1. Introduction

Photoemission in an atom or a molecule is one of the most fundamental processes in atomic physics and one of the key signatures of the quantum nature of atoms. For long time this process was regarded as instantaneous, but new advances in attosecond time resolved spectroscopy [1, 2] have shown that the emission of an electron after the absorption of an attosecond pulse takes a small but finite time. This natural response time which is intrinsic to every atom or molecule is often called delay in the photoemission [3, 4] or Wigner time [5, 6] and it is associated with the energy derivate of the dipole phase of the released electron.

The measurement of the Wigner time delay is a challenging task [3, 7]. While traditional experimental observables such as the cross section can measure the amplitude of the dipole matrix element, in order to provide a complete picture of the photoionization it is required to measure the amplitude and phase of the dipole matrix element. The attosecond metrology provide access to the phase of the dipole through new techniques such as the attosecond streaking [8], Reconstruction of Attosecond Beating By Interference of Two-photon transitions (RABBIT) [9] or Quantum Spectral Phase Interferometry for Direct Electron-wavepacket Reconstruction (QSPIDER) [7].

In 2010 Schultze and co-workers [10] measured the relative delay in photoemission from the 2s and 2p states of Ne using the streaking technique. The measurement showed a 21 as relative delay between these two states. In addition, the RABBIT technique [9] was used to measure a relative delay in photoemission between the 3s and 3p states in Argon [11]. These measurements triggered intense theoretical and experimental research about this topic. [4].

Recently, a measurement of the relative time delay in photoemission between direct and shake-up electrons has been used to measure the electron correlation in Helium [12]. In this paper the authors were able to access the effective dipole and ultimately reconstruct a bound electron wavepacket through the phase difference between the two participating states, demonstrating how these measurements can provide access into the dynamics of the ionization process.

The concept of delay in photoemission has been extended to small molecules such as ${\mathrm{H}}_2^{+}$ where the effect of the electron diffraction or the effect of the moving nuclei has been addressed [13, 14].

The measurement of the delay in photoemission has become an important tool in recent years to study the ultrafast electron dynamics of photoabsorption and in particular to study the effect of resonances in atoms [15–17] as well as in molecules. Shape resonances in molecules has been studied recently using the RABBIT technique [18–20] in diatomic molecules.

In this paper we look into the delay in photoemission in a small asymmetric molecule such as CO. In particular we study the effect of a resonance induced by the streaking field on the measured delay. While the streaking field is always considered an auxiliary tool which does not distort the intrinsic delay, we show that the IR laser can modify the delay in photoemission. We show this effect by comparing a measurement of the delay where only the attosecond pulse is used against the measurement of the same quantity with the streaking technique that uses the IR laser field.

We perform theoretical calculations to observe the evolution of the ionized and bound electrons. We use the Stereo Wigner Time Delay (SWTD) which we introduced before [21] to look into the left right asymmetry of ionized electrons. This is a very useful measurement which is very sensitive to small asymmetries of the molecule and provides rich insights of the ionization process and the Wigner Time Delay.

The paper is organized as follows: In Section II, we introduce the theory of Wigner time delay and a model of a small asymmetric molecule in cylindrical coordinates. This model is used to characterize the absorption of a single attosecond pulse from the HOMO and HOMO-1. In Section III, the Time Of Flight (TOF) technique is introduced to extract the Wigner time which provides a good estimation of the intrinsic delay [21]. In Section IV, we employ the streaking technique to recover the stereo Wigner time and describe the effects of the IR pulse within the measurement process. In Section V, we discuss the differences between the TOF and streaking techniques which takes into account the absorption of a SAP in the presence of a weak IR laser field.

2. Theory

In this section we will describe the concepts of Wigner Time Delay and introduce the model for the diatomic CO molecule which we will use to perform the simulations with the Time Dependent Schroedinger Equation (TDSE).

2.1. Winger time delay

The Wigner time delay [5] is defined as $\mathrm{\Delta }{t}_{\mathrm{W}}=\frac{\partial {\varphi }_l(E)}{\partial E}$ the partial energy derivative of the dipole phase, ϕ_l(E). This formula relates the phase ϕ_l(E) of the complex dipole matrix element to the response time of the system for the emission of an electron after the XUV photon absorption and depends on the final photoelectron energy E and angular momentum l.

