1. Introduction

Femtosecond laser pulses are unique tools to control multiphoton excitation processes in atoms and molecules [1–31]. The control over the corresponding initial-to-final state-to-state transition probabilities is achieved by shaping the pulse [32–34] to manipulate the interferences among the coherent manifold of state-to-state multiphoton quantum transitions photo-induced by the broadband ultrashort pulse. The pulse-shaping control knobs are generally the relative phase, amplitude, and polarization of different spectral components of the pulse. Here we focus on rational coherent control approach, where the femtosecond pulse shaping is based on the initial identification of the interfering transitions and their interference mechanism. Such an approach has been shown to be very powerful when the photoexcitation picture is available in the frequency domain [1–31], which is possible when a corresponding perturbative description of finite order is valid. The weak-field regime corresponds to a valid description by perturbation theory of the lowest non-vanishing order. For two-photon processes it is second-order perturbation theory, while for three-photon processes it is third-order perturbation theory.

Over the last decade weak-field multiphoton femtosecond coherent control has been demonstrated very successfully in many studies that have been conducted in the multi-cycle femtosecond regime [1–31], where the pulse duration of the (unshaped) transform-limited pulse is of many optical cycles. A natural way to further enhance the possible control is to enrich the variety of multiphoton transitions by a significant broadening of the exciting pulse spectrum up to an octave-spanning corresponding to the single-cycle regime [35–41]. There, the transform-limited pulse duration is of about one optical cycle. The shaping of such ultrabroadband pulses has recently been reported [40, 41]. As opposed to the multi-cycle regime, where the electric field is effectively determined only by the instantaneous temporal frequency and temporal envelope profile, in the single-cycle regime it is also strongly affected by the global phase between them. This phase is referred to as the carrier-envelope phase (CEP) and is also equal to the global phase of the spectral field at the different pulse frequencies. During the process of coherent spectral broadening and creation of single-cycle pulses, the CEP is experimentally stabilized to preset values [35–39]. The CEP has shown to play a significant role in multiphoton processes such as photoionization, high harmonics generation, and others [42–49].

Here we study for the first time weak-field multiphoton coherent control using shaped ultrabroadband femtosecond pulses having a bandwidth of the single-cycle regime. We investigate phase control of atomic bound-bound non-resonant multiphoton transitions. The CEP of the pulse, which in the multi-cycle regime does not play any control role, is shown here to be a new effective control parameter that its effect is highly sensitive to the spectral position of the ultrabroad spectrum. Rationally-positioned ultrabroad spectrum generally induces coherently several groups of multiphoton transitions from the ground to the excited state of the system: transitions involving only absorbed photons as well as Raman transitions involving both absorbed and emitted photons. For example, for a two-photon excitation process, one group is of transitions involving two absorbed photons and the other group is of Raman transitions involving one absorbed photon and one emitted photon. The intra-group interference is controlled by the relative spectral phases of the different frequency components of the pulse, while the inter-group interference is controlled jointly by the CEP and the relative spectral phases.

The corresponding effect and control principle are as follows. For N-photon excitation process, the global phase associated with all the transitions of N absorbed photons is N×CEP, while the global phase associated with all the transitions of M absorbed photons and N−M emitted photons is (2M−N) ×CEP. Thus, overall, the CEP contributes a value of 2(M−N) ×CEP to the relative phase between the excitation amplitudes photo-induced by these different groups. This value is added then to the relative phase between the excitation amplitudes as determined by the intra-group interferences taking place separately within each of the transition groups. The higher is the order of the multiphoton excitation (N), the smaller is the minimal ultrabroad bandwidth allowing such a control. In the multi-cycle regime, the spectrum is not broad enough to photo-induce several types of N-photon transitions; it can induce only a single type. Thus, no CEP effect is possible in the multi-cycle regime.

Specifically, the present study includes the investigation of non-resonant two- and three-photon excitation in a simple model system within the perturbative frequency-domain framework. Then, the developed intuition is applied to weak-field multiphoton excitation in atomic cesium (Cs), where the simplified model is verified by non-perturbative numerical solution of the time-dependent Schrödinger equation.

2. Two-photon theoretical perturbative description

Consider the simple model system shown in Fig. 1(a) , with a weak-field non-resonant two-photon transition from the ground state $|g〉$ of energy $E_g$ to the excited state $|f〉$ of energy$E_f$, with the corresponding transition frequency of ${\Omega }_{fg}=(E_f-E_g)/\hslash$. The $|g〉$ and $|f〉$ states are coupled via a manifold of intermediate states $|n〉$ (not shown in Fig. 1(a)) of the proper symmetry that are not accessed resonantly by the excitation. For a weak excitation pulse, the final amplitude $A_f^{(2)}$of $|f〉$ after the pulse is over is given by second-order time-dependent perturbation theory as [9, 10]

 

Fig. 1 (a) Two-photon excitation scheme in the femtosecond single-cycle regime. Indicated are the two-photon absorption transitions and two-photon Raman transitions. (b) Ultrabroad spectrum of single-cycle pulse centered at ${\Omega }_{fg}$ that provides two absorbed photons of frequencies around ${\Omega }_{fg}/2$ to the two-photon absorption transitions as well as two photons, one absorbed photon of frequency around $3{\Omega }_{fg}/2$ and one emitted photon of frequency around${\Omega }_{fg}/2$, to Raman transitions. (c) Temporal electric field of the TL single-cycle pulse with ${\omega }_0$ =12500 cm⁻¹ (800 nm) and different ${\varphi }_{CE}$. (d) Relative population of the excited state $|f〉$ induced by the TL single-cycle pulse with different values of ${\varphi }_{CE}$ and different carrier frequencies${\omega }_0$ (normalized to the case of${\varphi }_{CE}=$0).

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The spectral field, correlated to the temporal field via Fourier Transform, is given by $\tilde{\epsilon }(\omega )=\tilde{E}(\omega )\text{exp}\left(i{\varphi }_{CE}\right)+c.c.$, with $\tilde{E}(\omega )$ being the spectral complex envelope. It is given by $\tilde{E}(\omega )=\left|\tilde{E}(\omega )\right|\text{exp}\left[i{\Phi }_{\text{rel}}(\omega )\right]$, where $\left|\tilde{E}(\omega )\right|$ and ${\Phi }_{\text{rel}}(\omega )$ being, respectively, the spectral amplitude and relative phase of frequencyω. For the (unshaped) transform-limited (TL) pulse, ${\Phi }_{\text{rel}}(\omega )=$0 for anyω.

