Nonlinear Optics - Chapter 6 pot

Chapter 6 Nonlinear Optics in the Two-Level Approximation 6.1. Introduction Our treatment of nonlinear optics in the previous chapters has for the most part made use of power series expansions to relate the response of a material system to the strength of the applied optical field. In simple cases, this relation can be taken to be of the form ˜ P(t)= 0 χ (1) ˜ E(t) + 0 χ (2) ˜ E(t) 2 + 0 χ (3) ˜ E(t) 3 +···. (6.1.1) However, there are circumstances under which such a power series expansion does not converge, and under such circumstances different methods must be employed to describe nonlinear optical effects. One example is that of a sat- urable absorber, where the absorption coefficient α is related to the intensity I =2n 0 c|E| 2 of the applied optical field by the relation α = α 0 1 +I/I s , (6.1.2) where α 0 is the weak-field absorption coefficient and I s is an optical constant called the saturation intensity. We can expand this equation in a power series to obtain α =α 0  1 −(I/I s ) +(I/I s ) 2 −(I/I s ) 3 +···  . (6.1.3) However, this series converges only for I<I s , and thus only in this limit can saturable absorption be described by means of a power series of the sort given by Eq. (6.1.1). It is primarily under conditions such that a transition of the material system is resonantly excited that perturbation techniques fail to provide an adequate 277 278 6 ♦ Nonlinear Optics in the Two-Level Approximation description of the response of the system to an applied optical field. However, under such conditions it is usually adequate to deal only with the two atomic levels that are resonantly connected by the optical field. The increased com- plexity entailed in describing the atomic system in a nonperturbative manner is thus compensated in part by the ability to make the two-level approxima- tion. When only two levels are included in the theoretical analysis, there is no need to perform the sums over all atomic states that appear in the general quantum-mechanical expressions for χ (3) given in Chapter 3. In the present chapter we shall for the most part concentrate on the situation in which a monochromatic beam of frequency ω interacts with a collection of two-level atoms. The treatment is thus an extension of that of Chapter 4, which treated the interaction of a monochromatic beam with a nonlinear medium in terms of the third-order susceptibility χ (3) (ω = ω +ω − ω). In addition, in the last two sections of this chapter we generalize the treatment by studying nondegenerate four-wave mixing involving a collection of two-level atoms. Even though the two-level model ignores many of the features present in real atomic systems, there is still an enormous richness in the physical processes that are described within the two-level approximation. Some of the processes that can occur and that are described in the present chapter include saturation effects, power broadening, Rabi oscillations, and optical Stark shifts. Parallel treatments of optical nonlinearities in two-level atoms can be found in the books of Allen and Eberly (1975) and Cohen-Tannoudji et al. (1989) and in the reviews of Sargent (1978) and Boyd and Sargent (1988). 6.2. Density Matrix Equations of Motion for a Two-Level Atom We first consider the density matrix equations of motion for a two-level sys- tem in the absence of damping effects. Since damping mechanisms can be very different under different physical conditions, there is no unique way to include damping in the model. The present treatment thus serves as a starting point for the inclusion of damping by any mechanism. The interaction we are treating is illustrated in Fig. 6.2.1. The lower atomic level is denoted a and the upper level b. We represent the Hamiltonian for this system as ˆ H = ˆ H 0 + ˆ V(t), (6.2.1) where ˆ H 0 denotes the atomic Hamiltonian and ˆ V(t) denotes the