Holographic Angular Streaking of Electrons and the Wigner-Time Delay

For a circularly polarized single-color field at a central frequency of $2\omega$ the final electron momentum distribution upon strong field ionization does not carry any information about the phase of the initial momentum distribution. Adding a weak, co-rotating, circularly polarized field at a central frequency of $\omega$ gives rise to a sub-cycle interference pattern (holographic angular streaking of electrons (HASE)). This interference pattern allows for the retrieval of the derivative of the phase of the initial momentum distribution after tunneling $\phi^{\prime}_{\mathrm{off}}(p_i)$. A trajectory-based semi-classical model is introduced which links the experimentally accessible quantities and $\phi^{\prime}_{\mathrm{off}}(p_i)$. It is found that the differences in release time can deviate by more than 100\,as from multiples of half a cycle of the two-color field. Further, it is shown that a change in $\phi^{\prime}_{\mathrm{off}}(p_i)$ is equivalent to a displacement in position space $\Delta x$ of the initial wave packet after tunneling. This offset in position space allows for an intuitive interpretation of the Wigner time delay $\Delta \tau_W$ in strong field ionization for circularly polarized single-color fields.


INTRODUCTION
The appearance of a comb of discrete peaks in the energy distributions of electrons upon strong field ionization [1] of single atoms or molecules is well-known as above threshold ionization (ATI) [2][3][4].This quantization of energy can be interpreted as a consequence of energy conservation and the finite bandwith of the incident photons [5].Alternatively, the time-dependent electric field of the incident light can be considered: for a light pulse with multiple cycles, the periodic release of electron wave packets (grating in the time-domain) gives rise to equally spaced interference fringes in energy-space.Thus, it is obvious that the periodicity of the light's timedependent electric field and the value for the photon energy are two sides of the same coin [6].Upon strong field ionization by a single-color light field at a central frequency of 2ω the electron energy spectrum shows discrete energy peaks that are separated by ∆E = 2hω (T 390 = 2π 2ω , 2ω = 0.1168 a.u.corresponds to the frequency of a light field at a wavelength of 390 nm).These discrete electron energies appear as concentric rings in the electron momentum distribution in the plane of polarization.
Using two-color fields that consist of a high-intensity light field at a central frequency of 2ω and a low-intensity light field at a central frequency ω (typically the intensities differ by a factor of 100) leads to the appearance of sidebands between the energy peaks from the 2ω field [7].Light fields that consist of a fundamental and a second harmonic frequency that have the same helicity are referred to as co-rotating two-color (CoRTC) fields (see Fig. 1(a) for an example).In this case, the rings in the electron momentum distribution are modulated in intensity as a function of the angle in the plane of polarization [8].This results in an alternating half-ring pattern in momentum space.Fig. 1(b) shows a sketch of a typical alternating half-ring pattern.The origin of this pattern is a sub-cycle interference which has been observed experimentally and reproduced using saddle-point strong field approximation as well as by solving the time-dependent Schrödinger equation [8,9].Recently, Feng et al. [10] have succeeded to model the appearance of sidebands for linearly polarized light by considering semi-classical trajectories that have release times that differ exactly by T 390 which is one cycle of the 2ω field.
The generation of sidebands is closely related to reconstruction of attosecond harmonic beating by interference of two-photon transitions (RABBITT) [11][12][13].RAB-BITT can be used to access the Wigner time-delay [14].However, by definition RABBITT only treats two-photon transitions.Here, the perspective of interference of wave packets in momentum space [15,16] is chosen, which is referred to as holographic angular streaking of electrons (HASE).It will be shown that -by using the framework of HASE -the Wigner time-delay becomes accessible also in the multi-photon and tunneling regime by measuring final electron momentum distributions [17][18][19].
In this work a generalized trajectory based approach is presented that does not restrict the relative release times of the interfering trajectories to multiples of T 390 .The release time is the time at which the electron appears at the tunnel exit.The electron can possess an initial momentum after tunneling which is perpendicular to the direction of the laser electric field at the instance of tunneling.The basic idea is that there are two different release times within a single cycle of the CoRTC field that can lead to the same final electron momentum and that this gives rise to two path interference.

