# Breaking of reciprocity and the Pancharatnam-Berry phase for light scattered by a disordered cold-atom cloud

P.H.N Magnani, P.G.S Dias, M Frometa, M.A Martins, N Piovella, Robin

Kaiser, Ph.W Courteille, M Hugbart, Romain Bachelard, R.C Teixeira

► To cite this version:

P.H.N Magnani, P.G.S Dias, M Frometa, M.A Martins, N Piovella, et al. "Breaking of reciprocity and the Pancharatnam-Berry phase for light scattered by a disordered cold-atom cloud." Physical Review A, 2024, 110 (4), pp.L041302. 10.1103/PhysRevA.110.L041302. hal-04416900

HAL Id: hal-04416900

https://hal.science/hal-04416900v1

Submitted on 25 Nov 2025

**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_1.png" /></figure>
# Breaking of reciprocity and the Pancharatnam-Berry phase for light scattered by a disordered cold atom cloud

P. H. N. Magnani,¹ P. G. S. Dias,¹ M. Frometa,² M. A. Martins,¹ N. Piovella,³ R.

Kaiser,⁴ Ph. W. Courteille,² M. Hugbart,⁴ R. Bachelard,¹,⁴ and R. C. Teixeira¹

¹Departamento de Física, Universidade Federal de São Carlos,
Rodovia Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brazil

²Instituto de Física de São Carlos, Universidade de São Paulo - 13566-590 São Carlos, SP, Brazil

³Dipartimento di Fisica "Aldo Pontremoli", Università degli Studi di Milano, Via Celoria 16,
I-20133 Milano, Italy & INFN Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy

⁴Université Côte d'Azur, CNRS, INPHYNI, France

(Dated: January 11, 2024)

Collective effects on the light scattered by disordered media such as Anderson localization and coherent backscattering critically depend on the reciprocity between interfering optical paths. In this work, we explore the breaking of reciprocity for the light scattered by a disordered cold atom setup, taking advantage of the non-commutation of optical elements that manipulate the polarization of the interfering paths. This breaking of symmetry manifests itself in the reduction of the fringes contrast as the light scattered by the cloud interferes with that from its mirror image. We provide a geometrical interpretation in terms of the Pancharatnam-Berry phase, which we directly access from the fringes displacement. Our work paves the way toward the manipulation of path reciprocity and interference for light scattered by disordered media.

**Introduction.**— Symmetries are fundamental properties of a system, which determine the conserved quantities of its dynamics [1, 2], thus setting constraints on its evolution. In particular, the charge, parity and time reversal (CPT) symmetry is universal as it applies to all known forces. Yet an important aspect of this symmetry is that it holds for a system as a whole, and considering the full symmetry, rather than a single one of the three. A simple illustration can be found in classical optics: While the CPT symmetry holds for the scattering of light by particles if one monitors all degrees of freedom, it breaks down as one focuses, for example, on the coherently scattered light. Indeed, the scattering of electromagnetic energy in other modes is typically encapsulated in the imaginary part of the medium refractive index, thus inducing a dissipative nature to the system. The necessity to differentiate absorption, which as a matter of fact defines an arrow for time and prevents the time reversal symmetry, from other energy-preserving mechanisms which may break that symmetry, has led to the notion of reciprocity [3, 4].

Reciprocity, in optics and beyond, describes the similarity of the transformations undergone by two waves travelling along the same path, yet in opposite directions. The fields emerging from these two “reciprocal paths” can be made to interfere, and in disordered systems the constructive interference between such pairs of optical paths is at the origin of modifications to the classical diffusion of waves, such as the coherent backscattering effect [5–11] and the Anderson localization of waves [12, 13]. The vectorial nature of electromagnetic waves may actually lead to a reduction of the enhancement of the backscattered signal, as (de)polarization effects kick in [5–7, 14, 15]. Similarly, the presence

of multiple, coupled polarization channels for light has been reported to be detrimental to Anderson localization [16, 17], superradiance [18] and subradiance [19], or to the build up of a large refractive index in dense disordered media [20].

