Theory of single-photon bound–free transitions

J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 5141–5156. Printed in the UK

PII: S0953-4075(97)83553-5

Theory of single-photon bound–free transitions: extension of the pole approximation A G Kofman Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel Received 22 April 1997, in final form 12 August 1997 Abstract. A theory of single-photon ionization, detachment, and dissociation of quantum systems (atom, molecule, ion, etc) is presented. A unified treatment of excitation from the ground and excited states is performed, via a consideration of the rotating- and counter-rotatingwave terms on equal footing. The familiar pole approximation (PA) is extended to obtain higher-order corrections, which allows one to establish the PA validity conditions and provide a unified description of the cases of above- and below-threshold excitation. As a specific example, photodetachment of negative ions is considered. An exact long-time solution is obtained for the general case. A familiar model of photodetachment is used to validate the present PA formalism. The latter has allowed us to derive the PA validity condition, which holds for all values of the parameters, involving quite different situations. The applicability of the existing versions of the rotating-wave approximation has been checked.

1. Introduction The advent of laser sources of radiation made possible a systematic study of photoinduced atomic and molecular processes, such as photoionization [1] and photodissociation [2]. For moderate intensities and sufficiently short wavelengths of the field, photoionization of an atom or photodissociation of a molecule occurs upon absorption of one photon. Singlephoton bound–free transitions were studied for the cases of rectangular [1, 3–7] and smooth [1, 8–12] laser pulses. The case of rectangular pulses, which is considered here, has been studied rather thoroughly, due to its relative simplicity and close connections to such topics as quantum mechanics of unstable states [13, 14] and spontaneous decay [15–17]. In the case of a smooth continuum the perturbative approach (the Fermi golden rule) [18] or the more elaborate Wigner–Weisskopf (or pole) approximation [15] can be used. Therefore attention has been mainly attracted to cases, where excitation is in near resonance with sharp features in the continuum [1], such as a photoionization or photodetachment threshold or a structure due to an autoionizing state. In particular, such strong-coupling effects as nonexponential depletion dynamics, a dynamic threshold, population trapping, and bound dressed states have been revealed and discussed. Until now strong-coupling effects have been studied in the frame of simple models, which allow for exact solutions. Moreover, simplifying assumptions, such as the rotatingwave approximation (RWA) or its modification (called here RWA1), in which the counterrotating-wave interaction is allowed for by a field-dependent shift of the initial state [3], has been made. The RWA has been used with great success to describe strong, resonant interactions of light with transitions between discrete atomic levels [19]. However, since widths of material continua are usually of the order of or greater than the laser frequency, the c 1997 IOP Publishing Ltd 0953-4075/97/225141+16$19.50

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applicability of the RWA or even the RWA1 to bound–free transitions is not so obvious as in the case of bound–bound transitions. Even under simplifying assumptions, exact solutions are rarely possible and those obtained are involved and contain many parameters, which complicates the analysis. It would be desirable therefore to have a simple method to obtain the exact boundary between the weak-coupling regime, where simple, general solutions are possible, and the strong-coupling regime where a special consideration is required. In this paper a general theory of single-photon photoionization, photodetachment, and photodissociation is developed, which takes into account all the states of the quantum system of interest (atom, molecule, or ion), belonging both to the discrete and continuous spectra (section 2). Bound–free transitions from both the ground and excited states are treated in a unified manner. Rotating- and counter-rotating-wave interactions are treated on equal footing. The smoothing procedure is used to derive the basic equations. This approach is more general than the RWA or RWA1. Its validity condition is obtained. The pole approximation (PA) is a simple and general way of treating bound–free transitions. It is known to provide an exponential dynamics, with the decay rate scaling as the field intensity [20]. Due to the importance of this approach, it seems warranted to attempt to obtain higher-order corrections to the PA results. This is performed in section 3. As a result, a quantitative criterion of the PA is established. Moreover, a general solution for below-threshold excitation is derived. In section 4 the present PA formalism is applied to photodetachment of negative ions. The PA results are verified against more exact solutions. The PA formalism yields a dimensionless combination of relevant parameters (the coupling parameter), the value of which characterizes the coupling strength. Moreover, the RWA and RWA1 are discussed and shown to be insufficiently accurate in certain cases. Section 5 provides concluding remarks. 2. General theory The theory developed below is equally applicable to transitions in atoms, molecules, and ions, though, for definiteness, the quantum system under consideration below is often called the atom. Consider interaction between an atom and an external field E = ǫEe−iωL t + ǫE ∗ eiωL t E(t)

(2.1)

where ωL and ǫE(t) are the frequency and complex amplitude of the field respectively. We consider rectangular pulses, which are sufficiently long to contain many cycles of electromagnetic field, ωL T ≫ 1, where T is the pulse length. The Hamiltonian describing the field–atom interaction is E Hint = −dE · E(t).

