Quantum optical signatures in strong-field laser physics: Infrared photon counting in high-order-harmonic generation

We analytically describe the strong-field light-electron interaction using a quantized coherent laser state with arbitrary photon number. We obtain a light-electron wave function which is a closed-form solution of the time-dependent Schrödinger equation (TDSE). This wave function provides information about the quantum optical features of the interaction not accessible by semi-classical theories. With this approach we can reveal the quantum optical properties of high harmonic generation (HHG) process in gases by measuring the photon statistics of the transmitted infrared (IR) laser radiation. This work can lead to novel experiments in high-resolution spectroscopy in extreme-ultraviolet (XUV) and attosecond science without the need to measure the XUV light, while it can pave the way for the development of intense non-classical light sources.

Going beyond the reach of the semi-classical approach, we find that the quantum-optical properties of the HHG process are imprinted in measurable photon statistics of the transmitted IR laser field, thus accessing HHG dynamics does not require measuring the XUV radiation. This is a unique advantage of our work since our proposed measurements, dealing with high-resolution spectroscopy in XUV and attosecond science, can be performed in open air without the need for specialized optics/diagnostics required for the characterization of the XUV radiation. Additionally, it has been found that the interaction of strong laser fields with gas phase media leads to the production of non-classical high photon number light states.

Full-quantum theoretical description of light-electron interaction
The non-relativistic TDSE of an electron interacting with a single-mode long-wavelength linearly-polarized quantized light field of frequency ω reads (in atomic units) , respectively, q is the in-phase quadrature of the field 9,10 , and β π ω = c V 2 / is a constant determined by the quantization volume V, frequency ω, and light velocity c. The detailed derivation of the analytical solution of Eq. (1), termed quantum-optical Volkov wavefunction, will be given elsewhere. Here we provide the result, the validity of which can be checked by direct substitution into Eq. (1). Based on this, we then analyze its fundamental features and their consequences for the HHG process. The closed-form solution of Eq. (1) reads: The solution includes an arbitrary initial electron distribution ψ 0 (p) and an arbitrary initial photon number N 0 and the field phase θ. The wavefunction Ψ (p, q, t) provides the full quantum-optical description of the electron-light interaction. The term d(t)pq in the exponent renders the electronic and light degrees of freedom non-separable. In the high photon number limit where c 0 0 (where A 0 is the amplitude of the corresponding classically-described vector potential A = A 0 cos(ωt + θ)), Eq. (2) is simplified (see Methods) to Ψ ′ (p, q, t) maintaining all the quantum-optical properties of Eq. (1).
A crucial property of the quantum-optical Volkov wave function is that the matrix elements of any q-independent operator R coincide with the matrix elements obtained from using the well known semi-classical electron Volkov wave functions ψ V i.e.
where Ψ x is an arbitrarily chosen electron-light wave function and ψ x is the corresponding state of the electron in case of classically-described electromagnetic field. Thus, while Ψ ′ (p, q, t) goes beyond the semi-classical approach to completely describe the quantized electron and light interaction, it naturally reproduces the classical Volkov states after integrating over q. This has profound consequences for the description of HHG, since the well known results of the semi-classical models 4 can be retrieved, and more than that, utilized in our quantized-field approach.
In the particular case of HHG, Ψ x is the ground state of the system (ψ g ψ c ), R is the dipole moment r ( ), ψ x is the ground state of the atom (ψ g ) and ψ c is the initial coherent light state. Detailed description of the above considerations can be found in the Methods Section.
The calculation of the dipole moment in the high photon number limit demonstrates that the behavior of the electron in a strong laser field can be accurately described by semiclassical theories with negligible quantum corrections. However, our full quantum-optical approach can provide information about the IR laser field states during the interaction, inaccessible by the semi-classical theories. This information can be experimentally extracted utilizing XUV/IR-correlation approaches and/or balanced homodyne detection techniques [11][12][13] of the IR laser field transmitted from the harmonic generation medium.

