Quantum interference and collisional dynamics in excited bounds states revealed by time-resolved pump-high-harmonic-generation-probe spectroscopy

Konstantin E. Dorfman; Pengfei Wei; Jiayun Liu; Ruxin Li

doi:10.1364/OE.27.007147

1. Introduction

Time-resolved pump-probe spectroscopy [1] is a well established nonlinear optical tool for studies of ultrafast dynamics of matter. The pump pulse prepares the system in the superposition of electronic bound states [2, 3], whereas a weak probe provides a snapshot of the system at different delay times. The observation window into material response is governed by the pulse parameters: wavelength range and spectral bandwidth. Thus, to access electronic states 10 eV apart one needs to employ X-raypulses. Recent progress in designing table-top X-ray sources which employ high harmonic generation (HHG) allow to overcome this limitation thus allowing infrared lasers to monitor electronic states of high energy.

In the pump-HHG-probe (PHHGP) [4–9] a strong pump first ionizes the system while creating a superposition of valence electronic states. A strong probe pulse then induces ionization, free electron propagation guided bylaser field and consecutive recombination [10]. PHHGP has demonstrated various effects in both atoms and molecules revealing electron dynamics [11], molecular alignment [12] and chemical reactions [13]. Another type of picosecond dynamics is related to collisional dephasing prominent in atomic gas systems, which brings an important step for understanding the free electron dynamics and efficiency of the HHG generation. Multielectron theories have been developed to account for more complex systems [5, 14] which rely on sophisticated numerical algorithms. While the time-resolved PHHGP method has been established experimentally, the existing theoretical models lack the ability to describe consistently the dynamics of the excited bound states, chemical reactions and other transport effects. These dynamical processes are typically described in the weak field spectroscopy by using Liouville space formalism [1], superoperator algebra [15] and diagram techniques [16].

Fig. 1 Schematic of the PHHGP measurement. Strong pump pulse creates superposition of electronic states in the atom, which consists of the ground g, excited state e and continuum of state i. This dynamics undergoes dephasing due to collisions between neutral atoms and electrons governed by rate $γ_{b b}$ , ions and electrons determined by $γ_{b c}$ . After a delay T a probe laser pulse generates HHG in three-step process from the atom in superposition of electronic states which involvesionization and recombination pathways from bound-to bound $d_{m m^{'}}^{b b}$ and bound-to-continuum $d_{m m^{'}}^{b c}$ , $m, m^{'} = e, g$ electronic transitions. The resulting HHG spectra is recorded in the series of snapshots.

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Here we present a consistent analytical theory of PHHGP based on the density matrix formalism [17] which allows to describe dynamics of excited bound states in Liouville space by including self-consistently complex collisional dynamics and related material response. Liouville space description of atomic [18] and molecular [19] spectroscopies allows to use a reduced density matrix formalism by tracing over the environmental degrees of freedom. While the average decay rates have been previously investigated in the context of e.g. ground state depletion dynamics [10], the dephasing rates investigated in this work originate from the collisions between neutral and charged particles of the laser-driven plasma. As the result we predict rapid decay followed by slow recovery of the time-resolved PHHGP signal due to the two mechanisms of collisional dephasing of the bound-continuum coherence. Ellastic contributions from the ground and excited states are responsible for decay, while recovery is due to the quantum interference between the ground and excited state contributions. While the rapid decay has been extensively studied, the slow recovery in molecules [13], which occurs on a picosecond timescale, has not been described in detail. Furthermore, it has not been previously investigated in atomic due to the absence of ps dynamics typical for molecules. Our theory is able to simulate the experiment in Ar gas with great accuracy using the setup in [20, 21] and explain the slow recovery (~1 ns). Our approach is not limited to the atoms and can treat more complex molecular systems providing a unified treatment of general class of ultrafast signals with HHG-probe.

2. Bound-state dynamics in HHG

We consider a model of the atom represented by the ground and excited electronic states in a strong linearly polarized electric field of the laser shown in Fig. 1 described by the Hamiltonian

H = \frac{1}{2} v^{2} + V (r) - E (t) \cdot r,

where the first and second term correspond to kinetic and potential energy and the last term represents field-matter interaction. Without a loss of generality we assume a z-polarized light field

E (t) \cdot r = E (t) z

with Gaussian temporal envelope without invoking rotating-wave approximation

E (t) = \frac{E}{\sqrt{2 π} σ} e^{- \frac{t^{2}}{2 σ^{2}}} cos (ω_{0} t),

where

E

is the pulse amplitude, σ is the pulse duration, and ω₀ is the central pulse frequency. We further invoke the following approximations. First, the contribution to the evolution of the system of all bound states except the ground g and the excited state e can be neglected. Second, in the continuum, the electron can be treated as a free particle moving in the electric field with no effect caused by the atomic core potential

V (r)

