Echoes in Space and Time

Echo in mountains is a well-known phenomenon, where an acoustic pulse is mirrored by the rocks, often with reverberating recurrences. For spin echoes in magnetic resonance and photon echoes in atomic and molecular systems, the role of the mirror is played by a second, time-delayed pulse that is able to reverse the flow of time and recreate the original impulsive event. Recently, alignment and orientation echoes were discussed in terms of rotational-phase-space filamentation, and they were optically observed in laser-excited molecular gases. Here, we observe hitherto unreported fractional echoes of high order, spatially rotated echoes, and the counterintuitive imaginary echoes at negative times. Coincidence Coulomb explosion imaging is used for a direct spatiotemporal analysis of various molecular alignment echoes, and the implications to echo phenomena in other fields of physics are discussed.

Popular Summary

Molecules such as nitrogen or oxygen in the atmosphere are freely rotating and randomly oriented in space. However, if these molecules are excited by laser pulses that are short compared to their rate of rotation, they can be aligned in space: For a very short instant, the molecular axes point in a specific direction defined by the laser, and then the molecular orientation is randomized again. Our team has previously shown that if, after some delay, the molecules are exposed to another pulse, the second pulse serves as a sounding board, causing a series of periodic alignment echoes to reverberate for quite some time, just like an echo in the mountains. Here, we extend this finding and report a gamut of new echo phenomena, including fractional echoes that repeat themselves at fractions of the usual echo period, rotated echoes that shoot in a direction controlled by polarization of the laser pulses, and counterintuitive imaginary echoes that appear at negative times.

Using coincidence Coulomb explosion imaging of ${\mathrm{CO}}_2$ and ${\mathrm{N}}_2\mathrm{O}$ molecules in a vacuum chamber, we investigated how very short laser pulses could modify molecular angular distribution by applying a torque to them. The gases were excited by two short laser pulses (sometimes with different polarization angles), and a third strong laser pulse was used to explode the excited molecules and to image the angular distribution of the molecular fragments.

We expect that a generic mechanism behind the alignment echoes may have implications in many fields of physics, leading to a number of useful applications.

Figure 1

Experimental setup. A supersonic gas jet of ${\mathrm{CO}}_2$ molecules subject to a pair of polarization-skewed and time-delayed femtosecond laser pulses in an ultrahigh vacuum chamber of COLTRIMS is used to create molecular alignment echoes. By scanning the delay of an intense circularly polarized probe pulse, snapshots of the angular distribution of the molecular axis at various times are taken via Coulomb explosion.

Figure 2

Filamentation of the phase-space density distribution. For all parts of this figure, ${\mathrm{\Omega }}_1/\sigma =1$. (a) Shortly after the kick, $\sigma t=0.5$, the density distribution folds, resulting in a transient alignment along the direction $\theta =0$. The folded pattern is centrally symmetric with respect to the phase-space point $(\theta ,\omega )=(0,0)$ that is not affected by the kick. (b) On longer time scales, $\sigma t=5$, the probability density becomes wrinkled, and it develops multiple parallel filaments. (c) After a second kick is applied at $t=T$ (with $\sigma T=5$ and ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=1/3$), every filament in (b) folds in a manner similar to (a), giving rise to an alignment echo near $\theta =0$ at time $\tau \sim T$ after the second kick ($\sigma \tau =4.56$). (d) At $\tau \approx T/2$ ($\sigma \tau =2.293$), a fractional echo is formed.

Figure 3

Formation of the rotated echo. The polarization angles of the first and the second pulses, as well as the direction of the rotated echo are indicated by the three red dashed lines. The first pulse is applied at $t=0$, $\theta =0$, with ${\mathrm{\Omega }}_1/\sigma =1$. The second pulse is applied at $t=T$ (such that $\sigma T=5$), at crossing angle $\beta =0.4$ with respect to the first pulse, and with ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=1/3$. The rotated full echo is formed at $\tau \approx T$ (namely, when $\sigma \tau =4.56$), and at angle $\theta =0.8(2\beta )$.