THEORETICAL MODEL
The time-dependent electric field E(t) that is exemplary used throughout this work is illustrated in Fig. 2(a), together with the corresponding negative vector potential − A(t) which is given in Eq. 1. E(t) and A(t) are linked by Here, t is the time and ω is the angular frequency of light at a wavelength of 780 nm (the field amplitudes are E 390 = 0.04 a.u. and E 780 = 0.004 a.u.for the two-color field and E 390 = 0.04 a.u. and E 780 = 0.0 a.u.for the single-color field throughout this work).
In this paper, a two-step model is used: the electron is set free by tunnel ionization and then accelerated by the laser field.For the propagation after tunneling the Coulomb interaction is neglected.Thus, the final electron momentum p f is the sum of the negative vector potential at the time the electron tunnels − A(t) and the initial momentum p i after tunneling.The initial momentum p i must be perpendicular to the laser electric field E(t) at the instance of tunneling.(The initial momentum component perpendicular to the yz-plane (p x ) is set to zero in our model.)This leads to Eq. 2. If one considers only release times t within a single light cycle (0 as < t < T 780 , with T 780 = 2π/ω ≈ 2602 as), then there are two release times t 1 = t 2 that lead to the same final electron momentum p f .Each release time t n must fulfill Eq. 2 to ensure that the initial momentum along the tunneling direction (which is anti-parallel to the electric field at the instance of tunneling) is zero (trajectory number n ∈ 1, 2, 3, 4).
Here, p n defines the absolute value and the sign of the initial momentum after tunneling.The sign of p n is defined such that a positive value of p n corresponds to a case in which − A(t n ) and p i,n are parallel and therefore | p f | > | A(t n )| for the field geometry shown in Fig. 2(a).In full analogy, negative values of p n lead to Solving Eq. 2 numerically leads to two possible initial momenta ( p i,1 and p i,2 ) for every final electron momentum p f .This gives rise to a two path interference.The two possibles pathways to the same final electron momentum are illustrated in Fig. 2(b): For several final electron momenta the two possible vector potentials that lead to these final electron momenta are indicated by connecting the respective − A(t 1 ) and − A(t 2 ) (using blue lines) with the final electron momenta (indicated by blue dots).Interestingly, the lines for the dot that is labeled with I in Fig. 2(b) have a finite intermediate angle.Closer inspection reveals that this is the case for all pairs of lines except for the dot that is labeled with II in Fig. 2(b).
To be able to model intra-and inter-cycle interference on the same footing using a semi-classical model we consider four wave packets (release times t 1 , t 2 , t 3 = t 1 +T 390 , t 4 = t 2 + T 390 , p i,1 = p i,3 and p i,2 = p i,4 ).The phases of the four semi-classical trajectories (trajectory number n ∈ 1, 2, 3, 4) at the time t f = t 4 are modeled by using Eq. 3. See e.g.Ref. [20][21][22][23] for an overview regarding semi-classical trajectories.
Here, I p denotes the ionization potential (I p = 15.76 eV is used, which is the ionization potential of argon) and φ off,n is an offset phase (for the sake of simplicity this offset phase can considered to be zero until the discussion of Fig. 6).I p t models the phase evolution of the electron in its bound state.φ prop ( p f , t n , t f ) describes the change of the electron's phase after tunneling starting from the release time t until the final time t f .At the time t f the electron possesses the final momentum p f .Throughout this paper the final time is set to t f = t 4 .For every considered trajectroy (n ∈ 1, 2, 3, 4), the change in phase after tunneling is described by the integral of the electron's energy over time [20]: Here, m e = 1 a.u. is the electron's mass and h = 1 a.u. is the reduced Planck constant.For Eq. 4 it is used that the instantaneous momentum can be expressed by the difference of the instantaneous vector potential A(t) and the final momentum p f .Thus, the semi-classically modeled wave function at a given final electron momentum p f is given by: Here, B n is the amplitude which is given by the square root of the existence probability of the respective trajectory (for the sake of simplicity the amplitude B n can considered to be one until the discussion of Fig. 6).Finally, the experimentally accessible intensity in final electron momentum space is modeled by

