Complex rotation method

in the position and in velocity gauge, respectively.

Whilst time dependence has been eliminated in (2.44) and (2.45), there ap- pears a new quantum numberk. It can be shown [49,54] that, in the limit of large average photon number, there is a one-to-one correspondence between the quasienergy spectrum of the Floquet Hamiltonian and the spectrum of an atom dressed by a monochromatic coherent state of the quantized radiation field. In this semiclassical limit (referring to the mean energy of the driving field coherent state), the quantum number k counts the number of photons exchanged between the atom and the field, with respect to the expectation value of the photon number operator for the driving field coherent state.

2.4.2

Complex rotation method

In analogy to the spectrum of 3D helium [6], the spectrum of 2D helium consists of Rydberg series of states converging to single ionization thresholds (s. chapter 4). Therefore, even in the simplest case of the unperturbed atom, there are resonant or autoionizing states with finite life time embedded in the continuum of lower series. Now, an additional electromagnetic field

will couple these autoionizing states of the field-free atom and thus strongly modify the resonance structure of the system. This strongly enhances the effective density of states, due to the periodicity of the Floquet spectrum. Indeed, in this case, as can be seen from eq. (2.44) or (2.45), the external field induces a coupling of all states dressed withk photons to states dressed withk₋1 andk+1 photons. Consequently, also all bound states are coupled with the continuum of the atom.

To extract the resonance states and their decay rates we use complex rotation. This method is based on the analytic continuation of the resolvent operator or Green’s function into the complex plane. The applicability of the complex rotation method for the Coulomb potential is proven in [55], and for the Floquet operator in [56], while its mathematical foundations are outlined in [57–59]. These are far from being trivial and it is not the purpose of the present manuscript to reproduce them here – we only summarize the central results.

For a time-independent HamiltonianH (alikeH0 in eq. (2.2), or the Floquet

operator _HF, eq. (2.40)), the Green’s function is defined by

G(E) = 1

E−H, (2.46)

for E _∈ [60]. Hence, each discrete state of H corresponds to a pole of the Green’s function, while the continuum states induce a branch cut along the real axis. Therefore, there are two analytic continuations of the Green’s function in the complex plane, approaching the real axis from the upper or from the lower half plane, respectively:

G±(E) = 1

E_±i₋H, with →0

+_{. (2.47)}

The energies of the resonant states and their decay rates are given in terms of the complex poles Ei of G+(E), by Re(Ei) and −2Im(Ei), respectively.

To separate them from the continuum, we do not use explicitly the Green’s function. According to the complex rotation method, the poles of the resol- vent are isolated eigenvalues of a complex Hamiltonian, calledrotated Hamil-

tonian, which is derived from the original Hamiltonian H by

H(θ) =R(θ)HR(−θ), (2.48) whereθ is a positive real number, no greater thanπ/4, and

R(θ) = exp −θr·p+p·r 2 (2.49)

I I I1 2 3 ω b) Im(E) Re(E) continuum a) _Im(E) Re(E) continuum

Figure 2.4: Poles of the resolvent operator of the unperturbed Hamiltonian of 2D helium (a) and of the Floquet operator (b). The bound states in (a) are located on the real axis, while the resonance energies on the second Rie- mann sheet (dotted circles in (a) and (b)) have negative imaginary parts and need to be uncovered by analytic continuation of the corresponding resolvent operator. The triangles indicate the single ( ) and double ( ) ionization thresholds of the 2D helium atom, in (a), and the multiphoton ionization thresholds IN +kω, k ∈ ( ) and kω ( ) of the Floquet operator, in (b), connected to a branch cut along the real energy axis (bold black lines), which is due to the continuum states.

is the complex rotation operator. It is a hermitian operator with inverse

R(₋θ).

Under the action ofR(θ), coordinates and momenta transform like

R(θ)rR(−θ) = reiθ,

R(θ)pR(₋θ) = pe−iθ. (2.50) Hence, the rotated Hamiltonian is obtained after substituting r →reiθ _and

p _→ peiθ_{. H(θ) is no more hermitian, but complex symmetric, and its}

spectrum is complex and depends on θ. However, the spectrum of H(θ) has the following important properties:

(i) The bound spectrum of H is invariant under the complex rotation. (ii) The continuum states are located on half lines, rotated by an angle−2θ

around the ionization thresholds ofH into the lower half of the complex plane. In the specific case of the unperturbed 2D helium Hamiltonian (2.2), in analogy to the 3D case [61], the continuum states are rotated around the single ionization thresholds IN given by eq. (4.1). In the

case of the Floquet operator, they are rotated around the multiphoton ionization thresholdsIN +kω (k integer).

(iii) The resonance poles of G+_{(E) correspond to θ-independent complex}

eigenvalues ofH(θ) providedθhas been chosen large enough to uncover their position on the second Riemann sheet. The associated resonance eigenfunctions are square integrable [59], in contrast to the resonance eigenfunctions of the unrotated Hamiltonian. The latter are asymptot- ically diverging outgoing waves [45, 62].

The set of eigenvectors_|EiθiofH(θ) forms a complete basis of Hilbert space,

but it is not orthogonal with respect to the usual hermitian scalar product, due to the non-hermiticity of H(θ). One rather has [63]

hEiθ|Ejθi = δij

X

i

|EiθihEiθ| = 1, (2.51)

where_hEiθ|denotes the complex conjugate ofhEiθ|(i.e.,hEiθ|is the transpose

of_hEiθ|rather than its adjoint) and the sum is taken over all states including

the continua (which, in any numerical diagonalization on a finite basis set, are composed of discrete continuum states).

I I I1 2 3 2θ ω b) a) Im(E) Re(E) Im(E) Re(E) 2θ

Figure 2.5: Spectrum of (a) the complex rotated Hamiltonian of the unper- turbed 2D helium atom, and of (b) the complex rotated Floquet operator. The continua in both cases are rotated by an angle 2θ, around the ioniza- tion thresholds IN in (a), and around the multiphoton ionization thresholds IN +kω (k ∈ ) in (b). The continuous circles denote the exposed reso- nances. The dotted circles indicate the resonances that do not appear, but can be uncovered by further increasing the value of θ.

In the sequel of this thesis, we will be interested in the probability density of states of both the unperturbed 2D helium atom and of the Floquet operator of the corresponding driven system. For the unperturbed case, we need the projection operator _|E_ihE_| on a real energy state _|E_i in terms of the eigenstates of the rotated Hamiltonian, given by [64–66]

|E_ihE_|= 1 2πi X i R(−θ)|EiθihEiθ|R(θ) Eiθ −E − R(−θ)|EiθihEiθ|R(θ) Eiθ−E . (2.52)

In the case of the Floquet operator, we use the time evolution operator

rotated Floquet states and of the quasienergiesεj [65, 66], U(t2, t1) = X j,k1,k2 e−iεj(t2−t1)_eik1ωt1_e−ik2ωt2_R(−θ)|φk2 εj,θihφ k1 εj,θ|R(θ). (2.53)