Muga G. Time in Quantum Mechanics

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316 G.G. Paulus and D. Bauer

dependence of the position in time of the third pulse. This can be done by exciting

the atoms after the third reﬂection with yet another laser pulse, but now tuned to

the 6s

1/2

–6p

3/2

transition. The ﬂuorescence is detected by a photomultiplier. At

this point, the low-target gas pressure made possible by the twin-MOT becomes

essential. As one would expect, the transmission of the apparatus is very low. Out

of the 10

atoms captured in the lower MOT, just a few make it to the detection

interaction area. A few improvements in this respect are certainly possible and some

are in fact implemented in the work described in more detail in [63]. The sequence

of pulses is shown in Fig. 11.7.

Now it is straightforward to perform a double-slit experiment by switching on the

atom mirror twice at the second return of the atoms. This means that two interfering

de Broglie waves generated with a brief delay will emerge. Of course, the separation

of these two pulses should be shorter than the coherence time. Just like for diffrac-

tion with a single pulse as discussed above, the interference fringes can be recorded

by scanning the third pulse. Thus, the interference pattern is measured as a function

of the time of ﬂight (see Fig. 11.8). It is even possible to shift the fringe pattern.

In case one of the two pulses that form the double-slit is weaker than the other, the

atoms are reﬂected a little bit closer to the surface of the atom mirror. Therefore,

those atoms experience an additional phase shift. With Δφ = 2π ·Δ/Λ

, where

= h/(mgT) is the de Broglie wavelength close to the mirror surface, it can

easily be calculated that a difference in path length of Δ = h/(2mgT) ≈ 12 nm

results in destructive interference (Δφ = π). This means that a fringe shift by half

a period is observed, if the atom for one of the slits approaches the surface by 6 nm

closer than the other. A reduction in intensity of 5% for one of the atom mirror

pulses is sufﬁcient to realize this phase shift.

Fig. 11.7 Trajectories of the atoms following their release from the lower trap. The atoms bounce

like on a trampoline at the atom mirror which can be switched on and off by pulsing the laser that

creates the evanescent wave. The ﬁrst reﬂection by applying a laser pulse P

is used to narrow

down the velocity distribution, thus increasing the coherence time. The second reﬂection induces

diffraction in time. A double-slit in time can be realized by switching the atom mirror on twice

with a pair of pulses P

and P

, separated by a brief time lag. The atom mirror is switched on for

a third time by a laser pulse P

in order to probe the diffraction pattern. P

is shifted around the

nominal return time and the number of atoms that have successfully completed the entire trajectory

are detected (from Szriftgiser et al. [63])

11 Double-Slit Experiments in the Time Domain 317

Fig. 11.8 Interference fringes of cold atoms after reﬂection at a temporal double-slit. For the upper

panel both atom trajectories have the same length. For the lower panel, pulse P

was given a

slightly smaller intensity so that atoms reﬂected by this pulse would approach the prism surface a

few nanometers closer, thus giving rise to a fringe shift of half a period (from Szriftgiser et al. [63])

11.3.2 The Attosecond Double-Slit

Strong-ﬁeld ionization provides mechanisms for realizing a double-slit in time under

quite different conditions. In this case, the temporal slits are brief windows in time

within the cycles of the optical ﬁeld that induce ionization. The decisive fact is

that the phase (or time) within any optical cycle, at which ionization takes place,

determines the photoelectron momentum and thus its kinetic energy. This means

that the instant of ionization is mapped onto the momentum of the photoelectron.

Consequently, for more than one optical cycle, there is more than one possibility

to create photoelectrons of a given momentum and interference will be observed.

Few-cycle laser pulses consisting of two or even less optical cycles within their full

width at half maximum (FWHM) are particularly well suited to create double-slit

interferences [41].

In the present context, a laser ﬁeld is considered strong if optical ﬁeld ionization,

i.e., tunneling, is the leading process of ionization. As for regular ﬁeld ionization,

the barrier through which bound electrons may tunnel is formed by the laser ﬁeld

and the atomic potential. For linear ionization, this implies that the tunnel opens and

closes twice during one optical cycle. The transient nature of the potential barrier

suggests that a description of ionization based on tunneling can only be meaningful

318 G.G. Paulus and D. Bauer

if the frequency ω

of the electron that tunnels through the Coulomb barrier is high

as compared to the laser frequency ω = 2π/T [36, 37], i.e.,

γ :=

√

2m|E



< 1 . (11.29)

γ is the so-called Keldysh parameter, E

is the energy of the bound state from which

the electron starts, i.e., |E

| is the ionization potential, and

E is the amplitude of the

laser ﬁeld. The ponderomotive potential U

= e

/(4mω

) is the quiver energy of

an unbound electron in an electric ﬁeld oscillating with angular frequency ω and an

important scaling factor in strong-ﬁeld laser physics. Keldysh parameters γ ≈ 1 can

fairly easily be realized by exposing rare gas atoms to intense femtosecond lasers at

intensities around 10

W/cm

. A review of tunneling and multiphoton ionization

can be found in [55].