As described before, the measurement of such quantity is beyond the scope of the traditional observables. Only recently and thanks to the attosecond science tools this quantity has been measured successfully using the streak camera [10] and RABBIT [11] techniques. All these measurements consider the absorption of a single attosecond pulse or a train of attosecond pulses in the presence of a weak IR pulse.

The delay in photoemission Δt_obs measured in the laboratory is made of different contributions [22].

Where, Δt_W, is the Wigner time, Δt_atto, the phase of the attosecond pulse or attochirp [4,10], Δt_CLC, the Coulomb laser Coupling [23, 24] and, Δt_Pol, the polarization which could be produced by a permanent dipole of the system [25] or from the the laser field used in the measurements [22, 26]. Not all of these contributions will be present in each situation and some others might be relevant for the process. For instance, if the attosecond pulse is a Fourier limited pulse, then the term Δt_atto will be null. The inclusion of the laser field in the measurement introduces at least one term: Coulomb Laser Coupling, Δt_CLC. Several theoretical works [23, 24] estimate the value of this time which depends very weakly on the intensity. Another contribution from the laser field is Δt_Pol, which refers to the induced delay from the polarization of the initial state. The polarizations term Δt_Pol induced by the streaking field is usually small and this quantity is usually neglected, i.e. for deeply bound electrons and low intensity laser fields [4]. Further, a similar contribution from multi-electron effects has been reported to affect the measured delay in photoemission [4, 12, 27].

We study the delay in photoemission using the Stereo Wigner Time Delay which we introduced previously [21]. This technique measures the differences in Wigner time for electrons emitted to the left and right along a given direction with respect to the laser-field linear polarization. This time we look into this quantity in a 3D model of the CO asymmetric molecule where we integrate numerically the Time Dependent Schrödinger Equation (TDSE).

The SWTD technique has several advantages: First, a single orbital is self referenced which reduces the need to compare two different orbitals. Second, all the contributions to the observed delay, with the exception of the Wigner time, are equal for the left-right emitted electrons and therefore the CLC and the atto chirp contributions will be canceled with this method [21].

2.2. Diatomic asymmetric molecular model

We have built a model of the CO molecule which we use to integrate numerically the TDSE for different attosecond pulses in order to study the Wigner Time Delay. The model contains the most relevant features of the CO molecule to the absorption of the attosecond pulse.

As the nuclear dynamics is much slower than the electron within the short time duration of the laser pulses we use, the position of the nuclei is fix to the equilibrium internuclear distance.

We use the Single Active Electron (SAE) approximation and look into one orbital in each case. By orienting the molecular axis parallel to the XUV and laser fields, we restrict the geometry of the interactions which allow us to use cylindrical symmetry. Hence, only the coordinates (z, ρ) are considered to describe the system. The field-free Hamiltonian for our molecular model is (atomic units are used throughout this paper unless otherwise stated):

The potential energy, V₀(z, ρ), of the system reads:

The simulations are carried using our BALAS code [29] which solves the TDSE in cylindrical coordinates by implementing both the Crank-Nicholson method and Split Operator one [28, 30]. In this paper we use the Crank-Nicholson method which has an error proportional to the square of the time step O(Δt)². For the initial state computation, the imaginary time-step is Δt = −0.04i a.u. The length along the z- and the ρ-axis are L_z = 350 a.u., and L_ρ = 150 a.u., with a spacing Δz = Δρ = 0.2 a.u.

To mimic the shape and the number of nodes of the CO HOMO, we choose our initial state as the third excited orbital, of this potential which is depicted in Fig. 1. The orbital shape is the same than the HOMO for the CO molecule [31], therefore we refer to this orbital as the HOMO of the CO. Similarly, we refer to the second excited state of our model as the HOMO-1 and the fourth excited state of the model as the LUMO of CO.

 

Fig. 1 The cartesian 3D visualization of the asymmetric Coulomb potential (blue-red color scale) and the HOMO orbital (green-sangria color scale) are depicted as a function of the position (x, y, z). Z₁ And Z₂ denote the location of the nuclei charges for the ’C’ and ’O’, respectively. The red arrow points out the direction of the static dipole and the larger nucleus charge. The solid iso-volume of the electron density is shifted along y-axis to distinguish it from the potential. The potential and orbital iso-volumes are cut around an angle of 120° which allows us to view the change of the asymmetric potential and the electron density for different spatial volumes.

