Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays

TL;DR

This paper investigates quantum many-body effects in light transmission through planar atomic arrays, revealing how quantum fluctuations influence optical properties and atomic correlations at high densities and near saturation.

Contribution

It demonstrates that quantum effects are significant in light scattering from atomic arrays and introduces an enhanced semiclassical model that accurately reproduces these effects, enabling analysis of large ensembles.

Findings

Quantum effects are prominent at high densities and near saturation.

Enhanced semiclassical model accurately predicts light transmission.

Collective phenomena like Rabi splitting and suppressed reflection observed.

Abstract

We identify significant quantum many-body effects, robust to position fluctuations and strong dipole--dipole interactions, in the forward light scattering from planar arrays and uniform-density disks of cold atoms, by comparing stochastic electrodynamics simulations of a quantum master equation and of a semiclassical model that neglects quantum fluctuations. Quantum effects are pronounced at high atomic densities, light close to saturation intensity, and around subradiant resonances. We show that such conditions also maximize spin--spin correlations and entanglement of formation for the atoms, revealing the microscopic origin of light-induced quantum effects. In several regimes of interest, an enhanced semiclassical model with a single-atom quantum description reproduces light transmission remarkably well, and permits analysis of otherwise numerically inaccessible large ensembles, in which we observe collective many-body analogues of resonance power broadening, vacuum Rabi splitting, and significant suppression in cooperative reflection from atomic arrays.

Equations

$$\begin{gathered} {⟨{\hat{\psi }}_a^{†}(\mathbf{r}){\hat{\psi }}_b^{†}({\mathbf{r}}^{\prime }){\hat{\psi }}_c({\mathbf{r}}^{\prime }){\hat{\psi }}_d(\mathbf{r})⟩}_{\{{\mathbf{r}}_1,\dots ,{\mathbf{r}}_N\}} \\ =\sum _{j\ell (j\ne \ell )}{\rho }_{ad;bc}^{(j,\ell )}\delta (\mathbf{r}-{\mathbf{r}}_j)\delta ({\mathbf{r}}^{\prime }-{\mathbf{r}}_{\ell }), \end{gathered}$$

$$\dot{\rho }=$$

$$+\sum _{j\ell }{\gamma }_{j\ell }\left(2{\hat{\sigma }}_{-}^{(j)}\rho {\hat{\sigma }}_{+}^{(\ell )}-{\hat{\sigma }}_{+}^{(\ell )}{\hat{\sigma }}_{-}^{(j)}\rho -\rho {\hat{\sigma }}_{+}^{(\ell )}{\hat{\sigma }}_{-}^{(j)}\right),$$

$${\mathsf{H}}_{\text{sys},j}=-\hslash \Delta {\hat{\sigma }}_{ee}^{(j)}-{\mathbf{d}}_{eg}\cdot {\mathcal{E}}^{+}({\mathbf{r}}_j){\hat{\sigma }}_{+}^{(j)}-{\mathbf{d}}_{ge}\cdot {\mathcal{E}}^{-}({\mathbf{r}}_j){\hat{\sigma }}_{-}^{(j)},$$

$${\rho }_{ad;bc}^{(j,\ell )}\simeq {\rho }_{ad}^{(j)}{\rho }_{bc}^{(\ell )}.$$

$$\frac{\text{d}}{\text{d}t}{\rho }_{ge}^{(j)}=$$

$$-\text{i}\left(2{\rho }_{ee}^{(j)}-1\right)\sum _{\ell \ne j}\left({\Omega }_{j\ell }+\mathrm{i}{\gamma }_{\mathrm{j}\mathrm{\ell }}\right){\rho }_{ge}^{(\ell )},$$

$$\frac{\text{d}}{\text{d}t}{\rho }_{ee}^{(j)}=$$

$$+2\text{Im}\left[\sum _{\ell \ne j}{\rho }_{ge}^{(j)}\left({\Omega }_{j\ell }-\mathrm{i}{\gamma }_{\mathrm{j}\mathrm{\ell }}\right){{\rho }_{ge}^{(\ell )}}^{\ast }\right].$$

