Accurate photon echo timing by optical freezing of exciton dephasing and rephasing in quantum dots

Coherent nonlinear optics involving quantum emitters enables studies of advanced quantum phenomena. In solid-state ensembles, strong inhomogeneous broadening of optical transitions leads to rapid decay of macroscopic polarization due to dephasing, yet microscopic coherence can remain and be probed via collective phenomena such as superradiance and entanglement. Semiconductor quantum dots (QDs) are discrete, strong two-level-like emitters with long coherence times (limited by ~1 ns radiative decay at low temperatures), enabling ultrafast picosecond optical control, Rabi oscillations, adiabatic rapid passage, and Ramsey interferometry. However, ensemble-level phase control is challenging due to inhomogeneous broadening that induces dephasing even during ps excitations and yields complex coherent optical responses. Photon echo (PE) protocols can reverse dephasing in an inhomogeneous two-level system (TLS) ensemble, using a second pulse to invert phase distribution and rephase at t = 2τ12. Modifying phase evolution during dephasing or rephasing should therefore affect the PE timing. In magnetic resonance, complex pulse sequences (dynamic decoupling, spin locking) shape dephasing and generate multiple echoes, and in atomic systems polarization locking and echo shaping have been demonstrated. Deterministic PE timing control using off-resonant ac Stark pulses was shown in rare-earth nanophotonics by compressing atomic frequency combs. Yet, optical control of PE timing using resonant optical fields had not been explored, and QD echoes offer picosecond-scale control windows. In this work, a simple resonant method is proposed and demonstrated: applying an additional resonant control pulse of area 2πn during dephasing or rephasing freezes phase evolution, enabling robust advancement or retardation of PE timing in (In,Ga)As QD ensembles without stringent requirements on control pulse phase or exact timing.

Prior work established that inhomogeneous broadening in ensembles causes rapid macroscopic dephasing but microscopic coherence persists and can be refocused via photon echoes. In nuclear/electron spin resonance, elaborate pulse sequences (Carr–Purcell, Carr–Purcell–Meiboom–Gill, spin locking) control dephasing and produce multiple echoes over long timescales. In optical media such as atomic ensembles, polarization locking and field-inhibited dephasing demonstrated control over echo shapes and suppression of dephasing. In rare-earth-ion systems, ac Stark (off-resonant) control pulses modulate PE amplitude and have been used to deterministically control retrieval timing via optical frequency-comb compression in nanophotonic resonators. For semiconductor QD ensembles, previous studies reported that inhomogeneous broadening affects coherent transients during ps excitation, leading to non-Gaussian echo shapes and timing shifts for large pulse areas due to dephasing during excitation. However, PE timing control via resonant optical fields had not been considered before this study.

