Accurate measurement of a 96% input coupling into a cavity using polarization tomography
Pillar microcavities are excellent light-matter interfaces providing an electromagnetic confinement in small mode volumes with high quality factors. They also allow the efficient injection and extraction of photons, into and from the cavity, with potentially near-unity input and output-coupling efficiencies. Optimizing the input and output coupling is essential, in particular, in the development of solid-state quantum networks where artificial atoms are manipulated with single incoming photons. Here we propose a technique to accurately measure input and output coupling efficiencies using polarization tomography of the light reflected by the cavity. We use the residual birefringence of pillar microcavities to distinguish the light coupled to the cavity from the uncoupled light: the former participates to rotating the polarization of the reflected beam, while the latter decreases the polarization purity. Applying this technique to a micropillar cavity, we measure a output coupling and a input coupling with unprecedented precision.
To enhance the light-matter coupling, optical microcavities Vahala (2003) have been used to confine the electromagnetic field locally in small mode volumes. Among other solid-state cavities (photonic crystals Painter et al. (1999), microdisks Michler et al. (2000); Gayral et al. (1999)) micropillars Pelton et al. (2002) are excellent candidates for light matter interfacing: they provide high quality factors for small mode volumes, potentially low optical losses and high input and output coupling efficiencies. Loo et al. (2010); Arnold et al. (2014); Schlehahn et al. (2016); ? These structures have already commercial applications thanks to VCSELs Wiersig and Hentschel (2006) which are used, for instance, for optical fibre data transmission and laser reading/writing beams in DVD players. They are also used as efficient exciton-photon interfaces (such as polaritons Gutbrod et al. (1998); Bajoni et al. (2008)) and photon-phonon interfaces for high frequency phonons Fainstein et al. (2013); Anguiano et al. (2017).
A major potential application of micropillar devices lies in the development of a future quantum photonic network, which requires highly efficient interfaces between photons and artificial atoms. In this framework, a single photon should ideally couple and deterministically interact with a single artificial atom such as a semiconductor quantum dot. In the context of cavity QED with quantum dots, these devices have already allowed enhancing the spontaneous emission into the cavity modeGérard et al. (1998) and achieving the exciton-photon strong coupling regime Reithmaier et al. (2004). They allow the efficient extraction of indistinguishable single photons Ates et al. (2009); Somaschi et al. (2016); Ding et al. (2016) for quantum optics applications and they can potentially be used as a spin-photon interface for quantum computing Young et al. (2011). They have recently been used to filter a single photon from an optical pulse, thanks to an optical non linearity at the single photon level Loo et al. (2012); De Santis et al. (2017). This required in particular an excellent input coupling for the incoming photons Rakher et al. (2009), significantly better than the state of the art in photonic crystals.
To perform as efficient interfaces, micropillar cavities must not only confine the electromagnetic field, but also allow the efficient injection/collection of every photon into and from the micropillar cavity, via a careful optical alignment. In order to obtain an optimal input coupling, a free space gaussian beam must be matched both spatially and spectrally with the confined cavity field. However, the experimental measurement of this mode matching is evaluated by analyzing the far-field spatial profile of the input and cavity mode with an uncertainty higher than Loo et al. (2012).
In this Letter, we present a technique to measure accurately the input and output coupling efficiencies of a micropillar cavity. We reconstruct the polarization density matrix of the light reflected by the micropillar cavity using polarization tomography measurements Antón et al. (2017). This allows distinguishing between a pure polarization state and a general mixed polarization, and, as a consequence, identifying the respective contribution of the light coupled and uncoupled to the cavity. A record precision is obtained in the measurement of a high input coupling at and of a moderate output coupling (defined as the probability for a cavity photon to escape through the top mirror): .
In this experiment, we use a pillar microcavity Nowak et al. (2014); Somaschi et al. (2016) (see Fig. 1(a) ) consisting in a -GaAs cavity, positioned between two distributed Bragg reflectors, with 20 (30) pairs of alternating quarter-wavelength thick layers for the top (bottom) mirror. The difference in refractive index between GaAs and vacuum leads to a lateral confinement between the four ridges of the cavity in the central region, hereafter denoted micropillar.