Hence, $A_f^{(2)}$ is given in the frequency domain as

Two-photon absorption transitions involve two absorbed photons of frequencies ω and $\omega \text{'}={\Omega }_{fg}-\omega$ (satisfying $\omega ,\omega \text{'}<{\Omega }_{fg}$), while two-photon Raman transitions involve an absorbed photon of frequency ω (satisfying $\omega >{\Omega }_{fg}$) and an emitted photon of frequency$\omega -{\Omega }_{fg}$. Absorption of a photon contributes a global phase of ${\varphi }_{CE}$ to its corresponding transition amplitude, while a photon emission contributes a corresponding global phase of $-{\varphi }_{CE}$. Hence, as seen in Eq. (1), the two-photon absorption amplitude$A_{abs}^{(2)}$ has a global phase of $\text{2}{\varphi }_{CE}$, while the two-photon Raman amplitude$A_{Ram}^{(2)}$ has zero global phase. So, the CEP value strongly affects the corresponding inter-group interferences. The${\mu }_{abs}^2$, ${\mu }_{Ram,1}^2$ and${\mu }_{Ram,2}^2$ are, respectively, the effective non-resonant couplings of the two-photon absorption transitions and different two-photon Raman transitions [9, 10]. These couplings of the different groups of transitions generally have different values since each two-photon transition group accesses a different range of intermediate excitation energies (see the dashed lines in Fig. 1(a)) and thus involves different off-resonance detunings from the various intermediate states $|n〉$.

The possibility to simultaneously induce different types of two-photon transitions is a key feature of the ultrabroadband pulses of the single-cycle regime considered here as compared to the broadband pulses of the multi-cycle regime. For pulses with stabilized CEP ${\varphi }_{CE}$, the amplitudes associated with the different transition groups have relative phase of $\Delta \varphi =\text{2}{\varphi }_{CE}$ that dictates the nature of the interference between them. The observable considered here as the control objective of the CEP and relative spectral phase is the final population $P_f$ of state $|f〉$ that is given by$P_f={\left|A_f^{(2)}\right|}^2$.

3. Two-photon perturbative calculations for a model system

3.1 The model system

The model quantum system we consider here has two-photon transition frequency of ${\Omega }_{fg}$ =12500 cm⁻¹. When the ultrabroadband spectrum of the irradiating pulse is centered in the region of the two-photon transition frequency, i.e., ${\omega }_0\approx {\Omega }_{fg}$, it effectively provides photon pairs to both types of the two-photon transition groups: (i) Each two-photon absorption transition involves two absorbed photons with frequencies ${\omega }_1$,${\omega }_2\approx {\Omega }_{fg}/2$ that are symmetrically located around ${\Omega }_{fg}/2$, such that ${\omega }_1+{\omega }_2={\Omega }_{fg}$; (ii) Each two-photon Raman transition involves one absorbed photon of frequency ${\omega }_1\approx 3{\Omega }_{fg}/2$ and one emitted photon of frequency${\omega }_2\approx {\Omega }_{fg}/2$, such that ${\omega }_1-{\omega }_2\approx {\Omega }_{fg}$. Hence, we consider in our study pulses having their Gaussian spectrum centered in the region of ${\Omega }_{fg}=$12500 cm⁻¹. Their spectral intensity bandwidth (FWHM) is $\Delta \omega$=5425 cm⁻¹ corresponding to TL pulse duration of 2.7 fs, which is equal to one optical cycle of ${\omega }_0$ =12500 cm⁻¹ (800 nm). The spectrum of the 800-nm pulse and its TL temporal electric field with different ${\varphi }_{CE}$ values are schematically shown in Figs. 1(b) and 1(c).

As formulated and described above, the exact amplitudes contributed by the different two-photon transition groups ($A_{abs}^{(2)}$ and $A_{Ram}^{(2)}$) are determined, on one hand, by the values of the different two-photon couplings (${\mu }_{abs}^2$, ${\mu }_{Ram,1}^2$,${\mu }_{Ram,2}^2$) and, on the other hand, by the field of the pulse $\tilde{\epsilon }(\omega )$ and its different characteristics. Here, we assume for simplicity that${\mu }_{abs}^2={\mu }_{Ram,1}^2={\mu }_{Ram,2}^2$.

3.2 Results for two-photon CEP and relative-phase control

First, we study in this section the dependence of the two-photon excitation probability on the value of the CEP for the unshaped TL pulses of different carrier frequencies. The corresponding dependence of $P_{f,TL}$ on ${\varphi }_{CE}$, calculated using Eq. (1), is shown in Fig. 1(d) for different carrier frequencies${\omega }_0$, i.e., different positions of the pulse spectrum. Each trace is normalized by the corresponding value of $P_{f,TL}$ for zero CEP, i.e., $P_{f,TL}({\varphi }_{CE}=0)$. The observed dependence on ${\varphi }_{CE}$ is explained very simply as follows. The field $\tilde{E}(\omega )$of the TL pulse is real since ${\Phi }_{\text{rel}}(\omega )=$0. So, the intra-group interferences within $A_{abs,TL}^{(2)}$ and $A_{Ram,TL}^{(2)}$ are fully constructive and maximize the magnitude of each of them, while the inter-group relative phase between them is $2{\varphi }_{CE}$ with a corresponding periodicity of π. In the case of ${\varphi }_{CE}=0$ and π, the amplitudes $A_{abs,TL}^{(2)}$ and $A_{Ram,TL}^{(2)}$ have zero relative phase [see Eq. (1)] and, thus, their inter-group interference is fully constructive and maximizes $P_{f,TL}({\varphi }_{CE})$ for the given ${\omega }_0$. On the other hand, for ${\varphi }_{CE}=\pi /2$ and $3\pi /2$, $A_{abs,TL}^{(2)}$ and $A_{Ram,TL}^{(2)}$ have a relative phase of π and, thus, their inter-group interference is fully destructive and leads to the minimal value of $P_{f,TL}({\varphi }_{CE})$ for the given ${\omega }_0$.

