Abstract
We analytically describe the strongfield lightelectron interaction using a quantized coherent laser state with arbitrary photon number. We obtain a lightelectron wave function which is a closedform solution of the timedependent Schrödinger equation (TDSE). This wave function provides information about the quantum optical features of the interaction not accessible by semiclassical 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 highresolution spectroscopy in extremeultraviolet (XUV) and attosecond science without the need to measure the XUV light, while it can pave the way for the development of intense nonclassical light sources.
Introduction
Strongfield physics and attosecond science^{1,2,3,4} have been largely founded on the electron recollision process described by semiclassical approaches^{4} treating the electron quantummechanically and the electromagnetic field classically. This is because the high photon number limit pertinent to experiments with intense laser pulses appears to be adequately accounted for by a classicallydescribed electromagnetic wave, which is not affected by the interaction.
In the semiclassical approaches (known as threestep models) used for the discription of the HHG process, the electron tunnels through the stronglaserfielddistorted atomic potential, it accelerates in the continuum under the influence of the laser field and emits XUV radiation upon its recombination with the parent ion. Thus, the motion of the electron in an electromagnetic field is at the core of the recollision process. In the strongfield regime, this motion is well described by nonrelativistic semiclassical Volkov wavefunctions, obtained by solving the TDSE for a free electron in a classicallydescribed electromagnetic field.
Extending the semiclassical Volkov wave functions into the quantumoptical region is nontrivial and, to our knowledge, a closed form solution of the quantized TDSE with a coherentstate light field has never been obtained before. Although an accurate calculation of the properties of the XUV radiation emitted from a gas phase medium requires the consideration of the driving IR laser bandwidth and the propagation effects in the atomic medium, the fundamental properties of the interaction can be adequately explored with the singlecolor singleatom interaction, as has been done in the pioneering work of Lewenstein et al.^{4}. In this work we develop a quantizedfield approach for an ionized electron interacting with light field in a coherent state. We obtain a closedform solution of the TDSE, which contains complete information about the laserelectron quantum dynamics during the interaction, and use it to describe the HHG process. Differently than previous approaches^{5,6,7,8}, we describe the XUV emission as farfield dipole radiation by using an initially coherent laser state and the obtained closedform electronlaser wave function, named “quantumoptical Volkov wave function”. Our approach consistently extends the wellknown semiclassical theories^{4}, since from the obtained quantumoptical wave function we can retrieve the semiclassical Volkov wave functions by averaging over the light states. This is of advantage, since all the results of the semiclassical theory (like harmonic spectrum, electron paths, ionization times, recombination times, etc.) can be retrieved from and utilized in our quantizedfield approach.
Going beyond the reach of the semiclassical approach, we find that the quantumoptical 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 highresolution 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 nonclassical high photon number light states.
Fullquantum theoretical description of lightelectron interaction
The nonrelativistic TDSE of an electron interacting with a singlemode longwavelength linearlypolarized quantized light field of frequency ω reads (in atomic units)
where p and are the electron momentum and vector potential scalar operators along the polarization direction. The creation and annihilation operators are and , respectively, q is the inphase quadrature of the field^{9,10}, and 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 quantumoptical 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 closedform solution of Eq. (1) reads:
where the functions a(t), b(t), …., g(t), M(t) are given in terms of the parameters of Eq. (1) in the Methods Section. 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 quantumoptical description of the electronlight interaction. The term d(t)pq in the exponent renders the electronic and light degrees of freedom nonseparable. In the high photon number limit where N_{0} → ∞, β → 0 (V → ∞), and (where A_{0} is the amplitude of the corresponding classicallydescribed vector potential A = A_{0} cos(ωt + θ)), Eq. (2) is simplified (see Methods) to Ψ′(p, q, t) maintaining all the quantumoptical properties of Eq. (1).
A crucial property of the quantumoptical Volkov wave function is that the matrix elements of any qindependent operator coincide with the matrix elements obtained from using the well known semiclassical electron Volkov wave functions ψ_{V} i.e.
where Ψ_{x} is an arbitrarily chosen electronlight wave function and ψ_{x} is the corresponding state of the electron in case of classicallydescribed electromagnetic field. Thus, while Ψ′(p, q, t) goes beyond the semiclassical 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 semiclassical models^{4} can be retrieved, and more than that, utilized in our quantizedfield approach. In the particular case of HHG, Ψ_{x} is the ground state of the system (ψ_{g}ψ_{c}), is the dipole moment , 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 quantumoptical approach can provide information about the IR laser field states during the interaction, inaccessible by the semiclassical theories. This information can be experimentally extracted utilizing XUV/IRcorrelation approaches and/or balanced homodyne detection techniques^{11,12,13} of the IR laser field transmitted from the harmonic generation medium.
