Modern Spectroscopy: a graduate course. Chapter 1. Light-Matter Interaction

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1 Modern Spectroscopy: a graduate course. Chapter. Light-Matter Interaction. Semiclassical description o the light-matter eraction. Generally speaking, in spectroscopy we need to describe the light and matter as one complete system. Fully quantum description thereore would start with writing down a Hamiltonian or this system, which we can partition o the Hamiltonian or the material system, Hamiltonian or the light ield, and the light-matter eraction Hamiltonian: (.) H ˆ H ˆ ˆ ˆ M HL H Solving the ull quantum problem would mean solving the coupled equations o motion o quantum electrodynamics (QED) or the light ield and quantum dynamics (e.g., time-dependent Schrodinger equation) or the material system. This is generally a diicult task even or simple systems. In this course, we will adopt instead the semiclassical description o the lightmatter eraction, which obeys much simpler equations. That is, the light will be treated as classical electromagnetic (EM) ield described by Maxwell s equations (instead o the QED equations), while the matter will be described by quantum mechanics. Moreover, this will allow another simpliication: instead o having to simultaneously solve the coupled equations o motion or the light and matter variables (e.g., Maxwell equations coupled with the time-dependent Schrodinger equation or with quantum Liouville-von Neumann equation), we can break up the light-matter eractions o one-way steps: a Step. The light perturbs the quantum dynamics o the system such that the lightmatter eraction serves as perturbation to the material Hamiltonian (.) H ˆ H ˆ ˆ M H At this po, only the material EOMs need to be solved, we do not care about the changes in the state o the light resulting rom the eraction. The light aects the material system, but the material system does not aect the light. a This is because in the semiclassical description, we replace the quantum operators representing the light ield observables (e.g., its E-ield o vector potential vide inra) in H ˆ L and Ĥ o the total Hamiltonian H ˆ H ˆ H ˆ H ˆ with their expectation values, M L ˆ ˆ E( r, t) E( r, t) E( r, t), ˆ ˆ A( r, t) A( r, t) A( r, t), etc. E( r, t ) and A( r, t ) are numbers, not operators, and thereore commute with other operators in Ĥ. This allows to separate out the ield variables rom the EOM or the material system.

2 We go ahead and solve the quantum dynamics o the material system under Hˆ Hˆ ˆ M H (note that since light is an oscillatory EM ield, Ĥ is time-dependent). This will allow us to calculate (time-dependent) expectation values o physical observables characterizing motion o charged particles in the material, e.g. polarization. Step. Using the quantum dynamics o the system calculated in Step, we calculate time-dependent charge densities and currents in the material, plug them o the Maxwell equations as source terms, and calculate the EM ield radiated by the moving charges in the material. This is our spectroscopic signal. Scheme. A. Full quantum description o the light and matter. B. Semiclassical two-step description o the light-matter system. As any approximation scheme, the semi-classical description o the light-matter eraction has advantages and disadvantages/limitations. Pros:. The semiclassical equations o motion are much easier to solve than the ully quantum treatment.. Classical description o the EM ield is naturally connected to the conventional experimental methods or characterization/detection o the light signals: measurements o light ensity, requency, polarization state, pulse width, amplitude/phase, etc. 3. The semiclassical descrtiption allows a physically uitive picture o how the physical characteristics o the incoming light ield aect the quantum dynamics o the

3 system. This will in turn lead to an understanding o how to design a spectroscopic measurement to probe a particular aspect o the dynamics (e.g., how to use time delays between short laser pulses to measure relaxation phenomena, how to use polarization to measure molecule s orientation and rotational motion, etc.) 4. Multipolar expansion o the classical EM ield matter eraction leads to a convenient classiication o the eractions o electric dipole, magnetic dipole, electric quadrupole, etc., in order o diminishing eraction strength, and convenient (and physically transparent) selection rules based on symmetry. 5. As we shall see, the semiclassical ield-dipole eraction allows perturbative expansion o the quantum dynamics o the material system that naturally yields a hierarchy o linear and nonlinear optical processes, and shows what type o dynamics contributes to a particular order nonlinear signal. Cons:. Because we break the system-light eractions o two steps, the concept o the energy conservation does not naturally arise orm the semiclassical equations. Energy conservation is something that needs to be manually enorced in this treatment. For example, in Step, the energy o the material system will not be conserved since it may gain/lose energy rom/to the light ield, but we do not treat the state o the light in this step, so we do not account where the energy comes rom. It is possible to construct the theory to be consistent with the energy conservation, as it should, o course (e.g., we would get the missing energy back o the light ield in Step ), but it needs to be done by hand. (In a ully quantum picture, we would say that the light-matter eraction leads to annihilation/creation o a photon o the light ield, and corresponding raising/lowering o the quantum state o the material system, so the energy conservation would naturally arise rom the coupled light-matter equations o motion).. Classical description o the light ield is valid only or high photon population numbers, i.e. or light o moderate ensity. For extremely low light ensities such as or single-photon spectroscopy, quantum eects such as photon statistics, quantum noise, and photon entanglement become important and require ully quantum treatment. These eects are important in single-molecule spectroscopy and quantum optics, both emerging directions in molecular spectroscopy. 3. Separating the light variables rom the system EOMs leaves out some retardation eects (that is, dierent parts o the material system eract with the light at dierent moments in time because o the inite speed o light). Such eects arise in extended chromophore systems such as molecular aggregates and assemblies, and crystals. There are ways to included these eects in a more rigorous version o the semiclassical treatment, but they will not be considered in this course (See S. Mukamel, Ch. 6, 7) 3

