1. Introduction

Ionization of an electron in a system with bound discrete levels has been addressed many times. In this system with autoionizing levels formed by bound states, there occurs the competition among different ionization channels. In parallel to direct ionization, also indirect ionization based on bound discrete levels exists and contributes to the ionization process. Fano [1] has shown that, under suitable conditions, completely destructive interference between the direct and indirect ionization channels may be observed for one state in the continuum of free states. The existence of such a state in a long-time photoelectron ionization spectrum is referred as the presence of a Fano zero. The interference of two ionization paths in the vicinity of the Fano zero may also lead to a strong narrowing of spectral ionization peaks (confluence of bound-free coherences, [2, 3]). In general, the long-time photoelectron ionization spectra are influenced by many effects, including spontaneous emission of radiation [4–6], finite pump laser bandwidth [7] or collisions [6]. Additional interferences in photoelectron ionization spectra arise for the quantum low-light pumping and originate in the discrete energy levels of quantized fields [8, 9]. The Fano spectral zeros have been experimentally studied, e.g., in [10]. Ionization systems with discrete autoionizing levels can even exhibit transparency for ultra-short pulses [11] or slow down the propagating light [12]. The observed spectral profiles are asymmetric which is useful in interpretation of experimental spectral profiles [13]. For an extended list of references, see [6, 14–16]. We note that similar quantum interference effects have been observed also in semiconductor hetero-structures or photonic structures composed of waveguides and micro-resonators [17].

The photoelectron ionization spectra are considerably modified by the interaction with a neighbor system [18–21]. This interaction (e.g., dipole-dipole interaction in molecular condensates [22]) leads to more complex spectral profiles in which the Fano zeros are in general absent [20]. Only for weak electric-field amplitudes, there occur minima in photoelectron ionization spectra that behave similarly as Fano zeros and we call them Fano-like zeros. On the other hand, the so-called dynamical zeros [19] are observed in the conditioned photoelectron ionization spectra depending on the quantum state of the neighbor system. However, these dynamical zeros occur only once per the Rabi period belonging to oscillations of the neighbor system. The presence of a neighbor system may also considerably increase photoionization and recombination rates [21, 23].

Here, using a suitable canonical transformation, we formulate exactly the conditions for the existence of a Fano zero in an autoionization system interacting with a neighbor two-level atom. Photoelectron ionization spectra are then studied under these special conditions that lead to two distinguished spectral features: (i) suppression of spectral peaks close to the Fano zero and (ii) strong narrowing of spectral peaks lying not far from the Fano zero and caused by the confluence of bound-free coherences.

The structure of the article is as follows. A model Hamiltonian and a suitable canonical transformation are presented in Sec. II, in which conditions for the occurrence of a Fano zero are derived. Analytical formulas for the long-time photoelectron ionization spectra are written in Sec. III. Sec. IV contains a discussion of the long-time ionization spectral profiles containing Fano zeros. Conclusions are drawn in Sec. V.

2. System Hamiltonian and its Canonical Transformation

We consider the simplest possible Hamiltonian that describes an atom with only one autoionizing level. Its interaction with a neighbor two-level atom leads to energy transfer caused, e.g., by the dipole-dipole interaction (for the scheme, see Fig. 1). The autoionization atom b in a stationary optical field is described by the Hamiltonian Ĥ _b (h̄ = 1 is assumed, [24]):

 

Fig. 1 Scheme of the autoionization system b interacting with a two-level atom a. The excited state of atom a with energy E_a is denoted as |1〉_a. The excited bound [free] state of atom b with energy E_b [E] is named |1〉_b [|E〉]. Symbols μ_a, μ_b, and μ refer to the dipole moments between the ground states |0〉_a and |0〉_b and the corresponding excited states; α_L is the electric-field amplitude, V stands for the Coulomb configurational coupling between the states |1〉_b and |E〉. Energy transfer (dipole-dipole interaction) between states |1〉_a and |E〉 (|1〉_b) is quantified by J (J_ab). Double arrows indicate the participation of both electrons at atoms a and b in the interaction.