NUMERICAL EXAMPLES WITHOUT OFFSET PHASE
In the following, the newly introduced theoretical HASE-model is used to produce numerical results.In all examples two optical cycles of the two-color field are considered.The absolute value of the negative vector potential | − A(t)| is shown as a black line in Fig. 3(a) and the first half-cycle (0 as 2 is solved numerically.The release time t 1 is depicted as a function of the final electron momentum in Fig. 3(b).As expected, the release time increases monotonically with the angle in the plane of polarization [24].Fig. 3(c) shows the necessary initial momentum p i,1 as a function of the final electron momentum.Using Eq. 3 with an offset phase of zero (φ off,n = 0 rad) allows for the calculation of the phase as function of the final electron momentum (see Fig. 3(d)).
Fig. 3(e)-(h) are generated in analogy to Fig. 3(a)-(d) but here the release time is restricted to T 390 < t 2 ≤ 2T 390 .In Fig. 3(c) and (g) it can be nicely seen that the value of the initial momentum is zero at final momenta that coincide with the negative vector potential and increases (decreases) for increasing (lower) radial momenta.
In Fig. 4(a)-(d) the differences of the first and the second row in Fig. 3 are presented.The ionization times that are compared are highlighted in Fig. 4(a).The differences in release time t diff = t 2 − t 1 − T 390 are shown in Fig. 4(b) as a function of the final momentum.Strikingly, the difference is not zero but values of more than 100 as are found.The differences in the initial momentum p diff = p i,2 −p i,1 are shown in Fig. 4(c) as a function of final momentum and are as expected from the shape of the vector potential.The differences in final phase Φ diff = Φ 2 − Φ 1 are presented in Fig. 4(d) and show a slightly distorted circular symmetry (for a single-color laser field a perfect circular symmetric distribution would be observed).Fig. 4(e)-4(h) are analogous to Fig. 4(a)-4(d) but compare the third and the second half-cycle instead of the second and the first half-cycle (as visualized in Fig. 4(e)).Because of the definitions t 3 = t 1 + T 390 and p i,3 =p i,1 the results in Fig. 4(f) and 4(g) are the the same as Fig. 4(b) and 4(c) but with opposite sign.Fig. 4(d) and 4(h) are not trivially linked because the final electron phase has to be evaluated using Eq. 3.
The experimentally accessible quantity is Assuming that all four trajectories have the same amplitude (B n = 1) and, as before, using an offset phase of zero (φ off,n = 0) Eq. 5 can be evaluated for each final electron momentum p f .Each trajectory is released within one of the four half-cycles of the laser field (as illustrated in Fig. 5(a)).The result is presented in Fig. 5(b) as |Ψ simple | 2 and the expected alternating half-rings in final electron momentum space are reproduced.For comparison a single-color field at 390 nm with E 390 = 0.04 a.u. is evaluated that leads to the well-known ATI structure in final electron momentum space without any angular modulations or sidebands (see Fig. 5(d)).

NUMERICAL EXAMPLES WITH OFFSET PHASE
In this section the influence of a non-zero offset phase φ off,n is investigated.Assuming an offset phase that depends linearly on the initial momentum p i is modeled by φ off (p i ) = κp i (independent of the trajectory number n and for a real valued κ).using the following procedure: the angular distribution for every energy peak is analyzed separately by performing a Fourier transformation to extract the offset angle α from the phase of the lowest frequency component in Fourier space (not the DC component) (α is defined as shown in Fig. 6(c) and (d)).
In general, the relation φ off = κ holds only for a phase of the initial momentum distribution that is linear in p i .However, for any absolute value of the final electron momentum | p f | the two relevant initial momenta, that lead to this final momentum, differ by less than 0.15 a.u.(see Fig. 4(c) and (g)).Hence, the offset angle α can be used to infer the value of φ off (p i ) but represents not the exact derivative but approximates the deviate in in interval with a length of upto ∆p i = 0.15 a.u.(In principle the value of ∆p i could be further reduced by decreasing the intensity of the light field at at central frequency of ω, because this also reduces the difference of the minimal and the maximal value of the absolute value of the negative vector potential | − A|.) Inspecting Fig. 6(a) the choice of B(p i ) = 1 appears to be unrealistic.More realistically values of the amplitude distribution are shown in Fig. 6(b) using . Here, σ = 0.2 a.u.accounts for the width of the initial momentum distribution af-ter tunneling [25] and p offset = 0.2 a.u. is chosen to model a typical non-adiabatic offset [26][27][28].The result is shown as |Ψ lin,env | 2 in Fig. 6(d).As expected, this mainly affects the visibility of the inner and outer energy peaks (compare intensity envelopes of Fig. 6(c) and (d)).Interestingly, the rotation angles α are hardly affected (compare Fig. 6(e) and (f)).It can be concluded, that for typical [8,9,29] light intensities of CoRTC fields the derivative of the phase of the initial momentum distribution φ off can be inferred from the (experimentally accessible) offset angles α in final momentum space.
It is important to note that the sub-cycle dependence of the ionization rate [30] and Coulomb interaction after tunneling [22,31,32] are not included in the current analysis.The sub-cycle dependence of the ionization rate is expected to lead to similar deviations as the envelope of the amplitudes of the initial momentum distribution (see Fig. 6(a) and (b)).Coulomb interaction after tunneling will lead to significant changes of the alternating half-ring pattern.In principle it is difficult to distinguish changes that are due to φ off from changes that are due to Coulomb interaction after tunneling.

