Two-dimensional photon echo spectroscopy (2D PES) is a powerful technique capable of measuring inter- and intra-molecular coupling on an ultrafast time scale. In particular, 2D PES is advantageous in the presence of inhomogeneous broadening owing to the ability to rephase dipole oscillations resulting from a distribution of energies in the ensemble. This so-called static disorder is oftentimes responsible for the extremely broad spectral features in the linear absorption spectrum. Furthermore, 2D PES can identify electronic or vibrational coupling, which only manifest themselves as slight changes in peak position in the linear absorption spectrum, but are easily observable as off-diagonal peaks in the two-dimensional spectrum. In the presence of significant inhomogeneous broadening, these features are not directly observable in the linear absorption spectrum, making 2D PES an inherently higher resolution technique. When followed as a function of time, 2D PES allows various relaxation processes resulting from system-bath interaction to be resolved. For instance, an ultrafast Stokes shift shows up as a small shift of the center of the peak away from the diagonal. Energy transfer can also be measured by the growth of off-diagonal features in the 2D spectrum. The bath also modulates the site energies and coupling strength of and between chromophores or oscillators in the system, which has a marked effect on the lines shapes of diagonal and off-diagonal features.

Figure 1: Two-dimensional spectra at different waiting times, T. Taking a slice through the diagonal of the 2D spectrum approximately reproduces the linear absorption spectrum. The anti-diagonal line width is a measure of the homogeneous broadening. Off-diagonal features result from inter-molecular coupling and energy transfer processes. The peak shape encodes information on the system-bath interactions. Spectral diffusion, for instance, is a measure of the memory of the system frequencies during the waiting time period.

A detailed description of 2D optical spectroscopy is available in various review articles and books [ref]. Here we will focus on electronic, rather than vibrational, 2D PES, although the theory shares many similarities. The basic theory can be understood by examining the multiple response pathways that result from interactions of the pulses with the system. 2D PES is a special case of four-wave mixing, in which three pulses interact with the system to produce a signal field in a particular phase-matched direction (Figure 2). A more detailed description of nonlinear spectroscopy can be foundhere.

Figure 2: Pulse sequence consists of three pump pulses and a local oscillator field to recover the phase of the emitted signal in the appropriate phase-matched direction. The first pulse creates a single-quantum coherence, which evolves for a coherence time, τ. The second pulse creates a zero-quantum coherence, which for a single chromophore, is a population term. The system undergoes relaxation dynamics during the population time, T. After the third pulse, the system generates a macroscopic polarization emitted as an echo during the rephasing time, t. See text for more information.

In short, there are many ways in which the pulse can interact with the multi-level system in question. For instance, the pulse can cause absorption to an excited state or or it may cause stimulated emission. Alternatively, it can cause the system to reside in a super-position state between the ground and excited states or even between excited states on different molecules. To keep track of the large number of different possible transitions and states, it is useful to have a diagrammatic representation which acts as a book keeping method to follow the system at different moments during the pulse sequence. So-called double-sided Feynman diagrams are a convienent representation to follow the various response pathways that contribute to the emitted signal. The state of the system is expressed in terms of the density matrix formalism, where operators can act on either the bra or the ket states (see notes on quantum mechanics for a review of this notation). Consider the diagram shown in Figure 3. The interaction of the field with the dipole moment of the molecule results in an allowed transition to an excited state after the first pulse. After a time, \tau, the second pulse, depending on whether it interacts on the left or the right, can now cause stimulated emission back to the ground state or creation of a zero-quantum coherence, either between identical states (i.e. a population) or between excited states of different molecules (left diagram). This coherence then evolves for a waiting time, T, before application of a third pulse which stimulates the system back to a ground state population. A photon echo is formed when the phase evolution during the first period exactly cancels that during the last period. Therefore, this particular Feynman diagram represents a rephasing pathway which leads to a macroscopic polarization that results in a detectable signal field. If we now examine, what occurs in the first period, the coherence time, we see that the system undergoes phase evolution proportional to the energy difference between the ground state and the first excited state of the first chromophore, while in the last time period, the so-called rephasing period, it undergoes phase evolution proportional to the energy difference beween the first excited state of the second chromophore and the ground state. Upon double Fourier Tranformation of the signal, a cross-peak will appear at the position corresponding to these energy differences. If the excited state energies of the molecules are different and if the chromophores are coupled, then a non-zero cross-peak amplitude will result (right).