Quantumoptical description of the HHG process
Using the quantumoptical 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 laserelectron states having different ionization times t_{i}. In eq. 4 we assume that lightelectron 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 semiclassical theory^{4} we assume that a_{g}(t) ≈ 1, while all . 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 . In this case, the time dependent dipole moment is . As in the semiclassial theory^{4}, we neglect groundground and continuumcontinuum transitions, and consider only groundtocontinuum (and continuumtoground) transitions given by the matrix elements (and ).
Integrating over q, using Eq. (4) and integrating over p, we retrieve that which coincides with the expression given by the semiclassical theories. Thus, all the semiclassical results^{4,14,15,16}, in particular the short (S) and long (L) electron paths with electronic Volkov wave functions and 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 IRlaser 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 laserelectron 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 Ωfrequency 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 Ωfrequency emission is reduced and the average laser energy returns to the initial value (black dashed arrows in Fig. 1b). Among the infinite states, two are the dominant surviving the superposition, corresponding to the S and L electron paths, described by and , respectively. Correlated to them are the IRlaser states and (where and are the IR wave functions correspond to the S and L electron paths, respectively), as well as the Ωfrequency states and , respectively. This is consistent with the interpretation of recent experimental data^{17}. It is thus evident that by measuring quantum optical properties of the IRlight 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 semiclassical models^{1,2,3,4}.
Counting IR photons in HHG
The probability for measuring n photons in a noninteracting coherent light state is given by P_{n} = K_{n}^{2}, where K_{n} is a probability amplitude appearing in the expansion , in terms of photonnumber (Fock) states. In this expression, K_{n} is timeindependent, 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 and are the ionization and recombination times of the corresponding electron paths with momentum which lead to the emission of XUV radiation with frequency Ω_{i} (see Methods Section). The parameters , and are obtained using the 3step semiclassical 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 gasphase medium towards the emission of XUV radiation. In this case, where 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 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, , is superimposed on a large background which for reasons of simplicity is set , 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 and thus measure . The number of atoms interacting with the laser field is n_{int} and is the conversion efficiency of a singleXUVmode, which depends on the gas density in the interaction region. A rough estimation of ΔN 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 ΔN 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 25eV photon energy range)^{19,20,21,22,23}, it can be estimated that ΔN 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 , intermediate and high 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 for . 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, (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 . Importantly, in Fig. 2d we demonstrate that the absorbed IR photon number reveals the fundamental properties of the threestep semiclassical model: S and L paths lead to the emission of the same XUV frequency and degenerate to a single path in the cutoff region. Furthermore, as shown in Fig. 2e, the overall IR photon number distribution (red solid line) reproduces the wellknown XUV spectrum resulting from the semiclassical threestep model (blue dashed line), including the plateau and cutoff regions. Thus, we demonstrated that all known features of the semiclassical threestep 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 atomnumber dependence of P_{n}. While the number of IR photons absorbed by the system is proportional to , the width 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 nonclassical lightstates.
For multicycle laser field, the process is repeated every halfcycle of the laser period. In this case the probability distribution consists of a series of well confined peaks (Fig. 4a,b) appearing at positions and reflects the formation of well confined high order harmonics .
Additionally, the atomnumber 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 (for simplicity we consider only the case of ). Indeed, the harmonic spectrum can be obtained from the maxima of P_{n} centered at . The spacing between the maxima with for consecutive harmonics, and the width depend on and n_{int}. The resolving spectral power increases with n_{int} and for values of ΔN ~ 10^{9} photons, P_{R} can reach the values of ~10^{4}–10^{5} in the spectral range of 25 eV, which competes with stateoftheart XUV spectrometers. This is shown in Fig. 4c (right panel) where the probability distribution around 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 multicolor 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 multimode laser pulse.