HASE AND OFFSETS OF THE BOUND WAVE FUNCTION IN POSITION SPACE
As described above CoRTC fields can be used to experimentally measure φ off = ∂φ off ∂pi .The next big step is to understand the meaning of a phase gradient of the initial momentum distribution.
It appears as if the only position-space-information that is included in the model is that the initial momentum distribution is zero along the direction of the electric field at the instance of tunneling.However, a linear phase in momentum space corresponds to a shift in position space (complex momentum space and complex position space are linked by Fourier transformation).Let the position dependent wave function Ψ(x i ) be the Fourier transform of the momentum dependent wave function Ψ(p i ).Then a real valued position offset ∆x leads to the shifted position dependent wave function Ψ(x i ) = Ψ(x i + ∆x).The Fourier transform leads to the wave function Ψ(p i ).

Ψ(p
Fig. 7 illustrates the relation that is described in Eq. 6 by showing a wave function for the initial momentum distribution with a Gaussian distribution of the amplitudes that is centered around zero initial momentum.For a constant phase the Fourier transform of a Gaussian distribution would be another Gaussian distribution that is centered at zero.The linear phase in Fig. 7(a) (same phase dependence as in Fig. 6(a) and (b)) is reflected by an offset in position space by ∆x as it can be seen in Fig. 7(b).
Identifying p i ∆x/h = φ off allows to link ∆x = φ off h = κh for the case of a linear phase of the initial momentum distribution as discussed regarding Fig. 6.Thus, it can be concluded that the value of ∆x = φ off h can be interpreted as a measure of the displacement of the wave packet after tunneling in position space for the case of a linear phase of the initial momentum distribution.(Positive values of φ correspond to a displacement of the wave packet after tunneling in position space that is anti-parallel to − A(t) at the electron release time t.)This is the position space analogue to the much used  fact that tunneling acts like a filter on the initial bound state's wave function in momentum space [25,26,33].This insight has been used to infer fingerprints of the bound state's wave function in momentum space from the measured final state's momentum distribution (e.g. in Refs.[34,35]).Fig. 7 suggests that the measurement of the phase gradient in momentum space can be used as a probe of the amplitudes of the initial state's wave function in position space.In analogy to the filtering in momentum space, also the initial state's wave function in position space is considered as a filtered wave function of the bound state.Thus, HASE is a novel approach to access e.g.molecular structure and polarization states in position space.
Looking at the entire ionization process as a black box and only considering the continuum state allows for an illuminating insight: If Coulomb interaction after tunneling is neglected, any displacement of initial state leads to a displacement of final states by the same vector.The displacement is anti-parallel to the direction of the streaking momentum p streak = − A(t).Considering a single-color circularly polarized field and using the angular symmetry in the plane of polarization, a displacement of the initial positions by ∆x is equivalent to a continuum wave packet that leaves the black box with a time delay ∆t that can be described using the absolute value of the streaking momentum p streak = | A(t)| (see Eq. 7).
Here m e = 1 a.u. is the electron's mass, h = 1 a.u. the reduced Planck constant and p streak = | − A|.The basic idea how a displacement in position space corresponds to a time delay is illustrated in Fig. 8.In the next section it will be shown that this time delay is the same as the Wigner time delay.