In this work, we explore the breaking of reciprocity for light scattered by a disordered medium by introducing non-commuting polarizing optics on the path of the light. The light scattered by a large cold atom cloud is made to interfere with that from the cloud mirror image, yet the use of a birefringent mirror (M), along with a half waveplate (P), breaks the symmetry in the transformation that the polarization undergoes on each (reciprocal) path. The reciprocity breaking is monitored through the interference fringes from our mirror-assisted backscattering setup [21], and the reduction of their contrast is shown to derive directly from the lack of commutation of the above-mentioned polarizing optics. While this contrast is associated to the geodesic distance between the polarizations on the Poincaré sphere, the phase difference between them, also known as the Pancharatnam-Berry phase [22, 23], corresponds to the geodesic surface between the injected polarization and the scattered ones, as shown in Fig. 1: It is accessed in our experiment by measuring the fringes displacement, thus providing a geometrical interpretation to the breaking of reciprocity.

**Experimental Setup.**— Our cold atom setup consists of a cold cloud of $N \approx 3.10^7$ atoms of $^{88}$Sr, initially trapped in a magneto-optical trap with a temperature of 10 mK, see Fig. 2. The experiment is performed by turning off the trap, so the atomic cloud expands for a time of flight of 200 µs and reaches a $1/\sqrt{e}$ radius of $R = 1100$ µm. With an associated optical density $b_0 \approx 0.4$, single scattering is dominant. Then, the atoms are
<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Representation on the Poincaré sphere of the light polarization $\hat{\epsilon}_i$ and $\hat{\epsilon}_{ii}$, obtained after the action of the transformations $\hat{M}$ and $\hat{P}$ in direct or reverse order, respectively. The angles $\xi$ and $\gamma$ are defined by the operations $\hat{M}$ and $\hat{P}$, respectively, and the solid angle $\Omega$ is indicated in light yellow. (b) Schematic representation of the breaking of reciprocity between time-reversed paths, composed by non-commuting polarization operators $\hat{M}$ and $\hat{P}$.</figcaption></figure>

probed with a laser beam with a waist $w_0 = 2.1$ mm, so that the intensity is almost homogeneous throughout the cloud. The laser power $P_l = 0.2$ mW, resonant with the 461 nm broad transition of neutral Strontium ($\Gamma = 2\pi \times 30.5$ MHz), corresponds to a saturation parameter of $s_0 = 0.07$, so that most of the light is scattered elastically. The maximum pulse duration $t_p = 70$ µs is chosen such as to neglect the Doppler displacement of the transition due to the acceleration of the atoms during the pulse.

Our experiment is thus set in the single- and elastic-scattering regime, so the radiation from each atom can be decomposed as four optical paths, composed of the paths A, B, A' and B' shown in Fig. 2. Indeed, the light incoming on the atomic cloud may either:

(i) follow $A+A'+B$, i.e., cross the atomic sample without being scattered, be reflected on the 45° mirror (M) before being reflected back toward the half-waveplate (P), and finally be scattered by the atoms and reach the CCD,

(ii) follow $A+B'+B$, i.e., first be scattered by the atoms toward the plate P, then reflected back toward the 45° mirror M and eventually propagate until the CCD,

(iii) follow $A+B$, i.e., be directly backscattered by the atoms before being collected on the CCD,

(iv) follow $A+A'+B'+B$, i.e., cross the science chamber without being deviated, be reflected on the 45° mirror M before being reflected toward the plate P, and be backscattered by the atoms to meet again the plate P, be reflected toward the 45° mirror M and propagate until the CCD.

Here, (i) and (ii) are two reciprocal paths with very similar length, which interfere to build up fringes (according to the overlap between their polarization). In contrast, (iii) and (iv) correspond to a different optical path for

<figure><img src="image_3.png" /><figcaption>FIG. 2. Experimental setup. The laser light, resonant with the atomic sample, passes through a non polarising beam splitter (BS) before reaching the atomic cloud, which corresponds to path A indicated by a blue arrow. The transmitted beam passes through the 1:1 telescope composed of two lenses of same focal length, is reflected at the 45° mirror M, before being reflected back by a mirror with a small angle $\theta_0$, passing again through a lens, the 45° mirror M, and this time by a half-waveplate P before it reaches the other lens and then atoms; this stands for path A' (in blue). The scattered light by the atoms can propagate along the reverse path, where it passes the waveplate before the birefringent mirror and then travels back to the atomic cloud, which corresponds to path B' (in green); From the atomic cloud, the light follows the path B through the BS, to be reflected towards a CCD camera on which one can observe the fringes due to the interference of all the possible paths.</figcaption></figure>

each atom, thus contributing to a background light. Due to the presence of reciprocal paths, This mirror-assisted backscattering configuration [24] can be used to explore the light coherence in disordered atomic clouds [21, 25].

