F Rohde

M Almendros

C Schuck

J Huwer

M Hennrich

J Eschner

A diode laser stabilization scheme for 40 Ca + single ion spectroscopy

Keywords: numbers: 42.60.-v, 42.50.-p, 37.10.Rs

We present a scheme for stabilizing multiple lasers at wavelengths between 795 and 866 nm to the same atomic reference line. A reference laser at 852 nm is stabilized to the Cs D 2 line using a Doppler-free frequency modulation technique. Through transfer cavities, four lasers are stabilized to the relevant atomic transitions in 40 Ca + . The rms linewidth of a transfer-locked laser is measured to be 123 kHz over 200 ms with respect to an independent atomic reference, the Rb D 1 line. This stability is confirmed by the comparison of an excitation spectrum of a single 40 Ca + ion to an eight-level Bloch equation model. The long-term stability with respect to the same reference is measured to be 130 kHz over a periode of two hours. The high degree of stability is demonstrated by the measured Allan deviation around 10 -11 between 1 and 100 s.

Introduction

Since the first preparation of a single trapped Ba + ion [1], ion-trap experiments have become increasingly sophisticated. Ultra high precision spectroscopy has lead to singleion frequency standards [2,3] and today the most accurate clock is based on single-ion spectroscopy [4]. Trapped ions have also developed into very promising candidates for the implementation of schemes of quantum computation, and universal gate operations [5,6,7] as well as quantum algorithms [8,9,10] have been demonstrated. In quantum communication, first building blocks of a quantum network have also been realized with ions [11]. Besides these applications, ions have proven to be textbook-like model systems to study fundamental questions in quantum optics and quantum mechanics [12].

In such experiments the ions are optically cooled and manipulated by laser light. Depending on the chosen ion and the desired application, the lasers have to be frequencystabilized well below the typical frequencies involved such as the natural linewidth (≈ 20 MHz) of the optical transition that is to be driven, Zeeman splitting (≈ 10 MHz), or the vibrational frequency in the trap (≈ 1 MHz). Furthermore, data are typically collected over long periods of time, such as hours or even days, since experiments exhibit low count rates. A laser system used for single ion spectroscopy therefore faces the requirements of offering good frequency stability over both short and long periods of time.

In early experiments the choice of ion species was governed by the availability of laser sources, typically dye lasers, at the characteristic wavelengths of the ion. Technical progress and the availability of a broad spectrum of wavelengths has made robust and inexpensive diode lasers increasingly attractive. However, significant technical effort to frequency-stabilize these lasers has remained indispensable. A popular approach is to lock a laser to a passive cavity built from ultra-low expansion material and placed in a pressure-sealed container or in vacuum [13]. This eliminates the extreme sensitivity of the cavity resonance to pressure changes due to the dependence of the refractive index of air [14]. Nevertheless such cavities are not totally drift free, they only minimize the drift of the laser frequency mostly caused by temperature fluctuations in the environment. Other approaches use scanning transfer cavities in combination with a stable HeNe reference laser. This technique is slow compared to others as it is limited by the scanning frequency of the cavity [15]. An open cavity transfer scheme combining active piezo and temperature stabilization of the cavity length has been used to build a difference frequency spectrometer [14], whereby a custom-made pyrex cavity transfers the stability of a HeNe laser onto another laser with a cavity modulation technique.

A scheme similar to the one we will present here has been used for high-precision frequency measurements of the D 1 line of alkali atoms [16]: a reference laser is locked to the Rb D 2 line by saturated absorption spectroscopy. Its 10 kHz linewidth is transferred to a second laser of the same type via an in-vacuum ring cavity. This second laser was then used to measure the absolute frequency of the Rb D 1 line using a second saturated absorption spectroscopy set-up. The ultimate but also cost-intensive solution for stable referencing is the use of recently developed frequency combs. In high precision spectroscopy the use of frequency combs is becoming more and more common [17].

In this article we present a scheme for the stabilization of four diode lasers at wavelengths between 795 and 866 nm, that are needed to drive various transitions in 40 Ca + . All four lasers are stabilized with respect to the D 2 crossover line (F=3 → F = 3/4) of atomic Cs using a Pound-Drever-Hall (PDH) transfer lock [18,19,20]. The scheme employs a reference laser at the Cs wavelength and open low-finesse transfer cavities built from inexpensive commercially available components that avoid the need of a vacuum container and ultra low expansion materials. A combined temperature and piezo length stabilization of the cavities is controlled by self-built locking electronics. In general this method can be readily transferred to any desired wavelength by an appropriate coating of the cavity mirrors.

Stabilization scheme

The level scheme of 40 Ca + and the four relevant transitions are displayed in figure 1. All lasers are commercial grating-stabilized diode lasers. The laser at 397nm is a frequencydoubled grating-stabilized diode laser. The four lasers are stabilized using the same stabilization scheme that consists of a series of consecutive PDH locks.

In this scheme, a reference laser at 852 nm is locked to the Cs D2 line using frequency modulation at 20 MHz in combination with Doppler-free absorption spectroscopy in a Cs vapor cell. The stability of this laser is transferred to the other lasers by the use of open length-stabilized low-finesse cavities (section 3). Each transfer cavity is brought into resonance with both the reference laser and one of the 40 Ca + lasers at the same time by temperature tuning. A PDH lock stabilizes the length of the cavity with respect to the reference laser. Each 40 Ca + laser is then stabilized with another PDH lock to its respective transfer cavity. The advantage of this scheme is that not only the linewidth on a short time scale is reduced due to the lock to the cavity, but also long-term drifts Laser stabilization scheme for the 795 nm laser. The stability of a reference laser at 852 nm is transferred onto the 795 nm laser using transfer cavities. The elements used to stabilize the laser at 866 nm are equivalent to the ones used for the 795 nm laser and are not shown. All relevant elements of the scheme are described in detail in the text.

are eliminated due to the referencing to an atomic line.

As an example, figure 2 shows a schematic of the locking chain for the 795 nm laser, the master laser for the 397 nm system. A beam splitter sends part of the 852 nm reference beam to a Cs vapor cell and the remaining power to the transfer cavities. The error signal derived from the saturated absorption spectroscopy on the Cs cell is fed back onto a piezo controlling the length of the first cavity via a self-built electronic device for stabilization called "cavity locker" (section 4). The cavity locker has a PDH module that uses the 20 MHz oscillation signal from a current modulation of the reference laser to demodulate the absorption signal. A digitally implemented compensator controls the piezo of the cavity. At the same time a low frequency regulation stabilizes the temperature via a heating wire. The error signal from the light reflected from this first cavity is used to stabilize the reference laser to this cavity with the help of commercial electronics. The same cavity is used to stabilize a laser at 866 nm, as indicated in figure 2.

The transfer cavity of the 795 nm laser is stabilized to the reference laser employing the error signal derived from the reflected 852 nm beam. A second cavity locker, again using the local oscillator of the reference laser for demodulation, feeds the signal back onto heater and piezo of the cavity. The 795 nm laser itself is stabilized to the transfer cavity via its reflection signal. The three other lasers are stabilized, by means of additional transfer cavities and cavity lockers, in exactly the same way.

In the following, we describe the relevant elements of the stabilization scheme in detail.

Cavities

The confocal Fabry-Perot cavities are built from two mirrors ‡ (T = 99, 5%, radius = 30cm) with 15 cm distance and have 500 MHz free spectral range. Finesse and linewidth are measured to be 270 and 1.9 MHz, respectively.

The two mirrors are mounted at the two ends of an aluminium lens tube § as depicted in figure 3. Having a high temperature expansion coefficient, aluminium assures the temperature tunability of the mirror distance ( ∆L ∆T = 3.5 µm/ • C). A heating wire with 15 Ω resistance is wound around the tube and a PT100 temperature sensor is attached. The whole tube itself is wrapped in insulating material and placed inside a bigger aluminium container that is mounted on the optical table. One mirror is glued to an end piece of the tube that can be screwed in and out for course length adjustment. The other mirror is glued to an aluminium washer that is mounted in a two-piece teflon holder together with 3 piezo actuators and a rubber ring as shown in Figure 3. The holder is designed such that the aluminum washer with the mirror is pressed by the piezo stacks against the rubber ring ¶ that sits in between the aluminum washer and the outer teflon part. The piezo stacks are guided by three holes in the inner teflon part that sits inside the outer one. They are held by an end cap that is screwed onto the aluminium tube and pre-loads the flexible rubber ring. Thereby the piezos find sufficient resistance to compress the rubber ring thus changing the length of the cavity. A maximum voltage of ±10 V is applied to the piezos corresponding to a length change of about 2.5 µm and resulting in a shift of the resonance over about three free spectral ranges. Scanning over a wider frequency range is attained by changing the temperature.