Quantum-optical description of the HHG process
Using the quantum-optical Volkov wave functions Ψ ′ in high photon number limit, the time evolution of the HHG process is described by the following wave function where Ψ g = ψ g ψ c is the initial state of the system, ψ g is the ground state of the electron, ψ c is the initial coherent light state and Ψ ′ (p, q, t − t i ) are the continuum laser-electron states having different ionization times t i . In eq. 4 we assume that light-electron states (at any time t) can always be represented by a superposition of bound states (one bound state in our case) and continuum states. The amplitude of the bound state is a g (t), and the amplitudes of the continuum states are b i (t). In the same way as in the semi-classical theory 4 we assume that a g (t) ≈ 1, while In addition, we choose (as initial condition) that before the ionization a g (t 0 ) = 1 (where t 0 < t i ), so b i (t) = 0 for t < t i , and b i (t) = b i for t > t i . The complex amplitudes a g and b i satisfy the normalization condition Ψ Ψ = ∼ ∼ 1. In this case, the time dependent dipole moment is . As in the semi-classial theory 4  consider only ground-to-continuum (and continuum-to-ground) transitions given by the matrix elements 〈Ψ′ Ψ 〉 r g (and 〈Ψ Ψ′〉 r g ).
Integrating Ψ Ψ ∼ ∼ r over q, using Eq. (4) and integrating over p, we retrieve that ψ ψ ∝ 〈 〉 r t r ( ) V g which coincides with the expression given by the semi-classical theories. Thus, all the semi-classical results 4,14-16 , in particular the short (S) and long (L) electron paths with electronic Volkov wave functions ψ V S and ψ V L respectively, can be consistently used in the present approach. In a similar way, the same results can be obtained using the IR wave functions ψ l (see Method section).
A scheme which can describe the HHG process in the context of the present model is shown in Fig. 1. Although the quantization of the harmonic field is not required for this work and thus was not considered in the previous formalism, harmonic photons are included in Fig. 1 for a complete understanding of the process. Figure 1a shows the electron states in case of integrating over q and Fig. 1b shows the field states in case of integrating over p. The horizontal black lines in Fig. 1a,b are the initial states of the electron |ψ g 〉 and IR-laser field |ψ c 〉 with energy IP < 0 and W light (0), respectively. At t > 0 the system is excited (small red arrows) in an infinite number of entangled laser-electron Ψ′ i states (gray area), resulting in a reduction of the average laser energy (small downwards red arrows in Fig. 1b) and the enhancement of the average electron energy (small upwards red arrows in Fig. 1a). The Ω-frequency emission is taking place by constructive interference of ψ V states and recombination to the ground state (downwards red arrows in Fig. 1a). In this case the final laser energy remains shifted by ħΩ compared to the initial energy W light (0). When the ψ V states interfere destructively the probability of Ω-frequency emission is reduced and the average laser energy returns to the initial value (black dashed arrows in Fig. 1b). Among the infinite Ψ′ i states, two are the dominant surviving the superposition, corresponding to the S and L electron paths, described by ψ Ω V , S and ψ Ω V , L , respectively. Correlated to them are the IR-laser states ψ Ω l , S and ψ Ω l , L (where ψ l S and ψ l L are the IR wave functions correspond to the S and L electron paths, respectively), as well as the Ω-frequency states ψ Ω S and ψ Ω L , respectively. This is consistent with the interpretation of recent experimental data 17 . It is thus evident that by measuring quantum optical properties of the IR-light we can access the full quantum dynamics of the HHG process. Such properties, in particular photon statistics to be discussed next, are not accessible by semi-classical models 1-4 .

