Temporal and spectral dependence of photo-bleaching and -ionization

We shall mainly be concerned with the transition from $$ ∣g⟩=∣1s⟩$$ to $$ ∣b⟩=∣2p_{\pm }⟩$$ at ħω = 39.2 meV. The pump-induced change in probe transmission as a function of the time delay is shown in Fig. 2 for a variety of intensities and wavelengths on or close to this resonance. At the lowest intensity on resonance, the transient is positive (induced bleaching) and the recovery is exponential with time constant T₁ = 93 ± 1 ps. The dynamics are very different for off-resonance excitation and a clear intensity-dependent induced absorption (negative induced transmission) is visible, especially on the red side of resonance (Fig. 2).

Fig. 2

Pump-probe transients on or near the 1s–2p_± transition for different laser intensities.

The transient transmission change of the probe due to the pump pulse, ΔT is shown vs. t. They have been normalized to their global maximum and offset vertically. a Pump pulse energy (E_p) dependence on resonance (at laser photon energy ħω = 39.2 meV) for ^natSi:P. The solid line is an exponential decay fit. b As in a but off-resonance at ħω = 38.6 meV. c Dependence of the transients on photon energy (ħω) at a fixed pulse energy of E_p = 0.11 μJ. The solid lines are fits with Eq. (3) as described in the text.

The small-signal absorption spectrum of the sample is shown in Fig. 3a, showing the hydrogenic Lyman series. To illustrate the spectral dependence of the laser experiment more clearly, we fixed the time delay (at t = 15 ps) and swept the photon energy, as shown in Fig. 3b. Strongly asymmetric features are evident, with negative signals (induced absorption) on the red side, notably for 1s → 2p₀ and 1s → 2p_±. These negative signals are also visible in Fig. 2b, which was taken near the 1s → 2p_± transition with the laser red-shifted to the point that its spectrum does not overlap with the small-signal absorption (blue curve on Fig. 3a), close to the minimum in Fig. 3b. The induced absorption effect disappears on the timescale of the electronic relaxation back to the donor ground state, which rules out any thermal or purely optical interference effects.

Fig. 3

Spectral dependence of induced transmission and continuum structure with the Fano profile.

a The small-signal absorbance spectrum. The main transitions are labelled according to the excited states (all originating from the 1s(A₁) ground state). The laser spectrum corresponding to the off-resonance photon energy used in Fig. 2c is shown as a blue line. b Wavelength dependence of the pump-probe signal, ΔT/T, with fixed pump power (same as in Fig. 2c) and fixed time delay of t = 15ps between pump and probe as a function of photon energy, ħω. The solid curve is the result of a fit to the data using Eq. (1) as described in the Methods. c The recombination rate (blue symbols), 1/t_R, and bleaching (orange symbols), B, found from fitting the data in Fig. 2c. The solid curves are the results of a global fit using Eq. (1) for the ionization (blue line) and a Lorentzian for the bleaching (orange line). The top axis shows the detuning Δ in units of the width from the fit, Γ.

Extraction of probe absorption coefficients and decay constants

We now wish to extract from Fig. 2c the frequency dependence of the ionization density produced by the two-photon pumping, by separating the effects of the excitation/bleaching/relaxation from the photo-ionization/induced absorption/recombination³⁰ (see ‘Methods’). As illustrated in Fig. 1c, the pump generates excitation from $$ ∣g⟩$$ to $$ ∣b⟩$$ and two-photon ionization from $$ ∣g⟩$$ to $$ ∣c⟩$$. The population in the excited state, n_b, follows a rate equation of the form dn_b/dt = −n_b/T₁, which has an exponential decay solution $$ n_{\mathrm{b}}\left(t\right)=n_{\mathrm{b}}\left(0\right)\exp \left(-t/T_1\right)$$, where the time constant is T₁. The recombination of free electrons with ionized donors follows a rate equation of form $$ d{n}_{\mathrm{c}}/dt=-R{n}_{\mathrm{c}}{n}_{\mathrm{i}}=-R{n}_{\mathrm{c}}^2$$ where n_c is the density of free electrons and n_i is the density of ions, and n_i = n_c by charge conservation. R is the recombination rate coefficient³³. This differential equation has so-called reciprocal decay solutions of the form $$ n_{\mathrm{c}}\left(t\right)=n_{\mathrm{c}}\left(0\right){\left(1+t/t_{\mathrm{R}}\right)}^{-1}$$ where

