CUADRO 2/4 PRESUPUESTO

Wave packet dynamics in atoms and molecules

structure o f the radial distribution functions, presented as insets in figures 2.1 and 2.2. The vastly differing probability distributions over the electronic orbit, in the case o f electron wave packets, and the vibrational coordinate, in the case o f vibrational wave packets are important factors in determining the initial wave packet probability distribution. - 0.75 r / pm - 0.25 0 20 40 60 80 100 120 140 Time / ps

Figure 2.2 Main figure: recurrence spectrum o f a Rydberg electron w ave packet in the hydrogen atom, excited around n = 40 with a 1 ps bandwidth limited laser pulse. In a similar way to the vibrational wave packet, the periodicity o f the w ave packet is fully visible at the full revival, as well as second, third and fourth order partial revivals. In contrast to the vibrational wave packet, after just one classical oscillation, the electronic wave packet has started to disperse and after just two classical oscillations, a 5th order partial revival is visible. The inset shows the radial distribution function o f the o f the electronic wave packet immediately after its creation. Unlike the radial distribution function o f the vibrational wave packet which is spread out over the vibrational coordinate, it is localised at the inner turning point. Once created, the w ave packet travels to the outer turning point o f the Coulombic potential, where it w ill reach its maximum potential energy before being drawn back to the core.

To further illustrate the dynamics o f a Rydberg wave packet, figure 2.3 shows the probability density o f a radial wave packet

(2.29)

plotted as a function o f time t and radial coordinate r, where an and con are the amplitudes and angular frequencies o f the Rydberg states in the superposition.

70 60 50 40 C/5 a. <D B 20 10 u 0.05 0.10 0.15 0.20 r / pm

Figure 2.3 Time dependent radial distribution function o f a radial electron wave packet excited around n = 40 in the hydrogen atom, with a bandwidth limited 1 ps pulse. The dark sections indicate a high probability amplitude. At short times, the classical evolution o f the wave packet is apparent, as it is seen to oscillate between the core and the outer turning point at the classical period. At longer times, quantum mechanical behaviour manifests itself as the wave packet spreads and begins to exhibit non-classical behaviour.

Wave packet dynamics in atoms and molecules

The excitation parameters are the same as in the previous examples i.e. the wave packet is excited from the ground 1 s state to the 40p state o f atomic hydrogen, with a one picosecond, bandwidth limited laser pulse.

The wave packet is formed initially at the inner turning point, where it is localised in the radial coordinate. It then travels rapidly away from the positively charged nucleus towards the outer turning point o f the Coulomb potential. At this point, the kinetic energy o f the wave packet decreases and it becomes fairly localised at the edge o f the potential. The wave packet then travels rapidly back to the ionic core with a total round trip o f approximately 10 ps. It is apparent that after just one oscillation, the wave packet probability distribution has begun to broaden, a result of the anharmonicity o f the Coulomb potential. After approximately 20 ps the wave packet has dispersed over the whole o f the radial coordinate and the partial revivals begin to become apparent.

2 .3

W a v e p a c k e t i n t e r f e r e n c e

2 . 3 .1 Mu l t i p l e p u l s e e x c i t a t i o n

The above description o f wave packet excitation allows a straightforward extension to show the effect o f multiple pulses on the final state probability amplitudes. Assuming a sequence of k identical laser pulses, launched at times tk after the first pulse, the total electric field for the excitation is extended from (2.1) to:

E(t) = E 0 cos(m t)f(t)+ £ E0 cos[w(l - t k )}/■(< - tk) (2.30)

k

and the corresponding population amplitudes after the pulse sequence is

a_{n( 0 = - y n „ . ? H 1+ E exp ( 'V » )exp (imh )} (2.31)

For each individual excited state n, the term in square brackets in (2.31) oscillates rapidly at the angular frequency con -co m, which is the same frequency as the angular component o f the electric field by which that specific state was excited. From this argument, it is apparent how the slightly differing angular frequencies o f the various states in a wave packet superposition will oscillate in and out o f phase with one another, causing constructive and destructive interference in wave packet amplitude. These interference effects within the wave packet amplitude are discussed further in the following section.