The tunneling probability depends exponentially on the ﬁeld strength. As a con-

sequence, ionization is conﬁned to brief time intervals close to the maxima of the

oscillating electric ﬁeld strength of a laser pulse. To a large extent, the trajectory of

an electron (or rather an electron wave packet) released from the atom by tunneling

will be dominated by the laser ﬁeld because it is much stronger than the ﬁeld of the

ion for the biggest part of the trajectory.

Therefore, it is reasonable to model in a

ﬁrst approximation ionization by classical mechanics, i.e., by calculating classical

electron trajectories. Assuming, for the time being, a continuous, linearly polarized

laser ﬁeld of the form

E(t) =

E cos(ωt) (11.30)

and tunneling at a phase ωt

, one immediately obtains in atomic units (see also the

remark after Eq. (11.27) in Sect. 11.2.4)

v(t) =−

[

sin(ωt) −sin(ωt

)

]

(11.31)

for the velocity of the electron, if v = 0 is assumed for the electron at the instant of

tunneling. For the drift energy E

kin

of the photoelectron, i.e., the quantity measured

in an experiment, this results in

kin

= 2U

sin

(ωt

) . (11.32)

Equation (11.32) is quite an important result as it shows what has been announced at

the beginning: The kinetic energy is determined by the instant at which the electron

enters the continuum. In addition, we note that electrons tunneling at ωt

= 0 will

have zero drift energy, whereas those ejected into the ﬁeld at ωt

= π/2 will obtain

This is in fact the essence of the strong-ﬁeld approximation (SFA) to be introduced in Sect. 11.4.

11 Double-Slit Experiments in the Time Domain 319

the maximum kinetic energy of 2U

. Tunneling is, of course, much more likely at

the peak of the ﬁeld at ωt = 0, which explains that photoelectron spectra show

way more electrons at low energies than at high energies. This simple theory of

strong-ﬁeld ionization is known as the “simple man’s model” since the late 1980s

[65] and has been beautifully conﬁrmed in microwave experiments using Rydberg

atoms [25, 26].

The same result can be obtained taking advantage of conservation of the canoni-

cal momentum

can

= p + eA = const., (11.33)

where A is the vector potential and p = mv (= v in atomic units) is the kinetic

momentum. Using the same approximations as above, i.e., p(t

) = 0, remembering

that p(t →∞) ≡ p

drift

, and taking into account A(t →∞) = 0 for a pulse with no

DC component [48], the identity p

can

) = p

can

(∞) yields

drift

= eA(t

) . (11.34)

Thus, the electron momentum is given by the vector potential at the instant of ioniza-

tion. Or, reading this statement the other way round: Selecting electrons with a given

momentum is equivalent to selecting electrons leaving the atom at a well-deﬁned

phase. This result holds for arbitrary pulse forms, in particular also for few-cycle

pulses. For an illustration, see Fig. 11.9. Choosing the electron momentum implies

choosing the instant of ionization with sub-cycle resolution, i.e., on the attosec-

ond scale. For a given electron energy and emission direction (parallel to the laser

Fig. 11.9 Temporal variation of the vector potential A(t) for a few-cycle laser pulses with a −sine-

like temporal variation of the ﬁeld, i.e., E(t) =

E(t)cos(ωt + π/2). The temporal slits are given

by the condition p −eA(t

) = 0. For a −sine-like pulse, this leads to a double-slit in the negative

(since e =−|e|) direction and a single slit in the opposite direction. Each slit can be resolved into

a pair of slits (from Lindner et al. [41])

320 G.G. Paulus and D. Bauer

polarization), the temporal evolution of the ﬁeld of a few-cycle laser pulse can be

chosen such that for one emission direction, just a single optical (half-) cycle may

generate electrons of this energy, while for the opposite direction there are two such

(half-) cycles. This is in fact the essence of the attosecond double-slit.