Cavity Locker

The stabilization of the cavities to the reference laser is realized with a self-built electronic device called "cavity locker" [21]. The cavity locker consists of a PDH input ‡ Layertec § Thorlabs Piezomechanik PSt 150/2x3/7 ¶ inner diameter = 16 mm, thickness = 1.5 mm stage, a microcontroller + , a scanning unit, and outputs for low-and high-frequency feedback (figure 2). The PDH stage consists of an input amplifier, a mixer, a phase shifter, and a bandpass filter, and it demodulates the photo diode signal using the rf modulation signal of the reference laser current as local oscillator (figure 2). The PDH error signal which it produces is the input for the compensator which is implemented with software on the microcontroller. In control mode the low frequency output of the compensator drives the heating wire, and the high frequency output drives the cavity piezos. If the control loop is open, the piezo can be scanned by the scanning unit and the temperature is stabilized to the value defined by a temperature set point potentiometer.

The compensator implemented on the evaluation board consists of two parts, one for the piezos and one for the heating wire. In closed-loop operation the heater compensator is programmed to regulate the length of the cavity in such a way that the piezos remain in their mid position (figure 4). This assures that the piezo voltage is kept within ±10 V . Both compensators are programmed (in C language) and custom-designed after analyzing the response of the system, i.e. the cavity and its circuitry (figure 4). The piezo compensator consist of a combination of generic control techniques, e.g. dead beat [START_REF] Ogata | Modern Control Engineering[END_REF], integrators, etc., while the heater compensator is implemented as proportional and differential controller. The speed of the piezo compensator loop is limited by the eigenfrequency of the piezo actuators measured to be 3 kHz.

Characterization

Cs spectroscopy

The right part of figure 5 shows the set-up of the Doppler-free absorption spectroscopy on the Cs cell. The absorption signal detected with PD1 is shown in figure 6

line F=3 → F ′ co F ′ =2/3 F ′ =4 F ′ =2 F ′ =3 co F ′ =3/4 co F ′ =2/4

Characterization of the transfer lock

To estimate the absolute stability of the transfer locked lasers, one of them was compared with an independent atomic reference, a Doppler-free rubidium resonance in a gas cell. For this purpose a second Doppler-free absorption spectroscopy was set up and the master laser of the 397 nm SHG system was tuned to the D1 line of 85 Rb at 794.979 nm [START_REF] Banerjee | [END_REF]. In figure 5 this setup is shown.

The full Rb Doppler-free absorption signal obtained by scanning the frequency of the 795 nm laser is depicted in figure 7. Since the cell contains Rb isotopes

F'=2 F'=1 co F'=2 F'=3 co co F'=3 F'=2 F'=2 F'=1 co Figure 7.
Full spectrum of the Rb D 1 line, obtained via Doppler-free absorption spectroscopy. The frequency axis was calibrated using the literature values [START_REF] Steck | Rubidium 87 d line data[END_REF] of the 5 2 S 1/2 ground state hyperfine splitting of 87 Rb and of the transition frequency of the 5 2 S 1/2 to 5 2 P 1/2 transition of the same isotope; possible errors, for example due to pressure shifts in the recorded spectrum, are below the resolution of the display.

in their natural abundances, absorption lines for 85 Rb (72.2 %) and 87 Rb (27.8 %) are observed. The Doppler-free hyperfine structure is well resolved, and the various hyperfine transitions are easily identified using the splittings known from the literature [START_REF] Steck | Rubidium 87 d line data[END_REF][START_REF] Steck | Rubidium 85 d line data[END_REF].

The transfer lock is characterized by monitoring the Rb saturated absorption discriminant generated by a PDH circuit, which uses the modulation of the laser light at 20 MHz from the lock to its transfer cavity. The signal recorded with a fast photo diode (PD2) * behind the Rb cell is amplified, mixed down with a mixer and filtered by a bandpass filter † † (see figure 2 bottom).

With the whole stabilization chain locked, i.e. the master laser at 795 nm is locked via the transfer cavity to the stable reference laser, the laser was tuned to resonance with the strongest line in 85 Rb, the F = 3 → F = 2/3 crossover line. Scanning the AOM of figure 5 we identified the desired transition and set the frequency to resonance, i.e. to the point with the steepest slope of the discriminant. Figure 8 shows the discriminant of the F = 3 → F = 2/3 crossover line obtained by scanning the AOM.

The discriminant also serves as a frequency gauge for converting the voltage recorded with the oscilloscope into frequency. For this purpose the AOM is linearly scanned with ±2 MHz amplitude around the resonance. The recorded slope is fitted linearly to determine the conversion factor between frequency and voltage for each measurement.

To measure the short-term stability of the locking chain, the Rb saturated absorption discriminant was recorded for a fixed AOM frequency at three different times during one day over a time span of 200 ms with different sampling rates. voltage trace on the oscilloscope, rms deviations of 143 kHz, 123 kHz and 133 kHz were measured, respectively. These numbers do not take into account a possible variation of the Doppler-free signal itself, e.g. due to signal offset drifts, and therefore have to be understood as upper limits of the laser linewidth. Since the result has been obtained through measurement with respect to an independent reference, it can be assumed to apply equally to the Ca + spectroscopy. In order to measure the long-term stability of the setup, the mean value of the voltage of the Rb saturated absorption discriminant was recorded with a computer. A Labview program read out the voltage from the oscilloscope and stored the mean value over the time ∆τ = 1 s for a duration of 2 hours. The mean value (∆τ = 1 s) of the discriminant drifted by about one short-term rms deviation (∆t = 200 ms), i.e. about 130 kHz, during these two hours.

The main cause of instability are acoustic noise or pressure drifts, to which open transfer cavities are particularly sensitive: there is a remaining systematic error resulting from the different refractive indices of air at the wavelengths of the two lasers [14]. This differential refractive index change can not be compensated for with the presented scheme (i.e. without monitoring the atmospheric pressure in the laboratory) and represents its fundamental limitation. Since among the Ca + lasers and the Cs reference the difference in refractive indices is largest for 852 and 795 nm, the error for the 795 nm laser can be regarded as an upper limit for all lasers.

An analysis following Ref. [14] reveals that the observed drift is equivalent to a pressure change of 1 mbar. This is the order of magnitude of the daily cyclic variation of the atmospheric pressure for our latitudes, dynamic fluctuations apart. Since the pressure in our lab is not controlled, it is reasonable to assume that the frequency drift is caused by a pressure drift. This assumption, however, would have to be confirmed by a long-term measurement monitoring the ambient pressure in parallel. Such a measurement, on the other hand, would open the possibility of eliminating the pressure-induced error through appropriate feedback; without it, the sensitivity to pressure fluctuation is the main limitation of schemes using open transfer cavities.

Allan deviation

For a full characterization of the stability of the 795nm laser oscillator, the Allan deviation [START_REF] Allan | [END_REF] was calculated from the three short term measurements as well as from the long-term measurement. The four results are plotted in figure 9. The three measurements over 200 ms show good agreement between τ = 10 -4 and τ = 10 -2 s. The Allan deviation σ y decays with τ -1/2 over these two orders of magnitude, down to the 10 -11 region. This indicates the presence of white frequency noise caused in electronic components. It is expected that the accuracy of an oscillator increases with further integration time until other effects, such as long-term drifts, start to dominate [START_REF] Riehle | Frequency Standards[END_REF]. In accordance with this the Allan deviation for the long-term measurement rises again starting from τ = 10 2 s up to τ = 10 3 s. The slope is proportional to τ 1/2 indicating that the dominant noise type is a random walk of frequency noise caused by the environmental conditions. This is in agreement with the observation of high sensitivity to pressure fluctuations.

Long-and short-term measurements connect well for time scales of τ = 10 -2 to 10 1 s. Although there is data missing between 0.02 and 1 s, the available data is consistent with a flat behavior of σ y ∝ τ 0 and an expected flicker noise floor for intermediate frequencies.