(2.2)

Here dE is the dipole moment of the atom. Supposing that the atom is initially, at t = 0, in the bound (but not necessarily ground) state |ψg i with energy Eg , the wavefunction of the atom has the form X   Z ∞ X −iωL t dE |9(t)i = bg (t)|ψg i + e βEm (t)|ψEm i e−iEg t/¯h (2.3) βn (t)|ψn i + n

Ec

m

where |ψn i (n 6= g) are the wavefunctions corresponding to the discrete energy levels En , |ψEm i are the wavefunctions of the continuum states characterized by the energy E and the set of quantum numbers m, and Ec is the lower boundary of the continuous spectrum. We assume that the continuum wavefunctions are normalized by hψE ′ m′ |ψEm i = δ(E − E ′ )δmm′ .

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Inserting equation (2.3) into the Schro¨ dinger equation yields for bg (t) and βλ (t) (where λ stands for n and {E, m}) the following equations Z b˙g = −i dλ [∗λ,1 + ∗λ,−1 e−2iωL t ]βλ (2.4a) β˙λ = −i(ωλg − ωL )βλ − i[λ,1 + λ,−1 e2iωL t ]bg .

(2.4b)

E g i and Here ωλg = (Eλ − Eg )/¯h, λ,1 = −dEλg · ǫE/¯h, λ,−1 = −dEλg · ǫE∗ /¯h, dEλg = hψλ |d|ψ for any fλ Z Z ∞ X X dλ fλ = dE fEm + fn . (2.5) Ec

m

n

In deriving equations (2.4) only the transitions involving the state |ψg i were taken into account. Solving equation (2.4b) formally, one obtains that the amplitudes of the final states are given by Z t X ′ dt ′ ei(j ωL −ωλg )(t−t ) bg (t ′ ). (2.6) λ,j ei(1−j )ωL t βλ (t) = −i 0

j =±1

Hence the problem has been reduced to obtaining the quantity bg (t). Inserting equation (2.6) into (2.4a) results in the equation for bg (t), Z t Z X ′ ei(j −l)ωL t b˙g = − dt ′ dλ ∗λ,j λ,l ei(lωL −ωλg )(t−t ) bg (t ′ ) j,l=±1

≡−

X

0

ei(j −l)ωL t Ij l (t).

(2.7)

j,l=±1

Equation (2.7) can be treated by the method similar to that used in [5], as follows. Assume in the first approximation that bg (t) is sufficiently smooth, then the functions Ij l (t) are slowly varying, since the integration over time averages out fast oscillations. Therefore the terms with j 6= l on the right-hand side of equation (2.7) oscillate with the frequency 2ωL , giving rise to fast oscillating terms with small amplitudes in the function bg (t). On smoothing bg (t) over time intervals on the order of ωL−1 , the above terms with j 6= l can be eliminated, as follows. Integrating both sides of equation (2.7) over t from t to t + 1t, where 1t is such that ωL−1 ≪ 1t ≪ t, and taking into account that the functions Ij l (t) vary insignificantly over the interval 1t, one obtains   bg (t + 1t) − bg (t) 1 − e−2iωL 1t 1 − e2iωL 1t ≈ − I11 (t) + I−1,−1 (t) + I−1,1 (t) − I1,−1 (t) . 1t 2iωL 1t 2iωL 1t (2.8) The third and fourth terms on the right-hand side of equation (2.8) are much less than the first and second terms respectively and hence can be neglected in the first approximation. Let, for simplicity, either the dipole moment matrix elements dλg or the unit polarization vector eˆ = ǫE/|Eǫ | be real. Then equation (2.8) becomes Z t dt ′ cos[ωL (t − t ′ )]F (t − t ′ )bg (t ′ ). (2.9) b˙g = −2I 0

Here I = |Eǫ |2 is proportional to the field intensity and Z Z ∞ X dEA(E)e−iωEg t + An e−iωng t F (t) = dE AE e−iωEg t ≡ Ec

n

(2.10)

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where ωEg = (E − Eg )/¯h, ωng = (En − Eg )/¯h, X AE = A(E) + An δ(E − En )

(2.11)

n

is the spectral density of the coupling coefficient, and A(E) =

1 X E |dEm,g · e| ˆ 2, h ¯2 m

An =

|dEng · e| ˆ2 . 2 h ¯

(2.12)

Equation (2.9) takes into account all states of the quantum system as well as the counterrotating terms. The method of the derivation implies that equation (2.9) holds if the characteristic rate of change of bg (t), i.e. the magnitude of the field-induced (complex) energy shift h ¯ δg of the state |ψg i divided by h ¯ , is much less than ωL |δg | ≪ ωL .

