Attosecond and Angstrom Science

ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54

ATTOSECOND AND ANGSTROM SCIENCE HIROMICHI NIIKURA1,2 and P.B. CORKUM1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Tunnel Ionization and Electron Re-collision . . . . . . . . . . . . . . . . . . . . . . . 2.1. Tunnel Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Classical Electron Motion in an Intense Laser Field . . . . . . . . . . . . . . . . 2.3. Re-collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Quantum Perspective of the Re-collision Process . . . . . . . . . . . . . . . . . 3. Producing and Measuring Attosecond Optical Pulses . . . . . . . . . . . . . . . . . . 3.1. Producing Single Attosecond Pulses . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Attosecond Streak Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Measuring an Attosecond Electron Pulse . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Forming an Electron Wave Packet/Launching a Vibrational Wave Packet in H+ 2 4.2. Spatial Distribution of the Re-collision Electron Wave Packet . . . . . . . . . . 4.3. Time-Structure of the Re-collision Electron . . . . . . . . . . . . . . . . . . . . 4.4. Reading the Molecular Clock–the Vibrational Wave Packet . . . . . . . . . . . . 4.5. Confirming the Time-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The Importance of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Single, Attosecond Electron Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Attosecond Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Observing Vibrational Wave Packet Motion of D+ 2 . . . . . . . . . . . . . . . . 5.2. Laser Induced Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Controlling and Imaging a Vibrational Wave Packet . . . . . . . . . . . . . . . . 6. Imaging Electrons and Their Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Tomographic Imaging of the Electron Orbital . . . . . . . . . . . . . . . . . . . 6.2. Attosecond Electron Wave Packet Motion . . . . . . . . . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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

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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].

F IG . 1. Achieved laser pulse duration as a function of year.

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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 (∼1014 W/cm2 ) 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

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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 N2 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 N2 [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 ∼1011 Amperes/cm2 to match it.

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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. T UNNEL I ONIZATION 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/4πε0 r + eE(t)x,

E(t) = f (t) cos(ωt).

Here e is the charge on the electron, ε0 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 × 1014 W/cm2 . If the potential barrier is lower than the vertical ionization energy (IP ) 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 EBSI = IP4 /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

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F IG . 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 <1015 W/cm2 . 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 = IP /2UP using atomic units [38]. Here UP is referred to as the ponderomotive energy, given by UP = e2 |E|2 /4mω2 , where m is the electron mass, E is the strength of the electric field, such that UP [eV] = 9.34 × I [1014 W/cm2 ] × λ2 [µm2 ]. If γ  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 = 1014 W/cm2 , UP 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. C LASSICAL E LECTRON M OTION IN AN I NTENSE L ASER F IELD 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

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F IG . 3. The relation between time of the tunnel ionization (t0 ) and the re-collision time (tc ). 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|/mω2 (cos ωt0 − cos ωt) + ω(t0 − t) sin ωt0 + x(t0 ) where t0 is the time of ionization. The time of the re-collision (tc ) can be calculated as a function of t0 by x(tc ) = 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 (t0 < 0), then the ionized electron never returns to the parent ion. Recollision occurs only for 0  ωt0  π/2 and π  ωt0  3/2, modulo 2π. If tunneling occurs just at the peak of the laser field (t0 = 0), the electron returns to the parent ion after the one period. If t0 > 0, the electron re-collides

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at tc with the net kinetic energy that it gained from the laser field (and IP when the Coulomb potential is included in the calculation) as it traverses its trajectory. In the case of 0 < t0 < (17/180)π/ω, the kinetic energy at tc (re-collision energy) increases as t0 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 UP in the absence of the Coulomb potential. As we show later, the re-collision probability has a maximum value at this time also. For t0 > (17/180)π/ω, the re-collision energy and probability decrease as t0 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. R E - 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 UP is proportional to the square of the wavelength, the maximum recollision energy (∼3.17UP ) increases with laser wavelength for the same laser intensity. The maximum photon energy of the high harmonics is given by 3.17UP + 1.32IP [45]. For reference, at the laser intensity of I = 1.5 × 1014 W/cm2 ,

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F IG . 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. Q UANTUM P ERSPECTIVE OF THE R E - COLLISION P ROCESS 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 (ψb ) remains in the bound potential and the other (ψc ) 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 ∼ ¨ exp(−iωt) dt. From the spectrum, we can reψb |∂V /∂r|ψc  and d(ω) = d(t) construct ψb (Section 6.1) and its time-evolution with attosecond time-resolution (Section 6.2).

