| Sub-femtosecond, delayed nonlinear response of bound electrons in Kr. a-c, Global-phase spectrograms of the optical attosecond pulse recorded at peak-intensity settings of about 5 × 10 13 W cm −2 (a), 7 × 10 13 W cm −2 (b) and 9 × 10 13 W cm −2 (c) are shown in the left panels. Corresponding reconstructed spectrograms based on equation (1) are shown in the right panels. The colour bar represents spectral intensity in arbitrary units. d, e, Low-pass-filtered (0-8 eV) nonlinear dipoles obtained using the intensity settings in a-c (dark blue, orange, green lines, respectively) are shown along with the instantaneous response (black line) simulated for global-phase settings of the optical attosecond pulse of φ G ≈ 0 (d) and φ G ≈ π /2 rad (e). Standard errors of the mean for the delays (τ) indicated are evaluated from the reconstruction of three data sets recorded under identical experimental conditions. The dashed red lines are the normalized instantaneous intensities of the input electric fields. 

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... (PHz) evaluated from TDSE simulations (a, b) and our experiments in Fig. 4 (c, ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... apply the methodology presented above to retrieve the dynamics of the bound-electronic response, we supplemented our measurements with attosecond streaking of the driving field E(τ , φ G ). By studying the dependence of the VUV emission as a function of the intensity of the optical driver, we verified that bound-electronic nonlineari- ties dominate the response (see Methods). Figure 4 shows the meas- ured (left panels) and the corresponding reconstructed (right panels) spectro grams recorded at gradually increasing optical-field intensi- ties: about 5 × 10 13 W cm −2 (Fig. 4a); about 7 × 10 13 W cm −2 (Fig. 4b); and about 9 × 10 13 W cm −2 (Fig. 4c). The retrieved nonlinear dipoles (Fig. 4d) exhibit delays with respect to the instantaneous dipole (black line) and driving field E(τ , 0) (red dashed line) of τ ≈ 45 ± 4 as (Fig. 4a), τ ≈ 70 ± 5 as (Fig. 4b) and τ ≈ 115 ± 7 as (Fig. 4c). The data in Fig. 4d verify the attosecond control of the bound-electronic response induced by the field, and emphasize the non-trivial character of the induced dynamics, which is manifested by a non-uniform delay (chirp). The latter is highlighted by the delay values for consecutive half-field-cycles shown in Fig. 4e. These findings, which are in excellent agreement with the predictions of the TDSE simulations performed for identi- cal excitation fields, provide conclusive evidence for the feasibility of tracing and control of the nonlinear response of bound electrons on a sub-femtosecond timescale with high ...

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... d is chosen to match the nonlinear polarizability of Kr (refs 31, 32), ω 0 is the excitation energy ω ≈ 10 eV 0 and E(t) denotes the electric field of the optical attosecond pulse. To access the nonlinear component of the induced elec- tronic dipole moment at a given intensity of the driving field in both models, we perform a second calculation at a much lower (about six orders of magnitude) intensity. As a next step, we subtract the virtually linear dipole calculated at the lower intensity from the original one, after multiplying it by the corresponding ratio between the two intensities. The calculated global-phase spectrograms (spec- tral emission as a function of the global phase) using the adiabatic and TDSE models are shown in Extended Data Fig. 4a, b. In accordance with the discussion in the main text, the adiabatic model (Extended Data Fig. 4a) predicts uniform modulations of the spectral amplitude of the emitted spectral components as a function of the global phase φ G . In contrast, the spectrogram calculated using the TDSE model embodies the signatures of the delayed electronic response (Extended Data Fig. 4b) in the form of asynchronous amplitude modulations between differ- ent frequencies/energies of the emitted dipole. These features are present in the entire emitted spectrum, not only close to the resonant area (10-14 eV). We show that these effects can be used to extract the dynamics of the nonlinear response by reconstructing the global-phase (φ G ) spectrograms recorded in our ...