The strength of the classical model is the intuitive insight it provides. Hardly

more than the number and position of the solutions of p − eA(t

) = 0 for given

p has been used in order to explain double-slit behavior for strong-ﬁeld ioniza-

tion induced by few-cycle laser pulses. The respective solutions t

(p) in a quantum

model that will be discussed below in Sect. 11.4.3 are complex, thus allowing access

to classically forbidden electron energies. However, the symmetry of these solutions

stays the same and so do the results qualitatively.

Generation of few-cycle laser pulses is remarkably simple: Pulses of a conven-

tional femtosecond laser system are sent through a hollow ﬁber ﬁlled with a few

hundred millibar of Argon or Neon [51, 52]. Due to self-phase modulation the spec-

trum broadens and the pulses can be compressed to as few as 5 fs (FWHM). The

method works best for sufﬁciently powerful (1 mJ) and short (<25 fs) femtosecond

pulses to start with. The duration of the compressed pulse has to be compared with

the optical period of T = 2.5 fs for the typical wavelength of 800 nm of titanium-

sapphire femtosecond lasers. This means the temporal evolution of the ﬁeld E(t)

depends on the phase of the carrier wave with respect to the maximum of the pulse

envelope. This can be seen by writing E(t) as a product of envelope

E(t) and carrier

wave:

E(t) =

E(t) cos(ωt +ϕ) . (11.35)

ϕ is the so-called absolute phase, also known as carrier-envelope phase. Special

cases are cosine-like, sine-like, −cosine-like, and −sine-like pulses for ϕ = 0,

−π/2, π, and π/2. This phase determines the evolution of the ﬁeld and therefore its

measurement and stabilization is of pivotal importance for, e.g., attosecond laser

physics which takes advantage of processes evolving within fractions of optical

cycles. The absolute phase was ﬁrst detected in photoionization experiments [53]

and later measured in the same way [54]. Stabilization of the absolute phase has

ﬁrst been achieved in frequency metrology [59, 64, 35] and later also for ampliﬁed

laser pulses [5]. For the attosecond double-slit, the absolute phase determines the

number of slits for a given emission direction. More precisely, it determines the ion-

ization probability for each half-cycle by controlling their amplitude. The analogue

to a regular double-slit would be that the number and the width of the slits can be

controlled.

The phase-stabilized few-cycle laser pulses are focused onto a gas jet situated

in an electrically and magnetically shielded vacuum apparatus. Perpendicular to the

laser beam, two electron detectors are installed to the left and to the right of the laser

focus, thus forming a twin (or stereo) time-of-ﬂight (TOF) spectrometer. The setup

is depicted in Fig. 11.10. The laser polarization is linear and parallel to the axis of

the TOF spectrometers.

11 Double-Slit Experiments in the Time Domain 321

Fig. 11.10 (Color online) Stereo-photoelectron spectrometer. Two opposing electrically and mag-

netically shielded time-of-ﬂight spectrometers are mounted in an ultrahigh vacuum apparatus.

Argon atoms fed in through a nozzle from the top are ionized in the focus of a few-cycle laser beam.

The laser polarization is linear and parallel to the ﬂight tubes. Note that the laser ﬁeld changes

sign while propagating through the focus. Slits with a width of 250μm are used to discriminate

electrons created outside the laser focus region chosen. The slits can be moved from outside the

vacuum system. A photodiode and micro-channel plates (shown in gray (red online) and black)

detect the laser pulses and photoelectrons, respectively. A pair of glass wedges is used to optimize

dispersion and to change the absolute phase (from Paulus et al. [54])

Figure 11.11 displays measured electron spectra. In Fig. 11.11(a) the spectra

recorded at the left and the right detectors are shown for ±cosine-like and ±sine-

like pulses. A problem in presenting such spectra is that they quickly roll off with

increasing electron energy. This roll-off was eliminated by dividing the spectra by

the average of all spectra over the pulse’s phase. Clear interference fringes with

varying visibility are observed as expected from the discussion above. The highest

visibility is observed for −sine-like pulses in the positive (“right”) direction. For

the same pulses, the visibility is very low in the opposite direction. Changing the

phase by π interchanges the role of left and right as expected. The most straight-

forward explanation is to assume that, for −sine-like pulses, there are two slits and

no which-way information for the positive direction and just one slit and (almost)

complete which-way information in the negative direction. The fact that the inter-

ference pattern does not entirely disappear is caused by the pulse duration, which is

still slightly too long to create a perfect single slit.

