Research ArticleCONDENSED MATTER PHYSICS

Persistent coherence of quantum superpositions in an optimally doped cuprate revealed by 2D spectroscopy

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Science Advances  28 Feb 2020:
Vol. 6, no. 9, eaaw9932
DOI: 10.1126/sciadv.aaw9932
  • Fig. 1 Experimental scheme.

    (A) The geometry of the excitation is such that the three excitation pulses with wave vectors k1, k2, and k3 form three corners of a box with the LO on the fourth corner. The excitation is close to normal to the copper-oxygen plane, and the signal, which is overlapped with the LO, is measured in the reflected direction. (B) The shaped spectrum for each of the excitation pulses and the LO are shown together with the unshaped laser spectrum (black). (C) The convolution of k1 and k2 should determine the range of coherences that can be excited, shown in red. The data from the impulsive response from the sample (black squares) confirm this. (D) The first two nondegenerate pulses are overlapped in time, generating quantum superpositions between states separated by energies dependent on the pulse spectra shown in (C). The third pulse arrives at a controllable time, t2, later and probes the coherent evolution of these superpositions through the emission of the signal (red). (E) Two possible signal pathways are depicted for the case of discrete energy levels. The coherent superpositions that are probed can be excited by two “upward” transitions (pathway 1) or an “up” and “down” transition (pathway 2) similar to a coherent Raman-like process.

  • Fig. 2 Coherent dynamics of the low-energy excitations in LSCO.

    The temperature of the sample is Teffective ~ 50 K (see section S2), and LSCO is in the strange metal phase. (A) Real part of the spectrally resolved signal as a function of t2: the delay between second and third pulses. The oscillating amplitude originates from the phase evolution of the coherent superposition excited by the first two pulses. The peak signal at pulse overlap corresponds to 1500 photons/s [χ(3) ~ 2 × 10−6]. (B) The spectrally integrated amplitude plotted as a function of t2 shows the decay of the signal extending beyond 500 fs. The pulses cross-correlation is shown in gray. (C) Window functions used to isolate the response during pulse overlap (as wide as the cross-correlation of the pulses) and the response beyond pulse overlap. (D) The Fourier transform of the data with respect to t2 yields the 2D spectrum, where E2 corresponds to the coherence energy, i.e., the energy difference between the states in the quantum superposition. By windowing the data using the window functions shown in (C), the response at pulse overlap (E) and after pulse overlap (F) can be isolated. In (F), the diagonal peak shape corresponding to the extended signal becomes clear. In the insets of (D) to (F), a cross-diagonal slice of the respective 2D spectrum is plotted.

  • Fig. 3 Comparison with vibrational coherences and a broad nonresonant response.

    (A) Real part of the signal from IR813 laser dye, as a function of t2 and E3. The peak signal at pulse overlap corresponds to 3.3 × 106 photons/s [χ(3) ~ 2 × 10−4]. The response extends beyond 1500 fs with several oscillation frequencies. The normalized 2D spectrum (B) and projection on to the E2 axis (C) show discrete narrow peaks corresponding to the discrete vibrational modes. Here, each peak is broad along the E3 axis, matching the width of the third pulse. The signal amplitude beyond pulse overlap is about 10× larger for the molecular system (B) than for LSCO (Fig. 2F). (D) The real part of the signal from an LiNbO3 crystal as a function of t2 and E3 shows the nonresonant response, which decays within the pulse overlap range. The peak signal at pulse overlap corresponds to 1 × 105 photons per second [χ(3) ~ 2 × 10−5]. The normalized 2D spectrum (E) and projection on to the E2 axis (F) show broad peaks corresponding to the instrument response function. Further examination of the region beyond pulse overlap is shown in the Supplementary Materials. There is no evidence of any narrow diagonal peak in the nonresonant response.

  • Fig. 4 Model and simulations for coherent response in LSCO.

    (A) The simple single-particle model shows the first two pulses generating quantum superpositions between bands 1 and 2 (dashed colored arrows), with a distribution of coherence energies. The third pulse (solid black vertical arrows) drives the transition to band 3 with energies correlated to those in band 1 (black dotted lines), leading to emission (solid colored arrows), which is anticorrelated with the coherence energy. (B) The corresponding Feynman diagram beginning with the coherent superposition between states in bands 1 and 2 triggered by the first two light-matter interactions. In (C) to (F), we show the results of simulations based on this picture. (C) The evolution of the real part of the signal, (D) the 2D spectrum of the full response, (E) the 2D spectrum from the response during pulse, and (F) the 2D spectrum from contribution making up the extended signal without pulse overlap. The insets show the corresponding cross-diagonal slices.

  • Fig. 5 Polarization-dependent 2D spectra in LSCO.

    The temperature of the sample is Teffective ~ 50 K (see section S2), and LSCO is in the strange metal phase. The 2D spectra from the extended part of the signal for (A) cocircular, (B) colinear, (C) cross-circular, and (D) cross-linear excitation are shown. The narrow diagonal peak shape remains unchanged, and only the amplitude varies for the different polarization combinations.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/9/eaaw9932/DC1

    Section S1. Possible pathways for excitation of the 40 ± 20–meV coherent superpositions

    Section S2. Heating effects due to laser excitation

    Section S3. Emission at negative times

    Section S4. Correlations between E1, E2, and E3

    Section S5. Response from LiNbO3

    Fig. S1. Signal pathways.

    Fig. S2. Thermal effects.

    Fig. S3. Apparent signal at negative delays.

    Fig. S4. 3D spectra from LSCO and projection onto the 2D planes.

    Fig. S5. Windowed 2D spectra from LiNbO3.

    References (4451)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Possible pathways for excitation of the 40 ± 20–meV coherent superpositions
    • Section S2. Heating effects due to laser excitation
    • Section S3. Emission at negative times
    • Section S4. Correlations between E1, E2, and E3
    • Section S5. Response from LiNbO3
    • Fig. S1. Signal pathways.
    • Fig. S2. Thermal effects.
    • Fig. S3. Apparent signal at negative delays.
    • Fig. S4. 3D spectra from LSCO and projection onto the 2D planes.
    • Fig. S5. Windowed 2D spectra from LiNbO3.
    • References (4451)

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