Optically engineered quantum interference of delocalized wave functions in a bulk solid: The example of solid para-hydrogen

Local excitations of indistinguishable particles in a solid are quantum-mechanically superposed to give delocalized wave functions. Their interference is often so short-lived that it eludes observation and manipulation. Here we have actively controlled interference of delocalized vibrational wave functions in solid para-hydrogen produced by a pair of ultrashort laser pulses. The ultrafast evolution of their interference changes from almost completely constructive (amplification by a factor of $\sim$4) to destructive when we change the timing of those two laser pulses by only 4 fs. This active control serves as an experimental tool to investigate the spatiotemporal evolution of a wave function in a bulk solid.

Figure 1

Formation of the delocalized vibron in solid $p$-H${}_2$. (a) A band structure arises from the interaction among many indistinguishable $p$-H${}_2$ molecules, one of which is vibrationally excited. The red and blue spheres stand for the vibrationally excited and ground-state $p$-H${}_2$ molecules, respectively. (b) Every $p$-H${}_2$ molecule within a single crystal has an equal probability to be vibrationally excited. These locally excited states $|v_i=1\rangle$ ($i=1,...,N$) are coherently superposed to give a delocalized vibron state $|v=1,\mathbf{k}=0\rangle$.

Figure 2

Experimental scheme. (a) Snapshot of the $p$-H${}_2$ crystal grown in the cryostat. The dotted circle indicates the $p$-H${}_2$ crystal. (b) A sequence of the laser pulses employed in the present study to observe and control the interference of the delocalized vibron state $|v=1,\mathbf{k}=0\rangle$. The delay ${\tau }_{\mathrm{IRE}}$ between the first and second pairs of the pump and Stokes pulses is stabilized and controlled on the attosecond time scale, and the coherence which remains after the second pump-Stokes pair is observed by measuring the intensity of the anti-Stokes beam induced by the probe pulse.

Figure 3

Optical setup for a highly stabilized Michelson interferometer. DM, dichroic mirror; BS, beam splitter; SM, silver mirror.

Figure 4

Quantum interferogram of the vibron state $|v=1,\mathbf{k}=0\rangle$ delocalized in solid $p$-H${}_2$. The delay ${\tau }_{\mathrm{IRE}}$ between the first and second pump-Stokes pairs is scanned around (a) 10 ps and (b) 500 ps. The solid curves represent sine functions least square fitted to the measured interferograms shown with open circles. The intensities of the anti-Stokes beam arising from a single pump-Stokes excitation are shown with the dashed horizontal lines for reference. The full range of each abscissa has been calibrated with a reference interferogram of He-Ne laser beams measured simultaneously.

Figure 5

Feedback stabilization of the interferometer. The spectral interferogram of the two pump pulses produced from the interferometer was measured continuously for $\sim$6 min which corresponds to the full range of the ordinate (a) without the feedback control and (b) with the feedback control. The interpulse delay was set $\sim$100 fs.

Figure 6

Real-time observation of the actively controlled quantum interference of the vibron state $|v=1,\mathbf{k}=0\rangle$ delocalized in solid $p$-H${}_2$. The delay ${\tau }_{\mathrm{IRE}}$ between the first and second pump-Stokes pairs is different by 4 fs between the red and blue traces. The black trace shows a reference taken with the second pump-Stokes pair blocked. Each trace is an average of three repeated scans, and its vertical scaling is normalized by its intensities averaged over ${\tau }_{\mathrm{probe}}=0.7$– 484.3 ps before the arrival of the second IREP.