Fundamentals of Nonlinear Spectroscopy

            Nonlinear processes occur when the effects of a perturbation become large and the response to the perturbation no longer follows the perturbation’s magnitude. A common nonlinear phenomena occurs when the volume on a sound system is raised to levels where the capacity of the system’s audio speakers is exceeded and sound distortions are created. Here, the response of the speakers no longer follows the driving voltage. The distortions create new frequencies at the sum and differences of the sound frequencies as well as changing their amplitudes. Nonlinear optical processes are analogous. They occur when the electric field strength of light becomes comparable or larger than the electric fields in a sample so the polarization induced in the matter by the electric field of the light is distorted. It acquires new frequencies and the amplitude of the previous frequencies is modified. Rather than hearing new frequencies, one sees new colors at frequencies that correspond to the sum and differences of the exciting light’s frequencies.

            Molecular spectroscopy of vibrational and electronic states is largely based on the interaction of the electric field component of electromagnetic radiation fields with molecules through the induced molecular electric dipole moments. In contrast, nuclear magnetic resonance (NMR) is based on the interaction of the magnetic field component with the magnetic dipole moment of nuclear spins. There are many commonalities and differences between NMR and molecular spectroscopies. An important difference is the amount of interaction that occurs between the quantum states of matter and the thermal fluctuations of the environment. Nuclei are well-shielded from the thermal environment and have very sharply defined transitions between quantum states; electronic states interact strongly with the environment and have much broader transitions; vibrational states have interactions that are intermediate between the two cases.

 

Background         [back to the top]

            All spectroscopies are based on the polarization induced by light’s oscillating electric field. While the light field is on, the polarization is forced to oscillate at the frequency of the light. The oscillating polarization in turn emits light at the same frequency as the excitation light. This process is the driven emission. When the excitation field is first turned on, the polarization builds up over the quantum state’s lifetime until it reaches a steady state equilibrium polarization that provides a continuous driven emission. When the light field is off, the polarization can continue to oscillate at the frequency defined by the molecular states. The oscillating polarization will emit light (any oscillating charge distribution will emit light as long as it has a dipole moment) over a time period determined by the quantum states’ lifetime. This emission is called free induction decay (FID). The relative amount of driven and FID emission depends on the exciting field’s pulse-width, the relaxation rate of the excited molecular states, and how close the exciting field is to resonance with the molecular states. Nonlinear spectroscopy involves multiple interactions with the excitation fields and each interaction creates a driven emission and a FID. The nonlinear interactions are described either by a phenomenological relationship between a molecular polarization and the exciting electric field or a more detailed time dependent quantum mechanical model that describes the evolution of the states that are entangled during the nonlinear process.

 

Phenomenological Model of Nonlinear Spectroscopy           [back to the top]

             Figure 1 describes a phenomenological relationship where the polarization follows the electric field linearly for small fields but becomes nonlinear for larger fields as the fields distort the molecular structure and change the force constants characterizing the molecular bonds. At the highest intensities, the fields are strong enough to dislodge electrons from the molecules and induce a dielectric breakdown that is usually described as “zapping” your sample. In the figure, it occurs when the polarization approaches infinite values.

            When a complex relationship such as that shown in figure 1 is not understood, it is described by a Taylor series,

                                   (1)

where χ⁽ⁿ⁾ is the n^th order susceptibility tensor (they define the relationship between the polarizations of the electromagnetic fields and the nonlinear polarization’s polarization) and  is the total electric field from all of the light beams exciting the sample, . If the sample is isotropic, the polarization must reverse if the electric field reverses. Since the polarization created by the even terms in the series doesn’t reverse, these terms are unimportant for isotropic materials. They are very important for surfaces and interfaces where they persist and provide the techniques with exquisite selectivity for the interface. The first term, χ⁽¹⁾, describes linear processes while the remaining terms describe nonlinear processes. For example, if there are three exciting light beams,