Fano spectrum of photo-ionization due the pump

From Eq. (2), the inverse recombination time is simply proportional to the ionized electron density left behind by the pump and it is clear from the 1/t_R data (Fig. 3c) that the ionization is strongly asymmetric with a Fano-like spectrum. We therefore wish to extract the Fano shape index and width. It can be shown that B = (2σ_gb − σ_bc)n_b(0)L and C = (σ_cc − σ_gb)n_c(0)L (see ‘Methods’) where L is the sample thickness (Fig. 1c right), σ_gb is the absorption cross-section for the resonant excitation from $$ ∣g⟩\to ∣b⟩$$, σ_bc is the cross-section for photo-ionization from the excited state, and σ_cc is the FCA cross-section. The bound-free and free-free cross-sections, σ_bc,cc, are slowly varying with frequency (the standard Drude model for FCA produces σ_cc ∝ ω⁻²) and we take them to be constant over the range of each feature in Fig. 3b, c. We further assume that bound-free absorption is weak compared with resonant absorption, which simplifies B. It is clear from the B data (Fig. 3c) that the excitation has a simple, symmetric, resonant spectrum, and so we assume that σ_gb and the excited density n_b(0) have a Lorentzian spectrum of the same half-width γ. As mentioned above, we assume that n_c(0) is given by Eq. (1). With these assumptions,

Multi-photon ICS

We now consider the Fano theory¹ and the results just obtained for q. The classic Fano process is shown in Fig. 1a. The three states $$ ∣g⟩$$,$$ ∣a⟩$$,$$ ∣c⟩$$ are stationary states without the interaction V, which mixes a component of $$ ∣a⟩$$ into $$ ∣c⟩$$ producing a dressed state $$ ∣\mathrm{\Psi }⟩$$. The transition rate from $$ ∣g⟩\to ∣\mathrm{\Psi }⟩$$ produced by the dipole interaction D = eFr.ε for probe light with frequency ω, field amplitude F, and polarization ε may be found from the Golden Rule. The relevant matrix element is $$ ⟨g\mid d\mid \mathrm{\Psi }⟩\propto \frac{\mathrm{\Delta }}{E_{\mathrm{H}}}⟨g\mid d\mid k_{\omega }⟩+⟨g\mid d\mid a⟩⟨a\mid v\mid k_{\omega }⟩+⟨g\mid d{G}_{\omega }^{\left(k_{\omega }\right)}v\mid a⟩⟨a\mid v\mid k_{\omega }⟩$$ where d = r.ε/a_B, v = V/E_H, and $$ ∣k_{\omega }⟩\in ∣c⟩$$ is the unmixed continuum state at energy E_g + ħω, i.e., resonant with the excitation.

The three terms in the matrix element 〈g∣d∣Ψ〉 indicate the interfering pathways, $$ \mid g⟩\stackrel{D}{\to }\mid k_w⟩,\mid g⟩\stackrel{D}{\to }\mid a⟩\stackrel{V}{\to }\mid k_w⟩$$, and the last one is the total contribution via off-resonant states, $$ \mid g⟩\stackrel{D}{\to }\mid k\ne k_w⟩\stackrel{V}{\to }\mid a⟩\stackrel{V}{\to }\mid k_w⟩$$ produced by

In our experiment, ω is near to resonance with an allowed transition to a bound state and ionization requires two photons, as in MPICS¹⁵ (Fig. 1a). We used a single laser, so the induced Fano-coupling interaction is the one-photon perturbation produced by the same beam (V = D). The MPICS rate is found from the Golden Rule generalized for second-order processes, i.e., it now includes the Kramers–Heisenberg–Goeppert matrix element³⁴, $$ ⟨g\mid \left(d-d{G}_{\omega }^{\left(b\right)}d\right)\mid \mathrm{\Psi }⟩\propto ⟨g\mid d\mid b⟩⟨b\mid d\mid k_{2\omega }⟩-\frac{\mathrm{\Delta }}{E_{\mathrm{H}}}⟨g\mid d{G}_{\omega }^{\left(b\right)}d\mid k_{2\omega }⟩$$ neglecting a term that is the fourth order in d. The pathways are therefore $$ ∣g⟩\to ∣b⟩\to ∣k_{2\omega }⟩$$ and $$ ∣g⟩\to ∣k\ne b⟩\to ∣k_{2\omega }⟩$$. Now the ionization rate is

