ATTOSECOND AND ANGSTROM SCIENCE

ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54 ATTOSECOND AND ANGSTROM SCIENCE HIROMICHI NIIKURA 1,2 and P.B. CORKUM 1 1 National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, Canada K1A0R6 2 PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi-city, Saitama, Japan 1. Introduction........................................... 512 2. Tunnel Ionization and Electron Re-collision......................... 515 2.1. Tunnel Ionization..................................... 515 2.2.ClassicalElectronMotioninanIntenseLaserField... 516 2.3. Re-collision........................................ 518 2.4. Quantum Perspective of the Re-collision Process................... 519 3. Producing and Measuring Attosecond Optical Pulses.................... 520 3.1. Producing Single Attosecond Pulses.......................... 521 3.2.AttosecondStreakCamera... 521 4.MeasuringanAttosecondElectronPulse... 523 4.1. Forming an Electron Wave Packet/Launching a Vibrational Wave Packet in H + 2... 523 4.2. Spatial Distribution of the Re-collision Electron Wave Packet............ 524 4.3. Time-Structure of the Re-collision Electron...................... 527 4.4.ReadingtheMolecularClock thevibrationalwavepacket... 528 4.5.ConfirmingtheTime-Structure... 531 4.6.TheImportanceofCorrelation... 532 4.7.Single,AttosecondElectronPulse... 533 5.AttosecondImaging... 534 5.1. Observing Vibrational Wave Packet Motion of D + 2... 534 5.2.LaserInducedElectronDiffraction... 536 5.3. Controlling and Imaging a Vibrational Wave Packet.................. 538 6.ImagingElectronsandTheirDynamics... 539 6.1.TomographicImagingoftheElectronOrbital... 540 6.2.AttosecondElectronWavePacketMotion... 540 7.Conclusion... 545 8.References... 546 Abstract When a strong laser field ionizes atoms (or molecules), the electron wave packet that tunnels from the molecule moves under the influence of the strong field and can re-collide with its parent ion. The maximum re-collision electron kinetic energy depends on the laser wavelength. Timed by the laser field oscillations, the re-colliding electron interferes with the bound state wave function from which it 511 2007 Elsevier Inc. All rights reserved ISSN 1049-250X DOI: 10.1016/S1049-250X(06)54008-X

512 H. Niikura and P.B. Corkum [1 tunneled. The oscillating dipole caused by the quantum interference produces attosecond optical pulses. Interference can characterize both interfering beams their wavelength, phase and spatial structure. Thus, written on the attosecond pulse is an image of the bound state orbital and the wave function at the re-collision electron. In addition to interfering, the re-collision electron can elastically or inelastically scatter from its parent ion, diffracting from the ion, and exciting or even exploding it. We review attosecond technology while emphasizing the underlying electron ion re-collision physics. 1. Introduction Observing the internal motion of matter on an ever-faster time scale is one of the major aims of science. During the past few decades, optical science has dominated this quest. As shown in Fig. 1, during the 25 years following the invention of the laser, the pulse duration of optical pulses decreased from nanoseconds to a few femtoseconds. However, once the pulse duration reached 6 fs in 1986 [1], the record stood for the next 10 years. A 6 fs laser pulse at 600 nm is so short that the electric field oscillates only a few times in the pulse. Therefore, in order to reach the attosecond time scale (as, 10 18 s), a new approach based on a new physical mechanism was required. Although the minimum laser pulse duration remained fixed for the next 15 years, other aspects of laser technology improved, especially the technology of generating intense pulses [2]. Producing intense, well-controlled femtosecond pulses has proven to be a critical technology for attosecond science [3]. FIG. 1. Achieved laser pulse duration as a function of year.

1] ATTOSECOND AND ANGSTROM SCIENCE 513 If we apply the intense laser fields to gaseous atoms or molecules, then an attosecond photon or electron (when the electron is viewed from the perspective of its parent ion) pulses can be produced. The basic physics is tunnel ionization and electron re-collision [4]. The intense laser pulse ( 10 14 W/cm 2 ) transfers part of the bound electron wave function to the ionization continuum via tunnel ionization. In many ways tunneling is like a beam splitter for light, splitting the wave function in two. In the continuum, the newly formed electron wave packet is pulled away from the parent ion by the strong laser field but, when the laser field changes its sign, it can return to the parent ion with the high kinetic energy obtained from the laser field where it can recollide. In quantum mechanics, what we can know about an object depends upon how it is measured. The coherently re-colliding electron wave packet interferes with the remaining bound state electron wave-function and the dipole oscillation (or transition of the continuum electron back to the ground state where the which way information is lost) caused by this interference produces the coherent light in a short burst of radiation extending into the XUV. If we observe the radiation, we observe the interference. Since an electron wave packet that is born near any field maximum re-collides about 2/3 of a period later, the short burst of radiation is well-timed with respect to the laser field oscillation. Repeated over many 1/2 laser periods, a train of attosecond pulses, with correspondingly high harmonics of the fundamental, is generated. The spectrum of high harmonics is characterized by a long plateau region and cut-off [5]. Producing a single attosecond pulse instead of a train of pulses requires controlling a laser pulse, which in turn controls the electron recollision, so that it can only occur over a small fraction of one period of laser field oscillation. Attosecond optical pulse trains [6 8] and single attosecond optical pulses (250 as) [9 11] were first measured in 2001. Since that time, single attosecond pulses have been used to measure Auger decay dynamics of krypton [12,13] and to trace out the time-dependent electric field of a light pulse [11,14,15]. Spatial coherence of the high harmonics has been also measured [16,17]. The photon energy reaches to the water window [18,19]. From the spectrum of XUV radiation, we can obtain information of the highest occupied molecular orbital [20,21], internal attosecond electron wave packet motion [22], or the molecular vibrational motion of its parent ion [23]. Those are alternative approaches to attosecond measurement. If our observable is electrons instead of photons, then we know that the electron tunneled. In that case, interference between the bound and continuum parts of the wave function is not possible we know that the electron is not in the bound state. However, the continuum wave packet is still coherent and the electron can