4 . Classical EM ield. We begin with a brie review o the classical electrodynamics (it is recommended that you read a proper EM text or a complete coverage o this material e.g., Classical Electrodynamics by J. D. Jackson). The classical EM ield is completely described by two vector ields, the electric ield E( r, t ) and the magnetic ield B( r, t ), which obey the Maxwell s equations (here and below, we use the SI units) (.3.a) E Gauss s Law (.3.b) B Absence o magnetic monopoles (.3.c) B E Faraday s Law t E (.3.d) B J t Ampere s Law with Maxwell s correction where ( r) is the total charge density (ree and bound), Jr ( ) is the total current density (ree and bound). ε and μ are the electric permittivity and the magnetic permeability o vacuum. Note that the speed o light is c. Since E( r, t ) and B( r, t ) are not independent (connected by the Maxwell s equations), speciying all 6 vector components o the electric and magnetic ield overdetermines the state o the EM ield. To (partially) alleviate this, we roduce the vector potential A( r, t ) and the scalar potential ( rt, ) (a total o 4 parameters: A x, A y, A z, and Φ) such that (.4) A( r, t) E( r, t) ( r, t) t B( r, t) A( r, t) where c is the speed o light. (Note that writing the magnetic ield B( r, t ) as a curl o a vector ield A( r, t ) automatically satisies the second Maxwell s equation B, since A or any A( r, t ) ). However, speciying the 4 components o the potential is still overdetermined, and thus the vector and scalar potentials are not uniquely deined. Indeed, i we add gradient o an arbitrary unction ( rt, ) to A( r, t ) and at the same time subtract its time derivative orm ( rt, ), 4

5 (.5) A' A, ' t the electric and magnetic ields as deined by (.4) will remain the same. Indeed, A' A A E' ' E t t t t t B' A'( r, t) A( r, t) A( r, t) B (since or any ). The choice o ( rt, ) allows on to simpliy the EM equations or a particular situation. This is known as gauge ixing. For example, we can choose ( rt, ) such that (.6) A This is known as the Coulomb gauge (also known as the radiation or transverse gauge). As we demonstrate below, in vacuum (in the absence o charges and currents), this choice o gauge leads to only transverse electric and magnetic ields o the plane EM waves. Also note that Coulomb gauge implies that in the absence o charges, ( rt, ) (rom the Gauss s law, Eq. (.3.a)). (.6) For EM ield in vacuum, assuming Coulomb gauge, Eqs. (.4) become A E t B A The irst three o Maxwell s equations (.3 a-c) are automatically satisied, and substituting (.6) o the last one (.3d): E B c t yields the equation or A( r, t ) : A c A t which, ater transorming A A A A because o the Coulomb gauge choice, becomes the amiliar wave equation (.7) A A c t The solution is o course the transverse plane wave (.8) A r t A e c c i( kr t ) (, ).. 5