For the long-term measurement each data was averaged by the oscilloscope over 1 s, with no dead time between subsequent points. This assures that no fast oscillations were disregarded by the measurement apparatus. The three short term measurements were only sampled, i.e. each data point was recorded after the sampling period specified in figure 9. To make sure that there is no high frequency noise present, an additional Fast Fourier Transform (FFT) of the Rb saturated absorption discriminant was recorded with the oscilloscope. Figure 10 shows the FFT for a bandwith of 5 MHz. All noise with higher frequency than that is filtered by the bandpass filter (figure 2). The most prominent peak at 250 kHz with an amplitude of -65.4 dBm is still accounted for in the Allan deviation measurement with sampling rate 2 µs. For higher frequencies there is no significant noise present. Therefore sampling without averaging on the time scale of the short-term measurements is expected to not miss any higher-frequency noise contributions.

Single ion spectroscopy

To test the stabilization scheme, excitation spectra of single 40 Ca + ions have been recorded using the ion trap set-up described in [START_REF] Gerber | [END_REF]. The ion is continuously excited by two lasers at 397 nm and 866 nm, and the fluorescence light of the ion is collected by a high numerical aperture lens also described in [START_REF] Gerber | [END_REF], and detected with a PMT while the 866 nm laser is scanned over the resonance. The 397 nm laser is red-detuned to provide Doppler cooling. Figure 11 shows the excitation spectrum of a single ion with excitation under 45 • to the B-field direction. The 397 and 866 nm beams were set to vertical and horizontal polarization, respectively. Each data point represents the count rate on the PMT integrated over 100 ms. The complex structure of the spectrum is caused by two-photon (dark) resonances between the S 1/2 and the D 3/2 level. The eight Zeeman sublevels involved in the interaction with 397 and 866 nm light give rise to eight distinct dark resonances (dips in the excitation spectrum). Their positions and sizes depend on the magnetic field and the orientation of the polarizations of the exciting lasers with respect to the quantization axis. For the conditions of figure 11 some of the dark resonances are suppressed and in total five more or less pronounced dips are observed.

The red line is an eight-level Bloch equation model calculation [29] that was fitted to the data with a numeric (Matlab-based) fitting routine. The agreement with the model is very good exhibiting a reduced χ 2 of 1.8. The linewidths of the lasers extracted from the fit are δν 397 = 268 kHz for the blue laser and δν 866 = 1/2 • δν 397 = 134 kHz for the infrared laser. This is consistent with the linewidth measured with respect to Rb, as described in section 5.2.

Conclusions

We have described a laser frequency stabilization scheme for 40 Ca + single-ion spectroscopy with high short-and long-term stability. The residual rms deviation of 123 kHz is well below all significant transition linewidths of 40 Ca + and allows us to resolve well dark resonances in the excitation spectra. The Allan deviation proves high long-term stability limited only by pressure changes in the laboratory. The scheme is particulary well suited for measurements with low count rates that require stable conditions over a long time period. For the longest experimental run until the publication of this article, stable conditions for 40 hours were achieved. During this time no noticeable frequency drift on the MHz scale was observed.

The advantage in comparison with other transfer stabilization schemes lies in the use of open table-top cavities. The cavities are built mostly from off-the-shelf parts, no vacuum or ultra low expansion material is needed. A combined temperature and piezo length stabilization of the transfer cavities controlled by a micro controller is at the heart of the system.

Figure 1 .

 1 Figure 1. Level scheme of 40 Ca + . Diode lasers for the four indicated transitions are stabilized with the transfer lock. Light at 397 nm is produced by frequency doubling of a laser at 795 nm.

Figure 2 .

 2 Figure 2.Laser stabilization scheme for the 795 nm laser. The stability of a reference laser at 852 nm is transferred onto the 795 nm laser using transfer cavities. The elements used to stabilize the laser at 866 nm are equivalent to the ones used for the 795 nm laser and are not shown. All relevant elements of the scheme are described in detail in the text.

Figure 3 .

 3 Figure3. Cavity assembly consisting of a commercial lens tube system and a custommade teflon holder. The picture shows from left to right: end cap with hole for optical access and electric connections, three piezo stacks, teflon piece guiding the piezos, aluminium washer, rubber ring, first mirror, teflon piece holding the aluminium washer and rubber ring, aluminium tube with heating wire and temperature sensor, end piece for course adjustment, second mirror, mirror holder, threaded ring to fix the mirror holder in the end piece.

+Figure 4 .

 4 Figure 4. Block diagram of the control loop used to stabilize the cavities. The system consists of the cavity with its piezos and heater that control the length and the resulting PDH signal generated by the PDH module. The micro controller processes the error signal and controls the piezos and the heater. ADC and DAC stand for analog-todigital and digital-to-analog converter. The Pound-Drever-Hall (PDH) setpoint value is programmed to coincide with the zero crossing of the error signal. The piezo setpoint value is adjusted to the piezo mid position (0 V). If the control loop is closed, the heater compensator reacts according to the piezo signal and the piezo setpoint.

Figure 5 .

 5 Figure 5. Setup for the characterization of the transfer lock. The master laser of a frequency doubling stage at 795 nm is locked to a Doppler-free Cs resonance in a gas cell by three consecutive Pound-Drever-Hall stabilizations using a transfer cavity. To characterize the frequency stability of the laser a PDH error signal is derived from Doppler-free absorption spectroscopy on 85 Rb.

Figure 6 .

 6 Figure 6. Cs Doppler-free absorption spectroscopy signal (bottom) and corresponding error signal (top) for the F = 3 → F = 2, 3, 4 transitions and its cross-over lines. The abscissa shows absolute frequency.

Figure 8 .

 8 Figure 8. Error signal of the 85 Rb, F = 3 → F = 2/3 crossover line. The signal was recorded by scanning an AOM with the laser locked to an atomic Cs reference.The observed line broadening, compared to the natural line width of 6 MHz, is predominantly power broadening, as expected for the employed laser intensity and Rb pressure.

Figure 9 .

 9 Figure 9. Allan deviation for three short-term measurements with different sampling rate and 200 ms integration time, and one long-term measurement with 2h integration time. The dashed-dotted line (τ -1/2 ) indicates the dominating white frequency noise for high frequencies. The dashed line (τ 1/2 ) indicates the dominating random walk frequency noise for long time scales. The deviations are plotted for time scales allowing for at least 10 data points for each averaging time.

Figure 10 .

 10 Figure 10. Fast Fourier Transform (FFT) of the Rb error signal for a bandwidth of 5 MHz.

Figure 11 .

 11 Figure 11. Excitation spectrum of a single 40 Ca + ion in a linear Paul trap with excitation under 45 • to the B field and fluorescence detection in the direction of the B field. The red solid line is a calculation based on an eight-level Bloch equation model. From the fitted curve the experimental parameters are calibrated as follows: Rabi frequencies Ω 397 = 2π × 15 MHz, Ω 866 = 2π × 3.5 MHz, detuning ∆ 397 = 2π × -27 MHz and magnetic field B = 3.8 G. The fit also accounts for slight deviations of the laser polarizations from their ideal values (< 3 %).

Acknowledgments

We thank F. Dubin for technical support.

We acknowledge support from the European Commission (SCALA, contract 015714; EMALI, MRTN-CT-2006-035369), the Spanish MICINN (QOIT, CSD2006-00019; QLIQS, FIS2005-08257; QNLP, FIS2007-66944), and the Generalitat de Catalunya (2005SGR00189; FI-AGAUR fellowship of C.S.).


F Guzmán

L F Errea

Clara Illescas

L Méndez

B Pons

Calculation of total cross sections and effective emission coefficients for B 5+ collisions with ground-state and excited hydrogen

Keywords: numbers: 34, 70, +e, 34, 10, +x

Classical and semiclassical calculations of nl-resolved charge exchange cross sections in B 5+ collisions with H(n i ) are performed to compute effective emission coefficients for the n = 7 → n = 6 transition in B 4+ for plasma conditions typical of the ASDEX-U tokamak. For n i = 1, the value of the emission coefficient is larger than that obtained from ADAS database by a factor of two at energies of 10 keV/amu, but no differences are found at energies above 50 keV/amu. For n i = 2, our calculation yields emission coefficients close to those derived from ADAS data from low to high impact energies. The emission coefficients corresponding to B 5+ + H(n i = 3) collisions are of the same order of magnitude than those for n i = 2.

