PRL 110, 203002 (2013)
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PHYSICAL REVIEW LETTERS
Observing Rydberg Atoms to Survive Intense Laser Fields U. Eichmann,1,2,* A. Saenz,3 S. Eilzer,1 T. Nubbemeyer,1 and W. Sandner1,2 1
Max-Born Institut, D-12489 Berlin, Germany Institut fu¨r Optik und Atomare Physik, Technische Universita¨t Berlin, D-10623 Berlin, Germany 3 AG Moderne Optik, Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, D-12489 Berlin, Germany (Received 30 November 2012; published 13 May 2013) 2
The idea of atoms defying ionization in ultrastrong laser fields has fascinated physicists for the last three decades. In contrast to extensive theoretical work on atoms stabilized in strong fields only few experiments limited to intermediate intensities have been performed. In this work we show exceptional stability of Rydberg atoms in strong laser fields extending the range of observation to much higher intensities. Corresponding field amplitudes of more than 1 GV=cm exceed the thresholds for static field ionization by more than 6 orders of magnitude. Most importantly, however, is our finding that a surviving atom is tagged with a measure of the laser intensity it has interacted with. Reading out this information removes uncertainty about whether the surviving atom has really seen the high intensity. The experimental results allow for an extension of the investigations on the stabilization and interaction of a quasifree electron with a strong field into the relativistic regime. DOI: 10.1103/PhysRevLett.110.203002
PACS numbers: 32.80.Ee, 32.80.Rm
Survival of atoms in strong laser fields where already single-photon absorption is in principle sufficient to ionize was first predicted about a quarter century ago [1–3]. Since then the subject has been heavily discussed theoretically for the last two decades [4,5] with newly increased interest [6]. Different stabilization mechanisms [7] such as interference stabilization at lower intensities [8,9], adiabatic or dynamic stabilization [2,10–12], or strongly reduced ionization rates due to initial high angular momentum [13] have been theoretically suggested. These mechanisms have been, however, only partially confirmed in experiments using loosely bound Rydberg atoms exposed to relatively moderate laser intensities [14–17]. Moreover, ionization of Rydberg atoms in strong fields is expected to be neither following the picture of tunneling nor of a multiphoton process otherwise dealt with in strong field physics. The very successful semiclassical Keldysh theory for strongfield ionization [18] and with it, the Keldysh parameter, typically used to distinguish between a multiphoton or tunneling ionization process, are not applicable. Thus it is worthwhile to study the Rydberg strong-field ionization scenario in more detail at even higher intensities above 1015 W cm2 with special emphasis on the open question of whether proposed stabilization mechanisms remain valid at higher intensities. One of the key problems is, however, how to ensure that the atom has really seen the peak intensity and did not survive in the ‘‘shadow’’ of the laser light, the temporal or spatial outskirts. In this Letter we report experiments in which we expose Rydberg atoms to intense laser fields above 1015 W cm2 . Using a direct detection technique for surviving atoms we demonstrate experimentally the exceptional stability of Rydberg atoms. They withstand high laser intensities, where corresponding field amplitudes of more than 0031-9007=13=110(20)=203002(5)
1 GV=cm exceed the thresholds for static-field ionization by more than 6 orders of magnitude. Most strikingly, we are able to unravel that the intensity a surviving Rydberg atom has interacted with is encoded in the kinematics of the atom. To see how we can extract the intensity information from the kinematics we recall that strong-field excitation of He with linearly polarized light results in Rydberg states. These form a wave packet with principal quantum numbers centered around n 8 and with angular momentum states l distributed over a large range, essentially l < 10 [19]. Moreover, the excited atoms are accelerated in the gradient of the focused strong short-pulse laser field [20]. The deflection of atoms due to this acceleration can be measured using the experimental setup schematically shown in Fig. 1, also described elsewhere [19,20] and in the Supplemental Material [21]. In the experiments, we first produce the Rydberg wave packet with a linearly polarized laser pulse. After a variable time delay we apply a second elliptically polarized laser field. Sufficient elliptical polarization (ellipticity ¼ 0:66) ensures that no further excitation from the ground state takes place [19]. Consequently, any Rydberg atom which interacts with the second laser pulse and survives is additionally accelerated, provided it is located in a nonzero intensity gradient. Hence, the magnitude of acceleration, or more specific, the deflection of Rydberg atoms, verifies their interaction with the strong field. In Fig. 2(a) we show the Rydberg atom distribution on the detector after the first laser field with an intensity of 2:7 1015 W cm2 has interacted with the atomic beam. Without acceleration one would expect a pencil-shaped distribution along the laser beam propagation in z direction, with the focus at z ¼ 0. Clearly visible is the deflection of neutral excited helium atoms that occurs along the
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FIG. 1 (color online). Sketch of our experimental setup. The Mach-Zehnder interferometer [with beam splitter (BS) and =4 wave plate] provides two spatially well overlapping short laser pulses with variable time delay and linear and elliptical polarization, which are focused into the atomic beam. Surviving Rydberg atoms (either directly or their decay into a metastable level) are detected downstream with a position-sensitive microchannel plate.