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F IG . 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

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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. P RODUCING S INGLE ATTOSECOND P ULSES 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 S TREAK C AMERA 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

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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 Vi (1/2mVi2 = hω ¯ − IP where h¯ ω is the photon energy and IP is the ionization potential of the atom being ionized). Auger decay, or an inelastic scattering could equally produce an electron with velocity Vi . If photoionization occurs in the presence of a strong laser field, the electron gains an additional velocity from a strong laser field E(t) = E0 (t) cos(ωt):     V = Vi + eE0 (t)/mω sin(ωt) − eE0 (t0 )/mω sin(ωt0 ) where E0 (t) is the envelope of the laser field and t0 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 |Vi | > |(eE(t)/mω)|. The term (eE0 (t)/mω) sin(ωt) goes to zero after the optical pulse has gone, but the term (eE0 (t0 )/mω) sin(ωt0 ) 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.

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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 H2 . 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 H2 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. F ORMING AN E LECTRON WAVE PACKET /L AUNCHING A V IBRATIONAL WAVE PACKET IN H+ 2 Figure 6 is a plot of the important potential energy surfaces of H2 and its ions. Tunnel ionization launches an electron wave packet in the continuum. Using H2 as the parent molecule, it simultaneously launches a vibrational wave function + on H+ 2 (Σg ). The transition from H2 to H2 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 ++ [56]. Until the electron little population to the other excited state of H+ 2 or H2 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 (AΣu ) 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.

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F IG . 6. The potential energy surfaces of H2 and H+ 2 . Tunnel ionization launches a vibrational 2 wave function to the H+ 2 (X Σg ) state and produces an electron wave packet simultaneously. Until 2 the electron returns to the parent ion, the vibrational wave packet propagates on the H+ 2 (X Σg ) state 2 Σ ) vibrational wave (X with a vibrational period of ∼25 fs. Re-collision further promotes the H+ g 2 2 packet to the H+ 2 (A Σu ) 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 2 energy surface H+ 2 (X Σg ) is not affected by the laser fields. If the molecules are aligned parallel to the laser polarization direction, the potential energy surface + 2 2 of H+ 2 (X Σg ) is modified by the laser-induced coupling with H2 (A Σu ). 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. S PATIAL D ISTRIBUTION OF THE R E - COLLISION E LECTRON 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

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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 = Ey /Ex ) and given by dy = 5.14εE/mω2 where Ey and Ex 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|/ 2IP )1/2 = 5.6 Å/fs. Although argon has the same ionization energy as H2 , the observed value of the lateral initial velocity of H2 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.

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F IG . 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 × 1014 W/cm2 . 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.

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4.3. T IME -S TRUCTURE OF THE R E - COLLISION E LECTRON 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/cm2 ]—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 2

to 0.9 Å and the calculation used an area of 1 Å . However, the current density is 2 insensitive to the area used as long as the area  15 Å . We include the electron trajectories only with kinetic energy larger than the energy difference between + H+ 2 (X) and H2 (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 × 1014 W/cm2 and the pulse du-

F IG . 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 × 1014 W/cm2 , 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.

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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. R EADING THE M OLECULAR C LOCK – THE V IBRATIONAL 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

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F IG . 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.

F IG . 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.