**Breaking of reciprocity.**— The polarization of the reciprocal paths is manipulated by using polarizing optics between the cloud and the backscattering mirror (Fig. 2). The injected light has a linear polarization $\hat{\epsilon}_0 = \hat{x}$. M is a dielectric mirror that induces a phase delay $\xi$ on the $\hat{y}$ component (relative to the $\hat{x}$ component) of the light polarization. The fast axis of P is tilted from the $\hat{x}$ axis by an angle $\gamma/2$, which is our control parameter, inducing a reflection of the polarization with respect to the fast axis. Within the Jones formalism for polarising optics [26, 27], the matrices associated to M and P are, respectively:

$$ \hat{M} = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\xi} \end{bmatrix} ; \hat{P} = \begin{bmatrix} \cos \gamma & \sin \gamma \\ \sin \gamma & -\cos \gamma \end{bmatrix}, \quad (1) $$

in the $(\hat{x}, \hat{y})$ basis. Then, the polarizations emerging from each path are given by:

$$ \begin{aligned} \hat{\epsilon}_i &= \hat{P}\hat{M}\hat{\epsilon}_0, \\ \hat{\epsilon}_{ii} &= \hat{M}\hat{P}\hat{\epsilon}_0, \\ \hat{\epsilon}_{iii} &= \hat{\epsilon}_0, \\ \hat{\epsilon}_{iv} &= \hat{M}\hat{P}\hat{P}\hat{M}\hat{\epsilon}_0 = \hat{M}^2\hat{\epsilon}_0. \end{aligned} \quad (2) $$

Thus, the polarization of the two paths providing the background light, (iii) and (iv), remain along $\hat{x}$.
The observed fringes result from polarization effects and differences in optical path between the four paths. Indeed, accounting for the tilt between the incident beam after the beamsplitter (wavevector $k_0 = k(\sin\theta_0\hat{y} - \cos\theta_0\hat{z})$ and the backward direction of observation (wavevector $k = k(\sin\theta\hat{y} + \cos\theta\hat{z})$, we obtain the following field scattered by atom j:

$$ \vec{E}_j = E_s \left[ e^{ik(\cos\theta - \cos\theta_0)z_j} \hat{e}_i + e^{ik(\cos\theta_0 - \cos\theta)z_j} \hat{e}_{ii} + e^{ik(\cos\theta_0 + \cos\theta)z_j} \hat{e}_{iii} + e^{-ik(\cos\theta_0 + \cos\theta)z_j} \hat{e}_{iv} \right], \quad (3) $$

where $E_s$ is a prefactor which encapsulates the amplitude of the pump, the single-atom cross-section and the distance of the camera from the setup. Summing the fields from all atoms, and assuming a continuous Gaussian density and a small tilt $|\theta - \theta_0| \ll \theta_0$, we get [28]

$$ I_{\text{out}}(\theta) = N I_a \left( 1 + \frac{1}{2} \text{Re}[\langle \hat{e}_i, \hat{e}_{ii} \rangle e^{i\phi}] S_R(\theta) \right), \quad (4) $$

where the scalar product is defined as $\langle \hat{e}_i, \hat{e}_{ii} \rangle = \hat{e}_i \cdot \hat{e}_{ii}^*$, $\phi = 2\theta_0 kh(\theta - \theta_0)$ the angular variable describing the fringes, with $h$ the distance between the mirror and the center of the atomic cloud, and $S_R(\theta) = e^{-2(\theta_0 k R)^2 (\theta - \theta_0)^2}$ the fringes spatial envelope. Thus, while paths (iii) and (iv) contribute only to the background, the reciprocal paths (i) and (ii) interfere to provide fringes in an angle $|\theta - \theta_0| \lesssim 1/kR$. An example of these fringes and the surrounding background light can be observed in Fig. 2.