The modulation depth $M_{TL}$ obtained for a given ${\omega }_0$, defined as $M_{TL}=\left[\text{max}\left(P_{f,TL}({\varphi }_{CE})\right)-\text{min}\left(P_{f,TL}({\varphi }_{CE})\right)\right]/\left[\text{max}\left(P_{f,TL}({\varphi }_{CE})\right)+\text{min}\left(P_{f,TL}({\varphi }_{CE})\right)\right]$ or, equivalently, $M_{TL}^{}=\left[P_{f,TL}({\varphi }_{CE}=0)-P_{f,TL}({\varphi }_{CE}=\pi /2)\right]/\left[P_{f,TL}({\varphi }_{CE}=0)+P_{f,TL}({\varphi }_{CE}=\pi /2)\right]$, is determined by the amplitude ratio $r_{TL}=\left|A_{abs,TL}^{(2)}\right|/\left|A_{Ram,TL}^{(2)}\right|$ via the relation $M_{TL}^{}=2r_{TL}/(1+r_{TL}{)}^2$. The maximal value of $M_{TL}$=1 is obtained when $\left|A_{abs,TL}^{(2)}\right|=\left|A_{Ram,TL}^{(2)}\right|$, while the minimal value of $M_{TL}$=0 is obtained when $\left|A_{abs,TL}^{(2)}\right|$ =0 or $\left|A_{Ram,TL}^{(2)}\right|$ =0. Other values of $M_{TL}$, in between 0 and 1, are obtained for non-equal non-zero magnitudes of $\left|A_{abs,TL}^{(2)}\right|$ and $\left|A_{Ram,TL}^{(2)}\right|$. Here, as seen in Fig. 1(d), the modulation depth increases from close-to-zero value of $M_{TL}$≈0.05 at ${\omega }_0$ = 14500 cm⁻¹ (pink line) to $M_{TL}$≈0.25 at ${\omega }_0$=13500 cm⁻¹ (green line) to $M_{TL}$≈0.80 at ${\omega }_0={\Omega }_{fg}$=12500 cm⁻¹ (red line) and to its maximal value of $M_{TL}$=1 at ${\omega }_0$=11870 cm⁻¹ (black line), which is red shifted from ${\Omega }_{fg}$. Then, it decreases again to a value of $M_{TL}$=0.45 at ${\omega }_0$=10300 cm⁻¹ (blue line).

This observed behavior can be understood as follows. Analyzing Eq. (1) for a given ${\Omega }_{fg}$, it can be seen that in the case of a Gaussian TL pulse the magnitude of $\left|A_{Ram,TL}^{(2)}\right|$ (involving frequency differences) depends on the couplings ${\mu }_{Ram,1}^2$ and${\mu }_{Ram,2}^2$ and on the pulse spectral bandwidth $\Delta \omega$, however it is actually independent of the carrier frequency ${\omega }_0$. On the other hand, the magnitude of $\left|A_{abs,TL}^{(2)}\right|$ (involving frequency sums) is determined by the coupling ${\mu }_{abs}^2$ as well as by $\Delta \omega$ and ${\omega }_0$. Thus, for a given set of two-photon couplings and a given spectral bandwidth $\Delta \omega$, a change in ${\omega }_0$ leads to change only in $\left|A_{abs,TL}^{(2)}\right|$, while $\left|A_{Ram,TL}^{(2)}\right|$ remains unchanged, and thus to a change in the ratio $r_{TL}=\left|A_{abs,TL}^{(2)}\right|/\left|A_{Ram,TL}^{(2)}\right|$. Specifically, as the pulse spectrum is shifted to lower frequencies (i.e., shifted to the red), $\left|A_{abs,TL}^{(2)}\right|$ and $r_{TL}$ increase. In our model, due to the assumption of ${\mu }_{Ram,1}^2={\mu }_{Ram,2}^2={\mu }_{abs}^2$, one obtains that $\left|A_{abs,TL}^{(2)}\right|<\left|A_{Ram,TL}^{(2)}\right|$ and $r_{TL}$<1 for ${\omega }_0={\Omega }_{fg}$ and, thus, for any ${\omega }_0>{\Omega }_{fg}$. Hence, $M_{TL}<1$ for any ${\omega }_0\ge$12500 cm⁻¹ and the $M_{TL}$ value increases as ${\omega }_0$ decreases. As ${\omega }_0$ further decreases below ${\Omega }_{fg}$ and becomes more and more red-shifted from it, the values of $r_{TL}$ and $M_{TL}$ further increase (see above) up to the point of = 11870 cm⁻¹, where $\left|A_{abs,TL}^{(2)}\right|$ and $\left|A_{Ram,TL}^{(2)}\right|$ become equal and, thus, $r_{TL}$=1 and $M_{TL}$=1. Further red-shifting of ${\omega }_0$ leads to a further increase of $r_{TL}$ beyond the value of one ($r_{TL}$>1) and thus to a corresponding decrease of $M_{TL}$ below a value of one ($M_{TL}<1$), as indeed is the case for ${\omega }_0$=10300 cm⁻¹.

Quantitatively, based on Eq. (1), the carrier frequency ${\omega }_0$ at which $\left|A_{abs,TL}^{(2)}\right|=\left|A_{Ram,TL}^{(2)}\right|$ is given by ${\omega }_{0,M_{TL}=1}=\frac{1}{2}\left[{\Omega }_{fg}+\sqrt{{\Omega }_{fg}{}^2-8{\left(\frac{\Delta \omega }{\sqrt{8\ln 2}}\right)}^2\ln R_{coupling}}\right]$, where $R_{coupling}=\frac{{\mu }_{Ram,1}^2+{\mu }_{Ram,2}^2}{{\mu }_{abs}^2}$ is the ratio between the sum of the two-photon Raman couplings and the two-photon absorption coupling. As seen, the value of ${\omega }_{0,M_{TL}=1}$ and its shift from ${\Omega }_{fg}$ generally depends on ${\Omega }_{fg}$, $\Delta \omega$ and $R_{coupling}$. When $R_{coupling}$=1, one obtains ${\omega }_{0,M_{TL}=1}={\Omega }_{fg}$, i.e., the maximal modulation occurs when there is no shift of ${\omega }_0$ from ${\Omega }_{fg}$. Such a case is, for example, the case of ${\mu }_{Ram,1}^2={\mu }_{Ram,2}^2=\frac{1}{2}{\mu }_{abs}^2$. When $R_{coupling}$<1 or $R_{coupling}$>1, one obtains, respectively, ${\omega }_{0,M_{TL}=1}>{\Omega }_{fg}$ or ${\omega }_{0,M_{TL}=1}<{\Omega }_{fg}$, i.e., the maximal modulation occurs when there is, respectively, a blue or red shift of ${\omega }_0$ from ${\Omega }_{fg}$. The latter case corresponds to our model that includes the assumption of ${\mu }_{Ram,1}^2={\mu }_{Ram,2}^2={\mu }_{abs}^2$ corresponding to $R_{coupling}$=2. Indeed, we have obtained the maximal modulation for ${\omega }_{0,M_{TL}=1}$=11870 cm⁻¹, which is red-shifted from ${\Omega }_{fg}$.