514 H. Niikura and P.B. Corkum [1 elastically scatter [24,25] (and diffract) from its parent ion or can inelastically scatter from the ion. Attosecond electron pulses were first measured in 2002 [26]. Since then, electron pulses have been used to monitor the D + 2 vibrational dynamics with 200 attosecond and 0.05 Å precision [27,28], as well as the attosecond dynamics of double ionization in neon [29] and orientation dependence of the branching ratio of double ionization in N 2 between attosecond and slower dynamics [30]. Thus, attosecond science combines both optics and collision physics and opens new opportunities for both. Looking from the optics perspective the new technology produces the shortest duration optical pulses and the shortest wavelength coherent light that can be currently produced. In addition, anyone with an optics background will immediately recognize that interferometry can fully characterize an optical beam its spatial, frequency and phase characteristics. By analogy, measuring the photons produced by the electron interferometer, can fully characterize the electron both the bound state wave function and re-collision electron wave packet. From a collision physics perspective, attosecond science allows one to transfer optics concepts and methods to electrons. The field of a laser pulse can be used to time an electron ion collision to attosecond precision with respect to the laser field. This allows collision experiments to borrow pump-probe technology from optics the collision being either the probe to a photon pump or vice versa. In addition, if a collision leads to the rapid emission of one or more charged particles, then the strong laser field maps the time of release of the products onto the direction and energy of the electron. (Mapping is often called streaking referring to the attosecond streak camera [10,31,32] which we will briefly describe below.) Through collisions, ultrafast science may even extend its reach into measurements of the dynamics of atomic nuclei [27,33,34]. Thus attosecond science is truly a synthesis of optical and collision physics, each enhanced by the interplay with the other and the coherence of the process. Imaging the highest occupied molecular orbital of N 2 [20] is an example of the new opportunities that arise from this synthesis. This review will cover both attosecond electrons and photons. However we will place greater emphasis on the electrons since they are used in their own right and they are needed to produce attosecond photon pulses. In addition, attosecond electrons can be very efficiently used if the target atom or molecule of interest is consistent with re-collision, since we avoid the steps of generating an attosecond optical pulse, shining it to the target molecule and then observing the consequence. In a re-collision experiment electrons are delivered to their target with combined attosecond and angstrom precision. The probability of recollision is extremely high. An external source would need a current density of 10 11 Amperes/cm 2 to match it.

2] ATTOSECOND AND ANGSTROM SCIENCE 515 The chapter is organized as follows. Section 2 discusses tunnel ionization and electron re-collision using the semi-classical, three-step model. Sections 3 and 4 describe how attosecond optical and electron pulses are produced and characterized. Section 5 concentrates on how the attosecond measurement of the vibrational wave packet motion of D + 2 can be combined with measurement of the position of the wave packet. In Section 6 we discuss how electron interferometry can be used to measure the highest occupied molecular orbital of a small molecule and how the motion of the bound state electron wave packet can be observed. 2. Tunnel Ionization and Electron Re-collision The process of tunnel ionization and electron re-collision of a one-electron system in an intense laser field is fully described by the time-dependent Schrödinger equation [35]. However, in order to present an intuitive understanding of the process we use the semi-classical, three-step approach [4]. In this model, the tunnel ionization probability of an atom is calculated as a function of the laser intensity, the motion of the electron under influence of the field is treated as a classical particle ionized at a particular phase of the laser field, and the electron ion interaction is considered if the newly ionized electron returns to the ion. 2.1. TUNNEL IONIZATION The potential energy of the bound, single electron is the addition of the Coulomb potential from the ion core with the potential from the laser field: V = e/4πε 0 r + ee(t)x, E(t) = f(t)cos(ωt). Here e is the charge on the electron, ε 0 is permittivity of free space, the f(t) is the envelope function of the laser field, ω is the angular frequency of the laser field, and x is the coordinate. Figure 2(a) is a sketch of a 1-dimensional cut along the electric field direction through the center of a singly charged ion, evaluated for a constant electric field equivalent to the peak of the laser pulse at an intensity of 1 10 14 W/cm 2. If the potential barrier is lower than the vertical ionization energy (I P ) of the electron, then the electron is released in the ionization continuum according to classical physics (Barrier Suppression Ionization, BSI). The laser intensity where BSI occurs is given by E BSI = IP 4 /4 in atomic units [36]. However, before the laser intensity reaches that value, the bound state electron can tunnel through the potential barrier to the ionization continuum. Figure 2(b) shows the tunnel ionization rate calculated using an atomic ionization model that is tested widely against strong field experiments. It is often referred to as the ADK