**Non-commutation of the polarizing optics.** — As it can be seen in Eq. (4), the fringes contrast is given by the modulus of the overlap (that is, the scalar product) between the polarizations of the reciprocal paths (i) and (ii), which is calculated as

$$ \begin{aligned} \langle \hat{e}_i, \hat{e}_{ii} \rangle &= \langle \hat{P} \hat{M} \hat{e}_0, \hat{M} \hat{P} \hat{e}_0 \rangle \\ &= 1 - \langle \hat{M} \hat{P} \hat{e}_0, [\hat{P}, \hat{M}] \hat{e}_0 \rangle, \end{aligned} \quad (5) $$

where $[\cdot, \cdot]$ refers to the commutator. In other words, the interference between the reciprocal paths [see Eq. (4)] is reduced when the matrices of the polarizing optics stop commuting. In the case of M and P used in our experiment, the contrast reads

$$ C = |\langle \hat{e}_i, \hat{e}_{ii} \rangle| = \sqrt{1 - \sin^2(2\gamma) \sin^2(\xi/2)}. \quad (6) $$

The delay $\xi$ is determined by the dielectric coating of M, and chosen close to $\pi$ to maximize the reduction of the contrast ($\xi = (162 \pm 10)^\circ$ in our setup [28]), while $\gamma$ is used as a control parameter to explore the non-commutation between the optics, and subsequently the breaking of reciprocity between paths (i) and (ii).

With our experimental apparatus, we measure the scattered light profile at the CCD camera for different angles of the waveplate (i.e. for different values of $\gamma$). After over 1000 realizations of the experimental sequence described above, we obtain the light scattered as a function

of $\theta$, as shown in Fig. 3 (for details on the data analysis and fitting, see [28]). The contrast $C$ extracted from those fringes is presented in Fig. 4(a), and the minimum observed at $\gamma \approx \pi/4$ is consistent with the prediction from Eq. (6). Note that a better agreement with the experimental data is reached when two effects are accounted for, see plain red curve. First, the finite optical depth of the cloud is responsible for a larger signal from the background-building path (iii) as compared to the others, which in turn reduces the fringes contrast [21]. Second, the finite saturation parameter $s$ of the laser broadens the spectrum of the light scattered by the atoms, which in turn reduces slightly the contrast of the interference fringes [25]. The detailed modelling of these effects, and the associated fitting model, are described in [28].

<figure><img src="image_4.png" /><figcaption>FIG. 3. Colored lines: Interference fringes integrated along the azimuthal direction, normalized to the background intensity, and shifted vertically by one for visibility. Black continuous lines: Fitted fringe profiles, following Eq. (4). Dashed vertical line: center of the fringes' envelope, $\theta_0$, found by a global fit of the envelope to all curves.</figcaption></figure>

**Pancharatnam-Berry phase & Geometrical interpretation.** — It is clear from the fringes at Fig. 3 that, as the axis of P is rotated (i.e. as $\gamma$ is varied), the fringes suffer a reduction in contrast yet they are shifted at the same time. Indeed, if we now consider the phase $\delta$ from the scalar product between the polarizations,

$$ \langle \hat{e}_i, \hat{e}_{ii} \rangle = C e^{-i\delta}, \quad (7) $$

and incorporate the expression in Eq. (4), then it becomes clear that this phase actually translates into an angular displacement for the fringes. This shift corresponds to the Pancharatnam-Berry phase [22, 23], and
its geometrical interpretation on the Poincaré sphere is presented in Fig. 1: Starting from the injected polarization $\hat{e}_0 = \hat{x}$, the light is split into paths (i) and (ii), with its polarization turned into $\hat{e}_i$ and $\hat{e}_{ii}$, respectively, before being made to interfere. The solid angle $\Omega$ determined by the geodesic triangle $(\hat{e}_0\hat{e}_i\hat{e}_{ii})$ on the Poincaré sphere corresponds to twice the phase difference between the polarizations $\hat{e}_i$ and $\hat{e}_{ii}$ [22], that is, $\delta = -\Omega/2$.