When ΔN 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 IRlaser 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 shottoshot 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 quantizedfield approach which describes the strongfield lightelectron interactions using a quantized coherent laser state with arbitrary photon number. The description is based on the quantizedVolkov lightelectron wave function resulting from the closedform solution of TDSE. The obtained wave function provides information about the quantum optical features of the interaction, which are not accessible by the semiclassical approaches used so far in strongfield 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 highresolution 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 nonclassical IR light states. In general, this work establishes a promising connection of strongfield physics with quantum optics.
Methods
On the closedform solution of TDSE
In order to obtain a closedform solution of Eq. (1) of the main text of the manuscript, we consider as an initial state, a state where the electron is decoupled from the light i.e. to be a separable product of a coherent state of light and an arbitrary fieldindependent electron state ψ_{0}(p) in momentum representation, where is the parameter which introduces the light dispersion due to the presence of the electron^{24,25}, carries the information about the phase of the light θ, N_{0} is the average photon number of the initial (t = 0) coherent light state, and c_{0} is a normalizing constant.
The parameters appearing in Eq. (2) of the main text of the manuscript are
where , , m = 1 − Me^{2iωt}, , and C_{0} is normalization constant. From the general solution of Eq. (2) we can recover energy conservation, i.e. the instantaneous interaction energy of the electron is given by W_{e,int}(t) = W_{e}(t) − W_{e}(0) = W_{light}(0) − W_{light}(t), where W_{e}(0) is the initial kinetic energy of the electron, is the initial energy of the light field, is the field energy at any moment of time and . In the high photon number limit the qdependent part of the total wave function becomes exponentially small everywhere except the region around . Thus, Eq. (2) of the main text of the manuscript leads to
where now the parameters in the exponent are
and is normalization constant.
On the validity of Eq. (3)
Equation (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 , in the high photon number limit (where N_{0} → ∞, and ) the matrix element (where ) with the exponent to be proportional to . The integration over q i.e. , leads to = = = , where is the arbitrary state of system with classically described electromagnetic field and .
On the description of HHG using the IR wave functions
The results obtained by the semiclassical theories regarding HHG can be also obtained by integrating over p, using Eq. (3) and integrating over q. In this case the dipole moment is expressed in terms of the corresponding to the Volkovelectron states IR wave functions (where ψ_{g}(p) is a ground state of the electron in momentum representation, t_{V} are the ionization times Volkov electron paths contribute to the harmonic generation). The IR wave functions which correspond to the S and L electron paths are and , 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 noninteracting light field state Ψ is P_{n} = K_{n}^{2}, where K_{n} is a probability amplitude appearing in the expansion , in terms of photonnumber (Fock) states. In qrepresentation the Fock states are written as , where are Hermite polynomials. For coherent light states^{9}, the photon statistics are described by the Poisson distribution , wellapproximated by a Gaussian when . In case of HHG process, the probability distribution during the recollision process for a single path i of ionization time t_{i} and electron momentum p_{i}(t) which contributes to the production of XUV radiation with frequency the harmonic Ω_{i} is given by , where is determined through the expansion . c_{i}(t) are nindependent complex numbers proportional to the qindependent 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, (where A_{i}(t) is real), and the probability distribution reads , with
where and is the average number of photons during the recollision, with and Ω(t) = ((p^{2}(t)/2) − IP). When multiple paths contribute to the emission of multiple harmonics, , where 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, ) 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.
Additional Information
How to cite this article: Gonoskov, I. A. et al. Quantum optical signatures in strongfield laser physics: Infrared photon counting in highorderharmonic generation. Sci. Rep. 6, 32821; doi: 10.1038/srep32821 (2016).
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Acknowledgements
We acknowledge support by the Greek funding program NSRF and the European Union’s Seventh Framework Program FP7REGPOT201220131 under grant agreement 316165.
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I.A.G. obtained the closedform solution of the TDSE, contributed on the quantumoptical description of the HHG and manuscript preparation; N.T. performed the theoretical calculations shown in the figures and contributed on the data analysis; I.K.K. contributed on the quantumoptical description of the HHG and manuscript preparation; P.T. conceived the idea and contributed in all aspects of the present work except of solving the TDSE.
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Gonoskov, I., Tsatrafyllis, N., Kominis, I. et al. Quantum optical signatures in strongfield laser physics: Infrared photon counting in highorderharmonic generation. Sci Rep 6, 32821 (2016). https://doi.org/10.1038/srep32821
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