,                                         (2)

and one expands the term χ⁽³⁾  E³ , terms such as , , , , and  appear as well as other similar combinations. These terms in the polarization are responsible for different nonlinear processes such as frequency tripling, coherent anti-Stokes Raman Spectroscopy  (CARS), doubly vibrationally enhanced four wave mixing (DOVE FWM), and degenerate FWM, respectively. Each term can be associated with many different nonlinear processes depending on the experimental conditions. For example, the term associated with degenerate four wave mixing can also describe photon echo, transient grating, 2D-IR, optical Kerr effect, Raman gain and loss spectroscopy, as well as many other nonlinear processes. In each case, the oscillating polarization creates a new electromagnetic field at ω₄ that may be at a different frequency than any of the exciting fields (for example, ) or the same frequency (). In the latter case, the new field interferes with the exciting field to modify its amplitude. Since each example involves three exciting fields and an output field, the processes resulting from the  χ⁽³⁾  E³ term are called four wave mixing (FWM) since three waves mix to create the fourth wave.

            The output wave is created by a phased array of coherences that emit a directional beam. For example, the phase factor  describes the spatial and temporal oscillations of an FWM output polarization. The field that it launches  has a phase factor of , where  is the wave-vector of the output field, n₄ is the index of refraction, and  is a unit vector specifying the direction of the output beam. The wave-vectors of the nonlinear polarization () and the output field it creates () are necessarily the same since the indices of refraction are different in a dispersive medium.  This discrepancy causes the field created at one point in the sample to dephase from the fields created at other points on the sample so the electric fields are not in phase and intensity is lost. Phase matching techniques correct for this dephasing  so the output coherence and the field it creates are always in-phase and the output beam is intense and directional. Often, phase matching is performed by bringing the excitation beams into the sample at different angles so the vector addition results in phase matching. At the same time, the excitation beams are only overlapped over a limited region of space.

            As a second example, consider the second term in equation 1 when there are two excitation fields at ω₁ and ω₂. Expanding the  χ⁽²⁾  E² term, one sees that the polarization will have terms oscillating at , , , and . These terms are responsible for nonlinear processes such as second harmonic generation (SFG), sum frequency (SFG), difference frequency (DFG), and optical rectification, respectively. All are three wave mixing processes and all are selective for surfaces, interfaces, and other anisotropic situations. There are also higher order terms that correspond to five wave mixing, six wave mixing, etc. These terms become important at high light intensities. All odd wave mixing terms are surface and interface selective while all even order terms are important for all cases. Table 1 summarizes the excitation frequencies and output frequencies of the different nonlinear processes discussed in this chapter.

 

Quantum Mechanical Model for Nonlinear Spectroscopy           [back to the top]

            Electromagnetic radiation creates a polarization in a sample and that polarization will re-radiate light. This process forms the basis of molecular spectroscopy. The polarization on an individual molecule is created by the transition dipole moment of the molecule’s coherence. A coherence must be understood quantum mechanically. If a and b are two molecular quantum states and c_a and c_b are the quantum mechanical amplitudes for the molecule being in those states, then  and  measure the population of either state and  and  measure the coherence of both states. A coherence is a quantum mechanical entanglement of the two states created by the electromagnetic field perturbation. It is and the cross terms that result oscillate at . This coherence launches the electromagnetic fields that are responsible for molecular spectroscopy. The coherence’s imaginary term is responsible for absorption (i.e. the part of the polarization that is out-of-phase with the electromagnetic field) and the real term is responsible for refraction (i.e. the in-phase part of the polarization).

            Time dependent quantum mechanics describes the evolution of a molecular state as it becomes entangled with another state. Before the field is applied, only one state is populated so c_a=1 and c_b=0. Turning on an electromagnetic field then changes the amplitudes so c_a decreases while c_b increases until c_a=0 and c_b=1. Assuming state ω_ba > 0 (i.e. state b is more energetic than a), the molecule is absorbing energy from the field. The evolution continues though so now c_a increases while c_b decreases until c_a=1 and c_b=0.  The molecule is now emitting energy to the field, a process called stimulated emission. The intensity of the exciting field alternately dims during the absorption phase of the cycle and brightens during the stimulated emission phase. This cycling is optical nutation. The entire cycle from the initial state and back is called a Rabi oscillation and its period depends on the light intensity. It is analogous to pushing a child on a swing- the harder you push, the more rapidly the oscillation builds up. There are two time periods, the natural period of the child and swing and the time required to build-up the oscillation. Quantum mechanically, there are also two frequencies- the Rabi frequency that is intensity dependent and the Bohr frequency, ω_ba, defined by the molecular states. The Rabi frequency is  where μ is the transition dipole corresponding to the transition induced by the electromagnetic field, E.