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The charge parameters Z₁ and Z₂ are chosen to match as closely as possible the ionization potential, I_p, of HOMO for the CO (I_p ≈ 14 eV), and to provide the Coulomb asymptotic behavior of the potential Z/r. By considering those facts, we have found that Z₁ = 0.7 and Z₂ = 2.3 with the total charge, Z = Z₁ + Z₂, gives the best choice in the defined potential of Eq. (3). The internuclear distance is fixed to R = 2.2 a.u. These parameters yield an ionization potential for the HOMO of I_p = 0.58 a.u. (the experimental one is 0.51 a.u.) [31]. Figure 1 depicts the HOMO orbital and the potential in space. The asymmetry of the orbital and the potential can be observed in the figure.

Next, we describe the ionization of an electron from the HOMO of our potential model by a single attosecond pulse and introduce the Time of Flight method to extract the stereo Wigner time delay [21]. The Hamiltonian for the interaction of an molecular system with the single attosecond pulse in the dipole approximation using the velocity gauge reads:

For the first example the attosecond pulse has a gaussian envelope with pulse duration of FWHM= 10.6 a.u. (256 as), a central frequency of ω_X = 1.8 a.u. (49 eV), and intensity I₀ = 5.0 × 10¹² W/cm². Figure 2(a) shows an iso-volume of the bound orbital |Ψ₀(x, y, z, t_F)|² and the continuum electronic density of the wavefunction |Ψ_c(x, y, z, t_F)|² after the absorption of the attosecond pulse at time t = t_F = 80 a.u.

 

Fig. 2 a) Whole position electronic density of the wavefunction, Ψ (x, y, z), at time t = 80 a.u., after the absorption of a SAP. This XUV pulse is also depicted in the upper panel of a) in violet line. The central part shows the initial HOMO orbital and the continuum part of the electron position density is the outgoing electron wavepacket. In the plane y = 0 a projection of the whole electronic density shows the asymmetry of the continuum electron wavepacket. b) The continuum momentum electron density, |Ψ_c(p_x, p_y, p_z)|², consist of a shell with radius p₀ = 1.56 a.u.

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The emitted electron wave packet forms a spheric shell traveling outwards. It has three nodes along the z-direction, two of them from the original HOMO state and the third related to the photo absorption process. In the same figure, the wavefunction projection into the y = 0 plane show the mapping of the initial state asymmetry.

To compute the Photoelectron Momentum Distribution (PMD), we use a spatial mask at some time t_F after the end of the pulse where the ionized electron is far away from the Coulomb potential but still fully contained in the integration box. The free wavefunction is defined as Ψ_c(z, ρ) = f_mask(z, ρ)Ψ(z, ρ, t_F) where the mask is

Here a is an inner radius parameter. We then project the free electron wavepacket Ψ_c(z, ρ, t_F), into plane waves by means of the Fourier and Hankel transformations, $\mathrm{\Psi }\left(p_z,p_{\rho }\right)=\mathscr{H}ℱ\left[{\mathrm{\Psi }}_c\left(z,\rho ,t_{\mathrm{F}}\right)\right]$. These transformations have been implemented in our code BALAS and there computation is parallelized for multicore architectures which makes the computation very efficient. The Hankel transform and Fourier transform are performed by using the quasi-discrete Hankel transform [32] and FFT, respectively.

Figure 2(b) shows the electron wavefunction after the absorption of the attosecond pulse. The momentum distribution of the emitted electron wavepacket forms a shell centered at the momentum $p_0=\sqrt{2\left({\omega }_{\mathrm{X}}-I_p\right)}=1.56\mathrm{a}.\mathrm{u}.$. We observe that the photoelectron spectral width is proportional to the spectral bandwidth of the attosecond pulse and reflects also the asymmetry of the HOMO in the direction of the internuclear axis as expected.

