Coherent X-ray-optical control of nuclear excitons

Abstract

Coherent control of quantum dynamics is key to a multitude of fundamental studies and applications¹. In the visible or longer-wavelength domains, near-resonant light fields have become the primary tool with which to control electron dynamics². Recently, coherent control in the extreme-ultraviolet range was demonstrated³, with a few-attosecond temporal resolution of the phase control. At hard-X-ray energies (above 5-10 kiloelectronvolts), Mössbauer nuclei feature narrow nuclear resonances due to their recoilless absorption and emission of light, and spectroscopy of these resonances is widely used to study the magnetic, structural and dynamical properties of matter^4,5. It has been shown that the power and scope of Mössbauer spectroscopy can be greatly improved using various control techniques^6-16. However, coherent control of atomic nuclei using suitably shaped near-resonant X-ray fields remains an open challenge. Here we demonstrate such control, and use the tunable phase between two X-ray pulses to switch the nuclear exciton dynamics between coherent enhanced excitation and coherent enhanced emission. We present a method of shaping single pulses delivered by state-of-the-art X-ray facilities into tunable double pulses, and demonstrate a temporal stability of the phase control on the few-zeptosecond timescale. Our results unlock coherent optical control for nuclei, and pave the way for nuclear Ramsey spectroscopy¹⁷ and spin-echo-like techniques, which should not only advance nuclear quantum optics¹⁸, but also help to realize X-ray clocks and frequency standards¹⁹. In the long term, we envision time-resolved studies of nuclear out-of-equilibrium dynamics, which is a long-standing challenge in Mössbauer science²⁰.

Fig. 1. Schematic setup and samples.

a, A short synchrotron (SR) X-ray pulse is shaped into a double pulse using a resonant absorber acting as a delay stage, which we denote as the split-and-control unit (SCU). A fast displacement Δx of the SCU controls the relative phase ϕ between the two pulses corresponding to a relative delay Δt, thus forming a tunable X-ray double-pulse source. The double-pulses are used to coherently control the dynamics of the target nuclei. An exemplary dynamics is visualized via the nuclear magnetic transition dipole moment $⟨\hat{d}\left(t\right)⟩$ on a polar plot. b, Energy-level schemes and spectra of SCU absorber and target nuclei. For the coherent control, we tune the single resonance of the target nuclei to one of the two resonances of the SCU absorber (δ = 0).

Fig. 2. Experimental observation of the coherent control.

a, b, Time-and energy-resolved intensities recorded in the forward direction for two different double-pulse sequences corresponding to coherent enhanced emission (a) and enhanced excitation (b) of the target nuclei, respectively. The white areas at times t ≤ 15 ns reflect the detector dead time after the synchrotron excitation. c, Time-dependent intensity at relative detuning δ = 0, normalized to equal measurement times. The experimental data (dotted) exhibits the characteristic crossover (shaded areas) in the count rate between the two control cases at about 45 ns. The rapid oscillations are quantum beats due to the interference of light scattered at δ = 0 and δ = −63γ, with time-dependent visibility because the magnitude ratio of the two interfering contributions varies with time. The outlier dots are due to partial suppression of the time bins at the border of the dead time interval. The error bars indicate the photon shot noise. Corresponding theory curves are shown as lines.

Fig. 3. Time-dependent dipole moment of the target nuclei.

a, b, The modulus and the phase of the spatially averaged nuclear magnetic transition dipole moment reconstructed from the experimental data (solid lines). The dashed curves on top of the solid ones are the corresponding results calculated by averaging over the position-dependent dipole moment obtained from a full propagation analysis of the SCU pulse through the target. The accelerated decay in the case of enhanced emission (blue) and the enhanced excitation (orange) are clearly visible. A theoretical reference calculation without SCU is shown as the black line.

Fig. 4. Stability of the double-pulse sequence.

The Allan deviation σ_ϕ(τ) is in the approximately 40-mrad range for both SCU operation modes, corresponding to a temporal stability σ_ξ(τ) on the few-zeptosecond timescale. The shaded areas show the standard deviation error ranges and diagonal grid lines indicate $1/\sqrt{\tau }$ scaling. The green curve shows results for the coherent enhanced excitation, including an initialization time of 400 s in which the SCU motion exhibited systematic drifts causing phase deviations corresponding to the approximately 10-zs scale.

Extended Data Fig. 1. Theoretical predictions for the enhanced emission and enhanced excitation of the nuclear exciton.

