Quantum interactions between a laser interferometer and gravitational waves

LIGO’s detection of gravitational waves marks a first step in measurable effects of general relativity on quantum matter. In their current operation, laser interferometer gravitational-wave detectors are already quantum limited at high frequencies, and planned upgrades aim to decrease the noise floor to the quantum level over a wider bandwidth. This raises the interesting idea of what a gravitational-wave detector—or, more generally, an optomechanical system—may reveal about gravity beyond detecting gravitational waves from highly energetic astrophysical events, such as its quantum versus classical nature. In this paper we develop a quantum treatment of gravitational waves and their interactions with the detector. We show that the treatment recovers known equations of motion in the classical limit for gravity, and we apply our formulation to study the system dynamics, with a particular focus on the implications of gravity quantization. Our framework can also be extended to study alternate theories of gravity and the ways in which their features manifest themselves in a quantum optomechanical system.

Figure 1

Schematic of a second-generation laser interferometer gravitational-wave detector. (a) The full Michelson interferometer in its current configuration with power- and signal-recycling mirrors (PRM and SRM) and the two Fabry-Perot arm cavities. Here $L$ denotes the length of the arm cavity, and $l_{\mathrm{SRC}}$ denotes the length of the signal-recycling cavity (shown here not to scale). The arm cavities’ input mirror (ITM) has transmissivity $T$, and its end mirror (ETM) is perfectly reflective with $R=1$. For low frequencies $\mathrm{\Omega }$ of the GW wave such that $\mathrm{\Omega }L/c\ll 1$, and for $T\ll 1$, $l_{\mathrm{SRC}}\ll L$, the quantum inputs and outputs of the schematic in (a) can be mapped to those of a single one-dimensional Fabry-Perot cavity as shown in (b).

Figure 2

Representations of the GW interaction in the Newtonian versus TT gauges. In the Newtonian gauge, the gravitational wave exerts a tidal force $F_{\mathrm{GW}}$ so that the test-mass position is driven by both radiation pressure and gravitational-wave forces. In contrast, in the TT gauge, the GW interacts directly with the optical cavity mode, and the test-mass position is driven by the radiation pressure force alone. The two pictures are physically equivalent descriptions of the dynamics of a cavity whose mirrors fluctuate about their geodesic due to radiation pressure. In the presence of an incoming GW, the geodesics of the two mirrors deviate, and the time delay for a photon entering to be reflected back changes. In the Newtonian gauge, the change in time delay is reflected in the test-mass coordinate, while in the TT gauge this effect is directly accounted for by a phase shift in the cavity mode. The TT gauge viewpoint allows for a canonical description of the interaction.

Figure 3

Illustration of quantum coherent backaction effects on the cavity mode due to GW interaction in the presence of detuning. The GWs generated by ${\widehat{\alpha }}_1$ acts back on ${\widehat{\alpha }}_2$ in such a way that causes the field to beat coherently with itself. The above shows the case for red-detuning where $\mathrm{\Delta }>0$. The solid red line represents the cavity mode in the absence of backaction, while the dotted red line represents the contribution due to GW backaction. This effect is quantified by changes to the cavity’s effective damping and detuning rate so that $\tilde{\gamma }=\gamma +{\epsilon }_{\mathrm{GW}}\mathrm{\Delta }/2$ and $\tilde{\mathrm{\Delta }}=\mathrm{\Delta }-{\epsilon }_{\mathrm{GW}}\gamma /2$.