HASE AND THE WIGNER TIME DELAY
So far it has been shown that HASE is sensitive to changes in the slope of the phase of the initial momentum distribution φ off and that φ off is linked to offsets of the initial position distribution.In the following the question how φ off affects the phase of the final, semi-classically modeled electronic wave function Ψ( p f ) is investigated.
To answer this we assume that the complex valued wave function of the initial momentum distribution does not change if the weak field at the central frequency ω is switched off.(The CoRTC field with E 390 = 0.04 a.u. and E 780 = 0.004 a.u. is used to obtain the complex valued wave function of the initial momentum distribution.We assume that that the complex valued initial momentum distribution is the same for the slightly different single-color field with E 390 = 0.04 a.u.) The simple case of a single-color circularly polarized light field with E 390 = 0.04 a.u. is considered and the derivative of the initial phase φ off (p i ) is assumed to be known.In this case, the final electron momentum distribution is independent of the angle in the plane of polarization (see Fig. 5(d)).We now make use of the fact that changes in φ off (p i ) do not affect the trajectory or the probability |Ψ( p f )| 2 but only influence the final phase arg(Ψ( p f )).Initial and final momenta are unambiguously linked by p f = − A(t) + p i (see Eq. Obviously, this implies that the phase of the final electron momentum distribution also changes as a function of the electron energy E. This quantity is known as the Wigner time delay τ W which is defined by: Using the trajectory based HASE model and considering a single-color circularly polarized light field (as for Fig. 5(c) and (d)), the initial momentum is unambiguously linked with the final momentum.One can compare two scenarios: The first scenario uses φ off = 0 rad/a.u.leading to the semi-classically modeled wave function Ψ simple ( p f ) at a given time t f .In the second scenario an arbitrary phase of the initial momentum distribution φ off is used (the amplitudes B n can be shown to be irrelevant regarding τ W for a single-color circularly polarized light field) leading to the semi-classically modeled wave function Ψ delayed ( p f ) at the same time t f .The two semiclassically modeled wave functions are related by: As a result, the change of the Wigner time delay due to the phase of the initial momentum distribution is given by: ) Substituting energy with momentum leads to: The result in Eq. 11 is equivalent to the previously obtained expression for the delay time ∆t (see Eq. 7).This result allows one to gain very fundamental insight: the Wigner time delay for strong field ionization is related to an intuitive shift of the bound wave function in position space.

RECIPE FOR THE EXPERIMENTAL ACCESS TO CHANGES OF THE WIGNER TIME DELAY IN STRONG FIELD IONIZATION
The first step to access the Wigner time delay in strong field ionization is to conduct an experiment using a CoRTC field.High ratios of E 390 /E 780 decrease the visibilty of the sidebands [10] but also reduce the difference of the minimal and the maximal value of the absolute value of the negative vector potential | A(t)|.As described above, higher values of E 390 /E 780 lead to a more accurate mapping of the offset angle α and the phase of the initial momentum distribution after tunneling φ off .Typical values of E 390 /E 780 are close to 10 (see e.g.Refs.[8,9]).The amplitude of E 390 should be chosen such that the ionization channel that is investigated is not saturated.E 390 = 0.04 a.u. is a good choice for the single ionization of argon [28].
Second, the offset angles α have to be retrieved from the measured final electron momentum distribution for every energy peak.(In principle, the result of a theoretical calculation that solves the time-dependent Schrödinger equation on a grid could be used as well.In this case the absolute square of the final electronic wave function in momentum space should be analyzed.)The final electron momentum distribution is considered to be in polar coordinates with p x , p r = p 2 y + p 2 z and φ polar .Here, φ polar is the angle in the yz-plane (defined in the same was as α in Fig. 6(c) and 6(d)).p x should be restricted by |p x /p r | < √ 2 in order to make sure the considered electron momentum vectors are aligned in the plane of polarization.In a next step, the angular distribution for a given energy peak is analyzed.Let this angular distribution be represented by a vector X with N entries.The discrete Fourier transform Y is given by (where k ∈ {1, 2, ..., N }): Now, the offset angle for the selected energy peak is given by α = − arg (Y (2)) for the sidebands (energy peaks that vanish if E 780 is set to zero) and α = − arg (Y (2)) − 180 • for the ATI peaks (energy peaks that do not vanish if E 780 is set to zero).Now, α represents a distribution that is proportional to cos(φ polar − α − 180 • ) for sidebands and cos(φ polar − α) for ATI peaks.Third, the offset angles α can be related to the derivative of the phase of the initial momentum distribution φ off using Fig. 6(e) as a look up table.From Eq. 11 the change in Wigner time delay ∆t W can be directly obtained for an offset angle α.The result is presented in Fig. 9(a).However, this mapping of α to φ off and ∆t W can only be done if the following conditions are fulfilled: (i) the laser parameters and the value for the ionization potential I p are as in this work, (ii) the envelope of the amplitudes can be neglected (see Fig. 6(f) and Fig. 9(b)), (iii) the sub-cycle dependence of the ionization rate [30] can be neglected and (iv) Coulomb interaction after tunneling [32] can be neglected.Conditions (i)-(iii) are not problematic because one could simply rerun the simulations by solving Eq. 2 and Eq. 4 using new laser parameters, the appropriate ionization potential and an amplitude distribution of the initial momentum distribution that does not only depend on the initial momentum (B(p i )) but also on the electron release time (B(p i , t)).This time-dependence could be obtained from theory [30].Alternatively, B(p i , t) could be estimated directly from the envelope of the measured electron momentum distribution (see e.g.Refs.[9,24]).In general, the measured electron momentum distribution should be normalized by the envelope of the measured electron momentum distribution in order to remove the modulation of the intensity from the measured interference pattern.An addiational benefit of the extracted envelope of the measured electron momentum distribution is that it allows one to estimate the absolute orientation of the laser electric field (even if Coulomb interaction after tunneling is not neglected [32]).
Condition (iv) is usually not fulfilled.However, one can assume that the angular offset that is due to Coulomb interaction after tunneling α Coulomb only adds an additional offset angle to the offset angle that is due to φ off .Consequently, the measured offset angle α would have to be corrected for the offset angle α Coulomb by α corrected = α − α Coulomb .If one aims at measuring the absolute value of the Wigner time delay that is due to a non-zero offset in position space after tunneling, then the value of α Coulomb has to be known for every energy peak that is considered.An elegant alternative to circumvent condition (iv) is to compare two different ionization channels (e.g.different kinetic energy releases for molecular dissociation) and assume that both have the same value α Coulomb .The pairs of measured offset angles (α channel 1 and α channel 1 ) can be used to calculate the difference in the Wigner time delay for the two channels by ∆τ W,1,2 = ∆τ W,1 (α channel 1 ) − ∆τ W,2 (α channel 2 ).This is a good approximation if the mapping of ∆τ W to offset angle α is linear (which is the case for offset angles between 60 • and 120 • ).For this procedure the contribution of Coulomb interaction cancels out and allows for the experimental access of the difference in the Wigner time delay for the two ionization channels upon tunnel ionization.