Introduction

Information on plasma core characteristics is currently obtained by employing diagnostic beams, and these techniques will be also the main diagnostic tool of ITER ( [1] and references therein). In particular, in charge exchange recombination spectroscopy (CXRS) (see, e.g., [2]), neutral beams are injected in the plasma, and charge exchange (CX) occurs in collisions with plasma impurity ions. Although other atoms (He, Li) have been employed, we shall focus on the application of the diagnostics based on H (or D) beams. For this particular case, the basic process is:

H + A q+ → H + + A (q-1)+ (1)
The ion A (q-1)+ is formed in an excited state and the ensuing emission is employed to determine plasma density and temperature. For energetic beams, the CX reaction occurs in the plasma center, where the impurities ions are fully stripped, and therefore the basic process is a one-electron reaction.

The application of CXRS requires the knowledge of high-precision state-selective cross sections for populating excited states A (q-1)+ (n, l). These data are included in a collisional-radiative model that leads to the effective emission coefficients, which are afterwards applied in the diagnostic. The beam diagnostic techniques are therefore based on a set of atomic data, and those included in the Atomic Data and Analysis Structure (ADAS) database ‡ are often employed.

In previous works, we have calculated cross sections for several charge exchange reactions [3,4,5] in ion collisions with H(1s). However, a small fraction of excited atoms H(n i =2) can collide with the impurity ions; they are produced by interaction of ground state beam atoms with the plasma constituents. The main relevance of these excited atoms is the well-known fact that the CX cross section is in general larger than that for the corresponding process with ground state donors. In this respect, Hoekstra et al. [6] compared effective emission coefficients for collisions of several light ions with H(n i =1) and H(n i =2), and an example of the influence of excited donors on the measurements of impurity densities can be found in ref. [START_REF] Maggi | 30th EPS Conference on Controlled Fusion and Plasma Physics[END_REF]. Moreover, the work of ref. [8], on the CXRS diagnostics of B 5+ density in the Alcator C-Mod tokamak, has pointed out that the lack of n i =2 in the calculated effective emission coefficient leads to a misleading result that shows a spurious dependence on the beam energy. Also the lack of n i =3 yields errors of 15% in the measured densities.

The importance of Boron ions lies on the fact that using B to coat the first wall (boronisation) improves the energy confinement [9]. In the present work, we present effective emission coefficients for the reactions:

B 5+ + H(1s) → B 4+ (nl) + H + (2) B 5+ + H(n i = 2, 3) → B 4+ (nl) + H + (3)
The cross sections for reaction (2) were calculated in [5]. Preliminary calculations for collisions with H(2s) were reported in [10]. In the present work we present new ‡ http://adas.phys.strath.ac.uk partial cross sections for reactions (3); they have been calculated using the classical and semiclassical formalisms as explained in [3]. At high collision energies (E > 4keV/amu), we employ the Classical Trajectory Monte Carlo Method (CTMC). At low energies we have applied a semiclassical molecular method, and in particular, calculations with excited Hydrogen have been carried out as explained in [11]. Although the semiclassical results are relatively less important than the CTMC ones in view of their application in CXRS, since experiments take place at higher energies to those where the method is applicable, the comparison between CTMC and semiclassical results allows us to discuss the accuracy of our classical calculation. Besides we shall compare our results with the cross sections reported in ref. [6] which employed a CTMC method, and with those available in ADAS database, reported by the authors of ref. [6]. Classical calculations for collisions with H(n i =3) have been also carried out because of the potential relevance of these reactions in diagnostics [8]. In this respect it must be noted that cross sections and effective emission coefficients employed in ref. [8] for n i = 3 donors were obtained by scaling the CX cross sections from H(n i = 1, 2). At the collision energies typical of diagnostic beams (above 25 keV/amu), ionization is the dominant process. In the present work we have also evaluated, using the CTMC method, the total cross section for the reaction:

B 5+ + H(n i = 2, 3) → B 5+ + H + + e -, (4)
which is relevant to calculate beam stopping coefficients. Atomic units are employed unless otherwise stated.

Theory

2.1. Eikonal-CTMC formalism.

The CTMC treatment has been explained in previous publications and we shall only summarize it. First we assume (eikonal approximation) that the nuclei follow rectilinear trajectories with constant velocity v and impact parameter b (R = vt + b). It is known that this approximation is practically exact for collision energies above 100 eV. The electron dynamics is described by the statistical phase space distribution which satisfies the classical Liouville equation:

∂ρ(r, p, t; v, b) ∂t = -[ρ(r, p, t; v, b), H] ( 5 )
where H is the classical Hamiltonian for fixed nuclei:

H = p 2 2 - Z A r B - Z H r H (6)
with Z B,H the nuclear charges and r B,H the electron distances to both nuclei. In the CTMC method the electron distribution is discretized using N classical trajectories [12]:

ρ(r, p, t; v, b) = 1 N N j=1 δ(r -r j (t))δ(p -p j (t)) , (7)
and substitution of eq. ( 7) in eq. ( 5) yields the Hamilton equations for the individual trajectories r j (t), p j (t): ṙj = ∂H ∂p j ; ṗj = -∂H ∂r j (8) In general, the accuracy of capture and ionization cross sections basically depends on that of the initial distribution. In previous works we have employed either the microcanonical or the hydrogenic distributions. In the microcanonical one [12], all electronic trajectories have the energy of the initial quantum state = 0 . The hydrogenic distribution [13] is obtained by means of a linear combination of N microcanonical distributions with an average energy < >= 0 (e.g. [3]). For excited atomic states (H(n i =2)) a better description is achieved, as shown in figure 1, by using an initial gaussian distribution [14]:

ρ(r, p) = 2 -1/2 π -3 (-) 5/2 ρ( )δ(H -)d (9) with ρ( ) = K 1 e -K 2 (n H -2.2) 2 ; n H ( ) = Z H / √ -2 (10)
and where the constants K 1 and K 2 are determined by the conditions of normalization and mean energy < >= 0 . In practice, the integral in ( 9) has been evaluated numerically by including 60 values of in the interval -0.243 < ≤ -0.08336; these limits are those given by the criterion of Becker and McKellar [15], which defines the box in the phase space that corresponds to the n = 2 quantum level. In general, the box for the quantum level n is:

n(n -1/2)(n -1) ≤ n 3 H ≤ n(n + 1/2)(n + 1) (11)
where n H is the classical quantum number defined in eq. ( 10). The initial distributions for H(2s) and H(2p) have been obtained by retaining only those trajectories with angular momentum l c fulfilling 0 < nl c /n H < 1 and 1 < nl c /n H < 2, respectively, with n = 2 (see [15]). To illustrate the comparison of the classical distributions we plot in Fig. 1 the values of the radial components of coordinate and momentum distributions:

ρ C (r) = dp 2π 0 dφ π 0 dθ sin θ r 2 ρ 2s,2p (r, p) ( 1 2 ) ρ C (p) = dr 2π 0 dφ p π 0 dθ p sin θ p p 2 ρ 2s,2p (r, p) ( 1 3 )
and the corresponding quantities for the quantum mechanical distributions: 13), (15). For comparison purposes the microcanonical distribution for H(2s) is also shown.

ρ QM (r) = 2π 0 dφ π 0 dθ sin θ r 2 |Φ 2s,2p (r)| 2 (14) ρ QM (p) = 2π 0 dφ p π 0 θ p sin θ p p 2 | Φ2s,2p (p)| 2 (15)
It is important to note that, as for the case of H(1s) [14], the use of the gaussian distribution does not appreciably modify the momentum distribution, but notably improves the coordinate distribution with respect to the microcanonical one.

In the calculation we have employed a set of 2×10 5 electron trajectories and the Hamilton equations were solved up to t = t max =150. We apply an energy criterion to evaluate ionization, capture and excitation cross sections, by considering the electron energies with respect to both (H and B) moving nuclei. Namely, trajectories leading to ionization fulfill (the origin of the electronic coordinates is placed on the H nucleus):

H = p 2 /2 -1/r > 0; B = 1/2(p -v) 2 -Z B /|r -b -vt max | > 0 ( 16
)
while for capture we have: H > 0, B < 0, and for excitation: H < 0, B > 0. The probability for a process X reads:

P X (v, b) = dr dp ρ X (r, p, t max v, b) = N X /N (17)
where N X is the number of trajectories that lead to process X. Partial nl cross sections were obtained using the criterion of ref. [15]. The value of n associated to a given trajectory is obtained by applying the box definition of eq. ( 11) for the B 4+ ion; i.e., n

fulfills n(n -1/2)(n -1) ≤ n 3 B ≤ n(n + 1/2)(n + 1) with n B = Z B / √ -2 B .
Similarly, the value of l is obtained from the condition:

l ≤ nl c /n B ≤ l + 1 (18)

Semi-classical molecular expansion.

