}} \right)}{1+\sqrt{{{\left( 1+4\text{SNR} \right)}^{2}}+16\chi {{\prime }^{2}}\text{SNR}}-4\text{SNR}},\end{eqnarray} \tag{ 31 }$$

where $\text{SNR}=\eta \mu \left( 2N+2{{N}_{\text{BA}}}+1 \right)/\gamma$ defines the signal-to-noise ratio with which the combined thermal and back-action driven motion can be resolved over the measurement noise in the absence of driving. Since ${{N}_{\text{BA}}}\propto \mu$, the inclusion of backaction means that in the limit ${{N}_{\text{BA}}}\gg N+1/2$, the SNR becomes quadratic in μ rather than linear.

As SNR is increased, the effective measurement enhancement given by equation (31) passes through three regimes, as illustrated in figure 2 for three values of $\chi \prime$. For a strong drive ($\chi \prime \gg 1$), these regimes have simple, well-defined boundaries. The weak measurement limit, where the enhancement is maximized, ends when $\text{SNR}\approx \chi {{\prime }^{-2}}$. Beyond this is an intermediate region of non-zero but reduced gain, where the term $\chi {{\prime }^{2}}\text{SNR}$ is dominant in the denominator of equation (31). This corresponds to the amplified $\hat{Y}\,$ quadrature being well transduced above the measurement noise. Comparing to equation (16), this is also where the term $\mu {{V}_{Y}}$ becomes important and the effective measurement ceases to be dominated by the coherent parametric drive. Finally, when SNR exceeds $\chi {{\prime }^{2}}$, the direct measurement of the X quadrature is more efficient than the proxy measurement and ${{\mu }_{\text{eff}}}/\mu$ approaches 1.

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To utilize the full performance of the DMPA-based backaction evasion scheme, the effective measurement strength ${{\mu }_{\text{eff}}}$ must be large compared to γ, while the spurious heating ${{N}_{\text{BA}}}$ must be weak compared to the thermal noise. It is in the aforementioned intermediate regime that this occurs and the level of quantum squeezing is optimized. When $\chi \prime \gg 1$ this regime corresponds to an SNR of order unity, signifying that the thermal motion is barely transduced without the aid of the parametric drive. We can then simplify equation (31) to

$$\begin{eqnarray}&&\frac{{{\mu }_{\text{eff}}}}{\mu }\approx \frac{\chi \prime }{2\sqrt{\text{SNR}}}.\end{eqnarray} \tag{ 32 }$$

We see that in this intermediate regime, the enhancement is linear with $\chi \prime$, as is also seen in figure 2. This linear enhancement is in contrast to the weak measurement limit, where the enhancement scales as $\chi {{\prime }^{2}}$. Substituting the above expression into equation (17) and solving in the limit $\chi \prime \gg 1$, we get

$$\begin{eqnarray}&&{{V}_{X}}\approx \frac{\text{SN}{{\text{R}}^{3/4}}}{\sqrt{2\chi \prime }\eta \mu /\gamma },\end{eqnarray} \tag{ 33 }$$

in exact agreement with equation (23).