CONCLUSION
It is found that the release times in CoRTC fields are not spaced by multiplies of the period of the driving field T 390 but deviate by more than 100 as with respect to T 390 .Using the newly introduced semi-classical model, the numerically calculated final electron momentum distributions are analyzed.It is shown that the alternating half-ring pattern can be used to infer the derivative of the phase of the initial momentum distribution φ off from experimentally accessible quantities.φ off is related to a change in the Wigner time delay ∆τ W . Finally, an intuitive interpretation of the Wigner time delay for tunnel ionization is introduced that links the Wigner time delay to a displacement of the bound wave function in position space.
The contributions to the energy dependent offset an-gle α, that are due to Coulomb interaction after tunneling or a sub-cycle dependence of the ionization rate [30] are discussed qualitatively.One elegant possibility to minimize the relevance of Coulomb interaction after tunneling is to compare two different ionization channels with similar Coulomb interaction after tunneling.In future (coincidence) experiments, the offset angles α could be examined as a function of the kinetic energy release for molecular dissociation, the molecular orientation or the atomic species.Experiments with molecular oxygen that compare the alternating half-ring pattern in final electron momentum space for different kinetic energy releases were already done and show angular differences comparing the two ionization channels [29].
The experimentally accessible phase gradient of the initial momentum distribution could be used to access the bound wave function's amplitude distribution in position space along the direction that is perpendicular to the laser electric field at the instance of tunneling.Hence, HASE paves the road towards the measurement of molecular structure, polarization states and non-adiabatic fingerprints of tunnel ionization in position space with subcycle temporal resolution.

1 FIG. 1 .
FIG. 1.(a) shows the combined electric field E and the negative vector potential − A of a co-rotating two-color (CoRTC) field.The helicities of the two colors and the temporal evolution of E and − A are indicated with arrows.(b) shows a sketch of the alternating half-ring pattern in momentum space that is expected for a CoRTC field that is dominated by the light field at a central frequency of 2ω (colored regions indicate high intensity in final momentum space).