<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) Contrast derived from the experimental fringes (black squares), from the theoretical prediction (6) once the correction due to the optical thickness of the atomic cloud ($b_0 \approx 0.4$) is included (dashed curve), plus the correction due to the finite saturation parameter of the incoming beam ($s = 0.07$)(continuous curve). The red shaded area corresponds to the confidence interval, which accounts for the uncertainty on the measurement of $\xi$ and on the fitted rescaling factor to account for $b_0$. (b) Pancharatnam-Berry phase from the experimental fringes displacement (black dots) and from the theoretical prediction (8). The red shaded area corresponds to the confidence interval associated with the uncertainty on the measurement of $\xi$.</figcaption></figure>

From scalar product (7), we obtain the following equa-

tion for the geometrical phase:

$$ \tan \delta = \frac{\sin \xi \tan^2 \gamma}{1 + \cos \xi \tan^2 \gamma} \qquad (8) $$

The measurement of the Pancharatnam-Berry phase presents a fair agreement with the theoretical prediction, see Fig. 4(b). The substantial error bars for the experimental data of Fig. 4(b) stem from the uncertainty on the determination of the center of the envelope $\theta_0$. Finally, we note that the contrast also has a geometrical interpretation: It corresponds to $C = \cos(\zeta/2)$, where $\zeta$ is the geodesic distance between the polarizations $\hat{e}_i$ and $\hat{e}_{ii}$ on the Poincaré sphere [22].

**Conclusions.**— We have explored the breaking of optical path reciprocity in a mirror-assisted coherent backscattering setup. The fringes emerging from the light scattered by a disordered cold atomic cloud have their contrast reduced as the optical elements used stop commuting and the two interfering paths become nonreciprocal. The measurement of the contrast and angular displacement of the fringes finds their geometrical interpretation on the Poincaré sphere, encapsulated in, respectively, the geodesic distance between the two polarizations and their phase difference, as originally proposed by Pancharatnam and Berry to provide a measure of the difference between polarizations.

In full generality, these results explore the vectorial nature of light to control the reciprocity of two interfering paths, despite the persisting spectral and spatial coherence. This breaking of reciprocity in this setup mimics what happens in multiple scattering regime coherence phenomena, such as the CBS and Anderson localization, when time-reversal symmetry is broken [29–31] via e.g. the presence of multilevel Zeeman electronic structure in atoms [11, 32, 33] or depolarization due to the near-field scattered electric field of close atoms in high densities [16, 34]. Interestingly, while the application of a magnetic field can break the time-reversal reciprocity in some cases, it may also restore strong interference phenomena based on the reciprocity of paths, such as for CBS [35] or Anderson localization [36], or as in our setup when the contrast is restored for specific angles of the half waveplate.

Since our disordered sample is made of saturable atoms subjected to classical (thermal) and quantum (spontaneous emission) decoherence [37], as well as to collective [38, 39] or many-body effects [40], the role of reciprocity for the coherent light emission is usually hidden within the complexity of the system, but could be made explicit, and even controlled, through strategies such as those detailed here, with applications for example to random lasers [41]. Finally, for atoms interacting with a Fabry-Perot cavity, several reflection paths interfere to produce the cavity output; a small nonreciprocal operation, such as a Faraday rotation, on each roundtrip within the cavity could drastically change the spectrum of the system
and alter the lasing properties of extended-cavity diode lasers, with a potential to unveil rich new physics at the different cavity-matter interaction regimes.

