Probing few-excitation eigenstates of interacting atoms on a lattice by observing their collective light emission in the far field

The collective emission from a one-dimensional chain of interacting two-level atoms coupled to a common electromagnetic reservoir is investigated. We derive the system's dissipative few-excitation eigenstates, and analyze its static properties, including the collective dipole moments and branching ratios between different eigenstates. Next, we study the dynamics, and characterize the light emitted or scattered by such a system via different far-field observables. Throughout the analysis, we consider spontaneous emission from an excited state as well as two different pump-field setups, and contrast the two extreme cases of noninteracting and strongly interacting atoms. For the latter case, the two-excitation submanifold contains a two-body bound state, and we find that the two cases lead to different far-field signatures. Finally, we exploit these signatures to characterize the wave functions of the collective eigenstates. For this, we identify a direct relation between the collective branching ratio and the momentum distribution of the collective eigenstates' wave function. This provides a method to prove the existence of certain collective eigenstates and to access their wave function without the need to individually address and/or manipulate single atoms.

Figure 1

General structure of the analysis. (a) Shows the basic setup. The two-level atoms (bare transition energy ${\omega }_0$) are arranged on a one-dimensional lattice (lattice constant $a$) along the $z$ direction. The atoms' dipole moments are uniformly aligned and lie in the $x-z$ plane (in this sketch, we have exemplarily chosen an angle between the dipole moments and the atomic chain of $\theta =\pi /2$). The whole chain is coupled to a common reservoir indicated by the shading. In (b), the first considered pump-field configuration is shown, with an incoherent pump rate ${\left|\mathcal{P}\right|}^2$ applied at an excitation angle ${\beta }_{\mathrm{exc}}$. The pump's electric field polarization vector lies in the $x-z$ plane and is perpendicular to the driving field's wave vector $\mathbf{k}$. To separate the scattered field from the driving field, a detection out of plane (i.e., in the $y-z$ plane) could be performed. (c) Two-pump setup with excitation angles ${\beta }_1$ and ${\beta }_2$. The ratio of the driving fields' pump rates is ${\epsilon }^2$.

Figure 2

Complex dispersion relations for single-excitation states $|k\rangle$ (a) and two-excitation states $|K\nu \rangle$ (b). Note that the energies of the bound states are detached from the quasicontinuum of scattering states. The parameters chosen here are ${\lambda }_{\mathrm{at}}/a=0.5$ and $\theta =\pi /2$, resulting in ${\Gamma }_0/{\gamma }_0=1-0.637i$ and ${\Gamma }_1/{\gamma }_0=0.009-0.119i$. For ${\omega }_0\gg U\gg {\gamma }_0$, we can resort to a simplified level scheme of quasidegenerate bands (c).

Figure 3

(a) How does a two-excitation state $|K\nu \rangle$ decay into the submanifold of single-excitation states $\left\{\right|k\rangle \}$? This can be answered with the help of the matrix elements of the collective dipole moment operator, which are intimately linked to the relative wave function's momentum distribution $|{\overline{\eta }}_q^{\left(K\nu \right)}{|}^2$. (b) Momentum distribution $|{\overline{\eta }}_q^{\left(Kp\right)}{|}^2$ of the relative wave function for scattering states ($M=101$) for $U=0$ and (c) $U\gg {\gamma }_0$. (d) Momentum distribution $|{\overline{\eta }}_q^{(K,\mathrm{BS})}{|}^2$ of a bound state for $U\gg {\gamma }_0$ according to Eq. (37). (e), (f) Logarithmic plot of the scattering state's momentum distributions (b) and (c) [for better visualization, we have actually plotted $ln\left(|{\overline{\eta }}_q^{\left(Kp\right)}{|}^2+c\right)$ with $c={10}^{-3}]$.

Figure 4

Spontaneous emission patterns according to Eq. (84) for an initial state $|K=0,p\rangle$ as a function of the detection angle ${\beta }_{\det }$ and the emission wavelength ${\lambda }_{\mathrm{at}}$ in units of $a$ ($M=101$). Top row: $U=0$, bottom row: $U\gg {\gamma }_0$. From left to right: $pa/\pi =\frac{1}{2},\frac{1}{4},\frac{1}{8}$. Note the different scales of the color bars.

Figure 5

Spontaneous emission patterns as in Fig. 4 for an initial scattering state $|K=0,p=\pi /2a\rangle$ but for a fixed value of the emission wavelength ${\lambda }_{\mathrm{at}}/a=0.5$. In essence, this figure represents a cut of Figs. 4 and 4 along ${\lambda }_{\mathrm{at}}/a=0.5$. We have plotted the angle-dependent emission pattern for various atom numbers $M=51,99,199,399$. Note that the $x$ axis is discrete and that we have plotted each data point with a width $\delta {\beta }_{\det }\left(M\right)$ with respect to the $x$ axis.