3. Time of flight method

The Stereo Wigner Time Delay is a very sensitive measure of the molecular asymmetry and in this section, we will measure it using the Time of Flight (TOF) technique which we introduced before [21]. We choose to compare the ionized electron to the left and right of the molecule along the internuclear axis. The TOF concept is very simple: From the center of the initial state 〈z_HOMO〉, we track the time that the electron takes to reach a fix position to each side. Hence, two positions (left and right) along the internuclear z axis are defined. Then, we calculate the position expectation values, 〈z^(L,R)〉 by:

Here, Ψ_c(z, ρ, t), is the electron wavepacket ionized after the interaction with the attosecond pulse. The choice of this region (ρ_d = 1.0 a.u.) is tested for convergence and to ensure the robustness of the measurements. Assuming that the electron appears in the continuum at the position 〈z_HOMO〉 of the expected value of z of the HOMO state, two positions with respect to this point are defined.

We define the TOF delay, Δt_TOF = t_d − t₀, where, t₀ is the central time of the attosecond pulse envelope and t_d is the arrival time of the electron to the detection position. This arrival is measured by the electron position expectation values: $\left|〈z^{(\mathrm{L},\mathrm{R})}〉-〈z_{\text{HOMO}}〉\right|=z_d^{(\mathrm{L},\mathrm{R})}$. We obtain the TOF for left $(\mathrm{\Delta }{t}_{\text{TOF}}^{(\mathrm{L})})$ and right $(\mathrm{\Delta }{t}_{\text{TOF}}^{(\mathrm{R})})$ directions, thereby the difference between both times, at a certain position z_d = |〈z^(L)〉 − 〈z_HOMO〉| = |〈z^(R)〉 − 〈z_HOMO〉|. Then, the quantity $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{LR})}=\mathrm{\Delta }{t}_{\text{TOF}}^{(\mathrm{L})}-\mathrm{\Delta }{t}_{\text{TOF}}^{(\mathrm{R})}$ is referred as the Stereo Wigner Time Delay measured by the TOF.

As it can be inferred from the Eq. (1), the attosecond chirp and the CLC delay contributions are removed in the stereo measurement and therefore the $\mathrm{\Delta }{t}_{obs}^{(\text{LR})}$ corresponds to the differences in Wigner times only. In our previous paper [21], we have demonstrated that this measurement is equivalent to the exact left-right energy derivative of the dipole matrix element phase which defines the Stereo Wigner Time Delay.

Figure 3(a) shows a diagram of how the TOF delay is calculated within our model. The attosecond pulse is the purple curve and sets the initial time t₀ for the measurement. The left-right expectation values, |〈z^(L,R)〉 − 〈z_HOMO〉| are only well defined after the maximum of the pulse. They are shown in the green and blue curves of Fig. 3(a). Then, t_d is defined as the time when these curves cross the red line at |〈z^(L)〉 − 〈z_5σ〉| = z_d which correspond to the detection position. The inset shows the electron arrival to the reference position z_d. As the electron wavepacket is asymmetric, the emitted electrons to each side take different times to arrive to the detection position. This corresponds to different Wigner delays and therefore the stereo Wigner time is delay different from zero.

 

Fig. 3 Conceptual stereo time of flight delay. a) Stereo TOF delay technique computed via the numerical solution of the 3D-TDSE in case of the interaction of a SAP (violet line) with the CO HOMO. The blue and green lines depict the expectation position values, |〈z^(L,R)〉 − 〈z_HOMO〉|, as a function of time, for the emitted electron wavepackets on the left and right, respectively. The insets plot shows the stereo TOF delay, $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{TOF})}$, which is the difference on time between the left-right trajectories at a certain detection position z_d. In this case, the detection position is denoted by the red dashed horizontal line at z_d = 70 a.u. b) Delay in stereo photoionisation is calculated for a set of different XUV central frequency or central photoelectron energies.

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The parameters of the attosecond pulse to compute the results depicted in the Fig. 3 are the same as those employed in Fig. 2. The result from the measurement gives the value $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{LR})}=0.67\mathrm{a}.\mathrm{u}.$, for the photoelectron energy $E_0=p_0^2/2=1.21\mathrm{a}.\mathrm{u}.$

The same procedure is repeated for different central frequencies of the attosecond pulse which correspond to different photoelectron central energies $E_0=\frac{p_0^2}{2}$. In each case, the same analysis is made to obtain stereo TOF delays. Figure 3(b) shows the values of the SWTD using the TOF technique for different final photoelectron energies E₀. The variation of the SWTD is smooth over the region of final electron energies studied here.