radial intensity gradient of the focused laser beam. In the experiment we measure only the component in the x direction indicated in Fig. 2. The distribution of surviving Rydberg atoms after interaction with the second laser pulse is presented in Fig. 2(b). It indicates that the number of surviving Rydberg atoms is significant and that an additional acceleration occurs. This is particularly true for atoms stemming from the immediate laser focus at z ¼ 0. Taking the difference of the two distributions in Figs. 2(a) and 2(b) we can clearly identify additionally accelerated atoms, Fig. 2(c). We have measured excited He-atom distributions for three different intensities of the second laser pulse. To evaluate the fraction of Rydberg atoms surviving the strong laser field we exploit the position-sensitive detection of atoms and filter out only those atoms stemming solely from the center of the focus at z ¼ 0. The cuts are shown in Fig. 3. For the cases shown in Figs. 3(a) and 3(b), at intensities of the second laser of I2 ¼ 1:9 1015 Wcm2 and 3:0 1015 W cm2 , we find that the number of detected Rydberg atoms drops only by 7% and 9%, respectively, with regard to the number produced by the first laser alone. For the maximum laser intensity I2 ¼ 3:8 1015 W cm2 , shown in Fig. 3(c), the population drops by 15%. Given the high laser intensities applied to the atom, the loosely bound Rydberg electron turns out to be extraordinarily stable against ionization. We compare the cuts in Fig. 3 with model calculations for the atom distribution on the detector. We assume that
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FIG. 2 (color online). Distributions of excited He atoms on the position-sensitive detector. The number of impinging excited atoms on the detector is color coded. (a) Excited He-atom distribution on the detector for a single laser pulse with an intensity of I1 ¼ 2:7 1015 W cm2 . (b) Surviving excited He-atom distribution on the detector after applying a second elliptically polarized laser with an intensity I2 ¼ 3:8 1015 W cm2 and a time delay of about 500 fs. (c) The difference of the He-atom distributions in (a) and (b).
the acceleration of atoms is based on a quasifree electron quivering in the focused laser beam, which is nevertheless still bound to the ionic core after the laser pulse is over [20]. We use Monte Carlo techniques to simulate the distribution of excited He atoms on the detector. We first calculate the spatially resolved distribution of excited atoms in the laser focus after the first linearly polarized laser pulse. We take the spatial and temporal intensity distribution in the focused laser beam 2 2 2 2 to be Gaussian, Iðr; z; tÞ ¼ I0 ½w0 =wðzÞ2 e2r =wðzÞ et = . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ ¼ w0 1 þ z2 =z2R is the beam size, w0 is the beam waist, zR is the Rayleigh length, is the pulse duration. The yield of excited atoms in a small volume is assumed to be proportional to the tunnel-ionization yield (about 10%), which in turn is proportional to the local field strength F ¼ pffiffiffi I . As has been found by numerical calculations [19], the relative excitation yield can be sufficiently accurately calculated using a semiclassical tunnel-ionization formula [22,23]. The second elliptically polarized laser beam does not excite ground state atoms. Using the excitation probability distribution, we calculate (typically for 105 atoms) the momentum transfer to obtain a detailed momentum distribution of atoms. Thereby we assume for all excited Rydberg atoms the same dynamic polarizability / 1=!2 , which is based on a quasifree electron. ! is the photon angular frequency. To calculate the momentum transfer on an excited atom due to the first and second
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FIG. 3 (color online). Cuts through the experimental Rydberg density distribution at z ¼ 0. Black line (narrow peak, noisy curve): Rydberg atoms excited and accelerated with (a) I1 ¼ 1:5 1015 W cm2 , (b) I1 ¼ 2:3 1015 W cm2 , and (c) I1 ¼ 2:7 1015 W cm2 . Red line (broad peak, noisy curve): Rydberg atoms additionally accelerated with the second laser (a) I2 ¼ 1:9 1015 W cm2 , (b) I2 ¼ 3:0 1015 W cm2 , and (c) I2 ¼ 3:8 1015 W cm2 . The solid green and blue curves [(narrow peak, smooth curve) and (broad peak, smooth curve), respectively] in (a) and (b) are from model calculations. In (c) data points connected by interpolated curves are calculated using only certain focal regions: squares (blue) r < w0 =2 and circles (cyan) w0 =7 < r < w0 =2.
