Ab Initio Few-Mode Theory for Quantum Potential Scattering Problems

Few-mode models are a cornerstone of the theoretical work in quantum optics, with the famous single-mode Jaynes-Cummings model being only the most prominent example. In this work, we develop an ab initio few-mode theory, a framework connecting few-mode system-bath models to ab initio methods. We first present a method to derive exact few-mode Hamiltonians for noninteracting quantum potential scattering problems and demonstrate how to rigorously reconstruct the scattering matrix from such few-mode Hamiltonians. We show that, upon inclusion of a background scattering contribution, an ab initio version of the well-known input-output formalism is equivalent to standard scattering theory. On the basis of these exact results for noninteracting systems, we construct an effective few-mode expansion scheme for interacting theories, which allows us to extract the relevant degrees of freedom from a continuum in an open quantum system. As a whole, our results demonstrate that few-mode as well as input-output models can be extended to a general class of problems and open up the associated toolbox to be applied to various platforms and extreme regimes. We outline differences of the ab initio results to standard model assumptions, which may lead to qualitatively different effects in certain regimes. The formalism is exemplified in various simple physical scenarios. In the process, we provide a proof of concept of the method, demonstrate important properties of the expansion scheme, and exemplify new features in extreme regimes.

Popular Summary

The dynamics of many quantum systems of interest are too complex to be understood in their entirety. Fortunately, for many regimes of interest, such systems can often be modeled with a few dominant degrees of freedom. However, because these “few-mode” approaches are based on model assumptions, researchers have questioned the applicability to more extreme experimental platforms. These feature strong couplings within the system or leakage to the environment, and the resulting changes to the system dynamics have moved them into the focus of current research. Here, we promote few-mode models to a fundamental theory, thereby providing access to exact solutions in extreme parameter regimes.

Specifically, we first show how to derive few-mode Hamiltonians for noninteracting quantum potential scattering problems, and how to reconstruct the full scattering information using a first-principles input-output formalism. On this basis, we develop a systematic effective few-mode expansion for interactions, where the continuum in an open quantum system is boiled down to a few relevant degrees of freedom. Compared to standard phenomenological models, this ab initio treatment provides new predictions for qualitatively different phenomena in extreme parameter regimes.

From a more general perspective, our results make few-mode theory a rigorous tool for quantum scattering systems, allowing the extraction of relevant degrees of freedom or resonances. We therefore expect that our method will find applications for general scattering problems, and so further the exchange of concepts and methods between currently separated fields.

Figure 1

Schematic of the theoretical connections on noninteracting theories presented in this paper. The left-hand side represents the NMA to quantum potential scattering, where one can rigorously obtain a Hamiltonian from canonical quantization. The latter is conveniently expressed in terms of normal-mode operators $\widehat{c}$ (top left picture). The right-hand side represents the FMA, usually employed in terms of phenomenological models. There, a discrete set of system modes is coupled to a continuum of bath modes (top right picture), corresponding to operators $\widehat{a}$ and $\widehat{b}$, respectively, with coupling constant $W$ and loss rate $\mathrm{\Gamma }$. We show that the Hamiltonians on each side can be connected by a basis transformation. A common method to calculate scattering observables in the FMA, known as the input-output formalism, can further be connected to standard scattering theory by the inclusion of a background scattering contribution. Having shown that the Hamiltonians as well as the methods for calculating scattering observables can be rigorously connected allows the normal-mode and the few-mode approach to quantized potential scattering theory to be regarded as equivalent.

Figure 2

Schematic illustrating the part of the paper on interacting systems, in particular, the few-mode approximation and constructive approach to choosing few-mode bases for an ab initio effective few-mode expansion. After introducing system modes and bath modes (see also Fig. 1), the few-mode approximation consists of neglecting the interaction of the interacting subsystem (e.g., a two-level atom located inside the cavity with transition frequency ${\omega }_a$) with the bath modes. In the figure, an example of a two-mode basis is shown as the states inside the magenta shaded box. Ideally, the set of system modes is chosen exploiting physical insight into the system under study, to facilitate the modeling of the system with as few modes as possible. In the absence of any prior knowledge, a constructive approach can be used to determine a few-mode basis. For this approach, a locally complete basis (states inside the black box) is found as solutions to the Dirichlet boundary value problem in the potential region, which, in general, contains infinitely many modes. A few-mode basis is then given by a subset of the locally complete basis. Varying the number of modes in the few-mode basis and performing the few-mode approximation in each case yields a systematic expansion scheme.

