Quantum Lattice Contraction Induced by Transient Raman Process
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Quantum Lattice Contraction Induced by Transient RamanProcess
Tomobumi Mishina ∗ Department of Physics, Faculty of Science,Hokkaido University, Sapporo 060-0810, Japan (Dated: April 25, 2018)
Abstract
The lattice contraction that occurs in time resolved x-ray diffraction and electron diffractionexperiments is generally considered to be caused by photogenerated carriers. However, quantumcalculations with finite-time boundary conditions indicate that a transient Raman process directlyconnects optical transitions and lattice displacements. Lattice contraction and coherent phononsare well explained by the Raman process.
Newtonian
Force Classical
MechanicsCoherent Phonon nh W n h W h W ElectronicExcitationStokes Anti-Stokes ? (a)(b) FIG. 1. Fermi’s golden rule applied to the generation of coherent phonons. (a) SpontaneousRaman scattering from phonon of energy ~ Ω cannot excite the coherent phonon state centered atthe n th phonon state. (b) Other theoretical approaches consider Newtonian forces generated byelectronic excitation. i is H ( i ) = 12 m i n ~p i − q i (cid:16) ~A ext ( t ) + ~A i ( ~r i ) (cid:17)o + q i φ i ( ~r i ) , (1)where m i , q i , ~r i , and ~p i are the mass, charge, position and conjugate momentum of particle i , respectively. ~A ext ( t ) describes the uniform long-wavelength laser field. φ i ( ~r i ) and ~A i ( ~r i )are the scalar and vector electromagnetic potentials at the particle position caused by theother charged particles, respectively, and are given by φ i ( ~r i ) = 14 πε X j = i q j | ~r i − ~r j | , (2) ~A i ( ~r i ) = 14 πε c X j = i q j | ~r i − ~r j | d ~r j d t , (3)where ε and c are the dielectric constant and the speed of light in vacuum, respectively.The Hamiltonian (1) may be rewritten as H ( i ) = H ( i )0 + H ( i )d + H ( i )a , (4) H ( i )0 = 12 m i ~p i + X j = i πε q i q j | ~r i − ~r j | , (5) H ( i )d = − q i m i ~p i · ~A ext ( t ) + q i m i ~A ( t ) , (6)3 b) (c) A ext D tm el q el = m nuc q nuc A ext D t = A ext (a) m nuc q nuc A nuc D t m el q el A el D t D r el D r nuc D r el D r nuc FIG. 2. Electromagnetic interaction of a crystal lattice with a laser field. (a) Schematic drawingof lattice composed of nuclei and electrons. (b) Antiphase motion of a nucleus and electron causedby direct interaction with external vector potential. (c) Additional interaction enhances the latticedisplacement. H ( i )a = − q i m i (cid:16) ~p i − q i ~A ext ( t ) (cid:17) · ~A i ( ~r i ) + q i m i ~A i ( ~r i ) . (7)Equations (5)-(7) correspond to the free, the direct interaction, and the additional interactionHamiltonians, respectively. The equations of motion involving ~r i and ~p i for each chargedparticles are d ~r i d t = ~p i m i − q i m i ~A ext ( t ) − m i c X j = i q i q j πε | ~r i − ~r j | d ~r j d t , (8)d ~p i d t = − X j = i (cid:18) − c d ~r i d t · d ~r j d t (cid:19) ∂∂~r i q i q j πε | ~r i − ~r j | . (9)The second term on the right-hand side of Eq. (8) comes from the direct interaction Hamilto-nian (6) and causes the antiphase motion of electrons and nuclei at the optical frequency, asshown in Fig. 2(b). Conversely, the third term on the right-hand side of Eq. (8) comes fromEq. (7) and is responsible for the positive feedback between the electron and the nucleus, asshown in Fig. 2(c). This term enhances the lattice displacement. The third term gives the1/r long-range interaction network between the many charged particles, including the coreelectrons and the nuclei. The problem is how to quantize this many-body interaction.In quantum mechanics, ”divergence” occurs because of the uncertainty relation. TheFGR approximation [2] is the simplest and strongest way to exclude various processes notinvolved in actual observations by isolating only resonant transitions between eigenstates4ver infinite time. However, the rule threatens to exclude processes that can actually beobserved. Given the boundary conditions of finite time and space, ”divergence” causesdifferent types of processes [13–15]. The quantization of the electromagnetic Hamiltonianwith the boundary condition of finite space leads to the Aharonov-Bohm effect [16]. Theoptical