(2.13)

This approximation is called here the smoothed-dynamics approximation (SDA). Equation (2.9) can be solved by the Laplace transform, yielding ˆ b(s) = [s + K(s)I ]−1 (2.14) R∞ −st ˆ Here b(s) = 0 dt bg (t)e is the Laplace transform of bg (t) and the level-shift parameter K(s) = Kr (s) + Kcr (s)

(2.15)

Z

(2.16)

where Kr,cr (s) =

AE dE . s + i(ωEg ∓ ωL )

Note that the Laplace transform of equation (2.7) can be solved in terms of continued fractions [4, 7]. Then equation (2.14) is obtained formally, by neglecting certain factors in the solution [7]. The above derivation of equation (2.14) shows explicitly the physical assumptions leading to the SDA. 2.1. Discussion If |ψg i is the ground state, then Kr (s) and Kcr (s) can be considered as arising from the rotating- and counter-rotating-wave parts of the interaction Hamiltonian respectively. However, if |ψg i is an excited state, the terms in Kcr (s) corresponding to the states with En < Eg have generally resonant denominators. Such terms should be interchanged in Kr (s) and Kcr (s), if one wants to preserve the above interpretation of the latter parameters. This is important if one is to simplify the treatment of nonresonant terms, as in the following standard treatments (for convenience, the simplified approaches are discussed below for the case of the ground initial state |ψg i). In the RWA, the term Kcr (s) is dropped in equation (2.15). The simplest way to allow for the counter-rotating-wave interaction is to set [1, 3] Kcr (s) ≈ Kcr (0) = −iαcr

(2.17)

where αcr =

Z

AE dE . ωEg + ωL

(2.18)

Then equation (2.14) becomes

ˆ b(s) = [s − iαcr I + Kr (s)I ]−1 .

(2.19)

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Thus the effect of the counter-rotating part of the interaction Hamiltonian is reduced to a negative contribution −¯hαcr I to the AC-Stark shift of the state s. Below this approximation is called the RWA1. The limitations of the RWA and RWA1 are discussed below. In the subsequent analysis, both resonant and nonresonant terms are treated on equal footing, the starting point thus being the SDA result (2.14). Note that equation (2.14) can be interpreted in terms of an effective Hamiltonian with the rotating-wave interaction between the atom and the field. Indeed the term K(s) can be recast in the rotating-wave form, Z BE dE (2.20) K(s) = s + i(ωEg − ωL ) where BE = AE + AE−2¯hωL is the spectral density of the effective coupling coefficient. The latter relation shows that the effective atomic Hamiltonian involves, in addition to the states of the initial Hamiltonian, the same final states, but with the energies shifted upwards by 2¯hωL . BE can be recast as the sum of the contributions from the continuous and discrete spectra (cf equation (2.11)), X Bi δ(E − Ei ). (2.21) BE = B(E) + i

Here

B(E) = A(E) + A(E − 2¯hωL )

(2.22)

and X i

Bi δ(E − Ei ) =

X n

An [δ(E − En ) + δ(E − En − 2¯hωL )].

(2.23)

Note that the above theory excludes from consideration all transitions which do not involve the initial state. As a result, such phenomena as above-threshold ionization [21], interference of the initial state with neighbouring levels [22], and the AC-Stark shift of the final states are beyond the scope of this theory. These effects (at least, the two former ones) can be neglected for sufficiently weak and long laser pulses. Further discussion of them is beyond the scope of this paper. The problem is thus reduced to performing the inverse Laplace transform of equation (2.14). The details of the solution depend generally on the problem in question. However, in the case of the weak-coupling regime there is a general approach to the problem—the PA, which is considered in the following section. 3. Pole approximation In the case of a sufficiently smooth continuum one can use the Wigner–Weisskopf approximation or PA [1, 15]. In this section corrections to the PA will be obtained, which will allow us to establish exact validity conditions of the PA. Moreover, a unified description of the bound–free transitions for below- and above-threshold excitation will be obtained. We are interested in the exponential solution for bg (t). Proceeding similarly to the analysis of the Wigner–Weisskopf approximation for spontaneous decay [17, 23], we solve ˆ iteratively the equation for the poles of b(s) (cf equation (2.14)), s = −K(s)I.

(3.1)

Assuming that the field–atom interaction is sufficiently weak, in the zero approximation we obtain the position of the pole sp[0] = 0.

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In the next approximation one should insert s = sp[0] into the right-hand side of equation (3.1). Since the function (2.14) (and hence K(s)) is analytic in the half plane Re s > 0, one should perform the analytic continuation of K(s) to the imaginary axis by the limit [17] K(−iω) = lim+ K(η − iω) = π¯hBEg +¯h(ω+ωL ) − if (ω). η→0

Here f (ω) = αr (ωL + ω) + αcr (ωL − ω), where αr (ω) and αcr (ω), Z dEAE αr,cr (ω) = ωEg ∓ ω

(3.2)

(3.3)

are the ‘rotating’ and ‘counter-rotating’ contributions to

α(ω) = αr (ω) + αcr (ω). (3.4) P Here α(ω) = i,k αik (ω)eˆi∗ eˆk , where αik (ω) is the dynamic polarizability tensor of the atom [24]. Finally, one obtains sp[1] = (−aL + iαL )I.

(3.5)

aL = π¯hA(Eg + h ¯ ωL ).

