Nonperturbative time dependent solution of a simple ionization model.
We present a non-perturbative solution of the Schrödinger equation , written in units in which , describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large and small the energy distribution has sharp peaks at energies which are multiples of , corresponding to photon capture. We obtain small expansions that converge for all , unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in different emission processes.
The ionization of atoms and the emission of electrons from a metal, induced by an oscillating field, such as one produced by a laser, continues to be a problem of great theoretical and practical interest, see , , ,  and the references therein. This phenomena goes under the name of photo-emission. It was first explained by Einstein in 1905; an electron absorbs " photons" acquiring their energy, , which permits it to escape the potential barrier confining it. While the complete physics of these phenomena would involve quantization of the electromagnetic field and its interaction with matter, i.e. photons and relativity, the basic understanding is contained already in the semiclassical limit where the electromagnetic field is not quantized, expected to be valid when the density of photons is large ; for a mathematical derivation of this limit via Floquet states see . One then considers the solution of the non-relativistic Schrödinger equation in an oscillating field giving rise to a potential with period , , . Resonant energy absorption at multiples of then yields effects qualitatively similar to those of photons, in some regimes, see Fig. 3.
In units in which the Schrödinger equation has the form
Here describes the time-independent system assumed to have both discrete and continuous spectrum, and the laser field is modeled by a time periodic potential, . Typically, the latter is represented as a vector potential or a dipole field, e.g. , , .
Starting in a bound state of the reference hamiltonian , corresponding to the energy and expanding in generalized eigenstates, assuming is the only effective bound state, the evolution is given by
Physically, gives the probability of finding the particle in the eigenstate and is the probability density of the ionized electron in “quasi-free” states (continuous spectrum) with energies . It follows from the unitarity of the evolution that
Accordingly, if as , we say that the system ionizes completely.
When , a first order approximation ,  in the strength of (used very judiciously) gives emission into states with energy . Clever physics arguments also yield Fermi’s golden rule of exponential decay from the initial bound state , . These only hold approximately and only over some “intermediate” time scales as discussed in the sequel.
To deal with the case of transitions caused by large fields one needs to go to high order perturbation theory, which is complicated , . In fact, as we will explain, standard perturbation theory only produces a finite number of correct perturbative orders. To deal with larger fields one uses various "strong field" approximations due to Keldysh and others . For literature on strong field approximations see , , , . There, one uses scattering states strongly modified (Volkov states) by the oscillating field. We shall not consider that here but focus on getting a complete rigorous solution of (1) for a toy model which nevertheless exhibits many features of more realistic situations, see . We can then study carefully how photons show up in this semiclassical limit.
The model we study is a one dimensional system with reference Hamiltonian , whose mathematical properties are analyzed in , is
It has a single bound state
with energy and its generalized eigenfunctions are
Beginning at , when , we add a parametric harmonic perturbation to the base potential. For we have
(where we take for definiteness ) and look for solutions of the associated Schrödinger equation in the form (2). The full behavior of is very complicated despite the simplicity of the model. We expect the main feature of the evolution of to be universal for ionization by an oscillatory field.
As already noted this model has been studied extensively before. We refer the reader in particular to  where it was shown that, for all and , , i.e., we have complete ionization. We also investigated there both analytically and numerically the behavior of as a function of and showed qualitative agreement with experiments on the ionization of hydrogen-like atoms by strong radio frequency fields. In  we studied general periodic potentials and found the condition on the Fourier coefficients for complete ionization. There are (exceptional) situations where one does not get complete ionization. In  we showed ionization when the external forcing is an oscillating electric field. A large field approximation for this latter setting can be found in .
In this paper we introduce new methods which allow us to complete the analysis of this model for all : we obtain a rapidly convergent representation (in the form of a Borel summed transseries, or “multi-instanton expansion”) for the solution valid for all and and we find the distribution of energies of the emitted electrons as a function of . The latter, which was not done before, is where the "photonic" picture shows up most clearly. We will investigate this connection more explicitly in a separate article .
There are strong peaks of which for small and are centered near , see Fig. 1 for . The main peak corresponds to the absorption of one photon and approaches a Dirac distribution centered at in the limit followed by . Clearly, the discreteness of the emission spectrum in the above limit is a consequence of the periodicity of the classical oscillating field and does not require the concept of photons, see also , Footnote 1. We find that there are other (smaller) peaks emanating from the bottom of the continuous spectrum. For small these are centered near , see Theorem 4, (iv). We also obtain a perturbation expansion of the wave function for small in a form which is uniformly convergent for any , and which, in principle, can be carried out explicitly to any order.
It follows from our analysis that the predictions of the usual perturbation theory hold when , beyond which the behavior of the physical quantities is qualitatively different.
1.1. The Laplace transform and the energy representation
It was shown in  that
where satisfies the integral equation
It can be checked, , that the Laplace transform of
is analytic in the right half plane and satisfies the functional equation111A very similar functional equation can be obtained directly from the Schrödinger equation for .