As in the CTMC treatment the nuclei follow straight-line trajectories while the electronic motion is described quantum-mechanically by means of the wavefunction Ψ, solution of the equation:

H -i ∂ ∂t r Ψ(r, t; v, b) = 0 ; H = - 1 2 ∇ 2 - Z B r B - Z H r H (19)
The collision wavefunction Ψ is expanded as a linear combination of molecular orbitals (OEDMs, for One Electron Diatomic Molecules) {χ k } in the form:

Ψ(r, t; v, b) = e iU (r,t) k a k (t; v, b)χ k (r, R)e -i R t 0 ε k (t )dt (20)
where the OEDMS are eigenfunctions of H with energies ε k (R) and U is a common translation factor [16], defined in terms of the switching function proposed in ref. [17].

The molecular basis set includes 223 OEDMs, which are: the entrance channels of reactions ( 2) and ( 3), which dissociate into B 5+ + H(n=1,2), and the OEDMs dissociating into B 4+ (nl)+ H(1s) with 1 ≤ n ≤ 10 . The correlation diagram of the σ molecular orbitals is shown in fig. 2, where we plot the effective quantum numbers n k eff = 6/ √ -2ε k as functions of the internuclear distance R. The cross sections for CX and ionization from H(n i =2), σ(n i = 2) is obtained as:

σ(n i = 2) = 1 4 (σ(2s) + σ(2p z ) + σ(2p x ) + σ(2p y )) (21)
where the cross sections in the right hand side are obtained in the calculation where the initial states are the corresponding atomic orbitals defined in the rotating molecular frame, with Ẑ R and Ŷ perpendicular to the collision plane. The OEDMS 980 and 860 (labelled according to their united-atom limit) correlate as R → ∞ to the linear combinations √ 2(2s ± 2p z ), and the atomic orbitals are therefore obtained by combining these molecular orbitals. Accordingly, the initial amplitudes of the molecular orbitals are given by [18]:

a k (t = -t max ) = exp -i f k - -tmax 0 ε k dt + -∞ 0 E N 0 L 0 M 0 dt × χ k | Φ N 0 L 0 M 0 ( 22
)
where, in B 5+ + H(2s,2p) collisions:

f k = Z B d bv tan -1 b vt max . ( 23
)
d=3a 0 is the 2s-2p transition dipole moment and N 0 , L 0 , M 0 are the atomic quantum numbers (200 and 210 in the present case).

The contributions σ(2p x ) and σ(2p y ) are both evaluated by solving the system of differential equations with the system initially described by the OEDM 871. In the former case π OEDMs are coupled with σ and δ orbitals through rotational couplings, while in the latter case, where the system is initially described by a π molecular orbital, perpendicular to the collision plane, all non-adiabatic couplings with σ orbitals vanish. 5+ + H(1s) Substitution of equation ( 20) in [START_REF] Summers | The ADAS User Manual 2[END_REF] leads to a set of first-order differential equations for the expansion coefficients a k (t; v, b). CX and excitation cross sections are obtained by integrating the corresponding probabilities over the impact parameter:

B 5+ + H(n2) n B 4+ =2 n B 4+ =3 n B 4+ =4 n B 4+ =5 n B 4+ =6 n B 4+ =7 n B 4+ =8 n B 4+ =9 n B 4+ =10 Figure
σ B,H nlm (v) = 2π |a B,H nlm (t → ∞; v, b)| 2 bdb = 2π P B,H nlm (v, b)b db (24)
where the superscripts B, H indicate that the electron is asymptotically bound to the B nucleus (charge exchange) or H (excitation). The coefficients a nlm are obtained by applying the extrapolation procedure proposed in [18] and accounting for the Stark effect and the residual rotation of the internuclear axis from t = t max to t = ∞.

Results and discussion

Cross sections

The spectral lines usually employed in the CXRS diagnostics are those lying in the visible region of the spectrum. For the particular case of B(V) the diagnostic is normally based on the intensity of the n = 7 → n = 6 line with λ = 494.47 nm, and therefore the most important partial cross section is that for populating B 4+ (n=7), which is shown in fig. 3. The main limitation of the OEDM treatment is the overpopulation of the excited orbitals due to the ionization flux that remains trapped in the CX channels; this can be clearly noted in the shape of the CX cross section in fig. 3. Therefore the connection between classical and quantal partial CX cross sections must be carried out at energies where the ionization cross section is small compared to the capture cross sections. For instance, in the case of population of B 4+ (n = 7), the semiclassical data cannot be employed for energies larger than E 6 keV. Although the semiclassical results are not very significant in view of their application in CXRS, since experiments take place at higher energies to those where the method is applicable, they support the CTMC results and, in particular the initial distribution employed. In our calculation, the difference between the CX cross section from CTMC and OEDM results of fig. 3 is smaller than 5% at E = 4 keV/amu.

In order to obtain a smooth joining of both calculations, different energy ranges have been employed for different n. Similarly, recommended nl-partial cross sections have been built by joining the classical and semiclassical branching ratios r nl = σ nl σn at the energy that yields the smoothest joining. As a check of the procedure, we have found that |1l r nl | < 0.05. Our results are illustrated in fig. 4 for B 5+ +H(2p) → B 4+ (nl)+H + and B 5+ +H(n = 2) → B 4+ (nl)+H + CX collisions with n = 6, 7, 8. The σ(2p) and σ(n = 2) cross sections have respectively been computed according to the statistical averages (σ(2p x ) + σ(2p y ) + σ(2p z ))/3 and (3σ(2p) + σ(2s))/4 (see eq. ( 21)). Since σ(2s) ≈ σ(2p)/3, we find in Fig. 4 that σ(n = 2) ≈ σ(2p) for both nand nlpartial cross sections. One can further note in Fig. 4 that classical and semiclassical n-partial cross sections are smoothly joined, but the results are less smooth for the nlpartial cross sections corresponding to the lowest l values (l = 0, 1, 2), although these cross sections are very small with respect to those associated to high l values. Our recommended n-partial CX cross sections are shown in fig. 5, together with our results using a single microcanonical distribution, and the data included in ADAS. Our results are completely different from those of [6] (not shown in the figure). However the values stored in ADAS, which were obtained by the authors of ref. [6], agree with our values, which points to a mistake in the figure of that work. The comparison shows that total CX cross section for B 5+ collisions with H(n i = 2) is accurately evaluated down to E ≈ 2 keV/amu by employing an initial microcanonical distribution. Such an accurate behaviour of microcanonical calculations at low E is at first sight surprising, since previous CTMC calculations have shown that the initial condition has to be improved beyond the microcanonical framework to yield reliable cross sections (see, for instance, [3]). Nonetheless, most of these previous works have considered the ground-state H target, where it is necessary to improve the initial classical distribution by including a wide energy spread in the initial classical state in order to mimic underbarrier CX transitions from the entry channel [14]. The consequence of this procedure is that Total cross sections for CX into Be 4+ (n) and ionization in collisions of B 5+ with H(n=2). The lines are the recommended data obtained joining classical and semiclassical calculations. The triangles are the results of a classical calculation with a microcanonical initial distribution and the asterisks correspond to the data stored in ADAS, which are issued from the microcanonical CTMC calculations of [6].

the new distribution includes unphysically low energies, which lead to an unaccurate description of transitions to low n levels. In the present calculation for collisions with H(n i = 2), although the initial microcanonical distribution does not correctly reproduce the tail of the quantal spatial distribution (see fig. 1), it agrees with the quantal distribution for r ≤ 7, which explains the agreement of the total cross section. However, the corresponding partial cross sections for populating B 4+ (n = 6, 7, 8) of fig. 5 show noticeable differences with the recommended data.

Since Bespamyatnov et al. [8] have pointed out that excited donors H(n = 3) can significantly influence the determination of the impurity density, we have performed a CTMC calculation of CX and ionization cross sections in B 5+ + H(n = 3) collisions in the energy range 3 ≤ E ≤ 1000keV/amu. We have employed the microcanonical distribution, which has been found to yield accurate cross sections for E > 2keV/amu in collisions with H(n = 2). For the sake of conciseness, we only report in fig. 6 ionization and partial-n-CX cross sections. The most prominent feature of this figure is the spreading of the capture flux onto a broad range of B 4+ shells; this is a consequence of the initial states being degenerate with the high-lying B 4+ (n = 15) level, and their energies lie very close to a broad range of n-shells of B 4+ . Accordingly, semiclassical closecoupling calculations would require a prohibitively large number of basis orbitals. Since nl-CX partial cross sections are needed to evaluate the effective emission coefficients, they have been also computed in the CTMC calculation, and they are available from the authors upon request.