transition is also described by the vector potential, as in Eq. (6). By using thecommutator relation between the coordinates and the free Hamiltonian, the amplitude ofthe optical transition between two energy eigenstates | E i > and | E f > may be rewritten as D E f (cid:12)(cid:12)(cid:12) qm ~p · ~A ext (cid:12)(cid:12)(cid:12) E i E = ω fi ω D E f (cid:12)(cid:12)(cid:12) q~r · ~E ext (cid:12)(cid:12)(cid:12) E i E , (10)where ω fi and ω are the transition and optical frequencies, respectively, and ~E ext = − ∂ ~A ext /∂t is the corresponding electric field. In the FGR approximation, the momentum interaction isreplaced with the electric-dipole interaction because the amplitudes are identical in resonanceconditions. However, this assumption does not hold for many nonresonant transitions, whichoccur when time is finite [17]. The vector potential causes the quantum motion of thenuclei through the momentum operators without changing the total momentum, as shownin Fig. 2(c).Next we derive the effective Hamiltonian for the generation of coherent phonons. Thelattice Hamiltonian derived from Eq. (5) determines the energy eigenstates of the latticesystem, including the ground state, the electronic excited states, and the number of phononstates. The free Hamiltonian for the phonon mode isˆ H = ˆ P M + M Ω R , (11)where ˆ R , ˆ P , M , and Ω are the coordinate operator, conjugate-momentum operator, mass,and phonon energy, respectively. ˆ R is the linear combination of the displacements of nucleifrom their equilibrium positions. The interaction Hamiltonian derived from Eqs. (6) and (7)gives the dielectric energy of the lattice system. After averaging over an optical cycle, thecorresponding complex polarizability ˆ χ expressed in terms of phonon operators isˆ χ ( ˆ R, ˆ P ) = χ + ∂ ˜ χ∂R ˆ R + ∂ ˜ χ∂P ˆ P . (12)ˆ χ is also a function of the laser frequency ω and contains the effects of all optical transi-tions. The operator χ is responsible for the generating photoexcited charge carriers. Theinteraction Hamiltonian for the phonon mode isˆ V ( t ) = − (cid:18) ∂ ˜ χ Re ∂R ˆ R + ∂ ˜ χ Re ∂P ˆ P (cid:19) E ( t ) . (13)5f we define the phonon-annihilation operatorˆ a = r M Ω2 ~ ˆ R + i r M ~ Ω ˆ
P , (14)then the second-quantized effective Hamiltonian isˆ H eff ( t ) = ~ Ω2 (cid:0) ˆ a ˆ a † + ˆ a † ˆ a (cid:1) − (cid:0) Ξˆ a + Ξ ∗ ˆ a † (cid:1) E ( t ) , (15)where the complex coefficient isΞ = r ~ M Ω ∂ ˜ χ Re ∂R − i √ M ~ Ω ∂ ˜ χ Re ∂P = d ˜ χda . (16)Considering ˜ χ to be an analytic function of a , Eq. (16) can be thought of as a complexderivative, so the Cauchy-Riemann equations give ∂ ˜ χ Re ∂R = M Ω ∂ ˜ χ Im ∂P , ∂ ˜ χ Re ∂P = − M Ω ∂ ˜ χ Im ∂R . (17)In the interaction picture, the time evolution of the phonon wave function by the Schr¨odingerequation gives Ψ( t ) = T exp (cid:18) ~ Z tt ˆ V I ( t ′ ) E ( t ′ ) dt ′ (cid:19) Ψ( t ) , (18)where T is the time-ordering product andˆ V I ( t ) = e − ˆ H t − t ~ ˆ V ( t ) e ˆ H t − t ~ . (19)In the steady state, the perturbation integral averages to zero. If the interaction time t − t is less than 1 / Ω, the time evolution may be approximated asΨ( t ) ≈ exp (cid:18) − Ξˆ a + Ξ ∗ ˆ a † ~ Z tt E ( t ′ ) dt ′ (cid:19) Ψ( t ) . (20)The unitary operator in Eq. (20) corresponds to the displacement operator [7]. The coherentstate is generated by applying the operator to the vacuum state, which causes the transitionsfrom the many-number phonon states of the initial state to those of the final state, as shownin Fig. 3(a). The amplitude of each transition is very small but their sum could be very large.To evaluate the vibrational energy of the phonon mode, we consider the Heisenberg operatorand its expectation value on the ground state. Because the time derivative is obtained fromcommutation with the total Hamiltonian, the rate of the vibrational energy isd D (cid:12)(cid:12)(cid:12) ˆ H (cid:12)(cid:12)(cid:12) E d t = Ω (cid:18) ∂ ˜ χ Im ∂P P ( t ) + ∂ ˜ χ Im ∂R R ( t ) (cid:19) E ( t ) . (21)6 W h w (a) V i b r a ti on a l E n e r gy Optical Fluence (b)
InitialFinal
Number States
FIG. 3. (a) Optical transition and associated action of displacement operator on phonon-numberstates. (b) Vibrational energy of phonon as a function of optical fluence.
