Electromagnetically Induced Transparency Spectra of Ladder Four-Level System with Quantum Frequency Mixing

TL;DR

The paper addresses extending quantum frequency mixing (QFM) to a ladder four-level system to modify electromagnetically induced transparency spectra. It derives an effective Hamiltonian via multi-mode Floquet theory and reveals a secondary Autler-Townes splitting (double-ATS) with centers at $\Delta_c=\pm\Omega_L/2$ and splitting $\Omega_M$, enabling broadband resonant sensing without extra energy levels. A dual-Floquet drive adds two independent quantum interference mechanisms— Floquet-channel interference and loop interference—both tunable by a phase $\phi$ and drive strength $g$, manifesting as controllable peak spacing and linewidth asymmetry. The approach provides a new coherent-control paradigm with potential implementations in Rydberg atoms or superconducting circuits and supports phase readout for microwave fields with wide frequency coverage.

Abstract

In this paper, we generalized the quantum frequency mixing technology to a ladder-type four-level system and studied its effect on electromagnetically induced transparency spectra. We found a secondary splitting of Autler-Townes splitting in the probing field transmission spectra, which could be understood by the effective Hamiltonian derived with multi-mode Floquet theory. The Frequency mixing scheme developed here enables continuous tunablity of the resonant frequency between upper levels, which facilitates the broad band sensing of AC field. Furthermore, by introducing an additional periodic driving, we realize an effective model that two distinct quantum interference effects coexist: interference among Floquet channels and loop interference arising from closed coherent pathways. Both interference effects could be read out from the transmission spectra independently. The changing of the distance between double splitting peaks represents the interference of Floquet channels, while their asymmetric linewidth broadening is linked with the total effective phase of the loop. This not only provides complementary readout for extracting the phase of AC field, but also establishes a new paradigm for coherent control in multi-level quantum systems.

Figures (4)

• Figure 1: (a) Schematic diagram of quantum frequency mixing in the ladde four-level system, which consists of a ground state $\left| 1\right\rangle$, an excited state $\left| 2\right\rangle$ and two upper states $\left| 3\right\rangle$, $\left| 4\right\rangle$. The probe laser and the control laser are coupled to the transition $\left| 1\right\rangle \longleftrightarrow \left| 2\right\rangle$ and $\left| 2\right\rangle \longleftrightarrow \left| 3\right\rangle$. The two upper states are represented as the eigenstates in the $z$-axis of the Bloch sphere. The LO field along the $x$-axis induces the formation of the dressed states. Simultaneously, two far-detuning fields along the $x$-axis form an effective field along the $z$-axis through a quantum mixer, coupling the dressed states to form a new four-level system. (b) The transmission spectra when only the LO field exists (upper) and when an additional mixed-frequency field is resonant with two dressed states (lower). Parameters with the same unit: $\Omega_p/(2\pi)=\Omega_p/(2\pi)=2$, $\Gamma/(2\pi)=5$, $\Omega_L/(\pi)=80$, $\Delta_1/(2\pi)=1080$, $\Delta_2/(2\pi)=1000$, $\Omega_1/(2\pi)=108$, $\Omega_2/(2\pi)=10$.
• Figure 2: Transmission spectrum of scanning $\Delta_b$ for setting $\Delta_c=\Omega_L/2$. The scanning method is fixing $\Omega_b/\Delta_b=$0.1. $\Delta_s/(2\pi)=1000$,$\Omega_s/(2\pi)=10$, other parameters are same as in Fig. \ref{['fig1']} (b).
• Figure 3: (a) Floquet photon-assisted transition of the four-level system in the dressed-state picture. (b) The transmission spectrum of double-ATS for different Floquet sideband. Parameters with the same unit: $\Omega_p/(2\pi)=0.1$, $\Omega_c/(2\pi)=5$, $\Gamma/(2\pi)=5$, $\Omega_L/(2\pi)=\omega/(2\pi)=40$, $\omega_M/(2\pi)=80$, $\Omega_M/(2\pi)=\delta_M/(2\pi)=4$. $g=4.81$, $\phi=\pi/2$, $\phi_M=0$.
• Figure 4: The phase-tunable transmission spectra of double-ATS under three different conditions. (a) Floquet-channel interference only with $g=4.81$ ($\Omega_c^n\approx0$). (b) Loop interference only with $g=3.83$ ($\Omega_M^k,\delta_M^j\approx0$) . (c) Both Floquet-channel interference and Loop interference with $g=4$. Other parameters are the same as in Fig. \ref{['fig3']}. These parameters result in the Floquet photon number $n=-1, m=0,l=-3, k=1, j=-1$ for scanning center $\Delta_c=-0.5\omega$.