Probing weak dipole-dipole interaction using phase-modulated nonlinear spectroscopy

Phase-modulated nonlinear spectroscopy with higher harmonic demodulation has recently been suggested to provide information on many-body excitations. In the present work we theoretically investigate the application of this method to infer the interaction strength between two particles that interact via weak dipole-dipole interaction. To this end we use a full numerical solution of the Schrödinger equation with time-dependent pulses. For interpretation purposes we also derive analytical expressions in perturbation theory. We find one can detect dipole-dipole interaction via peak intensities (in contrast to line-shifts which typically are used in conventional spectroscopy). We provide a detailed study on the dependence of these intensities on the parameters of the laser pulse and the dipole-dipole interaction strength. Interestingly, we find that there is a phase between the first and second harmonic demodulated signal whose value depends on the sign of the dipole-dipole interaction.

Figure 1

(a) Arrangement of the particles and the laser polarization. (b) Illustration of two pulse pairs where $T_{\mathrm{Rep}}$ is the repetition time between pairs and $t_{21}$ is the pulse delay between the pulses within each pair.

Figure 2

Schematic diagram of collective energy levels of two particles (a) without and (b) with dipole-dipole interaction (only symmetric states are shown). The arrows indicate dipole-allowed transitions. The corresponding energy differences are also indicated. Note that we ignore direct interactions between two excited particles (like van der Waals interactions) which would lead to additional level shifts. In the table the collective eigenstates and the corresponding energies are specified.

Figure 3

The real (solid red line) and imaginary (dashed green line) parts of the 1HD (left column with $\kappa =1$) and 2HD (right column with $\kappa =2$) spectrum $S(\omega ,\kappa )$ in Eq. (11) for various dipole-dipole interactions. From top to bottom: $V_{ee}=0.0005$, 0.001, 0.005, and 0.01, and $V_{ff}=1.974V_{ee}$. The parameters of the pulse are $E_1=E_2=0.00384$, $\sigma =10.4$ ($\int A\left(t\right)dt=0.1$), and ${\delta }_{eg}\equiv {\omega }_{eg}-{\omega }_{\mathrm{L}}=-0.025$. Note the different intensity scales in each panel. All energies are in units of ${\omega }_0$.

Figure 4

Same as Fig. 3, but now for negative dipole-dipole interactions: $V_{ee}=-0.0005$, $-0.001$, $-0.005$, and $-0.01$, and $V_{ff}=1.974V_{ee}$.

Figure 5

Peak height (a) 1HD and (b) 2HD resonances corresponding respectively to ($S_e,S_f$) and ($S_{ee},S_{ef},S_{ff}$) signals [see, e.g., Eqs. (25), (29), and (30)]. The filled circles, stars, and squares are from numerical simulations while the solid, dashed, and dot-dashed lines from the Fourier transformation of analytical results [i.e., $S^{\left(1\right)}\left(t_{21}\right)$ and $S^{\left(2\right)}\left(t_{21}\right)$ in Eqs. (24) and (28)] of Sec. 4. Other parameters are the same as those in Fig. 3. The deviation between analytical and numerical calculations is caused by the approximation of nonoverlapping pulses made in the analytical calculations. We also performed calculations with shorter pulse length and found much better agreement. One example is shown in Appendix pp1. The thin dashed blue line and the empty triangles emerge from the solid blue curve via the transformation $-V_{ee}\to +V_{ee}$ and a sign change of the peak amplitude. The curves for $S_{ee}$ and $S_{ff}$ are symmetric with respect to the origin.

Figure 6

Peak height of (a) 1HD and (b) 2HD resonances corresponding respectively to 1HD ($S_e,S_f$) and 2HD ($S_{ee},S_{ef},S_{ff}$) signals versus laser pulse width. The filled circles, stars, and squares given by harmonic spectrum $S(\omega ,\kappa )$ in Eq. (11) are from numerical simulations, while solid, dashed, and dot-dashed lines are from the Fourier transformation of analytical results [i.e., $S^{\left(1\right)}\left(t_{21}\right)$ and $S^{\left(2\right)}\left(t_{21}\right)$ in Eqs. (24) and (28)] of Sec. 4. The dipole-dipole interaction is $V_{ee}=0.01$ and all pulses are normalized to $\int A\left(t\right)dt=0.1$. The other parameters are the same as those in Fig. 3.

Figure 7

Peak height of (a) 1HD and (b) 2HD resonances corresponding, respectively, to 1HD ($S_e,S_f$) and 2HD ($S_{ee},S_{ef},S_{ff}$) signals versus powers of the electric field strength. The filled circles, stars, and squares given by harmonic spectrum $S(\omega ,\kappa )$ in Eq. (11) are from the fully numerical simulation (Sec. 3) while the solid, dashed, and dot-dashed lines are given by the Fourier transformation of analytical results [i.e., $S^{\left(1\right)}\left(t_{21}\right)$ and $S^{\left(2\right)}\left(t_{21}\right)$ in Eqs. (24) and (28)] evaluated for Gaussian pulses. The dipole-dipole interaction is $V_{ee}=0.01$ and all other pulse parameters such as the pulse width are kept constant. The other parameters are the same as those in Fig. 3.

Figure 8

Peak height (a) 1HD and (b) 2HD resonances corresponding, respectively, to ($S_e,S_f$) and ($S_{ee},S_{ef},S_{ff}$) signals. The empty ($\sigma =6$) and filled ($\sigma =10.4$) circles, stars, and squares are from simulation while dashed ($\sigma =6$) and solid ($\sigma =10.4$) lines from the analytical results. Note that the results in Fig. 5 are reproduced here for comparison purposes.

Figure 9

Double-sided Feynman diagrams that contribute to the first harmonic (a, b) and second harmonic (c–n) demodulated signals. Below each diagram the resulting amplitude is given. A factor of 2 indicates that in the final state there are two particles excited, i.e., two photons will be emitted. Here the phase is defined as ${\varphi }_i^{jk\pm }=({\omega }_{j\mathrm{g}}\pm V_{kk}-{\omega }_{\mathrm{M}})t_i$ for demodulated signals while it becomes ${\varphi }_i^{jk\pm }=({\omega }_{j\mathrm{g}}\pm V_{kk})t_i-\left({\mathrm{\Omega }}_i{\tau }_m+{\varphi }_i^{\left(0\right)}\right)$ ($i=1,2$, $j=e,f$) before demodulation where only ${\varphi }_i^{j\pm }\left(={\varphi }_i^{jj\pm }\right)$ and ${\varphi }_i^{jk-}(j\ne k)$ are involved in the Feynman diagrams.