Next, as a benchmark case for the full phase control of the two-photon excitation, we consider shaping the pulses of different carrier frequencies and different CEPs with a relative spectral phase step of π, i.e., ${\Phi }_{\text{rel}}(\omega )=$0 for $\omega <{\omega }_{step}$ and ${\Phi }_{\text{rel}}(\omega )=$ π for $\omega \ge {\omega }_{step}$ [10]. The position of the phase step ${\omega }_{step}$ is scanned across the ultrabroadband pulse spectrum. The corresponding results for different values of ${\omega }_0$ in the range of moderate red-detuning to moderate blue-detuning from ${\Omega }_{fg}$ are shown in the different panels of Fig. 2 . For each value of ${\omega }_0$, the corresponding dependence of $P_f$ on ${\omega }_{step}$ is shown for the cases of ${\varphi }_{CE}$=0 (blue line) and ${\varphi }_{CE}$=π/2 (red line) as well as for the case of non-stabilized CEP (black line). The latter results from an average over all the different traces of $P_f$, each corresponding to a different value of ${\varphi }_{CE}$ within the range of 0 to 2π. All the traces presented in Fig. 2 for a given ${\omega }_0$ are normalized by the value of $P_f$ excited by the corresponding TL pulse having non-stabilized CEP.

 

Fig. 2 Coherent control of two-photon excitation in the model system: Excited state population induced by ultrabroadband pulses shaped with a spectral π phase step at different positions for several cases of carrier frequency ${\omega }_0$ and CEP ${\varphi }_{CE}$. Each trace is normalized by the population excited by the corresponding TL pulse having non-stabilized ${\varphi }_{CE}$.

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A feature to note of all the π-traces is that the asymptotic values of each trace, i.e., the $P_f$ values when ${\omega }_{step}$ is located far outside the spectrum either to the red or blue, are equal one to the other. It results from the fact that these two asymptotes correspond to a constant spectral phase of, respectively, π or 0 that is applied across the whole spectrum and thus is globally added to ${\varphi }_{CE}$. So, effectively they are equivalent to two unshaped TL pulses with a difference of π in their CEP values (${\varphi }_{CE}$ and ${\varphi }_{CE}$+π) that, as discussed above and shown in Fig. 1(d), both yield the same value of $P_f$. It also worth noting that the difference between the asymptotic values of the π-traces with different CEP values at a given ${\omega }_0$ also appears in the CEP-dependence results presented in Fig. 1(d).

A noticeable characteristic of the π-traces corresponding to ${\omega }_0$=10300 cm⁻¹, which are presented in Fig. 2(a), is the approximate symmetric double-well structure around ${\omega }_{step}={\Omega }_{fg}/2$ =6250 cm⁻¹. This feature can be explained as follows, considering the case of a spectrum having ${\omega }_0$ that is strongly red-detuned from ${\Omega }_{fg}$. Similar to the red-detuned TL case discussed above, also here in such a case one obtains that $\left|A_{abs}^{(2)}\right|>\left|A_{Ram}^{(2)}\right|$ and $P_f\approx {\left|A_{abs}^{(2)}\right|}^2$, i.e., the main contribution to $P_f$ originates from the two-photon absorption transitions. As previously observed for two-photon absorption in the multi-cycle regime [10], the π-trace corresponding to ${\left|A_{abs}^{(2)}\right|}^2$ has a double-well structure that is symmetric around ${\omega }_{step}={\Omega }_{fg}/2$ and its two minima of $P_f$=0 result from complete destructive interferences among the two-photon absorption transitions. This π-trace has no dependence on the CEP value. The deviations from such a perfect symmetry around ${\Omega }_{fg}/2$ as well as the CEP dependence, which are observed in the π-traces of ${\varphi }_{CE}$=0 and π/2 that are presented for ${\omega }_0$=10300 cm⁻¹, result from the still-existing non-zero small values of $A_{Ram}^{(2)}$ that interferes with $A_{abs}^{(2)}$. These deviations from the perfect symmetry are of opposite direction (along the axis of ${\omega }_{step}$) in these two CEP cases, hence they disappear in the corresponding π-trace of the non-stabilized CEP case.

A completely different characteristic is observed for the π-traces of ${\omega }_0$=13500 cm⁻¹ that are presented in Fig. 2(d). It is the symmetric wide double-well structure that is located around ${\omega }_0$. This feature can be explained as follows, in the same spirit of the above explanation for the red-detuned case but considering here the case of a spectrum that is strongly blue-detuned from ${\Omega }_{fg}$. Here, one obtains that $\left|A_{abs}^{(2)}\right|<\left|A_{Ram}^{(2)}\right|$ and $P_f\approx {\left|A_{Ram}^{(2)}\right|}^2$, i.e., the main contribution to $P_f$ originates from the two-photon Raman transitions. The π-trace corresponding to ${\left|A_{Ram}^{(2)}\right|}^2$ has a wide double-well structure that is symmetric around ${\omega }_0$ and its two minima of $P_f$=0 are located at ${\omega }_{step}={\omega }_0\pm ({\Omega }_{fg}/2)$ (here, ${\omega }_0\pm$6250 cm⁻¹), i.e., they are ${\Omega }_{fg}$ apart one from the other along the axis of ${\omega }_{step}$. They result from complete destructive interferences among the two-photon Raman transitions. Also here, this π-trace has no dependence on the CEP value. The CEP dependence observed for the different π-traces of ${\omega }_0$=13500 cm⁻¹ is due to the still-existing non-zero small values of $A_{abs}^{(2)}$ that interferes with $A_{Ram}^{(2)}$.