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F IG . 11. The kinetic energy distribution of H+ dissociating from H+ 2 for a 40 fs, 800 nm pulse having I = 1.5 × 1014 W/cm2 . 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 × 1014 W/cm2 , 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+ (A) are closed. Only the signal responsible for the re-collision is observed 2 in the higher kinetic energy region (squares). If the laser pulse duration is <10 fs and I > 5 × 1014 W/cm2 , then the other dissociation channel opens, referred as the double sequential ionization [61]. First, tunnel ionization of H2 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 H2 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 × 1014 W/cm2 . The peak is ∼6.5 eV for 8 fs. 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

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14 2 F IG . 12. The kinetic energy distribution of D+ dissociation from D+ 2 at I = 6 × 10 W/cm 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. C ONFIRMING THE T IME -S TRUCTURE 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 (XΣg ) by solving the time-dependent Schrödinger equation under field-free conditions. The initial wave packet is obtained from the H2 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 (AΣu ) 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

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+ + F IG . 13. (a) The kinetic energy distribution of H+ created from H+ 2 (Σu ) → H , H. H2 is produced by electron re-collision. The laser intensity is I = 1.5 × 1014 W/cm2 , 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. D2 is produced by the electron re-collision. The laser intensity is I = 1.5 × 1014 W/cm2 , 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. T HE I MPORTANCE OF C ORRELATION 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

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field (separated by the half optical period in time) are not coherent with each other. Therefore, the shape of the vibrational wave packet does not depend on the laser bandwidth or the laser pulse duration. In the next chapter we extend the use of correlated wave packets. We control the electron wave packet and use it to probe the vibrational wave packet motion. We achieve a combined resolution of 200 attoseconds and 0.05 Å. As you reflect on the past sections and read the next section, it is interesting to keep nuclear physics in mind. Can real-time measurements be extended to nuclear physics? However, first we show that we can achieve a single attosecond electron pulse.

4.7. S INGLE , ATTOSECOND E LECTRON P ULSE As we have seen in Section 3.1, one method of producing single attosecond optical pulses is to use a few-cycle laser pulse. In the same way, a few-cycle laser pulse can reduce multiple electron re-collisions shown in Fig. 8, favoring a single recollision. Figure 8(b) is a plot of the calculated current density for a 800 nm pulse with duration of 8 fs and I = 1.5 × 1014 W/cm2 . Compared to the pulse duration of 40 fs, the magnitudes of the current density after the first re-collision are suppressed. Since only three cycles are included in the laser pulse, the tunnel ionization and the re-collision probability depends on both the carrier-envelope phase and the crest of the laser field in the envelope. We calculate the motion of the electron trajectories ionized at different peaks of the laser pulse with the different carrier envelope phases, and average over them to obtain the curve in Fig. 8(b). We confirm the result of the calculation experimentally. Figure 13(b) shows the kinetic energy distribution of D+ measured by a 8 fs, 800 nm, I = 1.5 × 1014 W/cm2 laser pulse (linear polarization). We measure only the correlated fragments that have the same magnitude of momentum, but opposite direction. This specifies the dissociation potential only for the D++ 2 potential. We confirm it by comparing the spectrum measured with linearly and the elliptically polarized laser pulses. In the case of elliptical polarization, the signal counts of H+ disappear at energies >2 eV. Therefore, the spectrum is caused by electron re-collision. The dotted line in the spectrum is the calculated kinetic energy distribution using the first peak of the current density in Fig. 13(b) (8 fs). The solid line is the calculated spectrum using the current density in Fig. 13(a) (40 fs). The measured spectrum (squares) is in between them. The contribution of the second and third re-collision is reduced for this ∼8 fs laser pulse. In summary, we have characterized the re-collision electron wave packet using a molecular clock in H+ 2 . Since its pulse duration is <1 fs, the current density is large. As we noted at the beginning of the section, it is possible to dramatically improve this measurement by concentrating on the optical emission of the nonlinear

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medium. This emission gives us access to electron interferometry. However, from this example, it is clear that collision technology can be exploited for attosecond measurements. In the next section we extend this idea, showing that molecular structure can be measured simultaneously.

