Multi-spatial-mode effects in squeezed-light-enhanced interferometric gravitational wave detectors

Proposed near-future upgrades of the current advanced interferometric gravitational wave detectors include the usage of frequency dependent squeezed light to reduce the current sensitivity-limiting quantum noise. We quantify and describe the degradation effects that spatial mode-mismatches between optical resonators have on the squeezed field. These mode-mismatches can to first order be described by scattering of light into second-order Gaussian modes. As a demonstration of principle, we also show that squeezing the second-order Hermite-Gaussian modes ${\mathrm{HG}}_{02}$ and ${\mathrm{HG}}_{20}$, in addition to the fundamental mode, has the potential to increase the robustness to spatial mode-mismatches. This scheme, however, requires independently optimised squeeze angles for each squeezed spatial mode, which would be challenging to realise in practise.

Figure 1

The field emitted by the squeezer is reflected off a filter cavity to produce frequency dependent squeezed states. These states are injected into the interferometer through the signal recycling mirror. There are three potentially different spatial eigenbases in this setup: $U_{\mathrm{FC}}$ for the filter cavity (yellow background), $U_{\mathrm{SQZ}}$ for the squeezer (green), and $U_{\mathrm{IFO}}$ for the interferometer (blue), where the background colors indicate which basis that is used where. All the coherent laser power is in the fundamental mode of the interferometer basis.

Figure 2

The figure shows the quantum-noise-limited sensitivity for various levels of mode-mismatch between the interferometer and the filter cavity. The squeezer is kept mode matched to the filter cavity. This type of mode-mismatch creates a broad-frequency band squeezing degradation similar to an optical loss.

Figure 3

In contrast to Fig. 2, the squeezer is kept mode matched to the interferometer. Around the resonance frequencies of the involved spatial modes, we experience squeezing degradation due to coherent mode-scattering.

Figure 4

The figure shows the improvement in dB that we obtain by squeezing the vacuum fluctuations that enters through the signal recycling cavity. The dashed traces indicates the improvement when squeezing 3 spatial modes, and the solid lines indicate the improvement when squeezing 1 spatial mode. This is shown for two different levels of mode-mismatch.

Figure 5

The figure shows the effect of mode-mismatching the interferometer to the squeezer and the filter cavity. We see a broad-frequency band decrease in improvement that is similar to an optical loss.

Figure 6

The figure shows the effect of mode-mismatching the interferometer to the filter cavity and the squeezer. Initially, the vacuum noise in the spatial mode $U_0$ is squeezed by 10 dB, while the arbitrary higher-order mode $U_n$ contains pure vacuum noise. The noise fields in the two spatial modes mix after being subjected to the frequency and mode dependent rotation of the squeeze angle when reflected off the filter cavity.

Figure 7

The figure shows the effect of mode-mismatching the filter cavity to the squeezer and the interferometer. The noise fields in the two spatial modes mix twice, with a frequency and mode dependent rotation of the squeeze angle in between due to the filter cavity. Since the squeezer and the interferometer are mode matched, the two mixing operations are the inverse of each other.

Figure 8

The figure shows the effect of mode-mismatching the filter cavity to the squeezer and the interferometer. Due to coherent mode-scattering, we see squeezing degradations around the resonance frequencies for the involved spatial modes.

Figure 9

The figure shows the effect of mode-mismatching the Squeezer to the FC and the Interferometer. The noise fields in the two spatial modes mix before being subjected to the frequency and mode dependent rotation of the squeeze angle when reflected off the FC.

Figure 10

The figure shows Mach-Zehnder interferometers used to mix two quantum fields. In the left figure a squeezed vacuum field is mixed with pure vacuum, and in the right figure two independent squeezed vacuum fields are mixed. The photo detector indicates that the upper output path is the one of interest.

Figure 11

The figure shows the “probability density” for obtaining a certain quantum noise reduction when mixing a squeezed vacuum with pure vacuum (blue) and when mixing two squeezed vacuum fields (red). The squeeze angles have, for both distributions, been optimized to minimize the noise.

Figure 12

The figure shows the improvement in quantum-noise-limited sensitivity over the non-squeezed case, both for a single squeezed spatial mode (solid lines) and for multiple squeezed spatial modes (dashed lines). The blue and red traces are for mode-mismatch levels between the interferometer and the filter cavity of 5% and 15%, respectively. The squeezer is kept mode matched to the filter cavity. The black dotted trace indicates the improvement when the filter cavity and the output mode cleaner are near perfectly mode matched.