6 where c.c. indicate complex conjugate (remember, the ields and potentials are physical observables and thus must have real values!). The wavevector k deines the propagation direction and the wavelength,. The requency ω is connected to the wavevector through the dispersion relation k ck. The (scalar) amplitude o the vector potential is A, and its direction (polarization state) is speciied by a unit vector. The electric and magnetic ields o the plane EM wave calculated according to (.6) are (.9) E( r, t) i A e c. c. E e c. c. i( k r t ) i( k r t ) B( r, t) ( k ) ia e c. c. bb e c. c i( k r t ) i( k r t ) Here, the scalar amplitude and polarization direction o the electric ield are E and, and or the magnetic ield B and b. Note that, due to the Coulomb gauge choice, A i k A e c c, i( kr t ) ( ).. thereore k. That is, the polarization o the vector potential and o the electric ield is perpendicular to the wave propagation direction k (transverse wave). Furthermore, the magnetic ield polarization is b k, i.e. the magnetic ield is perpendicular to both the propagation direction and the electric ield (Fig. ). Lastly, the E E amplitudes o the electric and magnetic ields are related by c, i.e. B, B k c which gives you an idea why magnetic eractions are much weaker than the electric eractions. Figure..3 Charged particle eracting with EM ield. A particle with charge e moving with velocity v in an EM ield is subject to Lorentz orce (again, written in SI units) (.) F e( E v B) 6

7 Ater a somewhat lengthy derivation (given, e.g., in W.S. Struve), it can be shown that the eect o the Lorentz orce can be ully described by replacing the particle s momentum in the equations o motion with (.) p p' p ea This is true or both classical equations o motion and or quantum mechanics where the momentum is an operator ˆp i. In the EM ield, the momentum operator is replaced by p ˆ ' i ea. is (.) Thus, i the Hamiltonian or a particle o mass m moving in some potential Vr ( ) ˆ ˆ p H V ( r), m applying EM ield with vector potential A( r, t ) will change the Hamiltonian to (.3) pˆ ea Hˆ ield on V ( r) Hˆ ˆ H. m (Once again, we use Coulomb gauge, so ( rt, ) or the light ield and thus we do not have the e term due to the EM ield. There may be other electric charges aecting the motion o our particle, their potential is included in Vr. ( ) We are only erested in the eect o the EM ield on the motion o the particle). Thus, the particle-ield eraction Hamiltonian can be written as (.4) ˆ ˆ ˆ ield on ˆ ˆ e ˆ ˆ e H H H p ea p p A A p A m m m In this course, we will consider weak ield-matter eractions that can be treated perturbatively. In this case, the last term in (.4), which is quadratic in the ield, is much smaller than the irst two terms that are linear in the ield. We thereore neglect the last term. b Substituting ˆp ˆ i e H A A m i o (.4), Note that, in general, ˆp does not commute with A. However, working in the Coulomb gauge, we can easily prove (Homework problem) that A A, and thus b This can be done or most o the realistic cases encountered in spectroscopy, with the exception o strong laser ield-matter eractions when the eraction energy becomes comparable to the ernal energy o atoms/molecules, i.e. when the electric ield o the light wave is comparable to the ernal electric ield. Such light ensities can be achieved with emtosecond lasers. For example, a emtosecond laser pulse o μj ocused to a 5 μm spot size produces electric ield o. V/Å, i.e. comparable to the raatomic ields. 7

8 (.5) ˆ i e e H ˆ A A p m m It is straightorward to generalize result or a collection o N charged particles: N (.6) ˆ ei H ˆ A pi m i i This is the ield-matter eraction Hamiltonian in the semiclassical approximation..4 The electric dipole approximation In solving our time-dependent quantum dynamics or the system under the perturbed Hamiltonian H ˆ ˆ H, we will need to evaluate the matrix elements o the perturbation operator Ĥ, Hˆ i in some chosen basis set (usually, in the stationary states o the unperturbed Hamiltonian Hˆ n En n ). In order to do this, we need to consider the spatial dependence o (.7) So that (.8) Ĥ. For a plane wave, A( r, t ) is given by Eq. (.8), which we can modiy as i( k r t ) it 3 A( r, t) Ae c. c. Ae ik r ( ik r) ( ik r)... c. c.! 3! ˆ i e H i A i m i e i t A e i ik r i ik r i m! ( ) ( )... Now recall that k, where the wavelength o light λ is o the order o -4 Å or UV, 4-8 Å or the visible, and > 4 Å or IR light. On the other hand, the egration... i is over the region o space where the molecule s waveunction is non-zero, which is about the size o the molecule, - Å. Thus, ( k r) is a small number, o the order o - or less (unless it s a really deep-uv light and a really large molecule!). Thus, the series in powers o ( k r) in Eq. (.8) is rapidly converging. Leaving only the irst ( th -order) term in (.8) is reerred to as the electric dipole approximation (or reasons that will become apparent soon). This yields (.9) ˆ i e dipole i e t i H t ˆ i Ae i Ae p i m m 8