Effective emission coefficients

The integrated photon emissivity in the line of sight given by a transition decay in an impurity of nuclear charge Z after a CX collision is given by:

I Z-1 nn = q n i n→n N H (n i )N Z ds ( 25
)
where N H (n i ) is the number density of the donor (Hydrogen) in the energy level n = n i and N Z the number density of the ionized impurity (B 5+ in this work). q n i n→n are the effective emission coefficients; they contain the contributions of all the spontaneous nl-sublevels transitions and the cascade corrections from the CX populated levels that decay into the n level, calculated by means of a Generalized Collisional Radiative model [START_REF] Summers | The ADAS User Manual 2[END_REF][START_REF] Summers | [END_REF].

In order to discuss the relevance of our data in real practical conditions, and to compare with those already present in the ADAS database, we have used the ADAS implementation [START_REF] Summers | The ADAS User Manual 2[END_REF] to calculate ASDEX relevant effective coefficients; we have taken the plasma conditions of ASDEX discharges n. 19364-65 [START_REF] Maggi | Private Commuinication[END_REF]. We have used the electronic (T e ) and ionic (T i ) temperatures T e = T i =3keV; electronic density, N e = 5 × 10 13 cm -3 , and plasma ions density, N i = 4 × 10 13 cm -3 , with Z eff = 1.7. The magnitude of the magnetic field is 3 T. The emission coefficients for the 7→6 transition are shown in figure 7. There is a significant difference between the coefficients q 1 7→6 at E < 80 keV/amu which is due to the differences found in [5] between the corresponding cross sections; those stored in ADAS were obtained by means of an interpolation using results from atomic-orbital-close-coupling (AOCC) calculations of ref. [22] for E ≤ 50 keV/amu and unitarized-distorted-wave (UDW) calculations of ref. [23] for E > 50 keV/amu. Besides the deviations caused by the interpolation, the discrepancies between the ADAS and present cross sections can be related to the difficulties that face AOCC calculations in providing accurate cross sections for populating high-lying capture channels at low E. These channels are populated at small internuclear distances through rotational and radial transitions from lowest channels, and at large R's through a serie of adjacent (molecular) pseudocrossings (see Fig. 2). The AOCC method can be faced with difficulties in describing the molecularization of the electron cloud which comes into play in both small-and large-R mechanisms [24]. Reversely, two-center effects are well represented in both MOCC and CTMC frameworks which are presently used to build our recommended cross sections and calculate the emission coefficients. For E > 80 keV/amu, our computed and ADAS coefficents suitably coalesce since improved CTMC and perturbative methods such as UDW are known to yield similar CX cross sections (see, for instance, [25]).

Our effective emission coefficients q 2 7→6 are similar to those computed from ADAS data. Such an agreement was to be expected since the underlying cross sections stored in ADAS are issued from microcanonical CTMC calculations [6] which, as above mentioned, provide CX cross sections in good agreement with the present calculations (see Fig. 5).

The values q 3 7→6 , displayed in Fig. 7 as function of the impact energy E, are of the same order of magnitude than q 2 7→6 . This indicates that the relevance of H(n = 3) in the CXRS diagnostic will be relatively small, unless a very high beam excitation takes place.

Summary

We have evaluated CX cross sections in B 5+ + H(n i = 2) collisions using a close-coupling semiclassical molecular treatment and an eikonal-CTMC treatment with an improved initial distribution. In contrast with previous calculations, our results agree with the values obtained using a microcanonical distribution and with the CTMC values available in the ADAS database. This agreement is due to the fact that CX from excited H mainly occurs by means of overbarrier transitions whose description does not require the use of an improved initial CTMC state.

The effective coefficients for the B 4+ n = 7 → n = 6 emission after B 5+ + H(1s) CX show a significant deviation with respect to the ADAS values for E < 50 keV/amu. We traced back the root of this discrepancy to the inadequacy of the underlying cross sections stored in ADAS.

We have also evaluated CX cross sections in B 5+ + H(n i = 3) using the eikonal-CTMC treatment. The ensuing effective emission coefficient for ASDEX-U plasma parameters is of the same order of magnitude than the corresponding one for CX collisions with H(n i = 2) donor. We conclude that CX from H(n i = 3) could be relevant in high-density conditions, as those of the Alcator C-Mod tokamak [8]. 7→6 with n i = 1 -2 (dashed lines with squares), compared with those evaluated using the CX cross sections stored in ADAS (full lines). The line with triangles are the calculated values of q 3 7→6

Figure 1 .

 1 Figure 1. Comparison between the exact quantum mechanical initial distributions for H(2s) and H(2p) and the corresponding gaussian distributions (eqs. (13),(15). For comparison purposes the microcanonical distribution for H(2s) is also shown.

2 .

 2 Figure 2. Effective quantum numbers n k eff = 6/ √ -2E k as functions of the internuclear distance R for the OEDM of the BH 5+ quasimolecule.

Figure 3 .

 3 Figure 3. Total cross sections for CX into Be 4+ (n = 7) and ionization in collisions of B 5+ with H(n=2), calculated using CTMC and OEDM treatments.

Figure 4 .

 4 Figure 4. Partial cross sections for populating B 4+ (nl) sublevels with n = 8 (panels (a) and (b)), n = 7 (panels (c) and (d)) and n = 6 (panels (e) and (f)), in collisions of B 5+ with H(2p) (panels (a), (c) and (e)) and H(n = 2) (panels (b), (d) and (f)). The highest line on each panel is the corresponding σ n cross section obtained by adding the corresponding σ nl while the triangles are the values obtained by joining the classical and semiclassical values for σ n .

Figure 5 .

 5 Figure5. Total cross sections for CX into Be 4+ (n) and ionization in collisions of B 5+ with H(n=2). The lines are the recommended data obtained joining classical and semiclassical calculations. The triangles are the results of a classical calculation with a microcanonical initial distribution and the asterisks correspond to the data stored in ADAS, which are issued from the microcanonical CTMC calculations of[6].

Figure 6 .

 6 Figure 6. Total cross sections for CX into B 4+ (n) and ionization in collisions of B 5+ with H(n=3) evaluated using the CTMC method with a microcanonical initial distribution.

Figure 7 .

 7 Figure 7. Effective emission coefficients q ni7→6 with n i = 1 -2 (dashed lines with squares), compared with those evaluated using the CX cross sections stored in ADAS (full lines). The line with triangles are the calculated values of q 3

Acknowledgments

This work has been supported by the project ENE2007-62934 of the Ministerio de Ciencia e Innovación (Spain).


Zachary B Walters
email: zwalters@gmail.com

Olga Smirnova
email: olga.smirnova@mbi-berlin.de

Attosecond correlation dynamics during electron tunneling from molecules

Keywords: numbers: 42.50.Hz, 32.80.Rm, 33.80.Wz multichannel strong field ionization, attosecond multielectron dynamics, core rearrangement, molecular orbitals

We present analytical theory of strong field ionization of molecules, which takes into account rearrangement of multiple interacting electrons during the ionization process. We show that such rearrangement offers an alternative pathway to the ionization of orbitals more deeply bound than the Highest Occupied Molecular Orbital (HOMO). This pathway is not subject to the full exponential suppression characteristic of the direct tunnel ionization from the deeper orbitals. The departing electron produces an "attosecond correlation pulse" which controls the rearrangement during the tunneling process. The shape and duration of this pulse are determined by the electronic structure of the relevant states, molecular orientation, and the laser parameters.

Tunnel ionization in static electric fields is exponentially sensitive to ionization potential I p : ionization from deeper bound states is exponentially suppressed. Similar suppression is commonly expected during ionization in intense low-frequency laser fields [1]. It has thus been a natural and nearly universal assumption that only the least bound electron may escape the atom or molecule, leaving the resulting ion in its lowest electronic state. We show that the conventionally assumed exponential suppression of ionization rates is strictly valid only if the state of the ion is frozen during the tunneling process. The ionization dynamics change if the state of the ion is allowed to evolve, reflecting the interaction between the departing electron and the ionic core. We show that correlated electron dynamics during ionization provides a new pathway to the creation of excited states of the ion, which is not subject to the full exponential suppression accompanying direct tunneling from a more deeply bound state.