The measurement of the SWTD using the TOF technique has some further advantages. The delays computed with this method depend only on the parameters of the XUV pulse and the complex dipole transition matrix element of the molecule. In this technique, we do not use an auxiliary IR laser field and it is therefore a useful reference to evaluate the effects of the IR laser. Nevertheless, while the TOF technique is a good conceptual tool, it is not suitable as a measurement scheme in the lab. Therefore, we investigate the stereo Wigner delay with the streaking technique which is normally used in the experiment and compare the results.

4. Measurement of the SWTD with the streaking technique

The streaking technique [1] consists in the absorption of a single attosecond pulse in the presence of a weak and short IR pulse. The effect of the short IR laser field on the electron is to shift or streak the final momentum along the direction of the laser polarization at the ionization time.

If the absorption of the attosecond pulse occurs in the presence of the weak IR laser, the final momentum is then, p(τ) = p₀ − A_L(τ), where, τ, denotes the delay between the XUV attosecond pulse and the IR laser field

The streaking delay in photoemission, Δt_S, can be understood as the time-shift between the IR laser and the measured streaking trace. We have numerically solved the TDSE using the following Hamiltonian:

To produce the streaking trace, first, we obtain the final momentum distribution and integrate it over a small angle θ = 10° around the z-axis for the left-right emitted photoelectrons S_e(p_z, τ) [30]. Then, we scan on the XUV-IR time delay, τ, the streaking trace is then defined as a function of the momentum p_z and the delay τ

Figure 4 show the positive and negative momentum distributions along p_z with streaking traces for two different attosecond pulses polarized along the molecular axis z. In the panels (a–b) we use an attosecond pulse with ω_X = 2.5 a.u. (68.025 eV), FWHM= 10.0 a.u. (241.8 as). In panels (c–d) we show the case for a higher energy attosecond pulse ω_X = 4.0 a.u. (108.84 eV), FWHM= 6.25 a.u. (151.12 as), both cases with peak intensities I₀ = 2.0 × 10¹² W/cm². The IR laser pulse is six cycles long with a sin² envelope, frequency ω_L = 0.057 a.u., (800 nm of wavelength) and intensity I₀ = 2.0 × 10¹² W/cm².

 

Fig. 4 Momentum distribution of the photoelectron along the z direction as a function of the delay between the attosecond pulse and the streaking field. The black line shows the vector potential of the streaking field. (a) Positive (right) momentum for an attosecond with central energy ω_X = 2.5 a.u. (b) the corresponding negative (left) momentum. (c) Positive (right) momentum for an attosecond with central energy ω_X = 4.0 a.u. (d) the corresponding negative (left) momentum.

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The inspection of the streak traces from the HOMO orbital show several interesting features. Figure 4 shows the momentum distribution along the z axis for positive (a,c) momentum and (b,d) negative. The color scale shows that the amplitude of the distribution is different in both sides as expected from the asymmetry of the initial orbital. The most relevant information of the figure is the fact that after the streaking pulse has finished, the momentum distributions still changes as a function of the delay between pulses. This can be seen for all different streaking traces simulated with different central energies between ω_x = 1 − 4 a.u.

From the streaking field we can calculate the expected value of the momentum $〈p_{\tau ,z}^{(\mathrm{L},\text{R})}〉$, restricted to positive or negative momenta and compare these curves with the vector potential.

Figure 5 (a,b) shows the expected momentum distribution for left and right for ω_X = 2.5 a.u. and ω_X = 4 a.u. compared with the vector potential of the streaking field. At the beginning, for negative delays τ the three curves are similar. As expected, they show a delay between them, but around the middle of the IR field, the streaking curves are distorted and don’t follow the vector potential any more. After the streaking pulse is finished we observe oscillations in the expected momentum $〈p_{\tau ,z}^{(\mathrm{L},\text{R)}}〉$ with a well defined frequency. This occurs for all attoseconds pulses with different central energies considered.

 

Fig. 5 Expected values of the momentum 〈p_z〉 as a function of the delay between the attosecond pulse and the streaking field. The figure show the $〈p_z^{L,R}〉$ where the momentum is restricted positive (negative) values. The values of the central energy of the attoseconds are (a) ω_X = 2.5 a.u. and (b) ω_X = 4 a.u.