516 H. Niikura and P.B. Corkum [2 FIG. 2. (a) The potential energy that a bound state electron feels under the presence of a laser field. The electron can tunnel through the barrier of the combined Coulomb and laser interaction (tunnel ionization). (b) The ionization rate as a function of laser intensity calculated using the ADK model [37]. The abbreviation a.u. stands for arbitrary units throughout the manuscript. tunneling model named after the initials of the three authors of the paper [37]. The tunneling probability is highly non-linear as a function of the field intensity in the range <10 15 W/cm 2. Tunnel ionization is a valid approximation to multiphoton ionization when the electric field oscillates slowly compared to the time the electron spends below the barrier (tunneling time). The ratio between the electron s tunneling time and the laser period is defined as a Keldysh parameter, given by γ = I P /2U P using atomic units [38]. Here U P is referred to as the ponderomotive energy, given by U P = e 2 E 2 /4mω 2, where m is the electron mass, E is the strength of the electric field, such that U P [ev] =9.34 I [10 14 W/cm 2 ] λ 2 [µm 2 ].Ifγ 1, tunnel ionization dominates while, for γ 1, perturbation theory dominates. The terms multiphoton ionization and perturbation theory are often used interchangeably in strong field science, but of course tunnel ionization also involves many photons. For λ = 800 nm and I = 10 14 W/cm 2, U P is 6 ev. In general, for small to medium size molecules, the tunnel ionization probability of a molecule is suppressed compared to an atom with the same ionization potential [39 41]. If the molecular size approaches the dimensions of the electron oscillation, multielectron effects [42,43] must be considered. 2.2. CLASSICAL ELECTRON MOTION IN AN INTENSE LASER FIELD Since the tunnel ionization probability is non-linear with respect to the laser intensity, tunneling occurs near the peak of each laser cycle. After ionization, the electron wave packet propagates in the field, E. In a semi-classical approach, instead of treating the electron wave packet motion quantum mechanically, the

2] ATTOSECOND AND ANGSTROM SCIENCE 517 FIG. 3. The relation between time of the tunnel ionization (t 0 ) and the re-collision time (t c ). The re-collision time and energy relates to the laser phase. Two electron trajectories yields the same re-collision energy. The earlier is referred as a short trajectory, and the other is as a long trajectory. The re-collision time which provides the maximum re-collision energy is 2/3 of the optical period after the tunnel ionization. classical motion of the electron ionized at the phase of the laser field is calculated by solving Newton s equation. For simplicity, we assume that the Coulomb field can be neglected. This would be the case if the electric field of the laser is larger than the Coulomb field (strong field approximation) over most of the electron trajectory. In case of the linear polarization and with E(t) = E cos ωt, the position of the electron along the laser polarization as a function of time is give by x(t) = ( e E /mω 2)[ (cos ωt 0 cos ωt) + ω(t 0 t)sin ωt 0 ] + x(t0 ) where t 0 is the time of ionization. The time of the re-collision (t c ) can be calculated as a function of t 0 by x(t c ) = 0. Figure 3 shows the relation between ionization time and the re-collision time, schematically. If tunnel ionization occurs before the laser intensity reaches its peak (t 0 < 0), then the ionized electron never returns to the parent ion. Recollision occurs only for 0 ωt 0 π/2 and π ωt 0 3/2, modulo 2π. If tunneling occurs just at the peak of the laser field (t 0 = 0), the electron returns to the parent ion after the one period. If t 0 > 0, the electron re-collides

518 H. Niikura and P.B. Corkum [2 at t c with the net kinetic energy that it gained from the laser field (and I P when the Coulomb potential is included in the calculation) as it traverses its trajectory. In the case of 0 <t 0 <(17/180)π/ω, the kinetic energy at t c (re-collision energy) increases as t 0 increases. When ionization occurs at 17 degrees of the laser phase following the peak field, the re-collision energy reaches at its maximum value, 3.17 U P in the absence of the Coulomb potential. As we show later, the re-collision probability has a maximum value at this time also. For t 0 > (17/180)π/ω, the re-collision energy and probability decrease as t 0 increases. Thus, two classes of trajectories contribute the same re-collision energy each 1/2 period. The one that collides earlier is referred as a short trajectory, and the other as a long trajectory. In real atoms or molecules, the electron moves in the Coulomb field of the ion as well as the laser field. It attracts (Coulomb focusing) [43] the electron, modifying these statements a bit. Coulomb focusing increases the re-collision probability and modifies the time of re-collision. If we increase the ellipticity of the laser fields, then the electron is displaced along the direction of the minor axis of the ellipse. As the ellipticity increases, the electron can miss its parent ion. The re-collision probability drops rapidly with ellipticity [44]. 2.3. RE-COLLISION When the electron re-collides with its parent ion, a number of physical processes are induced, as is shown in Figure 4. The electron can scatter elastically [24,25]. In that case, if the parent ion is a molecule, the momentum distribution of the scattered electron (and the re-coil momentum of the ion) carries diffractive information of the molecular structure at the time of scattering. The electron can scatter inelastically. In that case, the ion is excited or further ionized. Inelastic scattering gives rise to the non-sequential double ionization or two-electron excitation [26 30]. The electron can interfere with its parent orbital (i.e. re-combine) [20 22]. In that case, the re-collision energy is converted to XUV radiation, producing attosecond pulses containing high harmonics of the fundamental. Because re-collision occurs within one optical cycle, molecular and electron dynamics can be probed with sub-laser-cycle time resolution using electron recollision. This is illustrated in Section 5 where we show how vibrational wave packet motion of D + 2 can be observed using the inelastic process [27,28]. In Section 6, we show how the electron wave packet motion can be observed using radiative re-combination (high harmonic generation) [22]. Since U P is proportional to the square of the wavelength, the maximum recollision energy ( 3.17U P ) increases with laser wavelength for the same laser intensity. The maximum photon energy of the high harmonics is given by 3.17U P + 1.32I P [45]. For reference, at the laser intensity of I = 1.5 10 14 W/cm 2,