As seen in Fig. 2 for ${\omega }_0$=11870 cm⁻¹ [Fig. 2(b)] and 12500 cm⁻¹ [Fig. 2(c)], the signatures of these large-detuning π-trace features also show up when the pulse spectrum is located in between these two extreme spectral detuning cases. However, the exact shape of the corresponding π-trace results from the overall detailed intra- and inter-group interferences of the phase-shaped $A_{abs}^{(2)}$ and $A_{Ram}^{(2)}$, where both of them are generally of significant magnitude.

Another general important feature of the single-cycle regime analyzed in the present work, i.e., when ${\omega }_0$ is in the region of ${\Omega }_{fg}$, is the fact that the final population $P_f$ excited by the unshaped TL pulse can sometime be exceeded by a properly shaped pulse. This is as opposed to the multi-cycle regime, where the (CEP-independent) $P_f$ value induced by the unshaped TL pulse can never be exceeded by any shaped pulse; At most, it can be matched. This single-cycle feature is illustrated in all the π-traces of ${\varphi }_{CE}$=π/2 (red lines) that are shown in Fig. 2, where, for example, the population excited by the shaped pulse of ${\omega }_{step}={\omega }_0$ is higher than the population excited by the unshaped TL pulse. As discussed above, the latter corresponds to the asymptotic value of the π-trace. For ${\varphi }_{CE}$=0 (blue lines), on the other hand, the relative population excited by the TL pulse is still the maximal one. Below we analyze in detail these cases.

In the case of the TL pulse of ${\varphi }_{CE}$, as also discussed above, the intra-group interferences within $A_{abs}^{(2)}$ and $A_{Ram}^{(2)}$ are fully constructive and thus maximize $\left|A_{abs}^{(2)}\right|$ and $\left|A_{Ram}^{(2)}\right|$, while the relative phase between $A_{abs}^{(2)}$ and $A_{Ram}^{(2)}$ is CEP dependent. For ${\varphi }_{CE}$=π/2, this relative phase is equal to π and thus the inter-group interference between the two amplitudes is completely destructive. For ${\varphi }_{CE}$=0, the inter-group relative phase is zero and the corresponding interference is fully constructive.

In the single-cycle case of ${\omega }_{step}={\omega }_0$, when ${\omega }_0$ is in the region of ${\Omega }_{fg}$, all the effectively contributing two-photon absorption transitions involve only pairs of photons that their frequencies are located near ${\Omega }_{fg}/2$, for which the π phase step contributes zero phase. Thus, the intra-group interferences within $A_{abs}^{(2)}$ are fully constructive and thus maximize $\left|A_{abs}^{(2)}\right|$, while the total phase of $A_{abs}^{(2)}$ is $\Phi (A_{abs}^{(2)})=2{\varphi }_{CE}+0=2{\varphi }_{CE}$ [see Eq. (1)]. On the other hand, all the effectively contributing two-photon Raman transitions involve pairs of photons that their frequencies are located in different spectral regions with respect to ${\omega }_{step}$ and thus the π phase step contributes a phase of 0 to one of them and a phase of π to the other. Hence, also the intra-group interferences within $A_{Ram}^{(2)}$ are fully constructive and maximize $\left|A_{Ram}^{(2)}\right|$, however the total phase of $A_{Ram}^{(2)}$ is actually $\Phi (A_{Ram}^{(2)})=$π [see Eq. (1)]. Hence, here, for ${\varphi }_{CE}$ = π/2, the relative phase between $A_{abs}^{(2)}$ and $A_{Ram}^{(2)}$ is zero and thus the inter-group interference between the two amplitudes is completely constructive. For ${\varphi }_{CE}$=0, the inter-group relative phase is π and the corresponding interference is fully destructive.

So, overall, as seen in the results of Fig. 2, one indeed obtains $P_{f,TL}({\varphi }_{CE}=\pi /\text{2)}<P_{f,shaped,{\omega }_{step}={\omega }_0}({\varphi }_{CE}=\pi /\text{2)}$ (red lines) and $P_{f,TL}({\varphi }_{CE}=0\text{)}>P_{f,shaped,{\omega }_{step}={\omega }_0}({\varphi }_{CE}=0\text{)}$ (blue lines).

4. Three-photon theoretical perturbative description and calculations for a model system

4.1 Theoretical description

In this section we study in detail the case of the non-resonant three-photon excitation. In the weak-field regime the final amplitude $A_f^{(3)}$of the excited state $|f〉$ is given by the third-order perturbation theory as

In the frequency domain it is given by

 

Fig. 3 (a) Three-photon excitation scheme in the femtosecond single-cycle regime. Indicated are the three-photon absorption transitions and three-photon Raman transitions. (b) Ultrabroad spectrum of the single-cycle pulse centered at ${\Omega }_{fg}$/2 provides three absorbed photons of frequencies around ${\Omega }_{fg}$/3 to the three-photon absorption transitions as well as three photons, two absorbed photon of frequency around 2${\Omega }_{fg}$/3 and one emitted photon of frequency around ${\Omega }_{fg}$/3, to the Raman transitions. (c) Relative population of the excited state $|f〉$ induced by the TL single-cycle pulse with different values of ${\varphi }_{CE}$ and different carrier frequencies ${\omega }_0$ (normalized to the case of ${\varphi }_{CE}=$ 0).