. Provided, that ponderomotive potential U_p is large enough, second approximation does not only hold for short range potentials, but also for long range potentials, as of hydrogen-like atoms. It further implies that the electron has gained such a large kinetic energy when returning to the nucleus, that the atomic core attraction is negligible. The present model is only applicable to higher harmonics with photon energies much larger than ionization potential

ℏ ω > I_{p}

. In general, both assumptions are justified, if the Keldysh parameter

γ = \sqrt{I_{p} / 2 U_{p}}

is smaller than one, thus

I_{p} < 2 U_{p}

. We also follow the approximation of

〈 v | V (r) | ψ 〉 = 0

, where v represent a free electron wavefunction. Omitting the potential contribution we neglect re-scattering effects. However the time-resolved PHHGP measurement reflects an integrated spectra which is dominated by the continuum-to-bound transitions, where the re-scattering does not play a significant role. For a strong field intensity

2.2 \times 10^{14}

W

/ c m^{2}

Keldysh parameter is

γ = \sqrt{E_{0} / 2 U_{p}} = 0.24

. The latter allows to use saddle-point method [22] under condition of small Keldysh parameter. Following the above approximation the wavefunction of the atomic system is given by

| ψ (z, t) 〉 = \sum_{m} e^{- i ω_{m} t} c_{m} (t) | ψ_{m} (z) 〉 + \int d v c_{i} (v, t) e^{- i ω_{v} t} | v 〉,

where c_m and c_i are amplitudes of the bound and ionized continuum states where v = v_z is a z-component of the velocity corresponding to z-polarized light,

| v 〉 = e^{i v z}

is a continuum eigenket,

ω_{m} = - I_{p, m}

is the ionization potential of the m-th state. Note, that the real HHG process contains contributions of both neutral atom as well as ion. Therefore, the correct eigenfunctions must contain relevant contributions. In the following calculations, however, the weakly ionized regime does not change eigenstates significantly compared to entirely neutral atom case. The Schrödinger equation for wave function components reads

{\dot{c}}_{m} (t) = i \sum_{n} μ_{m n}^{(b)} E (t) c_{n} (t) e^{i ω_{m n} t} + i \int d v μ_{m i} (v) E (t) c_{i} (v, t) e^{i (ω_{m} - ω_{v}) t}

\begin{matrix} {\dot{c}}_{i} (v, t) = - i (\frac{v^{2}}{2} - ω_{v}) c_{i} (v, t) - E (t) \frac{\partial c_{i} (v, t)}{\partial v} + i μ_{i m} (v) E (t) e^{- i (ω_{m} - ω_{v}) t} c_{m} (t), \end{matrix}

where

μ_{m n}^{(b)} = 〈 m | z | n 〉

is the bound-to-bound transition dipole moment and initial conditions are

c_{g} (0) = 1

,

c_{m \neq g} (0) = 0

and

c_{i} (v, 0) = 0

. Solving Schrödinger equation for ionized state we obtain

\begin{array}{l} c_{i} (v, t) = i \sum_{m} \int_{0}^{t} d t^{'} E (t^{'}) μ_{i m} (v + A (t) - A (t^{'})) \times e^{- i \int_{t^{'}}^{t} d t^{″} [(v + A (t) - A (t^{″}))^{2} / 2 - ω_{v}]} c_{m} (t^{'}) e^{- i (ω_{m} - ω_{v}) t^{'}} \end{array}

Assuming that the change of the amplitudes are small such that $c_{m} (t^{'}) ≃ c_{m} (t)$ one can substitute solution in Eq. (6) to Eq. (4) we obtain

{\dot{c}}_{m} (t) = i \sum_{n} [μ_{m n}^{(b)} + μ_{m n}^{(c)} (t)] E (t) c_{n} (t) e^{i ω_{m n} t}

where

μ_{m n}^{(c)} (t)

is the bound-to-continuum transition dipole moment defined by Eq. (31) obtained after saddle point momentum integration in Eq. (6). We therefore transformed the initial problem in full Hilbert space consisted of bound and continuum states to a reduced problem in the bound state Hilbert space alone.

The standard HHG treatment then prescribes to solve time-dependent Schrödinger equation (TDSE) [23, 24] and obtain the induced dipole acceleration corresponding to the HHG emission. To describe the pump-probe signals that capture various dynamical processes in the system it is more imperative to use density matrix (DM) formalism. The simplest semiclassical formulation [25] assumes that the observed quantum system is coupled to a classical bath that undergoes stochastic dynamics [26] due to e.g. various collisional processes that result in the dephasing of the atomic coherence and can be captured by time-resolved pump-probe signals. Similarly to the core electron stateelimination commonly used in coherent X-ray spectroscopy by introducing effective polarizability [27] we eliminate ionized continuum by introducing effective dipole moment given by Eq. (31). In order to include relaxation and dephasing in the system, one has to use Liouville space description for density matrix instead of wavefunction. The Liouville-von Neumann equation reads

\dot{ρ} = - i [H, ρ]

where density matrix elements are defined as

ρ_{m n} = c_{n}^{*} (t) c_{m} (t)

Before we obtain equations of motion for the bound-state density matrix components we discuss the dephasing processes in HHG in details and return to density matrix dynamics in Section IV.