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The Jaynes-Cummings Hamiltonian Ĥ _a describes a neighbor two-level atom a in a stationary optical field (for the symbols, see the caption to Fig. 1):

The dipole-dipole interaction [22] between the two-level atom a and the autoionization atom b is characterized by the Hamiltonian Ĥ _trans:

We need to construct a suitable canonical transformation of the overall Hamiltonian Ĥ = Ĥ _a + Ĥ _b + Ĥ _trans that reveals a state |E_F〉 in the free-state continuum that cannot be ionized. Such a state exists due to the possibility of mutual cancellation of several competing ionization paths. We will see later that a suitable canonical transformation may be constructed in the same way as it was originally done by Fano [1]. Fano considered an autoionization atom with one discrete excited level and found eigenstates that diagonalize the Coulomb configurational interaction between the discrete excited level and the continuum of free states. The dipole moment of optical interaction in the transformed basis then revealed a state in the continuum that cannot be ionized. We only apply this approach to the autoionization atom b. Subsequently, this transformation reveals additional constraints to the dipole-dipole interaction between the atoms a and b that preserve a Fano zero.

Following the work by Fano [1], we look for states |E) that are eigenstates of the following Hamiltonian Ĥ _diag:

These eigenstates |E) can be written as a linear superposition of the original states |1〉_b and |E′〉,

The Hamiltonian Ĥ _b introduced in Eq. (1) attains the following form in the basis spanned by the states |0〉_b and |E):

Also the Hamiltonian Ĥ _trans of the dipole-dipole interaction in Eq. (3) has to be transformed:

The dipole moment $\overline{\mu }$ derived in Eq. (12) equals zero if ɛ (E) = –q_b. Similarly, the coupling constant J̄ of dipole-dipole interaction vanishes if ɛ (E) = –q _trans. Provided that both the conditions are fulfilled, the state |E_F) inside the continuum cannot be ionized. The reason is that all four possible ionization paths mutually cancel. Such a state then forms a Fano zero found only for q_b = q _trans, i.e.,

The above derivation of conditions for the existence of a Fano zero can be interpreted as follows. The ionized free state |E_F) can be reached after the absorption of a photon either at atom a or atom b. If a photon at atom a is absorbed, there exist two ionization paths, both coming through the excited state |1〉_a. The subsequent dipole-dipole interaction leads either directly to the state |E_F) or excites the state |1〉_b. Then the state |E_F) is reached due to the Coulomb configurational interaction (V). Under the conditions written in Eq. (16), the two ionization paths mutually cancel. Similarly, if a photon is absorbed at atom b, two ionization paths compete. The first path describes the direct ionization, whereas the second one relies on the presence of the autoionizing level |1〉_b. Also in this case, the condition in Eq. (16) guarantees a completely destructive interference of both paths.

On assuming resonant pumping of atom a, this picture of competing ionization paths can even be rigorously formulated. The transformed coupling energies μ ̄(E) α_L and J̄(E) given in Eqs. (11) and (14) can be written in the form:

3. Long-Time Photoelectron Ionization Spectra

The photoelectron ionization spectra can be obtained after the solution of dynamics of the composed system described by the overall Hamiltonian Ĥ [20]. The long-time photoelectron ionization spectrum I ^lt can be conveniently written as a sum of two ionization spectra, $I_0^{\text{lt}}$ and $I_1^{\text{lt}}$, conditioned by the presence of neighbor atom a in the states |0〉_a and |1〉_a:

According to the formulas in Eq. (20), the long-time ionization spectrum I ^lt does not depend on time. On the other hand, the conditional long-time ionization spectra $I_0^{\text{lt}}$ and $I_1^{\text{lt}}$ exhibit harmonic oscillations. They can be observed using time-resolved spectroscopy [24] with the required resolution depending on the strength of electric-field amplitude. An example of these ionization spectra is shown in Fig. 2. As a Fano zero appears in the spectra shown in Fig. 2 close to the peaks, the effect of confluence of bound-free coherences [2, 14] leading to very narrow and high spectral peaks is observed.

