Quantum Electrodynamics of Intense Laser-Matter Interactions: A Tool for Quantum State Engineering

Intense laser-matter interactions are at the center of interest in research and technology since the development of high-power lasers. They have been widely used for fundamental studies in atomic, molecular, and optical physics, and they are at the core of attosecond physics and ultrafast optoelectronics. Although the majority of these studies have been successfully described using classical electromagnetic fields, recent investigations based on fully quantized approaches have shown that intense laser-atom interactions can be used for the generation of controllable high-photon-number entangled coherent states and coherent state superpositions. In this tutorial, we provide a comprehensive fully quantized description of intense laser-atom interactions. We elaborate on the processes of high-harmonic generation, above-threshold ionization, and we discuss new phenomena that cannot be revealed within the context of semiclassical theories. We provide the description for conditioning the light field on different electronic processes, and their consequences for quantum state engineering of light. Finally, we discuss the extension of the approach to more complex materials, and the impact to quantum technologies for a new photonic platform composed of the symbiosis of attosecond physics and quantum information science.

Popular Summary

For decades, the description of intense laser-matter interaction did not consider the quantum nature of the electromagnetic field. However, recent theoretical and experimental advances go beyond this semiclassical perspective and show that intense laser-matter interaction can be used to generate nonclassical and entangled field states. In this tutorial, we develop a full quantum mechanical theory of intense laser fields interacting with atoms and provide the operations needed for quantum state engineering of light. This allows, for instance, questions to be answered about the quantum state of the field for high-order harmonic generation or above-threshold ionization and the further consideration of the backaction onto the field itself. Then, conditioning experiments on the field after the interaction allows the generation of nonclassical field states.

In this tutorial, we first introduce the reader to the basic notions of the fully quantized light-matter interaction and particularly consider the case of atoms driven by ultrashort and intense laser fields. We then solve for the dynamics of the electromagnetic field modes conditioned on the process of high-order harmonic generation and above-threshold ionization. We obtain the quantum state of the field and can show the backaction of the interaction on the field itself. The quantum description of the field allows us to introduce quantum optical conditioning measurements, which then leads to the generation of nonclassical field states covering a wide spectral range. Furthermore, we present a comprehensive guide for conducting such experiments.

This connects strong laser-field physics with quantum optics. It provides a new direction of quantum state engineering, with implications in modern optical quantum technologies toward practical applications of quantum information science.

Figure 1

Depiction of the key quasiclassical strong-field processes, along with the quantum states of light and matter. (a) High-harmonic generation (HHG), (b) low-order above-threshold ionization (LATI) without rescattering events, (c) high-order above-threshold ionization (HATI) including rescattering events, (d) nonsequential double ionization (NSDI). The states of the driving field mode before and after interaction are given by $|{\alpha }_L⟩$ and $|{\alpha }_L+{\chi }_L⟩$, respectively. High-order harmonic modes, due to HHG, are given by $|{\chi }_q⟩$. The initial and final electronic states are given by $|g_{\alpha }⟩$ and $|{\mathbf{v}}_{\alpha }⟩$. To emphasize that the electrons in NSDI may be entangled [11], we have written them as a two-particle state. The quantum optical degrees of freedom may also be entangled with each other and the electronic states; however, for clearer labeling, we have written these separately.

Figure 2

Photon-number probability distributions for (a) two coherent states with $\alpha =7.95$ (red dashed curve) and $\alpha =8.73$ (black solid curve), and (b) a superposition of three coherent states of the form $|\varphi ⟩=a_1|7⟩+a_2|9⟩+a_3|10⟩$ with $(a_1,a_2,a_3)=(1.0,-1.0,0.75)$ (red dashed curve) and $(a_1,a_2,a_3)=(1.0,1.0,0.75)$ (black solid curve). The coefficients in the superposition are shown up to a normalization that, nonetheless, has been taken into account when doing the plots. In both plots, the mean photon number is $⟨n⟩\approx 63.2$ for the red dashed curve probability distribution and $⟨n⟩\approx 76.2$ for the black solid line.