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The oscillations also appear in the height of the momentum distribution peaks. As the delay τ between the attosecond and the IR laser pulse changes we observe the streak in the direction of the laser polarization but we also observe that the height of the peaks of the momentum distribution changes as a function of the delay. Each of the two peaks in the distribution change heights differently for different delays which is not normally observed in the calculated streak traces.

To recover the exact value of the time delay in the photoemission measured by the streaking technique, we take the Fourier Transform of, $〈p_{\tau ,z}^{(\mathrm{L},\text{R)}}〉$, with respect to τ. The stereo delay measurement to each side, $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\mathrm{L},\mathrm{R})}=\frac{{\varphi }^{(\mathrm{L},\mathrm{R})}\left({\omega }_0\right)}{{\omega }_0}$, is obtained as the value of the phase at the central frequency ω₀ [21]. We calculate the stereo streaking time delay, $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\text{LR})}$, as the difference between the time delays to each side.

As an example, for an attosecond pulse of central frequency ω_X = 1.5 a.u., we have obtained a SWTD with the streaking of $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\text{LR})}=2.77\mathrm{a}.\mathrm{u}.$, which is quite different to the SWTD with TOF $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{LR})}=0.65\mathrm{a}.\mathrm{u}.$ As both methods should measure the stereo Wigner time, this disagreement is not expected.

This difference between the Stereo Wigner time measured with the TOF and the Streaking technique can be observed for a large range of central frequencies of the XUV attosecond pulses which correspond to different final photoelectron energies. Figure 6 shows the Stereo Wigner Time Delay measured with streaking (orange curve) and the TOF (dark blue line) for a range of central frequencies ω_X = 1 − 4 a.u. of the attosecond pulse while maintaining all other parameters of the pulses the same. Although all of the extra contributions defined in Eq. (1) are removed by the stereo measurement, again, a large discrepancy is observed in the SWTD of the HOMO orbital measured with different techniques.

 

Fig. 6 The plot shows the SWTD $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{LR})}$ (blue line with circles) for the HOMO orbital as a function of the electron energy (E₀ = ω_X − I_p) using the TOF technique where no IR laser is present. It also show the SWTD $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\text{LR})}$ for the HOMO obtained with the streaking technique (yellow and red curves) which uses an auxiliary IR field. In the yellow curve we use the definition of the phase $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\mathrm{L},\mathrm{R})}=\frac{{\varphi }^{(\mathrm{L},\mathrm{R})}\left(-{\omega }_0\right)}{-{\omega }_0}$ and in red the definition $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\mathrm{L},\mathrm{R})}=\frac{{\varphi }^{(\mathrm{L},\mathrm{R})}\left({\omega }_0\right)}{{\omega }_0}$ which are both equivalent and provide the same result.

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The observation of distorted streaking traces in the HOMO orbital which leads different SWTD than those measured with the TOD are in stark contrast with the observations made for electrons starting in the HOMO-1 orbital. Figure 7 shows momentum distribution in p_z as a function of the delay τ, i.e the streaking trace for attoseconds pulses with ω_X = 2 and ω_X = 3 a.u. for an electron with the HOMO-1 as the initial state. In this case we observe a left right asymmetry in the streaking traces from the spatial asymmetry of the initial orbital HOMO-1 as expected. But in this case we do not observe any change in the momentum distribution after the streaking field has finished as expected in the streaking technique.

 

Fig. 7 Momentum distribution of the photoelectron from the HOMO-1 along the z direction as a function of the delay between the attosecond pulse and the streaking field. The black line shows the vector potential of the streaking field. (a) Positive (right) momentum for an attosecond with central energy ω_x = 2:0 a.u. (b) the corresponding negative (left) momentum. (c) Positive (right) momentum for an attosecond with central energy ω_x = 3:0 a.u. (d) the corresponding negative (left) momentum.

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By calculating the expected momentum in the z direction $〈p_{\tau ,z}^{(\mathrm{L},\mathrm{R})}〉$ restricted to positive and negative momentum (Fig. 8(a) and 8(b)) according to equation (8) we observe that the streaking curves are similar to the vector potential of the streaking field for all delays τ, therefore there is no distortion in this streaking trace. The curves show a clear delay between them and with the streaking vector potential as expected. Also, in all cases the expected momentum remains constant after the streaking field has finished. A look into the height of the momentum distribution peaks for the HOMO-1 orbital shows the expected streak in the polarization direction as a function of the delay but the height of the peaks remain equal for every delay as expected.