 

Fig. 2 Long-time intensity photoelectron ionization spectrum I ^lt (solid curve), steady-state intensities $I_0^{\text{st}}$ (solid curve with ○) and $I_1^{\text{st}}$ (solid curve with *), and magnitude I ^osc of intensity oscillations (solid curve with ▵). A Fano zero is located at (E_D – E_b)/Γ = −0.5; q_a = μ_a/(πμJ ^*), γ_a = π|J|², q_b = μ_b/(πμV^*), γ_b = π|V|², q _trans = J_ab/(πJV ^*), $\mathrm{\Omega }=\hspace{0.17em}\sqrt{4\pi \mathrm{\Gamma }}(Q+i)\mu {\alpha }_L$, Γ = γ_a + γ_b, Q = (γ_aq_a + γ_bq_b)/Γ. Spectra are normalized such that ∫ dEI ^lt(E) = 1; q_a = q_b = q _trans = 1, γ_a = γ_b = 1, Ω = 8, E_a = E_b = E_L = 1.

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Fano and dynamical zeros belong to important characteristics of these spectra. Whereas the frequency E_F of a Fano zero obeys the equation

4. Typical Shapes and Features of Long-Time Photoelectron Ionization Spectra

We discuss the behavior of long-time photoelectron ionization spectra I ^lt under the condition written in Eq. (15), i.e., when a Fano zero at the frequency given in Eq. (16) occurs. In this distinguished regime, there occur suppression of ionization around the Fano zero as well as strong narrowing of spectral peaks located not far from the Fano zero. As we want to emphasize the effects coming from the competition of the inter (V) and intra (J, J_ab) electron interactions at the atom b, we first consider the regime of γ_a ≈ γ_b. At the end of Sec. IV, we comment the case γ_a ≪ γ_b appropriate for molecular condensates. We divide the discussion into two parts depending on the mutual strength of stationary optical interactions at the atoms a and b.

We first consider comparable strengths of optical interactions at the atoms a and b, i.e. q_a ≈ q_b. The long-time ionization spectrum I ^lt is typically composed of two distinguished peaks with the center frequencies mutually shifted by the Rabi frequency δξ. This means that only one complex frequency Λ_l out of four occurring in Eq. (19) is located close to the real axis and determines the shape of spectrum I ^lt [see Fig. 3(a)]. Both the peaks move towards the lower frequencies E as the optical-interaction parameter Ω increases ( $\mathrm{\Omega }\hspace{0.17em}=\hspace{0.17em}\sqrt{4\pi \mathrm{\Gamma }}(Q+i)\mu {\alpha }_L$, for details, see the caption to Fig. 2). We note that parameter Ω is linearly proportional to the electric-field amplitude α_L as well as the optical dipole moment μ related to direct ionization of atom b. A peak disappears due to the destructive interference of ionization paths when it moves through the area around the Fano zero [see curves for Ω = 2 and 6 in Fig. 3(a)]. In this case, one peak dominates in the ionization spectra I ^lt. In the long-time ionization spectra I ^lt, there also occurs one Fano-like zero (observed only for weak electric-field amplitudes) with the frequency E_F–l = E_a – γ_a q_a [20]. The frequencies E_D of dynamical zeros depending on the optical-interaction parameter Ω are shown in Fig. 4. The frequencies E_D coincide for the conditional spectra $I_0^{\text{lt}}$ and $I_1^{\text{lt}}$ for resonant pumping of atom a. They typically occur in pairs, the annihilation and creation of a pair of frequencies is shown in Fig. 4(a). We note that the frequency E_D of one dynamical zero coincides with the frequency E_F–l of the Fano-like zero in the limit Ω → 0.