Full text

Light can mediate strong interactions between atoms, inducing strong position-dependent correlations, even in the limit of low light intensity, when the response (for the case of a simple level structure) is entirely classical. Such a correlated optical response can differ dramatically from that predicted by standard electrodynamics of continuous media, where resonant-light-induced dipole–dipole (DD) interactions between atoms are treated in an averaged sense Javanainen et al. (2014); Javanainen and Ruostekoski (2016). Beyond the limit of low light intensity, an isolated atom can scatter light quantum-mechanically, and quantum effects in the interactions of light with dilute atomic ensembles have been utilized in, e.g., quantum information protocols Hammerer et al. (2010). In strongly interacting dense systems the possible role of quantum and cooperative effects is less clear and has been the subject of long-standing debates Javanainen et al. (2014); Friedberg et al. (1973); Scully and Svidzinsky (2010); Röhlsberger et al. (2010); Guerin et al. (2017). A particularly promising system to explore and utilize strong light-induced DD interactions is a regular planar array of scatterers such as atoms. In the linear low-excitation limit these manifest, as shown both theoretically and experimentally, a wealth of phenomena, e.g., subdiffraction features Sentenac and Chaumet (2008); Lemoult et al. (2010); Jenkins and Ruostekoski (2012), nontrivial topological phases Perczel et al. (2017); Bettles et al. (2017), transmission varying from complete reflection to full transparency Tretyakov (2003); García de Abajo (2007); Jenkins and Ruostekoski (2013); Bettles et al. (2016); Facchinetti et al. (2016); Facchinetti and Ruostekoski (2018); Shahmoon et al. (2017), narrow resonances and subradiance Fedotov et al. (2010); Jenkins and Ruostekoski (2013); Yang et al. (2014); Facchinetti et al. (2016); Bettles et al. (2015, 2016); Jenkins et al. (2017); Plankensteiner et al. (2017); Asenjo-Garcia et al. (2017); Jen (2017); Guimond et al. (2019), as well as quantum technological applications Hebenstreit et al. (2017); Grankin et al. (2018) and other collective effects Olmos et al. (2013); Krämer et al. (2016); Yoo and Paik (2016); Wang et al. (2017); Yoo (2018); Zeytinoǧlu and İmamoǧlu (2018); Mkhitaryan et al. (2018).

We show that we can identify quantum effects in the light transmitted through planar arrays and uniform-density disks of cold and dense atomic ensembles. Many-body quantum correlations are induced by light-atom coupling, which, surprisingly, survive even strong many-body resonant DD interactions and atomic position fluctuations. Specifically, comparing the correlated optical response determined using the quantum master equation (QME) to simulations neglecting any quantum fluctuations between atomic levels in different atoms [referred to as the “semiclassical” equations (SCEs) Lee et al. (2016)], we systematically identify light-established quantum effects between atoms in the transmitted light as a function of atom confinement, density, and driving intensity. The effect of many-body quantum fluctuations on the scattering manifests most prominently at high densities when the light is close to saturation intensity, and especially significantly in the vicinity of subradiant resonances. We find that these conditions also produce maximal spin–spin correlations and entanglement of formation in the underlying atomic system, further confirming the role of many-body quantum correlations and entanglement in observing a difference in light transmission between the QME and SCEs models. Incorporating the single-atom quantum description of light emission into the semiclassical scattering, we can typically use SCEs also for incoherent scattering to qualitatively reproduce the full quantum scattering even in the regimes where quantum effects in coherent scattering were most pronounced, and elsewhere also quantitatively. SCEs therefore allow us to analyze cooperative transmission of light through large atomic arrays and disks beyond the limit of low light intensity, without needing to solve the full strongly-interacting quantum dynamics. Doing so, we find collective phenomena due to DD interactions that are many-body analogues of power broadening and vacuum Rabi splitting of atomic resonances in cavities Liberal et al. (2019); Khitrova et al. (2006), and demonstrate a significant effect of intensity on the transmission that may ultimately restrict the utilization of atomic arrays as highly reflective cooperative mirrors.

