OBE solutions in the rotating frame

Transcription

1 OBE solutions in the rotating frame The light interaction with the 2-level system is VV iiiiii = μμ EE, where μμ is the dipole moment μμ 11 = 0 and μμ 22 = 0 because of parity. Therefore, light does not couple to ρρ 22 or ρρ 11 (the populations in the excited and ground states) directly. In contrast, μμ 12 = 1 μμ 2 = 1 eerr 2 0. This matrix element oscillates at a frequency ωω 0 = ωω 2 ωω 1 even when it starts at zero (we get a superposition with time). The wave vector EE (t) is also rotating at the wave frequency ωω If ωω 0 ωω (resonance) the two vectors are aligned for some time, which gives a non-zero light matter interaction, and transitions from the lower to the upper level. When μμ and EE drift out of alignment, we have removal of energy from the system and stimulated transitions to the lower level. These are the Rabi oscillations. It is difficult to picture the time-evolution of these vectors, rotating at frequencies ωω 0 and ωω Going to the rotating frame (which rotates at the frequency ωω) removes the rotation of vector EE tt and replaces the rotation at ωω 0 of the vector P tt with a slower rotation at ωω 0 ωω

2 The mathematical correspondence between the time-evolution of a spin ½ in a variable external magnetic field and the time-evolution of PP tt for a 2-level system in a variable external electric field allowed us to apply the Pauli matrices σσ ii and gave an expression for the components PP xx,yy,zz in terms of components of the density matrix ρρ iiii The components of PP tt in the rotating frame are uu, vv, ww and are related to PP xx,yy,zz as ρρ 22 ρρ 11 = PP zz ww ρρ 12 + ρρ 21 = PP xx uu (the system response in-phase with EE ) ii ρρ 12 ρρ 21 = PP yy vv (the system response in-quadrature with EE ) The vector EE is stationary and oriented along uu Side note: vv describes absorption because Absorption dddddddddddddd dddd EEEEEE dddd and the time-derivative shifts the phase by ππ 2. With the neglect of terms rotating at 2ωω (the Rotating Wave Approximation), the OBE in the rotating frame in the new variables have the form uu = ωω 0 ωω vv uu TT 2 vv = ωω 0 ωω uu + kkee 0 tt ww vv TT 2 ww = kkee 0 tt vv ww ww eeee ττ Bloch vectors in the rotating frame Bloch vectors in the rotating frame are a useful representation of the time-evolution of a 2-level system polarization vector PP tt during the system s interaction with a wave EE tt Side note: Bloch vectors evolve on the unit sphere. To show this, take a pure state for simplicity. Then, uu 2 + vv 2 + ww 2 = cc 2 2 cc cc 1 cc 2 + cc 1 cc 2 cc. cc. + cc 1 cc 2 cc 1 cc 2 cc. cc. = 1 uu ww PP ww = 1 when all atoms are in the excited state vv ww = 1 when all atoms are in the ground state The Bloch vector starts in the ground state and evolves with its tip on the sphere surface.

3 Example The Rabi oscillation on resonance (ωω = ωω 0 Δωωωω = 0) continuously rotates the Bloch vector with its tip on the sphere along the longitude in the ww vv plane. On resonance, there is no rotation around the ww axis. If the resonance condition is not satisfied, the damping is relatively small, and we have a CW wave, the Bloch vector starts at the South pole and oscillates in latitude around the equator (the optical nutation). If the wave is a pulsed field of area 2ππ there is only one full rotation on resonance and more limited motions off-resonance as shown in the figure. [1968 McCall] Evolution of the Bloch vector under excitation by 2ππ pulses for various damping parameters ττ (the position of the poles is inverted).