laser pulse we suppose in a very good approximation that the intensity gradient in the focused laser beam is only radially oriented. Following Ref. [20] the momentum transfer pðz; rÞ is then given by pðz; rÞ ¼ pffiffiffiffi ½r =!2 wðzÞ2 Iðr; zÞS, where we time integrated over the complete laser pulse duration. This is possible since the atom does not move during the laser pulse. This formula holds for the momentum transfer in both laser pulses. S is a factor between 0 and 1 and the product S gives the effective pulse length for acceleration. All sets of measurements shown in Fig. 3 were reproduced with a consistent set of experimental parameters. For each calculated momentum transfer to an atom we chose the factor S from a Gaussian distribution at S ¼ 0:5 with a width of 0.2. This has been found to give overall the best agreement with the
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experimental data. Furthermore, we fitted by eye only the total ion yield for the cuts at z ¼ 0 for the single laser pulse data. Because of the good agreement of our measurements and the model calculations, we reveal the connection between momentum transfer and field strength or intensity more closely. At the focal plane at z ¼ 0 the momentum transfer p to the Rydberg atoms is proportional to the intensity gradient in the focus, p / r expð2r2 =w20 Þ. Highest momentum transfer takes place for those atoms located at r ¼ w0 =2, which corresponds to an intensity of 0:6 I0 . Since the inverse function of the momentum transfer is not bijective, atoms located at positions r > w0 =2 might gain the same momentum transfer as atoms located in the high-intensity region for r < w0 =2. Consequently, to provide a unique relation between laser intensity and momentum transfer we have to make sure that excited atoms produced in the first laser beam are located within w0 =2. That this is the case is shown in Fig. 3, where we repeated the calculation considering only the spatial region r < w0 =2. The two curves differ only by a few percent indicating that all detected atoms stem from the region r < w0 =2. Moreover, if we confine the calculation to the region w0 =7 < r < w0 =2 we clearly identify those atoms that survived intensities corresponding to 60% to 95% of the maximum intensity and are accelerated accordingly. Thus, although the acceleration for the highest intensities is zero, the method is capable of indicating the applied intensity in a range of 60% to 95% of the maximum intensity I0 . Finally we discuss the high survival probability of Rydberg atoms in strong laser fields. One might speculate whether the coherent Rydberg wave-packet dynamics along the laser polarization axis initiated after the first laser pulse might enhance the survival probability. We solve the time-dependent Schro¨dinger equation (TDSE) as described elsewhere [19] to obtain detailed information on the created Rydberg wave packet and on its ionization yield. We use alternatively a single-active-electron (SAE) approximation with a model potential introduced in Refs. [24,25] and a fully correlated calculation (see Supplemental Material [21]). Because of present limitations of the code, the TDSE is solved only for two linearly polarized laser pulses. The gradient of the laser field which might lead to ionization [26] is not included in the calculation. To decide whether the coherent wave-packet motion influences the survival rate of the Rydberg states, we perform the SAE calculations either by keeping the phases of the individual states after the first laser pulse or we set them to zero. Both calculations show no significant differences indicating that collective coherent processes do not play a role. We check this also experimentally by varying the time delay between the two lasers. Beside the region where the two laser pulses overlap and interference effects in intensity as well as in the polarization of the beam are dominant,
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FIG. 4 (color online). Calculated excitation and survival probability of Rydberg states with quantum number n. (a) Calculations are performed within the SAE approximation using basis states up to n ¼ 40 and l < 21. The pulse duration comprises 40 laser cycles with a cosine-squared-shaped pulse envelope. Black squares: Excitation with the first laser alone with an intensity of 2:7 1015 W cm2 ; surviving Rydberg states with quantum number n after the second laser pulse is applied with an intensity of 1:9 1015 W cm2 (red dots); 2:5 1015 W cm2 (green triangles). Total survival is 68%. The increase in the distribution towards n ¼ 40 is caused by the limited basis set. (b) Calculations are performed using a fully correlated two-electron basis set with states up to n ¼ 30 and l < 21. The pulse duration comprises 10 laser cycles with a cosine-squared-shaped pulse envelope. Black squares: Excitation with the first laser alone with an intensity of 1:5 1015 W cm2 . Red dots: surviving Rydberg states after the second laser pulse with an intensity of 1:3 1015 W cm2 . Total survival is 47%.
no time-dependent variations at larger time delays are detected. In Fig. 4(a) we show the n distribution produced after interaction with the first laser pulse and after the second laser pulse. Summing over all n states we find ionization of 32% of the Rydberg atoms. This is in good accord with the measured ionization of 15%. We note that the measured lower ionization could result from the elliptical polarization used in the experiment, instead of the linearly polarized one in the calculation. Nonlinear polarization might cause further reduction of close encounters of the electron with the core and thus reduced ionization rates.