3. Collisional dephasing

In order to include collisions we assume the random Stark shift to the energy difference $ω_{e g} \to ω_{e g} + δ ω_{e g} (t)$ . Ignoring atom-field interaction and pure lifetime broadening, in the Schrödinger picture the time evolution of the off-diagonal matrix element $ρ_{e g}$ reads

{\dot{ρ}}_{e g}^{(s)} = - i (ω_{e g} + δ ω_{e g} (t)) ρ_{e g}^{(s)},

where superscript s denotes Schr’ódinger picture. Integrating we obtain

ρ_{e g}^{(s)} (t) = exp [- i ω_{e g} t - i \int_{0}^{t} d t^{'} δ ω_{e g} (t^{'})] ρ_{e g}^{(s)} (0) .

Performing an ensemble average over the random variable $δ ω_{e g} (t)$ we need to evaluate $〈 \exp [- i \int_{0}^{t} d t^{'} δ ω_{e g} (t^{'})] 〉$ . Since the random variation $δ ω_{e g}$ is oscillating function and $〈 δ ω_{e g} (t) 〉 = 0$ and assuming rapid variation in $δ ω_{e g}$ compared to other changes with the scale $1 / γ$ we take

〈 δ ω_{e g} (t) δ ω_{e g} (t^{'}) 〉 = 2 γ δ (t - t^{'}) .

Assuming Gaussian random process we achieve

〈 \exp [- i \int_{0}^{t} d t^{'} δ ω_{e g} (t^{'})] 〉 = \exp [- γ t],

we arrive to the final averaged equation with respect to collisions

{\dot{ρ}}_{e g}^{(s)} = - (i ω_{e g} + γ) ρ_{e g}^{(s)} .

Note that the collisional dephasing affects only equation of motion for coherence and not populations, which is a reason why probability amplitude method is not a suitable approach for dealing with relaxation and nonunitary processes.

As a model system we consider an Argon plasma [28] similar to that studied in [20, 21]. To calculate plasma temperature we invoke a photoionization theory subject to the elliptically polarized pulse. Although we neglect ellipticity in the electron trajectory calculations, we include it in the plasma calculations. Due to ellipticity electron can gain a significant energy given by [29, 30]

k_{B} T_{e} ≃ 2 U_{p} ξ^{2} = 13 e V,

which is typical for such systems [31]. Here k_B is Boltzmann constant, and we used laser intensity of

I = 2.2 \times 10^{14}

W/cm² for a 800 nm laser, U_p is a ponderomotive energy and

ξ \sim 0.35

is the laser ellipticity. Electron density can be calculated from the ideal gas law (we will justify in the following)

n_{e} = \frac{{\bar{ρ}}_{i i} P}{k_{B} T_{0}},

where

{\bar{ρ}}_{i i} ≃ 0.095

is the ionized fraction which can be obtained from the exact numerical solution of Eqs. (6). Due to heavy numerical effort that is involved in integrating over the continuum one can instead invoke a simple estimate. The bound state population of the excited state

ρ_{e e} (t)

as a numerical solution of Eq. (29) yields

{\bar{ρ}}_{e e} = max (ρ_{e e}) ≃ 0.009

at its maximum (see Fig. 2). Note, that the depicted highly oscillating population behavior is typical for far off-resonant Rabi problem [32]. Numerical solution for the bound states is less expensive numerically as it already involves saddle-point result in Eq. (31). Using perturbative expansion of Eqs. (4) - (5) with respect to weak excited bound state population one can estimate

{\bar{ρ}}_{e e} \sim | ρ_{e g} |^{2}

whereas

{\bar{ρ}}_{i i} \sim | ρ_{e g} |

. Thus,

{\bar{ρ}}_{i i} \sim \sqrt{{\bar{ρ}}_{e e}} \sim 0.095

. Another physical argument that follows is that

{\bar{ρ}}_{e e} ≪ ρ_{i i}

. This is a reason that excited bound states are typically neglected in the HHG process. We finally estimate from Eq. (15) that for a room temperature gas at

P = 20

Torr we obtain

n_{e} = 6 \times 10^{16}

cm⁻³.

Fig. 2 Time-profile of the excited bound state population $ρ_{e e} (t)$ obtained from numerical solution of Eq. (29).