Counting IR photons in HHG
The probability for measuring n photons in a non-interacting coherent light state is given by P n = |K n | 2 , where K n is a probability amplitude appearing in the expansion Ψ = ∑ = ∞ K n n n 0 , in terms of photon-number (Fock) states. In this expression, K n is time-independent, since, as well known 18 , the photon probability distribution in a coherent state is constant within the cycle of the light field (Fig. 2a). When the coherent light state is interacting with a single atom towards the generation of XUV radiation, the probability distribution becomes time dependent, since Ψ ′ (p, q, t − t i ) is changing at each moment of time within the cycle of the laser field due the interaction with the ionized electron. In this case, the probability distribution is given by where Ω t i are obtained using the 3-step semi-classical model 4 .
In reality, an intense Ti:S femtoseond (fs) laser pulse with ~10 17 photons/pulse (which corresponds to N 0 ~ 10 12 photons/mode for a laser system based on a 100 MHz oscillator which delivers pulses of ~30 fs duration), interacts with gas-phase medium towards the emission of XUV radiation. In this case, where n a Q ( ) i atoms coherently emit XUV radiation with frequencies proportional (Q i = Ω i /ω) to the frequency of the IR laser, the interaction is imprinted in the photon number N of the IR field as , reflecting energy conservation (N abs is the number of those photons that do not lead to XUV emission during the recollision process). Since the signal of interest, n Q which for reasons of simplicity is set ′ ≈ N N 0 0 , a XUV/IR correlation approach and/or a balanced interferometer [11][12][13] is required in order to subtract the IR photon number N 0 from N and thus measure The number of atoms interacting with the laser field is n int and C Q ( ) i is the conversion efficiency of a single-XUV-mode, which depends on the gas density in the interaction region. A rough estimation of Δ N can be obtained by taking into account the typical harmonic conversion efficiency and gas density values used in a harmonic generation experiment. The propagation effects of the IR beam in the medium have not been considered, as the precise calculation of Δ N is out of the scope of the present work. Taking into account that for gas densities ~10 18 atoms/cm 3 the conversion efficiency is ~10 −4 (for Argon, Krypton, Xenon in the 25-eV photon energy range) [19][20][21][22][23] , it can be estimated that Δ N ranges from ~0 (for zero gas density) up to ~10 9 photons/mode (for gas density ~10 18 atoms/cm 3 ). Although the study is valid for all noble gases, in the following we will describe the HHG process considering Xenon atoms interacting with a coherent IR laser field in case of low number of emitting atoms. For a single recollision, the dependence of P n (t) on time during the process is shown in Fig. 2b,c . It is seen that in the time interval 0 < t < t i ≈ 300 asec where the ionization is taking place, the peak of the probability distribution is located at n = n peak ≈ 800. Since the ionization of one Xenon atom requires the absorption of n ≈ 8 IR photons, this value corresponds to the energy absorbed by 100 Xenon atoms. For t > t i the variation of the IR photon number n with time reflects the energy exchange between the IR laser field and the free electron. The peak of the probability distribution during the recollision is located at n = n peak = Ω i /ω, = Ω t t r ( ) i (Fig. 2d). This is due to the energy absorbed by Xenon atoms during the recollision process towards the emission of XUV radiation with frequency Ω i at the moment of recombination Ω t r ( ) i . Importantly, in Fig. 2d we demonstrate that the absorbed IR photon number reveals the fundamental properties of the three-step semi-classical model: S and L paths lead to the emission of the same XUV frequency and degenerate to a single path in the cut-off region. Furthermore, as shown in Fig. 2e, the overall IR photon number distribution (red solid line) reproduces the well-known XUV spectrum resulting from the semi-classical three-step model (blue dashed line), including the plateau and cut-off regions. Thus, we demonstrated that all known features of the semiclassical three-step model are imprinted in IR photon statistics.
We will now explore the new phenomena and potential metrological applications one can address utilizing IR photon statistics. To this end, we first elaborate on the atom-number dependence of P n . While the number of IR photons absorbed by the system is proportional to n a Q ( ) i , the width w t ( ) n a of the probability distribution is determined by Gaussian statistics, (Fig. 3a). However, the distribution is departing from the Gaussian statistics during the recollision process. This is clearly shown in Fig. 3b which depicts in contour plot the normalized probability distribution of Fig. 3a. For reasons of comparison, a Gaussian distribution is shown in Fig. 3c. The distortion of the probability distribution in Fig. 3b, more pronounced in the time interval 0.5T L < t < T L , is associated with energy/phase dispersion of the interfering electron wave packets in the continuum, alluding to the possibilities of producing non-classical light-states. and I l = 10 14 W/cm 2 . In the calculation, the electron momentum, the ionization and the recombination times have been obtained by the 3-step semi-classical model. (b) Contour plot of the normalized probability distribution of (a). (c) Contour plot of the normalized probability distribution which follows the Gaussian photon statistics. This has been calculated using a single electron path which contributes to the emission of a monochromatic XUV radiation with frequency Q i = ω/11. It is evident, that in case of reducing the number of emitting atoms form = n 100 a Q ( ) i (Fig. 2b,c)  For multi-cycle laser field, the process is repeated every half-cycle of the laser period. In this case the probability distribution consists of a series of well confined peaks (Fig. 4a,b) appearing at positions ω = = Ω  n Q / and reflects the formation of well confined high order harmonics  Q ( ). Additionally, the atom-number dependence of the IR photon distribution in the HHG process provides significant advantages for high resolution spectroscopy in XUV and attosecond science. In Fig. 4c (left panel) we show the dependence of the P n on the intensity of the laser field (I l = ε 0 |E 0 | 2 /2 ∝ N 0 ) and n for =  Fig. 4c (right panel) where the probability distribution around =  Q 15 has been calculated in case of recording the 799.95 nm, 800.00 nm and 800.05 nm IR modes of a Ti:S laser pulse. This measurement can be performed by collecting the photons of the IR modes of the spectrally resolved multi-color IR pulse. This can be done by means of an IR diffraction grating placed after the harmonic generation medium. This figure also depicts the broadening effects introduced in a measured distribution by the bandwith of the driving IR pulse in case of collecting more than one modes of the multi-mode laser pulse. When Δ N is reduced, the probability distribution is getting broader (Fig. 4d, left panel), while at the point where the probability distribution between the consecutive harmonics overlaps, an interference pattern associated with the relative phase between the consecutive harmonics appears in Fig. 4d (right panel). Additionally, the modulation of P n with the intensity of the laser field (clearly shown in the left panels of Fig. 4c,d) reflects the effect of the S and L path interferences in the context of Fig. 1, i.e. the maxima (minima) of P n versus N 0 correspond to those IR-laser intensities N 0 , for which Ψ V interferes destructively (constructively). These observations can be used for attosecond science and metrology, to be explored in detail elsewhere. Since the photon statistics measurements are sensitive to shot-to-shot fluctuations of the IR intensity, stable laser systems or IR energy tagging approaches are required in order to be able to record an "IR photon statistics spectrum". Additionally, in order to avoid the influence of the laser intensity variation along the propagation axis in the harmonic generation medium, a gas medium with length much smaller compared to the confocal parameter of the laser beam is required. Any influence of the intensity variation along the beam profile at the focus can be minimized (in case that is needed) using spatial filtering approaches where the IR photons of the specific area on the focal spot diameter can be collected.