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Inspecting the n-dependent survival (where we summed over all l states belonging to a fixed n quantum number) it is obvious from the calculations shown in Fig. 4(a) that states with n > 15 are hardly affected by the strong field and thus stable. A significant redistribution of n quantum numbers is not obvious. For this n region the pulse duration is much shorter than the Kepler orbit time k ¼ 2n3 . The very low ionization rates can be seen as an experimental confirmation of the stabilization of Rydberg states as proposed by Ref. [27] for low angular momentum states, where a field-independent ionization rate ¼ 1=2k has been derived. However, we find from our calculations that for higher l states no field independent rate is reached for intensities up to 1016 W cm2 . For states with 7 < n < 12 the laser pulse duration is comparable to the Kepler period. In this region resonancelike phenomena or redistributions are present so that no clear intensity dependence on the ionization of the n states exists. Interestingly, for states n < 7 a non-negligible survival rate is still visible. We have calculated systematically the ionization yield at a fixed laser intensity of 2 1015 W cm2 and at a fixed pulse duration for states with n ¼ 2 to n ¼ 20 with l from 2 to 10 in steps of 2 considering l < n. The pulse duration is larger than the Kepler orbit time for n < 7 and also the electron quiver amplitude of 76 a.u. is larger than the radial expectation values for n < 7. For these n quantum numbers we observe that the ionization yield never reaches 100%, even for the low l states, but it saturates or decreases slightly around the 80% ionization level. We attribute this to a dynamical stabilization, where the field dynamically reduces the overlap with the core region. We note that this can be regarded somewhat similar as the case, where ground state atoms survive strong laser fields in excited states [20,26]. To test whether electron correlation influences the low ionization rates, we repeat the calculations using a fully correlated two-electron basis. The results of the full twoelectron calculations shown in Fig. 4(b) confirm qualitatively the findings from the SAE model, but also indicate some differences especially for the lower lying states. In conclusion we have demonstrated experimentally in an unprecedented way the high survival probability of Rydberg atoms in strong laser fields. Most importantly, detecting the momentum transfer of the focused laser field to the surviving atom allows for a measure of the acting local laser intensity. We thus have clear and unique proof that Rydberg atoms indeed survive a strong laser field. Calculations show that even at highest intensities loworder photon processes dominate the ionization behavior. Furthermore, given the high survival of Rydberg atoms in strong laser fields, coherent manipulation with short-pulse lasers as well as efficient cascaded acceleration is envisaged. We thank Frank Noack for technical assistance with the laser. We acknowledge fruitful discussions with H. Reiss,
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M. Ivanov, O. Smirnova, W. Becker, and M. Vrakking. This work was partially supported by EU Initial Training Network (ITN) CORINF and the Humboldt Center for Modern Optics.
*[email protected] [1] W. C. Henneberger, Phys. Rev. Lett. 21, 838 (1968). [2] M. Pont, N. R. Walet, M. Gavrila, and C. W. McCurdy, Phys. Rev. Lett. 61, 939 (1988). [3] J. H. Eberly and K. C. Kulander, Science 262, 1229 (1993). [4] M. Gavrila, J. Phys. B 35, R147 (2002). [5] A. M. Popov, O. V. Tikhonova, and E. A. Volkova, J. Phys. B 36, R125 (2003). [6] F. Morales, M. Richter, S. Patchkovskii, and O. Smirnova, Proc. Natl. Acad. Sci. U.S.A. 108, 16906 (2011). [7] A. Scrinzi, N. Elander, and B. Piraux, Phys. Rev. A 48, R2527 (1993). [8] M. V. Fedorov and A. Movsesian, J. Phys. B 21, L155 (1988). [9] O. V. Tikhonova, E. A. Volkova, A. M. Popov, and M. V. Fedorov, Phys. Rev. A 60, R749 (1999). [10] K. C. Kulander, K. J. Schafer, and J. L. Krause, Phys. Rev. Lett. 66, 2601 (1991). [11] K. Burnett, P. L. Knight, B. R. M. Piraux, and V. C. Reed, Phys. Rev. Lett. 66, 301 (1991). [12] H. R. Reiss, Laser Phys. 7, 543 (1997).