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We can now classify the plasma by calculating plasma parameter (using cgs units commonly used in plasma physics):

Λ = 4 π n_{e} λ_{D e}^{3},

where

λ_{D e} = {(k_{B} T_{e} / 4 π n_{e} e^{2})}^{1 / 2}

is a radius of the Debye screening. The corresponding coupling parameter that represent the ratio of the Coulomb energy to thermal energy

Γ = \frac{E_{C o u l o m b}}{k_{B} T_{e}} = \frac{e^{2}}{4 π ϵ_{0} k_{B} T_{e}} {(\frac{4 π n_{e}}{3})}^{1 / 3}

For the parameters of the experiment $Λ ≫ 1$ and $Γ ≪ 1$ which corresponds to weakly coupled ideal, hot and diffuse plasma with densely packed Debye sphere. In this case ideal gas law in Eq. (15) is an appropriate approximation. The plasma frequency is given by

ω_{p} = {(\frac{4 π e^{2} n_{e}}{m_{e}})}^{1 / 2} ≃ 1.45 \times 10^{6} s^{- 1},

which means that field inhomogeneity will not affect the electron motion since

ω_{c 0} ≫ ω_{p}

for optical field.

We can now calculate various collision rates. There are several collision mechanisms that can affect the dephasing. First is due to collisions between atoms both in their ground states described by a rate

2 π ν_{a, a}^{(g)} = 2 π n_{0} σ_{0} {(\frac{k_{B} T_{0}}{m_{0}})}^{1 / 2} ≃ 9.6 \times 10^{8} s^{- 1},

where we used the concentration of Ar atoms at 20 Torr to be

6.4 \cdot 10^{17}

cm⁻³, and neutral atom cross section for Argon

σ_{0} = π r_{0}^{2}

for radius of Argon atom

r_{0} = 71

pm. If one of the atom is in the excited state then the collisional rate in Eq. (19) has to be multipled by

{\bar{ρ}}_{e e}

which yields

2 π ν_{a, a}^{(e)} = 8.6 \times 10^{6}

s⁻¹.

Another dephasing mechanism is due to collisions between free electrons and atoms that are in the ground state given by

2 π ν_{e l, a}^{(g)} = 2 π n_{0} σ_{0} {(\frac{k_{B} T_{e}}{m_{e}})}^{1 / 2} ≃ 1.3 \times 10^{11} s^{- 1} .

If the colliding atom is in its excited state, the corresponding collisional frequency is given by

2 π ν_{e l, a}^{(e)} = 2 π {\bar{ρ}}_{e e} n_{0} σ_{0} {(\frac{k_{B} T_{e}}{m_{e}})}^{1 / 2} ≃ 1.2 \times 10^{9} s^{- 1} .

If there is a significant contribution from ions then electron-ion collision can have a significant contribution:

ν = 4 π N_{i} \frac{Z_{i}^{2} e^{2}}{μ^{1 / 2} {(k_{B} T)}^{3 / 2}} log Λ

where the reduced mass

μ = (m_{e} m_{i}) / (m_{e} m_{i})

is m_e for electrons and

m_{i} / 2

for ions, Z_i = 1 for

A r^{+}

. For temperature

T_{e} = T_{i}

electron-ion collisions are given by

\begin{array}{l} 2 π ν_{e l, i} = 2 π \cdot 2.91 \cdot 10^{- 6} n_{e} \log (Λ_{e l, i}) T_{e}^{- 3 / 2} ≃ 1.7 \times 10^{11} s^{- 1}, \end{array}

and Coulomb logarithm is given by

log (Λ_{e l, i}) = 23 - log (\frac{N_{e}^{1 / 2} Z_{i}}{T_{e}^{3 / 2}}),

We now get back to calculation of the HHG signal. The dipole moment for HHG is defined by

\begin{array}{l} z (t) = 〈 ψ (z, t) | z | ψ (z, t) 〉 = \sum_{m \neq n} μ_{m n}^{(b)} c_{m}^{*} (t) c_{n} (t) + \sum_{n} \int d v μ_{i n} (v) c_{i}^{*} (v, t) c_{n} (t) + c . c . \end{array}

where the first term characterize bound-to-bound while the second term represents bound-to-continuum contribution to HHG emission. Now using Eq. (12) we can recast an HHG dipole moment averaged over collisions:

\begin{array}{l} z (t) = 〈 ψ (z, t) | z | ψ (z, t) 〉 = \sum_{m \neq n} μ_{m n}^{(b)} ρ_{n m} (t) e^{- γ_{b b} t} + \sum_{n} \int d v μ_{i n} (v) ρ_{n i} (v, t) e^{- γ_{b c} t} + c . c . \end{array}

Because coherence $ρ_{n i}$ is governed primarily by collisions between free electrons and nonparental ions, as well as electrons with neutral atoms, the bound-to-continuum dephasing rate is given by

γ_{b c} = 2 π (ν_{e l, i} + \frac{1}{2} (ν_{e l, a}^{(g)} + ν_{e l, a}^{(e)})) = 2.3 \times 10^{11} s^{- 1},

where for simplicity we take average contribution from the ground and excites state. The corresponding bound-to-bound component is determined by

γ_{b b} = 2 π ν_{e l, a}^{(e)} = 1.2 \times 10^{9} s^{- 1},

since atom-atom collision in this case can be neglected. We therefore have two distinct time scales that is determined by two different physical mechanisms.