(3.6)

αL = α(ωL ) = αr + αcr

(3.7)

Here aL = π¯hB(Eg + h ¯ ωL ) = π¯h[A(Eg + h ¯ ωL ) + A(Eg − h ¯ ωL )]. Since the initial state is assumed to be below the continuum edge, Eg < Ec , the latter equation yields Furthermore, in equation (3.5)

where αr = αr (ωL ). One can obtain higher-order corrections to the solution (3.5), by continuing to solve equation (3.1) iteratively. To this end, one can use the analytic continuation of K(s) to the left half plane, Re s < 0, given [23] by the Taylor expansion around a point s = −iω, ∞ X (is − ω)n (3.8) [π¯hBE(n)g +¯h(ω+ωL ) − if (n) (ω)] K(s) = n! n=0

where the superscript ‘(n)’ stands for the nth derivative with respect to ω. In equation (3.8), we choose ω = isp[0] = 0. As a result, e.g., the third-order approximation yields   1 d2 [z 3 (ω)] 1 d[z 2 (ω)] [3] . (3.9) + sp = i z − 3! dω2 2 dω ω=0

Here z(ω) = iK(−iω)I and z = z(0) = (αL + iaL )I . The higher-order corrections become increasingly complicated. 3.1. Validity conditions The approximation (3.5) for the pole is satisfactory if the higher-order corrections are negligibly small. Consider first the second-order correction to (3.5) (see equation (3.9)), sp[2] = sp[1] [1 − dz(0)/dω]

(3.10)

|dz(0)/dω| ≪ 1.

(3.11)

|Re[sp[1] dz(0)/dω]| ≪ aL I.

(3.12)

Equation (3.10) approximates (3.5) if

However, if aL is small, but nonvanishing, 0 < aL ≪ |αL |, the second-order correction can affect the transition rate. In this case for the applicability of (3.5), in addition to (3.11), one should require that

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Now we require that the third-order correction be much less than the second-order one. In view of equations (3.9) and (3.11), this yields the inequality zd2 z (0)/dω2 ≪ dz(0)/dω.

(3.13)

This condition can be more restrictive than equation (3.11), e.g. when z(ω) is a sum of a large, slowly varying term and a small, fast varying term. Here we will simplify equation (3.13), by writing d2 z(0)/dω2 ∼ [dz(0)/dω]/ l0 , where l0 is the characteristic length of change of dK(−iω)/dω in the vicinity of ω = 0. We assume that l0 ∼ min |ωL − ξu,l |, where Eg + h ¯ ξl and Eg + h ¯ ξu are the nonanalytic points in the effective coupling coefficient BE , nearest to Eg + h ¯ ωL from below and above, ξl < ωL < ξu . Finally, inequalities (3.11)–(3.13) yield, respectively, the following validity conditions of the approximation (3.5) (aL 6= 0): ′ (a) |αr′ − αcr |I + |aL′ |I ≪ 1;

(b) |αL aL′ |I ≪ aL ;

(c) |ωL − ξu,l | ≫ aL I + |αL |I.

(3.14)

Here the primes denote derivatives with respect to ωL . In the case aL = 0 only inequalities (3.11) and (3.13) should be taken into account, yielding† ′ (a) |αr′ − αcr |I ≪ 1;

(b) |ωL − ξu,l | ≫ |αL |I.

(3.15)

Equations (3.14) and (3.15) show that the PA holds if the spectral density of the coupling coefficient (i.e. the weighted density of states) is sufficiently smooth in the vicinity of Eg + h ¯ ωL and the field intensity is sufficiently low. 3.2. Dynamics of the depletion Assuming that bg (t) can be approximated by the contribution of a single pole sp , the inverse st ˆ transform of (2.14) is given by the residue of b(s)e at the pole [25], ! −1 dK esp t (3.16) I bg (t) ≈ 1 + ds s=sp which exhibits exponential depopulation dynamics. The first-order approximation yields (accounting for equation (3.2)) ′ bg (t) ≈ [1 + (αr′ − αcr + iaL′ )I ]−1 e(−aL +iαL )I t .

(3.17)

The present approximation implies inequality (3.14a) (or (3.15a)), which allows us to recast equation (3.17), without a loss of accuracy, as ′ bg (t) ≈ [1 − (αr′ − αcr + iaL′ )I ]e(−aL +iαL )I t .

(3.18)

The population of the initial state is then ′ P (t) ≡ |bg (t)|2 ≈ [1 − 2(αr′ − αcr )I ]e−2aL I t .

(3.19)

The above exponential dynamics, equations (3.18) and (3.19), does not hold for very short and very long times. The short-time behaviour may often have a minor importance, since, as shown below, it describes the decay of a small portion of the initial-state population. However this very fact also allows us to obtain the short-time dynamics if necessary, by setting bg (t ′ ) ≈ 1 on the right-hand side of the integro-differential equation (2.9) and † Note that conditions (3.14c) and (3.15b) guarantee that the solution sp of equation (3.1) is inside a vicinity of s = 0 that does not include nonanalytic points. The latter is necessary if one wants to use the Taylor expansion (3.8) (with ω = 0) to obtain sp .

