Probing optically silent superfluid stripes in cuprates

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Science  02 Feb 2018:
Vol. 359, Issue 6375, pp. 575-579
DOI: 10.1126/science.aan3438
  • Fig. 1 Phase diagram for La2–xBaxCuO4.

    SC, SO, and CO denote the bulk superconducting, spin- and charge-ordered (striped), and charge-only–ordered phases, respectively. Tc, Tso, and Tco are the corresponding ordering temperatures. TLT denotes the orthorhombic-to-tetragonal structural transition temperature. The samples examined in this study are x = 9.5, 11.5, and 15.5% (dotted lines). Further, a schematic stripe-ordered state is shown wherein the tan stripes depict the charge rivers and the gray stripes depict the antiferromagnetic insulating region (inset). Figure adapted with permission from (1).

  • Fig. 2 Third harmonic from homogeneous superconductor.

    (A) Linear and nonlinear reflectivity of La2–xBaxCuO4, where x = 9.5%, measured at T = 5 K with ωpump = 450 GHz. The linear reflectivity displays a Josephson plasma edge at ωJP0 = 500 GHz (0.5 THz), whereas the nonlinear reflectivity shows a red shift of the edge and third-harmonic generation (red shading). (B) Temperature dependence of nonlinear reflectivity for x = 9.5%. The third-harmonic peak disappears above Tc = 34 K. (C) Linear and nonlinear reflectivity for x = 15.5%, measured with ωpump = 700 GHz. (D) Temperature dependence of third-harmonic generation in x = 15.5%. Third-harmonic generation (red shading) disappears above Tc = 32 K. (E) Third-harmonic electric field strength (normalized to the highest signal) plotted as a function of the incident electric field strength [definition explained in section S1 of (9)] measured at T = 5 K from the x = 9.5% sample. The third-harmonic field displays a cubic dependence on the incident field strength. (F) Temperature dependence of the third-harmonic amplitude (normalized to the measurement at T = 5 K) from the x = 9.5 and 15.5% doping. The superfluid density [(ωσ2(σ→0)] (normalized to the measurement at T = 5 K) extracted from the linear optical properties of the x = 9.5% sample is also shown. All of the quantities vanish above Tc. Red shading indicates the bulk superconducting phase.

  • Fig. 3 Simulated nonlinear reflectivity for homogeneous superconductivity.

    Simulations at x = 9.5 and 15.5% doping. (A) Simulated space (x, not to be confused with the symbol for doping concentration)– and time (t)–dependent order parameter phase [θ(x, t)] obtained by numerically solving the sine-Gordon equation on x = 9.5% samples [see section S3 of (9)]. The equation makes use of equilibrium superfluid density extracted from the linear optical properties and assumes excitation with terahertz pulses of shape and strength used in the experiment. The horizontal dotted lines indicate the spatial coordinate x at which the line cuts are displayed (lower panel). (B) Simulated order parameter phase (A) after frequency filtering centered at 3ωpump with its corresponding line cut (lower panel). (C) Simulated reflectivity in the linear (E = 0.1 kV/cm) and the nonlinear (E = 80 kV/cm) regime. The third-harmonic generation component is highlighted (red shading). (D to F) Same as in (A) to (C) but for x = 15.5%.

  • Fig. 4 Third-harmonic generation in the striped phase.

    (A) Nonlinear frequency-dependent reflectivity measured in the striped x = 11.5% samples recorded for three different field strengths at T = 5 K (<Tc = 13 K). (B) Electric field dependence of the third-harmonic amplitude for T = 5 K. (C) Temperature dependence of the nonlinear reflectivity for T > Tc = 13 K. (D) Temperature dependence of the third-harmonic signal (normalized to the highest field measurements at T = 5 K). (E) Schematic of the order parameter phase in a pair density wave condensate. The black arrows represent the superconducting order parameter phase at each lattice point. Interlayer tunneling from perpendicularly aligned superfluid stripes is equivalent to a checkerboard lattice of alternating π/2 and –π/2 phase Josephson junctions. Such a lattice has tunneling currents of Ic and –Ic flowing at the neighboring junctions at equilibrium (thick red lines). (F) Excitation modes of the PDW indicating the ϕ0 and ϕπ modes (see text). The shaded region under the black arrow represents the phase excursion from the equilibrium geometry (δθ0 and δθπ). Corresponding current fluctuations δI0 and δIπ produced by such excitations are also depicted (thin red lines). (G) Calculated nonlinear current response for the unit cell of (E) after application of a single-cycle optical pulse centered at 500 GHz frequency [see section S5 of (9) for details].

Supplementary Materials

  • Probing optically silent superfluid stripes in cuprates

    S. Rajasekaran, J. Okamoto, L. Mathey, M. Fechner, V. Thampy, G. D. Gu, A. Cavalleri

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