4. HHG spectra

We now get back to the DM formalism to describe evolution of the bound-state components of the density matrix based on evolution of bound state amplitudes in Eqs. (7) which now include the two dephasing mechanisms described by the rates (27) - (28):

{\dot{ρ}}_{m n} = - γ_{m n} ρ_{m n} - i \sum_{k} (μ_{m k} (t) ρ_{k n} - μ_{k n} (t) ρ_{m k} E) (t),

where m, n, and k run over the bound states of the system and the generalized dipole moment is

\begin{array}{l} μ_{m n} (t) = μ_{m n}^{(b)} (1 - δ_{m n}) e^{- [i ω_{m n} + γ_{b b}] t} \\ + \frac{1}{2} (μ_{m n}^{(c) *} (t) + μ_{n m}^{(c)} (t)) e^{- [i ω_{m n} + γ_{b c}] t} . \end{array}

with

ω_{m n} = ω_{m} - ω_{n}

. Here the first term

μ_{m n}^{(b)} = 〈 m | z | n 〉

represents the bound-to-bound dipole transition, whereas the second

μ_{m n}^{(c)} (t)

given by

\begin{array}{l} μ_{m n}^{(c)} (t) = i \int_{0}^{\infty} d τ {(\frac{π}{ϵ + i τ / 2})}^{1 / 2} E (t - τ) \times μ_{i m}^{*} (p_{s t} - A (t)) μ_{i n} (p_{s t} - A (t - τ)) e^{- i S_{s t} (t, τ)}, \end{array}

Where $μ_{i m} (v) = 〈 v | z | m 〉$ is bound-to-continuum transition dipole moment; $〈 v |$ represents free space electron wavefunction), $A (t) = - \int_{0}^{t} d t^{'} E (t^{'})$ and the stationary value of momentum is given by $p_{s t} = \frac{\int_{t - τ}^{t} d t^{″} A (t^{″})}{τ}$ and quasiclassical action is $S_{s t} (t, τ) = - ω_{n} τ + \int_{t - τ}^{t} d t^{″} \frac{1}{2} {(p_{s t} - A (t^{″}))}^{2}$ . Since not all the population remains in the ground state and depletion becomes important, we calculate ground and excited state populations $ρ_{g g} (t)$ and $ρ_{e e} (t)$ , respectively. Dephasing rate $γ_{m n}$ in Eq. (29) in the form of two rates $γ_{b c}$ and $γ_{b b}$ correspond to the electron collisions with ions and ground state atoms and with excited atoms, respectively. This collisional dynamics can be described as fluctuation of the transition energy (See [32] and Section 3). Interplay of the two collisional mechanisms described above can be monitored by a pump-probe pulse sequence, where the pump first initiates the electronic dynamicswhich includes charge density (electronic population) and electronic polarization (coherence) between bound and continuum states. Evolution of populations and coherences is consequently probed by the probe pulse delayed by T which produces HHG radiation. The HHG spectra is defined as

S (ω) = | \int d t \ddot{z} (t) e^{i ω t} |^{2} .

Integrated over all spectral components the time-resolved signal provides a snapshot dynamics of the electronic dynamics as a function of the interpulse delay T

S (T) = \int_{0}^{\infty} d t | z (t) + z (t + T) |^{2},

where the induced dipole moment is given by

z (t) = 2 Re [\sum_{m, n} μ_{m n} (t) ρ_{n m}] .

Fig. 3 Simulation of the snapshot HHG spectra of Argon for three-step model (top two rows) and including excited bound state (bottom two rows). Different panels correspond to interpulse delay $T = 100$ fs, 10 ps, 80 ps, and 200 ps.

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Fig. 4 PHHGP signal vs interpulse delay T, experiment [21] - red, theoretical model in Eq. (33) - blue. Inset - same but for a model with the single bound (ground) state.

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5. Simulations

5.1. Snapshot spectra of PHHGP

Figure 3 displays a series of snapshots of HHG spectra generated in Argon gas with pump and probe pulses centered at 800 nm with intensity $2.2 \times 10^{14}$ W/ $c m^{2}$ at different delays T. Argon’s ground 3p state has energy $ω_{g} = - 15.771$ eV corresponding to ionization potential. We first simulate a model without excited bound states. When calculating the transition dipole moment for the bound to continuum transition, the analytical integral is evaluated using Riemann summation. To get the HHG spectrum, the discrete Fourier Transform of the dipole moment is computed. Since the highest frequency observable on the spectrum is twice the sampling frequency, the time interval is chosen such that the highest observable frequency corresponds to thrice the cutoff frequency. As such, the time interval used for the calculation based on the cutoff frequency is $0.0036$ fs.