Figure 4

Echoes for positive and negative time. Two pulses are applied at $\tau /T=-1$ and at $\tau =0$. Real echoes are seen at $\tau /T=1$, 2, 3 after the second pulse, and imaginary echoes are visible at $\tau /T=-2$, $-3$, i.e., at negative propagation time. The plotted quantity $⟨\cos (2\theta )⟩$ [Eq. (3), with $n=1$] is related to the standard alignment factor $⟨{\cos }^2\theta ⟩$ by the standard trigonometric identity $\cos (2\alpha )=2{\cos }^2\alpha -1$.

Figure 5

Time-dependent angular distributions of full, $1/2$ and $1/3$ echoes, as indicated between the white dashed lines in (a) 2.2 ps—3.5 ps, (d) 1.9 ps—3.0 ps, and (g) 1.4 ps—1.9 ps, experimentally measured and theoretically simulated for ${\mathrm{CO}}_2$. (a) Carpet of the angular distribution of full echo for parallel input pulses, $P_1$ and $P_2$, with $T=3\mathrm{ps}$ time delay between them. (b) and (c) Polar plots of the alignment and anti-alignment regions of the distribution for the full echo, showing vertical and horizontal directionality, respectively. (d)-(f) The same, for the $1/2$ fractional echo, with $T=5\mathrm{ps}$ time delay. (g)-(i) The same for $1/3$ fractional echoes with $T=5\mathrm{ps}$ time delay. In all cases, the solid red line is the quantum-mechanically simulated distribution.

Figure 6

Experimentally measured and theoretically simulated time-dependent angular distributions of the rotated echo in ${\mathrm{CO}}_2$. $P_1$ and $P_2$ arrive at $-3\mathrm{ps}$ and 0 ps, respectively, with a crossing angle of $\beta =-20°$. (a) Angular distribution of the rotated echo as a function of the probe time delay (see the region of 2.2–3.5 ps between the white dashed lines). (b) Polar plots of the rotated echo at 3 ps. Panels (c) and (d) show quantum simulations, and panels (e) and (f) show classical simulations.

Figure 7

Experimentally measured and theoretically simulated time-dependent angular distributions of the imaginary echo in ${\mathrm{CO}}_2$. $P_1$ and $P_2$ arrive at $-2.6\mathrm{ps}$ and 0 ps, respectively. (a) A “carpet” describing the angular distribution of the imaginary echo as a function of the probe time delay (see the region of 36 ps–38 ps between the white dashed lines). (b) The alignment factor $⟨{\cos }^2\theta ⟩$ as a function of the probe delay (red shows the revival of the response to the first pulse, blue that of the second pulse, green depicts the revival of the first full echo, and purple presents the imaginary echo). (c) Polar plots of the imaginary echo around 36.6 ps (blue shows the experiment and red the simulation).

Figure 8

Experimentally measured and theoretically simulated time-dependent angular distributions of the imaginary echo in ${\mathrm{N}}_2\mathrm{O}$. $P_1$ and $P_2$ arrive at $-3\mathrm{ps}$ and 0 ps, respectively. (a) A “carpet” describing the angular distribution of the imaginary echo as a function of the probe time delay (see the region of 32.5–34.5 ps between the white dashed lines). (b) The alignment factor $⟨{\cos }^2\theta ⟩$ as a function of the probe delay (red shows the revival of the response to the first pulse, blue that of the second pulse, green depicts the revival of the first full echo, and purple presents the imaginary echo). (c) Polar plots of the imaginary echo around 33 ps (blue is for experiment, red for simulation).

Figure 9

Quantum simulation of time-dependent angular distributions of full, $1/2$, and $1/3$ echoes. (a) Angular distribution of full echo in the case of a parallel condition with a time delay of 3 ps between $P_1$ and $P_2$. (b,c) Alignment and anti-alignment of full echo in polar plots. Panels (d)–(f) and (g)–(i) show the same for $1/2$ and $1/3$ fractional echoes with a time delay of 5 ps between $P_1$ and $P_2$, respectively.

Figure 10

Classical simulation of time-dependent angular distributions of full, $1/2$, and $1/3$ echoes. (a) Angular distribution of full echo in the case of a parallel condition with a time delay of 3 ps between $P_1$ and $P_2$. (b,c) Alignment and anti-alignment of full echo in polar plots. Panels (d)–(f) and (g)–(i) show the same for $1/2$ and $1/3$ fractional echoes with a time delay of 5 ps between $P_1$ and $P_2$, respectively.