Under the conditions of this experiment, each argon atom emits at most one

electron, whose various options of how to reach a given ﬁnal state lead to interfer-

ence. For sine-like pulses, these options correspond to a double-slit in time in one

direction and to a single-slit in the other and are created for each atom separately

by the few-cycle laser pulse. Therefore, even though there is more than one argon

atom in the laser focus, the experiment operates under single-electron conditions.

On the scale of the electron’s de Broglie wavelength, other atoms are far away and,

322 G.G. Paulus and D. Bauer

Fig. 11.11 (Color online) Photoelectron spectra of argon measured with 6-fs laser pulses at an

intensity of 1 ×10

W/cm

as a function of the absolute phase. Panel (a) displays the spectra for

±sine- and ±cosine-like laser ﬁelds. The gray (red online) curves are spectra recorded with the left

detector (negative direction), while the black curves relate to the positive direction. For ϕ = π/2

the fringes exhibit maximum visibility for electron emission to the right, while in the opposite

direction minimum fringe visibility is observed. In addition, the fringe positions are shifted. Panel

(b) displays the entire measurement where the fringe visibility is coded in false colors (online).

The fringe positions vary as the phase ϕ of the pulse is changed. This causes the wave-like bending

of the stripes in these ﬁgures. Both panels, in principle, show the same information because a

phase shift of π mirrors the pulse ﬁeld in space and thus reverses the role of positive and negative

direction. However, the data shown were recorded simultaneously but independently where the

phase φ was varied between 0 and 2π (from Lindner et al. [41])

11 Double-Slit Experiments in the Time Domain 323

moreover, randomly distributed. This is in contrast to the double-slit in space where

the beam has to be sufﬁciently dilute to ensure a one-electron measurement.

The fringe pattern exhibits an envelope. From Fig. 11.11 a width of this envelope of

about four fringes is inferred. Just as for a double-slit experiment, the width of this

envelope can be associated with the width of the slits. However, Fig. 11.9 suggests

that what is seen here is not the width of the slit. Rather, each slit can be resolved

into a pair of slits whose separation is inversely proportional to the width of the

envelope.

It might be worth reﬂecting that this version of the double-slit experiment has

quite some remarkable features. Besides the fact that it is realized in the time–energy

domain, there are slits being windows in time of attosecond duration. These “slits”

can be opened or closed by changing the temporal evolution of the ﬁeld of a few-

cycle laser pulse by controlling the absolute phase. At any given time there is only a

single electron in the double-slit arrangement. In addition, the presence and absence

of interference may be observed for the same electron at the same time by choosing

the direction of observation.

11.3.3 More Slits

Fringes in photoelectron spectra recorded under strong-ﬁeld conditions are not new

at all. In fact, strong-ﬁeld photoionization is also known as above-threshold ioniza-

tion (ATI), and the signature of ATI spectra is a series of peaks separated by the

photon energy. The effect has ﬁrst been observed in 1979 [1]. The conventional

interpretation of the effect is an extension of multiphoton ionization. Accordingly,

the energy position E

of the sth ATI peaks is predicted to be given by a generalized

Einstein law E

= (n +s)ω −|E

|−U

. Here, n is the minimum number of photons

necessary to subdue the ionization threshold which is raised by U

due to light

shift [15]. However, it is hardly possible to generalize this picture to few-cycle laser

pulses. For these the notion of ponderomotive level shifts is questionable. Moreover,

the varying contrast of the fringes as well as the fringe shifts in dependence of the

absolute phase are difﬁcult to explain.

It is much easier to extend diffraction in the time domain from few-cycle to longer

pulses. In place of the double-slit one would then speak of a grating in time, as dis-

cussed in Sect. 11.2.3. The respective temporal windows for photoelectron emission

in a given direction are separated by the optical period T . Therefore, the separation

of the ATI peaks by ω is an immediate consequence. With slightly more effort,

the ATI peak shift due to the ponderomotive shift of the ionization potential can be

explained. For this, the phases of the trajectories given by S/ have to be computed.

Hereby S =

Ldt is the classical action of the trajectory calculated by integrating

the Lagrange function L along the trajectory. For constructive interference, trajec-

tories launched during subsequent optical cycles must have a phase difference of

an integer multiple of 2π. This is the case when the generalized Einstein law is

fulﬁlled.

324 G.G. Paulus and D. Bauer

The times t

at which the temporal slits are open for a given photoelectron

momentum are determined by Eq. (11.34). However, t

determines not only the drift

momentum of the photoelectrons via Eq. (11.34) but also the probability with which

such an electron is ionized in the ﬁrst place. For tunneling the ionization probability

depends exponentially on −1/|E(t

)|.