The experimental setup is presented in Fig. 1(b): the pillar microcavity is kept inside a helium gas exchange cryostat at approximately 10 K. The cavity is excited by a tunable continuous wave laser with 1 MHz spectral linewidth. The polarisation of the incident light is set with a polarizer and waveplates controlling an adjustable polarization state . The spatial shape of the free space incoming beam is controlled with a beam adapter in order to focus it into the micropillar surface. Therefore, we can adapt the coupling efficiency of the external field to the cavity mode. Given a total incident light intensity , the intensity that couples to the micropillar cavity mode is ; the intensity corresponding to uncoupled light is . In the following, we denote the reflected intensity associated to light coupled into the mode, and the reflected intensity associated to light that was not coupled into this mode, as illustrated in Fig. 1(a). The coupled light is reflected with reflectivity coefficient denoted , so that . The uncoupled light is entirely reflected as long as the focused incident beam is smaller than the micropillar surface, so that . It will be the case in the following experiments.
To perform a complete polarization tomography as also illustrated in Fig. 1(b), the reflected beam is separated in two orthogonally-polarized components in various polarization bases, using calibrated waveplates and a Wollaston polarizing prism. The input and output field intensities are measured with avalanche photodiodes. By adjusting the waveplates of the polarization analyzer, we measure the intensities and in the horizontal/vertical polarization basis, the intensities and in the diagonal/anti-diagonal polarization basis, and and in the right-handed/left-handed circular polarization basis. For a given set of orthogonally polarized intensities , we define the corresponding Stokes component as . This method Antón et al. (2017) allows measuring the density matrix of the polarization state, and representing it in the Poincaré sphere as a vector with coordinates of , and , ranging between and . The norm of the Poincaré vector is the purity of the polarization density matrix, equal to 1 for a pure polarization state. Thus we are able to reconstruct the polarization density matrix of the reflected light and represent it in the Poincaré sphere.
The fundamental mode energy of the cavity is ; it is splitted due to a small geometrical ellipticity, leading to linearly polarized horizontal (H) and vertical (V) modes. We first excite the device with horizontally (respectively vertically) polarized light (). The blue (red) curve in Fig. 1(c) displays the total reflectivity as a function of the laser-cavity detuning, evidencing a cavity mode splitting .With this notation, the resonance energies are given by and for the horizontally-polarized and vertically-polarized cavity modes, respectively. In this figure as in the following ones, the experimental points are compared to a theoretical fit that will be described later on. The reflectivity curve presents a Lorentzian reflectivity dip with linewidth ( ). We can also see that the Lorentzian dip does not reach zero reflectivity value, as would be obtained for a perfect input coupling and a top mirror output coupling of . Indeed, the reflectivity coefficient for each mode is ?:
So the reflectivity of the mode is where is the cavity mode energy and the cavity linewidth: it can go down to zero at if . In general, however, the total reflectivity takes into account both coupled and uncoupled reflected light: . As the reflectivity values provides a single constraint for two unknown variables, we cannot deduce the contributions of and unambiguously without complementary information. For example, in the case of Fig. 1(c), the minimal reflectivity of can be explained by and or by and (or ). We can only deduce that and .
For the same incoming polarization, (resp. ) and for each laser wavelength, we measure the Stokes parameter for the reflected light as a function of (Fig. 1(d)). We obtain for all wavelengths that (resp. ) showing that (resp. ). This shows that given an incident polarization or , the reflected light is also a pure polarization state .
A strikingly different situation appears if we excite the cavity with a diagonal polarization , away from its cavity polarization axis: we obtain the reflectivity curve displayed in Fig. 2(a) and the complete set of Stokes parameters , and described in Fig. 2(b). In the absence of uncoupled light, i. e. , the total reflectivity would be described by and the reflected polarization state by , rotated from as . A polarization rotation is indeed observed, as shown by the variation of the Stokes parameters , and as function of the input laser energy. However, the reflected output is not a pure polarization state, as shown in Fig. 2(c), displaying a polarization purity below unity when . This behavior is also complementarily illustrated in Fig. 2(d), where the polarization density matrix is displayed in the Poincaré sphere for various values of . When is detuned from , the reflected polarization is close to the incoming one (), but notably different when . In Fig. 2(d), the experimental points are encoded in a colorscale representing the corresponding polarization purity, which decreases down to when . Therefore, the reflected light polarization can not be considered as a pure polarization state.