4 Rate equations Rate equations are an intuitive model of a time-dependent process The standard rate equation is ddnn 2 = RR nn dddd 2 nn 1 nn 2 = ddnn 1 ττ dddd The term RR nn 2 nn 1 > 0 and increases nn 2, the term nn 2 < 0 and decreases nn ττ 2 ddnn 2 = ddnn 1 when nn dddd dddd 1 + nn 2 = cccccccccc. [the detailed balancing of the two-level system in this case] The rate equation can be recast in terms of the inversion ww = nn 2 nn 1 as dddd = 2dddd 2 = 2RRRR nn 2 nn 1 +nn 2 +nn 1 = 2RRRR ww+1 dddd dddd ττ ττ Because the term 2RRww depends on the inversion ww, rate equations can describe saturation effects, just like the steady-state OBE solution (previous lecture) Rate equation are applied in the derivation of Einstein s coefficients AA 21, BB 21, BB 12 and they work for lasers above threshold Advantages: simpler than OBE Disadvantages: these expressions neglect coherence and all factors depending on the phases Expressions for transition rates RR are part of all graduate-level quantum mechanics textbooks For a 2-level system we have cc 1 nn 2 2ππ VV 2 ħ nnnn ρρ EEnn tt tt (1 st order perturbation theory) Then, dd 1 cc 2 dddd nn nn = ww ii [nn] = 2ππ VV 2 ħ nnnn ρρ EEnn (Fermi s golden rule) When VV tt = VVee iiωωωω + VV + ee iiωωωω (a light wave) we get cc 1 nn = ii tt VVnnnn tt ddtt = ωω nnnn ωω nnnn + ωω + (ωω nnnn ωω nnnn ωω) The transition rate is ww ii [nn] = 2ππ VV nnnn 2 ρρ EE nn EEnn EE ii ωω (for absorption) and ww ii [nn] = 2ππ VV iiii + 2 ρρ EE nn EEnn EE ii +ωω (for stimulated emission) This gives the detailed balancing EEEEEEEEEEEEEEEE rrrrrrrr ii [nn] DDDDDD aaaa [nn] = 0 AAAAAAAAAAAAAAAAAAAA rrrrrrrr nn [ii] DDDDDD aaaa [ii] This approach is valid if VV iiiiii 1 tt (perturbation theory valid) and 1 t Δνν, where Δνν is the width of DOS ρρ EE nn (replacing final states with smooth DOS) If the RE are the 1 st order perturbation theory solution in QM, can they be obtained in the PT of OBE? The answer is yes. We obtain RE in 2 nd order PT of OBE because the order counting in OBE PT is in powers of EE, while in QM PT it is in powers of EE 2.

5 OBE vs. rate equations and the free induction decay The OBE reduce to the rate equations when TT 2 ττ pp, ττ (Note: there is no condition on the relation between ττ pp and ττ) Proof: when TT 2 ττ, the parameters uu and vv will quickly reach the steady-state To find this state we set uu = 0, vv = 0 or ωω 0 ωω vv uu TT 2 = 0 and ωω 0 ωω uu + kkee 0 tt ww vv TT 2 = 0 This system of equations can be solved as vv = kkee 0 TT 2 1+ ωω 0 ωω 2 TT 2 2 ww and uu = kkee 0 ωω 0 ωω TT 2 With this solution, the 3 rd OBE becomes ww = kk2 EE 0 2 TT 2 1+ ωω 0 ωω 2 TT 2 2 ww ww+1 equivalent to the rate equation with 2R = kk 2 EE 0 2 TT 2 1+ ωω 0 ωω 2 TT 2 2 ττ 2 1+ ωω 0 ωω 2 TT 2 2 ww (with ww eeee = 1), which is As TT 2 decreases, coherence effects diminish and the time evolution can be described with rate equations OBE TT 2 decreases RE Similarly, if we are interested in the time evolution of the light pulse, rather than in that of the 2-level system, it is possible to reduce the Maxwell field equations to a rate equation. This reduction to RE is not always justified. That phases can continue to evolve coherently in a material long after material stopped radiating (as would be represented by a term ww+1 in the rate equations) has been experimentally observed in free induction decay ττ The material stops radiating long before the dipole oscillation itself is damped because individual dipoles drift out of phase. This shows that the dephasing time can be much longer than the apparent decay time (the dipoles have not decayed to the ground state, they only stopped radiating, as if they did). The energy continues to be stored in the material system and can be recovered, as demonstrated by such effects as the photon echo. The radiative efficiency is related to the mixed ensemble ρρ 21 (tt). FID shows how its time-evolution can be very different from the much longer decay of cc 1 tt cc 2 tt of each individual atom. This is one example of how ρρ 21 cc 1 cc 2 in mixed ensembles. FID opened to door to the many coherent effects that require OBE.