5. Attosecond Imaging There are at least four ways that strong field and attosecond science can be used to image molecular structure. These will be the subject of this and the following chapter. First, we show how inelastic scattering of the re-colliding electron allows us to trace wave packet dynamics in D+ 2 . Then we discuss how a re-collision electron that elastically scatters from its parent molecule is, in fact, diffracting from it. Thus, molecular structure is encoded in the above threshold ionization (often called ATI) spectrum of the scattered electron. We complete the chapter by exploiting a third method of measuring molecular structure. We use Coulomb Explosion Imaging to measure vibrational wave packets in D2 , and monitor the influence of a control pulse that guides dissociation. Section 5 focuses on measuring the position of the atoms in a molecule. Section 6 turns to imaging electrons—both their wave function (static structure) and their wave packet motion (dynamical structure). By the end of these two sections it will be clear that strong field and attosecond science allows structural determination in partnership with dynamics. This combination goes to the heart of what we might wish to know about quantum systems. The determination of structure and dynamics is a unique contribution that attosecond science has to offer science as a whole. 5.1. O BSERVING V IBRATIONAL WAVE PACKET M OTION OF D+ 2 In Section 4 we demonstrated that ionization launched a vibrational wave packet. Although we do not know when it was launched in laboratory time, we did not need this information in order to measure the electron wave packet. All that we needed was the knowledge that the electron and vibrational wave packets were launched together. If we modify our perspective on the measurement, we can think of the electron wave packet as a probe of the vibrational dynamics. From this point of view, we have an additional tool to bring to bear on the measurement. Once launched, the electron wave packet is controlled by the laser field. If we change the field, we modify the motion of the electron wave packet. We show that this control can be used to measure the position of the vibrational wave packet as a function of time with combined 200 attosecond, 0.05 Å resolution. This section, therefore, introduces the first experiment that uses attosecond technology for molecular imaging.

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F IG . 14. The observed kinetic energy distribution of D+ produced from D+ 2 by the electron re-collision at four different laser wavelengths (square data points). By shifting the laser wavelength from 800 nm to 1850 nm, the re-collision time increases from 1.7 fs to 4.2 fs.

As is seen in the next two sections, imaging and dynamics develop hand-in-hand in attosecond technology. In some ways our approach is analogous to conventional pump-probe methods. The pump is the tunnel ionization of D2 . It produces both the vibrational wave packet on D+ 2 (X) and the electron wave packet in a correlated fashion. Recollision between the D+ 2 vibrational wave packet and the electron wave packet is the probe. Since the electron wave packet returns to the parent ion for the first time at ∼2/3 of the optical period, the pump-probe delay can be changed by changing the laser wavelength. We use the laser wavelength of 800 nm, 1200 nm, 1530 nm, and 1850 nm, where the corresponding re-collision time is 1.7 fs, 2.7 fs, 3.4 fs, and 4.2 fs, respectively. The position of the vibrational wave packet at the time of the re-collision is determined by the kinetic energy distribution of D+ .

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F IG . 15. The measured peak position of the vibrational wave packet on the D+ 2 (XΣg ) state as a function of time (squares). The solid line is the average position of the wave packet calculated by solving time-dependent Schrödinger equation. A 200 attoseconds and 0.05 Å resolution is achieved.

Figure 14 is a plot of the observed kinetic energy distributions at four different wavelengths. In each case, we subtracted the signal counts measured by the elliptically polarized laser pulse from those measured by the linear polarized laser pulse. As the laser wavelength increases, the spectrum shifts to lower kinetic energy. This agrees with the qualitative prediction that the vibrational wave packet propagates to larger internuclear separation as the re-collision time becomes longer. We evaluate the average motion of the vibrational wave packet on D+ 2 (X) from the measured spectrum. First, we reflect the measured kinetic energy spectrum + to the D+ 2 (A) state and back to the D2 (X) state including the dependence of + the cross section between D2 (X) and D+ 2 (A) on the internuclear separation. In Fig. 15, we plot the peak position of the reflected spectrum as a function of the re-collision time (squares). Figure 15 includes the calculated average motion of the vibrational wave packet (solid line). In the measured time range (1.7–2.4 fs), the vibrational wave packet motion is almost linear because it is very short range compared to one vibrational period of ∼25 fs. So the vibrational wave packet motion is measured with 0.05 Å and 200 attosecond resolution. In Fig. 14, we plot the contribution of D+ 2 (A) to the dissociation as triangles. In the lower energy side, the contribution of the second and third re-collision or other excited states should be included. Recent results of the quantum mechanical calculation agree with our calculation and the observed spectra [63]. 5.2. L ASER I NDUCED E LECTRON D IFFRACTION Section 5.1 emphasized opportunities that arise because the re-collision electron inelastically scatters from its parent ion. It also scatters elastically. It was recog-