If we use the HOMO-1 molecular orbital as initial state and compute the Stereo Wigner Time Delay using the TOF and the streaking methods we obtain a very good agreement. While in a real molecule the attosecond pulse might ionize several states through different channels, we perform this numerical experiment considering only one electron in the HOMO-1 orbital to highlight the fact that when the resonance is not present, the TOF and the Streaking method produce the same SWTD as expected.

We perform this comparison using the central XUV attosecond pulse frequency of ω_X = 2.3 a.u. (62.6 eV), a pulse has a gaussian envelope FWHM= 250 as, with a the peak intensity of I₀ = 5 × 10¹² W/cm², and CEP of $\frac{\pi }{2}$. From this measurement, we have obtained a stereo Wigner delay or TOF delay of $\mathrm{\Delta }{t}_{\text{TOF}}^{(\text{LR})}=-1.37\mathrm{a}.\mathrm{u}.$ From the stereo streaking measurement of the asymmetric delay, we have extracted $\mathrm{\Delta }{t}_{\mathrm{S}}^{(\text{LR})}=-1.3\mathrm{a}.\mathrm{u}.$ This good agreement with the TOF measurement is in complete agreement with our previous demonstration in [21].

This good agreement for the case of HOMO-1 orbital rules out any numerical problem as a source of the observed discrepancies between the TOF and the streaking methods in the HOMO results. Therefore we look into the physical reason for the disagreement in the effect of the streaking IR laser which is not present in the TOF method.

In our molecular model for the CO molecule, the energy difference between the HOMO and the LUMO is ΔE = E_LUMO − E_HOMO = 0.06807 a.u., which is close to the photon energy of the IR laser ω_L = 0.057 a.u., therefore a resonant transition from HOMO to the LUMO is possible with one photon from the streaking field.

Figure 9 shows the population of the HOMO and LUMO as a function of time when the molecule interacts only with the IR laser pulse. The laser parameters are the same as those used in the streaking trace of the HOMO. Initially all the population is in the HOMO but slowly the electron is transferred to the LUMO to reach ~ 10% of the whole probability at the end of the pulse. The transfered population can be very efficient even though the intensity of the IR laser is low. The population transfer creates a coherent superposition of bound states, the population in the LUMO is significant only after the middle of the IR pulse. Therefore if the attosecond is located at an XUV-IR time-delay τ > 0, it will probe the coherent wavepacket created by the streaking IR laser.

 

Fig. 8 Expected values from the HOMO-1 orbital of the momentum 〈p_z〉 as a function of the delay between the attosecond pulse and the streaking field. The figure show the $〈p_z^{L,R}〉$ where the momentum is restricted to positive (negative) values. The values of the central energy of the attoseconds are (a) ω_x = 2 a.u. and (b) ω_x = 3 a.u.

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Fig. 9 (Population of the HOMO (blue solid line) and LUMO (orange solid line) bound orbitals as a function of time. In the black solid line, we have depicted the temporal shape of the IR laser electric field.

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This is the reason for the observed temporal distortions of Fig. 5 and as well as the source of discrepancy between the SWTD measured with streaking and the TOF. This observation is similar to that reported in the literature [18] where a resonance, in that case a shape resonance, alters significatively the measured time delay. In this case the resonant absorption produces a much larger delay in photoabsorption.

We can remove the effect of the resonant population transfer by using a different wavelength for the streaking field. When we use a 4 cycles long streaking field of 200 nm, I₀ = 5 × 1011 W/cm² and ω_X = 2:0 a.u., I_X = 10¹³ W/cm² and 8 cycles in the attosecond pulse, we have found that the resulting SWTD is Δt_S ≈ 0.877 a.u. As it is expected, this streaking delay is in reasonable agreement with the result from the TOF technique, Δt_TOF = 0.75 a.u., for the same XUV attosecond pulse. In this case, the streaking field spectrum is far from the resonance and the field do not distort the initial state significantly.

Another test to confirm the resonant population transfer by the streaking laser is the calculation of the final LUMO population as a function of the IR intensity. We have calculated it for an intensity range of three orders of magnitude by considering only the interaction of the 800 nm laser pulse. We have found that the of intensity vs population in the LUMO state in the log-log scale fits to a straight line with a slope equal to one. This confirms the one photon resonant transition process.