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[13] F. Benvenuto, G. Casati, and D. L. Shepelyansky, Phys. Rev. A 45, R7670 (1992). [14] R. R. Jones and P. H. Bucksbaum, Phys. Rev. Lett. 67, 3215 (1991). [15] L. D. Noordam, H. Stapelfeldt, D. I. Duncan, and T. F. Gallagher, Phys. Rev. Lett. 68, 1496 (1992). [16] J. H. Hoogenraad, R. B. Vrijen, and L. D. Noordam, Phys. Rev. A 50, 4133 (1994). [17] N. J. van Druten, R. C. Constantinescu, J. M. Schins, H. Nieuwenhuize, and H. G. Muller, Phys. Rev. A 55, 622 (1997). [18] L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965). [19] T. Nubbemeyer, K. Gorling, A. Saenz, U. Eichmann, and W. Sandner, Phys. Rev. Lett. 101, 233001 (2008). [20] U. Eichmann, T. Nubbemeyer, H. Rottke, and W. Sandner, Nature (London) 461, 1261 (2009). [21] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.203002 for details on the experimental setup and numerical calculations. [22] V. S. Popov, Phys. Usp. 47, 855 (2004). [23] M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. Phys. JETP 64, 1191 (1986). [24] Y. V. Vanne and A. Saenz, J. Mod. Opt. 55, 2665 (2008). [25] A. Lu¨hr, Y. V. Vanne, and A. Saenz, Phys. Rev. A 78, 042510 (2008). [26] E. Wells, I. Ben-Itzhak, and R. R. Jones, Phys. Rev. Lett. 93, 023001 (2004). [27] M. V. Fedorov and O. V. Tikhonova, Phys. Rev. A 58, 1322 (1998).
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PHYSICAL REVIEW LETTERS
Observing Rydberg Atoms to Survive Intense Laser Fields U. Eichmann,1,2,* A. Saenz,3 S. Eilzer,1 T. Nubbemeyer,1 and W. Sandner1,2 1
Max-Born Institut, D-12489 Berlin, Germany Institut fu¨r Optik und Atomare Physik, Technische Universita¨t Berlin, D-10623 Berlin, Germany 3 AG Moderne Optik, Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, D-12489 Berlin, Germany (Received 30 November 2012; published 13 May 2013) 2
The idea of atoms defying ionization in ultrastrong laser fields has fascinated physicists for the last three decades. In contrast to extensive theoretical work on atoms stabilized in strong fields only few experiments limited to intermediate intensities have been performed. In this work we show exceptional stability of Rydberg atoms in strong laser fields extending the range of observation to much higher intensities. Corresponding field amplitudes of more than 1 GV=cm exceed the thresholds for static field ionization by more than 6 orders of magnitude. Most importantly, however, is our finding that a surviving atom is tagged with a measure of the laser intensity it has interacted with. Reading out this information removes uncertainty about whether the surviving atom has really seen the high intensity. The experimental results allow for an extension of the investigations on the stabilization and interaction of a quasifree electron with a strong field into the relativistic regime. DOI: 10.1103/PhysRevLett.110.203002
PACS numbers: 32.80.Ee, 32.80.Rm
Survival of atoms in strong laser fields where already single-photon absorption is in principle sufficient to ionize was first predicted about a quarter century ago [1–3]. Since then the subject has been heavily discussed theoretically for the last two decades [4,5] with newly increased interest [6]. Different stabilization mechanisms [7] such as interference stabilization at lower intensities [8,9], adiabatic or dynamic stabilization [2,10–12], or strongly reduced ionization rates due to initial high angular momentum [13] have been theoretically suggested. These mechanisms have been, however, only partially confirmed in experiments using loosely bound Rydberg atoms exposed to relatively moderate laser intensities [14–17]. Moreover, ionization of Rydberg atoms in strong fields is expected to be neither following the picture of tunneling nor of a multiphoton process otherwise dealt with in strong field physics. The very successful semiclassical Keldysh theory for strongfield ionization [18] and with it, the Keldysh parameter, typically used to distinguish between a multiphoton or tunneling ionization process, are not applicable. Thus it is worthwhile to study the Rydberg strong-field ionization scenario in more detail at even higher intensities above 1015 W cm2 with special emphasis on the open question of whether proposed stabilization mechanisms remain valid at higher intensities. One of the key problems is, however, how to ensure that the atom has really seen the peak intensity and did not survive in the ‘‘shadow’’ of the laser light, the temporal or spatial outskirts. In this Letter we report experiments in which we expose Rydberg atoms to intense laser fields above 1015 W cm2 . Using a direct detection technique for surviving atoms we demonstrate experimentally the exceptional stability of Rydberg atoms. They withstand high laser intensities, where corresponding field amplitudes of more than 0031-9007=13=110(20)=203002(5)