2 .2 .2 Ra m s e y f r in g e s

Both the experimental and theoretical work presented in this thesis rely on the ability to monitor the excited state population, with the ultimate aim o f gaining complete control over the final state composition o f the wave packet, and the resulting dynamics. When considering wave packet interference, the last term in (2.31) is the most important. The square o f this term is proportional to the excited state population and oscillates between 0 and 4. The term in the first exponential in (2.31) is a rapidly oscillating term (cotk ), which is the product o f the laser frequency and the delay between the pulses. This term is associated with the phase difference between the wave packets. The second term (A ntk), where A n = con -co m -co is the ‘detuning’ angular frequency, represents a much slower oscillation responsible for the period of wave packet evolution and is associated with the overlap o f the wave packets. See reference [12] for a detailed derivation of these terms. This is represented graphically in figure 2.4, which is a plot o f the total excited state population , versus the delay time between the two pulses. In this example, taken from [13], a sequence o f two identical bandwidth limited pulses excites a superposition o f p Rydberg states from the ground Is level in the hydrogen atom. The pulses are one picosecond long ( r = 1 ps) and have a central frequency corresponding to excitation o f the n = 40 level in hydrogen. The Ramsey fringes

Wave packet dynamics in atoms and molecules

depicted in figure 2.4 have been calculated with a scaled down optical frequency, so that both the oscillations at the optical frequency and the wave packet envelope are clear. In this particular example, the optical frequency was scaled by a factor o f 2000. In reality, there are around 3000 optical cycles per picosecond. The overall envelope in figure 2.4 shows the oscillation o f the radial wave packet. At I = 0 , excitation from the Is state creates a wave packet localised in the core region. The wave packet travels up the Coulombic potential and reaches its maximum potential energy at the outer turning point, at ~5 ps, before being drawn back to the core with high kinetic energy. The first complete evolution o f the wave packet, Tcl, lasts for ~10ps. The energy difference between adjacent Rydberg states scales as « -3, therefore the higher energy electronic states oscillate more rapidly than the lower energy states and the wave packet distribution spreads, as can be seen in the broadening o f the wave packet envelope.

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Figure 2.4 A plot o f the total population o f hydrogen atom Rydberg states after excitation by a sequence o f two identical, one picosecond pulses. The w ave packet is created at t = 0 , (a) and reaches the outer turning point o f its motion at approximately 5 ps, (b). The first evolution o f the wave packet, Td , takes approximately 10 ps, (c). In reality, there are many more oscillations at the optical frequency than depicted - to make the figure clearer, the optical frequency has been scaled down by a factor o f 2000.

The rapidly oscillating terms can be explained in terms o f the delay, tk, between the two pulses. For t = 0 , the pulses overlap and the excited state distribution will be at a maximum. If the second pulse is delayed by a time t = tk + , where t^ « r p and

tf =(f)/ CO corresponds to an optical phase shift <f> with reference to the optical frequency

o) , there will be interference between the two light pulses as they are still overlapped

spatially. First, consider the case when $ - n , which corresponds to a delay o f half an oscillation o f the optical field. This delay causes the two pulses to be n out o f phase with one another, and as the atom is not exposed to any field at this point, there will be no population in the excited state. Conversely if </) = 2 n , there will be constructive interference between the light pulses resulting in maximum excitation o f the atom. Therefore in this region close to t - 0 , point (a) on figure 2.4, constructive and destructive interference between the two light pulses determine the final excited state population and correspondingly peaks and dips, or “fringes” are observed in the total excited state population, , at the optical frequency.

Now consider the case where the two optical pulses are temporally separated by a time t » Tcl/ 2 , where Tcl = In/co - the classical oscillation period o f the system.. The first pulse creates a wave packet which travels to the outer turning point, at which point the second wave packet is excited. In the current example, Tcl/ 2 » 5 ps. The two wave packets do not overlap spatially and therefore cannot interfere with one another; in this case the total Rydberg population around this time is simply the incoherent sum o f the contributions from each pulse. This is illustrated in region (b) in figure 2.4.

The final scenario to consider is when the two pulses are temporally separated by one orbit period, t = Tcl « 10 ps, (c) in figure 2.4. This can be explained by likening the behaviour o f the atom to that o f an interferometer: The first pulse creates a wave packet. This wave packet travels to the outer turning point and returns to the core, at which point a second pulse launches a second wave packet in the core region. These two wave packets interfere constructively or destructively depending on the optical phase difference between the two excitation pathways. Consequently, just as at t = 0, interference fringes are observed in the final Rydberg state population at the optical

Wave packet dynamics in atoms and molecules

frequency. These fast oscillations are referred to as Ramsey fringes [12]: the first pulse creates some excited state population, whilst the phase o f the second delayed pulse determines whether this excited state population in enhanced or diminished.

