Coherent control of collective nuclear quantum states via transient magnons

Abstract

Ultrafast and precise control of quantum systems at x-ray energies involves photons with oscillation periods below 1 as. Coherent dynamic control of quantum systems at these energies is one of the major challenges in hard x-ray quantum optics. Here, we demonstrate that the phase of a quantum system embedded in a solid can be coherently controlled via a quasi-particle with subattosecond accuracy. In particular, we tune the quantum phase of a collectively excited nuclear state via transient magnons with a precision of 1 zs and a timing stability below 50 ys. These small temporal shifts are monitored interferometrically via quantum beats between different hyperfine-split levels. The experiment demonstrates zeptosecond interferometry and shows that transient quasi-particles enable accurate control of quantum systems embedded in condensed matter environments.

Fig. 1. Nuclear-level scheme with magnon excitation.

(A) Level scheme of a⁵⁷Fe nucleus with the transition from the ground ∣g⟩ to the excited state ∣e⟩ at 14.4 keV. The magnetic hyperfine-field B_hf splits the nuclear transition by the Zeeman effect. The four circularly polarized transitions are shown that are excitable in our experimental geometry. A static magnetic field H aligns the magnetization M and the hyperfine field in the sample. When a coherent magnon∣m⟩ is excited in the magnetic material at its resonance ω_m, the magnetization precesses around the external field H. In that case, the hyperfine field is reduced, leading to a shift of the nuclear transitions. (B) Energy spectrum of the circularly polarized nuclear transitions in a permalloy film. Black shows the transitions for a static hyperfine field of B_hf = 27.9 T. Blue shows the shifted transitions for a hyperfine field reduced by 3.0 T as can be induced by magnons. The frequency difference between the transitions with same polarization is Ω.

Fig. 2. Nuclear dynamics with magnon excitation.

(A) An ⁵⁷Fe nucleus is excited by an x-ray pulse and subsequently emits a γ-ray photon. The emitted photon is in a superposition state of the four nuclear dipole transitions oscillating at different frequencies, here only shown for two frequencies of same polarization (green and gray). An RF burst with delay Δt_p excites the ferromagnetic magnon (blue) and leads to a nanosecond shift of the resonance line as shown in Fig. 1. (B) The oscillation of the nuclear transition dipole moment of a single transition at frequency ω_γ is shown in gray. During the period indicated by the blue bar, the magnon is excited and the oscillation frequency is shifted by Δω_γ. Afterward, the oscillation frequency is the same as before, but shifted by Δφ (magenta). This process happens for all four transitions although with different phase shifts.

Fig. 3. Quantum beat phase shift.

(A) Schematic phase relation of dipole radiation emitted from the nucleus at transitions 1 and 4 (same polarization) on the zeptosecond time scale. Their interference leads to a quantum beat over nanoseconds [black in (C)] at the quantum beat frequency Ω = ω₄ − ω₁. (B) Phase relation at a later time when a relative zeptosecond phase shift Δφ is generated (magenta) compared with the undisturbed oscillation (gray and green). The red curve in (C) shows the quantum beat resulting from a zeptosecond phase shift Δφ generated by the magnon, which is applied over the nanosecond period indicated by the blue bar. The zeptosecond phase shift is visible as a temporal shift Δt_Ω in the quantum beat on the nanosecond time scale.

Fig. 4. Data for continuous magnon excitation.

(A) Quantum beat pattern without excitation (black) and excitation powers of 13, 19, and 23 dBm (blue), from bottom to top. Graphs are offset for clarity. The corresponding fits to the nuclear quantum beats are shown in red and result in hyperfine field reductions of −1.5, −2.3, and − 6.5 T, respectively. (B) Electrical RF transmission of the stripline showing the ferromagnetic resonance at 1.98 GHz at an external magnetic field of 5 mT, which is continuously excited during the measurements shown in (A). (C) Hyperfine field distribution in permalloy, as derived from the data without excitation. a.u., arbitrary units.

Fig. 5. Power dependence of pulsed magnon excitation.

(A) Quantum beat pattern without excitation (black) and with a pulsed magnon excitation of 22.3 dBm for a duration of 15 ns (red). The blue bar indicates the ontime of the magnon. (C) Color plot of the quantum beat pattern in dependence of the magnon excitation power. (B) Amplitude and (D) phase of the Fourier transform of the quantum beat pattern shown in (C). (E) Measured temporal shift and phase shift of the quantum beat and the associated hyperfine field reduction. Red dots show the phase shift determined from the Fourier transform. (F) Phase shift and temporal shift of the single transitions, labeled as in Fig. 1, for the right circularly (red) and left circularly (black) polarized transitions.

Fig. 6. Timing dependence of pulsed magnon excitation at 22.3 dBm.

(A and B) Quantum beat patterns in dependence of the RF pulse delay between a 10-ns transient magnon and the nuclear excitation. The lines in (A) indicate the induced temporal shift after magnon excitation. Graphs are offset for clarity. In (B), the dashed lines indicate the 10-ns magnon duration and its relative timing to the x-ray pulse at time zero. For positive delays, the magnon is excited after the nuclear exciton. (C and D) Quantum beat patterns in dependence of the pulse duration that excites the magnon. In (C), blue bars exemplarily indicate the ontime of the magnon. Graphs are offset for clarity. In (D), the magnon is already excited when the nuclear exciton is triggered. The dashed line indicates when the magnon excitation stops.