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Three-photon absorption transitions involve three absorbed photons of frequencies ${\omega }_1$, ${\omega }_2$ and ${\omega }_3={\Omega }_{fg}-{\omega }_1-{\omega }_2$ (satisfying ${\omega }_1,{\omega }_2,{\omega }_3<{\Omega }_{fg}$), while three-photon Raman transitions involve two absorbed photons of frequency ${\omega }_1$, ${\omega }_2$ (satisfying ${\omega }_1+{\omega }_2>{\Omega }_{fg}$) and an emitted photon of frequency ${\omega }_1+{\omega }_2-{\Omega }_{fg}$. Hence, the global phases of the amplitudes $A_{abs}^{(3)}$ and $A_{Ram}^{(3)}$ are, respectively, 3${\varphi }_{CE}$ and 2${\varphi }_{CE}-{\varphi }_{CE}={\varphi }_{CE}$, resulting in a relative phase between them of $\Delta \varphi =2{\varphi }_{CE}$. The ${\mu }_{abs}^3$,${\mu }_{Ram,1}^3$,${\mu }_{Ram,2}^3$ and${\mu }_{Ram,3}^3$ are, respectively, the effective non-resonant couplings of the three-photon absorption transitions and different three-photon Raman transitions.

The observable considered here as the control objective of the CEP and relative spectral phase is the final population $P_f$ of state $|f〉$ that is given by$P_f={\left|A_f^{(3)}\right|}^2$.

4.2 The model system

The model system we consider has three-photon transition frequency of ${\Omega }_{fg}$=25000 cm⁻¹. In the three-photon excitation case, the rational choice of the excitation pulse spectrum is with a carrier frequency ${\omega }_0$ in the region of one-half of the three-photon transition frequency, i.e., ${\omega }_0\approx {\Omega }_{fg}/2=$12500 cm⁻¹. As illustrated in Fig. 3(b), such spectral position effectively provides significant intensity at frequencies around ${\Omega }_{fg}/3$ as well as around $2{\Omega }_{fg}/3$. Hence, as desired for CEP control, it generally yields significant magnitude to both $A_{abs}^{(3)}$ and $A_{Ram}^{(3)}$. The three-photon absorption amplitude $A_{abs}^{(3)}$ results from transitions involving three photons of ${\omega }_1$,${\omega }_2$,${\omega }_3\approx {\Omega }_{fg}/3$ with ${\omega }_1+{\omega }_2+{\omega }_3={\Omega }_{fg}$, while the three-photon Raman amplitude$A_{Ram}^{(3)}$ results from transitions involving three photons of ${\omega }_1$,${\omega }_2\approx 2{\Omega }_{fg}/3$ and ${\omega }_3\approx {\Omega }_{fg}/3$ with ${\omega }_1+{\omega }_2-{\omega }_3={\Omega }_{fg}$. For consistency, we use here for the three-photon case the same pulse of the two-photon case, i.e., a pulse of 5425-cm⁻¹ bandwidth corresponding to a 2.7-fs TL pulse that is of a single cycle at ${\omega }_0$=12500 cm⁻¹ (800 nm). Generally, it worth noting that the higher is the order of the multiphoton excitation, the smaller is the minimal bandwidth of the ultrabroad spectrum that is required for observing a CEP control effect. This is important for some experimental configurations, where the pulse-shaping implementation is somewhat easier with a reduced bandwidth. Regarding the effective three-photon couplings, similar to the two-photon excitation case, we assume here for simplicity that ${\mu }_{abs}^3={\mu }_{Ram,1}^3={\mu }_{Ram,2}^3={\mu }_{Ram,3}^3$.

In order to avoid here an additional source of interferences, the excitation amplitude induced via one-photon transition at frequency ${\Omega }_{fg}$, which is described theoretically by the first-order perturbative term, is not included in our model. This is justified since, as seen in Fig. 3(b), this spectral component falls far outside the main spectral region, with a corresponding intensity that is about three orders of magnitude lower than the intensity at ${\omega }_0={\Omega }_{fg}/2$. Thus, experimentally the spectral tail located in the region of ${\Omega }_{fg}$ can easily be eliminated by proper amplitude shaping of the pulse [32–34].

4.3 Results for three-photon CEP and relative phase control

The results for the phase control of the three-photon excitation are shown in Figs. 3(c) and 4 . The former presents the CEP control results for unshaped TL pulses, while the latter presents the results for the full phase control utilizing both the CEP and relative spectral phase. Conceptually and qualitatively, the three-photon results are very similar to the control results of the two-photon case [Figs. 1(d) and 2], hence in the following we will only describe in short some of the features of the three-photon results.

 

Fig. 4 Coherent control of three-photon excitation in the model system: Excited state population induced by ultrabroadband pulses shaped with a spectral π phase step at different positions for several cases of carrier frequency ${\omega }_0$ and CEP ${\varphi }_{CE}$. Each trace is normalized by the population excited by the corresponding TL pulse having non-stabilized ${\varphi }_{CE}$.

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The dependence of the three-photon excited population $P_{f,TL}$ on the ${\varphi }_{CE}$ value for the unshaped TL pulses is shown in Fig. 3(c) for different carrier frequencies ${\omega }_0$. Each trace is normalized by the corresponding value of $P_{f,TL}({\varphi }_{CE}=0)$. With a TL pulse, the magnitudes of $A_{abs,TL}^{(3)}$ and $A_{Ram,TL}^{(3)}$ are maximal due to fully constructive intra-group interferences within each of them, and the inter-group relative phase between them is $2{\varphi }_{CE}$ having a corresponding periodicity of π. With ${\varphi }_{CE}=$0 and π, $A_{abs,TL}^{(3)}$ and $A_{Ram,TL}^{(3)}$ have zero relative phase [see Eq. (2)] and, thus, their inter-group interference is constructive leading to the maximal $P_{f,TL}({\varphi }_{CE})$ for the given ${\omega }_0$. With ${\varphi }_{CE}=$π/2 and 3π/2, $A_{abs,TL}^{(3)}$ and $A_{Ram,TL}^{(3)}$ have a relative phase of π and, thus, their inter-group interference is destructive and $P_{f,TL}({\varphi }_{CE})$ is the minimal one for the given${\omega }_0$.

