Probing macroscopic quantum states with a sub-Heisenberg accuracy

Significant achievements in high-sensitivity measurements will soon allow us to probe quantum behaviors of macroscopic mechanical oscillators. In a recent work [Phys. Rev. A 80, 043802 (2009)], we formulated a general framework for treating preparation of Gaussian quantum states of macroscopic oscillators through linear position measurements. To outline a complete procedure for testing macroscopic quantum mechanics, here we consider a subsequent verification stage which probes the prepared macroscopic quantum state and verifies the quantum dynamics. By adopting an optimal time-dependent homodyne detection in which the phase of the local oscillator varies in time, the conditional quantum state can be characterized below the Heisenberg limit, thereby achieving a quantum tomography. In the limiting case of no readout loss, such a scheme evades measurement-induced back action, which is identical to the variational-type measurement scheme invented by Vyatchanin et al. [JETP 77, 218 (1993)] but in the context for detecting gravitational waves. To motivate macroscopic quantum mechanics experiments with future gravitational-wave detectors, we mostly focus on the parameter regime where the characteristic measurement frequency is much higher than the oscillator frequency and the classical noises are Markovian, which captures the main feature of a broadband gravitational-wave detector. In addition, we discuss verifications of Einstein-Podolsky-Rosen-type entanglement between macroscopic test masses in future gravitational-wave detectors, which enables us to test one particular version of gravity decoherence conjectured by Diósi [Phys. Lett. A120, 377 (1987)] and Penrose [Gen. Rel. Grav. 28, 581 (1996)].

Figure 1

A schematic plot of a Wigner function $W(x,p)$ (left) and the corresponding uncertainty ellipse (right) for the covariance matrix ${\mathbf{V}}^{\mathrm{cond}}$ (which can be viewed as a projection of the Wigner function). The center of the plot is given by the conditional mean $(x^{\mathrm{cond}},p^{\mathrm{cond}})$. The Heisenberg limit is shown in a unit circle with radius given by the zero-point fluctuation $\sqrt{\hslash /(2m{\omega }_m)}$. For a pure Gaussian conditional quantum state, the area of the ellipse [i.e., $\pi det{\mathbf{V}}^{\mathrm{cond}}/(2m{\omega }_m){}^2$] is also equal to that of the Heisenberg limit. Therefore, the uncertainty product $det{\mathbf{V}}^{\mathrm{cond}}$ can serve as an appropriate figure of merit for quantifying purity of a quantum state.

Figure 2

A schematic plot of the random walk of the conditional quantum state (i.e., its Wigner function) in phase space. Its center is given by the conditional mean $[x^{\mathrm{cond}}(t),p^{\mathrm{cond}}(t)]$, with its uncertainty given by conditional variances $V_{\mathit{xx},\mathit{pp},\mathit{xp}}^{\mathrm{cond}}$. To verify the prepared conditional quantum state, the only knowledge that the verifier needs to know is classical information of the conditional mean provided by the preparer if the noises are Markovian.

Figure 3

A schematic plot of the uncertainty ellipses of reconstructed states with the same prepared Gaussian quantum state but different levels of verification accuracy, which shows the necessity of a sub-Heisenberg accuracy. The center of the plot is given by the conditional mean $(x^{\mathrm{cond}},p^{\mathrm{cond}})$. The shaded regimes correspond to the verification accuracy. The Heisenberg limit is shown by a unit circle. The dashed and solid ellipses represent the prepared state and the reconstructed states, respectively.

Figure 4

Values of reconstructed Wigner functions on the $p=0$ plane [i.e., $W_{\mathrm{recon}}(x,p=0)$] for a single-quantum state, obtained at different levels of verification accuracy. The solid curve shows the ideal case with no verification error. Dashed and dotted curves correspond to cases with verification errors of $1/4$ and $1/2$ of the Heisenberg limit, respectively. The negative regime (shaded) or the nonclassicality vanishes as the verification error increases. This again manifests the importance of a sub-Heisenberg verification accuracy.

Figure 5

A schematic plot of the system (upper left panel) and the corresponding space-time diagram (right panel) showing the timeline of the proposed MQM experiment (see Sec. 3a for detailed explanations). In this schematic plot, the oscillator position is denoted by $\stackrel{\wedge }{x},$ which is coupled to the optical fields through radiation pressure. The ingoing and outgoing optical fields are denoted by ${\stackrel{\wedge }{a}}_{1,2}$ and ${\stackrel{\wedge }{b}}_{1,2}$ with subscripts $1,2$ for the amplitude and phase quadratures, respectively. In the space-time diagram, the world line of the oscillator is shown by the middle vertical line. For clarity, ingoing and outgoing optical fields are represented by the left and right regions on the different sides of the oscillator world line, even though, in reality, optical fields escape from the same side as where they enter. We show light rays during preparation and verification stages in red and blue. In between, the yellow shaded region describes the evolution stage with the light turned off for a duration of ${\tau }_E$. The conditional variance of the oscillator motion is represented by the shaded region alongside the central vertical line (not drawn to the same scale as the light propagation). At the beginning of the preparation, the conditional variance is dominated by that of the initial state (orange). After a transient, it is determined by incoming radiations and measurements. Right after state preparation, we show the expected growth of the conditional variance due to thermal noise alone, and ignoring the effect of back-action noise, which is evaded during the verification process. The verification stage lasts for a duration of ${\tau }_V$, and it is shorter than ${\tau }_F$, after which the oscillator will be dominated by thermalization.

Figure 6

Rotation and diffusion of a highly position-squeezed conditional quantum state prepared by a strong measurement with ${\Omega }_q\gg {\omega }_m$. The initial-momentum uncertainty will contribute an uncertainty in the position comparable to the initial position uncertainty when the evolution duration ${\tau }_E~{\tau }_q$.

Figure 7

A schematic plot of time-dependent homodyne detection. The phase modulation of the local oscillator light varies in time.

Figure 8

Optimal filtering functions $g_1$ (solid curve) and $g_2$ (dashed curve) in the presence of Markovian noises. We have assumed ${\Omega }_q/2\pi =100$ Hz, ${\zeta }_x={\zeta }_F=0.2$, $\eta =0.01,$ and vacuum input ($q=0$). For clarity, the origin of the time axis has been shifted from ${\tau }_E$ to 0.

Figure 9

The uncertainty ellipse for the added verification noise in the presence of Markovian noises. We assume ${\zeta }_x={\zeta }_F=0.2$, vacuum input (dashed curve), ${\zeta }_x={\zeta }_F=0.2,$ and 10dB squeezing (dotted curve). For contrast, we also show the Heisenberg limit as a unit circle and the ideal conditional quantum state as a solid ellipse.

Figure 10

A schematic plot of advanced interferometric GW detectors for macroscopic entanglement between test masses as a test for gravity decoherence. For simplicity, we have not shown the setup at the bright port, which is identical to the dark port.

Figure 11

Logarithmic negative $E_{\mathcal{N}}$ as a function of the evolution duration ${\tau }_E$, which indicates how long the entanglement survives. The solid curve corresponds to the case where ${\Omega }_F/2\pi =20\mathrm{Hz}$ and the dashed curve is for ${\Omega }_F/2\pi =10\mathrm{Hz}$. To maximize the entanglement, the common mode is 10 dB phase squeezed at $t>{\tau }_E$ and $t<0$ while the differential mode is 10 dB amplitude squeezed at $t<0$ and switching to 10 dB phase squeezed at $t>{\tau }_E$.