Dynamically Dressed States of a Quantum Four-Level System

TL;DR

This work addresses how a four-level quantum dot system exhibits dynamically dressed states under pulsed two-photon excitation. It combines high-resolution spectroscopic measurements with a four-level Hamiltonian and numerically exact simulations to capture the time-dependent emission spectrum and phonon interactions. A key finding is that sidebands emerge only on the biexciton-to-exciton transition and their number, detuning, and motion depend on the pulse area and pulse duration. These results generalize the dressed-state concept to multi-level, pulsed driving and point to new routes for deterministic and non-classical light generation from dynamically dressed systems.

Abstract

In this work, we experimentally and theoretically study the dressed-state emission of the biexciton-exciton cascade in a semiconductor quantum dot under pulsed, resonant, two-photon excitation. Building on the well-characterized steady-state dressed emission of the four-level system, we examine its dynamic counterpart under pulsed, resonant excitation, addressing both experimental observations and theoretical modeling. Here we report several sidebands emerging from the biexciton-to-exciton transition, whose number and spectral width depend on the excitation pulse duration and the effective pulse area, while no sidebands emerge from the exciton-to-ground-state transition. Since the biexciton state population follows a nonlinear pulse area function, sidebands with a small spectral nonlinearity result. Detuning- and time-dependent measurements provide deeper insight into the emission properties of the dressed states. They show that side peak emission only occurs in the presence of the excitation pulse. Moreover, when the system is excited by a Gaussian-shaped laser pulse, side peak emission takes place sequentially.

Figures (10)

• Figure 1: (a) Level scheme of the biexciton-exciton cascade. Double-sided arrows indicate the laser field. Arrows pointing downward indicate emission by radiative decay. We excite the system with horizontally polarized laser light and collect only vertically polarized photons in a cross-polarized RF setup. (b) Measured emission spectrum after TPE of the biexciton. The biexciton and exciton emission energies are separated by the biexciton binding energy $\mathrm{E_b}$. (c) Energy diagram of the biexciton-exciton cascade in the rotating frame of the excitation laser. At $\mathrm{t=5\ps}$, the system is excited by a strong cw laser, resulting in the dressing and mixing of states. There are six allowed transitions, indicated by the arrows, when the system is driven on one polarization branch while detecting the other. Dark gray arrows indicate transitions from $\mathrm{\ket{XX} \rightarrow \ket{X_V}}$, light gray arrows indicate transitions from $\mathrm{\ket{X_V} \rightarrow \ket{G}}$. (d) Measured dressed emission spectrum of the biexciton-exciton cascade under strong cw excitation. Six optical transitions (dark and light gray background areas) and the imperfectly suppressed excitation laser are observed. (e) Dressed energy diagram of the biexciton-exciton cascade under pulsed excitation. The dressing of the states occurs only during the presence of the pulse, varies in time and follows the temporal shape of the excitation pulse. Due to a time-dependent population of the excited states a modulated emission probability exists. (f) Measured dressed emission spectrum of the biexciton-exciton cascade under strong pulsed excitation in a semi-logarithmic scale. A one-sided multi-peak structure red-detuned from the biexciton transition appears.
• Figure 2: Logarithmically plotted intensity of measured and simulated power-dependent emission spectra. Rabi oscillations between the $\mathrm{\ket{G} \leftrightarrow \ket{XX}}$ transition are driven by pulsed resonant TPE for pulse durations of (a) 14p, (b) 10p, and (c) 6p. In addition to the biexciton and exciton emission lines, multiple sidebands emerge from the biexciton transition. As the excitation power increases, an increasing amount of side peaks appear which shift towards lower energies, resulting in greater detuning from the biexciton emission line. The strongly quenched side peak emission emerging from the exciton-to-ground-state transition is only visible in the theoretical calculations with a pulse duration of (a) 14p. With decreasing excitation pulse duration the dressed part of the spectrum broadens, and the number of side peaks decreases within a given power range.
• Figure 3: (a) Rabi rotations of the $\mathrm{\ket{G} \leftrightarrow \ket{XX}}$ transition driven by resonant TPE with Gaussian pulses of duration of $\mathrm{14\ps}$ (red) and $\mathrm{4\ps}$ (dark teal). The first $\mathrm{\pi}$ rotation requires the same average excitation power regardless of the pulse length. With increasing excitation power, the Rabi rotations begin to diverge. For both Rabi rotations, a nonlinear behavior with the square root of the excitation power is observed. (b) Pulse areas of Rabi rotations for six different pulse durations ranging from $\mathrm{14\ps}$ down to $\mathrm{4\ps}$ are plotted as a function of the square root power. For the $\mathrm{4\ps}$ pulse, the pulse area is almost linear with the square root of the excitation power. For longer pulses, the nonlinear and pulse length dependent nature of TPE becomes visible in an upward slope. All curves show the same average excitation power for the first $\mathrm{\pi}$ rotation. With decreasing pulse duration, the number of $\mathrm{\pi}$ rotations decreases for the same power range.
• Figure 4: Measured and simulated emission spectra under detuned cw and pulsed excitation. (a) A strong cw laser is tuned across the TPE resonance over a range of $\mathrm{\pm 0.21m\eV}$. Six emission lines appear. The dressed part of the spectrum follows the detuning of the excitation laser, while at resonance the shifting lines anticross with the non-shifting lines. (b) A pulsed excitation laser with a pulse length of 14p and a pulse area of $\mathrm{6\pi}$ is tuned across the TPE resonance by $\mathrm{\pm 0.51m\eV}$. Under pulsed excitation, the biexciton and exciton transition lines maintain their transition energies. For $\mathrm{\Delta < 0}$, the side peaks follow the detuning of the excitation laser. For $\mathrm{\Delta >0}$ the side peaks asymptotically approach the biexciton emission line, with the innermost side peak continuing to follow the laser detuning.
• Figure 5: (a) Time-resolved measurements of the side peaks $\mathrm{S_1}$ (red), $\mathrm{S_2}$ (yellow), and $\mathrm{S_3}$ (blue). The measured temporal width of the side peaks is short compared to the biexciton and exciton emission lifetimes and follows the temporal profile of a Gaussian pulse with a small additional tail at the falling edge. The side peaks appear sequentially in time at $\mathrm{t_1}$ (red arrow), $\mathrm{t_2}$ (yellow arrow), and $\mathrm{t_3}$ (blue arrow). (b) Simulated time-integrated emission spectrum of the biexciton excited via TPE by a $\mathrm{14\ps}$ Gaussian pulse (gray) with a pulse area of $\mathrm{10\pi}$, plotted as a function of time. Five side peaks (blue markers) start to detach from the biexciton emission line, while at $\mathrm{t_1}$, $\mathrm{t_2}$, and $\mathrm{t_3}$ (red markers) the side peaks $\mathrm{S_1}$, $\mathrm{S_2}$, and $\mathrm{S_3}$, respectively, start to gain strength and lock in place.