In terms of the modulation depth $M_{TL}$, also here, a spectrum that is largely red or blue shifted from ${\Omega }_{fg}/2$=12500 cm⁻¹ leads to a magnitude $\left|A_{abs,TL}^{(3)}\right|$ that is much larger or much smaller, respectively, than $\left|A_{Ram,TL}^{(3)}\right|$. Such detuned spectrum supports effectively only one group out of the two groups of three-photon transitions. The resulting ratio $r_{TL}=\left|A_{abs,TL}^{(3)}\right|/\left|A_{Ram,TL}^{(3)}\right|$ is, respectively, either much smaller or much larger than 1, and thus the modulation depth $M_{TL}$ in both cases is small [see blue and green lines in Fig. 3(c) for${\omega }_0$=10500 and 13500 cm⁻¹]. As seen in Fig. 3(c), the maximal modulation depth of $M_{TL}$=1 and the corresponding minimal TL population of zero, obtained when $\left|A_{abs,TL}^{(3)}\right|=\left|A_{Ram,TL}^{(3)}\right|$, occur here at ${\omega }_0$=11750 cm⁻¹ (black line), i.e., at a moderate red shift from ${\Omega }_{fg}/2$. The cause for this maximal-modulation red-shift effect is qualitatively similar to the one of the two-photon excitation case. It is the combination of the following facts: (i) $r_{TL}$<1 for ${\omega }_0={\Omega }_{fg}/2$ due to our model assumption of ${\mu }_{abs}^3={\mu }_{Ram,1}^3={\mu }_{Ram,2}^3={\mu }_{Ram,3}^3$, and (ii) $r_{TL}$ increases as ${\omega }_0$ increases.

Figure 4 presents the results for the full phase control of the three-photon excitation using shaped pulses of different carrier frequencies and different CEPs applied with a relative spectral phase step of π. The different panels of Fig. 4 present the results for different values of ${\omega }_0$. For each value of ${\omega }_0$, the corresponding dependence of $P_f$ on the phase step position ${\omega }_{step}$ is shown for the cases of ${\varphi }_{CE}$=0 (blue line) and ${\varphi }_{CE}$=π/2 (red line) as well as for the case of non-stabilized CEP (black line). All the π-traces presented for a given ${\omega }_0$ are normalized by the value of $P_f$ excited by the corresponding TL pulse having non-stabilized CEP.

As in the two-photon case, the asymptotic values of each π-trace are equal one to the other. It results from the fact that the two asymptotes correspond to a constant spectral phase of 0 or π that is globally added to ${\varphi }_{CE}$ and, thus, they correspond to two unshaped TL pulses with a difference of π in their CEP values. As shown in Fig. 3(c), such two TL pulses induce the same population$P_f$. The difference between the asymptotic values of the π-traces with different CEP values at a given ${\omega }_0$ also appears in the CEP-dependence results of Fig. 3(c).

When ${\omega }_0$ is far detuned to the red from ${\Omega }_{fg}$/2 [Fig. 4(a)], as in the TL case, the dominant contribution to the total excited amplitude $A_f^{(3)}$ is from the absorption part $A_{abs}^{(3)}$. Then, the resulting π-trace is similar to the one obtained for the three-photon absorption case in the multi-cycle regime [10], with three zero-value minima located around two peaks of small magnitude. On the other hand, when ${\omega }_0$ is far detuned to the blue from ${\Omega }_{fg}$/2 [Fig. 4(d)], the dominant contribution to $A_f^{(3)}$ comes from the Raman part $A_{Ram}^{(3)}$ leading to a π-trace that is of similar structure to the red-detuned case but much wider. As the CEP value affects only the inter-group interferences, when one of the group amplitudes $A_{abs}^{(3)}$ or $A_{Ram}^{(3)}$ dominates, the π-trace is expected to have only weak dependence on the CEP value. Indeed, as seen in Figs. 4(a) and 4(d), in these two highly-detuned cases there is only a small difference between the π-traces obtained with ${\varphi }_{CE}$=0 and π/2.

When ${\omega }_0\approx {\Omega }_{fg}$/2 [Figs. 4(b) and 4(c)], the exact shape of the corresponding π-trace is determined by the overall detailed intra- and inter-group interferences of the phase-shaped $A_{abs}^{(3)}$ and $A_{Ram}^{(3)}$, where both of them are generally of significant magnitude. For every three-photon transition that contributes to the absorption or Raman amplitudes, the position of the π-step determines the relative phases of the three involved photons and, hence, the way this three-photon transition interferes with the other transitions of his group. All such intra-group interferences determine the magnitudes of $A_{abs}^{(3)}$ and $A_{Ram}^{(3)}$, and, together with the CEP, they also determine the phases of $A_{abs}^{(3)}$ and $A_{Ram}^{(3)}$. All in all, these resulting amplitudes yield the final outcome of $P_f$.

Last, it is important to note that also the three-photon results exhibit the general important feature of the single-cycle regime that, as opposed to the multi-cycle regime, the final population excited by the TL pulse can sometime be exceeded by a properly shaped pulse. This is illustrated in the π-traces of ${\varphi }_{CE}$=π/2 (red lines) presented in Figs. 4(a) and 4(b).

5. Non-perturbative calculations for cesium (Cs) atom

In this section, the control strategy presented above is applied to atomic Cs, demonstrating it for its two excited states $|f_1〉\equiv$5d${}_{3/2}$ and $|f_2〉\equiv$5d${}_{5/2}$ that are populated by two-photon excitation from the ground state $|g〉\equiv$6s. The corresponding two-photon couplings between the ground and excited states are provided by the manifold of p-states of Cs. As opposed to our model system analyzed above, in a real atom, depending on its specific structure, the effective two-photon couplings ${\mu }_{abs}^2$, ${\mu }_{Ram,1}^2$, and ${\mu }_{Ram,2}^2$ generally have different magnitude and might also have different sign. The Cs excitation scheme is shown in Fig. 5 , with examples of the two-photon absorption and Raman transitions. The results for the two-photon excitation of the Cs atom irradiated by weak unshaped (TL) and shaped ultraboradband pulses have been calculated by the numerical propagation of the time-dependent Schrödinger equation using the fourth-order Runge-Kutta method. The calculations have considered the atomic states 6s to 9s, 6p to 8p, and 5d to 7d, with all their fine-structure states. The data of the states and the corresponding transition dipole moments are based on Refs [50–53].