Figure 3

In (a) we consider a linearly polarized electromagnetic field with a sinusoidal squared envelope, central frequency $\omega =0.057$ a.u., field amplitude $E=0.053$ a.u., and $n_{\mathrm{cyc}}=5$ cycles. The orange dashed curve shows the electric field while the blue solid curve the vector potential. In (b) we show the dependence of $|{\alpha }_{\mathbf{k},\mu }|$ on the frequency mode.

Figure 4

High-harmonic generation spectrum obtained from the evaluation of Eq. (48) using the Qprop software [119], when a hydrogen atom in the ground state ($I_p=0.5$ a.u.) interacts with a linearly polarized electromagnetic field with a sinusoidal squared envelope that has ten cycles, driving frequency $\omega =0.057$ a.u., and electric field amplitude $E=0.053$ a.u. The harmonic yield extends until the 21st harmonic, which constitutes the cutoff frequency of the spectrum, that is, the maximum energy that can be gained by the electron when accelerating in the continuum.

Figure 5

Behavior of the shifts $|\chi (t_1)|$ [in (a)] and $|\delta (T,t_1,{\omega }_k,\mathbf{p})|$ [(b) $p=0$ and (c) $p=0.93\sqrt{U_p}$] over ionization time $t_1$. The different curves in each of the plots represent contributions to different harmonic modes ($n_{\mathrm{harm}}$). In particular, the blue solid curve corresponds to the fundamental mode $n=1$, the orange dashed curve to the second harmonic $n=2$, and the green dash-dot curve to the third harmonic $n=3$. In general, the displacement from $|\chi (t_1)|$ is, at least, 2 orders of magnitude smaller than that coming from $|\delta (T,t_1,{\omega }_k,\mathbf{p})|$, which furthermore gets enhanced for increasing values of the canonical momentum $\mathbf{p}$.

Figure 6

Real (a) and imaginary (b) parts of the shift of the fundamental mode $\delta (T,t_1,{\omega }_k,\mathbf{p})$ for values of the canonical momenta $p=0.93\sqrt{U_p}$ (blue solid curve) and $p=-0.93\sqrt{U_p}$ (orange dashed curve). The different behaviors obtained for positive and negative momenta are a consequence of the carrier-envelope phase (CEP) (which is the phase between the carrier wave and the pulse envelope) of the employed laser field, such that a change in $\pi$ of this phase interchanges the behavior obtained for positive and negative momenta.

Figure 7

The mean value of the photon number (75) of the fundamental mode at the end of the pulse. Here, we use $\alpha =7i$ to reproduce the form of the considered field. In order to avoid reaching the numerical limit of the machines used for the computation, we have multiplied the field constant by 0.2 (see Appendix pp7 for details). We consider a field with a sinusoidal squared envelope with $\omega =0.057$ a.u., five cycles of duration, and field amplitude $E_0=0.053$ a.u.

Figure 8

Photon-number probability distribution (74) for positive (upper row) and negative (lower row) values of the canonical momentum. Specifically, we use (a) $p=0.05\sqrt{U_p}$, (b) $p=0.14\sqrt{U_p}$, (c) $p=0.32\sqrt{U_p}$, (d) $p=-0.05\sqrt{U_p}$, (e) $p=-0.14\sqrt{U_p}$, and (f) $p=-0.32\sqrt{U_p}$. We consider a field with a sinusoidal squared envelope with $\omega =0.057$ a.u., five cycles of duration, and field amplitude $E_0=0.053$ a.u. The vertical gray dashed line shows the location of the maximum photon-number probability for the initial coherent state used in the numerical analysis, i.e., $\alpha =7i$.

