GaAs exciton and multiexciton states. (a) Exciton ladder illustrating the ground state (|0⟩), the two exciton states (|1⟩), the three biexciton states (|2⟩), and the four triexciton states (|3⟩). The approximate energies of each level are listed. The binding energies of the multiexcitons are about 1–3 meV. (b) The absorption spectrum has two peaks at about 1540 meV and 1547 meV for the H and L excitons, respectively.
Experimental apparatus for coherent multidimensional spectroscopy. (a) The apparatus contains four essential components: a laser producing femtosecond pulses (red box), a spatial beam shaper producing a user-defined 2D geometrical arrangement of beams (blue box), a spatiotemporal pulse shaper capable of independently delaying and phase-shifting each pulse in the set of beams (green box), and finally a signal detector which in this case is a spectrometer with a CCD array (grey box). The three-beam geometry is shown for clarity. For additional clarity, the final lens which collimates the signal and LO and the waveplates used to control the polarization of each beam are not shown. (b) A more detailed depiction of the apparatus following the same color-coding scheme as above but illustrating a four-beam, Y-shaped geometry. The spatial beam shaper is composed of two lenses and a 2D spatial light modulator (SLM 1). The spatiotemporal pulse shaper imparts delays and phase shifts to the pulses in the beams by constructing sawtooth phase grating patterns on the second 2D SLM (SLM 2). After lens L5, the beams have the geometry defined by SLM 1 with relative pulse timings and phases (or more general specified amplitude and phase profiles) defined by SLM 2. The signal is generated in the same direction as the LO; this beam is isolated by the iris. Their frequency-dependent interference fringes are read by a CCD detector after diffraction by the grating (G2) in the spectrometer. The focusing mirror in the spectrometer is not illustrated. (c) The beam geometry at three points in the pulse shaper. The top arrangement illustrates propagation throughout most of the device. The middle is the beam arrangement at the plane of the cylindrical lens (CL) in the pulse shaper. The bottom arrangement shows how the beams are refocused on the SLM 2 surface so that a single frequency component encounters one vertical column of pixels, regardless of which beam is considered. (d) The SLM 2 plane is focused by lens L4 to the plane of the pick-off mirror (M), which is adjusted to send only the desired first-order diffraction order from the SLM 2 vertical grating pixel patterns toward the sample. The light grey lines illustrate the incoming beam and the black lines illustrate the outgoing beam.
Phase stability of the COLBERT spectrometer. The data presented here were measured on a device with a phase stability of λ/88 over 14 h (black line). A recent improvement increased the phase stability to λ/157 (red line). The two traces are offset from each other for clarity. The inset shows an enhanced view of 2 h of the measurement.
The available grating periods up to 700 μm and their associated 800-nm diffraction angles for one dimension of the 2D SLM are shown for 1 μm (blue dots), 12 μm (black diamonds), and 24 μm (open red squares) pixel sizes. Their respective largest diffraction angles are about 24°, 2°, and 1°. The SLM pixel size restricts the possible grating periods, which in turn restricts the available diffraction angles. Smaller pixels allow more accurate placement of the beams. For the 150 cm focal length lens used here, the minimum angle needed to clear the input beam is 0.08°; the maximum angle, 0.95°, is governed by the radius of the output lens (2.5 cm). As the pixel size decreases toward 1 μm, the available angles approach a continuum.
Two-dimensional Fourier beam shaping. (a) An 80 pixel by 80 pixel portion of the phase pattern encoded in the beam shaping SLM; each pixel is 24 μm × 24 μm. This specific pattern is used to generate a Y-shaped beam geometry,31 and the red area illustrates the focused input beam. The beam waist diameter is about 1 mm, corresponding to 40 pixels. The geometry is rotated by about 15° so that each beam encounters a different vertical region of the spatiotemporal pulse shaper. The green box is a 10 pixel by 10 pixel portion which clearly shows the pixelation of the pattern. (b) The calculated real space beam pattern after recollimation by the output lens. The position and intensity of each beam can be controlled. The wavevectors indicated are those for a fifth-order measurement discussed below. (c) Photograph of scatter from an index card of the experimental beam geometry with clearly visible relative intensity differences. Unwanted, low-intensity diffraction orders were blocked.