where

Implicit summation of the non-resonant matrix element

The matrix element for the pathway via off-resonant intermediate states produced by D⁽²⁾, i.e., $$ ⟨g\mid d{G}_{\omega }^{\left(b\right)}d\mid k_{2\omega }⟩$$ appearing in Eqs. (6) and (7), has the form 〈ψ∣d∣k_2ω〉, where $$ ∣\psi ⟩=G_{\omega }^{\left(b\right)}d∣g⟩$$. The sum inside the operator (Eq. (5)) appearing here is over an infinite number of intermediate states. In a similar way to the problem of two-photon absorption between bound states, implicit summation^20,21,35 can reduce this difficulty substantially. In this technique $$ ∣\psi ⟩$$ becomes the solution of partial differential equation (PDE) that does not require construction of $$ G_{\omega }^{\left(b\right)}$$. Here we used a new variant (see ‘Methods’), which takes account of the excluded state. In this case, $$ ∣\psi ⟩$$ is the solution of

The calculation for $$ ∣b⟩=∣2p⟩$$ in hydrogen produces 〈g∣d∣b〉 = 0.74, $$ ⟨b\mid d\mid k_{2\omega }⟩=0.38,⟨g∣d{G}_{\omega }^{\left(b\right)}d∣k_{2\omega }⟩=8.5$$, which are all of order unity, and therefore it is apparent from Eq. (7) that to achieve q near unity F must be of order F_a. There have been no previous experimental observations of a Fano lineshape in MPICS, because F_a = 5.1 GV/cm; thus, unless the laser is extremely intense, q is very large and the resonance reduces to very narrow Lorentzian.

As a first approximation for Si:P, the hydrogen result may be scaled to the hydrogenic donor using E_H = e⁴m/ħ²κ² and a_B = ħ²κ/e²m (where the symbols have their usual meaning) and inserting the appropriate dielectric constant and the effective mass. We use no other parameters (adjustable or otherwise). Taking a_B = 3.2 nm and E_H = 40 meV²⁷ produces F_a = 130 kV/cm, so defining $$ q=-F_1^2/F^2$$ and Δ₀ = − qΓ, where F₁ is the field required to achieve unity ∣q∣ and Δ₀ is the blue shift of the Fano window (which is independent of field), we find unity q expected at F₁ = 36 kV/cm and the zero is expected at ħΔ₀ = 1.4 meV.

Experiment

The silicon sample was cut in the [100] direction from a crystal grown by the float zone technique. The P-donor concentration was near n_P = 2 × 10¹⁵ cm⁻³ (same sample as in ref. ²⁷). The small-signal absorbance spectrum of Fig. 3a was taken at 4.2 K using Fourier transform infrared spectrocopy with 0.01 meV resolution. The laser used was the Free Electron Laser FELIX, which emits tunable picosecond pulses of high intensity. Its spectrum is approximately Gaussian, and the central frequency and r.m.s. width were recorded during the experiment with a spectrum analyser. An example synthetic spectrum reconstructed from this is shown in Fig. 3a. The laser repetition frequency was 25 MHz.

The optical setup was a standard pump-probe experiment. For a detailed description of this arrangement, see refs. ^28–30. The sample was mounted in vacuum on a cryostat cold-finger at 5 K. THz pulses from a Free Electron Laser were focussed through room-temperature, thin-film windows. The r.m.s. radius of the beam intensity was 0.5 mm. The intensity of the probe beam was kept 15 times less than the pump. The small-signal absorption spectrum of the sample is shown in Fig. 3a. The half-width of each absorption line is ~0.15 meV, which was therefore somewhat narrower than the laser spectrum half-width of 0.19 meV used in the pump-probe experiment. The laser is near bandwidth-limited, so the pulses have a half-width duration of t_p = 2.5 ps.