            The picture assumes that the molecule is sufficiently isolated that the oscillations remain synchronized with the exciting field. If synchronization is maintained, the molecule’s emission is temporally coherent, i.e. its phase at future times can be predicted. Interactions with the thermally fluctuating environment, however, causes decoherence of the molecular states as the phase relationships of the polarization to the field become scrambled. This process is dephasing. We label this dephasing rate, Γ. Condensed phase dephasing rates for NMR are ~10³ sec^-1, vibrational states are ~10¹² sec^-1, and electronic states are ~10¹⁴ sec^-1. Since the Rabi frequencies are usually much lower, the evolution of the states does not proceed very far into a Rabi period before dephasing occurs so absorption usually dominates in molecular spectroscopies and most spectroscopies are incoherent. This situation is quite different from NMR where the Rabi frequency exceeds the dephasing rate so fully coherent experiments are easily performed. Typically, the ground state amplitude, c_g ~ 1, and an excited state amplitude, c_a <<1. If one starts from the ground state, , the interaction with the field induces a transition from the ground state population given by to the emitting coherence, . The relationship between and  in the steady state limit is where  is the detuning factor of the field’s frequency from the molecular frequency. Since normally , one can work in the perturbative limit where each interaction with the field changes just one of the entangled quantum states. Note that this simple equation already contains the essence of absorption spectroscopy and refraction. Remembering that absorption and refraction are described by the imaginary and real parts of the coherence, we see that  gives  and for the frequency dependences of absorption and refraction, respectively. The two dependences are related by the Kramers-Kronig transform.

 

            The character of a nonlinear process is defined by the sequence of coherences and populations that are created by successive interactions with the excitation fields. Let’s first start with the simplest case and follow the sequence of coherences and populations when a system starting in the ground state interacts with a resonant electromagnetic field that promotes transitions between states g and a. The entangled states resulting from each step will be described by the subscripts in the corresponding density matrix element, ρ_ij. Absorption and refraction are described by a single interaction, gg→ag, i.e. the field interacts with the ground state population, gg, to create an quantum entanglement of states a and g. The ag coherence creates an output field at the same frequency as the excitation field so the new field and the exciting field are superimposed and interfere so the net field forming the output is smaller than the input field (absorption) and phase shifted (refraction). Figure 2a diagrams the transitions between the energy levels expressed by gg→ag.

            There are three types of coherences that are important for nonlinear spectroscopy. In analogy to NMR, an ag coherence is an example of a single quantum coherence. A second excitation can create an excitation on the ket resulting in an (a+b),g coherence or on the bra resulting in an ab coherence. These coherences are called a double quantum or zero quantum coherence.

 

Free Induction Decay and Quantum Beating               [back to the top]

            Absorption spectroscopy is associated with the steady state value of the ag coherence during an excitation that is long compared to the dephasing time, so the phase relationships between different molecular coherences is lost and coherent emission is not observed. The re-emission intensity from the excited molecules scales as the number density, N. If the excitation pulse is shorter than the dephasing time, the molecular coherences will still retain their proper phases and the re-emission from all the molecules will in-phase and coherent. The emission will be intense (the intensity will scale as N ²) and directional. It will decay exponentially at the rate defined by dephasing. This process is the free induction decay and it is the coherent transient created by the pathways in figure 2a.