We have repeated the measurements of Stereo Wigner Time Delay with streaking in case of HOMO for lower intensities to observe the effect of the final population in the LUMO towards the extracted delay in photoemission. One could expect that for low IR intensities the LUMO population should decrease and the stereo streaking delay would come closer to the stereo TOF delay. This is not the case. The variation of stereo streaking delay is small over three order of magnitude of the IR streaking field strength and where the LUMO population goes from the 30 to 0.1%. The coherent bound wave packet driven by the IR pulse largely modifies the streaking measurement even for low intensities.

This behavior can be used to measure some features of the coherent bound wavepacket created by the IR streaking field. As mentioned before, the expected value of the momemtum distribution continues to oscillate after the streaking field has finished. This is the result of the bound electron created by the streaking field being probed by the attosecond pulse. The phase of this oscillation is directly related to the relative phase between the HOMO and LUMO orbitals which establish uniquely most of the information of the bound wavepacket that was formed. This provides an important insight for the reconstruction of this bound electron wavepacket.

As the Stereo Wigner Time Delay is a very sensitive measure of the asymmetry in the molecule it could be used to study the formation of bound wapackets and to access the relative phase between the states forming these wavepackets.

We now look at the results of the streaking measurement for the HOMO orbital and ω_X = 2.0 a.u. with the same parameters of the vector potential as before. Fig. 10 shows the shifted expected value of the electron momentum along the z direction, 〈p_z〉 ± p₀ for electrons emitted to the left and right together with the streaking vector potential for that measurement. In this figure we have included the scaled cosine of the relative phase Δϕ_r between the states HOMO and LUMO obtained obtained by projecting over these state during the the time propagation of the TDSE. The bound wavepacket created by the streaking field is absolutely determined by the amplitude and relative phase Δϕ_r of the two states HOMO and LUMO forming the wavepacket. As the violet curve in figure shows, the cos(Δϕ_r) matches exactly with the behavior of the 〈p_z〉 − p₀ curve which means that the relative phase can be extracted from the measurement and the bound wavepacket reconstructed.

 

Fig. 10 Expected value of the momentum distribution compared to the cos(Δϕ_r) relative phase of the HOMO and LUMO obtained with the simulation of IR only. The phase difference between these two states can be obtained with the streaking technique

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5. Conclusions

The Wigner time delay or delay in photoemission is a very accurate measure of the dipole matrix element phase in molecules but its experimental observation represent an important challenge.

In this paper we report the effects of a resonant transitions in the measurement of the Stereo Wigner time delay for a reduced model of the asymmetric CO molecule. To this end we make use of the numerical integration of the TDSE and the concept of SWTD which presents several advantages for the measurement.

While the weak IR laser field is always considered as an auxiliary field in the measurement of the Wigner time delay with the streaking technique, we show a case where this field induce a resonant transition which distorts the measurement.

For the HOMO orbital of our CO model molecule, the measurements of the STWD with and without the IR laser field provide different results. This is explained because in our model the energy gap between the HOMO and LUMO matches the laser frequency.

For an electron starting in the HOMO-1 the resonance does not exist and the streaking technique coincides with the TOF technique as expected.

By further examination of the resonant case in the HOMO orbital we have found that regardless of the population transfered to the LUMO orbital the measurement of the SWTD shows the same difference with the TOF measurement even with populations as low as 0.1%. This means that the measurement of the SWTD is highly sensitive even though the population is low.

More importantly, analyzing the results of the streaking measurement in the HOMO orbital we have found that the oscillations of the expected value of the momentum 〈p_z〉 after the end of the streaking field coincides with the cosine of the relative phase Δϕ_r between the HOMO and LUMO orbitals. This is an important finding because this relative phase Δϕ_r between the HOMO and LUMO determines the bound wavepacket created by the streaking field and this information could be measured experimentally.

The observed behavior depends on energy separation of the HOMO and LUMO orbitals in our reduced model of CO. But this effect could be observed in real molecules as the energy difference between the HOMO and LUMO is often close to the photon energies in IR lasers [33]. Or equivalent, a streaking field can be chosen to match the energy difference and observe this effect.

This paper shows then the important applications of the SWTD in small asymmetric molecules exploiting the unique characteristics of the attosecond pulses.