Calculated two-dimensional diffraction-based pulse shaping patterns. The broadband pulse is dispersed spectrally across the horizontal dimension. One column of pixels is defined to be the carrier frequency ω0; this column of pixels does not change for delays and chirps. (a) Stripe with all phase parameters set to zero. (b) A π phase offset was applied, shifting the entire sawtooth pattern up by half a period, and shifting the phase of the optical pulse by π. This is a rare situation in which ω0 changes phase. (c) The pulse was delayed by the linear phase sweep of slope Δϕ/Δω = 500 fs. The pulse had no phase offset, as defined by the phase at ω0. (d) Same as previous but a delay of 10 fs using four groups of binned pixels. (e) The pulse has no phase offset or time delay, but was given a small amount of quadratic chirp. (f) The pulse was given a time delay of 100 fs and the same chirp as in (e).
Greyscale to phase calibration of the spatial light modulator (SLM). (a) The input beam transmits through a beam splitter (BS), and the two first-order diffractions from the phase mask (PM) are focused by the cylindrical lens (CL) onto the surface of the SLM. One half of the SLM surface remains at a greyscale value of zero while the other half is scanned. At each of the 256 values, an isolated fringe of the beam reflected by the BS is measured by the detector (det). (b) Measured greyscale to phase change function (black dots) and its fit to a fifteenth-order polynomial (solid red line). (c) From the polynomial fit, the function relating greyscale to phase, g(ϕ), is calculated and plotted. This specific SLM is capable of about 2.5π phase modulation for 800 nm light. The green dashed line is a linear reference to show the slight nonlinearity of the function. The two SLMs have different responses and so must be calibrated individually.
The wavelength to pixel calibration is performed with the sample removed and all beams except the LO blocked. (a) The position of a three-pixel-wide stripe, centered at pixel x n , is scanned across the SLM surface, usually every five pixels. At each position, the wavelength corresponding to the maximum of the sharp spectral peak, λ n , is measured. (b) Maximum wavelength values are collected across the SLM surface. (c) The wavelength values are interpolated to a frequency coordinate and fit to a fifth-order polynomial. This function is then inverted to the desired frequency-to-pixel function: λ(x) → x(ν).
Measuring the global phase offset (ΔΦ) of the emitted signal to phase a 2D spectrum. (Top) Spectra around the H exciton wavelength measured for varying values of the phase of one excitation pulse. (Bottom) Measured (black squares) and cosine fit (red line) to the integrated phase profile. The fit to several cycles of the cosine function indicates that ΔΦ = 1.2 radians.
Spectral interferometry algorithm. (a) Spectral fringes caused by interfering the signal with the LO in the spectrometer. The τ1 = τ2 = 0 spectrum of the emitted signal (black dashed line) and the LO spectrum (black solid line) are subtracted from their interference spectrum (red solid line) to leave only the cross term (blue solid line). While the H exciton is the strong feature at 806 nm, the L exciton feature—which is barely visible at 802 nm in the signal spectrum—is enhanced in the interference and cross term spectra. Signals from the substrate are visible near 820 nm. (b) The cross term is interpolated to a frequency axis. (c) The cross term is then Fourier transformed to the time domain (blue and black lines), and a filter (red line) is applied to select only the positive-time component of the signal (black line). Only the real part of the complex signal is displayed. (d) The remaining signal is Fourier transformed back to the frequency domain and converted to an energy unit. The amplitude (blue line) and phase (green line) of the resulting complex-valued signal are displayed.
Result of a third-order rephasing experiment (τ2 = 0) using co-linearly polarized pulses in the BOXCARS geometry. (a) The unprocessed data show coherent oscillations at the H and L exciton emisson energies of 1540 and 1546 meV, respectively. Phase cycling reduced signals due to scattered light, eliminated homodyne contributions, and amplified the signal. (b) One step in the spectral interferometry procedure is to Fourier transform the emission frequency dimension to the time domain. At this point in the procedure, we plot the amplitude of the complex-valued time domain signal. (c) The final amplitude spectrum after finishing the spectral interferometry procedure along the emission dimension and Fourier transforming the scanned time dimension. Cross peaks between the H and L excitons indicate their coupling. The real part of the spectrum provides additional information about many-body interactions.16
The pulse timing sequences used to create the spectra described in Sec. IV. We use letters rather than numbers to label the beams to avoid suggesting that the beam labels relate to their time ordering. The LO interacts 1 ps before the final excitation field in all measurements. (a) Third-order rephasing scan where the conjugate field, E a , is the first field to interact with the sample. The second time period is often not scanned in 2D measurements, that is τ2 = 0 or some finite value. The spectrum of the signal emitted during is measured in the spectrometer. (b) Third-order two-quantum nonrephasing scan in which τ2Q is scanned and τ1 = 0 or some specific value. The conjugate field now interacts after both nonconjugate fields have interacted. (c) Fifth-order two-quantum rephasing scan. The two conjugate fields, E a and E b , interact first to create two-quantum coherences during time period τ2Q . These coherences are rephased at half their rate of dephasing during . (d) Fifth-order three-quantum non-rephasing scans contain triexciton–ground-state three-quantum coherences during time period τ3Q . The two conjugate fields now interact after the three nonconjugates.