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integrating both sides of the equation from 0 to t. The short-time behaviour was analysed in such a way for the problem of spontaneous decay [26]. In terms of bound–free transitions, the results of [26] yield that the above exponential dynamics is valid for t ≫ δ −1 , where δ is the characteristic length of change for AE . Taking into account that |aL′ |I . aL I /δ and ′ |I . |αL |I /δ, one obtains that the initial-state amplitude bg (t) changes insignificantly |αr′ −αcr for short times, t . δ −1 , in the frame of the present approximation (see equations (3.14a, b) and (3.15a)). For above-threshold excitation (aL 6= 0), the above dynamics can become invalid at very long times, due to an incomplete depopulation of the initial state and/or power-law tails [1, 3]. Nevertheless, in the region defined by equation (3.14), depopulation for most atoms occurs in the time interval where the above exponential dynamics is valid (see also figure 1, broken curves).

Figure 1. The initial-state population P (t) = |bg (t)|2 as a function of the dimensionless time 2aL I t for different values of the coupling parameter κ under above- and near-threshold excitation. The cases of a (a) broad and (b) narrow continuum are shown. Full and dotted curves are calculated with the RWA1 solution (4.9). Broken curves show the PA prediction (3.21) (in the coordinates chosen the PA curves coincide for all values of κ). The parameters chosen are (a) ωc = 2β, 10 = 0.1β; (b) ωc = 200β, 10 = 10β.

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3.3. Above-threshold excitation Equations (3.18) and (3.19) are applicable both above and below the threshold of the bound– free transitions. Above the threshold equations (3.18) and (3.19) can be further simplified, yielding the familiar PA results bg (t) ≈ e(−aL +iαL )I t

(3.20)

P (t) ≈ e−2aL I t .

(3.21)

and

As follows from equation (3.21), the coefficient aL is proportional to the cross section of the bound–free transitions. The initial-state population (3.21) depends on the fluence of the field, which is proportional to I t. The imaginary component of the exponent in equation (3.20), which describes the ACStark shift [27] of the state |ψg i, is often neglected in common treatments of photoinduced bound–free transitions. However it can affect significantly the reaction-product spectrum, as follows. The spectrum is defined as the t → ∞ limit of the reaction-product energy distribution in the continuum [1]. In this case the term with j = 1 dominates in equation (2.6). Taking into account equation (3.20), one obtains the familiar Lorentzian spectrum X 1 h ¯ aL I |βEm (t)|2 ≈ . (3.22) S(E) ≡ lim 2 t→∞ ¯ ωL − Eg + h ¯ αL I )2 π (¯haL I ) + (E − h m

The integral of this distribution equals one, which corresponds to the fact that in the present approximation, the ultimate decay of the initial state is complete. The width and shift of the spectrum are proportional to the field intensity. The FWHM width of the spectrum equals the decay rate times h ¯ (cf equation (3.21)). The shift of the spectrum, which is intensity dependent and thus easily measurable, equals the AC-Stark shift of the initial state. Note that the decay rate 2aL I and the energy (AC-Stark) shift −¯hαL I of the initial state can also be obtained from perturbation theory [18, 24]. The above expressions (3.18) and (3.19) include the lowest-order corrections to the results (3.20) and (3.21). The ′ | ≫ aL /δ, since approximation (3.19) can be appreciably better than (3.21), if |αr′ − αcr ′ ′ −1 in this case, in an interval δ ≪ t . |αr − αcr |/aL the depopulation described by the pre-exponential factor dominates the depopulation, due to the exponential. 3.4. Below-threshold excitation If there is a gap between the continuum A(E) and the discrete states, as e.g. in the case of photodetachment of negative ions, then for the below-threshold excitation, ωL < (Ec − Eg )/¯h ≡ ωc , one gets aL = 0, and equations (3.18) and (3.19) yield ′ )I ]eiαL I t . bg (t) ≈ [1 − (αr′ − αcr

(3.23)

′ )I. P (t) ≈ P ≡ 1 − 2(αr′ − αcr

(3.24)

and

Here, as follows from equations (2.11), (2.18), (3.7) and (3.3), Z ∞ X A(E) dE An ′ ± . αr,cr = ± 2 2 Ec (ωEg ∓ ωL ) n (ωng ∓ ωL )

(3.25)

′ ′ > 0 and P < 1, as < 0. Therefore αr′ − αcr Equation (3.25) shows that αr′ > 0 and αcr one should expect. Now the reaction yield 1 − P is proportional to the field intensity,