            A short light pulse will have a large band of frequencies, so it is possible to excite different coherences such as shown in figure 2b. Quantum beating occurs since the FID of the polarization now has two emission frequencies, one from the and one from the coherences, so they interfere and beat in time at the difference frequency. A fourier transformation of the quantum beats gives a frequency domain spectrum of the frequency differences between all the coherences. The beating persists as long as the interfering polarizations maintain their phase coherence, T₂=.

 

Rabi Oscillations, Optical Nutation, and Dynamic Stark Effects           [back to the top]

            If the coherence created by the interaction with a field (labeled ag) does not dephase during the time associated with the Rabi oscillations (i.e. Γ_ag << Ω), the interaction with the field can continue for a complete Rabi cycle. In the perturbative limit, successive interactions would create the following evolution of the quantum states starting from the ground state- gg→ag→aa→ga→gg. Wave mixing energy level (WMEL) diagrams show these transitions so one can see the resonances involved. The WMEL diagram for the Rabi cycle appears in figure 2c. In the WMEL diagrams we will use, the solid arrows show ket-side transitions and the dotted arrows show bra-side transitions. Up arrows correspond to photon annihilation operators of the electric field and cause absorption at the intensity level and down arrows correspond to creation operators that cause emission. Time starts at the left and increases to the right. The wavy arrow at the end represents the output emission.

            The first two resonances in figure 2c correspond to the absorption of a photon creating the excited state described by the population, ρ_aa, while the last two steps correspond to a stimulated emission returning the molecule to the ground state. If one measures the light emerging from the sample, one would observe it alternately growing dimmer and then brighter at the Rabi frequency corresponding to absorption and stimulated emission, respectively. This process is called optical nutation. It is not often observed experimentally since the dephasing rate is usually fast compared to the Rabi frequency.

            Take particular note that this description for creating an excited state population, aa, is quite different from the usual energy level diagram where absorption is represented by a single arrow from state g to state a. Such diagrams show the changes in populations that occur at the intensity level (i.e. the square of the electric fields). In our case, the excited state population, aa, results from two interactions with the electric field; the two arrows show the evolution of the states at the quantum mechanical amplitude level. The reason for worrying about the electric field and quantum state amplitude is that the phase of the electric fields and coherences is required to describe the interference effects that are important in nonlinear processes. These effects are not important if the measurement is incoherent so that quantum mechanical phase is unimportant.

            The 4 interactions shown above for the complete Rabi cycle of absorption and stimulated emission is appropriate in the perturbative limit. The interactions are written in Dirac’s bra and ket notation where the first index (a in ρ_ag) is the ket and the second index (g) is the bra and the coherence is then . The optical nutation sequence written above is represented by the diagram in figure 2d. The sequence in figure 2c would correspond to starting at the gg population and proceeding clockwise through four interactions with the electric field to return to gg.

            Figure 2d shows arrows going in both the clockwise and counter-clockwise directions. All possible interactions will occur and one must consider all of them to correctly describe any nonlinear process. These multiple transitions become important in describing the dynamic Stark effects that occur at very high intensities where  so many interactions can occur within the dephasing time. In this regime, one observes a broadening of the transitions because of the increased rates into and out of states and line splitting that result from the entanglement of the photons with the molecular states. These diagrams are useful because they show the relationships between multiple pathways that produce the same final coherences. Multiple pathways will interfere at the quantum mechanical amplitude level and are responsible for important effects in nonlinear spectroscopy.

 

Role of Vacuum Fluctuations in Spectroscopy             [back to the top]

            Although a vacuum is usually considered to be the absence of anything, it is actually quite busy. Quantum mechanics finds that when you quantize the electromagnetic field, the absence of a field in a vacuum is not allowed by the uncertainty principle since a vacuum would then have a well-defined energy, namely 0. This result is identical to the quantum mechanical simple harmonic oscillator that also has a finite energy when it is unexcited. Therefore, the lowest energy state for a photon field still contains fluctuating fields. Energy cannot be extracted from these fields since they represent the lowest possible energy state but they can stimulate transitions that will add energy to the photon field. As we shall see, these vacuum fluctuations that stimulate transitions play key roles in such processes as stimulated Raman, incoherent Raman, and fluorescence spectroscopies.

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