Data analysis

The rate equations for densities undergoing the relaxation processes shown in Fig. 1c (middle) are:

The effect of the pump is governed by non-classical processes, but the probe is weak, and (away from t = 0 when it is simultaneous with the pump) its absorption may be described classically. The probe transmission for a sample of thickness L is

We now have a functional form for the transients of Fig. 2. It is described by six independent parameters: three dimensionless transmission coefficients (A, B, C), two timescales (t_R, T₁), and the fraction of recombination that occurs into the excited state (ρ). Of these, T₁ and ρ are expected to be constants of the system, while the other four depend on n_c0,b0, i.e., on the degree of excitation by the pump, which varies with intensity and photon energy. We therefore fitted each of the transients in Fig. 2c with T₁, ρ as global variables, while A, B, C, t_R were different for each trace. We found that ρ = 0, in which case ΔT reduces to Eq. (3).

Implicit summation theory

The calculation of q in Eq. (7) requires $$ ⟨g\mid d{G}_{\omega }^{\left(b\right)}d\mid k_{2\omega }⟩$$. Let $$ d∣g⟩=∣{\psi }^{\prime }⟩$$ and $$ ⟨g\mid d{G}_{\omega }^{\left(b\right)}d\mid k_{2\omega }⟩=⟨\psi \mid d\mid k_{2\omega }⟩$$. Therefore, $$ ∣\psi ⟩=G_{\omega }^{\left(b\right)}∣{\psi }^{\prime }⟩$$. By inspection,

Parameters of the MPICS lineshape

In the main text, we present our result with a semi-classical description of the field. This allows a simple and intuitive connection between the original Fano resonance and the multi-photon induced Fano resonance. Here we detail the formal derivation of the formulae, which is best done when the field is treated as quantum¹⁵.

The Hamiltonian for the system of an atom interacting with a laser mode with frequency ω is

Let there be initially n photons in the field and let ω be close to the resonance between $$ ∣g⟩$$ and $$ ∣b⟩$$. The initial atom-field state $$ ∣g,n⟩$$ can be excited to the continuum state $$ ∣{\kappa }_{2\omega },n-2⟩$$ by either a sequential single photon excitation through $$ ∣b,n-1⟩$$ or a direct two-photon excitation process through the non-resonant intermediate states $$ ∣j\ne b,n-1⟩$$. The normalization for $$ ∣{\kappa }_{2\omega }⟩$$ is as usual for continuum states, $$ ⟨\kappa \mid {\kappa }^{\prime }⟩=\delta \left(E_{\kappa }-E_{{\kappa }^{\prime }}\right)$$, so that $$ ∣{\kappa }_{2\omega }⟩$$ is related to $$ ∣k_{2\omega }⟩$$ in the main text by $$ ∣k_{2\omega }⟩=\sqrt{E_{\mathrm{H}}}∣{\kappa }_{2\omega }⟩$$. As the discrete state $$ ∣b,n-1⟩$$ has the same energy as the continuum state $$ ∣{\kappa }_{2\omega },n-2⟩$$ and is coupled to it by H_I, it can be seen as a pseudo-autoionizing state¹⁵. This coupling produces the dressed Fano state^1,15:

The ∑ symbol in $$ {\mathcal{G}}_{2\omega }^{\left({\kappa }_{2\omega }\right)}$$ includes both the summation over the discrete states and the integration over the continuum states. As κ_2ω is in the continuum the exclusion κ ≠ κ_2ω should be understood as taking the principal value of the integration part. Due to the normalization for the continuum states, the wavefunction of $$ ∣{\kappa }_{2\omega }⟩$$ in real space has the unit of $$ {\left(\left[\mathrm{energy}\right]\times {\left[\mathrm{length}\right]}^3\right)}^{-1/2}$$, and hence Γ has the unit of energy as it should.