1 GV=cm exceed the thresholds for static-field ionization by more than 6 orders of magnitude. Most strikingly, we are able to unravel that the intensity a surviving Rydberg atom has interacted with is encoded in the kinematics of the atom. To see how we can extract the intensity information from the kinematics we recall that strong-field excitation of He with linearly polarized light results in Rydberg states. These form a wave packet with principal quantum numbers centered around n 8 and with angular momentum states l distributed over a large range, essentially l < 10 [19]. Moreover, the excited atoms are accelerated in the gradient of the focused strong short-pulse laser field [20]. The deflection of atoms due to this acceleration can be measured using the experimental setup schematically shown in Fig. 1, also described elsewhere [19,20] and in the Supplemental Material [21]. In the experiments, we first produce the Rydberg wave packet with a linearly polarized laser pulse. After a variable time delay we apply a second elliptically polarized laser field. Sufficient elliptical polarization (ellipticity ¼ 0:66) ensures that no further excitation from the ground state takes place [19]. Consequently, any Rydberg atom which interacts with the second laser pulse and survives is additionally accelerated, provided it is located in a nonzero intensity gradient. Hence, the magnitude of acceleration, or more specific, the deflection of Rydberg atoms, verifies their interaction with the strong field. In Fig. 2(a) we show the Rydberg atom distribution on the detector after the first laser field with an intensity of 2:7 1015 W cm2 has interacted with the atomic beam. Without acceleration one would expect a pencil-shaped distribution along the laser beam propagation in z direction, with the focus at z ¼ 0. Clearly visible is the deflection of neutral excited helium atoms that occurs along the
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FIG. 1 (color online). Sketch of our experimental setup. The Mach-Zehnder interferometer [with beam splitter (BS) and =4 wave plate] provides two spatially well overlapping short laser pulses with variable time delay and linear and elliptical polarization, which are focused into the atomic beam. Surviving Rydberg atoms (either directly or their decay into a metastable level) are detected downstream with a position-sensitive microchannel plate.
radial intensity gradient of the focused laser beam. In the experiment we measure only the component in the x direction indicated in Fig. 2. The distribution of surviving Rydberg atoms after interaction with the second laser pulse is presented in Fig. 2(b). It indicates that the number of surviving Rydberg atoms is significant and that an additional acceleration occurs. This is particularly true for atoms stemming from the immediate laser focus at z ¼ 0. Taking the difference of the two distributions in Figs. 2(a) and 2(b) we can clearly identify additionally accelerated atoms, Fig. 2(c). We have measured excited He-atom distributions for three different intensities of the second laser pulse. To evaluate the fraction of Rydberg atoms surviving the strong laser field we exploit the position-sensitive detection of atoms and filter out only those atoms stemming solely from the center of the focus at z ¼ 0. The cuts are shown in Fig. 3. For the cases shown in Figs. 3(a) and 3(b), at intensities of the second laser of I2 ¼ 1:9 1015 Wcm2 and 3:0 1015 W cm2 , we find that the number of detected Rydberg atoms drops only by 7% and 9%, respectively, with regard to the number produced by the first laser alone. For the maximum laser intensity I2 ¼ 3:8 1015 W cm2 , shown in Fig. 3(c), the population drops by 15%. Given the high laser intensities applied to the atom, the loosely bound Rydberg electron turns out to be extraordinarily stable against ionization. We compare the cuts in Fig. 3 with model calculations for the atom distribution on the detector. We assume that
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FIG. 2 (color online). Distributions of excited He atoms on the position-sensitive detector. The number of impinging excited atoms on the detector is color coded. (a) Excited He-atom distribution on the detector for a single laser pulse with an intensity of I1 ¼ 2:7 1015 W cm2 . (b) Surviving excited He-atom distribution on the detector after applying a second elliptically polarized laser with an intensity I2 ¼ 3:8 1015 W cm2 and a time delay of about 500 fs. (c) The difference of the He-atom distributions in (a) and (b).