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F IG . 16. Ratios of momentum distributions for an electron wave packet diffracted by a diatomic core. (a) Diffraction of a Gaussian wave packet with its initial momentum matching the effective cutoff momentum at I = 1 × 1014 W/cm2 . (b) Laser induced diffraction at the same intensity.

nized very quickly after the idea of the re-collision electron was introduced that elastic scattering of the short wavelength (∼1 Å wavelength) electron requires that the electron diffracts and therefore yields structural information on its parent ion at the moment of re-collision [24]. Thus, controlled by the laser field, the re-collision electron becomes a probe of molecular structure that is almost as convenient to use as an optical probe. Since the re-collision electron is delivered to an atom or molecule with very high timing precision and high current density, re-collision allows diffraction from molecules undergoing chemical dynamics on any time scale relevant to atomic motion in molecules. Here we show one theoretical result in Fig. 16 [25]. There are other theoretical studies [24,65]. Experiments exploiting laser induced electron diffraction have not yet been demonstrated. Figure 16(b) shows a calculated diffraction pattern found by solving the 3-D time dependent Schrödinger equation for a two atom, one electron molecule [25] with its internuclear axis aligned perpendicular to the laser field. Plotted is the ratio of the electron probability scattered parallel to the internuclear axis to that perpendicular to the axis. The diffraction pattern is clearly evident in the image. Figure 16(a) is for an external, monoenergetic electron source. It is included for reference. Analyzed in this manner, the diffraction pattern is as clearly identified with laser induced diffraction as with conventional diffraction. By concentrating on atoms and molecules we may have given the reader the mistaken impression that re-collision is important only for moderate intensity optical pulses. The impression is incorrect. While we will not review it here, by intensities of 1022 W/cm2 (just now becoming experimentally feasible) the re-

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collision electron has sufficient energy to interact elastically or inelastically with the nucleus. It may be used to stimulate and probe nuclear dynamics or to probe nuclear structure by re-collision [33].

5.3. C ONTROLLING AND I MAGING A V IBRATIONAL WAVE PACKET The intense laser pulse can influence the bonding electron of the molecule to change the bond strength of the molecule. Therefore, controlling the laser field can control vibrational motion. In this section we show how the vibrational wave packet of D+ 2 is controlled and offer another approach to its measurement during propagation. Figure 17 is a plot of the potential energy surfaces in the presence (dotted lines) ++ and the absence (solid lines) of the laser fields of D+ 2 together with D2 and D2 . When an intense laser field is applied parallel to the molecular axis, the ground

F IG . 17. The potential energy surface of D+ 2 under the presence (dotted lines), I = 2 × 1014 W/cm2 and the absence (solid lines) of the laser field together with D2 and D++ 2 . The laser pulse is parallel to the molecular axis. The laser field couples two potential energy surface of D+ 2 (XΣg ) and (AΣu ) states, leading the bound (upper, dotted line) and dissociative (lower, dotted line) potentials. Using three 8 fs laser pulses, control and imaging of molecular bond dissociation is possible [68]. The first laser pulse launches the vibrational wave packet to the D+ 2 state by tunnel ion++ ization. The delayed, control pulse modifies the potentials. The third pulse further ionizes D+ 2 to D2 , leading the correlated D+ . From the kinetic energy distribution of D+ , we obtain the time-evolution of the square of the wave packet on D+ 2 at the time of probing.