An appealing feature of light scattering from cold atoms Balik et al. (2013); Chabé et al. (2014); Pellegrino et al. (2014); Sheremet et al. (2014); Kwong et al. (2014); Jennewein et al. (2016); Kwong et al. (2015); Bromley et al. (2016); Jenkins et al. (2016); Guerin et al. (2016); Saint-Jalm et al. (2018) is that light-mediated strong DD interactions can establish correlations between atoms at fluctuating positions, which are most simply described using atomic field operators for the ground and excited states $\hat{\psi}_{g,e}(\mathbf{r})$. Hence, $\langle\hat{\mathbf{P}}^{+}(\mathbf{r})\rangle=\langle\mathop{\hat{\psi}_{g}^{{\dagger}}(\mathbf{r})}\mathbf{d}_{ge}\mathop{\hat{\psi}_{e}(\mathbf{r})}\rangle$ denotes the light-induced atomic polarization, where $\mathbf{d}_{ge}=\mathbf{d}_{eg}^{*}$ is the dipole matrix element, and the populations are $\langle\mathop{\hat{\psi}_{g}^{{\dagger}}(\mathbf{r})}\mathop{\hat{\psi}_{g}(\mathbf{r})}\rangle$ and $\langle\mathop{\hat{\psi}_{e}^{{\dagger}}(\mathbf{r})}\mathop{\hat{\psi}_{e}(\mathbf{r})}\rangle$. Because of the DD interactions, the polarization and populations also depend on two-body correlations $\langle\mathop{\hat{\psi}_{a}^{{\dagger}}(\mathbf{r})}\mathop{\hat{\psi}_{b}^{{\dagger}}(\mathbf{r}^{\prime})}\mathop{\hat{\psi}_{c}(\mathbf{r}^{\prime})}\mathop{\hat{\psi}_{d}(\mathbf{r})}\rangle$, where $a,b,c,d\in\{g,e\}$, representing the correlations in the optical response of an atom at $\mathbf{r}$ given the presence of a second atom at $\mathbf{r}^{\prime}$. These in turn depend on three-body correlations, etc., resulting in a hierarchy of correlation function equations of motion Ruostekoski and Javanainen (1997); Morice et al. (1995). In a cold, dense ensemble this hierarchy can significantly and nonperturbatively modify the scattering behavior, invalidating attempts to truncate it Javanainen et al. (2014). This is a key ingredient in, e.g., Anderson localization of light, which has been a subject of considerable controversy and debate Skipetrov and Sokolov (2014); Sperling et al. (2016).

A numerical device for solving this correlation function hierarchy is to treat the atoms as discrete point particles, meaning for a particular configuration of atomic positions $\{\mathbf{r}_{1},\dots,\mathbf{r}_{N}\}$ that two-body correlation functions take the form

[TABLE]

where $\rho_{ad;bc}^{(j,\ell)}$ denote correlation functions of the internal atomic energy levels only Lee et al. (2016). We then solve the internal atom dynamics at discrete positions, and the new correlation functions simply emerge from the $N$-body density matrix $\rho\equiv\rho_{{\{\mathbf{r}_{1},\dots,\mathbf{r}_{N}\}}}$. This evolves according to QME

[TABLE]

where the collective scattering is represented by the dispersive $\Omega_{j\ell}$ and dissipative $\gamma_{j\ell}$ DD interactions, the single-atom half-width at half-maximum (HWHM) linewidth by $\gamma_{jj}=\gamma$, and $\hat{\sigma}_{+}^{(j)}=(\hat{\sigma}_{-}^{(j)})^{{\dagger}}=|e\rangle_{j}\mbox{}_{j}\langle g|$. For simplicity, we consider two-level atoms and the Hamiltonian

[TABLE]

where $\bm{\mathbf{\cal E}}^{+}=(\bm{\mathbf{\cal E}}^{-})^{*}$ is the positive-frequency-component of the frequency $\omega$ laser field, detuned from the atomic resonance frequency $\omega_{ge}$ by $\Delta=\omega-\omega_{ge}$ and $\hat{\sigma}_{ee}^{(j)}=|e\rangle_{j}\mbox{}_{j}\langle e|$. Spatial correlations are numerically synthesized by ensemble-averaging over stochastic realizations of atomic positions sampled from the density distribution Javanainen et al. (1999); Lee et al. (2016). Solving Eq. (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays) for large systems is numerically taxing, although few-atom ensembles already demonstrate many-body effects in their spectra Jones et al. (2017).

