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We performed non-collinear DFWM experiments as illustrated in Fig. 1, using THz pulses from the free electron laser FELIX, both on and off resonance with the 1s → 2p transitions in Si: P and Si: Bi at 10 K. We chose this geometry because it enables the measurement of dynamical relaxation and dephasing times needed to make detailed theoretical comparisons, under identical experimental conditions. It is very difficult to obtain clean beam profiles with low diffraction in the THz regime, and great care was taken in avoiding apertures and optical imperfections in order to obtain them, as shown in Fig. 1. Care was also taken to accurately calibrate absolute pulse energies. It may be seen immediately from the relative strength of the output beam (k3) in Fig. 1 that the DFWM process is very efficient.
In the plane-wave limit (i.e., for infinitely long pulses and infinitely broad beams), the complex polarisation amplitude of the generated beam (${{\mathcal{P}}}_{3}$) is related to the complex field amplitudes of the input beams (${{\mathcal{F}}}_{1, 2}$) inside the material by
$$ {{\mathcal{P}}}_{3}={\epsilon }_{0}{\chi }^{(3)}{{\mathcal{F}}}_{1}^{* }{{\mathcal{F}}}_{2}^{2} $$ (1) i.e. the intensity of the output is determined by χ(3). The definition of χ(3) in Eq. (1) suggests that, for a pulsed experiment, the internal pulse energies Ei of the three beams ki (Fig. 1) are related by
$$ {E}_{3}={E}_{1}{E}_{2}^{2}/{E}_{{\rm{c}}}^{2} $$ (2) where Ec is a constant that is inversely proportional to χ(3)L and L is the sample thickness. Ec defined by Eq. (2) is a critical pulse energy at which the output would become equal to the inputs, and we generally stay well below this limit so as to avoid the need to consider higher-order non-linear effects.
We varied E1 keeping the ratio E2/E1 fixed, as shown in Fig. 2, and a clear cubic dependence is observed at low pulse energy. The resulting values of Ec are shown on Fig. 2 and given in Table 1. At high intensity, a saturation occurs for resonant cases, due to an intensity-dependent reduction in dephasing time15, which reduces χ(3).
Fig. 2 Internal DFWM conversion efficiency for different samples, both on and off resonance.
The different doping densities (nD) and samples are given in the legend. Each curve is labelled by either the laser photon energy ( ω in meV) or the resonant transition being excited. The ratio between pump pulse energies x = E2/E1 was kept constant in each case: values of x are given on each data set. The data are very close to cubic ($\hbar$ as expected), and the solid lines are fits to the low intensity portion. The fitted values of Ec are also indicated. For the high density Si: P sample, only one intensity was measured at each laser frequency and a cubic dependence (dashed lines) is shown for comparison with the other measurements${E}_{3}={x}^{2}{E}_{1}^{3}/{E}_{{\rm{c}}}^{2}$ Si: P Si: Bi ħω (meV) 32.5 34 36.7 39.2 42.5 64.5 - (2p0) - (2p±) (3p±) (2p±) T R T T R R R L (mm) 0.6 0.5 0.5 0.6 0.5 0.5 1 nD 10 1 1 10 1 1 3.4 x 1.6 4.7 5.6 2.3 4.6 3.9 4.9 r0 (mm) 0.53 0.6 0.6 0.53 0.6 0.6 0.64 Ec (μJ) 2.7 4.9 32.3 1.1 1.4 2.1 0.17 f 3 28 6.1 6.2 310 27 310 χexpt(3)L 0.13 0.8 0.025 0.58 27 1.5 160 χexpt(3) 0.22 1.6 0.05 0.96 54 2.9 160 χexpt(3)/nD 0.022 1.6 0.05 0.096 54 2.9 46 μeg (e.nm) - 0.37 - - 0.71 0.32 0.34 ħ/T1 (μeV) - 11 - - 5 3.9a 19 ħ/T2 (μeV) - 26 - - 26 109 44 ħ/T2* (μeV) - 115 - - 115b 194 165 χtheory(3)/nD 0.0024 100 0.015 3100 23 18 ħω is the photon energy, and labels R and T refer to resonant and transparent excitations. Values of μeg are all taken from ref. 29. All values for T1, 2 were found from photon echo and pump-probe performed under the DFWM conditions, except: ataken from ref. 21. All values of the half-width, ħ=T2*, were found from the small-signal absorption spectrum, except: bassumed equal to the 2p0 halfwidth. x is the ratio of the intensities of the pump pulses from Fig. 2. L is the sample thickness and r0 is the spot size. The dimensionless factor f, which is unity for zero loss and infinitely long pulses, appearing in Eq. (3) (and described in detail in the text), was found from integrating the propagation equations. The experimental values of Ec were extracted from Fig. 2. Values of nD are given in units of 1015 cm−3; χ(3)L in units of 10−16 m3 V−2; χ(3) in units of 10−13 m2 V−2; and χ(3)/nD in units of 10−34 m5 V−2. Theoretical predictions are from Eq. (4), and for off-resonance excitation at 36.7 meV, the 2p± contribution was used (it has much higher μeg4) while at 32.5 meV we used the 2p0 contribution (it has much smaller Δ) Table 1. Third-order susceptibility for Si: P and Si: Bi both on and off-resonance
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Away from resonance and in the limit of long pulses, the relationship between Ec (given on Fig. 2) and χ(3) has a straightforward dependence on the geometry and pulse duration. For short pulses, the dynamics are important, and on resonance there is loss that attenuates the pumps and the output, which must also be taken into account. We integrated the equations describing propagation of light through a lossy non-linear medium for the case of inhomogeneous broadening and finite pulse durations to find χ(3) from Ec (see Supplementary Materials Section Ⅳ). In this case, the conversion from the experimental Ec of Fig. 2 to the value of χ(3) is, for a beam with a Gaussian spatial profile,
$$ {E}_{{\rm{c}}}={3}^{3/4}\sqrt{2\pi }{n}^{2}{\lambda }_{0}{r}_{0}^{2}{t}_{0}f/{Z}_{0}{\chi }^{(3)}L $$ (3) where n is the refractive index (which we took to be n = 3.4), λ0 is the free-space wavelength, Z0 is the characteristic impedance of free space, and r0 and t0 are the root mean square (r.m.s.) beam radius and pulse duration, respectively (at which the intensity has fallen by $1/\sqrt{e}$).