2] ATTOSECOND AND ANGSTROM SCIENCE 519 FIG. 4. Processes caused by re-collision: (1) inelastic scattering, (2) excitation or double ionization, (3) double excitation, and (4) radiative re-combination (high harmonic generation). Since the electron returns within one optical laser cycle, dynamics of molecules or electrons can be probed using these processes with attosecond time precision. the maximum re-collision energy at 800 nm is 31 ev while it is 190 ev at 2000 nm. 2.4. QUANTUM PERSPECTIVE OF THE RE-COLLISION PROCESS The semi-classical three-step approach that we have just introduced is evident in the quantum mechanical approach of Lewenstein et al. [45]. In the semi-classical approach, we have regarded the electron wave packet as the sum of the electron trajectories ionized at different laser phases. Of course, there is nothing in this process that destroys the coherence of the re-collision electron with respect to its parent orbital. Thus, we refer to the analogy with optical interferometry (upper panel in Figure 5). From a quantum perspective, tunnel ionization splits a bound state electron wave packet into two, one (ψ b ) remains in the bound potential and the other (ψ c ) propagates in the ionization continuum (lower panel in Fig. 5). Re-collision recombines them. At the time of the re-combination, coherent interaction between two wave packets induces the electron s dipole moment which generates the radiation (high harmonics). The spectrum of the high harmonics is given by a Fourier transformation of the dipole acceleration, d(t) ψ V/ r ψ ψ b V/ r ψ c and d(ω) = d(t)exp( iωt)dt. From the spectrum, we can reconstruct ψ b (Section 6.1) and its time-evolution with attosecond time-resolution (Section 6.2).

520 H. Niikura and P.B. Corkum [3 FIG. 5. Quantum perspective of the tunnel ionization and the re-collision process. Lower panel: Tunnel ionization splits the bound electron wave function into two, one remains in the bound state and the other propagates in the continuum. At the time of re-collision, two parts of the wave-function coherently interact and the dipole induced by their interaction produces high harmonics. The high harmonic spectrum contains information of both bound and continuum electron wave-function. This process is analogous to an optical interferometer (upper panel). 3. Producing and Measuring Attosecond Optical Pulses Attosecond optical pulses are produced during the electron ion re-collision occurring in an intermediate density gas. Essential to the process is the coherence of the electron wave packet with the wave function from which it has tunneled. At the single atom level, coherence ensures that, when the electron re-collides, it can interfere with the bound portion of the wave function. At the multi-atom level coherence plays another role. It ensures that each atom in a gas interferes in an identical fashion, synchronized by the fundamental pulse. That is, high harmonic generation is phase matched just like other nonlinear optics processes are also phase matched. Synchronized re-collisions produce attosecond optical pulses. The characteristics of attosecond optical pulses are largely imposed by the electrons. The optical pulses are chirped (except at the cut-off) because the electron pulses are chirped. The electrons are perfectly phased with the laser field and therefore so are the photons that they produce. Comparing attosecond optical and electron pulses, the conversion efficiency from laser light to high harmonic photons is 10 6 for mid-plateau photons in argon (considerably lower for helium and neon). As we shall see below, in many

3] ATTOSECOND AND ANGSTROM SCIENCE 521 ways the electron pulses are more intense. They also have a much shorter wavelength. It is the short wavelength that offers the potential for imaging the structure of matter. However, the electrons are only seen as attosecond bursts by their parent atom while the photons can be transported out of the system. Ideas for how to generate attosecond pulses are more than a decade old. The main hold-up that kept attosecond pulses out of the lab was finding a method of measuring them. We begin by briefly reviewing the two methods that are used to produce isolated attosecond pulses. Then we move to the attosecond streak camera [31,32], one of the approaches to attosecond pulse duration measurements. We choose the streak camera measurement because it provides attosecond time resolved measurement in collision physics as well. We refer the reader to other approaches to characterize the attosecond optical pulses [46 49]. 3.1. PRODUCING SINGLE ATTOSECOND PULSES In a multi-cycle laser pulse, attosecond optical pulses are generated at every half laser cycle. If we select an electron trajectory so that the electron re-collision occurs one time during the laser pulse, then single and isolated attosecond optical pulses can be generated. Two approaches have been proposed so far. One uses a laser pulse whose polarization changes rapidly during the pulse so that the polarization is circular at the rising and falling part of the pulse while it is a linear in the middle range of the pulse [50]. Since the electron re-collision probability decreases rapidly with ellipticity, only in the middle range of the pulse can the attosecond optical burst be generated effectively. Another approach uses few-cycle, carrier-envelope phase stabilized laser pulses where only the middle part of the laser pulse has a sufficient intensity to ionize a gas. Adjusting the carrier-envelop phase of the 5 fs, 800 nm laser pulses, the electron trajectories that contributes to the re-collision can be restricted to only one path near the cut-off region. Using this approach, isolated attosecond optical pulses have been produced for the first time [10,11]. If one combines the carrier-envelope phase stabilized, few-cycle laser pulse with time-dependent polarization techniques, reduction of the attosecond pulse duration to about one atomic unit seems possible [51,52]. 3.2. ATTOSECOND STREAK CAMERA The key to measuring the duration of attosecond optical pulses has been to produce a photo-electron replica of the attosecond pulse and then to measure it. There are two ways to produce a replica pulse. It can be accomplished by using atomic photoionization the atom being a photocathode appropriate for attosecond technology or by using the re-collision electron an already existing