In the limit of low light intensity, where the excited state population vanishes, the internal level correlations, such as those described by $\rho_{ad;bc}^{(j,\ell)}$ in Eq. (1), also vanish for two-level atoms. The stochastic electrodynamics simulations are then formally exact Lee et al. (2016); Javanainen et al. (1999), reproducing the many-atom spatial correlations, which are identical to those occurring in the classical electrodynamics of coupled linear electric dipoles. Beyond the limit of low light intensity, the full dynamics of Eq. (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays) can be greatly simplified by factorizing the internal atomic level correlation functions:

[TABLE]

Following the formalism of Lee et al. (2016) we then obtain coupled nonlinear equations

[TABLE]

Note the relatively small number of equations $2N$ compared to the full quantum system size $2^{N}$. This formalism has been applied to the modeling of pumping of atoms in dense clouds Machluf et al. (2019), and has also been extended to cavity QED Lee et al. (2017).

Spatially correlated scattering between different atoms is accounted for in Eqs. (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays) and (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays) via $\Omega_{j\ell}$ and $\gamma_{j\ell}$ (for $\Omega_{j\neq\ell}=\gamma_{j\neq\ell}=0$ they reduce to the independent-atom optical Bloch equations). In the limit of low light intensity the ensemble-averaged response of SCEs coincides with the exact classical electrodynamics; beyond this limit, the model incorporates nonlinear internal level dynamics of the atoms. However, because of the factorization in Eq. (4), they cannot account for many-body quantum entanglement between different atoms’ internal levels. Finding situations in which the predictions of SCEs observably differ from the full QME solution therefore identifies light-induced quantum effects in the transmitted light. Conversely, regimes where quantum fluctuations are minimal allow for the simulation of much larger systems than are accessible with QME, and also test the validity of related approaches in other contexts, based, e.g., on mean-field approximations, intensity expansions, or truncations of the correlations Krämer and Ritsch (2015); Sutherland and Robicheaux (2017); Zhang and Mølmer (2019); Henriet et al. (2019); Parmee and Cooper (2018).

We begin by calculating the coherent and incoherent forward transmission, $T_{\mathrm{coh}}$ and $T_{\mathrm{inc}}$ (Figs. 2 and 3) Sup , through planar square arrays and thin disks of $N=4$ atoms (Fig. 1). The array could be realized, e.g., by an optical lattice Gross and Bloch (2017) or dipole traps Nogrette et al. (2014); Bernien et al. (2017). Unless otherwise stated, we consider lattice spacing $a=0.25\lambda$ and disk radius $R=0.28\lambda$. Physically, we calculate the far-field light intensity in the same mode as the driving field $\bm{\mathbf{\cal E}}^{\pm}$, integrated over the polar angle $\sin\theta\lesssim 0.24$ Sup . We account for the fluctuations in atomic positions due to finite trap confinement by ensemble-averaging over many stochastic realizations of position configurations Jenkins and Ruostekoski (2012); Sup . We unambiguously identify quantum effects in coherent transmission from the difference between the full quantum and semiclassical transmissions $T_{\mathrm{coh}}^{\mathrm{QM}}-T_{\mathrm{coh}}^{\mathrm{SC}}$. Since the coherent scattering of a single atom is classical, this difference is due solely to many-body quantum correlations in the atomic response.

