超快光學(xué)ultrafastoptis題目選擇

1、Theory of Ultrafast SpectroscopyorFeynman Diagrams Made Simplecw monochro-matic beamsdelta-functionpulsesMedium to be studied Nonlinear-Spectroscopic Experiments:Limiting CaseswhereorTime DomainFrequencyDomainorUltrashort laser pulses are an intermediate case.Ultrashort laser pulses are really short

2、, so they appear to be time-domain experiments waiting to happen. But, unlike true d-function pulses, they have finite bandwidth.So they can be resonant or nonresonant. This will be the key.timeUltrashort pulses have large, but finite, bandwidth. So experiments using them can be resonant or nonreson

3、ant.In addition, ultrashort-pulse experiments can be“nearly resonant.” This involves much more complex formulas. We wont treat this case.Also, ultrashort-pulse experiments can be nonresonant for some input pulses and resonant for others. We can treat this case.Ultrafast-Spectroscopy Experiments: An

4、Intermediate Case Nonresonant frequency domain Resonant time domainabababagbQuick quantum-mechanical derivationNonlinear-optical Feynman diagrams in the frequency domaincw experimentsExample and tricksNonlinear-optical Feynman diagrams in the time domaindelta-function-pulse experimentsExample and tr

5、icksFeynman diagrams for experiments with simultaneous time- and frequency-domain characterultrashort-pulse experimentsThe “ultrashort-pulse domain”Examples and tricksFeynman Diagrams Made SimpleA Feynman diagram can be interpreted in the time or frequency domains.Time domainFrequency domaindelta-fu

6、nction pulse inputs at times, t1, t2, and t3 cw beam inputs of frequencies, Each diagram corresponds to a term in a complex sum. We use such diagrams because theyre easier to remember than the actual equation.In both domains, the particular ordering of the pulse times or beam frequencies is referred

7、 to as a time-ordering.Many time-orderings contribute to the total response/susceptibility.a, b, g, and drepresent states.A Sneak Preview of Feynman Diagramsdagabt3t1t2dagabw3w1w2Treat the medium quantum-mechanically and the light classically.Assume negligible transfer of population due to the light

8、.Assume that collisions are very frequent, but very weak: they yield exponential decay of any coherenceUse the density matrix to describe the system.Effects that are not included in this approach: saturation, population of other states by spontaneous emission, photon statistics.Semiclassical Nonline

9、ar-OpticalPerturbation TheoryIf the state of a single two-level atom is:The density matrix, rij(t), is defined as:When laser beams with different k-vectors excite the atom, rij(t) tends to have a spatially sinusoidal variation.A grating is said to exist if aa(t) or bb(t) is spatially sinusoidal,A co

10、herence is said to exist if rab(t) or rba(t) is spatially sinusoidal.The density matrix abwaa(t) or bb(t) are the population densities of states a and b. For a many-atom system, the density matrix, rij(t), is defined as:where the sums are over all atoms or molecules in the system.The density matrixS

11、implifying:The diagonal elements (gratings) are always positive, while the off-diagonal elements (coherences) can be negative or even complex. So cancellations can occur in coherences.Why do coherences decay?A coherence is the sum over all the atoms in the medium. The collisions dephasethe emission,

12、 causingcancellation of the total emitted light, typically exponentially.Grating and coherence decay: T1 and T2A grating or coherence decays as excited states decay back to ground.A coherence can also cancel out if each atom has different phase.The time-scales for these decays to occur are:Grating r

13、aa(t) or rbb(t): T1 “relaxation time”Coherence rab(t) or rba(t): T2 “dephasing time”The measurement of these times is the goal of much of nonlinear spectroscopy!Collisions dephase; so, except in dilute gases, T2 T1.The Liouville equation for the density matrix is: (in the interaction picture)which c

14、an be formally integrated:which can be solved iteratively:Note that i.e., a “time ordering.”Nonlinear-Optical Perturbation TheoryExpand the commutators in the integrand:Consider, for example, n = 2:Thus, contains 2n terms.Perturbation Theory (continued)Now, V is the perturbation potential energy due

15、 to the lightand is of the form, , where E is the total light electric field.But V is in the interaction picture, so we have:where:Note that U(t) U*(t) = U(t-t) is also in the interaction picture:Dividing out these U(t)s yields:Notice that time propagates from to to t along two different paths.Pertu

16、rbation Theory (continued) So a typical term looks like: So a typical term (in second order) is:But, in nth order, the E-field is typically the sum of n input light fields:As a result, each of the above type of terms expands into many terms.Allowing each field to occur only once yields n! as many mo

17、re.Thus, in nth order, there are 2nn! terms!Perturbation Theory (continued) Use diagrams!Consider two input beams and this second-order term, noting that time propagates from t0 to t along two paths:How do we remember all these terms?timeNow expand in terms of the atomic eigenstates:For our second-o

18、rder term, for example:Computation of the number of terms now is an exercise left to the studentPerturbation Theory (contd)we find:w1w2Doing the integralsNow, to go further, well consider limiting cases.Dipole moment matrix elements at the ith beam polarization Set t0 = 0 The Frequency Domain: cw be

19、amsBefore we evaluate these integrals, we must include dephasing.Every time a transition frequency occurs, we must subtract off the dephasing rate for that transition. This is the usual method for adding width to a transition. Thus:Including dephasingw1w2This addition comes from a complex analysis t