the acceleration of atoms is based on a quasifree electron quivering in the focused laser beam, which is nevertheless still bound to the ionic core after the laser pulse is over [20]. We use Monte Carlo techniques to simulate the distribution of excited He atoms on the detector. We first calculate the spatially resolved distribution of excited atoms in the laser focus after the first linearly polarized laser pulse. We take the spatial and temporal intensity distribution in the focused laser beam 2 2 2 2 to be Gaussian, Iðr; z; tÞ ¼ I0 ½w0 =wðzÞ2 e2r =wðzÞ et = . qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðzÞ ¼ w0 1 þ z2 =z2R is the beam size, w0 is the beam waist, zR is the Rayleigh length, is the pulse duration. The yield of excited atoms in a small volume is assumed to be proportional to the tunnel-ionization yield (about 10%), which in turn is proportional to the local field strength F ¼ pffiffiffi I . As has been found by numerical calculations [19], the relative excitation yield can be sufficiently accurately calculated using a semiclassical tunnel-ionization formula [22,23]. The second elliptically polarized laser beam does not excite ground state atoms. Using the excitation probability distribution, we calculate (typically for 105 atoms) the momentum transfer to obtain a detailed momentum distribution of atoms. Thereby we assume for all excited Rydberg atoms the same dynamic polarizability / 1=!2 , which is based on a quasifree electron. ! is the photon angular frequency. To calculate the momentum transfer on an excited atom due to the first and second
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FIG. 3 (color online). Cuts through the experimental Rydberg density distribution at z ¼ 0. Black line (narrow peak, noisy curve): Rydberg atoms excited and accelerated with (a) I1 ¼ 1:5 1015 W cm2 , (b) I1 ¼ 2:3 1015 W cm2 , and (c) I1 ¼ 2:7 1015 W cm2 . Red line (broad peak, noisy curve): Rydberg atoms additionally accelerated with the second laser (a) I2 ¼ 1:9 1015 W cm2 , (b) I2 ¼ 3:0 1015 W cm2 , and (c) I2 ¼ 3:8 1015 W cm2 . The solid green and blue curves [(narrow peak, smooth curve) and (broad peak, smooth curve), respectively] in (a) and (b) are from model calculations. In (c) data points connected by interpolated curves are calculated using only certain focal regions: squares (blue) r < w0 =2 and circles (cyan) w0 =7 < r < w0 =2.
laser pulse we suppose in a very good approximation that the intensity gradient in the focused laser beam is only radially oriented. Following Ref. [20] the momentum transfer pðz; rÞ is then given by pðz; rÞ ¼ pffiffiffiffi ½r =!2 wðzÞ2 Iðr; zÞS, where we time integrated over the complete laser pulse duration. This is possible since the atom does not move during the laser pulse. This formula holds for the momentum transfer in both laser pulses. S is a factor between 0 and 1 and the product S gives the effective pulse length for acceleration. All sets of measurements shown in Fig. 3 were reproduced with a consistent set of experimental parameters. For each calculated momentum transfer to an atom we chose the factor S from a Gaussian distribution at S ¼ 0:5 with a width of 0.2. This has been found to give overall the best agreement with the
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experimental data. Furthermore, we fitted by eye only the total ion yield for the cuts at z ¼ 0 for the single laser pulse data. Because of the good agreement of our measurements and the model calculations, we reveal the connection between momentum transfer and field strength or intensity more closely. At the focal plane at z ¼ 0 the momentum transfer p to the Rydberg atoms is proportional to the intensity gradient in the focus, p / r expð2r2 =w20 Þ. Highest momentum transfer takes place for those atoms located at r ¼ w0 =2, which corresponds to an intensity of 0:6 I0 . Since the inverse function of the momentum transfer is not bijective, atoms located at positions r > w0 =2 might gain the same momentum transfer as atoms located in the high-intensity region for r < w0 =2. Consequently, to provide a unique relation between laser intensity and momentum transfer we have to make sure that excited atoms produced in the first laser beam are located within w0 =2. That this is the case is shown in Fig. 3, where we repeated the calculation considering only the spatial region r < w0 =2. The two curves differ only by a few percent indicating that all detected atoms stem from the region r < w0 =2. Moreover, if we confine the calculation to the region w0 =7 < r < w0 =2 we clearly identify those atoms that survived intensities corresponding to 60% to 95% of the maximum intensity and are accelerated accordingly. Thus, although the acceleration for the highest intensities is zero, the method is capable of indicating the applied intensity in a range of 60% to 95% of the maximum intensity I0 . Finally we discuss the high survival probability of Rydberg atoms in strong laser fields. One might speculate whether the coherent Rydberg wave-packet dynamics along the laser polarization axis initiated after the first laser pulse might enhance the survival probability. We solve the time-dependent Schro¨dinger equation (TDSE) as described elsewhere [19] to obtain detailed information on the created Rydberg wave packet and on its ionization yield. We use alternatively a single-active-electron (SAE) approximation with a model potential introduced in Refs. [24,25] and a fully correlated calculation (see Supplemental Material [21]). Because of present limitations of the code, the TDSE is solved only for two linearly polarized laser pulses. The gradient of the laser field which might lead to ionization [26] is not included in the calculation. To decide whether the coherent wave-packet motion influences the survival rate of the Rydberg states, we perform the SAE calculations either by keeping the phases of the individual states after the first laser pulse or we set them to zero. Both calculations show no significant differences indicating that collective coherent processes do not play a role. We check this also experimentally by varying the time delay between the two lasers. Beside the region where the two laser pulses overlap and interference effects in intensity as well as in the polarization of the beam are dominant,
