Ultimate limits of exoplanet spectroscopy: A quantum approach

One of the big challenges in exoplanet science is to determine the atmospheric makeup of extrasolar planets and to find biosignatures that hint at the existence of biochemical processes on another world. The biomarkers we are trying to detect are gases in the exoplanet atmosphere such as oxygen or methane, which have deep absorption features in the visible and near-infrared spectrum. Here we establish the ultimate quantum limit for determining the presence or absence of a spectral absorption line, for a dim source in the presence of a much brighter stellar source. We characterize the associated error exponent in both the frameworks of symmetric and asymmetric hypothesis testing. We found that a structured measurement based on spatial demultiplexing allows us to decouple the light coming from the planet and achieve the ultimate quantum limits. If the planet has intensity $\epsilon \ll 1$ relative to the star, we show that this approach significantly outperforms direct spectroscopy yielding an improvement of the error exponent by a factor $1/\epsilon$. We find the optimal measurement, which is a combination of interferometric techniques and spectrum analysis.

Figure 1

(a) The spectrum of the star and planet under $H_0$: no absorption line is present. (b) Under $H_1$, an absorption line is present in the exoplanet's atmosphere.

Figure 2

Two most common physical implementations of exoplanet spectroscopy: (a) The transit method spectroscopy under $H_1$: exoplanet passes in front of their star, and an absorption line is imprinted onto the exoplanet's atmosphere. The relative intensity of the planet to the star is $\epsilon /(1-\epsilon )\ll 1$, and the absorption coefficient of the planet's atmosphere of the wavelengths of interest is $\eta$. (c) The star-planet system is spatially separated, and the relative intensity is $\epsilon /(1-\epsilon )\ll 1$. The absorption coefficient of the planet's atmosphere is $\eta$. In general the values of $\epsilon$ and $\eta$ can be different for the transit and spatially separated cases [32] because the stellar light goes through different thicknesses of the exoplanet's atmosphere.

Figure 3

(top panel) Error exponents for symmetric hypothesis testing, plotted vs the overlap ${\left|\kappa \right|}^2$. Green solid line shows the ultimate quantum limit, expressed by the quantum Chernoff exponent (QCE) (calculated numerically). Yellow dotted-dashed line shows the classical Chernoff exponent (CCE) for direct imaging (DI) for a Gaussian PSF, calculated numerically from Eq. (33). The red circles indicate the classical Chernoff exponent for a nearly optimal SPADE-assisted spectral measurement in Eqs. (42) and (43). (bottom panel) Error exponents for asymmetric hypothesis testing. Green solid line shows the ultimate quantum limit expressed by the quantum relative entropy (QRE). Yellow dotted-dashed line shows the classical relative entropy (CRE) of direct imaging for a Gaussian PSF, calculated numerically from Eq. (34). The red circles show the classical relative entropy for the nearly optimal SPADE-assisted spectral measurement in Eqs. (42) and (43). The parameters of the model are $\epsilon =0.01, W=1, \mathrm{\Delta }=0.2W, \eta =0.5$.

Figure 4

A multimode waveguide can be used as Hermite-Gaussian mode sorter [13, 14]. Light entering the device can be described as a combination of Hermite-Gaussian modes, each input mode is demultiplexed and sent to a separate mode, which is coupled into a spectrum analyzer. The star is aligned with the center of the waveguide. At the output, the light is put into spectrum analyzers where its frequency is measured.

Figure 5

The error exponent for asymmetric hypothesis testing, plotted vs the number of detected photons $n$, computed from Eq. (49) for $\alpha =0.05, \epsilon =0.05, \eta =0.5, W=1$, and $\mathrm{\Delta }=0.2W$ shown on a log-log scale. We show ${\left|\kappa \right|}^2=0.3$ (purple dotted-dashed line), ${\left|\kappa \right|}^2=0.4$ (teal solid line), ${\left|\kappa \right|}^2=0.7$ (blue dotted-dashed line), and ${\left|\kappa \right|}^2=1$ (red dashed line). The straight horizontal lines indicate the asymptotic behavior as $n\to \infty$, and the curved lines show ${\vartheta }_{\text{QSL}}$.