(The square root is understood to be positive on , and analytically continued on its Riemann surface. 222In previous papers we used instead of and a different branch of the square root; with these changes the formulas agree.)
2. Main results
2.1. Results for general
For all and ,
(i) is bounded and is analytic in the closed right half plane, except for
where it is analytic in ;
(ii) in the open left half plane has exactly one array of simple poles located at
and the residues can be calculated using continued fractions, see §3.3, and satisfy
Away from the line of poles, is bounded in the left half plane. The functions and are analytic in in a neighborhood of ;
The following is the non-perturbative (arbitrary coupling) form of the decay of the bound state.
(ii) Similarly, the function is a Borel summed transseries
where the have the same properties as the .
For all we have
2.2. Perturbation theory: results for small .
In this section we assume that . See Note 5 regarding .
Notation: In the rest of the paper “” denote functions analytic and vanishing at .
Assume . Let as in Theorem 1 (ii).
(i) For , we have, for small enough,
With the least integer for which we have
(ii) The residues (see (12) for arbitrary ) satisfy
Furthermore, as ,
where the are defined in (14).
(iii) As a function of , is analytic for small and real-analytic for .
With as in (i), on the scale we have333This is the Fermi Golden Rule of exponential decay of the bound state valid for small amplitudes and moderately large times.
(iv) The distribution of energies satisfies
If for some we have , which means that poles are close to branch points, there is a smooth transition region where and change from to . We will not analyze this intricate transition in the present paper.
3.1. Organization of the paper and main ideas
We are interested in obtaining rapidly convergent expansions for and for all and . To achieve this we study in great detail the singularity structure of . We prove in particular that has exactly one array of evenly spaced poles, for , and one array of branch points, for . Their location and residues determine, via the inverse Laplace transform, the transseries representation of and . To show this rigorously for all we first establish these facts for small using compact operator techniques; we then extend them for arbitrary by devising a periodic operator isospectral with the one of interest, whose pole structure can be analyzed by appropriate complex analysis tools.
The proof of Theorem 1 (i) is found in §3.2, (ii) and (iii) in §3.5, (12) in §3.7.1. The functional equation (9) is rewritten as a parameter dependent equation on and analyzed with compact operator techniques.
Section §3.3 contains results and notations used further in the paper.
Theorem 4 (i) is proved in §3.4, (ii) in §3.8 and (iii), (iv) in §3.10. For small the position of the poles is found from a continued fraction representation described in §3.3. The information about the poles for larger relies on the analysis of a periodic compact operator isospectral to the main one and zero-counting techniques. Theorem 2 is proved in §3.9.1.
3.2. Proof of Theorem 1(i)
It turns out that the pole of at has no bearing on the regularity of the solutions, as the equation can be regularized in a number of ways. One is presented in detail in . A simpler way is presented in §3.2.1.
It is convenient to discretize (21). With the notation
and setting , , we obtain the difference equations with parameters
or, in operator notation,
3.2.1. Regularization of the operator
We rewrite (23). Let
where . Now , and are pole-free in the closed lower half plane (analyticity of the solution in the upper half plane is known, see the beginning of §1.1). Note that is a multiple of .
Extension. It is convenient to remove the restriction in (22) on , and allow .
(i) The operator is compact in . It is linear-affine in , and analytic in except for a square root branch point at .
(ii) For , has the properties of listed above.
(i) For compactness, note that is a composition of two shifts and multiplications by diagonal operators whose elements vanish in the limit (all its coefficients, see (26), are ). Noting that has a pole at and a branch point at , the analyticity properties are manifest. The proof of (ii) is similar. ∎
The homogeneous equation
has no nontrivial solution if . By the Fredholm alternative (27) has a unique solution which has the same analyticity properties as .
In particular, is analytic if and on each segment ; at it is analytic in .
The proof is given in ; for completeness, we sketch the argument in the Appendix.
3.3. Further properties of the homogeneous equation
The general theory of recurrence relations  shows that the homogeneous part of (23) has two linearly independent solutions, one that grows like and one that decays like for , and two similar solutions for ; the one that decays at is different from the one that decays at , unless is in the spectrum of . Since we need more details, we reprove the relevant claims. The main results are given in Corollaries 14, 15.
In this section it is convenient to work with the continuous equations (21). Its homogeneous part is
Lemma 12 shows the existence of a solution of (29) which goes to zero as for not too large, with tight uniform estimates for all , and of a similar solution for . Lemma 13 shows existence of such solutions for any , providing estimates only for large enough.
Looking for a solution that decays for large we define
and obtain from (29)
where is the nonlinear operator
Similarly, looking for solutions which decay for the ratio
As usual, a domain in is an open, connected subset. denotes the open lower half plane in .
(for a suitably small ). Consider the Banach space of functions continuous in the strip , with the sup norm. Let denote the Banach space of continuous functions in
We denote by the class of functions which are real-analytic in for all and in , continuous on and with possible square root branch points at .
