Few-photon transport in many-body photonic systems: A scattering approach

We study the quantum transport of multiphoton Fock states in one-dimensional Bose-Hubbard lattices implemented in QED cavity arrays (QCAs). We propose an optical scheme to probe the underlying many-body states of the system by analyzing the properties of the transmitted light using scattering theory. To this end, we employ the Lippmann-Schwinger formalism within which an analytical form of the scattering matrix can be found. The latter is evaluated explicitly for the two-particle, two-site case which we use to study the resonance properties of two-photon scattering, as well as the scattering probabilities and the second-order intensity correlations of the transmitted light. The results indicate that the underlying structure of the many-body states of the model in question can be directly inferred from the physical properties of the transported photons in its QCA realization. We find that a fully resonant two-photon scattering scenario allows a faithful characterization of the underlying many-body states, unlike in the coherent driving scenario usually employed in quantum master-equation treatments. The effects of losses in the cavities, as well as the incoming photons' pulse shapes and initial correlations, are studied and analyzed. Our method is general and can be applied to probe the structure of any many-body bosonic model amenable to a QCA implementation, including the Jaynes-Cummings-Hubbard model, the extended Bose-Hubbard model, as well as a whole range of spin models.

Figure 1

(a) Proposed method to probe the structure of bosonic many-body models as implemented in QCA simulators. Photons traveling in the left waveguide are injected into the array and are transported through the device to the right waveguide. In this work, the QCA is assumed to realize the Bose-Hubbard model, but other models such as the Jaynes-Cummings-Hubbard model, spin models, or the extended Bose-Hubbard model can also be realized [9, 17]. The injected photons scan through the many-body eigenstates of the simulated model and if they are fully resonant to the many-body states as illustrated in (b), then the full information of the relevant states is mapped out faithfully in the output spectra and correlation functions.

Figure 2

Energy-level diagram of the two-site Bose-Hubbard QCA, where the bare-cavity energies and the coupled-mode energies of the cavities are shown. One of the resonant two-photon excitation paths satisfying Eq. (4) is illustrated by the arrows on the left, while the off-resonant path for identical input photons is shown on the right.

Figure 3

Bound-state contribution in the scattering matrix element, $\overline{S_{RR}}$, is shown as a function of $\delta$, (a) for $V^2=0.01$ (blue solid curve) and $V^2=0.25$ (orange dashed curve), and (b) with detuning of total incident energy and nonlinearity for $V^2=0.25$. (c) The resonant conditions of $\mathrm{\Delta }k\left(\mathrm{\Delta }p\right)$ in $|S_{RR}{|}^2$ for $\delta =0,2{\epsilon }_{\pm }^{\left(1\right)}$, and ${\epsilon }_{0,\pm }^{\left(2\right)}$ for $U=5$ and $V^2=0.25$. (d) The respective eigenstate excitation amplitudes at the corresponding two-photon energy resonances with increasing $U/J$ when $\mathrm{\Delta }k=\delta -2{\epsilon }_{-}^{\left(1\right)}$ (solid curve) and $\mathrm{\Delta }k=0$ (dashed curve) for $V^2=0.04$. All units are defined with respect to $J$. Panel (c) clearly depicts the resonance condition written in Eq. (4), while panel (d) shows how the desired eigenstates are more efficiently excited when this condition is met (solid curves) as opposed to the off-resonant case (dashed curves).

Figure 4

Left- and right-hand columns : $\mathrm{\Delta }k=\delta -2{\epsilon }_{-}^{\left(1\right)}$ and $\mathrm{\Delta }k=0$. Two-photon transmission ($P_{RR}$) is shown as a function of the two-photon detuning $\delta /J$ for different photon-photon interaction strengths $U/J=0,1,5$ in (a) and (e). The probabilities $P_{LL},P_{LR}$, and $P_{RR}$ are also shown as a function of $U/J$ for two-photon eigenenergies $\delta ={\epsilon }_{0,\pm }^{\left(2\right)}$ in (b)–(d) and (f)–(h). Weak waveguide-cavity coupling and the narrow bandwidth of initial photons are assumed: $V^2/J=0.04$ and $\sigma /J=0.005$. Note the difference in the behavior of scattering probabilities as a function of $U/J$ when two-particle states are probed fully resonantly with different energy photons via the one-particle manifold (left column) compared to the case where a virtual (off-resonant) one-photon absorption is required (right column). In addition, in the former case, transmission is generally significantly larger, which makes this approach experimentally more efficient (see text for more details).

Figure 5

Left- and right-hand columns : $\mathrm{\Delta }k=\delta -2{\epsilon }_{-}^{\left(1\right)}$ and $\mathrm{\Delta }k=0$. Second-order intensity correlation function $g_{RR}^{\left(2\right)}$ is shown (a),(c) as a function of two-photon detuning $\delta /J$ for different photon-photon interaction strengths $U/J=0,1,5$ and (b),(d) as a function of $U/J$ for two-photon eigenenergies $\delta ={\epsilon }_{0,\pm }^{\left(2\right)}$. The same parameters are chosen as used in Fig. 4. We highlight here the direct mapping of the correlations of the many-body state ${\epsilon }_{-}^{\left(2\right)}$ onto the transmitted light $g_{RR}^{\left(2\right)}$ [the green dotted line in (b)], as the former monotonically approaches the Mott-like state $|1,1\rangle$ with increasing $U/J$. This does not hold in the identical-photons case, however, where $g_{RR}^{\left(2\right)}$ first increases before it dips down to follow the correlations of the state. The same behavior is also found in the coherent-driving scenario [16]. (See the detailed discussion in Sec. 3b regarding the rest of the states and regimes, and differences between the two approaches.)

Figure 6

Second-order intensity correlations for the transmitted light as a function of the two-photon detuning $\delta /J$ for $U/J=1,5$, obtained from the scattering approach with the identical two-photon input (blue solid line) and a coherent-state input (orange thick-dashed line), and also from the master-equation formalism (green dashed line). For the latter, $\mathrm{\Omega }=V\overline{n}=0.0002$ and $\gamma =V^2=0.04$ are used.

Figure 7

For the fully resonant case, two-photon transmissions ($P_{RR}$) and second-order correlations as a function of the total energy $\delta /J$ with ${\gamma }_{\mathrm{bath}}/J=0$, 0.02, and 0.04 when $V^2/J=0.04$ and $U/J=1$.

Figure 8

(a) $g_{\mathrm{initial}}^{\left(2\right)}\left(0\right)$ as a function of $M_2$, constructed from different values of momentums $k_1$ and $k_2$ for initial two photons. (b) $g_{\mathrm{initial}}^{\left(2\right)}$ as a function of $\delta$ when $\mathrm{\Delta }k=\delta -2{\epsilon }_{-}^{\left(1\right)}$.

Figure 9

$g_{RR}^{\left(2\right)}$ as a function of detuning $\delta /J$ for $U/J=1$ when (a) $\mathrm{\Delta }k=\delta -{\epsilon }_{-}^{\left(1\right)}$ and (b) $\mathrm{\Delta }k=0$. Parameters are the same as in the main text.