Optical spectroscopy gains a third dimension

Two-dimensional spectroscopy has long been used to study couplings and dynamics in material and chemical systems. In its simplest form, a 2D spectrum measures the change in a sample’s response to a probe frequency ω _b that results from excitation at a pump frequency ω_a. If, for example, ω_a and ω _b are both absorption lines of the same molecule into two different excited states, then absorption at ω_a depletes the ground state and reduces the sample’s ability to absorb at ω_b.

The off-diagonal peaks in a 2D spectrum can therefore aid in interpreting a complicated 1D spectrum that contains many overlapping peaks from different components. One can also use 2D spectroscopy to study changes in the sample that occur during the time between the pump and the probe. For instance, 2D optical spectra can provide information about the flow of energy through the light-harvesting complex of a photosynthesizing organism. (See Physics Today, July 2005, page 23 .) A 2D nuclear magnetic resonance (NMR) measurement offers a look at how nuclear spin migrates in a large molecule, from which one can deduce which parts of the molecule are spatially close to one another. (See the article by Clare Grey and Robert Tycko, Physics Today, September 2009, page 44 .) And 2D NMR or IR spectroscopy can be used to study the time scales of processes such as ions hopping from site to site within a crystal, water molecules moving in a biological system, and the breaking and reforming of hydrogen bonds in liquid water.

Typically, to generate a simple 2D spectrum, one pumps the sample not with monochromatic light but with two short, broadband pulses separated by a variable time delay t ₁. The first pulse excites all the resonant modes in the sample, and the second either enhances or suppresses each oscillation, depending on how close t ₁ is to an integral number of cycles of the oscillation. The single-frequency probe is replaced by a third pulse, which induces the sample to emit a coherent signal. That pulse may be paired with a so-called local-oscillator pulse, weaker than the others, which beats against the signal field and allows it to be recorded as a function of time t ₃ or frequency ω₃. Repeating the measurement with different values of t ₁ and Fourier transforming that time-domain signal gives the 2D spectrum as a function of two frequencies, ω₁ and ω₃.

Now, Keith Nelson and colleagues at MIT have produced 3D optical spectra, in which the signal is a function of ω₁, ω₃, and also ω₂, corresponding to the time t ₂ between the two pulse pairs. ¹ Applying their technique to semiconductor quantum wells—a system in which electrons are excited from nondegenerate valence-band states into the same conduction-band states—they found that important quantum pathways whose signals were indistinguishable in a 2D spectrum could often be separated in a 3D spectrum.

Going through a phase

For the Fourier-transform method to work, the pulses must have definite phase relationships that are stable to within about 1% of a wavelength. It makes sense, therefore, that multidimensional spectroscopy had its origins in nuclear magnetic resonance (NMR), in which the RF waveforms used to excite and probe the sample can be generated on the fly with precisely controlled phase. Application to IR spectroscopy is also relatively straightforward.

But in the optical regime, phases need to be kept stable to within about 5 nm, which is a challenge. The usual method of adjusting the delays between the pulses involves passing each pulse through a separate set of optical components, which leaves the system vulnerable to noise from vibrations and air currents. Spectroscopists were able to stabilize the phase relationships between the first two pulses and the last two, but the lack of phase coherence between the pulse pairs limited the measurements they could perform.

In 2007 Nelson and colleagues presented a fully coherent 2D optical spectrometer, in which the phases of all four pulses are mutually related. ² To control the pulse timing, they used a diffraction grating and a 2D spatial light modulator (SLM). The grating separated each pulse into its component frequencies, and the SLM adjusted the relative phase of each frequency, so that when the components were reflected back onto the grating and reassembled, they would constructively interfere at the sample at the desired time. Because different parts of the same grating and the same SLM are used for all the pulses, mechanical vibrations don’t destroy the phase coherence.

The MIT apparatus allowed the researchers to perform optical measurements that were only previously possible in NMR and in the IR regime. Not only could they scan t ₁ and plot a spectrum as a function of ω₁ and ω₃, but they could also scan t ₂ to yield spectra ³ as a function of ω₂ and ω₃. And, as they’ve now demonstrated, they can scan both t ₁ and t ₂ to produce a full 3D spectrum. ¹

Excitons and biexcitons

Nelson and colleagues demonstrated their system’s 3D capabilities by looking at gallium arsenide quantum wells—layers of GaAs 10 nm thick—separated by equally thin layers of Al_0.3Ga_0.7As, as shown in figure 1(a). In bulk GaAs, an optical excitation can promote an electron from the valence band to the conduction band; at low enough temperatures, the electron and the nascent hole can form a bound state, or exciton, with a hydrogenic wavefunction.