 

Fig. 3 Long-time photoelectron ionization spectra I ^lt for (a) q_a = q_b = 1 and (b) q_a = 100, q_b = 1 for different values of optical-interaction parameter Ω: Ω = 0.5 (solid curve with ○), Ω = 1 (solid curve), Ω = 2 (solid curve with *), Ω = 4 (solid curve with ▵), and Ω = 6 (solid curve with ♦). The Fano zero occurs at (E_F – E_b)/Γ = −0.5; q _trans = 1, γ_a = γ_b = 1, E_a = E_b = E_L = 1.

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Fig. 4 Normalized frequencies (E_D – E_b)/Γ of dynamical zeros as they depend on optical-interaction parameter Ω for resonant pumping of atom a; values of parameters are given in the caption to Fig. 3.

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If optical interaction of the neighbor atom a dominates that of the autoionization system b (q_a ≫ q_b), two pairs of peaks build up the long-time ionization spectrum I ^lt [see Fig. 3(b)]. Peaks belonging to one pair are mutually shifted by the Rabi frequency δξ and represent the Autler-Townes doublets. Two peaks at the outer edges belonging to different pairs stick out as the optical-interaction parameter Ω increases; the one at the left-hand side moves towards lower frequencies, whereas the peak at the right-hand side shifts towards higher frequencies. On the other hand, the remaining two peaks in the middle approach each other and merge together for larger values of the optical-interaction parameter Ω. As a consequence, the ionization spectrum I ^lt consists of only three peaks for larger values of the optical-interaction parameter Ω. The spectral widths of all peaks increase with the increasing value of parameter Ω. The frequencies E_D of dynamical zeros as functions of parameter Ω are depicted in Fig. 4(b) in this case. We note that one dynamical zero that coincides with the Fano-like zero for Ω → 0 [(E_D – E_b)/Γ = −50] is not displayed in Fig. 4(b).

Considering molecular condensates as potential candidates for the observation of the described effects, the strengths of dipole-dipole interaction and Coulomb configuration interaction dictate the regime characterized by γ_a ≪ γ_b. If comparable values of the dipole moments of molecules a and b are considered, we arrive at the inequality q_a ≫ q_b. Under these conditions, the influence of the neighbor molecule a is observed only for smaller values of the optical-interaction parameter Ω such that the strengths of the dipole-dipole interaction and the optical dipole interaction at molecule a are comparable. This leads to the long-time photoelectron ionization spectra composed of three peaks located in the near vicinity of the frequency E_b (see Fig. 5). These peaks are suppressed for larger values of the optical-interaction parameter Ω that give an asymmetric spectral peak typical for the Fano model. As the values of optical-interaction parameter Ω increase, the asymmetric peak moves towards the lower frequencies crossing the frequency of Fano zero. There occurs a strong narrowing of the spectral peak in the vicinity of the Fano zero caused by the confluence of bound-free coherences [2] (see Fig. 5).

 

Fig. 5 Long-time photoelectron ionization spectra I ^lt for different values of optical-interaction parameter Ω: (a) Ω = 5 × 10⁻⁴ (solid curve) and Ω = 3 × 10⁻² (solid curve with *, only the central part is shown) and (b) Ω = 1 (solid curve with *), Ω = 2.5 (solid curve) and Ω = 4 (solid curve with ▵). The Fano zero occurs at (E_F – E_b)/Γ ≈ −1; q_a = 100, γ_a = 1 × 10⁻⁴, q_b = γ_b = 1, q _trans = 1, E_a = E_b = E_L = 1.

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5. Conclusions

A Fano zero in the long-time photoelectron ionization spectrum of an autoionization system interacting with a neighbor system has been revealed under special conditions that relate to the strengths of optical dipole and dipole-dipole interactions. Depending on the strength of optical dipole interaction of the neighbor atom, the ionization spectra are composed of one or two pairs of Autler-Townes peaks mutually shifted by the Rabi frequency of the neighbor atom. However, the number of peaks may be reduced for stronger optical interaction either because of the destructive interference in the area around the Fano zero or due to the merging of two peaks together. Moreover, dynamical zeros have been observed in the long-time conditional photoelectron ionization spectra.