Conclusions
Concluding, we have developed a quantized-field approach which describes the strong-field light-electron interactions using a quantized coherent laser state with arbitrary photon number. The description is based on the quantized-Volkov light-electron wave function resulting from the closed-form solution of TDSE. The obtained wave function provides information about the quantum optical features of the interaction, which are not accessible by the semi-classical approaches used so far in strong-field physics and attosecond science. The approach has been used for the description of HHG in gases. We have found that the quantum optical features of the HHG can be unraveled by measuring the photon statistics of the IR laser beam transmitted from the gas medium without the need of measuring the XUV radiation. This is a unique advantage of the work since our proposed measurements, dealing with high-resolution spectroscopy in XUV and attosecond science, can be performed without the need for specialized XUV equipment (gratings, mirrors, high vacuum conditions etc.). Additionally, we have found that the HHG process in gases can lead to non-classical IR light states. In general, this work establishes a promising connection of strong-field physics with quantum optics.

Methods
is the field energy at any moment of time and = = Ψ Ψ +N t N a a ( ) . In the high photon number limit the q-dependent part of the total wave function becomes exponentially small everywhere except the region around ≈ q N 2 0 . Thus, Eq. (2) of the main text of the manuscript leads to  (3) of the main text of the manuscript can be proved in the following way (since the origin of time t can be arbitrary chosen, for simplicity and without loss of generality we set θ = 0). For an arbitrary Ψ x (p, q, t) and R p t ( , ), in the high photon number limit (where

On the validity of Eq. (3). Equation
, respectively, with t S and t L being the ionization times of the short and long electron paths.
On the calculations of the IR probability distribution. The probability to measure n photons in a non-interacting light field state Ψ is P n = |K n | 2 , where K n is a probability amplitude appearing in the expansion Ψ = ∑ = ∞ K n n n 0 , in terms of photon-number (Fock) states. In q-representation the Fock states are written as . c i (t) are n-independent complex numbers proportional to the q-independent part of the Ψ ′ and θ i = ω(t − t i ) is the phase of the laser field at the moment of ionization. In the high photon number limit, → ∼ i i denotes the electron paths contribute to the emission of the Ω i frequency. Since the probability distribution during the recollision is located at (n = n peak = Ω i /ω, = Ω t t r ( ) i ) the above expression of P n and Φ i can be further simplified by omiting the time t. This is very useful for calculating the dependence of P n on the intensity of the laser field as is shown in Fig. 4.