One would think that the importance of excited electronic states of the ion during ionization could be easily verified by examination of photoelectron spectra. Indeed, different ionization channels, corresponding to different I p 's should lead to photoelectron peaks with different energy. While this is indeed the case for one-photon ionization, highly non-linear nature of strong-field processes make observation of the alternative ionization channels quite challenging in the strong field regime [2].

The first experiment [3] aimed at observing ionization channels with higher I p during strong-field ionization of molecules has been performed with H 2 . However, the large (about 20 eV) energy needed to excite the H + 2 ion resulted in a vanishingly small (10 -6 ) excited state population [3]. An important problem arising in such experiments is the need to separate the creation of excited ions during and after ionization. Recent studies of strong field ionization of hydrocarbon molecules [4] solved this problem by using covariance technique, correlating energies in the photoelectron spectra with the ionic fragments. These experiments have unambiguously demonstrated the contribution of several ionization channels corresponding to the population of different electronically excited states during ionization. Together with an avalanche of alternative experimental evidence [5,6,7,8,9,[START_REF] Leone S Ultrafast | Attosecond Transient Absorption with High Order Harmonics[END_REF][START_REF] Arissian | [END_REF], the body of new experimental data poses challenges to the theory of strong field ionization.

While strong field ionization of atoms is well understood (see e.g. [13,[START_REF] Popruzhenko | [END_REF]) in the single channel limit, a quantitative explanation of single channel ionization of molecules or an accurate treatment of multiple coupled channels represent problems at the frontier of strong field theory (see e.g. [14,15,16,17,18,[START_REF] Becker | Physical Review[END_REF]20,21,22,23]). The importance of channel coupling due to the laser field has been emphasized theoretically in [9,20,23]. Ref. [9] suggested the importance of electron correlations during tunnelling and provided the first evidence of non-trivial dynamics induced by this coupling.

We present a multichannel generalization of the strong field ionization theory originally proposed in the classic papers of Popov, Perelomov and Terent'ev (PPT) [13]. We include channel coupling induced by electronic interactions between the departing electron and the electrons left behind in the molecule. Interactions with the departing electron produce an "attosecond correlation pulse" which may drive transitions between ionic states during the tunnelling process. The shape and duration of this pulse are determined by (i) the electronic structure of the relevant states and (ii) the parameters of the laser field initiating ionization. The resulting transitions present alternative pathways to removing an electron from more deeply bound orbitals, which are not subject to the full exponential suppression of the direct pathway.

The full Hamiltonian of an N electron molecule interacting with a laser field is given by

H N = T N e + V N C + V N ee + V N l , V N C = ∑ i=N m,i=1 |R m -r i | -1 , V N ee = ∑ N i̸ =j |r i -r j | -1 , V N l = ∑ i E(t)•r i ,
where the atomic nuclei are frozen at positions R m , index m runs over the nuclei, superscript N indicates the number of electrons involved, T N e is the electron kinetic energy operator, V N C describes the Coulomb potential of the nuclei, V N ee describes the electron correlations, V N l describes interaction with the laser field. It is convenient to define the Hamiltonian of the ion in the laser field

H N -1 = T N -1 e +V N -1 C +V N -1 ee +V N -1 l ,
and the Hamiltonian for a single electron interacting with the laser field and the molecular ion H e = H N -H N -1 . The exact solution for the N electron wavefunction of a molecule, which starts from the ground electronic state Ψ N g (r), can be written as [24]:

Ψ N (r, t) = - ∫ t 0 ∫ t 0 dt ′ dt ′′ U N (t, t ′′ )V ee (t ′′ )U N 1 (t ′′ , t ′ )V N l (t ′ )U N 0 (t ′ , 0)Ψ N g (r) - -i ∫ t 0 dt ′ U N 1 (t, t ′ )V N l (t ′ )U N 0 (t ′ , 0)Ψ N g (r) + U N 0 (t, 0)Ψ N g (r). (1)
Here

V ee = V N ee -V N -1
ee describes the interaction of the departing electron with all the electrons in the ion. N-electron propagators U N 0 and U N 1 are determined as

i U N i = H N i U N i , (i = 0, 1)
, where H N 0 is the field-free Hamiltonian of the molecule

H N 0 = H N -V N l and H N 1 = H N -V ee .
Since ionization occurs in the presence of a strong laser field, it is necessary to introduce basis states which incorporate the effects of the laser field. Therefore, we write propagators in equation ( 1) using the basis of field-dressed (Floquet) electronic states of the ion |n N -1 t ⟩ and channel specific one-electron continuum states |k n t ⟩. The evolution of the (N -1)-electron Floquet states is governed by the Hamiltonian H (N -1) . The Hamiltonian for the single electron includes a channel specific Hartree potential

V n H = ⟨n N -1 t |V N ee |n N -1
t ⟩, the Coulomb potential of the nuclei, and the laser field. Note that the Hartree potential is defined for field-dressed states of the ion and includes polarization of the core by the laser field.

Ionization amplitudes are found by projecting the N electron wavefunction (1) onto a basis of (N -1)-electron ionic states and single electron continuum states. The integrals are calculated using the saddle point method, generalizing the approach of Ref. [13] to the multi-channel case. Just as in Ref. [13], the effect of the ionic core is included in the eikonal approximation, and the saddle points of these integrals are found in the zeroth order of the iterative saddle point method [28,29]. The saddle point equations define a trajectory followed by the departing electron:

(k f + A(t 0 )) 2 /2 = -I p , r 0 (t) = ∫ t t 0 (k f + A(τ ))dτ
, where the laser electric field E(t) ≡ -∂A(t)/∂t. These conditions describe an electron which begins tunnelling at (complex) time t 0 and exits the subbarrier region at a time of birth on the real axis. †

The final result for the ionization amplitude is the sum of direct and indirect terms:

⟨k m f m (N -1) t |Ψ N (r, t)⟩ = a (d) mm (k m f ) + ∑ n a (i) mn (k m f ). (2)
The direct and indirect ionization amplitudes are:

a (d) m (k f , t) = b mm (t, t 0 )a (P P T ) m (k f , t), b mm = ⟨m (N -1) t |U N -1 (t, t 0 )|m (N -1) t 0 ⟩, ( 3
)
a (i) mn (k f , t) = c mn (k f , t, t 0 )a (P P T ) n (k f , t), c mn = -i ∫ t t 0 dt ′′ ⟨m N -1 t ′′ |V ee (r 0 (t ′′ ))|n N -1 t ′′ ⟩ (4)
Here V ee (r 0 (t ′′ )) denotes electron-electron correlation operator V ee with the coordinate of the tunnelling electron substituted by the saddle-point subbarrier trajectory. The matrix elements from this operator calculated between the laser-dressed (Floquet) states of the ion

|n N -1 t ⟩, |m N -1 t
⟩, determine the correlation potential:

V corr mn (t) = ⟨m N -1 t |V ee (r 0 (t))|n N -1 t ⟩, (5)
i.e. the effective field that induces transitions in the ion [30,31]. Both direct and indirect ionization amplitudes are proportional to the standard PPT single-channel amplitude a

(P P T ) l (k f , t), corresponding to the parent channel |l t ⟩. The direct amplitude a (d)
m (k f , t) corresponds to the traditional single channel ionization amplitude, in which an electron ionizes and propagates in the channel specific Hartree potential without causing the ion to change its state. The factor b mm (t, t 0 ) reflects a correction to the conventional tunneling amplitude due to laser-induced channel interaction. In the quasistatic limit, this corresponds to the Stark shift. In our analysis, the time integration is performed using complex time, and b mm (t, t 0 ) may be interpreted as the norm of Floquet states following non-Hermitian dynamics in the complex interval between t and t 0 [START_REF] Moiseev | [END_REF].

Our focus is the indirect amplitude a (i) mn (k f ) equation ( 4), which reflects changes in the ionic state due to electronic correlation during ionization. At time t ′′ , interaction with the tunneling electron causes the ion to change its state from |n⟩ to |m⟩. The expression for the off-diagonal matrix element c mn shows that after t ′′ , the tunneling electron propagates in the Hartree potential of the new state. Naturally, all t ′′ between t 0 and t contribute to c mn . The channel coupling is described in the lowest order with respect to electron correlations and in all orders with respect to the laser field.