The spectrum shown in Fig. 3(a) for $T = 100$ fs consists of two regions of odd harmonics. Low harmonics from n = 1 to $n = 10$ correspond to the spectral range below ionization potential $n < I_{p} / ω_{0}$ . High harmonics form a flat plateau between $n = 11$ and the cutoff at $n = 37$ . At $T = 10$ ps HHG spectrum looks very similar to that at $T = 100$ fs as shown in Fig. 3(b). At $T = 80$ ps the magnitude of the plateau drops down by four orders of magnitude as depicted in Fig. 3(c). Distinct harmonics disappear at $T = 200$ ps as shown in Fig. 3(d).

Addition of the excited 4s bound state with energy $ω_{e} = - 4.037$ eV changes the spectra dramatically. Figure 3(e) for $T = 100$ fs shows that the spectrum consists of three parts. Low harmonics between n = 1 and n = 7, corresponding to $ω_{e} - ω_{g}$ bound-to-bound transition. High harmonics plateau between $n = 17$ and $n = 37$ . Finally middle range harmonics between n = 7 and $n = 17$ is characterized by two strong peaks at even harmonics $n = 14$ and $n = 16$ . Usually even harmonics are suppressed due to the symmetry. Furthermore, the standard HHG process at a given intensity level corresponds to linear harmonic response. Thus, the nonlinear response that gives rise to $H 14$ and $H 16$ must be explained in more details. We first note, that the function $f_{m m^{'}} (t) = μ_{i m}^{*} (p_{s t} - A (t)) μ_{i m^{'}} (p_{s t} - A (t - τ)) E (t - τ)$ in Eq. (31) satisfies $f_{m m} (t + π / ω_{0}) ≃ - f_{m m} (t)$ , $m = e, g$ due to the central symmetry of the atomic potential and the corresponding properties of the ground and excited state wavefunctions. The latter equality which becomes exact for CW field is a consequence of the fact that the same dipole moment $μ_{i m}$ enters $f_{m m^{'}}$ twice and therefore the symmetry properties are fully determined by the field $E (t^{'})$ . In addition to the terms with $m = m^{'}$ there exist cross-terms due to quantum interference between the pathways $g \to i \to e$ and $e \to i \to g \sim μ_{i e} μ_{i g}$ which have a different symmetric properties since they represent s and porbitals which consequently break the symmetry and give rise to the even harmonics in the spectra. Second, it is important to note, that even harmonics remain stronger contributions to HHG spectra than fundamental H1 harmonics at longer interpulse delay times. This unexpected feature occurs due to several collisional dephasing timescales. The odd harmonics including $H 1$ , which correspond to elastic ground-to-ground state process, are governed by the collisional dephasing with the atoms/ions in the ground state. Even harmonics, in contrast, arise from the interference e.g. when initial state is the ground state while the final state is excited bound state. These cross terms have longer dephasing time governed by a combination of atoms in both excited and ground states which are responsible for the recovery of the PHHGP signal at longer times. Therefore, the nonlinear regime becomes dominant at longer delay, when elastic contribution to the spectra is diminished whereas inelastic interference terms remain dominant even at longer timescales. In addition, if we look at harmonic $H 7$ , which corresponds to a direct bound-to-bound transition, it also remain dominating at longer time delays with much higher intensity than $H 1$ since the corresponding dephasing timescale isgoverned by $γ_{b b}^{- 1}$ which is much longer than the electron-atom collisions.

Furthermore, the τ integration in Eq. (31) evaluated by a saddle point method has the leading contribution from the point corresponding to zero velocity: $p_{s t} (t, τ) - A (t - τ) = 0$ [10]. While this equation is approximate, as it assumes semiclassical action (I_p = 0) it illustrates the effect qualitatively. In our simulations $I_{p} \neq 0$ [33] and the physics is more complicated. The magnitude of the relevant electron kinetic energy scale corresponds to H14 and H16 $p_{s t}^{2} / 2 = | A {(t - τ_{0})}^{2} / 2 | = n ω_{0}$ where τ₀ is the stationary point. The large amplitudes of the H14 and H16 are thus a consequence of the interference in ionization-recombination pathways corresponding not to the electrons that return to the nucleus, but to those with zero initial velocity [10]. While we increase the delay T the amplitudes of three strong peaks at 7, 14 and 16 harmonics remain unchanged while the high harmonics plateau becomes more pronounced at $T = 10$ ps as shown in Fig. 3(f) with noticeable contributions from even harmonics due to the broken symmetry. Further increase of the delay leads to the suppression of the plateau height similar to that shown in Fig. 3(g) and becomes negligibly small at $T > 200$ ps as shown in Fig. 3(h). On the other hand the low order harmonics $n < 7$ become very well distinguished at longer delays together with the distinct ones at $n = 7, 14, 16$ . An important note regarding re-scattering processes and Cooper minima [34] contribution is that while its addition may change the HHG spectra by few orders of magnitude in some range of harmonics, it will not change the following fact. The leading contribution of the quantum interference terms that manifest as few strong even harmonics (many orders of magnitude difference) remain a dominant process in the long term system dynamics described by the integrated HHG which is not very sensitive to the individual phases of harmonics. Furthermore, one expect that the snapshot spectra in Figs. 3(e)-3(h) will be reduced back to odd harmonics spectrum in Figs. 3(a)-3(d) at lower intensities, where nonlinear effects, which cause even harmonics effects will subside.