522 H. Niikura and P.B. Corkum [3 attosecond replica pulse. Here we concentrate on photoionization. It is used in most attosecond metrology experiments so far. In general, a process that was able to produce attosecond pulses is a good place to look for measurement. The attosecond streak camera [10,11,31,32] exploits the phase dependent drift energy transferred to a photoelectron by a strong laser field. This energy depends on the phase at the birth of the electron and it remains after the optical pulse is terminated. We can characterize the photoelectron by its velocity V i (1/2mVi 2 = hω I P where hω is the photon energy and I P is the ionization potential of the atom being ionized). Auger decay, or an inelastic scattering could equally produce an electron with velocity V i. If photoionization occurs in the presence of a strong laser field, the electron gains an additional velocity from a strong laser field E(t) = E 0 (t) cos(ωt): V = V i + ( ee 0 (t)/mω ) sin(ωt) ( ee 0 (t 0 )/mω ) sin(ωt 0 ) where E 0 (t) is the envelope of the laser field and t 0 is the moment that the photoelectron is released into the laser field. Here we have assumed no re-collision has occurred. This is ensured if V i > (ee(t)/mω).theterm(ee 0 (t)/mω) sin(ωt) goes to zero after the optical pulse has gone, but the term (ee 0 (t 0 )/mω) sin(ωt 0 ) remains, labeling the time of birth of the photoelectron into the laser field. Since, in re-collision physics, an attosecond optical pulse is perfectly phased with the laser field, the photoelectron velocity distribution depends on the range of times over which the electron can be released into the laser field. A long pulse releases electrons over a long time interval, while a short pulse has a very short range of release times. Therefore, the photoelectron spectrum is smeared more by the field for a long pulse than for a short one. At the optimum phase (the attosecond pulse placed at a field maximum) the attosecond streak camera is capable of resolving 70 attosecond transformed limited pulses [53]. A non-transform limited pulse is easier to resolve than a transform limited pulse. Scanning the phase, any attosecond optical pulse can be fully characterized [53,54]. During the past few years it has become apparent that all of the measurement technology developed for visible laser pulses can be transferred to attosecond optical pulses. Thus, the measurement problem is fully solved for attosecond optical pulses. It is interesting, however, that the solution has been to transfer visible technology to the XUV with only one small change. The measurement is performed on a photoelectron replica rather than on the pulse itself. This contrasts with the underlying technology of attosecond pulse generation which is a major departure from the ultrafast technology that preceded it. An alternate approach would be to measure the re-collision electron a preexisting replica of the optical pulse. This requires developing radically different technology for metrology. We now turn our attention to this seemingly more complex task.

4] ATTOSECOND AND ANGSTROM SCIENCE 523 4. Measuring an Attosecond Electron Pulse In this section, we borrow collision physics techniques to characterize the recollision electron wave packet seen from the parent ion. We determine the recollision probability (current density) as a function of time using inelastic scattering in H 2. Since the electron pulse duration is mapped onto the optical pulses, this is a first step towards a new, uniquely attosecond, measurement technology. However it does not allow the precision of the streak camera yet. We include it for two important reasons. First, it shows how attosecond metrology can make use of collision physics techniques (and vice versa). We use inelastic scattering for the measurement. Second, it shows the important role that correlation can play in attosecond science. For our measurement, tunnel ionization of H 2 produces two correlated wave packets, the electron and the vibrational wave packet. We use the vibrational wave packet to clock the time and intensity (current density) of the re-colliding electron wave packet. Correlated measurements extend the range of technology of ultrafast science and will allow ultrafast methods to be used in completely new areas of science, such as nuclear dynamics [27,33,34]. Full characterization of the re-collision electron as complete as any optical measurement has just been achieved [55]. It is too early to be included it in a review. However, it is clear that the key to full characterization of the re-collision electron is interferometry. From general principles we know that interferometry allows all aspects of the interfering waves to be measured. The only uncertainty is the details of how the measurement can be performed. 4.1. FORMING AN ELECTRON WAVE PACKET/LAUNCHING A VIBRATIONAL WAVE PACKET IN H + 2 Figure 6 is a plot of the important potential energy surfaces of H 2 and its ions. Tunnel ionization launches an electron wave packet in the continuum. Using H 2 as the parent molecule, it simultaneously launches a vibrational wave function on H + 2 (Σ g). The transition from H 2 to H + 2 is essentially (but not quite) vertical since the tunnel ionization probability is only slightly dependent on the internuclear co-ordinate (through the co-ordinate dependence of the ionization potential). It is confined to a single potential surface because tunnel ionization transfers very little population to the other excited state of H + 2 or H++ 2 [56]. Until the electron returns to the parent ion, the vibrational wave packet moves on the H + 2 X potential. Inelastic scattering caused by re-collision promotes the vibrational wave packet to the H + 2 (AΣ u) state or other excited states, leading to H + fragments. The kinetic energy of the fragments indicates the internuclear separation at the time of the electron re-collision. Using the vibrational wave packet motion as a molecular clock, we can evaluate when the electron re-collides with the parent ion.