An alternative, and perhaps simpler, perspective on this type o f wave packet interference is obtained by Fourier transforming the two pulses. If we consider one pulse with a Gaussian profile, the Fourier transform would yield a Gaussian frequency spectrum. The Fourier transform o f two Gaussian pulses, separated by a delay o f the order o f a wave packet oscillation, gives the same frequency spectrum, but with a modulation superimposed on it. These modulations shift as a function o f the delay between the pulses so if the peaks overlap in frequency with the eigenfrequencies o f the states in the excitation, those particular states will become populated. If the delay between the two pulses is equal to Tcl, the spacing o f the peaks in the frequency domain spectrum correspond to the spacing between the states in the original excitation, so that the peaks coincide with the excitation frequencies o f the Rydberg states. If the delay between the pulses is equal to Tcl +tt , this has the effect o f flipping the phase o f the

second optical pulse by n and hence, phase shifting the modulation in the frequency spectrum by n . In this case, the dips in the frequency spectrum correspond to the excitation frequencies o f the Rydberg states and no Rydberg population is excited. The spectral profiles obtained by Fourier transforming a pulse pair, with delays corresponding to (a) 0, (c) Tcl , (d) Tcl + n are illustrated in figure 2.5. In this example the excitation is made from the 40/? <- Is transition in the Hydrogen atom by a pair o f identical, bandwidth limited, one picosecond optical pulses. For reference, the positions o f the Hydrogen Rydberg levels are also presented along the top o f the plot.

An interesting scenario occurs when the pulses are separated by Tcl/ 2 , illustrated in figure 2.5(b). In this case the peaks in the frequency spectrum coincide with the excitation frequencies o f alternate states in the superposition, enabling the creation o f a wave packet that consists o f only odd or even states. Even more interesting is the situation that arises then the pulses are separated by 3 7 ^ /2 , illustrated in figure 2.5(e). This situation is similar to the case where the pulses are separated by Tcl! 2 , in

that alternate Rydberg states are excited, but the frequency profile is in fact very different. 35 36 37 38 39 40 4 1 4 2 44 46 48 n •40 -30 -20 -10 0 10 20 30 40 (a) (b) (c) (d) (e) Wavenumber spread / cm'

Figure 2.5 A plot o f the spectral profiles obtained from Fourier transforming o f pair o f bandwidth lim ited, optical pulses separated by (a) 0, (b) Tc l/ 2 , (c) Tch (d) Td + 7t and (e) 2>Td / 2 . The pulses are one picosecond long and have a central frequency corresponding to the 40/? < - Is transition in atomic Hydrogen. The frequency spectra are all plotted on the same intensity scale. The Hydrogen Rydberg states are plotted at the top o f the figure for reference.

As can be seen in figure 2.5(b), Fourier transforming a pair o f pulses separated by Td l 2 results in a frequency spectrum where every peak and dip occurs at a Rydberg state frequency. In contrast, the Fourier transform o f a pair o f pulses separated by 3 7 ^ /2 results in a frequency spectrum where every third peak and every third dip corresponds to a Rydberg state frequency. This has implications for the coherent control of wave packets composed o f interleaved series, as the peaks and dips can be made to coincide with the frequencies o f specific series. This type o f control is employed in the

Wave packet dynamics in atoms and molecules

experiments discussed in chapter 5, where a pair o f bandwidth limited laser pulses with carefully designed delays are used to control the angular momentum composition o f an electron wave packet in atomic sodium, by modifying the population o f s and d Rydberg states.

2 .3

C

This chapter has provided an introduction to the formation and dynamics o f atomic and vibrational wave packets. Through the optical Ramsey method and the corresponding frequency domain analysis, a description o f the phase evolution o f the wave packet is presented. Knowledge o f the evolution o f wave packets has allowed the formation of intuitive coherent control schemes and provided a basis for the theory and experiments presented in chapters 4 and 5.

2 .4

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1. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics. Wiley, 1977. 2. B. H. Bransden and C. J. Joachain, Physics o f Atoms and Molecules, 2nd ed.

Prentice Hall, 2003.

3. X. Chen and J. A. Yeazell, Phys. Rev. A 56 (3), 2316 (1997). 4. R. W. Robinett, Phys. Rep. 392, 1 (2004).

5. G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34 (2), 34 (1986).

6. J. Wals, H. H. Fielding, and H. B. V. van den Heuvell, Physica Scripta T58, 62 (1995).

7. J. Parker and C. R. Stroud, Phys. Rev. Lett. 56 (7), 716 (1986). 8. I. S. Averbukh and N. F. Perelman, Phys. Lett. A 139 (9), 449 (1989). 9. J. H. Hanney and M. V. Berry, Physica D 1, 267 (1980).

10. I. S. Averbukh and N. F. Perelman, Sov. Phys. Usp. 34 (7), 572 (1991). 11. G. Alber and P. Zoller, Phys. Reps. 199 (5), 231 (1991).

12. L. D. Noordam, D. I. Duncan, and T. F. Gallagher, Phys. Rev. A 45 (7), 4734 (1992).

Experimental techniques fo r the creation, observation and control o f electronic and vibrational wave packets

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