To obtain the incoherently scattered light $\langle\delta\hat{\mathbf{E}}_{\mathrm{d}}^{-}(\mathbf{r})\,\delta\hat{\mathbf{E}}_{\mathrm{d}}^{+}(\mathbf{r})\rangle$, we write the scattered light field as $\hat{\mathbf{E}}_{\mathrm{d}}^{+}=\langle\hat{\mathbf{E}}_{\mathrm{d}}^{+}\rangle+\delta\hat{\mathbf{E}}_{\mathrm{d}}^{+}$, where $\delta\hat{\mathbf{E}}_{\mathrm{d}}^{+}$ denotes the fluctuations Meystre and Sargent III (2007). This yields incoherent transmission Sup for which quantum behavior also is isolated by $T_{\mathrm{inc}}^{\mathrm{QM}}-T_{\mathrm{inc}}^{\mathrm{SC}}$. We can improve the semiclassical incoherent model, without increasing the computational complexity, by adding the single-atom quantum description of incoherent light emission for all the atoms. In a single realization of stochastic atomic positions, the incoherent scattering contribution to intensity from independent quantum-mechanical atoms $\propto\sum_{j}A_{j}(\langle\hat{\sigma}^{(j)}_{ee}\rangle-|\langle\hat{\sigma}^{(j)}_{+}\rangle|^{2})$, where $A_{j}$ encapsulates the light propagation effects Meystre and Sargent III (2007); Sup . Augmenting the semiclassical model with this single-atom quantum description integrated over the sample yields the incoherent transmission $T_{\mathrm{inc}}^{\mathrm{SAQ}}$. The many-body quantum effects of the incoherent signal are then encapsulated in $T_{\mathrm{inc}}^{\mathrm{QM}}-T_{\mathrm{inc}}^{\mathrm{SAQ}}$.

In Fig. 2(b), we identify many-body quantum fluctuations in the coherent transmission ($T_{\mathrm{coh}}^{\mathrm{QM}}-T_{\mathrm{coh}}^{\mathrm{SC}}$) that increase with increasing DD interaction [Fig. 3(a,c)], reaching normalized residuals of over $10\%$ at $a=0.25\lambda$ and $I\simeq I_{\text{sat}}$ (when the dipole amplitudes are greatest), where ${I=2\epsilon_{0}c|\bm{\mathbf{\cal E}}^{+}(\mathbf{r}=\mathbf{0})|^{2}}$ is the total intensity of the incident laser field at the focus and $I_{\text{sat}}=4\pi^{2}\hbar\gamma c/3\lambda^{3}$ is the saturation intensity. Strikingly, quantum effects constitute over $30\%$ of the signal in the vicinity of the narrow subradiant resonant shown in the inset of Fig. 3(c). This may be due to the enhanced dipole magnitude of subradiance Bettles et al. (2016), the antisymmetry of the collective dipolar eigenvector [see diagram in Fig. 2(a)], or the rapidly varying Fano interference. It is remarkable that even for a fully random disk quantum effects on the scattering are not washed out but can produce residuals between the models of a few percent.

Conversely, as in Fig. 2(e,f), once $I\gtrsim I_{\text{sat}}$ the incoherent transmission is almost entirely dominated by quantum fluctuations ($T_{\mathrm{inc}}^{\mathrm{SC}}\to 0$ 111See also Fig. S1 of Sup ). However, once we incorporate the single-atom quantum description into the scattering and therefore transmission $T_{\mathrm{inc}}^{\mathrm{SAQ}}$, the difference becomes much smaller and the many-body quantum fluctuations are, as with the coherent scattering, maximal around $I\sim I_{\text{sat}}$. Hence, using the improved model $T_{\mathrm{inc}}^{\mathrm{SAQ}}$, it is possible, even for incoherent scattering, to obtain excellent qualitative, and frequently quantitative agreement with the full quantum scattering. For example, in Fig. 3(c) the difference between $T_{\mathrm{coh}}^{\mathrm{QM}}$ and $T_{\mathrm{coh}}^{\mathrm{SC}}$ is less than $5\%$ for $a\gtrsim 0.4\lambda$ or $I/I_{\text{sat}}\gtrsim 16$.