Figure 9

Schematic illustration of the operation principle of the experimental approach. Unit 1: laser beam delivery (LBD) system. Unit 2: target area (TA). The intense laser-atom interaction is shown in the context of the electron recollision picture. The ATI photoelectrons are emitted in the direction of the polarization of the driving IR field, while the generated high harmonic copropagates with the IR field. Unit 3: IR attenuation and conditioning (A-C) on HHG and/or ATI processes. Here $|{\alpha }_L⟩$ is the initial coherent state of the driving field and $|{\alpha }_L+{\chi }_L⟩$, $|\alpha +\chi ⟩$ are the states of the IR field after the interaction and attenuation, respectively; the $|{\chi }_q⟩$ correspond to the coherent states of the harmonic modes; $|{\varphi }_c^{(\mathrm{IR})}⟩$ and $|{\chi }_c^q⟩$ are the field states conditioned on HHG or ATI processes for the IR and high-harmonic field states, respectively, which correspond to a coherent state superposition. We note that the detailed expressions of the light states are given in Sec. 4. Unit 4: a typical scheme of the QT method for the characterization of state $|{\varphi }_c⟩$, with state $|{\alpha }_r⟩$ of the local oscillator reference field, with a controllable phase shift $\varphi$. BS is a beam splitter, the PD are identical IR photodiodes used from the balanced detector, and $i_{\varphi }$ is the $\varphi$-dependent output photocurrent difference used for the measurement of the electric field operator and the light state characterization via the reconstruction of the Wigner function.

Figure 10

A more detailed scheme of unit 2. Here $L_1$ is an IR focusing lens. The arrangement with the TOF spectrometers can be used for measuring the ATI spectra associated with positive and negative electron momenta. HS is a harmonic separator that transmits the IR field and reflects the high harmonics towards the XUV detectors. Al is a thin aluminum metal filter that transmits the harmonics with $q\ge 11$ (harmonics in the plateau and cutoff region of the spectrum). In this way we eliminate the contribution of low-order harmonics that are mainly produced by multiphoton processes. Additionally, the presence of a metal filter is an experimental requirement for blocking any residual part of the IR beam reflected by the HS. The lens $L_2$ is used to collimate the IR beam after the interaction. With $S_{\mathrm{ATI}}^{(\mathrm{pos},\mathrm{neg})}$ and $S_{\mathrm{XUV}}$ we denote the integrated, over a defined region of the spectra, ATI and HHG signals, respectively. When using multicycle driving laser fields, the use of only one TOF is sufficient as $S_{\mathrm{ATI}}^{\text{(pos)}}\approx S_{\mathrm{ATI}}^{\text{(neg)}}$.

Figure 11

Calculated HHG and ATI spectra using the Qprop software [119]. Panels (a) and (b) respectively show the HHG and ATI spectra produced by the interaction of xenon atoms with a linearly polarized laserpulse, with a sinusoidal squared envelope, of intensity $8\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and about $30$-fs duration. In (b) the photoelectron spectra corresponding to direct and rescattered photoelectrons with positive ($p>0$) and negative ($p<0$) momenta are shown with different colors. Panels (c) and (d) respectively show the ATI spectra for negative and positive electron momenta produced by the interaction of xenon atoms with a linearly polarized laser pulse of intensity $8\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and about $5$-fs duration. The CEP effect is depicted in the high-energy part of the ATI spectra of the negative and positive photoelectrons. The ATI spectra corresponding to the direct and rescattered electrons are shown with different colors. In all graphs the total ATI spectrum (black solid curve), which includes the contribution of the direct and rescattered electrons, has been shifted by a factor of 100 for visualization reasons. The gray shaded areas depict an example of the controllable (in the width and their position in the spectra) integrated areas of the $S_{\mathrm{ATI}}^{(\mathrm{pos},\mathrm{neg})}$ and $S_{\mathrm{XUV}}$.

Figure 12

A schematic illustration of unit 3. Here $F$, BS, and $A$ respectively represent a neutral density filter, a beam splitter, and an aperture; ${\mathrm{PD}}_{\mathrm{IR}}$ is an IR photodetector; $S_{\mathrm{XUV}}$, $S_{\mathrm{ATI}}^{(\mathrm{pos},\mathrm{neg})}$, $S_{\mathrm{IR}}$, and $S_{\text{IR0}}$ are the signals used by the quantum spectrometer (QS) to condition the $|\alpha +\chi ⟩$ state on the HHG or ATI processes.