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being independent of the pulse width T (for T ≫ δ −1 ). Equations (3.23) and (3.24) are the long-time limits of the depletion dynamics, valid for t ≫ δ −1 . Recall, however, that if necessary, the excitation dynamics can be obtained in the whole time domain in the PA region of validity (see the remark after equation (3.19)). Consider the case when the negative ion has only one bound state |ψg i. Then the sum in equation (3.25) drops. One can see that near the threshold, ωc − ωL ≪ ωc , the counterrotating contribution is negligible, |αcr | ≪ αr . However, when excitation is deeply inside ′ ∼ αr′ . Hence equation (3.24) differs significantly the gap, ωc − ωL ∼ ωc , one obtains −αcr ′ = 0. Moreover, if ωL ≪ ωc , from the results of the RWA and RWA1, which imply αcr equation (3.24) yields 1 − P ≈ 4αr′ |ωL =0 I

(ωL ≪ ωc )

(3.26)

which is twice the value which follows from the RWA or RWA1. Note that for excitation below threshold, the final reaction yield obtained here results from abrupt switching on and off of the field [1, 6]. The further the excitation frequency is from the threshold, the stricter are the requirements for the nonadiabaticity of the pulse. Nevertheless there is no fundamental prohibition for an experimental test of the above results. 4. Photodetachment 4.1. Long-time solution Negative ions have an energy gap between the bound states and the continuous spectrum [20]. We assume for simplicity that the ion has no excited discrete states. A laser tuned near or below the continuum edge Ec generally produces an incomplete depopulation of the initial state, due to the existence of a stable dressed state [1, 14]. The long-time behaviour can be obtained in the frame of the SDA as follows. Recall that the SDA solution (2.14) becomes essentially identical with the respective RWA and RWA1 results (see, e.g., equation (2.19)), if one substitutes the effective coupling coefficient BE by AE (cf equations (2.16) and (2.20)). Hence, noting that BE , like AE , has a gap at E < Ec , one can use general results obtained for photodetachment in the frame of the RWA or RWA1 [1] (see also [23]). In particular, one obtains that the long-time solution is given by equation (3.16), where sp = −iω0 is the purely imaginary root of equation (3.1). This root corresponds to a stable dressed state with energy Ed = Eg + h ¯ (ωL + ω0 ). There exists at most one such state and it should lie inside the gap, i.e. Ed < Ec or ωL + ω0 < ωc , where h ¯ ωc = Ec − Eg > 0 is the electron binding energy. In view of equation (3.2), the latter condition reduces the equation for ω0 to ω0 + [αr (ωL + ω0 ) + αcr (ωL − ω0 )]I = 0.

(4.1)

Finally, if ω0 exists one obtains that the long-time solution is nonvanishing and has the form ′ (ωL − ω0 )]I }−1 e−iω0 t bg (t) = {1 + [αr′ (ωL + ω0 ) − αcr

(t → ∞). (4.2)

This solution extends the previous treatments [3, 6, 1] beyond the RWA1. 4.2. Model As shown by Wigner [28], just above the threshold A(E) ∼ (E − Ec )l+1/2 , where l is the angular momentum of the continuum states. Here we consider the s-wave continuum and

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choose the model [3]

√ G E − Ec AE = A(E) = θ(E − Ec ) (4.3) ¯ β) π¯h1/2 (E − Ec + h where G and β determine the oscillator strength and the width of the continuum, respectively, and θ( ) is the unit step function. From equations (2.16) and (4.3) one obtains [3, 23] G G Kcr (s) = √ Kr (s) = √ √ ; √ . (4.4) is + 10 + i β is − 2ωc − 10 + i β Here 10 = ωL − ωc is the detuning of the laser from the continuum edge. In the frame of the present model one can determine the parameters in equations (4.1) and (4.2), as well as other parameters characterizing photodetachment. In particular, (p ωL > ωc β/(10 + β) (4.5) αr = αr (ωL ) = G p p −1 ωL < ωc . ( |10 | + β)

To obtain equation (4.5) we used the relation αr (ωL ) = −ImKr (0) (cf equation (3.2)) and equation (4.4). Equation (4.5) and the relation αcr (ωL ) = αr (−ωL ) yield .p p  2ωc + 10 + β . (4.6) αcr = αcr (ωL ) = G Differentiation of expressions (4.5) and (4.6) results in  p  − β/(10 + β)2 , ′ ′ −1 p αr = αr (ωL ) = G  p p −2  2 |10 | |10 | + β ,

ωL > ωc (4.7) ωL < ωc

and

G ′ ′ αcr = αcr (ωL ) = − √ √ √
2 . 2 2ωc + 10 2ωc + 10 + β

(4.8)

Note that αr (ωL + ω0 ) and αr′ (ωL + ω0 ) in equations (4.1) and (4.2) follow from the lower expressions in equations (4.5) and (4.7). The calculation of the depletion dynamics simplifies significantly when the RWA1 (2.17) holds. The inverse transform of equation (2.19) accounting for (4.4) was obtained in [1, 23] and in our notation reads √ 3  X √  yj2 + βyj 2 i10 t (4.9) bg (t) = e eiyj t erfc −yj it √ 2 j =1 3yj + 2 βyj + 1

where 1 = 10 − αcr I is the effective detuning from the continuum edge, erfc( )√is the complementary error function [25], and yj are the roots of the equation (y 2 +1)(y + β) = GI . 4.3. The P A results