The dimensionless factor f appearing in Eq. (3) depends on the loss and also the pulse shape and duration relative to the dynamical timescales of the system. The equation defines f in such a way that f = 1 when the loss is negligible (which is our case when far from resonance) and in the monochromatic limit of very long pulses with Gaussian temporal profile (t0 ≫ T1, 2, i.e. pulse duration much larger than the population decay, T1, and dephasing time, T2, of the system). For negligible loss but with pulses that are very short compared with the inverse line-width, then f becomes of order T1/t0, which can evidently be larger than unity (effectively replacing t0 in Eq. (3) with T1 because now the atomic polarisation ${{\mathcal{P}}}_{3}$ lasts much longer than the drive pulses). For significant loss, f becomes very large and sample thickness dependent.
Using perturbation theory for temporally overlapping, weak beams within the two-level approximation1, and averaging over the distribution for a Gaussian (fully inhomogeneously broadened) line, we calculated values of f for our experimental circumstances. See Supplementary Materials for more details. The results are shown in Table 1. As expected, the off-resonant values of f in Table 1 are of order unity and are not significantly affected by the details of the model chosen. They are slightly greater than unity primarily because of the short pulses. The on-resonance values of f in Table 1 are large primarily because of the loss. The two-level model is expected to give a reasonably good estimate of f in resonant cases because there is one dominant transition: the one shown in Fig. 1a1.
The experimental values of Ec from Fig. 2 along with the calculated f have been converted to values of ${\chi }_{\, \text{expt}\, }^{(3)}$ in Table 1.
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We now obtain theoretical estimates for χ(3) to compare with the experimental results. Silicon donors at low temperature are hydrogen-like, with a series of levels and orbitals closely resembling the Rydberg series 1s, 2p0, etc.16. The energies are scaled down and the orbital sizes scaled up, thanks to the small effective mass and large dielectric constant. The large orbitals give a commensurately large dipole moment, and this has a very large influence on non-linear optical coefficients.
Using the same two-level model mentioned above, the following limits may be found (see Supplementary Materials) for the contribution per bound electron in the vicinity of its resonance:
$$ \frac{{\chi }^{(3)}}{{n}_{{\rm{D}}}}\approx \frac{{\mu }_{{\rm{eg}}}^{4}}{{\epsilon }_{0}{\hbar }^{3}}\times \left\{\begin{array}{ll}{T}_{1}{T}_{2}{T}_{2}^{* }, &\, \text{if}\, \ {{\Delta }}=0\\ {T}_{1}{T}_{2}^{-1}{{{\Delta }}}^{-3}, &\, \text{if}\, \ | {{\Delta }}| \gg {T}_{2}^{* -1}\end{array}\right. $$ (4) where nD is the donor concentration, $\hbar$Δ is the detuning from resonance in energy and ${\mu }_{{\rm{eg}}}=e| \langle {\psi }_{e}|{\bf{r}}|{\psi }_{g}\rangle.{\boldsymbol{\epsilon }}|$ is the component of the dipole moment transition matrix element between ground and excited states along the polarisation direction, ϵ. The total dephasing time ${T}_{2}^{* }$ is defined by the total absorption line half-width in energy, $\hbar /{T}_{2}^{* }$, which was obtained from the small-signal absorption spectrum. The population relaxation time, T1, was obtained by performing a pump-probe experiment17, and the homogeneous dephasing time, T2, was obtained using a photon echo experiment1, 15. The results are shown in Table 1. These time-resolved experiments were performed with the same set-up that was used for the main DFWM experiment, simply by varying the delay between the beams and changing the position of an iris after the sample. This ensures that times T1, 2 were obtained under the same experimental conditions as Fig. 2. The calculated values of χ(3)/nD in the approximation of Eq. (4) are shown in Table 1 as ${\chi }_{\, \text{theory}\, }^{(3)}/{n}_{{\rm{D}}}$. These predictions from the two-level model may be expected to give reasonable order of magnitude estimates, but it should be noted that the intermediate states and permutations neglected in the approximation of Eq. (4) can give both positive and negative contributions. Earlier work on theoretical prediction of χ(3) for silicon donors has included an infinite number of all possible intermediate states but not the dephasing and decay (T1, T2 and ${T}_{2}^{* }$)13.