Figure 6

Spontaneous emission patterns according to Eq. (85) (for $t_{\mathrm{ret}}=0$) emerging from an initial bound state $|K,\mathrm{BS}\rangle$. (a) Emission pattern (for $K=0$) normalized to the single-dipole pattern as a function of the detection angle ${\beta }_{\det }$ and the emission wavelength ${\lambda }_{\mathrm{at}}$ (in units of $a$). As ${\lambda }_{\mathrm{at}}/a$ increases, less Bragg orders become visible and the width of the emission peaks increases. (b) Emission pattern as in (a) but for ${\lambda }_{\mathrm{at}}/a=0.45$ as a function of the detection angle and the center-of-mass wave number $K$. The full emission pattern taking into account the single-dipole pattern is shown in (c)–(f) (we have plotted $G^{\left(1\right)}{r}^2/{\xi }^2{\left|d\right|}^2$ for $K=0$ as a 3D spherical plot, the color bar indicates the emission strength). The patterns exhibit a toroidal-like structure (which is a remnant of the single-dipole pattern) with lobes resulting from the properties of the bound state's momentum distribution. The angle between the single-atom dipole moment $\mathbf{d}$ (denoted by the blue line) and the atomic chain (dashed black line) is (c) $\theta =\pi /2$, (d) $\theta =\pi /4$, (e) $\theta =\pi /8$, and (f) $\theta =\pi /3$.

Figure 7

Normalized emission pattern emerging from a bound state $|K=0,\mathrm{BS}\rangle$ as in Fig. 6, but for different strengths of the atom-atom interaction $U$ (atomic dipole moments perpendicular to the atomic chain). Regions that are “dark” for any angle correspond to parameters for which the criterion for the existence of a bound state is not satisfied.

Figure 8

Driving the 1D lattice of two-level atoms with an incoherent drive, as sketched in Fig. 1, reduces the Hilbert space to the levels shown here. Since the external pump “imprints” the wave number $k_P$ on the atomic system, the relevant Hilbert space comprises only two-excitation states with $K=2k_P$.

Figure 9

An external, incoherent field with two spatial Fourier components [as depicted in Fig. 1] drives the atomic system and “imprints” the wave numbers $k_1$ and $k_2$ (the ratio of the driving fields' pump rates is ${\epsilon }^2$). The reduced level scheme for this two-pump excitation setup comprises single-excitation states $|k_1\rangle$ and $|k_2\rangle$, and two-excitation states $|2k_1,\nu \rangle$, $|2k_2,\nu \rangle$, and $|k_1+k_2,\nu \rangle$. The figure shows the situation for the case $k_1\ne k_2$. Single-excitation levels that are only populated via spontaneous emission from two-excitation states are not shown here since they would only be relevant for detection angles ${\beta }_{\det }^{\left(1\right)}\ne {\beta }_1,{\beta }_2$ (which we do not consider).

Figure 10

Relative difference of the normalized far-field intensity for zero and nonzero second driving field, i.e., $\delta {\overline{G}}_{\mathrm{NL}}^{\left(1\right)}({\beta }_1,{\beta }_2,\epsilon )$ according to Eq. (103) (in the limit of many atoms). We have chosen ${\lambda }_{\mathrm{atom}}/a=0.5$ and ${\beta }_2=arcsin({\lambda }_{\mathrm{at}}/a)$. The signatures for the two cases of noninteracting $[U=0$, (a)] and strongly interacting atoms $[U\gg {\gamma }_0$, (b)] are clearly different. See the main text for the identification and discussion of the individual features.

Figure 11

Spectral signature $\delta S({\beta }_1,{\beta }_2,\epsilon ,\omega ={\omega }_0+U)$ of a bound state as a function of the detection angle according to Eq. (106). As the bound states' momentum distribution has a maximum at the argument $q=0$ (see Sec. 2c2), we observe peaks around (see also the discussion in Sec. 4c1) ${\beta }_1/\pi \simeq 0,0.17,0.5$ [for ${\lambda }_{\mathrm{at}}/a=0.5$, (a)]. For ${\lambda }_{\mathrm{at}}/a=0.3$ (b), we have ${\beta }_1/\pi \simeq 0,0.10,0.20,0.36$.

Figure 12

Normalized photon-photon correlation function $g^{\left(2\right)}[{\beta }_1,{\beta }_2=arcsin({\lambda }_{\mathrm{at}}/a)]$ according to Eq. (111) in the limit of many atoms [such that the last line in Eq. (111) vanishes]. For $U=0$ (blue line), the correlation function exhibits a constant value of $\frac{2}{3}$, except for angles where $q=0$ is realized (red crosses). For $U\gg {\gamma }_0$ (black line), the $g^{\left(2\right)}$ function varies smoothly with respect to the angle ${\beta }_1$, but is also interrupted when $q=0$ (red crosses). At $q=0$, the correlation function always has a value of $\frac{1}{6}$ (denoted by the red dashed line) for both $U=0$ and $U\gg {\gamma }_0$. The green dashed lines indicate reference values from Refs. [57, 58, 59, 60, 61], referring to a two-atom system (see text for discussion).