5.2. Integrated time-resolved PHHGP signal

A separate experimental study has been performed in Shanghai Institute of Optics and Fine Mechanics [21]. We compared our theoretical model with the experimental data of PHHGP by plotting the integrated HHG spectra in Eq. (33) vs the delay T. Figure 4 shows that the overall HHG yield rapidly decreases reaching its minimum at around 80 ps and then slowly recovers reaching the steady value (around 20%) at around 600 ps. SFA using single ground state can predict the rapid decay behavior as shown in the inset in Fig. 4 and is unable to explain the recovery process. While observing a significant contribution of H14 and H16 in the interference term in HHG spectra in Fig. 3 and the corresponding recovery process in time-resolved measurement in Fig. 4 signifies the role of quantum interference in the electron dynamics and its interplay with collisional dynamics.

While the electron collisions with ions and ground state atoms occur at a rate $γ_{b c} = 2.3 \times 10^{11} s^{- 1}$ the collisions with excited atoms occur at a rate $γ_{b b} ≪ γ_{b c}$ . It represents the elastic process since initial and final states of the atom before and after the probe pulse are the same. Slow recovery is thus due to the interference of these two contributions that manifests as an inelastic process (initial and final state before and after the probe are different). One can further investigate the phase matching which has an additional term due to the excited bound state of the atom which contributes to the recovery mode. In addition we justify the choice of the particular 4s to achieve the best fitting. Note, that although the integrated signal is not commonly used in HHG, it is a standard technique to characterize time-resolved measurements. The fact that this signal may be dominated by the excite-to-bound state transitions results in the strong excitedbound state contributions which is not detected otherwise.

6. Discussion

In the present model we kept the pump pulse ellipticity for calculation of electron temperature, while neglecting it in the dipole moment calculations, since the integrated spectra is not very sensitive to the parameters of the individual electron trajectory. Therefore, although the ellipticity may bring an additional dimension to electron dynamics it won’t change the main effect related to quantum interference contribution and long-time dynamics described by the integrated HHG spectra.

In summary, we have demonstrated that the excited bound state dynamics in atomic gas can be monitored by time-resolved PHHGP signal where this information stored in matter systems can be retrieved at long interpulse delays in the form of the dissipation and relaxation dynamics. The present result is a step towards fully microscopic description of nonlinear HHG spectroscopy signals that target to measure electron and energy transfer dynamics in complex matter systems which occur on multiple timescales. Indeed, integrated HHG spectra of Ar discussed in this work has a significant contribution from several even harmonics (e.g., H14 and H16) due to quantum interference of ionization and recombination pathways, which makes the temporal evolution of the signal unique and strongly dependent on the collisional process. While these processes are detrimental for standard single-pulse HHG, they serve as a fingerprint for spectroscopic studies of a particular atomic and molecular systems. PHHGP signal has thus a potential to become a high precision measurement tool for extracting this information on the ultrashort timescales without using expensive accelerator based systems. In the future work we will consider other possible experimental observables in addition to the present measurements [20, 21] e.g. emission phase and more detailed spectrum structure which may provide an additional information. For instance the present technique can be a sensitive probe for valence electronic state Raman signatures [27] due to the large bandwidth of the HHG light while using the conventional IR or visible light sources. We further expect that inclusion of larger number of bound states will create more rich dynamics with larger number of quantum pathways of ionization and recombination and their interference that would imprint its phases into the time-resolved signals.

Funding

Science and Engineering Research Council at the Singapore Agency for Science, Technology and Research X-ray Photonics Program (1426500053); Zijiang Endowed Young Scholar Fund, East China Normal University; Overseas Expertise Introduction Project for Discipline Innovation (111 Project, B12024).

Acknowledgments

We would like to thank Dr. Ying Zhang and Dr. Qijie Wang for fruitful discussions.

