Propagation of intense squeezed vacuum light in non-linear media

Abstract

Recent developments in quantum light engineering have enabled the use of infrared bright squeezed vacuum (BSV) femtosecond pulses in highly nonlinear optics, particularly strong field physics and high-harmonic generation. However, theoretical studies were focused on the microscopic interaction with a single atom, neglecting the crucial macroscopic aspect of light propagation through the media. This raises a key question: How does BSV propagates in strongly light-driven nonlinear media and how this affects the generation of non-linear optical signals? We address this question by introducing a fully quantized framework that accounts for the propagation in gas media. We find that atomic ionization caused by strong BSV fluctuations and the associated infrared photon losses introduce decoherence effects that can substantially limit the propagation length in the medium, reduce the harmonic yield, and decrease the number of emitted harmonics at high intensities. However, these effects are not detrimental. We identify conditions under which propagation-induced decoherence is minimized while the generated harmonics remain clearly detectable--an issue of particular importance for future studies exploring the connection between strong-field physics and quantum optics. Our results lay the foundation for future studies of BSV in strong-field physics, nonlinear optics, and ultrafast science, and establish a basis for exploring its propagation through all states of matter in a fully quantized framework.

Figures (7)

• Figure 1: Schematic of the propagation of intense IR BSV in a gas medium. Multiphoton ionization (MPI), tunneling ionization (TI), high-harmonic generation (HHG), and IR scattering (Scat.) are the processes that typically occur during propagation. $\ket{\text{BSV}}_{\text{in}}$ and $\hat{\rho}_{\text{out}}$ depict the IR light states entering and exiting a gas medium of length $L_{m}$, respectively. $N_{q}$ indicates the generated harmonic photon number.
• Figure 2: Atomic ionization in a BSV field. (a) Schematic of the interaction between BSV light pulses and a single Ar atom. $Q(E_{\alpha})$ represents a marginal of the Husimi distribution of BSV light with a mean intensity $\langle I\rangle = 1.3 \times 10^{14} \text{W}/\text{cm}^{2}$. For a coherent state of the same intensity $Q(E_{\alpha})$ is narrowly peaked at $E_{\alpha} \approx 3 \times 10^{8}$ (red-dashed line). (b) Dependence of the mean ionization probability $\langle Y_{i} \rangle$ on $\langle I\rangle$ of a BSV field (black line). For comparison, the dependence of $\langle Y_{i} \rangle$ on $\langle I\rangle$ for interactions with coherent light states is shown in blue.
• Figure 3: Decoherence of BSV in non-linear media. (a) Dependence of the Wigner function $W(X_{1},X_{2})$ of the light state on the photon losses during the propagation in the medium. Here, $\hat{X}_{1}=(\hat{a}^\dagger+\hat{a})$ and $\hat{X}_{2}=-i(\hat{a}^\dagger-\hat{a})$ are the optical quadrature operators. The left panel corresponds to the $W(X_{1},X_{2})$ of an initial BSV state $\ket{\text{BSV}}_{\text{in}}$, with $A=0$ ($0\%$ losses) and squeezing parameter $r$=2. The middle and right panels show the $W(X_{1},X_{2})$ for $A=1/8$ ($12.5\%$ photon losses) and $A=1/2$ ($50\%$ photon losses), respectively. (b) Dependence of product of variances of the optical quadratures on the photon losses minus the Heisenberg limit. The white dashed line depicts the $12\%$ photon losses that we define as the border above which the BSV state losses its quantumness, i.e., for $A<1/8$ the BSV preserves the initial quantum features $\ket{\text{BSV}}_{\text{in}}$ and for $A>1/8$ the state changes to a mixed state $\hat{\rho}_{\text{out}}$. (c) Dependence of $A$ on $\langle I \rangle$ at different values of $L_{m}$ for $\rho_{\text{at}}=10^{18}$ atoms per cm$^3$. (d) Dependence of $L_{m}^{\text{(BSV)}}$ on $\langle I \rangle$ for $\rho_{\text{at}}=10^{18}$ atoms per cm$^3$.
• Figure 4: Propagation of BSV in non-linear media and HHG. The upper panels ((a), (b)) refer to the harmonic yield generated when a BSV and coherent fields interact with a single Ar atoms. The lower panels ((c), (b) and (e)) refer to the harmonic yield exiting an Ar gas medium when the interaction occurs with BSV and coherent fields. (a) Dependence of the mean photon number of the 15th harmonic $\langle N_{15} \rangle$ on $\langle I\rangle$ of a BSV field (black line). For comparison, the dependence of $\langle N_{15} \rangle$ on $\langle I\rangle$ for interactions with coherent light states is shown with a blue line. (b) HHG spectra generated by BSV (black line) and coherent (blue line) fields at $\langle I\rangle = 1.3 \times 10^{14} \text{W}/\text{cm}^{2}$. (c) Dependence of $\langle N_{15} \rangle$ on the medium length $L_{m}$ generated by BSV (black line) and coherent (blue line) fields without taking into account the BSV decoherence effects in the medium. (d) Dependence of $\langle N_{15} \rangle^{(\text{Coh.})}/\langle N_{15} \rangle^{(\text{BSV})}$ on $\rho_{\text{at}}$ calculated taking into account the BSV decoherence in the medium. The calculation has been conducted for $L_{m}=2L_{a}$ where the BSV preserves its quantumness. For the coherent states the calculation has been conducted for $L_{m}>\frac{5}{2} L_{a}$. (e) HHG spectra generated by BSV (black line) and coherent states (blue line) after propagation of $L_{m}\approx L_{m}^{\text{(BSV)}}\approx 2 L_{a}$ and $L_{m}>\frac{5}{2} L_{a}$, respectively. $N_{q}^{\text{(BSV)}}$ indicates the harmonic photon number generated by the state $\hat{\rho}_{\text{out}}\approx \dyad{\text{BSV}}$.
• Figure 5: $Q(\alpha)$ function and harmonic yields of various harmonics computed for different mean intensities of the driving BSV light. In all cases, we considered a pulse with a $\sin^2$-envelope, 13 fs of duration and central wavelength $\lambda = 800$ nm. The atomic species under consideration is Argon ($I_p=0.58$ a.u.).