Figure 3

Schematic of the different d.o.f. and scattering connections in the Feshbach projection formalism for potential scattering problems. The full scattering matrix $S$ can be used to relate asymptotically free states or operators to each other. The background scattering $S_{\mathrm{bg}}$ arises from a basis transformation of the free states into the bath states. The bath states scatter via $S_{\mathrm{res}}$ on the system states, which span part of the region where the scattering potential $V(r)$ is nonzero. Similarly on the operator level, the scattering between bath operators that are coupled to the system is given by the input-output scattering matrix $S_{\mathrm{io}}$. To obtain the full scattering matrix, a background scattering contribution ${\tilde{S}}_{\mathrm{bg}}$ has to be applied.

Figure 4

Model potential with two barriers. In the Maxwell case, this model corresponds to the Ley-Loudon model for a two-sided Fabry-Perot cavity [101], and the solid blue curve shows the spatial refractive index distribution. For simplicity, in the calculation, the thin-mirror limit $d\to 0$ is considered, with $n_0\to \infty$ such that $\eta =n_0^{2}d$ remains finite [95, 101]. In the cavity, the first two perfect-cavity modes ${\chi }_1$ and ${\chi }_2$ are shown as magenta curves. For the Schrödinger case, the solid blue curve indicates the potential energy, which defines a tunneling problem.

Figure 5

Transmission spectra for a Fabry-Perot cavity as a function of the mirror quality $\eta$. The top row shows the case in which the system comprises the single mode $\lambda =8$. The bottom row shows corresponding results with the system consisting of the modes $\lambda \in \{7,8,9,10\}$. In both cases, the left column illustrates the full transmissivity of the system. The middle and right columns show the input-output ($S_{\mathrm{io}}$) and the background ($S_{\mathrm{bg}}$) contributions, respectively. The full result can be obtained either from standard methods such as a transfer matrix formalism also known as Parratt’s formalism [102, 103] or as a product of the input-output and background scattering matrices.

Figure 6

Quantum optical parameters calculated via the ab initio few-mode theory. The upper row shows the transmission coupling strength ${\kappa }^{(T)}=2\pi |\mathcal{W}{|}^2$, the mode frequency shift $\mathrm{\Delta }$, and the resonance width $\gamma$ as a function of the frequency and mirror quality $\eta$. The parameters are as in the upper row in Fig. 5, with the system comprising the single mode $\lambda =8$. The magenta curves indicate the width of the resonance as a function of $\eta$. The lower row shows cuts through the upper panels at fixed $\eta =0.01$, 0.1, 0.19 (left to right, corresponding to the transition from a bad to a good cavity) indicated as dotted lines in the upper panel. The respective widths of the modes are indicated as shaded magenta regions, defined as twice the value of $\gamma$. The vertical dashed lines indicate the bare center frequency of the mode (black) as well as the actual center frequency (magenta).

Figure 7

Transmission spectra for a Schrödinger (left column) and a Maxwell (right column) Fabry-Perot cavity, with different sets of system modes (top, $\lambda =1$; middle, $\lambda \in \{1,2\}$; bottom, $\lambda \in \{1,2,\dots ,100\}$). In each case, the full (solid blue curve), input-output (dashed red curve), and background (dotted orange curve) transmissivity are shown. For both wave equations, each added system mode transfers a resonance peak from the background to the input-output contribution, such that, in the many-mode case featuring 100 modes, the input-output result alone agrees with the full transmissivity (bottom panels). For sharp resonance peaks, the input-output result captures the behavior in the relevant energy range, if the corresponding modes are included (top and middle left panels). For overlapping resonances, the background contribution is crucial even in the vicinity of the resonance peak (top and middle right panels).

Figure 8

Linear transmission spectra of a coupled atom-cavity system. The cavity is chosen as in Fig. 5, and the transition frequency of the atom with $d=0.01$ is chosen resonant with the ninth empty cavity mode at each mirror quality $\eta$. (a) shows the full transmissivity calculated using the linear dispersion theory as a reference. (b) shows the input-output part without the background contribution which is shown in (c), each calculated with three system modes (${\mathrm{\Lambda }}_Q=\{{\chi }_7,{\chi }_9,{\chi }_{11}\}$). (d)–(f) show slices at $\eta =0.289$, 0.124, and 0.011, respectively. They correspond to a transition from strong coupling (d) via weak coupling (e) to a regime with negligible cavity confinement (f). In the entire range, the ab initio few-mode result ($S_{\text{few}}=S_{\mathrm{bg}}{S}_{\mathrm{io}}$, red dash-dotted line) agrees well with the full result from linear dispersion theory ($S$, blue solid line), with good convergence already found using a single mode ${\chi }_9$ for (d) and (e) and using three modes ${\mathrm{\Lambda }}_Q$ for (f). In the weak and strong coupling regimes, the input-output term alone is sufficient to model the interacting system (d),(e). But in the regime of strongly overlapping modes and weak confinement, background scattering plays a crucial role (c), such that input-output scattering alone gives a vastly different line shape from the full result (f).