Consider now the present problem in the frame of the above PA formalism. Equations (3.6) and (4.3) yield √ G 10 θ(10 ). (4.10) aL = 10 + β On differentiating equation (4.10) with respect to ωL , one obtains G(β − 10 ) θ(10 ). (4.11) aL′ = √ 2 10 (10 + β)2

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4.3.1. Above-threshold excitation For above-threshold excitation, 10 > 0, the PA solution for the depletion dynamics is given in the lowest order by equations (3.20) and (3.21), where the parameters are given by equations (3.7), (4.5), (4.6), and (4.10). Equation (3.14) now yields the following validity condition of the PA (see appendix) GI κ≡ ≪ 1. (4.12) √ |10 | |10 | + β Here a dimensionless parameter κ is defined, the smallness of which guarantees the validity of the PA. In the frame of the PA, one can obtain the SDA condition by setting |δg | ∼ (aL + αL )I in (2.13), which yields p (4.13) GI ≪ ωL |10 | + β.

For large detunings, 10 ∼ ωL & ωc , conditions (4.12) and (4.13) are equivalent, whereas for 10 ≪ ωc (4.13) is weaker than (4.12). Hence, in the frame of the SDA, the strong coupling is achievable only near the threshold, i.e. for 10 ≪ ωc . Another feature of large detunings, 10 ∼ ωL & ωc , is that now αcr ∼ aL + αr , i.e. the contribution to the spectral shift due to the counter-rotating interaction is significant (cf equation (3.22)). Moreover, for 10 ≫ β this contribution dominates the shift. Hence for large detunings the RWA fails to describe correctly the spectral shift, whereas the RWA1 yields the correct result. The failure of the RWA is consistent with the fact that for excitation well above the threshold the frequency of the radiation is actually comparable with the width of the material transition, which makes the validity of the RWA dubious. 4.3.2. Below-threshold excitation. Below the threshold, 10 < 0, the PA yields equations (3.23) and (3.24), with the parameters given by equations (3.7) and (4.5)–(4.8). The PA condition (3.15) now coincides with (4.12) (see appendix). Condition (4.12) should be compared with the SDA restriction (2.13), where now |δg | = αL I . This again yields condition (4.13). Hence one concludes that for both above- and below-threshold excitation in the frame of the SDA the PA condition can be violated (and, respectively, strong coupling may be achievable) only under near-threshold conditions, i.e. for |10 | ≪ ωc . Thus when the laser is tuned deeply inside the gap, |10 | ∼ ωc , only the weak-coupling regime is possible in the frame of the SDA. In this case equations (4.7) and (4.8) show that ′ ∼ αr′ , which means a failure of the RWA1 (cf section 3.4). In particular, in view of −αcr the second equation (4.7), expression (3.26) now takes the form 2GI 1−P = √ (ωL ≪ ωc ). (4.14) √
2 √ ωc ωc + β 4.4. Results and discussion

Before we make numerical calculations, it is helpful to note that typically [20] h ¯ ωc ∼ 0.5– 1 eV and the width of the continuum is of the order of several electron volts. Taking into √ account that for the model (4.3) the FWHM width of the continuun is 8 3¯hβ ≈ 13.9¯hβ, one obtains that, as a rule, β ∼ ωc (a broad continuum). However, when the negative ion has a shallow virtual level [29], the narrow-continuum case β ≪ ωc can also occur. Figure 1 shows the photodetachment dynamics for above-threshold excitation, under the near-threshold condition, 10 ≪ ωc , for the cases of a broad and narrow continuum. While the former case implies that 10 ≪ β ∼ ωc , the latter case allows any relation between 10 and β. In figure 1(b) the case β ≪ 10 ≪ ωc is shown. As follows from figure 1, for both

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Figure 2. The ultimate photodetachment yield 1 − P as a function of the coupling parameter κ for below-threshold excitation under near-threshold conditions. The cases of a (a) broad and (b) narrow continuum are shown. The calculations have been made with the SDA solution (4.2), the RWA1 solution (4.9), the RWA solution, equation (4.9) with αcr = 0, and the PA prediction (3.24). The parameters chosen are (a) ωc = 2β, 10 = −0.1β; (b) ωc = 200β, 10 = −10β.

cases, when κ ≪ 1, the solution (4.9) is close to the PA prediction (3.21), the agreement between the two results improving with the decrease of κ. In particular, for κ = 0.01 the curves calculated by (4.9) practically coincide with the PA curves in figure 1. Figures 2 and 3 show the ultimate photoelectron yield 1 − P (t → ∞) for the below-threshold case. An analysis of figures 2 and 3 yields that, under quite different conditions, the PA solution (3.24) always provides a good approximation to the exact SDA solution for small κ. This holds both near the threshold (figure 2) and well inside the gap (figure 3). The relative error of the PA result decreases almost linearly with κ for small κ. Thus small values of κ (i.e. κ . 0.1) indeed correctly indicate applicability of the PA for both above- and below-threshold excitation and for all allowed values of the relevant parameters. Note that applicability of the PA means that the field–atom coupling is weak. The opposite statement, which implies that for κ & 1 the coupling is strong, is not always correct. Indeed, in the case shown in figure 2(a), unlike in figures 1 and 2(b), for κ = 1 the coupling is still weak, since then 1 − P ≈ 0.1 ≪ 1. A more thorough analysis of this fact is beyond the scope of this paper.