Ionization probability and drift momentum

can be disentangled by inducing ionization with attosecond XUV pulses. Attosec-

ond pulses are much shorter than an optical period in the visible (VIS) or near-

infrared (NIR) spectral range. If used for ionizing atoms exposed to an additional

optical ﬁeld in the VIS or NIR, they can promote an electron at a deﬁned phase

ωt

into the optical ﬁeld. At that instant, the kinetic energy of that photoelectron is

E(t

) = ω

XUV

−|E

|. This is different from the situation discussed in Sect. 11.3.2

where we had E(t

) = 0. Nevertheless, also in this situation it is possible to calculate

the photoelectron’s drift momentum using Eq. (11.33):

drift

=±

2mE(t

) + eA(t

) . (11.36)

If this process is repeated in subsequent optical cycles, interference of the electron

wave packets created in each cycle can be expected. Due to the periodicity with T ,

the spacing of the fringes must be equal to the photon energy ω.

It might seem to be exceedingly difﬁcult to implement the idea just described

experimentally because one needs to synchronize the XUV attosecond pulses with

the optical ﬁeld with attosecond precision, which corresponds to sub-micron spa-

tial precision. Fortunately, the prevalent method for generation of attosecond pulses

offers an intrinsic solution to this problem. XUV attosecond pulses can be gen-

erated by high-harmonic generation (HHG) in gases. HHG was discovered in the

1980s [24]: An intense NIR femtosecond pulse is focused on rare gases at a density

of 10

–10

−3

. Depending on intensity and other experimental parameters, all

odd harmonics up to a few hundred orders may be produced. The mechanism of

this process is not very relevant to the present topic. Brieﬂy, it can be explained

by a simple extension of the classical model of strong-ﬁeld ionization discussed in

Sect. 11.3.2. Electron wave packets leaving the atom near the peak of the NIR ﬁeld

may be driven back to the ion core. Depending on the phase at which the wave

packet was launched, the kinetic energy of the electrons upon return may be as high

as 3.17U

. In case the electron recombines about 3/4 T later, short-wavelength radi-

ation with ω

XUV

< 3.17U

+|E

| is created [40, 17] and XUV attosecond pulses

are emitted collinear to the NIR femtosecond laser pulses in each half-cycle. For

reviews on HHG, see [61], and for one on re-colliding electrons in general [7]. The

The tacit assumption is that tunneling is an instantaneous process. This assumption may be

wrong. In that case, one would need to distinguish between the time t

when the electron enters

the tunnel and the time t

when the electron leaves the tunnel and the trajectory begins to evolve

in the laser ﬁeld. The ﬁeld strength relevant for calculating the ionization probability would be

E(t

), while the drift momentum would be given by eA(t

). Obviously, this complication would

not alter the mechanisms discussed so far. The only consequence would be a change of the shape

of the spectrum.

11 Double-Slit Experiments in the Time Domain 325

Fig. 11.12 (Color online) Attosecond pulse trains synchronized to an NIR laser ﬁeld are used to

create electron wave packets by ionizing rare gas atoms. Successive electron wave packets have a

delay of one optical cycle. The ionization probability can be controlled by shifting the attosecond

pulses with respect to the NIR laser ﬁeld. The electron wave packets interfere and create a diffrac-

tion pattern. Panel (c) displays experimental results obtained for XUV–NIR delays of 0, T/4, T/2,

and 3T/4 (from Mauritsson et al. [47])

re-collision process explains why the train of attosecond pulses are synchronized

with the optical ﬁeld by which they are created.

In order to choose t

, one has to delay the attosecond pulses with respect to the

optical ﬁeld. This is certainly technically demanding, but possible. The remaining

problem is that the attosecond pulses are created every half-cycle, i.e., for different

signs of the ﬁeld which results in a π phase shift for consecutive attosecond pulses.

Also this problem can be solved by using a two-color femtosecond laser pulse [46].

The idea of realizing a multiple slit diffraction experiment in this way has been

realized recently [47], see Fig. 11.12. Admittedly, the purpose of that experiment

reaches beyond the present discussion by addressing potential applications outside

the scope of this chapter.

11.4 Strong-Field Approximation and Interfering

Quantum Trajectories

In our theoretical analysis of time slits in Sect. 11.2, we tacitly assumed that wave

packets somehow appear in the continuum and evolve freely afterward. Such a

dynamics does not correspond to a unitary time evolution. Instead, the wave function

Muga G. Time in Quantum Mechanics - Vol. 2

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