By the usual properties of the Laplace transform, is analytic in the upper half plane. Since below we are interested in the properties of in the lower half plane it is convenient to place the branch cuts in the upper half plane. Later, in §3.11, when we deform the contour of an inverse Laplace transform (in it is horizontal, in the upper half plane), the points on the curve are moved vertically down, yielding a collection of vertical Hankel contours 444A Hankel contour is a path surrounding a singular point, originating and ending at infinity, . around the branch points and residues. For this particular purpose, placing the cuts in the upper or lower half plane can be seen to be equivalent.
(i) For , the operator defined in (30) is contractive in the ball in ; the contractivity factor is as .
Thus (30) has a unique fixed point . Also, is analytic in for and satisfies as .
(ii) The operator is contractive in a ball in .
Let us state first a more general result, valid for all (where now the dependence of on is made explicit):
(i) For any fixed and large enough the operator in (30) is contractive in the ball
and thus it has a unique solution, which is analytic in .
Similar estimates hold for (31).
(ii) For large ,
(iii) For large in the lower half-plane,
Proof of Lemma 12..
(i) We note that the minimum of in is implying for small enough.
We first show that leaves the ball invariant. A straightforward estimate shows that if then is well defined on the ball and
The contractivity factor is obtained by taking the sup of the norm of the Fréchet derivative of with respect to :
Under the assumptions in the Lemma, we have
The same analysis goes through in the space of functions which are of class for and analytic in for , in the joint sup norm, in and , proving joint analyticity in , except for the mentioned square root branch points. To show that are square root branch points, we return to the representation of (22). For simplicity of presentation assume . We repeat the arguments above, now in the space of functions of the form where are analytic near , in the norm .
Moreover, since the only singularities of are a pole of order one at and a square root branch point at , a similar analysis shows that is of class .
Straightforward estimates in (30) show that for small we have
(ii) We note that hence . The rest of the proof is as for (i). ∎
Proof of Lemma 13..
We note that for any and , we have, for large enough
We use Pringsheim’s notation for continued fractions
Iteration of (30) yields a continued fraction
which is convergent for , and . For small , the rate of convergence is .
Similarly, iteration of (31) yields a convergent continued fraction
Convergence of the continued fraction, by definition, means that the th truncate of the continued fraction, that is , converges to the fixed point . Since zero is in the domain of contractivity of , convergence follows directly from Lemma 12. The norm of the Fréchet derivative of is implying the last statement. Convergence of (39) is similar. ∎
As , there is a solution of (23) with which is ; a second, linearly independent solution, has the property , that is, such a solution grows factorially. A similar statement holds as .
The first part follows from the fact that . For the second part, one looks as usual for a second solution in the form and notes that satisfies a first order recurrence relation that can be solved in closed form in terms of .
3.4. Proof of Theorem 4 (i): location of the singularities for small
By Proposition 9 the resolvent can only be singular if , which we will assume henceforth. By Remark 7 we can then work with the simpler operator . We place branch cuts in the upper half plane, see Remark 11.
There is a such that for all complex with the following hold.
(i) There exists a unique in the strip so that .
More precisely, where
for some analytic at zero.
(ii) For we have555 Note that for we have .
where is real, given by (40), and is real, given by
There is a such that for small enough equation (26) has a unique solution for any in the strip with dist. In particular, for such , Ker.
As mentioned above, we can additionally assume that , hence invertibility of is equivalent to Ker, which is equivalent to Ker.
Let (where now does not depend on ). If is such that , then . But straightforward estimates show that , if is large enough and dist. This implies that is contractive and thus . ∎
The functions are meromorphic in for in the open lower half plane and .
Let be fixed and be the fixed point provided by Lemma 13, analytic in for . Using the recurrence relation (30) can be continued to a meromorphic function for all with smaller real part (the coefficients in (30) are meromorphic except for square root branch points on the real line).
Similarly, can be continued to a meromorphic function. ∎
Proof of Theorem 16 (i).
Let be given by Proposition 18. The value(s) of for which Ker are those for which
as discussed in §3.3. By Lemma 17 any such has the form with . We now show that for small there is exactly one solution of (43) in the strip . (Note that using the assumption that which ensures that the poles and the branch points do not coincide.)
Expanding , respectively , in a power series in we have
Finally, note that if equation (43) had a solution of the form then would also be a solution, but this is ruled out by the uniqueness of solutions. ∎
Proof of Theorem 16 (ii).
As in the proof of Corollary 14, the th truncate of the continued fraction defining , that is , converges to the fixed point . Similarly, , converges to the fixed point .
Note that, for small , and all other and, inductively, and . Therefore
it follows that all with have a power series expansion in with real coefficients for . Then so does the denominator in the right of (3.4), as well as the denominator on the left, truncated to terms.
We now look for a solution to
in the form where with and all real.
A simple calculation gives