References

1. S. Mukamel, Principles of nonlinear optical spectroscopy, 6 (Oxford University, 1999).

2. J. H. Shirley, “Solution of the schrödinger equation with a hamiltonian periodic in time,” Phys. Rev. 138, B979–B987 (1965). [CrossRef]

3. F. I. Gauthey, B. M. Garraway, and P. L. Knight, “High harmonic generation and periodic level crossings,” Phys. Rev. A 56, 3093–3096 (1997). [CrossRef]

4. M. Spanner and P. Brumer, “Probing electron transfer within alkali-metal halides via high-order harmonic generation,” Phys. Rev. A 78, 033840 (2008). [CrossRef]

5. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009). [CrossRef] [PubMed]

6. O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009). [CrossRef]

7. D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012). [CrossRef] [PubMed]

8. S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015). [CrossRef]

9. J. Marangos, “Development of high harmonic generation spectroscopy of organic molecules and biomolecules,” J. Phys. B: At. Mol. Opt. Phys. 49, 132001 (2016). [CrossRef]

10. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]

11. P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015). [CrossRef] [PubMed]

12. P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013). [CrossRef]

13. H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010). [CrossRef] [PubMed]

14. S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006). [CrossRef] [PubMed]

15. U. Harbola and S. Mukamel, “Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions,” Phys. Rep. 465, 191–222 (2008). [CrossRef]

16. S. Mukamel and S. Rahav, “Chapter 6 - ultrafast nonlinear optical signals viewed from the molecule’s perspective: Kramers-heisenberg transition-amplitudes versus susceptibilities,” in “Advances in Atomic, Molecular, and Optical Physics,” vol. 59 of Advances In Atomic, Molecular, and Optical Physics, P. B. E. Arimondo and C. Lin, eds. (Academic Press, 2010), pp. 223 – 263.

17. N. Rohringer and R. Santra, “Multichannel coherence in strong-field ionization,” Phys. Rev. A 79, 053402 (2009). [CrossRef]

18. F. Schuller, The Liouville Space Formalism in Atomic Spectroscopy, Heron Press science series (Heron Press, 2002).

19. W. Parson, Modern Optical Spectroscopy: With Exercises and Examples from Biophysics and Biochemistry (SpringerBerlin Heidelberg, 2007). [CrossRef]

20. X. Yuan, P. Wei, C. Liu, Z. Zeng, Y. Zheng, J. Jiang, X. Ge, and R. Li, “Enhanced high-order harmonic generation from excited argon,” Appl. Phys. Lett. 107, 041110 (2015). [CrossRef]

21. P. Wei, M. Qin, K. E. Dorfman, X. Yuan, C. Liu, Z. Zeng, X. Ge, X. Zhu, Q. Liang, B. Yao, Q. J. Wang, H. Li, J. Liu, Y. Zhang, S. Y. Jeong, G. S. Yun, D. E. Kim, P. Lu, and R. Li, “Probing electron-atom collision dynamics in gas plasma by high-order harmonic spectroscopy,” Opt. Lett. 43, 1970–1973 (2018). [CrossRef] [PubMed]

22. M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005). [CrossRef]

23. M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982). [CrossRef]

24. J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992). [CrossRef] [PubMed]

25. K. E. Dorfman, B. P. Fingerhut, and S. Mukamel, “Time-resolved broadband raman spectroscopies: A unified six-wave-mixing representation,” The J. Chem. Phys. 139, 124113 (2013). [CrossRef] [PubMed]

26. R. Kubo, “Stochastic liouville equations,” J. Math. Phys. 4, 174–183 (1963). [CrossRef]

27. M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017). [CrossRef] [PubMed]

28. S. Mori and O. Sueoka, “Excitation and ionization cross sections of he, ne and ar by positron impact,” J. Phys. B: At. Mol. Opt. Phys. 27, 4349–4364 (1994). [CrossRef]

29. D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015). [CrossRef]

30. D. Kartashov and M. N. Shneider, “Femtosecond filament initiated, microwave heated cavity-free nitrogen laser in air,” J. Appl. Phys. 121, 113303 (2017). [CrossRef]

31. T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996). [CrossRef] [PubMed]

32. M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University, 1997). [CrossRef]

33. O. Smirnova and M. Ivanov, Multielectron High Harmonic Generation: Simple Man on a Complex Plane (John Wiley & Sons, Ltd, 2014), chap. 7, pp. 201–256.

34. J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011). [CrossRef]

Quantum interference and collisional dynamics in excited bounds states revealed by time-resolved pump-high-harmonic-generation-probe spectroscopy

Abstract

1. Introduction

2. Bound-state dynamics in HHG

3. Collisional dephasing

4. HHG spectra

5. Simulations

5.1. Snapshot spectra of PHHGP

5.2. Integrated time-resolved PHHGP signal

6. Discussion

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (34)

Optics Express