Tunable absorption spectrum splitting in a pulse-driven three-level system

TL;DR

This work investigates how a three-level ladder system's absorption spectrum can be engineered by driving the bottom transition with a periodic train of $\pi$ pulses and probing the $|2\rangle\leftrightarrow|3\rangle$ transition with a weak field. Using a density-matrix master equation in the rotating-wave approximation and numerical integration, the authors show the absorption spectrum features an Autler-Townes–like doublet with main peaks separated by $\pi/\tau$, where $\tau$ is the inter-pulse delay, and this separation is largely independent of the pulse carrier frequency. The spectrum remains robust to variations in pulse width (even broad pulses) and exhibits asymmetry and peak-shift when finite detuning $\Delta_1$ is introduced, with imperfect pulses further perturbing peak positions. These results demonstrate flexible spectral modulation of a three-level system via realistic pulse protocols, offering a pathway toward tunable absorption spectroscopy and alternative quantum-memory implementations in such systems.

Abstract

When a two-level system is driven on resonance by a strong incident field, its emission spectrum is characterized by the well-known Mollow triplet. If the absorption from the excited state, in this continuously driven two-level system, to a third, higher energy level, is probed by a weak field, the resulting absorption spectrum features the Autler-Townes doublet with two peaks separated by the Rabi frequency of the strong driving field. It has been shown that when the two-level system is instead driven by a periodic pulse sequence, the emission spectrum obtained has similarities with the Mollow triplet even though the system is only driven during the short application time of the pulses and is allowed to evolve freely between pulses. Here, we evaluate the absorption spectrum of the three-level system in the ladder/cascade configuration when the bottom two levels are driven by a periodic pulse sequence while the transition between the middle and the highest level is probed by a weak field. The absorption spectrum displays similarities with the Autler-Townes doublet with frequency separation between the main peaks defined by the inter-pulse delay. In addition, this spectrum shows little dependence on the pulse carrier frequency. These results demonstrate the capacity to modulate the absorption spectrum of a three-level system with experimentally achievable pulse protocols.

Figures (7)

• Figure 1: Schematic representation of a three-level system in a ladder/cascade configuration. The Autler-Townes splitting is observed in the absorption from the first excited state $(|2\rangle)$ to the top excited state $(|3\rangle)$, probed by a weak field of Rabi frequency $\Omega_2$, when the bottom transition $(|1\rangle \leftrightarrow |2\rangle)$ is driven continuously on resonance by a strong continuous drive of Rabi frequency $\Omega_1$ (a). Here, we will evaluate the $(|2\rangle \leftrightarrow |3\rangle)$ absorption spectrum when the $(|1\rangle \leftrightarrow |2\rangle)$ is driven by a periodic sequence of $\pi$ pulses produced by periodically applying on the transition a field of Rabi frequency $\Omega_1$ for time $t_{\pi} = \pi/\Omega_1$ (b).
• Figure 2: Population of the second excited state ($|3\rangle)$ as a function of time, $\rho_{33}(t)$, when the $(|1\rangle \leftrightarrow |2\rangle)$ transition is driven by a periodic sequence of $\pi$ pulses with period $\tau = 0.3$. The pulse carrier frequency is resonant with the transition, $\Delta_1 = 0$, and the pulses are due to a driving field of Rabi frequency $\Omega_1 = 50$ that is periodically applied for time $t_{\pi}=\pi/\Omega$. $\rho_{33}(t)$ is shown for $\omega = 0$ (solid green line), $\pi/2\tau$ (dashed blue line) and $3\pi/2\tau$ (dashed red line). Time is measured in units of $2/\Gamma_2$
• Figure 3: Population of the second excited state or absorption spectrum of the $(|2\rangle \leftrightarrow |3\rangle)$ transition when the $(|1\rangle \leftrightarrow |2\rangle)$ is driven by a periodic sequence of $\pi$ pulses with period $\tau$ the pulse is due to a driving field of Rabi frequency $\Omega_1 = 50$ that is applied for time $t_{\pi}=\pi/\Omega_1$. Absorption spectrum shown for detuning $\Delta_1 = 0$ under pulse sequences of different periods: $\tau = 0.1$ (black dashed line), $\tau = 0.2$ (dashed blue line), $\tau = 0.3$ (dashed green line), $\tau = 0.4$ (dotted red line). The solid lines show the continuously driven system with frequency $\pi/\tau$ and all other parameters identical to the pulse-driven case. All energies are measured in units of $\Gamma_2/2$.
• Figure 4: Separation between the main peaks as a function of $1/\tau$ for $\Delta_1 = 0$. The red line and symbols represent the data while the dashed blue line represents the linear fit with slope $\pi$ confirming the dependence on $\tau$. All energies are measured in units of $\Gamma_2/2$.
• Figure 5: Population of the second excited state or absorption spectrum of the $(|2\rangle \leftrightarrow |3\rangle)$ transition when the $(|1\rangle \leftrightarrow |2\rangle)$ is driven by a periodic sequence of imperfect pulses accomplishing a $0.8 \pi$ rotation (dashed green line), $0.9 \pi$ rotation (dashed red line), $1.0 \pi$ rotation (solid black line), $1.1 \pi$ rotation (dashed blue line), $1.2 \pi$ rotation (dashed purple line), with period $\tau = 0.3$ and $\Omega_1 = 50$. The detuning with the pulse carrier frequency is $\Delta_1 = 0$. All energies are measured in units of $\Gamma_2/2$.