Quantum-state preparation and macroscopic entanglement in gravitational-wave detectors

Long-baseline laser-interferometer gravitational-wave (GW) detectors are operating at a factor of $\sim 10$ (in amplitude) above the standard quantum limit (SQL) within a broad frequency band (in the sense that $\Delta f\sim f$). Such a low-noise budget has already allowed the creation of a controlled 2.7 kg macroscopic oscillator with an effective eigenfrequency of 150 Hz and an occupation number of $\sim 200$. This result, along with the prospect for further improvements, heralds the possibility of experimentally probing macroscopic quantum mechanics (MQM)—quantum mechanical behavior of objects in the realm of everyday experience—using GW detectors. In this paper, we provide the mathematical foundation for the first step of a MQM experiment: the preparation of a macroscopic test mass into a nearly minimum-Heisenberg-limited Gaussian quantum state, which is possible if the interferometer’s classical noise beats the SQL in a broad frequency band. Our formalism, based on Wiener filtering, allows a straightforward conversion from the noise budget of a laser interferometer, in terms of noise spectra, into the strategy for quantum-state preparation and the quality of the prepared state. Using this formalism, we consider how Gaussian entanglement can be built among two macroscopic test masses and the performance of the planned Advanced LIGO interferometers in quantum-state preparation.

Figure 1

(Color online) Schematic plot of the planned Advanced LIGO [5] interferometer: a power- and a signal-recycled Michelson interferometer with cavities in the arms and a homodyne detection scheme at the dark port.

Figure 2

(Color online) Example noise spectral densities (in arbitrary units) of a Markovian measurement process observing a free mass: the quantum-noise spectral density at different values of the measurement frequency ${\Omega }_q$ as well as the spectral densities corresponding to a classical force noise and a sensing noise (marked in the plot). The gray-shadowed region marks the classical-noise SQL beating which we have chosen to have ${\Omega }_x/{\Omega }_F=5$.

Figure 3

(Color online) Test-mass squeezing normalized with respect to the conditional ground state of a free mass for (upper panel) optimal measurement frequency ${\Omega }_q={\Omega }_{\text{cl}}\equiv \sqrt{{\Omega }_F{\Omega }_x}$ and different homodyne detection angles (increasing from bright to dark color) and (lower panel) for different measurement frequencies (increasing from bright to dark color) at phase quadrature detection including in both cases a certain classical-noise budget: we have chosen ${\Omega }_x/{\Omega }_F=5$.

Figure 4

(Color online) Logarithmic negativity versus ${\Omega }_x/{\Omega }_F$ maximized with respect to ${\Omega }_q^{\text{c}}$ and ${\Omega }_q^{\text{d}}$ using phase quadrature detection (dashed red line) as well as additionally maximized with respect to ${\zeta }^{\text{c}}$ and ${\zeta }^{\text{d}}$ (solid purple line). In both cases no laser noise is assumed, i.e., with $S_{a_1{a}_1}=S_{a_2{a}_2}=1$. At some positions optimal parameter values for ${\Omega }_q^{\text{c}}$, ${\Omega }_q^{\text{d}}$, ${\zeta }^{\text{c}}$, and ${\zeta }^{\text{d}}$ are given in the plot.

Figure 5

(Color online) Test-mass uncertainty product $U$ (solid red line) compared to the entanglement between test mass and cavity mode (dashed blue line) both versus the dimensionless ratio ${\Omega }_q^{\text{cav}}/\gamma$ through which both quantities are totally described. No classical noise is present. Free mass limit is used, i.e., ${\gamma }_m={\omega }_m=0$, and the phase quadrature $\zeta =0$ is detected. Entanglement is quantified by the logarithmic negativity.

Figure 6

Contour plot of the normalized test-mass uncertainty product given by $U=2/\hslash \sqrt{V_{xx}{V}_{pp}-V_{xp}^2}$ versus ${\Omega }_q^{\text{cav}}/\gamma$ and ${\omega }_m/\gamma$ for quantum noise only. Again phase quadrature $\left(\zeta =0\right)$ is detected.

Figure 7

(Color online) Test-mass uncertainty product $U$ versus classical sensing noise and without force noise (upper panel) as well as versus classical force noise without classical sensing noise (lower panel) both for different examples of ratios between measurement frequency and bandwidth. Free mass limit is used, i.e., ${\omega }_m=0$, as well as phase quadrature detection $\zeta =0$.

Figure 8

Contour plot of the normalized test-mass uncertainty product given by $U=2/\hslash \sqrt{V_{xx}{V}_{pp}-V_{xp}^2}$ versus the ratio between detuning $\Delta$ and bandwidth and ${\Omega }_q^{\text{cav}}/\gamma$ for quantum noise only. Free mass limit is used, i.e., ${\omega }_m=0$, as well as phase quadrature detection $\zeta =0$.

Figure 9

(Color online) Upper panel: seismic noise pre-estimated by the simulation tool Bench [66] (red curve) and a fit by rational function (gray smooth line). Lower panel: straw man classical-noise budget of an advanced interferometric GW detector. Seismic (violet, follows $\sim 1/f^{1}0$), suspension thermal (blue, follows $\sim 1/f^{5/2}$), and internal thermal (green, follows $\sim 1/f^{1/2}$, dashed for different cutoff frequencies) noise spectra are shown as well as the total noise (black)—including Markovian quantum noise with phase quadrature readout. For the thermal noise sources we employed a Padé approximation that generated the $1/f$ spectra between 0.1 and 2 Hz.

Figure 10

(Color online) Uncertainty product of the conditional state versus sensing noise cutoff frequency for three different sensing noise levels. Force noise contributions are the same as in Fig. 9. The second-order moments formally diverge when downshifting the cutoff frequency.

Figure 11

(Color online) Upper panel: spectral densities of main noise sources present in the Advanced LIGO detector: seismic, suspension thermal, and internal thermal noise (as marked), as well as two examples of the quantum noise (magenta solid and purple dashed lines). Pre-estimated classical non-Markovian noise budget is fitted by rational functions with characteristic spectra as in Fig. 9. Lower panel: effective occupation number ${\mathcal{N}}_{\text{eff}}$ of conditional state (of the differential mirror mode) versus effective detuning $\lambda$ and effective bandwidth $ϵ$. The dots mark the Advanced LIGO broadband configuration state (magenta) and lowest occupation number state (purple, within the two contours) when $\zeta =0.7\pi$.

Figure 12

(Color online) Upper panel: spectral densities of main noise sources present in an improved Advanced LIGO detector. Lower panel: effective occupation number ${\mathcal{N}}_{\text{eff}}$ of the differential mirror mode’s conditional state versus effective detuning $\lambda$ and effective bandwidth $ϵ$ for phase quadrature detection. The dot marks the lowest occupation number state.