Figure 3: Left - Rephasing pathway for 2D PES. Middle - Energy level diagram for two coupled chromophores. Right - Corresponding peak position in the 2D spectrum after Fourier transformation with respect to the coherence and rephasing time periods. Excited-state absorption pathways (not shown) also give rise to signal at this position in the spectrum, but with opposite sign. In the absence of electronic coupling, these two pathways exactly cancel and no cross-peak amplitude is observed. Other pathways (not shown) give rise to diagonal peaks and below-diagonal cross-peak.

2D PES in the visible and near-IR region of the spectrum has been demonstrated by a variety of methods. Here we will only discuss the passive phase-stabilization method pioneered by Hybl, Cowan, and Brixner. The major challenge in 2D PES compared to other four-wave mixing techniques or third-order experiments in general is that throughout the experiment phase stability must be maintained between pulses 1 and 2 during the coherence period and between pulse 3 and the local oscillator. Otherwise, a random phase error is introduced into the signal which is indistinguishable from the phase that contains that spectral information. Such phase considerations become more relevant at higher frequencies since the period of oscillation becomes shorter. Small differences in the path length traversed by each pulse is then translated into larger changes in phase and sebsequent loss of signal and spectral resolution. Therefore, maintaining phase stability between pairs of pulses represents the largest single challenge in successful implementation of point-by-point 2D PES - that is, sequences in which parametric sampling of at least one Fourier variable (e.g. coherence period) occurs. A clever method by which to achieve phase stability passively, i.e. without an active feedback loops, is to employ common mirrors in a compact geometry. Common mirrors insure that any mechanical vibration experienced by the optic is translated equally to each pulse such that the relative phase between pairs remains unchanged. A compact geometry insures that temperature gradients, which lead to changes in the index of refraction and concomitant changes in the pulse phase are minimized. The optical layout shown in Figure 4 satisfies these conditions. There are two additional important parts of the design, which have proved critical to achieving good phase stability. The first involves the use of a diffractive optic to create the pulse pairs. The main advantage of the diffractive optic is that the resultant pulse pairs are inherently in phase with one another since they are essentially copies of the original pulse (m = +/- 1 orders). Another important but often overlooked advantage is that by imaging the focused spot on the diffractive optic onto the sample, one eliminates the effects of _ filtering - which occurs when the local oscillator does not overlap with the signal field for all frequencies (see section of phase matching). Finally, the use of glass wedges instead of optical delay lines to generate the requisite delays is an important technological feat. A pair of glass wedges in an anti-parallel configuration act like a two gears that transmits torque. A large motion along one direction translates into a small motion n the approximately perpendicular direction for a small wedge angle without changing the direction of propagation of the pulse. This leads to very precise positioning of the wedges to a small fraction of the carrier wavelength.

Figure 4: Point-by-point experimental implementation of 2D optical spectroscopy. Two time-delayed pulses are focus onto a transmissive diffractive optic to generate two pairs of phase locked beams. These are routed through common mirrors and focused onto the sample. Pairs of wedges aligned in an anti-parallel orientation create pulse delays that define the 2D pulse sequence. Passive phase stabilization is insured by the use of the diffractive optics, common focusing mirror, and proper alignment of the wedge face to the input beam. The emitted third-order signal is isolated from the excitation pulses by a spatial filter before being spectrally dispersed off a diffraction grating and imaged onto a linear CCD detector.

1. S. Mukamel, Principles of nonlinear optical spectroscopy. Oxford series in optical and imaging sciences ; 6 (Oxford University Press, New York ; Oxford, 1995), pp. xviii, 543 p.

2. J. D. Hybl, A. A. Ferro, D. M. Jonas, Two-dimensional Fourier transform electronic spectroscopy. Journal of Chemical Physics 115, 6606 (Oct 8, 2001).

3. T. Brixner, T. Mancal, I. V. Stiopkin, G. R. Fleming, Phase-stabilized two-dimensional electronic spectroscopy. Journal of Chemical Physics 121, 4221 (Sep 1, 2004).

4. M. L. Cowan, J. P. Ogilvie, R. J. D. Miller, Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes. Chemical Physics Letters 386, 184 (Mar, 2004).

5. V. Volkov, R. Schanz, P. Hamm, Active phase stabilization in Fourier-transform two-dimensional infrared spectroscopy. Opt Lett 30, 2010 (Aug 1, 2005).