20、hat takes into account collisions.Now we can do the integrals in the various cases.Evaluating a single second-order term for monochromatic fields yields:and where:whereThe factor of is the population density of the initial state.The factors of are dipole-moment matrix elements between the states a a

21、nd b for the polarization of beam k.The denominators contain the line shape-the dynamical information.The Frequency DomainThe (-1) occurs in terms with an odd number of V(t)s to the right ofw1w2Draw two vertical line segments.2.Draw a rightward-pointing diagonal arrow for each input field. Upward-po

22、inting arrows correspond to absorbed photons, and downward-pointing photons correspond to emitted photons. Choose an ordering for these “interactions,” and also choose which side each should appear on. Label each interaction with a light frequency. 3. Write in states (a, b, c, d, e) at the base and

23、just above each interaction.Every possible diagram of this form corresponds to a term in the expression for c(n), where n is the number of interactions.This diagram corresponds to:Drawing Feynman Diagrams in the cw LimitDiagramInclude a factor of 1 if there are an odd number of interactions on the r

24、ight:Include a factor of the initial population density of the state at the base of the diagram:Drawing Feynman Diagrams in the cw LimitPiece of diagramAt each interaction, we write down a dipole-moment matrix element:“(1)” means “for thepolarization of beam 1”After each interaction (reading upward)

25、, we write a resonant denominator of the form:whereDrawing Feynman Diagrams in the cw LimitDiagramThe contribution to c(n) is the product of all factors shown below:Resonant denominator:Resonant denominator:Resonant denominator:Resonant denominator:Matrix elements:The population density of the state

26、 at the base:Two interactions on the right (a factor of 1 for each): (-1)2Interpreting Feynman Diagrams in the cw LimitLinear optical problems involve only one photon:Resonance frequency LightfrequencyDephasing timeThis is just the well-known complex Lorentzian line shape, whoseeven (imaginary) comp

27、onent is the absorption coefficient and whose odd(real) component is the refractive index.Example: Linear OpticsThe Absorption Coefficient and Refractive Index vs. FrequencyFirst consider the process, and include only the most resonant, and hence strongest, terms.For example, consider difference-fre

28、quencygeneration,Maximally resonant denominators:Anti-resonant denominators:How do you know which diagrams to include?w1w212-wave mixingSignal frequency: Example: Higher-Order Wave MixingA 12-Wave-Mixing Feynman DiagramAll denomi-nators are maximally resonant.Unfortunately, there are about 10,000 mo

29、re such diagrams to considerNow, suppose that the input light is a sum of delta-function pulses.The relevant variables arenow the pulse relativedelays, We can now write Feynman diagrams for this class of processes, but wemust label the interactions with times, rather than frequencies.Every possible

30、diagram of this formcorresponds to a term in the expressionfor the response, R(n), where n is the number of interactions.Drawing Feynman Diagrams in the Time Domaintimet2t1tt1t2t4t3The Time Domain: d-function pulsesNote that the result is a product of propagators.The integrals are now even easier.As

31、 before, include a factor of 1 if there are an odd number of interactions on the right:As before, include a factor of the initial populationdensity of the state at the base of the diagram.Also, as before, at each interaction, we write downa dipole-moment matrix element:Instead of resonant denominato

32、rs, we write simple exponential propagators:Interpreting Time-Domain Feynman diagramst1t2t4t3As before, linear optical problems involveonly one photon and use the same diagram:where . Dropping the subscript, 1: Resonance frequencyDephasing timeThis is just the well-known fact that the molecules osci

33、llate at their own frequency, emitting “free-induction decay,” and dephase exponentially. The Fourier transform of this response is the complex Lorentzian line shape, whose even component is the absorption coefficient and whose odd component is the refractive index.Example: The Linear ResponseExcita

34、tion pulseProbe pulseObserve change in probe-pulse energy vs. delay,The observed signal vs. delayis complex, with threecomponents:Example: The Excite-Probe ExperimentPhoton echo(PFID)Coherence spikeExcited-State decayDelay, The excite-probe experiment is a third-order process,with the excitation pul

35、se providing two photons:The Excite-Probe Experiment (contd)Photon echo(PFID)Coherence spikeExcited-state decayDelay, Only state a is populated initially.SignalpulseProbepulseExcitePulse(s)Ultrashort pulses have finite bandwidth and finite pulse length. Can we define Feynman diagrams for nonlinear-o

36、ptical experiments with them?All the integralsare of the form:For ultrashort pulses, two important cases yield simple results.Yes!The intermediate domain: ultrashort pulsesCase 1. Resonant excitation: Set E(t) = time domainCase 2. Nonresonant excitation: Set E(t) = constant frequency domainThe Ultra

37、short-Pulse DomainPurely resonant ultrashort-pulse experiments are pure time-domainexperiments, and we use the time-domain Feynman diagrams.Purely nonresonant ultrashort-pulse experiments are pure frequency-domain experiments, and we use the frequency-domain Feynman diagrams.What about experiments t

38、hat are resonant at some steps and nonresonant at others?Nonresonant steps,So label with frequenciesResonant steps, solabel with timesThe Ultrashort Pulse Domain (contd)Yes!Doing the integrals, we see that time-domain steps yield the same factors,But frequency-domain (nonresonant) steps yield slightly different denominators (we must take into account the existing coherence from the previous time