Nuclear dynamics under the action of different double pulses, calculated with the simplified model introduced in the Methods section ‘Nuclear resonant scattering’, which neglects one of the SCU resonances such that no quantum beats appear. a, b, The magnitude (a) and phase (b) of the spatially averaged nuclear magnetic transition dipole moment, respectively. The excitation pulse induces a nuclear excitation at t = 0, and the panels show the subsequent dynamics induced by the control pulse. The opposite phase between the excitation and control pulses leads to enhanced emission, followed by subsequent coherent re-excitation (blue). Equal phases induce enhanced excitation (orange). The black dashed line indicates the dipole response in the absence of the SCU. c, The total intensity emitted in the forward direction. The shaded areas indicate a crossover in the dominating intensity.

Extended Data Fig. 2. Dipole dynamics induced in the target by the propagating SCU pulse.

Absolute value Abs($⟨\hat{d}⟩$) of the target’s magnetic transition dipole moments as a function of time t after excitation and depth/position x in the target of length L. The results are obtained from solving the Maxwell−Bloch equation as explained in the Methods section ‘Propagation effects in the target’. a−c, The cases without SCU (a), coherent enhanced emission (b) and coherent enhanced excitation (c). All panels share the same colour scale for Abs($⟨\hat{d}⟩$), and the parameters are as in Fig. 3.

Extended Data Fig. 3. Dipole dynamics at the target entry, middle, and exit.

Absolute value Abs($⟨\hat{d}⟩$) and phase Arg($⟨\hat{d}⟩$) of the target’s magnetic transition dipole moments as function of time t after excitation in thin slices at the entry, in the middle, and at the exit of the target. The results are obtained from solving the Maxwell−Bloch equation as explained in the Methods section ‘Propagation effects in the target’. The parameters and the curve styles are as in Fig. 3. The three colours indicate the cases without SCU (black), coherent enhanced emission pulse sequence (blue) and the coherent enhanced excitation sequence (orange).

Extended Data Fig. 4. Three absorber motions to illustrate the event-based detection.

a, Three SCU absorber motions used to illustrate the capabilities of the event-based detection. b, c, The magnitudes (b) and phases (c) of the magnetic transition dipole moments induced in the target nuclei owing to the double pulses generated by the respective motions, or in the absence of an SCU.

Extended Data Fig. 5. Event-based detection and time-dependent intensities.

a, b, Theoretical predictions for the resonant time-dependent intensity using the three motions in Extended Data Fig. 4. a, These intensities for the three motions essentially coincide. b, This panel illustrates this further with the differences between the time-dependent intensities for the motions 1 and 2 as well as motions 1 and 3, respectively. c, d, Corresponding theoretical predictions for our event-based spectroscopy technique. c, The relative difference (I₂ – I₁)/(I₁ + I₂) between the 2D spectra of motions 1 and 2, with dimensionless values as indicated by the colour scale. The rich interference structures and high visibility show that the motions can clearly be distinguished. d, Intensity differences corresponding to the results in b, but energy-resolved at sections with different detunings δ through the 2D data, where δ is defined in Fig. 1. Each detuning δ leads to characteristic strong interference patterns, further illustrating that the motions can easily be distinguished using the event-based detection.

Extended Data Fig. 6. Evaluation of the Allan deviation.

To evaluate the Allan deviation, the full data set is split into N samples of duration τ. Each sample comprises a 2D time- and energy-resolved spectrum, to which we fit spectra obtained using the SCU motion x₀(t) modified via an error model with a parameter A. This yields the best-fit parameter A_i for each sample i, which is proportional to the phase error ϕ_i and the corresponding temporal error ξ_i as explained in the methods. From these deviations, the Allan deviation can be calculated using equation (31).

Extended Data Fig. 7. Comparison of different noise models.

Allan deviation in the case of coherent enhanced excitation obtained for the three noise models we employed. The uncertainty according to the linear noise model exceeds that of other models by about a factor of three. Shaded areas indicate the standard deviation ranges and diagonal grid lines indicate $1/\sqrt{\tau }$ scaling.

Extended Data Fig. 8. Influence of detector dead time.

The orange curve shows the Allan deviation for the case of enhanced emission and a linear noise model. The fluctuations at intermediate sampling times are due to dead time in the data acquisition system. In the purple curve these drops in the count rate were artificially avoided by assuming a constant count rate instead. Consequently, the fluctuations vanish. Shaded areas indicate the standard deviation ranges and diagonal grid lines indicate $1/\sqrt{\tau }$ scaling.

Extended Data Fig. 9. Systematic deviations throughout the initialization phase.

The figure shows the deviations ϕ_i and ξ_i obtained for sampling times τ = 200 s throughout the full measurement period for the case of coherent enhanced excitation. We note that in this figure, temporally overlapping samples were analysed in order to trace the time evolution of the deviation throughout the initial phase of the measurement. The first 400 s or so during the initialization period of the measurement time exhibit systematic drifts corresponding to the approximately 10-zs scale. Shaded areas indicate the standard deviation ranges.