Quantum enhanced estimation of diffusion

Momentum diffusion is a possible mechanism for driving macroscopic quantum systems towards classical behavior. Experimental tests of this hypothesis rely on a precise estimation of the strength of this diffusion. We show that quantum-mechanical squeezing offers significant improvements, including when measuring position. For instance, with $10\mathrm{dB}$ of mechanical squeezing, experiments would require a tenth of proposed free-fall times. Momentum measurement is better by an additional factor of three, while another quadrature is close to optimal. These have particular implications for the space-based MAQRO proposal—where it could rule out the spontaneous collapse theory due to Ghirardi, Rimini, and Weber—as well as terrestrial optomechanical sensing.

Figure 1

A pictorial representation of the measurement of wave-packet expansion which forms part of the MAQRO proposals [27, 28]. (a) A particle is initially trapped, (b) then released, (c) the free particle wave function expands, more rapidly with a localization term, and (d) the localization rate can be inferred through position measurements. Expansion as depicted in two spatial detections is for illustrative purposes; we analyze only one independent spatial dimension.

Figure 2

Precision of estimating momentum diffusion from wave-packet expansion for MAQRO parameters (Table 1). Dashed lines denote a squeezing of $10\mathrm{dB}$. The optimal homodyne and fundamental limit lines overlap until around $\mathrm{\Lambda }\sim {10}^{20}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. Three years' data collection with $t=100\mathrm{s}$ yields $\nu \sim {10}^6$ repetitions.

Figure 3

Minimum detectable collapse rate for three years of observation with a $100\mathrm{nm}$ radius sphere of mass $5.5\times {10}^9\mathrm{u}$, with other parameters as Table 1. The minimum required collapse rate given is based on the criteria of Ref. [7] to ensure macroscopic objects rapidly collapse to classical states. The magenta dot represents the values originally proposed by Ghirardi et al. [15].

Figure 4

Ratio of quantum Fisher information against classical Fisher information for optimal homodyne quadrature $[F(\mathrm{\Lambda };{\theta }_{\mathrm{opt}})/H\left(\mathrm{\Lambda }\right)]$, plotted for ${\kappa }_{\text{th}}=1$ and $r=0$. The red rectangle is representative of the MAQRO parameter regime.

Figure 5

Ratio of quantum Fisher information against classical Fisher information for heterodyne detection $\left[F\right(\mathrm{\Lambda })/H(\mathrm{\Lambda }\left)\right]$, plotted for ${\kappa }_{\text{th}}=1$ and $r=0$. The red rectangle is representative of the MAQRO parameter regime.

Figure 6

Ratio of classical Fisher information for heterodyne detection against classical Fisher information for homodyne detection of the optimal quadrature, plotted for ${\kappa }_{\text{th}}=1$ and $r=0$. The red rectangle is representative of the MAQRO parameter regime.

Figure 7

Bounds plotted for a $r_{\mathrm{s}}=100\mathrm{nm}$ sphere of mass $5.5\times {10}^9\mathrm{u}$ with values otherwise as Table 1. The minimum required collapse rate given is based on the criteria of Ref. [7]. The magenta dot represents the values originally proposed by Ghirardi et al. [15].

Figure 8

Bounds plotted for a $r_{\mathrm{s}}=100\mathrm{nm}$ sphere of mass $5.5\times {10}^9\mathrm{u}$ with values otherwise as Table 1. The minimum required collapse rate given is based on the criteria of Ref. [7]. The magenta dot represents the values originally proposed by Ghirardi et al. [15].