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Figure 3. The ultimate photodetachment yield 1 − P as a function of the coupling parameter κ. Here the excitation is well inside the gap. The calculations have been made by the same equations as in figure 2. The parameters chosen are ωc = 2β and 10 = −β.

Figures 2 and 3 also provide a test of the RWA and RWA1 against the SDA results. In figure 2 the RWA1 approximates the SDA well, providing generally an improvement with respect to the RWA, though for a weak coupling, both the RWA and RWA1 yield a good approximation to the SDA. However, for excitation well inside the gap (figure 3), both the RWA and RWA1 significantly underestimate the photodetachment yield, in agreement with the discussion in section 3.4. 5. Summary In the preceding sections the problem of single-photon bound–free transitions from ground and excited states with rectangular pulses has been considered. An expression for the Laplace transform of the depletion dynamics, which takes into account all the states of the quantum system in question, has been obtained in the frame of the SDA. The terms stemming from the rotating and counter-rotating interactions enter the SDA expressions on equal footing. The SDA is more general than various versions of the RWAs, such as the RWA and RWA1. The validity condition of the SDA has been obtained. The problem has been treated, in the frame of the SDA, with the help of the PA. The PA has been extended to take into account higher-order corrections, with the help of the Taylor expansion of the level-shift parameter, as in equation (3.8). In particular, this has allowed us to obtain: (i) quantitative validity conditions for the PA, (ii) corrections to the exponential solution for above-threshold excitation, and (iii) the reaction yield for belowthreshold excitation. Moreover, the short-time, nonexponential dynamics has been shown to be obtainable in the region of validity of the PA. The general formalism developed above has been applied to photodetachment of

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negative ions. The long-time solution has been derived for the general case. This extends the previous results valid for the RWA and RWA1. The present PA formalism has been applied to a previously known model. In particular, a simple validity condition of the PA results has been derived. This condition is applicable for all values of the parameters, including quite different situations, such as the cases of a broad and narrow continuum, above- and below-threshold excitation, and near- and far-from-threshold conditions. It has been shown that, in the frame of the SDA, the intermediate- and strong-coupling regimes are possible only for near-threshold excitation, whereas far from the threshold the PA and SDA regions of validity actually coincide. The PA results have been verified numerically against more exact solutions. Applicability of the RWA and RWA1 has been also checked. Both approximations yield rather good results under near-threshold conditions, the RWA1 being typically more accurate than the RWA. In contrast, for excitation well above the threshold the RWA describes the intensity-dependent shift of the photoelectron spectrum at best by the order of magnitude, the RWA1 still being valid. Finally, for excitation well below the threshold both the RWA and RWA1 fail. Acknowledgments I wish to acknowledge useful discussions with Professor M Shapiro. Appendix. Derivation of the PA condition (4.12) Consider first the case of above-threshold excitation, 10 > 0. Accounting for (3.7) and ′ | ∼ |αr′ |, equations (3.14a, b) are equivalent to (4.5)–(4.8) and the fact that now |αr′ − αcr GI ≪1 |aL′ |I + |αr′ |I ∼ √ 10 (10 + β) √ GI β ′ ≪1 αr |aL |I /aL ∼ 10 (10 + β) GI αcr |aL′ |I /aL ∼ √ √ ≪ 1. 210 ( 2ωL + β)

(A.1a) (A.1b) (A.1c)

Summing equations (A.1a) and (A.1b) yields inequality (4.12). Inequality (A.1c) is generally weaker than (4.12) and hence can be dropped. Now in condition (3.14c) ξl = ωc and ξu = 2ωL + ωc . It is easy to see that equation (4.12) is equivalent to the inequality 10 ≫ aL I + αL I , which guarantees the validity of inequality (3.14c). Thus for 10 > 0 the PA criterion is given by inequality (4.12). ′ . αr′ (see the For below-threshold excitation, 10 < 0, taking into account that −αcr second equation (4.7) and (4.8)), equations (3.15a) and (4.7) yield p GI ≪ |10 |(|10 | + β). (A.2) Condition (3.15b), where now ξl = −∞ and ξu = ωc , becomes αL I ≪ |10 |. The latter is equivalent to inequality (4.12), which is more stringent than (A.2). Hence the PA criterion is now given by inequality (4.12). References [1] Raczy´nski A and Zaremba Z 1993 Phys. Rep. 235 1

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[22] [23] [24] [25] [26] [27] [28] [29]

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