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+ (D+ 2 XΣg ) and excited states (D2 AΣu ) are coupled by the laser field. This coupling shifts the energy levels (Stark shifts). Applying the laser field at an appropriate time during one vibrational period, one can control the D+ 2 vibrational wave packet motion, dissociation and further ionization [66–68]. If the vibrational wave packet moves to the shorter bond distance when the laser pulse is applied, then the vibrational wave packet motion is slowed. On the other hand, if the vibrational wave packet moves towards the longer bond distance when the laser pulse is applied, then the vibrational wave packet is accelerated. If the vibrational wave packet reaches at outer classical turning point (∼2.5 Å) when the laser pulse is present, then the vibrational wave packet of D+ 2 can propagate towards larger internuclear separation beyond the field-free classical outer turning point (bond softening dissociation [69]). If the vibrational wave packet reaches critical distances where the binding electron is localized to one of the protons when the laser field is still present, then further ++ ionization of D+ and a pair of the fragments (en2 is enhanced leading to D2 hanced ionization [70]). We have demonstrated experimental control over the dissociation of D+ 2 by using a series of intense, few-cycle (8 fs) laser pulses [68]. The first laser pulse launches the vibrational wave packet to D+ 2 state by tunnel ionization. The delayed, control pulse modifies the potentials. The third pulse further ionizes D+ 2 + + to D++ 2 , leading to correlated D . From the kinetic energy distribution of D , we obtain the time-evolution of the square of the wave packet on D+ 2 at the time of probing. Using this approach, we have imaged the shape of the vibrational wave packet as a function of time when it undergoes field-induced dissociation. We have observed that the vibrational wave packet separates into two parts, one remains in a bound state and the other propagates to the ionization continuum with a few-femtosecond time-resolution. This is an imaging of molecular bond breaking as it occurs.

6. Imaging Electrons and Their Dynamics In Section 5 we discussed how elastic or inelastic scattering can yield images of the positions of the atoms in a molecule. These images can be obtained with attosecond precision. Once this technology developed, it will be possible to image atomic positions and their changes on any time-scales using the molecular equivalent of time-lapse photography. Now we show that it is also possible to image electrons and their dynamics.

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6.1. T OMOGRAPHIC I MAGING OF THE E LECTRON O RBITAL We have described how tunneling splits the ground state wave function, ψg , launching a coherent wave packet in the continuum, ψc . It is possible to think of tunneling as a beam splitter for the wave function. We have then described how the electron wave packet moves coherently in the continuum first away from the molecule and then back. It is possible to think of this motion as a delay line in an electron interferometer. During the re-collision the two parts of the electron wave function interfere just as light might interfere in an interferometer. We fully characterize optical pulses with shearing interferometry (SPIDER [62]) by observing the harmonic emission. Equivalently, we can fully characterize the wave function through the high harmonics that are emitted during the electron ion re-collision. Regarding the continuum wave packet as a sum of the  plane waves ψc = a(k) exp(ikx − iwt), the potential can be readily seen by considering that the Fourier transform of the radiating dipole is given by:    d(ω) ∼ a(k) ψg r  exp(ikx) where a(k) is the current density of the re-collision electron with momentum k, and x is the co-ordinate along the direction or the laser polarization. Aside from a(k), d(ω) is a one-dimensional spatial Fourier transform times the initial wave function. Measuring d(ω) for different projections of the wave function measures the full wave function. The transform d(ω) can be obtained from the spectrum of high harmonics (|d(ω)|2 ) by assuming an appropriate spectral phase. Experimentally it is achieved by measuring the spectrum of the high harmonics from different molecular alignments with respect to the laser polarization. Intense infrared pulses with pulse durations shorter than the rotational period makes coherent superposition of the rotational levels [71,72]. It allows field-free control of the molecular alignment. We refer the reader to a recent paper for a detailed description of how “orbital tomography” is accomplished [20], as well as for a discussion of recent experiments involving high harmonic generation with aligned molecules [21,73–75].