FIG. 2 .
FIG. 2. (a)shows the combined electric field E(t) and the negative vector potential − A(t).The blue dots in (b) indicate possible final electron momenta p f .Each indicated value of p f is connected by two blue lines to the values − A(t1) and − A(t2).t1 and t2 are the two release times that have been obtained from solving Eq. 2 numerically.The vectors of the blue lines reflect the corresponding initial momenta pi,1 and pi,2.

2 FIG. 3 .
FIG. 3. (a) illustrates the absolute value of the negative vector potential | − A(t)| and highlights the first half-cycle (0 as < t ≤ T390) of the two-cycle light field in blue.| − A(t)| is shown as a black line in all panels.(b) [(c)] shows the electron release time [initial momentum] as a function of the final electron momentum.(d) depicts the final phase modulo 2π as a function of the final electron momentum.(e)-(f) show the same as (a)-(d) but using the second half-cycle (T390 < t ≤ 2T390).The black arrows in (b)-(d) and (f)-(h) indicate the temporal evolution of − A(t).
FIG. 4. (a) the relevant ionization times for the first row are highlighted.The first (second) half-cycle is highlighted in blue (green).(b) shows the differential release time of Fig. 3(b) and 3(f).(c) shows the differential initial momentum of Fig. 3(c) and 3(g).(d) shows the differential final phase comparing Fig. 3(d) and 3(h).(e)-(h) are analogous to (a)-(d) but compare the third and the second half-cycle instead of the second and the first half-cycle.
FIG. 5. (a) illustrates that electron release times from all halfcycles are considered.(b) shows the electron momentum distribution |Ψ simple | 2 with the expected alternating half-rings.(c) and (d) show the same as (a) and (b) but using a singlecolor field with E390 = 0.04 a.u.The black line represents the negative vector potential and the gray dashed line guides the eye and is the same in (b) and (d).

FIG. 6 .
FIG. 6.(a) [b] shows an initial momentum distribution with a constant [non-constant] amplitude B and a linear phase.(c) [(d)] shows the expected intensity in final momentum space |Ψ linear | 2 [|Ψ lin,env | 2 ] using the initial momentum distribution shown in (a) [(b)].(e) [(f)] shows the offset angles α that have been extracted from the data shown in (c) [(d)] for each energy peak separately (see text).

FIG. 7 .
FIG. 7. (a) shows a wave function for an initial momentum distribution with a Gaussian distribution of the amplitudes and a linear phase.(b) shows the Fourier transform of the distribution that is shown in (a).The black vertical lines guide the eye to emphasize the shift of the amplitudes in (b) that is proportional to the slope of the phase in (a) (see Eq. 6).

FIG. 8 .
FIG. 8. (a) schematically illustrates an initial bound state in position space xi that is perpendicular to the tunneling direction.(b) shows a sketch of the initial position distribution after tunneling xi (note that the width of the distribution might change during tunneling which is neglected here for the sake of simplicity).(b) depicts the position distribution along the same direction as in (a) directly after tunneling (xi) and for the final time (x f ).At the final time t f the position distribution x f is broadened due to the dispersion of the continuum wave packet.(c) shows the initial momentum distribution along pi and the corresponding final momentum distribution along p f which only differ by the momentum that is due to the streaking of the laser p streak .(d)-(f) show the same as (a)-(c) with the only difference that the initial bound state in position space is displaced by ∆x as illustrated in (d).This shifts all positions in (e) by the value of ∆x.The final momentum distributions in (c) and (f) are the same.The shift in position ∆x of a continuum wave packet with a given momentum p f allows to calculate the time delay of the scenario in (a)-(c) relative to the scenario in (d)-(f) using ∆t = me∆x p f .The temporal evolution of the tunneling direction is indicated in (a) and (d) by the arrow labeled with "evo". ∆t 2) and the value of | A(t)| is constant for a single-color field.Because of the symmetry of circularly polarized single-color light fields, a phase change at an initial moment with a given value of | p i | directly allows to quantify the phase change at a final momentum with a given value of | p f |.

FIG. 9 .
FIG. 9. (a) shows the change in Wigner time delay due to a linear phase of the initial momentum distribution for the case defined in Fig. 6 (a).This relates the change in Wigner time delay (for a single-color field with E390 = 0.04 a.u.) and the offset angles α (that are obtained using a two-color field with E390 = 0.04 a.u. and E780 = 0.004 a.u.).(b) is analogous to (a) but uses the amplitude distribution from Fig. 6 (b).