To explain this depolarization, we have to take into account the limited input coupling , which is the overlap between the spatial profiles of the incoming beam and the fundamental cavity mode. The light coupled to the cavity is indeed reflected in a pure polarization state with a reflected intensity . The uncoupled light is fully reflected with an unrotated polarization state , whose reflected intensity is (see Fig. 1 (a)). As an example, for , is given by and .
An important property of the coupled and uncoupled components and is that the corresponding optical beams have orthogonal spatial profiles, and thus their superposition does not lead to interference in the total intensity. Regarding the coupled component, its spatial profile is governed by the spatial shape of the fundamental cavity mode only. The spatial profile of the uncoupled component arises from the contribution of other modes, which all have spatial profiles orthogonal to that of the fundamental one. The intensities of the two beams thus sum up without interference: this leads to a total reflectivity and to a polarization density matrix given by:
In the latter case, the reflected light has two contributions ( and ) which incoherently superpose different polarizations, as when . In this case, the polarization purity is below unity as shown in Fig. 2(c). However, far from the cavity resonance, so that : thus , corresponding to an unrotated and pure polarization state.
As we now describe, thanks to the polarization tomography technique and to the theoretical model presented above, the input and output couplings can be measured accurately. Fig. 3(a) displays the dependence of the minimum polarization purity achieved at as a function of the input coupling. For high values of input coupling (i. e. ), the minimum polarization purity is very sensitive to small changes in . From Fig. 3(a), we see that for a given value of the minimal polarization purity, there are actually two possible values of . However, it is easy to discriminate which is the correct value of by looking at the amplitude of the polarization rotation induced in the Poincaré sphere (see Fig. 2(c)): for a low , most of the light is uncoupled and experiences no rotation, for a high , most of the light experiences a rotation of polarization. By fitting the reflectivity curves in Fig. 1(b), Fig. 2(a), and the polarization density matrix (which corresponds to the Stokes parameters in Fig. 2(b) and the Polarization purity in Fig. 2(c) or the Poincaré sphere in Fig. 2 (d), we can accurately and unambiguously estimate the value of the top mirror output coupling and the input coupling . Such a value corresponds to the best spatial overlap that we could experimentally achieve, by careful optical alignment, between the incoming free space optical beam and the spatial profile of the fundamental cavity mode. This overlap could potentially be improved up to 100%, by further shaping the incoming optical beam to exactly match the cavity mode profile.
Furthermore, by modifying the spatial size of the free-space incoming beam with the beam adapter (, see Fig. 1 (a)), we can vary . As illustrated in Figs. 3(b,c) this was performed for two other incident beam sizes and thus different experimental values of . Figures 3(b,c) display the reconstructed polarization density matrix of the reflected light, as in Fig. 2(d), with the same experimental conditions and the same input polarization . In order to fit the data, we use the same cavity linewidths and output coupling , which was determined thanks to the previous experiment and to the theoretical model of Eq. 2. The only parameter that is varied to fit the Poincaré sphere displayed in Fig. 3(b) and (c) is . The agreement between theory and experiment allows us to estimate a mode matching of and respectively.
In summary, we have demonstrated that polarization tomography is a robust technique to determine input and output couplings with high accuracy. Indeed, the polarization purity of the reflected light is very sensitive to a slight amount of uncoupled light. In addition, we use a simple theoretical model to interpret the data and accurately determine an experimental input coupling of and an output coupling of . In the context of photonic quantum networks, such high input couplings are crucial to the realization of deterministic photon-photon gates Koshino, Ishizaka, and Nakamura (2010); Shomroni et al. (2014), where a first photon must be coupled to the cavity to modify the state of a second one.
This work is supported by the Agence Nationale de la Recherche (ANR) (ANR- 458 12-BS10-0010, ANR-14-CE32-0012, and the “Investissements d’avenir” program Labex NanoSaclay ANR-10-LABX-0035); H2020 European 459 Research Council (ERC) Starting Grant QD-CQED (277885); 460 H2020 Marie Skłodowska-Curie Actions (MSCA) Marie Curie 461 Fellowship SQuaPh (702084); Labex (NanoSaclay).
- preprint: AIP/123-QED
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