6.2. ATTOSECOND E LECTRON WAVE PACKET M OTION The same approach can be used to measure the bound state electron wave packet motion. Consider the case where the electron wave packet is produced in the bound state by coherent superposition of the ground and excited electronic wave functions. In the presence of laser fields, part of the excited state wave function (with a lower ionization potential) tunnels to the ionization continuum. When the continuum electron wave packet returns to the parent ion, it can coherently interacts with both the ground and excited state wave function (Fig. 18). It generates

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F IG . 18. Schematic diagram showing how the continuum electron wave packet (ψc ) interacts with both the ground (ψg ) and excited state (ψe ) electron wave-function. Tunnel ionization produces the continuum electron wave packet from the excited state. When the continuum wave packet returns, it interacts coherently with both wave-function. The spectrum of the high harmonics contains information on the bound state wave packet motion.

the high harmonics whose spectrum contains the information of the wave packet motion, i.e., the phase relation between two bound state electron wave functions. We demonstrate this process by solving the one-dimensional Schrödinger equation. We assume the 1600 nm, 8 fs laser pulse whose carrier-envelope phase is adjusted so that only one re-collision dominates over the entire laser 2 2 pulse, E(t) √ = sin(ω(t − t0 )) exp(−(t − t0 ) /2σ ), where t0 = 8 fs and σ = 6/ 8 ln 2 fs. (Carrier-envelope phase stabilization has been achieved using 800 nm laser pulses [14,76–79] and carrier-envelope phase stabilized laser pulses will be achieved in the infrared using recent advances in laser technology, i.e., OPCPA [80].) In Figure 19(a), we plot the electric field as a function of time. We use a one-dimensional, one electron and two-center potential to adjust the level spacing and ionization potential: V = −e/(4πε0 (x − R/2)2 + a 2 ) −

e/(4πε0 (x + R/2)2 + a 2 ), where a is a smoothing parameter, and R is the internuclear distance between two nuclei. The initial wave function is defined as a coherent sum of the ground and excited electronic wave functions: ψ(t) = ψ0 + ψ1 exp(−i(Et/h¯ + φ)), where E is the energy difference between the ground (ψ0 ) and excited (ψ1 ) electronic states, and φ is the initial phase difference. Solving the time-dependent Schrödinger equation, we calculate time¨ evolution of the electron wave packet as well as the dipole acceleration, d(t). The ¨ Fourier transform of d(t) yields the spectrum of high harmonics. Figure 19(b) is the calculated spectrum when only the first excited state is populated, and (c) is the spectrum when the ground and the excited states are equally initially populated. The vertical ionization energy is 32.8 eV for the ground and 18.6 eV for the first excited state (E = 14.2 eV, R = 0.4 Å and a = 0.65 Å), respectively. The period of the electron wave packet is 290 attoseconds. With a laser intensity of 1 × 1014 W/cm2 , the electron wave function tunnels from the excited state only to the ionization continuum. The spectrum (b) is continuous since only one re-collision occurs during the laser pulse. On the other hand, the

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F IG . 19. (a) The laser field profile that we use in the calculation (1600 nm, 8 fs). The carrier-envelop phase is adjusted so that only one re-collision dominates. (b) The calculated high harmonic spectrum when only the excited state is populated initially. Because of the single electron re-collision, the spectrum is continuous. (c) The calculated high harmonic spectrum when the ground and the excited states are equally populated (E = 14.2 eV). The periodic dips in the spectrum reflect the bound state electron wave packet motion.

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F IG . 20. The relation between the photon energy of the high harmonics and time. The solid line in the middle panel is a plot of frequency chirp of the high harmonics that connects the photon energy and time. Left panel is a plot of the laser field and the bottom is the same high harmonics spectrum as Fig. 19(c).