524 H. Niikura and P.B. Corkum [4 FIG. 6. The potential energy surfaces of H 2 and H + 2. Tunnel ionization launches a vibrational wave function to the H + 2 (X2 Σ g ) state and produces an electron wave packet simultaneously. Until the electron returns to the parent ion, the vibrational wave packet propagates on the H + 2 (X2 Σ g ) state with a vibrational period of 25 fs. Re-collision further promotes the H + 2 (X2 Σ g ) vibrational wave packet to the H + 2 (A2 Σ u ) state or the other states, leading to dissociation. The kinetic energy of the H + fragment indicates the time of the re-collision. Since the laser field is present throughout the measurement, to simplify the interpretation of the vibrational wave packet motion, we require that the potential energy surface H + 2 (X2 Σ g ) is not affected by the laser fields. If the molecules are aligned parallel to the laser polarization direction, the potential energy surface of H + 2 (X2 Σ g ) is modified by the laser-induced coupling with H + 2 (A2 Σ u ). Therefore, we select the kinetic energy distribution of H + dissociating from the parent molecule aligning perpendicular to the laser polarization. With the molecule perpendicular to the laser field the vibrational motion is a clock that can time the electron re-collision. 4.2. SPATIAL DISTRIBUTION OF THE RE-COLLISION ELECTRON WAVE PACKET After the tunnel ionization, the electron wave packet spreads in all three directions. In the direction perpendicular to the laser field, spreading occurs because of the initial lateral velocity dv that the electron acquires as it exits the tunnel. In the direction of the laser polarization the shear imposed by the laser field is

4] ATTOSECOND AND ANGSTROM SCIENCE 525 responsible mainly for spreading of the electron wave packet. The lateral velocity determines the re-collision probability for linearly polarized light. We estimate the value of the dv by measuring the ellipticity dependence of the re-collision yield of H + 2. Figure 7(a) shows schematically how the electron wave packet is influenced by the ellipticity. For linear polarization, the electron wave packet moves along the x axis, but if the light is elliptical with its minor axis of the laser fields along the y axis, then the wave packet is pushed laterally. The electron offset of the classical trajectory from the ion core (dy) at the time of re-collision with the maximum re-collision energy caused by the laser ellipticity is proportional to the ellipticity (ε = E y /E x ) and given by dy = 5.14εE/mω 2 where E y and E x are the components of the laser fields in each direction, respectively [44]. If we observe double ionization (or high harmonic generation) then the lateral initial velocity compensates for this offset. By measuring the strength of the double ionization signal as a function of ellipticity, we measure dv.it is by dv = dy/dt, where dt is the time between the tunnel ionization and the re-collision. Figure 7(b) is a plot of the signal counts of H + produced by the electron recollision as a function of the laser ellipticity. At each ellipticity, we measure the kinetic energy spectrum of H + and integrate the signal counts at >4 ev (see Section 4.4). The figure includes the data points when the main laser polarization axis is parallel to the molecular axis (circles) and perpendicular to the molecular axis (triangles). To keep the tunnel ionization probability the same, we maintain the laser intensity of the main polarization axis and increase the intensity of the minor axis. In either cases, the re-collision probability has its maximum value at ε = 0 and decreases as the ellipticity increases. The measured data points are well-fitted by the Gaussian curve (solid line). Taking the 1/e width of the curve, we estimate the average spatial distribution of dx = 9 Å and the average lateral initial velocity of dv = 5.0 Å/fs for parallel to the molecular axis, dy = 7.7 Å and dv = 4.2 Å/fs for the perpendicular case. Therefore, the re-collision electron wave packet is a nano-beam with the diameter of 15 Å at the maximum re-collision time. For comparison, we show the ellipticity dependence curve of the non-sequential double ionization probability of argon measured by the same laser conditions (squares). The data points are also well-fitted by Gaussian curve and the 1/e average initial lateral velocity is dv = 5.4 Å/fs. The lateral initial velocity agrees with the prediction of the atomic tunneling theory that gives dv = ( E / 2I P ) 1/2 = 5.6Å/fs. Although argon has the same ionization energy as H 2, the observed value of the lateral initial velocity of H 2 is smaller than the value for argon. Molecular tunneling theory [40,41] or recent study of high harmonic generation [21,57,58] may find the origin of the differences.

526 H. Niikura and P.B. Corkum [4 FIG. 7. (a) Until the re-collision, the electron wave packet spreads spatially. In the case of a linearly polarized laser pulse, the electron wave packet moves along its polarization axis by the laser field and spreads vertically by the lateral initial velocity at the time of tunnel ionization. An elliptically polarized laser pulse pushes the electron wave packet away from the parent ion, leading to the smaller re-collision probability. (b) The ellipticity dependence of the number of H + ions produced by re-collision when the main axis of the laser polarization is parallel (circles) and perpendicular (triangles) to the molecular axis for a 40 fs, 800 nm pulse having I = 1.5 10 14 W/cm 2. For comparison, the ellipticity dependence of Ar + ionization yield due to the re-collision is also plotted (squares). The data points in each case are well-fitted by the Gaussian curve (solid or dotted lines). The upper axis plots the distance of the electron from the parent ion at the time of re-collision with maximum re-collision energy.