Result of a third-order rephasing experiment using cross-linearly polarized pulses in the BOXCARS geometry. The τ2 delay of 667 fs ensured that pulse overlap effects were negligible. In this polarization configuration, many-body interactions except pure biexcitons are suppressed. The spectra are plotted using 25 linearly spaced contours. (Top) The amplitude spectrum shows that the HH biexciton-exciton emission shoulder is enhanced relative to the other peaks compared to co-linear polarization. (Bottom) The real part of the spectrum shows the absorptive character of the peaks now that many-body interactions are not distorting their lineshapes.
Result of a third-order two-quantum experiment (τ1 = 0 fs) using cross-linearly polarized pulses in the BOXCARS geometry. The spectrum was adjusted to excite only H excitons. In this polarization configuration, all many-body interactions except pure biexcitons are suppressed. (Left) The amplitude spectrum clearly shows that the HH biexciton peak is red-shifted from the diagonal by an amount equal to its biexciton binding energy, Δ HH . Biexciton-exciton emission is visible as a shoulder to the main exciton–ground-state emission. (Right) The real part of the spectrum has nodes that are perpendicular to the diagonal, indicating the nonrephasing character of the spectrum.28,68,125
Result of a third-order one-quantum 3D rephasing experiment using co-circularly polarized pulses in the BOXCARS geometry. Only the real parts of the projections are displayed; the amplitudes were presented previously.29 (Top) The (ℏω1, ) projection shows the dispersive character of the four peaks. In this polarization configuration, pure biexciton-exciton emission is prohibited, thus there is no shoulder on the H diagonal peak. Although mixed biexciton-exciton emission is possible, it is not observed under these excitation conditions. (Left) The (ℏω2, ℏω1) projection has intense features on the ℏω2 = 0 energy level due to population relaxation pathways. Quantum beats between the two single excitons result in the peaks above and below the diagonal at approximately ±8 meV. (Right) The (ℏω2, ) projection shows similar features. In both projections involving ℏω2, the quantum beat peaks and the peaks at ℏω2 = 0 are dispersive. The contours were plotted carefully to eliminate excessive noise. The top spectrum was plotted using 25 contours from −1 to −0.025 and 25 contours from +0.025 to +1. The two spectra involving ℏω2 were plotted similarily, but over ranges of ±(0.1 to 0.0075).
Result of a fifth-order two-quantum rephasing experiment (τ1 = 0 fs) using co-circularly polarized pulses in the three-beam geometry. In this polarization configuration, unbound-but-correlated exciton pairs and mixed biexciton coherences are observed during the two-quantum time period while pure biexcitons are suppressed. (Top) The amplitude spectrum shows peaks due to unbound H pairs, mixed biexcitons, and unbound L pairs at their respective two-quantum energies. Rapid dephasing causes the peaks to blend together. (Bottom) The real part of the spectrum over only the unbound H pair and mixed biexciton features shows the dispersive character of the unbound H pair peak and the phase twisted, absorptive, lineshape of the mixed biexciton–ground-state coherence.
Result of a fifth-order three-quantum experiment (τ1 = τ2 = 0 fs) using co-linearly polarized pulses in the three-beam geometry. All four triexciton–ground-state coherences are observed during the three-quantum time period. Their Fourier-transform-related spectral peaks, amplitude shown here, allow their binding energies to be extracted. Only the binding energy of the HHH triexciton is indicated, Δ HHH = 1.8 ± 0.2 meV.
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