Conceptual approach: A TLS ensemble with transition frequency distribution (Gaussian, center ω0, broadening Δ) is driven by a two-pulse PE sequence (first pulse area θ1 ≈ π/2 at t=0; second pulse area θ2 ≈ π at t=τ12). An additional resonant control pulse with area Θc = 2πn and duration tc is applied during either the dephasing interval (0 < t < τ12, pre-pulse) or the rephasing interval (τ12 < t < 2τ12, post-pulse). On the Bloch sphere, for strong resonant driving (ΩR ≫ Δ), detuned TLS undergo rotations with generalized Rabi frequency Ω = √(ΩR^2 + Δ^2) around Ω = (ΩR,0,Δ), resulting in near-closure of 2π cycles and suppressed phase accumulation (freezing) compared to free evolution. Freezing reduces effective dephasing or rephasing during tc, shifting the PE time earlier (pre-pulse) or later (post-pulse) approximately by the control duration. Experimental setup: A single layer of self-assembled (In,Ga)As QDs was embedded at the antinode of a planar λ-microcavity (distributed Bragg reflectors: GaAs/AlAs pairs, 5 top and 18 bottom; layer thicknesses 68/82 nm). QD density: 1.8 × 10^9 cm−2. A Si δ-doping layer (8 × 10^9 cm−2) 10 nm below the QDs provides resident electrons, yielding charged QDs. The cavity exhibits a photonic mode spanning 1.343–1.362 eV (910–923 nm) with FWHM ≈ 7.4 meV (quality factor ≈ 200), overlapping the QD ensemble emission. The sample was cooled to 2 K in a bath cryostat. Laser and pulse parameters: A mode-locked Ti:sapphire laser (75.75 MHz repetition rate) provided excitation, control, and reference pulses with tunable central wavelength, pulse duration ≈ 2.5 ps (from autocorrelation), and spectral width ≈ 0.6 meV (FWHM), narrower than the cavity mode. Typical pulse areas: θ1 ≈ π/2 (I1 ≈ 4 nJ cm−2), θ2 ≈ π (I2 ≈ 23 nJ cm−2); control pulse Θc around 2π (Gaussian temporal shape). Spot sizes: first pulse ≈ 400 μm diameter; second and control pulses ≈ 250 μm; all beams focused on the same position. Geometry and polarization: Degenerate transient four-wave mixing (FWM) in reflection was used. Pulses with wavevectors k1 (incidence ~3°) and k2 (~4°) generated a PE detected along 2k2 − k1. A control pulse (k3) arrived as a pre- or post-pulse with set delays relative to the first or second pulse. Polarization configuration selected echoes from charged QDs: first pulse horizontal (H), second and control vertical (V); the H-polarized FWM signal was detected. Delay times were controlled by mechanical translation stages. Heterodyne detection: The PE field interfered with a reference pulse; scanning the reference delay tref yielded the temporal profile of the FWM electric field (detection proportional to the modulus of the cross-correlation of the emitted field with the reference). Data acquisition: Two-pulse echo temporal profiles were recorded versus τ12 and pulse areas. Control pulses were inserted as pre- or post-pulses with typical delays of ~33 ps (pre) or ~27 ps (post) relative to the adjacent excitation pulse. PE durations and timing shifts were extracted from Gaussian fits and peak positions of the heterodyne signals. Theoretical modeling: Optical Bloch equations (OBEs) for a TLS ensemble described microscopic polarization pij(t) and excited-state occupation nii(t), including relaxation with T2 (coherence) and T1 (population), inhomogeneous broadening (Gaussian weight G(ω) with FWHM Δω ≈ 7.5 meV), and spatial averaging due to finite Gaussian beam profiles (exp(−r^2/σg^2)). The macroscopic polarization P(r,t) was obtained by summing over 1500 TLS frequencies spanning −15 to +15 meV (resolution 0.02 meV), weighted by G(ω), multiplied by dipole moment d12, and integrating over radial coordinates r with Gaussian weighting. The detected signal Psignal(t) was computed as the temporal convolution of the spatially averaged polarization Paverage(t) with the normalized reference pulse εref(t). Numerical integration used a fourth-order Runge–Kutta scheme with time step 0.01 ps; spatial radii from 0.05σg to 3.5σg with step 0.05σg; pulse area scans with ΔΘ = 0.05. Experimental T2 = 710 ps and T1 = 360 ps were used as inputs. Simulations reproduced Rabi oscillations, non-Gaussian echo shapes at larger θ1, and control-induced timing shifts.

