Observing collisions beyond the secular approximation limit

Abstract

Quantum coherence plays an essential role in diverse natural phenomena and technological applications. The unavoidable coupling of the quantum system to an uncontrolled environment incurs dissipation that is often described using the secular approximation. Here we probe the limit of this approximation in the rotational relaxation of molecules due to thermal collisions by using the laser-kicked molecular rotor as a model system. Specifically, rotational coherences in N₂O gas (diluted in He) are created by two successive nonresonant short and intense laser pulses and probed by studying the change of amplitude of the rotational alignment echo with the gas density. By interrogating the system at the early stage of its collisional relaxation, we observe a significant variation of the dissipative influence of collisions with the time of appearance of the echo, featuring a decoherence process that is well reproduced by the nonsecular quantum master equation for modeling molecular collisions.

Fig. 1. Schematic representation of the experimental set-up.

N₂O gas molecules contained in a room-temperature high-pressure gas cell are impulsively aligned by two time-delayed 800 nm pulses P₁ and P₂. The induced rotational dynamics is measured through the time-dependent birefringence experienced by a 400 nm probe pulse. The detection uses two photodiodes connected head-to-tail to a lock-in amplifier delivering a signal proportional to the part of the probe field that has been depolarized by the aligned molecules. The image reproduced on the computer screen illustrates the alignment echo signal produced by the two time-delayed strong laser kicks; QWP, quarter-wave plate; WP, Wollaston prism.

Fig. 2. Rotational structures of N₂O aligned by two laser kicks.

a Alignment signal recorded in pure N₂O gas at low pressure by scanning the temporal delay between the first aligning pulse P₁ (at t = 0) and the probe pulse over slightly more than the full rotational period T_R = 40.4 ps of the molecule. The peaks identified by P₁ and P₂ correspond to the transient alignment signals produced by the two pulses separated by the delay τ₁₂. The main echo is generated at t = 2τ₁₂, with the secondary echo observable at t = 3τ₁₂, and the imaginary echo produced at T_R/2-τ₁₂ (equivalent features also appearing at times shifted by +T_R/2). In addition to echoes, other transients corresponding to the standard half and full alignment revivals of P₁ (at T_R/2 and T_R) and P₂ (at T_R/2 + τ₁₂ and T_R + τ₁₂) are also observed. b Alignment traces of N₂O diluted in He measured at various densities around the main echo at 2τ₁₂ for τ₁₂ = 2.18 ps. The amplitudes S of the alignment structures are measured from peak to dip. c Amplitudes of the half revival (open red circles), full revival (full blue circles), and of the main echo for five different values of the delay τ₁₂ versus the gas density d multiplied by the time of observation t_R (T_R/2 or T_R) and 2τ₁₂ for the revivals and echo, respectively, expressed in picosecond amagat (ps.amagat, with 1 amagat = 2.687 × 10²⁵ mol. m⁻³) units. The lines indicate the best exponential fits.

Fig. 3. Time constants of collisional dissipation of N₂O.

The blue circles with error bars (representing two standard deviations of the mean) are the density-normalized decay time constants τ_E of the echoes deduced from the measurements of the alignment signal recorded at various N₂O(4%) + He(96%) gas densities and fixed delays τ₁₂ between the two pulses. The dashed lines denote the results of simulations conducted by solving the density matrix equations for molecules impulsively aligned by two short laser pulses and interacting with each other through collisions. The green and black dashed lines have been obtained using the standard Bloch equations (i.e., using the secular approximation) with initial N₂O rotational populations corresponding to temperatures of T = 100 K and 295 K, respectively. The red dashed line represents the results obtained, for populations associated with T = 100 K (see text), using the nonsecular Redfield equations, i.e. including all relaxation terms in Eq. (3).

Fig. 4. Secular and nonsecular effects.

a Relative modification of the ${\rho }_{J_0{M}_0,J_0+2M_0}\left(t\right)$ coherence for J₀ = 15 induced by collisional transfers from the coherences ${\rho }_{J_1{M}_1,J_1+2M_1}\left(t\right)$ with J₁ = 17 (black), J₁ = 19 (red), J₁ = 21 (green), and J₁ = 23 (blue), all normalized to unity at t = 0. This simple modeling (see Supplementary Note 2) of coherence transfers considering only few rotational states around the most populated state of N₂O at 300 K reveals that during the short time evolution of the system, exchanges between coherences are efficient, which slow down the decay of the alignment factor with respect to what would be obtained using the secular approximation that neglects these collisional transfers and only takes the losses into account. b, c Computed real and imaginary parts of the ${\rho }_{JM=0,J+2M=0}\left(t\right)$ coherences for N₂O in He at 33 amagat, obtained using the nonsecular (full lines) and secular (dashed lines) models.