Figure 9

Convergence behavior of the few-mode expansion in the strong coupling (a),(c) and multimode strong coupling (b),(d) regimes. (a)–(d) show few-mode spectra at different numbers of system modes ($N_{\text{modes}}\in \{1,3,10,255\}$; see the legend). The solid blue line shows linear dispersion spectra as a benchmark ($S^{(T)}$). In the strong coupling case, the convergence is fast, with a single mode being sufficient (c), as expected. At multimode strong coupling, the qualitative behavior is already captured in a single-mode description, but for quantitative agreement more than 100 modes of the generic mode basis are necessary (d). The convergence in the two cases can be quantified by the few-mode (FM) deviation Eq. (89). (e) shows this quantity as a function of the mode number ($N_{\text{modes}}+1$). The strong coupling case (blue dots) shows the expected behavior. In the multimode strong coupling case (mmsc, orange crosses, green hatches), the convergence is much slower, and, for low mode numbers, the convergence depends on the order in which the modes are added to the few-mode system space, as explained in the main text.

Figure 10

One-dimensional double cavity potential. The $Q$-space basis modes are chosen as solutions of the Dirichlet problem in the left cavity (the figure shows mode ${\chi }_9$ in magenta as an example). This choice of basis appears particularly suitable for describing local light-matter interactions in the left cavity but not in the right cavity. An atom is placed at the center of the left cavity, chosen resonant with the ninth cavity mode.

Figure 11

Linear transmission spectra for the double Fabry-Perot cavity in Fig. 10 as a function of the central mirror refractive index, calculated using the ab initio few-mode theory with a single system mode (${\chi }_9$). (a) shows the empty cavity without an atom. A transition between different regimes featuring an avoided crossing is observed. In (b), results of the interacting system with an atom at the center of the first cavity are shown. The atom’s resonance frequency is chosen resonant with the empty cavity spectral peak, indicated by the magenta dots. A splitting of the resonant cavity mode is observed, with interesting features in the transition regions. The shown few-mode results are found to agree well with linear dispersion theory in both cases.

Figure 12

Quantum optical quantities corresponding to the atom-cavity spectra in Fig. 11. The Purcell-enhanced linewidth ${\gamma }_S$ of the atom and its cavity-modified Lamb shift ${\mathrm{\Delta }}_{\mathrm{LS}}$ as well as ${\kappa }_{\text{atom}}^{(T)}=2\pi |[{\underset{\_}{\underset{\_}{\mathcal{W}}}}^{†}(\omega ){\underset{\_}{\underset{\_}{\mathcal{D}}}}^{-1}(\omega ){\underset{\_}{g}}^{*}{\underset{\_}{g}}^T{\underset{\_}{\underset{\_}{\mathcal{D}}}}^{-1}(\omega )\underset{\_}{\underset{\_}{\mathcal{W}}}(\omega ){]}_{1,0}|$ are shown. These quantities are the atomic line analogs of the empty cavity couplings shown in Fig. 6 and are computed using the ab initio few-mode theory with a single system mode (${\chi }_9$). (d)–(f) show slices at $n_{\mathrm{mid}}=2.7$, 7.0, and 15.0, respectively, demonstrating the transition between different regimes and that even the single-mode theory is able to include effects beyond the isolated resonance approximation.

Figure 13

Nonlinear spectra as a function of the driving strength, corresponding to each of the regimes in Fig. 12. (a)–(c) show results for $n_{\mathrm{mid}}=2.7$, 7.0, and 15.0, respectively. The cavity is driven from one side, such that ${\underset{\_}{b}}^{(\text{in})}=b_{\text{in}}(\begin{array}{c}1\\ 0\end{array})$, and the corresponding transmission spectra are shown. At $b_{\text{in}}=0$, the spectrum coincides with the linear interaction spectra shown in Fig. 11 at each $n_{\mathrm{mid}}$. At large $b_{\text{in}}$, the corresponding empty cavity spectra in Fig. 11 are approached due to saturation of the atom excitation. The atom resonance frequency for each spectrum is chosen as before and is indicated by the magenta arrows.

Figure 14

A cavity geometry whose Schrödinger analog is solved exactly by Domcke [71] using the Feshbach projection formalism.