Ultra-high phase-locked harmonic generation from magnetic transversal con-ﬁnement of electrons.

. The technological reﬁnements on high-power laser systems of Petawatt class unveil scenarios for light-matter interaction beyond the laser-plasma perspective. In this contribution we explore the possibility of assisting high-harmonic generation (HHG) with the strong magnetic ﬁeld associated with one of those intense sources. Recently, there has been an interest in developing schemes to employ intense laser beams to deﬁne spatial volumes in which strong magnetic ﬁelds are found isolated from the electric ﬁeld. In this conditions, the magnetic ﬁeld can be used to assist the harmonic generation from atoms from standard drivers. We demonstrate that, using the proper interaction geometry, the magnetic ﬁeld conﬁnes the transversal dynamics of the continuum electrons allowing, on one side, to enhance the e ﬃ ciency of the electron rescattering that produces the harmonic radiation and, on the other side, to excite the electron transverse dynamics and to convert this energy into phase-locked harmonic photons of few hundreds of eV.


Introduction
The development of structured high-power ultrafast laser beams [1] offers a unique opportunity to explore novel possibilities in HHG.Among the schemes proposed for the generation of intense B-fields, as the use of femtosecond relativistic interactions and plasma physics, some interesting schemes use structured fields to produce intense oscillating magnetic fields isolated from the electric field in limited spatial volumes [2].This isolation is also found naturally in a standing-wave configuration, were the nodes of the electric field are placed at the antinodes of the magnetic field.
In this article we identify a novel regime in which HHG assisted by the magnetic field associated to highpower laser beams leads to the generation of highfrequency radiation beyond the conventional limit, together with the production of isolated near Fourier-limited attosecond pulses.We propose to drive HHG in an atomic gas by a moderately intense circularly polarized laser beam, assisted by an intense B-field.Such B-field, up to tens of kT, can be obtained from structured beams or stationary configurations of state-of-the-art Petawatt laser systems (see Fig. 1).

Magnetic dressing of the HHG.
The Hamiltonian that rules the interaction of an groundstate electron in a hydrogen atom, interacting with a driving circular-polarized field, with vector potential A d (t), * e-mail: lplaja@usal.esand assisted by an oscillating magnetic field B a (t) in the z direction is given in atomic units by As becomes apparent from this equation, in this scenario, the assisting B-field dynamically confines the electron in the transverse coordinate, therefore structuring the continuum into a discrete set of eigenstates, i.e. the assisting B-field creates an effective quantum wire (QW) structure for the detached electrons.The transverse confinement has two main roles.One one hand precludes de electron to depart from the atom in the transverse space, therefore increasing the probability of recollission with the parent ion, once the electron is driven back it by the driv- ing field.On the other hand, the time-dependent curvature of the transverse harmonic magnetic trap can parametrically excite the electron to high energy oscillation states.The transverse dimension, therefore, acts as an energetic reservoir for the continuum electron, whose energy can be released at the moment of recollision, giving rise to highenergy harmonic photons.

Results.
We show in Fig. 2a the comparison between the HHG spectra obtained with the standard geometry, a linear-polarized driver and no magnetic field, and with our scheme of a magnetic-field asisted HHG driven by a cicularly-polarized laser.In both cases the driver intensity is 1.6 × 10 14 W/cm 2 at 800 nm wavelength.For our scheme, the assisting field amounts to and amplitude of 2.8 × 10 4 T with the oscillation period corresponding to a 1.6 µm wavelength intense laser.It becomes apparent that the magnetic field dressing has a substantial role in the extension of the harmonic cut-off to higher harmonic photon energies, in the form of a secondary plateau.Fig. 2b plots the attosecond pulse corresponding to this highenergy plateau, showing a duration near the Fourier limit, thus indicating an almost chirp-free generation of ultrashort high-freqeuncy radiation.In the magnetic-assisted setup, the spectrum is extended to 310 eV, hundreds of harmonic orders above the cutoff energy (black arrow).Remarkably, the radiation arises as a few-cycle near Fourierlimited 27 as full-width half maximum pulse.
To understand the underlying physics, we have developed a unidimensional model for the transverse component of the free-electron wavepacket, considering an electron initially tunnel-ionized from the atom (Fig. 2c).Note that the transverse dynamics corresponds to that of the excited harmonic oscillator, and it is basically given by a well-localized semiclassical wavepacket.From the interplay between the trapping potential and the transverse component of the CP driver, the wavepacket follows a series of transverse breathing cycles.If the maximum wavepacket compression is synchronized with the ion recollision time, the effective rescattering time is minimized leading to the chirp-free emission.

Figure 1 .
Figure 1.The proposed scheme for the HHG assisted by a strong fast oscillating B-field.The B-field results from two counterpropagating Petawatt laser beams.

Figure 2 .
Figure 2. (a) Comparison between the HHG spectra with the standard scheme (orange) with a LP driver and no B-field, and in the proposed scheme with a CP driver and a 2.8 × 10 4 T Bfield at 1.6 µm.The driver intensity in both cases is |E 0 | 2 = 1.6 × 10 14 W/cm 2 at 0.8 µm.(b) Pulse obtained filtering out harmonics below 108 eV for the orange spectrum in (a).The pulse exhibits 27 as full-width half maximum (FWHM) duration, near the Fourier limit (FL).(c) Lateral breathing dynamics for the free wavepacket from the interplay between the trapping potential and the transverse component of the CP driver.