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FIG. 4 (color online). Calculated excitation and survival probability of Rydberg states with quantum number n. (a) Calculations are performed within the SAE approximation using basis states up to n ¼ 40 and l < 21. The pulse duration comprises 40 laser cycles with a cosine-squared-shaped pulse envelope. Black squares: Excitation with the first laser alone with an intensity of 2:7 1015 W cm2 ; surviving Rydberg states with quantum number n after the second laser pulse is applied with an intensity of 1:9 1015 W cm2 (red dots); 2:5 1015 W cm2 (green triangles). Total survival is 68%. The increase in the distribution towards n ¼ 40 is caused by the limited basis set. (b) Calculations are performed using a fully correlated two-electron basis set with states up to n ¼ 30 and l < 21. The pulse duration comprises 10 laser cycles with a cosine-squared-shaped pulse envelope. Black squares: Excitation with the first laser alone with an intensity of 1:5 1015 W cm2 . Red dots: surviving Rydberg states after the second laser pulse with an intensity of 1:3 1015 W cm2 . Total survival is 47%.
no time-dependent variations at larger time delays are detected. In Fig. 4(a) we show the n distribution produced after interaction with the first laser pulse and after the second laser pulse. Summing over all n states we find ionization of 32% of the Rydberg atoms. This is in good accord with the measured ionization of 15%. We note that the measured lower ionization could result from the elliptical polarization used in the experiment, instead of the linearly polarized one in the calculation. Nonlinear polarization might cause further reduction of close encounters of the electron with the core and thus reduced ionization rates.
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Inspecting the n-dependent survival (where we summed over all l states belonging to a fixed n quantum number) it is obvious from the calculations shown in Fig. 4(a) that states with n > 15 are hardly affected by the strong field and thus stable. A significant redistribution of n quantum numbers is not obvious. For this n region the pulse duration is much shorter than the Kepler orbit time k ¼ 2n3 . The very low ionization rates can be seen as an experimental confirmation of the stabilization of Rydberg states as proposed by Ref. [27] for low angular momentum states, where a field-independent ionization rate ¼ 1=2k has been derived. However, we find from our calculations that for higher l states no field independent rate is reached for intensities up to 1016 W cm2 . For states with 7 < n < 12 the laser pulse duration is comparable to the Kepler period. In this region resonancelike phenomena or redistributions are present so that no clear intensity dependence on the ionization of the n states exists. Interestingly, for states n < 7 a non-negligible survival rate is still visible. We have calculated systematically the ionization yield at a fixed laser intensity of 2 1015 W cm2 and at a fixed pulse duration for states with n ¼ 2 to n ¼ 20 with l from 2 to 10 in steps of 2 considering l < n. The pulse duration is larger than the Kepler orbit time for n < 7 and also the electron quiver amplitude of 76 a.u. is larger than the radial expectation values for n < 7. For these n quantum numbers we observe that the ionization yield never reaches 100%, even for the low l states, but it saturates or decreases slightly around the 80% ionization level. We attribute this to a dynamical stabilization, where the field dynamically reduces the overlap with the core region. We note that this can be regarded somewhat similar as the case, where ground state atoms survive strong laser fields in excited states [20,26]. To test whether electron correlation influences the low ionization rates, we repeat the calculations using a fully correlated two-electron basis. The results of the full twoelectron calculations shown in Fig. 4(b) confirm qualitatively the findings from the SAE model, but also indicate some differences especially for the lower lying states. In conclusion we have demonstrated experimentally in an unprecedented way the high survival probability of Rydberg atoms in strong laser fields. Most importantly, detecting the momentum transfer of the focused laser field to the surviving atom allows for a measure of the acting local laser intensity. We thus have clear and unique proof that Rydberg atoms indeed survive a strong laser field. Calculations show that even at highest intensities loworder photon processes dominate the ionization behavior. Furthermore, given the high survival of Rydberg atoms in strong laser fields, coherent manipulation with short-pulse lasers as well as efficient cascaded acceleration is envisaged. We thank Frank Noack for technical assistance with the laser. We acknowledge fruitful discussions with H. Reiss,
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M. Ivanov, O. Smirnova, W. Becker, and M. Vrakking. This work was partially supported by EU Initial Training Network (ITN) CORINF and the Humboldt Center for Modern Optics.
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