4] ATTOSECOND AND ANGSTROM SCIENCE 527 4.3. TIME-STRUCTURE OF THE RE-COLLISION ELECTRON We have obtained the initial velocity that the electron acquires on tunneling. With it we can calculate the time-structure of the re-collision electron wave packet seen from the parent ion using the semi-classical three-step model. We regard the electron wave packet as a sum of the electron trajectories ionized at different laser phases and positions and calculate those motions under the Coulomb potential combined with the laser fields by solving the Newton s equation. We calculate the equivalent current density [Amperes/cm 2 ] that is, the ratio of the number of the electron trajectories returning to the parent ion per unit time and unit area with the total number of the electron trajectories. The bond distance of H + 2 is assumed to 0.9 Å and the calculation used an area of 1 Å 2. However, the current density is insensitive to the area used as long as the area 15 Å 2. We include the electron trajectories only with kinetic energy larger than the energy difference between H + 2 (X) and H+ 2 (A) at a bond distance of 0.9 Å, as the trajectories that contributes the re-collision. Figure 8(a) is a plot of the calculated electron equivalent current densities as a function of time at a laser intensity of 1.5 10 14 W/cm 2 and the pulse du- FIG. 8. The calculated equivalent current densities as a function of time for a laser pulse duration of (a) 40 fs and (b) 8 fs (I = 1.5 10 14 W/cm 2, 800 nm). Panel (c) is a schematic plot of the relation between the return time and the laser phase. After ionization, the electron wave packet returns several times during the 40 fs laser pulse while multiple re-collision probability is suppressed for the 8 fs laser pulse.

528 H. Niikura and P.B. Corkum [4 ration of 40 fs (800 nm). The electron returns at 2/3 of the laser period (T ) with a maximum re-collision probability and returns several times at 5/2T, 7/4T and so on. The upper panel of the Fig. 8(c) shows the relation between the re-collision time and the laser phase schematically. The re-collision probability decreases drastically after the first peak, but the probability remains relatively high for a few-femtoseconds. That is because Coulomb focusing, that keeps the electron wave packet near the ion core [59]. The first peak of the equivalent current density contains 50% of all re-collision probability and its duration is <1 fs. We can use the first peak as a probe for observing the dynamics in the parent ion. 4.4. READING THE MOLECULAR CLOCK THE VIBRATIONAL WAVE PACKET Figure 9 is a schematic of the laser set up. The 800 nm, 40 fs, 0.8 mj laser pulse is generated by the chirp pulse amplified, Ti:Sapphire laser system. Depending on the aim of the experiments, we guide the laser pulse into different optical systems. We put the laser pulse directly into the vacuum chamber with attenuating the pulse intensity for a 800 nm, 40 fs laser pulse. For longer laser wavelengths (used in Section 5), we convert the wavelength by the optical parametric amplifier (OPA, TOPAS). The tuning wavelength range is 1200 1550 nm for the signal output, and 1700 2100 nm for the idler output. To generate a few-cycle, 8 fs (800 nm) pulse, we couple 400 µj of the laser output into a hollow core fiber filled with argon gas [60,61] where self-phase modulation broadens the bandwidth to 650 900 nm. The output pulse from the fiber is compressed by six reflections on two pairs of the chirped mirrors. We further compensate the chirp by inserting thin quartz plates in the optical path before the vacuum chamber. Each pulse is focused by a parabolic mirror (5 cm focal length) in the vacuum chamber. The duration of the few-cycle laser pulse is measured by SPIDER [62]. We measure the kinetic energy distribution of the fragment (H + ) by timeof-flight (TOF) mass spectrometry. The ions are accelerated to the direction of multi-channel plates by a DC electric field (1600 V) applied between two electrodes separated by 3 cm (Fig. 10). A 1-mm hole in the middle of the electrode selects the fragments dissociating from the parent ion aligning parallel to the TOF axis. The detection angle is 8 degrees for 8 ev of H +. Therefore, if the laser polarization is parallel (perpendicular) to the TOF axis, then we observe the kinetic energy distribution of the fragments dissociating from the parent ion parallel (perpendicular) to the laser polarization. Dissociation of H + 2 can be identified by measuring the kinetic energy distribution of fragment H +. Figure 11 shows the measured kinetic energy distribution when the laser polarization is parallel (a) and perpendicular (b) to the molecular

4] ATTOSECOND AND ANGSTROM SCIENCE 529 FIG. 9. Schematic diagram for the laser setup. The chirp pulse amplified Ti:Sapphire laser system generates 800 nm, 0.8 mj, and 40 fs laser pulses. The wavelength is shifted to longer values by an optical parametric amplifier (OPA, TOPAS). The 8 fs laser pulse is generated by optical fiber compression techniques. FIG. 10. Schematic diagram for the time-of-flight (TOF) mass spectrometer. The ions are accelerated to the multi-channel plate (MCP) by a DC electric field applied between two electrodes. A 1 mm hole on the electrodes selects alignment of the parent ion which produces the fragments detected by the MCP. At the laser polarization vertical to the TOF axis, we select the kinetic energy distribution of the fragments whose parent ion is aligned perpendicular to the laser polarization axis.