*Acknowledgements*.— R.C.T., R.B. and Ph.W.C. acknowledge funding from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) through Grants Nos. 2018/23873-3 and 2022/00261-8; R.C.T. and R.B. acknowledge funding from FAPESP through Grants No. 2018/15554-5 and 2023/03300-7; R.C.T., Ph.W.C. acknowledge funding from FAPESP through Grant No. 2013/04162-5; Ph.W.C. acknowledges support from the project CAPES-COFECUB Ph879-17/CAPES 88887.130197/2017-01; P.G. S.D., R.K., M.H., R.C.T., R.B. and Ph.W.C. acknowledge funding from ANR + FAPESP (projectQuaCor ANR19-CE47-0014-01/ FAPESP 2019/13143-0) and project STIC-AmSud (Ph879-17/CAPES 88887.521971/2020-00) and CAPES-COFECUB (CAPES 88887.711967/2022-00); R.K. and M.H. acknowledge funding from French National Research Agency (ANR) (project PACE-IN ANR19-QUAN-003-01), from European Unions Horizon 2020 research and innovation programme in the framework of Marie Skodowska-Curie HALT project (grant agreement No 823937), and financial support of the Doeblin Federation; R.K. acknowledges support from the European project ANDLICA, ERC Advanced Grant Agreement No. 832219; M.H. and R.B. acknowledge support by the French government, through the UCAJ.E.D.I. Investments in the Future project managed by the ANR (ANR-15-IDEX-01).

[1] E. Noether, Invariante variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 235 (1918).

[2] E. Noether, Invariant variation problems, Transport Theory and Statistical Physics 1, 186 (1971).

[3] R. J. Potton, Reciprocity in optics, Reports on Progress in Physics 67, 717 (2004).

[4] R. Carminati, J. J. Sáenz, J.-J. Greffet, and M. Nieto-Vesperinas, Reciprocity, unitarity, and time-reversal symmetry of the s matrix of fields containing evanescent components, Phys. Rev. A 62, 012712 (2000).

[5] Y. Kuga and A. Ishimaru, Retroreflectance from a dense distribution of spherical particles, J. Opt. Soc. Am. A 1, 831 (1984).

[6] P.-E. Wolf and G. Maret, Weak localization and coherent backscattering of photons in disordered media, Phys. Rev. Lett. 55, 2696 (1985).

[7] M. P. Van Albada and A. Lagendijk, Observation of weak localization of light in a random medium, Phys. Rev. Lett. 55, 2692 (1985).

[8] K. M. Yoo, G. C. Tang, and R. R. Alfano, Coherent backscattering of light from biological tissues, Applied Optics 29, 3237 (1990).

[9] M. I. Mishchenko, On the nature of the polarization op-

position effect exhibited by saturn's rings, The Astrophysical Journal 411, 351 (1993).

[10] A. Tourin, A. Derode, P. Roux, B. A. van Tiggelen, and M. Fink, Time-dependent coherent backscattering of acoustic waves, Phys. Rev. Lett. 79, 3637 (1997).

[11] G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura, and R. Kaiser, Coherent backscattering of light by cold atoms, Phys. Rev. Lett. 83, 5266 (1999).

[12] P. W. Anderson, Absence of diffusion in certain random lattices, Physical review 109, 1492 (1958).

[13] E. Akkermans and G. Montambaux, Mesoscopic physics of electrons and photons (Cambridge university press, 2007).

[14] E. Akkermans, P. E. Wolf, and R. Maynard, Coherent backscattering of light by disordered media: Analysis of the peak line shape, Phys. Rev. Lett. 56, 1471 (1986).

[15] M. J. Stephen and G. Cwilich, Rayleigh scattering and weak localization: Effects of polarization, Phys. Rev. B 34, 7564 (1986).

[16] S. E. Skipetrov and I. M. Sokolov, Absence of anderson localization of light in a random ensemble of point scatterers, Phys. Rev. Lett. 112, 023905 (2014).

[17] C. E. Máximo, N. Piovella, P. W. Courteille, R. Kaiser, and R. Bachelard, Spatial and temporal localization of light in two dimensions, Phys. Rev. A 92, 062702 (2015).

[18] M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Physics Reports 93, 301 (1982).

[19] A. Cipris, R. Bachelard, R. Kaiser, and W. Guerin, van der waals dephasing for dicke subradiance in cold atomic clouds, Phys. Rev. A 103, 033714 (2021).

[20] F. Andreoli, M. J. Gullans, A. A. High, A. Browaeys, and D. E. Chang, Maximum refractive index of an atomic medium, Phys. Rev. X 11, 011026 (2021).