6 A review of pump-probe experiments Non-resonant experiments ( optics ): ultrafast diffraction, melting, etc. Resonant experiments ( spectroscopy ): atomic physics, phonons, excitons in dielectrics or semiconductors, electrons in metals, etc. A strong pump pulse starts the time-evolution. A delayed weak probe pulse interacts with the system in a way that depends on the orientation of the polarization vector PP(tt). Usually the pump and probe pulses have similar durations and they are both modelled with either OBE or RE depending on how ττ pppppppp, ττ pppppppppp compare to the material dephasing time TT 2. Typically, the goal is to obtain the dephasing time TT 2 and decay time ττ from the measurements. For instance, we may look at the damping of the oscillations (dephasing time) or of the average trace (decay time). It is not always best to use the shortest pulse because a shorter pulse has a larger BW. The relatively small energy of the TiS (1.5 ev) can be increased with high-harmonic generation giving 2ωω, 3ωω, etc. X-ray fs pulses can be generated with a strong laser pulse on a moving target. The SLAC facility also outputs fs X-ray pulses. On the left is a sketch of a pump-probe setup Next are several examples of pumpprobe experiments.

7 Transient absorption and reflectivity and coherent phonons If the change in transmission is measured ΔTT TT = ee Δαααα 1 Δαααα SSSSSSSSSSSS Beam-splitter Lens Probe Detector Sample Pump beam Delay stage ωω aa,pppppppp ωω bb,pppppppp [1997Merlin] Optical phonons have a wavelength comparable to the unit cell of the lattice, and correspondingly a frequency in the TTTTTT range. Acoustic phonons have a much smaller frequency and a larger wavelength. A transient absorption or reflectivity signal may be due to an incoherent process that can be modelled with rate equations (e.g. fluorescence) or a coherent process (e.g. Raman, which also changes the incident frequency) It is called stimulated when both photons at ωω aa,bb come from the same pulse, which is possible if the pulse has a broad spectrum. When ττ pp < TT ppppppppppp it is called Impulsive stimulated Raman scattering. 2 The missing energy ωω aa,pppppppp ωω bb,pppppppp is left in the material and excites a phonon wave which can be detected with a probe pulse.

8 Time-resolved MOKE in insulators and half-metals Transmission (Faraday) and reflection (Kerr) of light off a magnetic material gives a rotation of polarization and light ellipticity. When the result of the light-matter interaction is a flip of electron spins, it can be viewed as a stimulated Raman process and a generation of magnons (spin-density waves), just like generation of coherent phonons. The time-evolution can be described in a process in which the pump pulse modifies the local effective magnetic field. The spins precess under the torque of this new field, a precession that can be measured with timeresolved Faraday rotation, where a delayed probe pulse detects the position of the spins as a function of time delay. Magnetic semiconductors and insulators have relatively long dephasing times and give strong coherent signals. [2009Kimel] for HoFeO3 [2010Liu] for different orientations of Co2MnSi halfmetal

9 Time-resolved 2-photon photoemission in metals Experiments on metals face the additional experimental difficulty of a short absorption length. Large number of interacting particles must also be considered in the models. Because of the very fast dephasing that results, coherent effects can be seen only when the electron wave functions are relatively decoupled from the environment. Vacuum level Probe photon Pump photon Real or virtual intermediate state [1996Hertel] and the lifetime of electrons above the Fermi level [2000Fauster] and coherent effects in image states

10 Ultrafast demagnetization and all-optical switching in metals Several unexpected results were obtained recently in measurements of the magnetization time evolution in metals Ultrafast demagnetization and all-optical switching in GdFeCo [2014Hashimoto] (AOS considered impossible before 2007) Ultrafast demagnetization in Ni [2012Roth] (considered impossible before 1996) Calculation of the magnetization time evolution during AOS [2014Oniciuc] Note: here is plotted the time evolution of MM tt for the transition metal and rare-earth atoms in the magnetic alloy GdFeCo, not the Bloch vector PP tt