In the quantum wells, there are two types of excitons, as shown in figure 1(b): The valence electrons’ total angular-momentum quantum number j equals 3/2, and when angular momentum is quantized in the direction perpendicular to the layers, the states with m_j = ±3/2 are about 8 meV more energetic than the states with m_j = ±1/2. Excitations from the ±3/2 states produce heavy-hole excitons (so called because of the holes’ larger effective mass, and denoted in the figures by a double arrow ⇑), and excitations from the ±1/2 states produce light-hole excitons (denoted by a single arrow ↑). Excitons of either type can be spin up or spin down, depending on whether m_j of the electron in the conduction band is +1/2 or −1/2. The type and spin of the excitons can be partially controlled by using circularly polarized light.

Not only can the electrons and holes form bound states, but pairs of excitons can bind to form biexcitons, with wave-functions analogous to those of hydrogen molecules. To form a biexciton, two excitons—whether heavy, light, or one of each—must be opposite in spin. Bi-excitons can’t be created with a single photon or pulse. Studying them with multidimensional spectroscopy requires full coherence among all four optical pulses.

Pathways and peaks

Figure 2 shows data from a 3D spectrum for which all four pulses had the same circular polarization: Spin-up heavy excitons and spin-down light excitons could be formed, but not spin-down heavy excitons or spin-up light excitons. The plots in the figure are all projections of the 3D spectrum onto two axes, integrated over either all of the third axis (figures 2(a) and 2(b)) or part of it (figure 2(c)). The axis labels E ₁, E ₂, and E ₃ are the energies that correspond to the frequencies ω₁,ω₂, and ω₃.

Figure 2(d) shows six quantum pathways that are visible in the spectrum. In each of them, starting from the ground state, the first pulse creates an exciton—more accurately, a “coherence,” or coherent superposition of the exciton state and the ground state, that oscillates with frequency determined by the energy of the exciton. The second pulse excites the one-exciton coherence into a two-exciton coherence; if the two excitons are opposite in spin, they form a biexciton, and if they are parallel in spin, they form a so-called interaction-induced coherence with frequency equal to twice that of a single exciton. The third pulse takes the two-exciton coherence back down to a one-exciton coherence, which radiates a signal whose frequency is recorded on the ω₃ axis.

Figure 2(a) shows the spectrum as a function of E₂ and E ₃—one of the 2D measurements that first became possible with the MIT group’s fully coherent spectrometer. On the E ₃ axis, pathways i, ii, and vi, which end up in a light-exciton coherence, are seen as peaks at the light-exciton energy of 1.5495 eV; the other three pathways have peaks at the heavy-exciton energy of 1.5416 eV. On the E ₂ axis, the peaks display the energies of the two-quantum coherences formed during time t₂ : Pathways i, ii, iii, and iv all have the energy of the mixed biexciton, which means that pathways i and ii are indistinguishable, as are pathways iii and iv.

Figure 2(b) shows a different projection of the same data along the E ₂ and E ₁ axes. Pathways i and ii are separated, but now they’re superimposed on pathways iii and iv, respectively. Also, pathway v, the interaction-induced coherence with two heavy excitons—by far the most intense feature in the spectrum due to the large transition dipole moment of the heavy excitons—obscures the adjacent peak from pathways i and iii.

But 3D spectroscopy offers a way to get an unimpeded view of pathways i and ii, as shown in figure 2(c). There, E ₂ is shown only over the range around the mixed-biexciton energy indicated by the shaded box in figure 2(b), and more importantly, E ₃ is integrated only over the range around the light-exciton energy shown by the box in figure 2(a). Since the peaks from pathways iii, iv, and v fall outside that range, all that remain are the peaks from pathways i and ii, which are well separated on the E ₁ axis, so their intensities and linewidths can be measured directly.

Notes Nelson, “If you want to resolve the different contributions to the signal, it’s essential to scan the additional dimension.” Indeed, Nelson and company have been working on adding even more dimensions, using chains of five or seven pulses rather than three, in order to look at coherences of more than two excitons.