9 The matrix element pˆ i can be evaluated easily by using ˆ m p i Hˆ, r (Homework, Problem ). The result is (.) ˆ dipole i H i E i which is the amiliar expression or the ield-dipole eraction, where er is the it dipole moment operator, the E-ield o the light is E( r, t) Ee c. c, E i A, E E and is the transition requency between the initial state i and the inal i state. For a collection o particles, we obtain the same expression by simply summing over the particles, with er now being the collective dipole moment o the system. i i i Although we will not consider the higher multipole terms in this course, or your reerence, the second term in the expansion (.8) yields the magnetic dipole and electric quadrupole eractions (see derivation in W. S. Struve). Physically, the electric dipole approximation means that we neglect the spatial variation o the EM ield s amplitude over the region o space occupied by the molecule. In other words, we assume that dierent parts o the molecule experience the same electric ield o the light wave. The dipole approximation is thereore also known as the local approximation, meaning that the light eraction with a molecule at position r is determined only by the E-ield at that po in space. This approximation may thus break down or extended systems, e.g. metals, semiconductors, or molecular assemblies, where charge carriers and/or elementary excitations can travel over much larger distances than the typical molecular size. In such systems, nonlocal eractions such as electric quadrupole may become important..5 The transition dipole and selection rules We can also re-write Eq. (.) as (.) ˆ dipole i H i E i As we shall see in Chapter 3, or a system initially existing in a stationary state i and exposed to the perturbation ˆ dipole H, this matrix element determines the probability o the transition to the inal state via the Fermi Golden rule expression, i ˆ dipole P H i We thereore see that the transition probability (or a given light ield strength) is governed by the matrix element 9

10 (.) i i known as the transition dipole moment. Using symmetry considerations, it is sometimes possible to determine that the transition dipole between two states is zero, that is the egral in (.) vanishes, without even perorming the egration. For example, or a one-dimensional case, μ=ex is an odd unction o position. I i and waveunctions are both even or both odd, the egrand in i is odd and the egral vanishes. In these cases, we say that the i transition is orbidden in the electric dipole approximation. I the transition dipole does not strictly vanish by symmetry, the transition is dipole-allowed. Here are some examples o the electric dipole selection rules.. Particle in a box, xa V( x), x, x a Energy levels and eigenstates: nx, sin,,,... ma a a En n n n i or even even and odd odd transitions. (Homework Problem). Harmonic oscillator V ( x) m x i only or =i±, i.e. the selection rule is Δv=±. (Homework Problem) Anharmonic oscillator. Introducing anharmonicity, e.g. V '( x) x x relaxes the harmonic oscillator selection rules and allows overtone transitions Δv=±, etc. (depending on the anharmonicity Homework Problem). e 4. Hydrogen atom Vr () 4 r. The stationary states are spherical harmonics times the radial unctions (associated Laguerre polynomials o r). Stationary states are deined by 3 quantum numbers: n, l, m. For light E-ield polarized along z-axis, * *,, i, i, i sin n cos l l m nili limi n l m n l m d d r drr Y er R Y

11 By representing cosθ in terms o spherical harmonics, the angular part o the egral yields the ollowing selection rules: i only i l m. 5. Gerade-ungerade (g-u) symmetry. For atoms and molecules that have inversion symmetry, their stationary states are classiied as either g (symmetric) or u (antisymmetric) with respect to inversion operation. Because er has u symmetry, the transition dipole vanishes or g g and u u transitions. Only u g and g u transitions are allowed..6 Oscillator strength We can get a sense o the absolute magnitudes o the transition dipoles or dierent systems (e.g., atoms and molecules, electronic, vibrational, or rotational transitions) by comparing them to the transition dipole o one agreed upon reerence system. By convention, such reerence system is the transition o an electron in a 3D harmonic potential. The transition dipole is (.3) 3he 3 8 me (the oscillator orce constant is chosen to give transition requency ν this way we have a consistent reerence or transitions at dierent requencies). This transition is said to have oscillator strength o. Any transition dipole can be compared to μ (calculated at the same requency), and the ratio is called the oscillator strength or that transition: (.4) i i It can be shown that any transition involving one electron has oscillator strength. In act, there is a strict sum rule or single-electron transitions (i.e. transition where only one electron in the system changes its quantum state): the sum o oscillator strengths or transitions originating in one particular state i and ending in all possible inal states is, (.5) i Strong electronic transitions (e.g., chromophore dye molecules) have oscillator strengths close to. Forbidden electronic transitions such as singlet-triplet (e.g., phosphorescence) have oscillator strength - or lower. The transition dipole and the oscillator strength are o course connected to the phenomenological Einstein coeicients or the transition:

12 (.6) B B 4 e mehc A 4 e mehc g 8 e 3 mec g (where g and g are the degeneracies o the two states). g g