Because t 0 satisfying the saddle point conditions will in general be complex, propagation of a state with positive I p to the real axis yields the familiar exponential suppression of tunneling with respect to energy. However, the expression for c mn involves not one but two propagation steps, from t 0 to t ′′ and from t ′′ to the real axis. Since the ion may change its state at time t ′′ due to the correlation potential, the indirect channel will have an exponential suppression which is intermediate between those of the more-and less deeply bound states. Excitations occurring at t ′′ near t 0 will be more † For the laser field E(t) = E 0 cos(ωt) and ionization at the maximum of the laser field r 0 (t) = E 0 /ω 2 (1 -cosh(ωt)) [13].

suppressed than excitations which occur near the real time-axis, i.e. close to the exit from the barrier. The indirect ionization mechanism thus represents a pathway to the ionization of deeply bound orbitals which is not subject to the full exponential energy suppression of the direct pathway. Neglecting laser-induced dynamics in the ion for simplicity, the ratio of indirect and direct pathways leading to the same final state of the ion m is given by: e ∆Ipτ ∫ τ 0 dt ′′ V corr mn (t ′′ )e -∆Ipt ′′ , where V corr mn (t) = ⟨m|V ee (r 0 (t))|n⟩ is a matrix element of electron-electron correlation operator along the subbarrier trajectory calculated between the field-free states of the ion, ∆I p is the energy gap between these states, τ is the imaginary part of t 0 , refereed to as tunneling time.

To gauge the importance of the indirect ionization pathway, we calculated the ratio of indirect ionization amplitude to the direct amplitude for the parent channel:

a (i) mn (k f ) = a (i) mn (k f )/a (d) n (k f ) = c mn (k f )/b nn (t, t 0 ). ( 6
)
for N 2 and CO 2 molecules. Following convention [6,9], we name the essential channels X, A and B in order of increasing ionization potential. Here we focus on indirect ionization pathways leading to excitation of the cation upon removing the electron from HOMO and thus contributing to ionization from deeper molecular orbitals. Deexcitation pathways, leaving the cation in the ground state upon removal of electron from deeper orbitals and thus contributing to ionization signal from HOMO will be considered elsewhere. Calculations were performed for an 800 nm laser with intensity I=1.7 10 14 W/cm 2 for electrons leaving the molecule at the peak of the electric field. The vertical ionization potentials for channels X, A and B are known spectroscopically: 15.6 eV, 16.9 eV, 19.1 eV for N 2 and 13.8 eV, 17.3 eV, 18.1 eV for CO 2 . To include the effects of the laser field, we use complete active space self-consistent field (CASSCF) values for the transition dipole matrix elements between ionic states calculated in [6,9]. For N 2 , the dipole coupling vectors between the ionic states are d CAS XA = (0.25, 0, 0), d CAS XB = (0, 0, 0.72), while for CO 2 , they are d CAS XA = (0, 0, 0.46), d CAS XB = (0.27, 0, 0). The first component in the brackets is perpendicular and the last is parallel to the molecular axis. Note that the Hartree-Fock approximation often overestimates the strength of dipole couplings: for the XA transition in CO 2 by a factor of 3.5, for the XB transition in N 2 by a factor of 2.2. Using the Hartree Fock dipoles thus lead to a strong overestimation of laser-induced dynamics in molecular ions N 2 and CO 2 . Because the correlation potential, calculated in the Hartree-Fock approximation as

V corr mn (t) = ∫ dr ϕ * m (r)ϕn(r)
|r 0 (t)-r| , n, m = X, A, B, asymptotes to V mn (t) ≈ d mn /r 2 0 for large r 0 , it is likely that this correlation potential will lead to an overestimation of correlation effects. To achieve the correct asymptotic value of the correlation potential, we scale the value of the correlation potential by the ratio of CASSCF and the Hartree-Fock transition dipoles.

Because the indirect channel results from electronic correlation, it is strongly affected by the electronic structure of the ion in the field of the laser. This may be seen in three ways. First, the enhanced exponential suppression of the excitation pathway from t ′′ to the real time-axis means that highly energetic excitations, or excitations which occur at large imaginary times, are strongly suppressed. Second, the sign of the correlation potential may change as the electron passes through an orbital node while exiting the molecule. Third, laser-induced polarization may lead to substantial modification of orbital geometry. All of these effects may be seen by comparing CO 2 to N 2 at appropriate molecular orientations.

The effects of exponential suppression and laser-induced polarization may be seen by comparing N 2 aligned at 90 • to the laser electric field to CO 2 aligned at 0 • . At both orientations, only the X and A channels contribute to ionization. AX | = 0.5 for CO 2 , reflecting exceedingly high value of d HF XA in the Hartree-Fock approximation. A similar artefact might be responsible for strong laserinduced coupling between X and A channels in CO 2 reported in Ref. [23].

In reading Figs 1,2, note that tunneling starts at (imaginary) τ = 13 a.u. and the (imaginary) time decreases reaching real time-axis at t=0, and hence the correlation potential evolves from right to left. The "later" the transition from channel X to channel A occurs, the smaller is the effect of exponential suppression, with the correlation potential at early times (t ∼ τ ) suppressed the most.

The orbital nodal structure may strongly affect the temporal shape of the correlation potential. This effect may be seen by rotating CO 2 molecules 30 • away from the laser polarization axis allowing for ionization into the B channel. The correlation potential V corr BX (Figure 2(a), black curve) changes sign as the electron passes through a node in the B orbital. In contrast, there is no nodal plain in A orbital for the electron leaving the molecule at 30 • to the molecular axis and the sign of the correlation potential V corr AX remains the same throughout (see Fig. 2(a), red curve). Because of the node in the field-free V corr BX (t), the laser-dressed V corr BX (t) remains small in the initial moments after ionization (Fig. 2(b), black solid curve). Since indirect ionization amplitude is controlled by the temporal structure of the laser-dressed correlation potential V BX (t), the effect of the node is to inhibit a This paper provides first theoretical analysis of attosecond dynamics of electron correlation during tunnel ionization. Calculations of indirect ionization amplitude are essentially analytic, with the ratio of indirect to direct channel given by simple integral of the correlation potential. Since multichannel strong field ionization is a key to recording ultrafast multielectron dynamics in molecules [6,9], its understanding will have broad impact on control and imaging of these dynamics on attosecond time scale. We gratefully acknowledge stimulating discussions with M. Yu. Ivanov and SAW grant of the Leibniz society.

Figure 1 .

 1 Figure 1. (a) Field-free correlation potentials V corr AX (t) vs (imaginary) time for CO 2 aligned at 0 o (dashed black) and N 2 aligned at 90 o (dashed red). Solid curves show the effect of exponential suppression V corr AX (t)e -∆I AX

  Fig 1(a) shows the correlation potential calculated between field free Hartree Fock states, scaled to agree with CASSCF dipoles at long range. Dashed lines show the correlation potential, while solid lines show the correlation potential multiplied by the suppression factor e -∆I AX p t . Due to the much smaller energy energy gap ∆I AX p between states X and A in N + 2 , the correlation potential experiences much less suppression in N 2 than in CO 2 . The effects of laser polarization may be seen by comparing the field free correlation potential of Fig 1(a) to the one calculated using field-dressed states shown in Fig 1(b). Here the closely spaced levels of N 2 result in strong orbital mixing, while the more widely spaced orbitals of CO 2 are less affected. After accounting for both polarization and energy suppression, the indirect amplitude for N 2 , | a (i) AX | = 0.28, is almost two times larger than for CO 2 , | a (i) AX | = 0.17. The non-scaled correlation potential (proportional to the Hartree-Fock dipole in long range) yields | a (i)

Figure 2 .

 2 Figure 2. (a) Field-free Vcorr AX (t) (red), V corr BX (t) (black) and (b) laser dressed V corr AX (t) (red), V corr BX (t) (black) correlation potentials vs (imaginary) time for CO 2 aligned at 30 o .Ionization starts at τ =13 a.u.

  BX for CO 2 yielding | a (i) BX | = 0.11.

  V corr AX (t) vs (imaginary) time for CO 2 aligned at 0 o (dashed black) and N 2 aligned at 90 o (dashed red). Solid curves show the effect of exponential suppression V corr AX (t)e -∆I AX p t vs (imaginary) time for CO 2 (black) N 2 ( red). Ionization starts at τ =13 a.u. (b) Laser-dressed correlation potentials V corr AX (t) with (solid) and without (dashed) exponential suppression for CO 2 oriented at 0 o ( black) and N 2 oriented at 90 o (red)