Up until now, we have identified many-body quantum effects in the transmitted light. These originate from the light-induced quantum correlations between internal levels of different atoms that do not satisfy the factorization assumption of the SCEs, given in Eq. (4). We explicitly show these induced spin–spin correlations and many-body entanglement of formation (in the latter case, for an analogous configuration of a pair of atoms, using the formalism of Wootters (1998)) in Figs. 4 and 5, respectively. The conditions under which the spin–spin correlations and the entanglement of formation are maximal do indeed match the conditions for enhanced quantum effects in the light transmission in Figs. 2 and 3.

Working in the conditions in which the quantum effects on the light scattering are minimal, we can employ SCEs [Eqs. (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays) and (Quantum and Nonlinear Effects in Light Transmitted through Planar Atomic Arrays)] by neglecting quantum fluctuations to analyze the coherent transmission through much larger ensembles, for which the full QME is inaccessible. In Fig. 6 we show how the transmission lineshapes of a $10\times 10$ array significantly differs from the Lorentzians of independent atoms. For a single atom the linewidth is power broadened to $\gamma_{\mathrm{PB}}(I)=\gamma\sqrt{1+I/I_{\text{sat}}}$. In the interacting case, the coherent lineshape is also power broadened but, depending on whether the linewidth of the dominant symmetric collective eigenmode $\upsilon$ at low light intensity Sup is subradiant ($\upsilon<\gamma$) or superradiant ($\upsilon>\gamma$), it will also be narrower or broader [Fig. 6(b)], respectively, than $\gamma_{\mathrm{PB}}(I)$. There is no analogous broadening of the incoherent lineshape, however 222See Fig. S2 of Sup . Furthermore, the many-body lineshape also exhibits a dip or “hole burning” on resonance [Fig. 6(b)]. This dip is analogous to vacuum Rabi splitting Liberal et al. (2019); Khitrova et al. (2006), where the interatomic DD coupling has now taken the role of the cavity coupling, and, while it only occurs for sufficiently high density, it can interestingly still exist even in the fully random ensemble.

A key feature of general subwavelength-spaced resonant emitter arrays is that they can exhibit perfect reflection Tretyakov (2003); García de Abajo (2007), which may typically be modeled using point-dipole scatterers Jenkins and Ruostekoski (2013); Shahmoon et al. (2017); Bettles et al. (2016); Facchinetti et al. (2016). Dipolar planar arrays can act as cooperative antennae Adamo et al. (2012), with applications to quantum information processing Grankin et al. (2018), making understanding nonlinear transmission essential. We calculate this for large arrays in Fig. 6(c), and find that the reduction in the extinction as a function of light intensity is considerable — although less prominent with smaller spacings. This may ultimately restrict the applications of atomic arrays as highly reflective cooperative mirrors to weak light intensities only.

To conclude, by comparing SCEs and QME, we have identified light-induced spin–spin correlations and quantum entanglement in the light transmitted through planar arrays and disks which survive both position fluctuations and strong DD interactions. At narrow subradiant resonances, quantum fluctuations can be over $30\%$. Outside these resonances, provided we improve the model by incorporating the single-atom quantum description, SCEs typically still reproduce, also for incoherent scattering, the full quantum behavior at least qualitatively. This provides a methodology to calculate transmission of light through large arrays, consisting of hundreds of atoms, which can exhibit striking many-body phenomena (even without any quantum effects) reminiscent of single-atom power broadening and vacuum Rabi splitting. The existence of many-body quantum effects despite strong driving, high densities, and even with significant atomic position fluctuations is surprising. It suggests that optical quantum information processing in atomic ensembles Hammerer et al. (2010) need not necessarily be restricted to dilute systems. Subradiant resonance narrowing has now been experimentally observed in the transmitted light through an optical lattice of atoms in a Mott-insulator state in the classical limit of low light intensity Rui et al. . Several of our findings could also be verified in this setup by increasing the intensity of the incident light. The presence, even in uniform disks, of many-body effects, that are attracting considerable interest Javanainen et al. (2014); Javanainen and Ruostekoski (2016); Saint-Jalm et al. (2018); Keaveney et al. (2012), further relaxes the conditions necessary for their experimental observation.