Figure 13

(a) $(S_{\mathrm{XUV}},S_{\mathrm{IR}})$ photon-number distribution (gray points). The mean values of $S_{\mathrm{XUV}}$ and $S_{\mathrm{IR}}$ are normalized to 1. Here $W_j$ shows the width of the distribution. The red points show the selected points along the anticorrelation diagonal of width $w_{\mathrm{ant}}$. The distribution is created by keeping the energy stability of the driving field at the level of approximately $1\mathrm{\%}$, and after subtracting the electronic noise from each laser shot. (b) Probability of absorbing IR photons ($P_{\mathrm{IR}}$) towards HHG. The arrows depict the positions of the peaks in the multipeak structure of $P_{\mathrm{IR}}$. Figure reproduced from Ref. [92].

Figure 14

An optical arrangement that leads to the entangled coherent state $|{\psi }_f⟩$. Here $|{\alpha }_0⟩$ is the initial coherent state of the driving IR field; $|{\alpha }_{1,2}⟩$ are the coherent states after the beam splitter ${\mathrm{BS}}_1$; $|{\alpha }_2{e}^{i\phi }⟩$ is the state of the field after the phase shifter that introduces a phase $\phi$ in $|{\alpha }_2⟩$; HHG and QS are the HHG areas and the QS arrangements; $|{\psi }_f⟩$ is the final state after the last beam splitter (${\mathrm{BS}}_2$); ${|{\varphi }_c^{(1/2)}⟩}_{\mathrm{HHG}}$ are the optical “catlike” states created after conditioning on HHG. State $|{\psi }_f⟩$ can be controlled by the variables ${\chi }_1$, ${\chi }_2$, and $\phi$.

Figure 15

Quantum tomography of an optical coherent state $|\alpha ⟩$. In the left panel we show the calculated homodyne detection signal $x_{\varphi }$, while the corresponding Wigner function centered at $|\alpha |=2$ is shown in the right panel.

Figure 16

Quantum tomography of optical “catlike” CSS [(a) and (b)] and “kittenlike” CSS [(c) and (d)] states created when the HHG and the QS are switched on. The left panels show the calculated homodyne detection signal $x_{\varphi }$ and the right panels the corresponding Wigner functions centered at $|\alpha |$. The calculations are preformed using state $|{\varphi }_c⟩=|\alpha +\chi ⟩-{\xi }_{(\mathrm{IR})}|\alpha ⟩$ (with $|\alpha |=2$) created by conditioning on the HHG process. The value of $|\chi |$ is set at 0.8 for the optical “catlike” state [(a) and (b)] and reduced to 0.1 for the “kittenlike” state [(c) and (d)].

Figure 17

(a) Dependence of the error amplitude of a reconstructed Winger function of a coherent state on $k_c$, using a homodyne trace with about ${10}^4$ points for $0\le \varphi \le \pi$. (b) Dependence of the accuracy of measuring the mean photon number on the mean photon number of the light state. (c) Dependence of the Wigner function amplitude error on the number of points of the measured homodyne detection signal $x_{\varphi }$ for $k_c\approx 3.7$. (d) Dependence of the accuracy of measuring the mean photon number of the light state on the number of points of the measured homodyne detection signal $x_{\varphi }$ for $⟨n⟩=3$. In all graphs, the red solid line is a 15-point running average of the data (gray points). Panels (a) and (b) are reproduced from Ref. [92].

Figure 18

(a) Dipole moment expectation value obtained from the numerical calculations under the SFA approach. (b) Vector potential used to obtain the dipole moment shown in (a). The pulse shown has $n_{\mathrm{cyc}}=5$ cycles.

Figure 19

Behaviors of (a) ${\chi }_{{\mathbf{k}}_L}(t)$ and (b) $C_{\mathrm{HH}}(p=2U_p,T,t)$ with a time interval of $\mathrm{\Delta }t=1.0$ (solid black curve), and the function obtained from the interpolation (dashed orange curve). For the interpolation plot, we considered a time step $\mathrm{\Delta }t=0.01$.