Following Armstrong et al.¹⁵ (and correcting a sign error), the transition rate from the ground state $$ ∣g,n⟩$$ to the Fano eigenstate $$ ∣\chi ⟩$$ can be obtained by Fermi’s Golden Rule generalized for second-order processes³⁴

As $$ {\mathcal{G}}_{\omega }^{\left(j\right)}$$ has units of [energy]⁻¹, $$ H_{\mathrm{I}}^{\left(2\right)}$$ and H_MPICS have units of energy and [energy]², respectively. Whereas the first two terms in H_MPICS are second order in H_I, the third term is fourth order in H_I and involves non-resonant continuum states (through $$ {\mathcal{G}}_{2\omega }^{\left({\kappa }_{2\omega }\right)}$$). Likewise, the energy shift of the resonance, the last term in the equation for Δ above, is second order in H_I and also involves $$ {\mathcal{G}}_{2\omega }^{\left({\kappa }_{2\omega }\right)}$$, whereas the others are zeroth order. We neglect these higher order terms in our calculation below.

Now, one can use the identity $$ a_{\omega }∣n⟩=\sqrt{n}∣n-1⟩$$ to eliminate the field component in the matrix elements. Noting that the electric field amplitude is $$ F={\left(n\hslash \omega /2{ϵ}_0{ϵ}_{\mathrm{r}}{L}^3\right)}^{1/2}$$ and n is very large, and defining $$ ∣k_{2\omega }⟩=\sqrt{E_{\mathrm{H}}}∣{\kappa }_{2\omega }⟩$$, we obtain Eq. (8) in the main text:

Fig. 5

The MPICS wavefunction.

Ψ(r) ∝ b(r) + θk_2ω(r) is shown for $$ ∣b⟩\equiv ∣2p_{\pm }⟩$$ in hydrogen, in the xz plane, with z chosen to be the axis of the field polarization, at different values of ∣θ∣. For θ = 0, Ψ(r) ∝ b(r), while for θ = ∞, Ψ(r) ∝ k_2ω(r). The colour indicates the sign, and the brightness indicates the amplitude, normalized to the maximum in each case.

The rate R now has the form of Eq. (6) given in the main text:

As mentioned in the main text, for $$ ∣b⟩\equiv 2p$$ in H, we find that 〈g∣d∣b〉 = 0.74. Due to selection rules the continuum state may have angular momentum l = 0 or l = 2. We find that the background two-photon ionization rate, R₀, to the l = 2 continuum state is 30 times larger than the rate to the l = 0 continuum state, and hence we neglect the latter. Then $$ ⟨b\mid d\mid k_{2\omega }⟩=0.38,⟨g∣d{G}_{\omega }^{\left(b\right)}d∣k_{2\omega }⟩=8.5$$ and hence $$ \mathrm{\Gamma }=0.45E_{\mathrm{H}}{F}^2/F_{\mathrm{a}}^2$$ and $$ q=-0.074F_{\mathrm{a}}^2/F^2$$, so that the zero in the Fano lineshape is at Δ₀ = − qΓ = 0.033E_H (independent of field, and therefore robust against fluctuations) and the peak is at $$ {\mathrm{\Delta }}_p=\mathrm{\Gamma }/q=-6.1E_{\mathrm{H}}{F}^4/F_{\mathrm{a}}^4$$.

In silicon, if we simply scale the hydrogen result as mentioned in the main text, then E_H = 40 meV, so we expect the zero at Δ₀ = 1.4 meV, and F_a = 0.13 MV/cm so with our experimental conditions F is ~22 kV/cm, and the peak is at Δ_p = − 0.2 meV. These conditions imply θ ≈ 0.09 at the zero and the peak rate when θ ≈ − 0.01, and the corresponding dressed excited state wavefunctions $$ ∣\mathrm{\Psi }⟩$$ for these values of θ are shown in Fig. 5. At zero detuning when $$ ∣\mathrm{\Psi }⟩=∣b⟩$$, the photo-ionization is obviously purely step-wise via the discrete state using one-photon excitation. At very large detuning the excitation is purely to the continuum via two-photon excitation (where the intermediate state is off-resonant). At the zero (which is field independent) the amplitudes of the two contributions are comparable, so that the matrix elements with the ground state are equal and opposite. For the relatively small electric field we used, the wavefunction at peak ionization has a quite small component of the continuum and is dominated by the pathway through the bound state.