- Demonstrated resonant optical control of photon echo (PE) timing in (In,Ga)As quantum dot ensembles by freezing dephasing/rephasing with an additional ~2π control pulse. - With θ1 ≈ π/2, θ2 ≈ π, and a resonant Θc ≈ 2π control pulse, the PE peak was advanced (pre-pulse) or retarded (post-pulse) by ~5 ps relative to t = 2τ12, despite the control pulse duration being ~2.5 ps. The shift is robust to the exact arrival time of the control within the dephasing or rephasing windows and independent of the control optical phase. - The detected PE temporal width remained essentially unchanged (FWHM ≈ 6.5 ps in the heterodyne signal, corresponding to ~4 ps PE duration), while the amplitude decreased by ~50% under control, mainly due to damping of Rabi oscillations. - Measured T2 ≈ 0.7 ns from P_PE ∝ exp(−2τ12/T2), independent of θ1 (π/2, 3π/2, 5π/2), indicating weak excitation-induced dephasing; exciton lifetime T1 ≈ 360 ps. - Two-dimensional scans versus θ1 showed up to two full Rabi flops in PE amplitude and significant PE timing shifts at larger θ1 due to dephasing during excitation; extended OBE simulations quantitatively reproduced these features. - Control pulse shifts were additive: two pre-pulses doubled the advancement; two post-pulses doubled the retardation; combining pre- and post-pulses with equal areas canceled the net timing shift (confirmed by OBE simulations). - Larger total shifts were achieved for higher-intensity sequences: for θ1 ≈ 3π/2 with Θc ≈ 2π pre-pulse, the PE appeared ~8 ps earlier than for θ1 = π/2 without control, exceeding the ~4 ps PE duration. - Dependence on control area exhibited Rabi flopping; for weak control (Θc < π), PE amplitude quenched with slight delay shifts; near Θc ≈ 2π, maximal timing shifts occurred while retaining echo shape. - Off-resonant control (ac Stark regime) in the model yielded reduced timing shifts and rapid PE amplitude suppression with increasing detuning and control intensity.

The results validate that a resonant 2π control pulse can effectively freeze the phase evolution of an inhomogeneously broadened exciton ensemble, slowing dephasing (pre-pulse) or rephasing (post-pulse). This modifies the rephasing condition after the second pulse, deterministically advancing or delaying the PE peak by a time set by the effective strong-drive interval. The effect is insensitive to the precise arrival time of the control within the dephasing or rephasing windows and to its optical phase, simplifying implementation. Extended OBE simulations including inhomogeneous broadening and spatial averaging reproduce the observed Rabi oscillations, echo timing shifts, amplitude changes, and additive behavior for multiple control pulses. The preserved PE temporal width indicates that under strong resonant control (ΩR ≫ Δ), timing can be tuned without distorting the echo envelope. The approach offers ultrafast (picosecond-scale) timing control beyond previous off-resonant or slow-control schemes. Practical constraints include reduced efficacy and amplitude under detuning (ac Stark regime) and amplitude reductions due to Rabi damping at high drives. The demonstrated robustness and bidirectionality make the protocol attractive for timing adjustments in optical quantum memory protocols and for advanced manipulations (e.g., pulse splitting via spin selection in multilevel schemes).

This work introduces and experimentally demonstrates a simple, resonant method to control photon echo timing in quantum dot ensembles by optically freezing exciton dephasing/rephasing with 2π control pulses. The PE peak can be advanced or delayed by several picoseconds (up to ~5 ps for single control and ~8 ps in combined conditions), with minimal change to echo shape and robustness against control timing and phase. The magnitude of the timing shift is set by the duration and strength of the control pulse, and shifts add for multiple pulses while pre- and post-pulses cancel each other. The technique enables accurate, ultrafast timing control suitable for quantum technologies, including timing corrections in photon-echo-based quantum memories. Future research could engineer longer or tailored control pulses for larger shifts, explore multilevel (V/Λ) schemes to split echoes into polarization-resolved components for wave-function interferometry, and optimize pulse shapes to minimize amplitude loss while maximizing timing control under realistic inhomogeneous conditions.

- The PE amplitude decreases (≈50% in experiments) under strong control, attributed mainly to damping of Rabi oscillations. - Effective timing control relies on strong resonant driving (ΩR ≫ Δ); with detuning (off-resonant ac Stark regime), model results show reduced timing shifts and rapid PE amplitude suppression as control intensity increases. - Inhomogeneous dephasing during finite-duration excitation pulses can distort echo shapes and shift timing at large excitation areas, necessitating careful choice of θ1 to avoid non-Gaussian transients. - The precision of timing shifts is limited by control pulse parameters (duration, temporal shape, intensity), requiring stable pulse generation for sub-picosecond accuracy. - Demonstrated shifts (~5–8 ps) were limited by available pulse durations and intensities; larger shifts require tailored longer or shaped strong control pulses.