530 H. Niikura and P.B. Corkum [4 FIG. 11. The kinetic energy distribution of H + dissociating from H + 2 for a 40 fs, 800 nm pulse having I = 1.5 10 14 W/cm 2. The polarization of the laser pulse is (a) parallel and (b) perpendicular to the molecular axis. The square data points in both panels are the signal counts measured using a linearly polarized laser pulse. The peaks of 0.5 ev and 2.5 ev are produced by bond softening dissociation and enhanced ionization, respectively. At the ellipticity of 0.3, the signal with energies >4 ev in (b) disappears (open circles). The difference of the signal between the square and the open circles is responsible for the electron re-collision. axis at the laser intensity of 1.5 10 14 W/cm 2, with wavelength of 800 nm and pulse duration of 40 fs (FWHM). The peak in (a) 0.5 ev and 3 ev is caused by the bond softening dissociation and the enhanced ionization, respectively. Contributions of the re-collision are found at >4 ev, but cannot be seen in this vertical scale. If the laser polarization is vertical to the molecular axis, then the signal due to enhanced ionization disappears since the potential energy surfaces of H + 2 (X) and H + 2 (A) are closed. Only the signal responsible for the re-collision is observed in the higher kinetic energy region (squares). If the laser pulse duration is <10 fs and I>5 10 14 W/cm 2, then the other dissociation channel opens, referred as the double sequential ionization [61]. First, tunnel ionization of H 2 produces the H + 2 vibrational wave packet at the leading edge of the pulse. Next, further ionization of H + 2 occurs in the vicinity of the peak of the laser pulse that leads to H ++ 2 before the vibrational wave packet reaches the classical outer turning point. The kinetic energy distribution of the correlated H + fragments indicates the time between first and the second ionizations. Here again, the vibrational wave packet motion on H + 2 is used as a molecular clock. Figure 12 is a plot of the kinetic energy spectrum of D + with a pulse duration of 8 fs 800 nm and I = 6 10 14 W/cm 2. The peak is 6.5 evfor8fs.this method allows us to check the pulse duration without optical techniques. Re-collision is the fastest pathway of all. To identify the kinetic energy distribution of H + caused by re-collision, we measure the kinetic energy spectrum at both linear (square data points in Fig. 11(b)) and elliptical polarization (circle data points in Fig. 11(b), ellipticity ε = 0.3). If the ellipticity of the laser pulse

4] ATTOSECOND AND ANGSTROM SCIENCE 531 FIG. 12. The kinetic energy distribution of D + dissociation from D + 2 at I = 6 1014 W/cm 2 and a pulse duration of 8 fs. increases, then the ionized electron wave packet is pushed away from the parent ion and the re-collision probability decreases. As is mentioned earlier, at the ellipticity of 0.3, re-collision between H + 2 and the electron becomes impossible. From the Fig. 11(b) the energetic fragments >4 ev are caused by electron re-collision. 4.5. CONFIRMING THE TIME-STRUCTURE Using the molecular clock based on H + 2 vibration, we experimentally confirm the time structure obtained by the semi-classical calculation. Figure 13(a) also contains a plot of the observed kinetic energy distribution of H + (squares). We measured the distribution for the case of linear and elliptical polarization (ε = 0.3), and subtract the signal counts measured by the linearly polarized pulse from those measured by the elliptically polarized pulse. To compare the experimental results with calculations, we predict the kinetic energy distribution of H + using the current density shown in Fig. 8. Specifically, we calculate the vibrational wave packet motion on H + 2 (XΣ g) by solving the time-dependent Schrödinger equation under field-free conditions. The initial wave packet is obtained from the H 2 ground state vibrational wave function weighted by the tunnel ionization probability that depends on the internuclear separation. Assuming that the vibrational wave packet is excited to the H + 2 (AΣ u) state with the excitation probability according to the current density, we calculate the kinetic energy distribution of H +. The dotted line in Fig. 13(a) is the calculated kinetic energy distribution when only the first peak of the current density is included, and the dashed line is the distribution when only the third peak of the current density is included. The solid line includes all five peaks. The first peak is separated from the third peak by

532 H. Niikura and P.B. Corkum [4 FIG. 13. (a) The kinetic energy distribution of H + created from H + 2 (Σ u) H +, H. H + 2 is produced by electron re-collision. The laser intensity is I = 1.5 10 14 W/cm 2, the pulse duration is 40 fs (800 nm) and the laser polarization is perpendicular to the molecular axis. The three curves are calculated results using the current density shown in Fig. 8(a). The dotted curve is produced by the first re-collision, the dashed curve is produced by the third re-collision, and the solid line includes all five peaks. (b) The kinetic energy distribution of D + created from D ++ 2 D +, D. D + 2 is produced by the electron re-collision. The laser intensity is I = 1.5 10 14 W/cm 2, the pulse duration is 8 fs (800 nm) and the laser polarization is perpendicular to the molecular axis. The two curves are calculated results using the current density shown in Fig. 8. The dotted curve is produced by the first re-collision in Fig. 8(b) and the solid line includes all five peaks in Fig. 8(a). The experimental results are consistent with a single re-collision with a small satellite pulse. 2.7 fs, and then the differences can be resolved in the kinetic energy spectrum. In the upper axis of Fig. 13(a), we plot the time scale converted from the kinetic energy distribution of H + with a molecular clock. The observed spectrum agrees well the calculated spectrum. The dotted vertical line in the figure (8.2 ev) is the kinetic energy if the dissociation of H + 2 occurs just after the tunnel ionization (t = 0). These results indicate that the re-collision electron wave packet contributing to the excitation is well localized spatially and temporally. Recent quantum mechanical calculations agree with the results of our calculation [63,64]. 4.6. THE IMPORTANCE OF CORRELATION We have just described a measurement of the electron packet in which we achieve a time resolution of 1 fs. We achieved this in spite of using a 40 fs laser pulse. How is this ultrafast measurement without ultrashort pulses possible? In our case it is possible because the vibrational wave packet and the electron wave packet were correlated (strictly speaking they were entangled). Because of the entanglement, vibrational wave packets that are launched at different peaks of the laser