Multiphoton absorption in optical gratings for matter waves

We present a theory for the diffraction of large molecules or nanoparticles at a standing light wave. Such particles can act as a genuine photon absorbers due to their numerous internal degrees of freedom effecting fast internal energy conversion. Our theory incorporates the interplay of three light-induced properties: the coherent phase modulation due to the dipole interaction, a nonunitary absorption-induced amplitude modulation described as a generalized measurement, and a coherent recoil splitting that resembles a quantum random walk in steps of the photon momentum. We discuss how these effects show up in near-field and far-field interference schemes, and we confirm our effective description by a dynamic evaluation of the grating interaction, which accounts for the internal states.

Figure 1

(a) Sinusoidal visibility (23) of KDTLI as a function of the Talbot parameter $L/L_{\mathrm{T}}$, i.e., the grating distance over Talbot length $L_{\mathrm{T}}$. The case of a pure phase grating (solid blue line, ${\varphi }_0=\pi$ at ${\sigma }_{\mathrm{abs}}=0$) is compared to an absorptive molecule with $n_0=1$ (shaded area). A classical random-walk description of absorption (dashed red line) does not match the correct quantum prediction. (b) Visibilities (25) of the constituent conditional interferograms corresponding to $\ell =0,1,$ and 2 photon absorptions (solid, dashed, and dash-dotted lines, respectively). The shaded area represents the unconditional visibility as in (a). The opening fraction is $f=0.42$ both plots.

Figure 2

(a) and (b) show the predicted conditional $\ell$-dependent fringe patterns (24) underlying the visibilities in Fig. 1. They are plotted as a function of the lateral shift $x_{\mathrm{s}}$ of the third grating at two fixed Talbot parameters $L/L_{\mathrm{T}}=3.25$ and $4.25$. Each panel contains the stacked conditional interferograms for molecules absorbing $\ell =0$, 1, and 2 photons (thin lines, lower mean signal for greater $\ell$ values), as well as the weighted sum over all $\ell$ (unconditional interferogram, thick line on top).

Figure 3

Sketch of a symmetric far-field configuration for molecular diffraction at a standing-wave grating. A collimated beam of molecules, as produced by an incoherent pointlike source in combination with a collimation slit of width $D$ at distance $L$, is diffracted at a standing laser wave with grating period $d$. The resulting far-field interference pattern can be recorded by a spatially resolving detector in distance $L$ to the laser grating.

Figure 4

Far-field interferograms on the screen plane behind a laser grating for nonabsorbing (shaded area) and for absorbing (solid line) molecules, assuming state-indiscriminate detection. (a) and (b) correspond to two different absorption strengths, $n_0=2$ and $n_0=10$, respectively. The shaded curves represent diffraction at a pure phase grating ($n_0=0$) at ${\varphi }_0=2.5$. The screen coordinate is given in units of the expected separations $\mathrm{\Delta }x$ of the coherent diffraction maxima. (c) Conditional far-field interferograms contributing to the unconditional fringe signal in (a); the curves are shifted vertically for better illustration. The thin lines from top to bottom are the result of evaluating (28) for $\ell =0,1,2,3$ photon absorptions, respectively. The thick line represents the unconditional result depicted in (a), i.e., the incoherent sum over all contributions. For this plot, we assume a finite detector resolution of $0.1\mathrm{\Delta }x$ and a collimator width of $D/d=10$.

Figure 5

(a) Sinusoidal visibility as a function of the Talbot parameter $L/L_{\mathrm{T}}$ for molecules with different excited-state parameters. For the solid curve, we assume that the ground-state values of polarizability and absorption cross section (corresponding to $n_0=1.5$ and ${\varphi }_0=1.25n_0$) remain the same no matter how many photons are absorbed. The dashed and the dash-dotted curve correspond to an increase of the polarizability and of the absorption cross section, respectively, by a factor of $1.5$ upon absorption of the first photon. (b) Same visibilities as a function of laser power, i.e., for varying $n_0$ at a fixed Talbot parameter of $L/L_{\mathrm{T}}=2.2$. For reference, the red dot marks the same spot $(L/L_{\mathrm{T}},n_0)$ in both panels. A significant difference between the curves appears at high laser powers.

Figure 6

Scheme of a three-level Rabi model, where the laser drives the transition between the ground state $|0\rangle$ and the excited state $|1\rangle$ at a detuning $\mathrm{\Delta }$ off resonance. The excited state with lifetime $\tau$ can decay without emission into a dark metastable state $|2\rangle$.

Figure 7

(a) Probability density for the transmission of ground-state molecules through a standing wave of period $d={\lambda }_{\mathrm{L}}/2$ as a function of position. The shaded area corresponds to the conditional probability for zero absorptions, taken from the ladder model (26) with $n_0=1.2$. An evaluation of the coherent three-level Rabi model yields the blue solid line, assuming ${\mathrm{\Omega }}_0{t}_{\mathrm{L}}=4\pi$ (i.e., a $4\pi$ pulse length at the antinodes) and a long excited-state lifetime of $\tau =t_{\mathrm{L}}$. (b)–(e) Numerical results for the KDTLI fringe signal in the presence of molecular Rabi oscillations, according to the three-level model on resonance ($\mathrm{\Delta }=0, \tau =t_{\mathrm{L}}$). We use a setup with $L=2L_T$ grating separation, and $f=0.1$ opening fraction for G1 and G3; only ground-state molecules are detected. The antinode intensity is chosen such that it amounts to an effective pulse length ${\mathrm{\Omega }}_0{t}_{\mathrm{L}}$ increasing from $2\pi –8\pi$ in (b)–(e), respectively. We observe higher-order fringes emerging with each Rabi cycle.