spectrum in Fig. 19(c) shows periodic intensity dips that reflect the bound state wave packet motion. The photon energy of the high harmonics can be converted to time from ionization to re-collision. The re-collision energy sweeps (short trajectory) until it reaches its maximum energy at ∼2/3 of the optical period, (∼3.4 fs for 1600 nm) and then decreases (long trajectory). In that case the photon energy is uniquely related to the time. The solid line in the middle panel of Fig. 20 is a plot of the frequency chirp of the high harmonics together with the spectrum (lower panel) and the laser field (left panel). This analysis shows the intensity dips in the spectrum (therefore destructive interference) appear when the bound state wave packet counter-propagates against the incoming continuum electron wave packet. Thus, attosecond pulse generation maps bound state electron wave packet motion onto the spectrum. It is a single laser shot measurement. A pump-probe approach could be used. A pump pulse produces the bound state electron wave

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packet, and the delayed carrier-envelope phase stabilized, short laser pulse probes the wave packet motion. In the mapping approach, it is not necessary to change the delay between pump and probe since the dynamics is mapped in the spectrum. However, if we change the delay, then the intensity dips slide in photon energy. Figure 21(a) is a plot of the calculated spectrum as a function of the delay at the laser field of 3 fs, 800 nm, 1 × 1014 W/cm2 . Increasing the delay, the spectrum

F IG . 21. (a) The intensity map of the calculated high harmonic spectrum as a function of the pump-probe delay (upper panel). The lower panel shows the spectrum zero delay. We assume a laser pulse characterized by 1×1014 W/cm2 , 800 nm, and 3 fs. (b) The radiation intensity at photon energy of 33 eV as a function of the pump-probe delay.

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shifts periodically. Taking a particular photon energy and changing the delay, the bound state wave packet motion is also observed. Figure 21(b) shows the radiation intensity at ∼33 eV as a function of the pump-probe delay. The period of the curve agrees with the period of the electron wave packet motion. The pump-probe approach is useful when the bound state electron wave packet motion is slower than the mapping time window (photon energy) of the high harmonic spectrum. Finally, if the electron wave packet loses its coherence, then the depth of the dips decreases. The decay of the electron wave packet motion can be also observed. We have shown that the attosecond bound electron wave packet motion is mapped onto the spectrum of attosecond pulses. An alternative approach is to use attosecond optical pulses as a probe of the bound state electron wave packet [81].

7. Conclusion Attosecond technology is a radical departure from the ultra-fast technology that preceded it. Although it relies heavily on the advances of femtosecond technology, attosecond science is a mixture of electron and photon physics where their mutual coherence can play an important role. This mixture offers optical science the potential to spatially resolve electron and vibrational wave functions of atoms, molecules and solids. Even measurements within the atomic nucleus may be possible [27,33,34]. Spatial resolution is available on any time scale up to and including attoseconds. In this review we have presented concepts and experimental techniques for attosecond measurement. There are two key features. One is that intense laser fields can split the molecular (or atomic) system coherently into two or more parts and control them. First, we have demonstrated that the tunnel ionization of H2 produces the correlated vibrational and electron wave packets (Sections 4 and 5). Controlling one of the wave packets is a measure of the other wave packet’s motion with attosecond precision. We use the inelastic scattering between two wave packets to measure one of the wave packet’s motion. We have also shown that tunnel ionization splits the bound state electron wave function to the two wave packets, one remaining in the bound and other propagating in the continuum (Section 6). One (or both) of the wave packets can be characterized by measuring their dipole interaction which appears as the high harmonic generation. We have shown how this process is essentially electron interferometry. The other key feature is that the laser phase provides a well-defined attosecond clock. The streak method for measuring an attosecond optical pulse relies on the attosecond clock. (Section 3). Measuring the vibrational wave packet motion also uses the fact that the electron re-collides with its maximum re-collision energy

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at a well-defined time (Section 5). Finally, measurement of the attosecond bound state wave packet motion makes use of the relation between the photon energy and the time of re-collision (Section 6). It has always been clear that attosecond science would allow dynamics measurements to be extended from the femtosecond to the attosecond time scale. However, as you have seen, this single statement does not capture the essence of the attosecond science. Attosecond science, at its core, borrows much from collision science. That is why we chose the title “Attosecond and Angstrom Science”. “Attoseconds and Angstrom Science” represents an inextricable mixture of both optical and collision science.

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