Equations (17) are used to simplify Eq. (21). The equation shows that the increase invibrational energy is supplied by the photon absorption induced by the lattice displacement.The absorption is essential to generating coherent phonons and was already reported in earlyexperiments in coherent phonons [3]. The amplitude of the coherent phonon is proportionalto the optical fluence and the vibrational energy of the phonon is proportional to the squareof the amplitude, so the vibrational energy is proportional to the square of the fluence, asshown in Fig. 3(b). This dependence of the vibrational energy on the optical fluence is fullyexplained by the induced absorption. Comparing Eq. (21) with the corresponding classicaloptical absorption gives the relationΩ (cid:18) ∂ ˜ χ Im ∂P P + ∂ ˜ χ Im ∂R R (cid:19) = ω (cid:18) ∂χ Im ∂P P + ∂χ Im ∂R R (cid:19) . (22)The feedback process shown in Fig. 3(a) between optical absorption and lattice displace-ment is neglected in the classical theory and is indicated by the factor ω/ Ω. By using theHeisenberg operator, the differential equation of the expectation value of the annihilationoperator is (cid:18) dd t + i Ω (cid:19) a = − Ξ ∗ ~ E ( t ) . (23)Integrating this equation from the equilibrium state gives a ( t ) = − Ξ ∗ ~ Z t E ( t ′ ) e iΩ( t − t ′ ) dt ′ . (24)Applying Eqs. (14), (16), (17), and (22) to Eq. (24) and using the impulsive condition7 d (a) (b) FIG. 4. (a) The ideal lattice consists of atomic layers with cross sectional area S and latticeconstant d . (b) Photoexcitation causes the quantum lattice to contract without the photogenerationof carriers. t < / Ω, the momentum P and displacement R are approximated as P ( t ) ≈ ∂χ Re ∂R ω Ω Z t E ( t ′ ) dt ′ ,R ( t ) ≈ ∂χ Im ∂R ωM Ω Z t E ( t ′ ) dt ′ , (25)where ∂χ Re /∂R and ∂χ Im /∂R are the real and imaginary parts of the classical Ramantensor. These parts correspond to the processes formerly assigned to ISRS [4] and DECP[9], respectively.Finally, we apply the effective Hamiltonian to the ideal lattice system composed of atomiclayers with a cross-sectional area S and lattice constant d , as depicted in Fig. 4(a). If weassume that R and P are the displacement of the interlayer distance and its conjugatemomentum, the Hamiltonian of the monoatomic layer is H L = P ρSd + Y Sd (cid:18) Rd (cid:19) − ωSd Ω (cid:16) ε Re ∂R R + ε Re ∂P P (cid:17) I ( t ) cn b , (26)where ρ , Y , and ε are the density, elastic constant, and relative dielectric constant, re-spectively. The total optical polarizability χ , optical intensity I ( t ), and the correspondingphonon frequency are, respectively, χ = Sdεε , I ( t ) = ε n E ( t ) c/n b , Ω = p Y /ρ/d, (27)where n b is the index of refraction of the crystal. We consider the change in interlayerdistance in the impulsive limit ( t < / Ω). In addition, because the acoustic velocity v =Ω d , acoustic propagation is limited by the lattice constant d , so that each atomic layer is8echanically decoupled. Replacing the parameter of Eq. (25) with the parameter of thelattice Hamiltonian Eq. (26), the lattice displacement is R ≈ ∂ε Im ∂R ωd Y cn b Z t I ( t ′ ) dt ′ . (28)For simplicity, the index n b of refraction is held constant. If the imaginary Raman tensor isnegative, the equation gives a quantum lattice contraction, as shown in Fig. 4(b). By usinga lattice strain u = − R/d , a vacuum laser wavelength λ = 2 πc/ω , and an optical fluence F = Z t I ( t ′ ) dt ′ , (29)the useful expression for the associated stress σ is σ = Y u ≈ n b ∂ε Im ∂u πFλ . (30)A fluence of 10mJ / cm at 800 nm corresponds to 2 πF/λ of 0.785 GPa. The remaining partof Eq. (30) is a dimensionless quantity of order unity.In conclusion, we derive herein the effective Hamiltonian describing an optical-lattice in-teraction by using the complex Raman tensor and applying a finite-time boundary condition.The optical transition and lattice displacement are tightly linked by the transient interactioncaused by the vector potential so that the lattice-induced absorption takes place. The non-Newtonian momentum interaction explains the lattice contraction and the coherent-phonongeneration formerly attributed to DECP. For further progress in this field, we encouragethe lattice contraction and the induced photoabsorption to be verified qualitatively andquantitatively by experiment.The author thanks Professor Kenzo Ishikawa for encouragement and stimulating discus-sions. ∗ [email protected][1] C. V. Raman and K. S. Krishnan, Nature , 501 (1928).[2] P. A. M. Dirac, Proc. Roy. Soc. A114 , 243 (1927).[3] J. R. Salcedo, A. E. Siegman, D. D. Dlott, and M. D. Fayer, Phys. Rev. Lett. , 131 (1978).[4] S. D. Silvestri, J. G. Fujimoto, E. P. Ippen, E. B. GambleJr., L. R. Williams, and K. A.Nelson, Chem. Phys. Letters , 146 (1985).
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