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... d is chosen to match the nonlinear polarizability of Kr (refs 31, 32), ω 0 is the excitation energy ω ≈ 10 eV 0 and E(t) denotes the electric field of the optical attosecond pulse. To access the nonlinear component of the induced elec- tronic dipole moment at a given intensity of the driving field in both models, we perform a second calculation at a much lower (about six orders of magnitude) intensity. As a next step, we subtract the virtually linear dipole calculated at the lower intensity from the original one, after multiplying it by the corresponding ratio between the two intensities. The calculated global-phase spectrograms (spec- tral emission as a function of the global phase) using the adiabatic and TDSE models are shown in Extended Data Fig. 4a, b. In accordance with the discussion in the main text, the adiabatic model (Extended Data Fig. 4a) predicts uniform modulations of the spectral amplitude of the emitted spectral components as a function of the global phase φ G . In contrast, the spectrogram calculated using the TDSE model embodies the signatures of the delayed electronic response (Extended Data Fig. 4b) in the form of asynchronous amplitude modulations between differ- ent frequencies/energies of the emitted dipole. These features are present in the entire emitted spectrum, not only close to the resonant area (10-14 eV). We show that these effects can be used to extract the dynamics of the nonlinear response by reconstructing the global-phase (φ G ) spectrograms recorded in our ...

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... d is chosen to match the nonlinear polarizability of Kr (refs 31, 32), ω 0 is the excitation energy ω ≈ 10 eV 0 and E(t) denotes the electric field of the optical attosecond pulse. To access the nonlinear component of the induced elec- tronic dipole moment at a given intensity of the driving field in both models, we perform a second calculation at a much lower (about six orders of magnitude) intensity. As a next step, we subtract the virtually linear dipole calculated at the lower intensity from the original one, after multiplying it by the corresponding ratio between the two intensities. The calculated global-phase spectrograms (spec- tral emission as a function of the global phase) using the adiabatic and TDSE models are shown in Extended Data Fig. 4a, b. In accordance with the discussion in the main text, the adiabatic model (Extended Data Fig. 4a) predicts uniform modulations of the spectral amplitude of the emitted spectral components as a function of the global phase φ G . In contrast, the spectrogram calculated using the TDSE model embodies the signatures of the delayed electronic response (Extended Data Fig. 4b) in the form of asynchronous amplitude modulations between differ- ent frequencies/energies of the emitted dipole. These features are present in the entire emitted spectrum, not only close to the resonant area (10-14 eV). We show that these effects can be used to extract the dynamics of the nonlinear response by reconstructing the global-phase (φ G ) spectrograms recorded in our ...

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... Data Fig. 4c shows a global-phase spectrogram simulated for a sin- gle-cycle pulse using the TDSE Kr model. In this regime of single-cycle pulses, the global-phase spectrogram does not show discernible variation over φ G , which is experimentally verified (see Fig. 3e-g). Ionization-free strong field polarization of bound electrons. An essential inno- vation introduced by using optical attosecond pulses is their unique capability to drive nonlinear dynamics in quantum systems without inducing a substantial degree of ionization or excitation, that is, without greatly altering the original sys- tem. In experiments where the polarization of the system is to be probed, both excessive ionization and excitations markedly modify the system, resulting in a considerable degree of 'contamination' in the emitted signal from the new atomic entities; such contamination is challenging to resolve both experimentally and ...

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... Data Fig. 7a-c shows representative, synthetic, global-phase spectrograms for three values of the parameter dt (dt = 0 (instantaneous response), dt = 20 as and dt = 30 as), generated by equation (1) in the spectral range of our experiments (see Fig. 4). The synthetic spectrograms highlight the capability of the model to capture key features of the experimental spectrograms (for example, Figs 1 or 4), such as the profoundly asynchronous modulation of the emission in the range 7-8 eV and the weakening of the amplitude in the range 6-6.5 eV-both are the result of dynamic nonlinear interference between delayed and instantane- ous terms in equation ...

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... results of this study are summarized in (Extended Data Fig. 8a, b), in which the original and the reconstructed spectrograms are shown. The reconstruction parameters in equation (1) Intensity dependent nonlinear delay. We study the delayed nonlinear bound- electronic response over a wide range of intensities ((2-8) × 10 13 W cm −2 ). The evaluated delays between the instantaneous and the TDSE-simulated dipoles as function of the driver field intensity are shown in Extended Data Fig. 9a; the delays between the instantaneous dipoles and those reconstructed from our measured spectrograms (see Fig. 4) are shown in Extended Data Fig. 9b. 2 10 -1 10 0 1000 750 6 00 500 429 375 3 33 300 273 2 50 230 Wavelength ...