 

Fig. 5 Weak-field two-photon femtosecond coherent phase control of Cs in the single-cycle regime. Upper panel: Excitation scheme of Cs (fine structure splitting is not shown), with examples of two-photon absorption and Raman transitions. Panels (a) and (b): Relative population of the excited states 5d${}_{3/2}$ [panel (a)] and 5d${}_{5/2}$ [panel (b)] induced by TL single-cycle pulses of ${\omega }_0$ = 16667 cm⁻¹ ($\Delta \omega$ = 7350 cm⁻¹) having different values of ${\varphi }_{CE}$. Panels (c) and (d): Relative population of the excited states 5d${}_{3/2}$ [panel (c)] and 5d${}_{5/2}$ [panel (d)] induced by such ultrabroadband pulses applied with ${\varphi }_{CE}$ = 3.3 rad and shaped with a spectral π phase step at different positions. Indicated are the different state-to-state transitions associated with the different resonance-mediated enhancement peaks of the π-traces (see text).

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Similar to the model results presented above, the Cs results also show high degree of CEP control using unshaped TL pulses, with a high sensitivity to the value of the pulse carrier frequency ${\omega }_0$ as compared to the value of the two-photon transition frequency. The transition frequencies corresponding to the Cs states 5d_3/2 and 5d_5/2 are, respectively, ${\Omega }_{f_1,g}$ = 14499.3 cm⁻¹ and ${\Omega }_{f_2,g}$ = 14596.8 cm⁻¹. We have found that the most effective CEP control over the Cs two-photon excitation occurs with ${\omega }_0$ = 16667 cm⁻¹ (600 nm), where the single-cycle TL pulse is of 2-fs duration and bandwidth of $\Delta \omega$=7350 cm⁻¹. The corresponding results of the CEP control with such 600-nm single-cycle unshaped TL pulses are shown for the final populations of the Cs states 5d${}_{3/2}$ and 5d${}_{5/2}$ in, respectively, Figs. 5(a) and 5(b). As seen, there is a strong dependence of the final populations of both states on the ${\varphi }_{CE}$ value, also here, with a periodicity of π. The maximal modulation depth of $M_{TL}$=1 is obtained for the 5d_5/2-state population [Fig. 5(b)], while a very high modulation depth of $M_{TL}$=0.83 is obtained for the 5d_3/2-state population [Fig. 5(a)]. As also seen in Figs. 5(a) and 5(b), the minimal CEP-dependent TL populations of both states 5d_5/2 and 5d_3/2 are obtained for ${\varphi }_{CE}$=0.16 rad and ${\varphi }_{CE}$=0.16+π=3.3 rad, indicating that at these CEP values the inter-group interference between the corresponding overall TL two-photon absorption and Raman amplitudes is the most destructive one. These CEP values are different from the corresponding model-system values of ${\varphi }_{CE}$=π/2 and 3π/2 due to the resonance-mediated nature [11, 20] of some of the two-photon transitions involved in the Cs excitation with the ultrabroadband pulses. Such resonance-mediated transitions do not exist in the non-resonant model excitation analyzed above.

The Cs results also exhibit effective full phase control using the CEP and relative spectral phase of the pulse. As an illustrative example, Figs. 5(c) and 5(d) show the results for, respectively, the 5d${}_{3/2}$ and 5d${}_{5/2}$populations that are controlled with the above ultrabroadband pulses (${\omega }_0$=16667 cm⁻¹, $\Delta \omega$=7350 cm⁻¹) applied with CEP of ${\varphi }_{CE}$=3.3 rad and shaped with a π phase step. One main feature is that, as seen, setting the π-step position in the central spectral region of the pulse (i.e., ${\omega }_{step}\approx {\omega }_0$) converts the above-mentioned destructive inter-group interference of the TL case into a constructive one and leads to an enhancement of the excited populations. Also this effect is indeed predicated by the above model-system results.

Additional prominent features of the π-traces presented in Figs. 5(c) and 5(d) are the several enhancement peaks appearing there. They also originate from the above-mentioned resonance-mediated nature [11, 20] of some of the two-photon transitions induced in the ultrabroadband excitation of Cs. When ${\omega }_{step}$ is set exactly at a spectral frequency that is equal to a transition frequency between the initial ground state (6s) and an intermediate state (np) or between an intermediate state (np) and the final state (5d${}_{3/2}$ or 5d${}_{5/2}$), the two-photon excitation probability is significantly enhanced. As previously explained [11], such an enhancement results from the interferences among two-photon transitions that are near-resonant with the intermediate state and their corresponding detunings are of different signs. These (intra-group) interferences are of destructive nature when induced by the TL pulse, while they become to be constructive when induced by a pulse shaped with a properly-positioned π phase step. The different state-to-state resonant transitions leading to the different resonance-mediated enhancement peaks in the π-traces of the 5d_3/2 and 5d_5/2 populations are indicated in Figs. 5(c) and 5(d). It worth noting that the 5d_3/2 and 5d_5/2 populations are not equally enhanced even when the same intermediate state is involved. For example, the resonance-mediated enhancement of the 5d_3/2 population is either larger or smaller than the enhancement of the 5d_5/2 population when the involved intermediate state is either 6p or 7p, respectively. The corresponding reason is the difference in the effective two-photon couplings associated with the different excitation pathways, which in the above example are 6s-6p-5d and 6s-7p-5d.

6. Conclusions

To summarize, we have shown theoretically the power of weak-field multiphoton femtosecond coherent control in the single-cycle regime. In this regime, for a rationally chosen pulse spectrum, the CEP of the pulse serves as a powerful control parameter in addition to the relative spectral phase. For weak-field N-photon excitation process, the ultrabroadband pulse induces several groups of initial-to-final N-photon transitions, where each group corresponds to a different combination of the number of absorbed photons (M) and the number of emitted photons (N-M). The intra-group interference is phase controlled by the relative spectral phase, while the inter-group interference is phase controlled jointly by the CEP and relative spectral phase. Specifically, here, we have revealed and illustrated the corresponding control mechanism in non-resonant two- and three-photon excitation of a simple model system using frequency-domain perturbative analysis. Then, the phase control mechanism of the single-cycle regime has been confirmed also for weak-field two-photon excitation of atomic cesium (Cs) studied by non-perturbative numerical solution of the time-dependent Schrödinger equation. The same mechanism generally applies also to multiphoton excitations of higher order. We expect this work to serve as a basis for a new line of femtosecond coherent control experiments, most prominently for selective control in excitation scenarios that simultaneously involve several channels.