[21] P. H. Moriya, R. F. Shiozaki, R. C. Teixeira, C. E. Máximo, N. Piovella, R. Bachelard, R. Kaiser, and P. W. Courteille, Coherent backscattering of inelastic photons from atoms and their mirror images, Physical Review A 94, 053806 (2016).

[22] S. Pancharatnam, Generalized theory of interference, and its applications, Proceedings of the Indian Academy of Sciences - Section A 44, 247 (1956).

[23] M. Berry, The adiabatic phase and pancharatnam's phase for polarized light, Journal of Modern Optics 34, 1401 (1987).

[24] J. J. Greffet, Backscattering of s-polarized light from a cloud of small particles above a dielectric substrate, Waves in Random Media 3, S65 (1991).

[25] P. G. S. Dias, M. Frometa, P. H. N. Magnani, K. R. B. Theophilo, M. Hugbart, P. W. Courteille, and R. C. Teixeira, Mirror-assisted backscattering interferometry to measure the first-order correlation function of the light emitted by quantum scatterers, Phys. Rev. A 104, 053716 (2021).

[26] R. C. Jones, A new calculus for the treatment of optical SystemsI description and discussion of the calculus, Journal of the Optical Society of America 31, 488 (1941).

[27] E. Collett, Equation summary, in Field Guide to Polarization (SPIE, 1000 20th Street, Bellingham, WA 98227-0010 USA, 2009) pp. 114–123.

[28] See Supplemental Material at [URL will be inserted by publisher].

[29] T. Micklitz, C. A. Müller, and A. Altland, Echo spectroscopy of anderson localization, Physical Review B 91,
064203 (2015).

[30] K. Müller, J. Richard, V. V. Volchkov, V. Denechaud, P. Bouyer, A. Aspect, and V. Josse, Suppression and reversal of weak localization through control of time-reversal symmetry, Physical Review Letters **114**, 205301 (2015).

[31] C. Hainaut, I. Manai, J.-F. Clément, J. C. Garreau, P. Szriftgiser, G. Lemarié, N. Cherroret, D. Delande, and R. Chicireanu, Controlling symmetry and localization with an artificial gauge field in a disordered quantum system, Nature communications **9**, 1382 (2018).

[32] C. A. Müller, T. Jonckheere, C. Miniatura, and D. Delande, Weak localization of light by cold atoms: The impact of quantum internal structure, Physical Review A **64**, 053804 (2001).

[33] G. Labeyrie, C. Miniatura, C. A. Müller, O. Sigwarth, D. Delande, and R. Kaiser, Hanle effect in coherent backscattering, Physical review letters **89**, 163901 (2002).

[34] L. Bellando, A. Gero, E. Akkermans, and R. Kaiser, Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic hamiltonian, Physical Review A **90**, 063822 (2014).

[35] O. Sigwarth, G. Labeyrie, T. Jonckheere, D. Delande, R. Kaiser, and C. Miniatura, Magnetic field enhanced

coherence length in cold atomic gases, Physical review letters **93**, 143906 (2004).

[36] S. E. Skipetrov and I. M. Sokolov, Magnetic-field-driven localization of light in a cold-atom gas, Physical Review Letters **114**, 053902 (2015).

[37] P. Lassègues, M. A. F. Biscassi, M. Morisse, A. Cidrim, P. G. S. Dias, H. Eneriz, R. C. Teixeira, R. Kaiser, R. Bachelard, and M. Hugbart, Transition from classical to quantum loss of light coherence, Physical Review A **108**, 042214 (2023).

[38] W. Guerin, M. O. Araújo, and R. Kaiser, Subradiance in a large cloud of cold atoms, Physical review letters **116**, 083601 (2016).

[39] A. Cidrim, T. S. do Espirito Santo, J. Schachenmayer, R. Kaiser, and R. Bachelard, Photon blockade with ground-state neutral atoms, Physical Review Letters **125**, 073601 (2020).

[40] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. Yavuz, T. Walker, and M. Saffman, Observation of rydberg blockade between two atoms, Nature Physics **5**, 110 (2009).

[41] D. S. Wiersma, The physics and applications of random lasers, Nature physics **4**, 359 (2008).