energy of interaction of the atom with the electromagnetic field. We denote the energies 6.2. Density Matrix Equations of Motion for a Two-Level Atom 279 FIGURE 6.2.1 Near-resonant excitation of a two-level atom. of the states a and b as E a = ¯ hω a and E b = ¯ hω b . (6.2.2) The Hamiltonian ˆ H 0 can thus be represented by the diagonal matrix whose elements are given by H 0,nm =E n δ nm . (6.2.3) We assume that the interaction energy can be adequately described in the electric dipole approximation, in which case the interaction Hamiltonian has the form ˆ V(t)=−ˆμ ˜ E(t). (6.2.4) We also assume that the atomic wave functions corresponding to states a and b have definite parity so that the diagonal matrix elements of ˆμ vanish—that is, we assume that μ aa =μ bb =0 and thus that V aa =V bb =0. (6.2.5) The only nonvanishing elements of ˜ V are hence V ba and V ab , which are given explicitly by V ba =V ∗ ab =−μ ba ˜ E(t). (6.2.6) We describe the state of this system by means of the density matrix, which is given explicitly by ˆρ =  ρ aa ρ ab ρ ba ρ bb  , (6.2.7) 280 6 ♦ Nonlinear Optics in the Two-Level Approximation where ρ ba =ρ ∗ ab . The time evolution of the density matrix is given, still in the absence of damping effects, by Eq. (3.3.21) as ˙ρ nm = −i ¯ h  ˆ H, ˆρ  nm = −i ¯ h  ˆ H ˆρ  nm −  ˆρ ˆ H  nm  = −i ¯ h  v (H nv ρ vm −ρ nv H vm ). (6.2.8) We now introduce the decomposition of the Hamiltonian into atomic and in- teraction parts (Eq. (6.2.1)) into this expression to obtain ˙ρ nm =−iω nm ρ nm − i ¯ h  ν (V nν ρ νm −ρ nν V νm ), (6.2.9) where we have introduced the transition frequency ω nm = (E n −E m )/ ¯ h.For the case of the two-level atom, the indices n, m,andν can take on the values a or b only, and the equations of motion for the density matrix elements are given explicitly as ˙ρ ba =−iω ba ρ ba + i ¯ h V ba (ρ bb −ρ aa ), (6.2.10a) ˙ρ bb = i ¯ h (V ba ρ ab −ρ ba V ab ), (6.2.10b) ˙ρ aa = i ¯ h (V ab ρ ba −ρ ab V ba ). (6.2.10c) It can be seen by inspection that ˙ρ bb +˙ρ aa =0, (6.2.11) which shows that the total population ρ bb +ρ aa is a conserved quantity. From the definition of the density matrix, we know that the diagonal elements of ˆρ represent probabilities of occupation, and hence that ρ aa +ρ bb =1. (6.2.12) No separate equation of motion is required for ρ ab , because of the relation ρ ab =ρ ∗ ba . Equations (6.2.10) constitute the density matrix equations of motion for a two-level atom in the absence of relaxation processes. These equations pro- vide an adequate description of resonant nonlinear optical processes under conditions where relaxation processes can be neglected, such as excitation with short pulses whose duration is much less than the material relaxation times. We next see how these equations are modified in the presence of relax- ation processes. 6.2. Density Matrix Equations of Motion for a Two-Level Atom 281 FIGURE 6.2.2 Relaxation processes of the closed two-level atom. 6.2.1. Closed Two-Level Atom Let us first consider relaxation processes of the sort illustrated schematically in Fig. 6.2.2. We assume that the upper level b decays to the lower level a at a rate  ba and therefore that the lifetime of the upper level is given by T 1 = 1/ ba . Typically, the decay of the upper level would be due to spon- taneous emission. This system is called closed, because any population that leaves the upper level enters the lower level. We also assume that the atomic dipole moment is dephased in the characteristic time T 2 , leading to a transition linewidth (for weak applied fields) of characteristic width γ ba =1/T 2 . ∗ We can describe these relaxation processes mathematically by adding decay terms phenomenologically to Eqs. (6.2.10); the modified equations are given by ˙ρ ba =−  iω ba + 1 T 2  ρ ba + i ¯ h V ba (ρ bb −ρ aa ), (6.2.13a) ˙ρ bb = −ρ bb T 1 − i ¯ h (V ba ρ ab −ρ ba V ab ), (6.2.13b) ˙ρ aa = ρ bb T 1 + i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.13c) The forms of the relaxation terms included in these equations will be justified in the discussion given below. One can see by inspection of Eqs. (6.2.13) that the condition ˙ρ bb +˙ρ aa =0 (6.2.14) is still satisfied. Since Eq. (6.2.13a) depends on the populations ρ bb and ρ aa only in terms of the population difference, ρ bb − ρ aa , it is useful to consider the equa- tion of motion satisfied by this difference. We subtract Eq. (6.2.13c) from ∗ In fact, one can see from Eq. (6.3.25) that the full width at half maximum in angular frequency units of the absorption line in the limit of weak fields is equal to 2γ ba . 282 6 ♦ Nonlinear Optics in the Two-Level Approximation Eq. (6.2.13b) to find that d dt (ρ bb −ρ aa ) = −2ρ bb T 1 − 2i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.15) The first term on the right-hand side can be rewritten using the relation 2ρ bb = (ρ bb −ρ aa ) +1 (which follows from Eq. (6.2.12)) to obtain d dt (ρ bb −ρ aa ) =− (ρ bb −ρ aa ) +1 T 1 − 2i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.16) This relation is often generalized by allowing the possibility that the popu- lation difference (ρ bb −ρ aa ) (eq) in thermal equilibrium can have some value other than −1, the value taken above by assuming that only downward spon- taneous transitions could occur. This generalized version of Eq. (6.2.16) is given by d dt (ρ bb −ρ aa ) =− (ρ bb −ρ aa ) −(ρ bb −ρ aa ) (eq) T 1 − 2i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.17) We therefore see that for a closed two-level system the density matrix equa- tions of motion reduce to just two coupled equations, Eqs. (6.2.13a) and (6.2.17). In order to justify the choice of relaxation terms used in Eqs. (6.2.13a) and (6.2.17), let us examine the nature of the solutions to these equations in the absence of an applied field—that is, for V ba = 0. The solution to Eq. (6.2.17) is  ρ bb (t) −ρ aa (t)  = (ρ bb −ρ aa ) (eq) +  ρ bb (0) −ρ aa (0)  −(ρ bb −ρ aa ) (eq)  e −t/T 1 . (6.2.18) This equation shows that the population inversion [ρ bb (t) − ρ aa (t)] relaxes from its initial value ρ bb (0)−ρ aa (0) to its equilibrium value (ρ bb −ρ aa ) (eq) in a time of the order of T 1 . For this reason, T 1 is called the population relaxation time. Similarly, the solution to Eq. (6.2.13a) for the case V ba =0 is of the form ρ ba (t) = ρ ba (0)e −(iω ba +1/T 2 )t . (6.2.19) 6.2. Density Matrix Equations of Motion for a Two-Level Atom 283 We can interpret this r esult more directly by considering the expectation value of the induced dipole moment, which is given by  ˜μ(t)  = μ ab ρ ba (t) +μ ba ρ ab (t) = μ ab ρ ba (0)e −(iω ba +1/T 2 )t +c.c. =  μ ab ρ ba (0)e −iω ba t +c.c.  e −t/T 2 . (6.2.20) This result shows that, for an undriven atom, the dipole moment oscillates at frequency ω ba and decays to zero in the characteristic time T 2 , which is hence known as the dipole dephasing time. For reasons that were discussed in relation to Eq. (3.3.25), T 1 and T 2 are related to the collisional dephasing rate γ c by 1 T 2 = 1 2T 1 +γ c . (6.2.21a) For an atomic vapor, γ c is usually described accurately by the formula γ c =C s N +C f N f , (6.2.21b) where N is the number density of atoms having resonance frequency ω ba , and N f is the number density of any “foreign” atoms of a different atomic species having a different resonance frequency. The parameters C s and C f are coefficients describing self-broadening and foreign-gas broadening, re- spectively. As an example, for the resonance line (i.e., the 3s → 3p transi- tion) of atomic sodium, T 1 is equal to 16 nsec, C s = 1.50 × 10 −7 cm 3 /sec, and for the case of foreign-gas broadening by collisions with argon atoms, C f = 2.53 × 10 −9 cm 3 /sec. The values of T 1 , C s ,andC f for other transi- tions are tabulated, for example, by Miles and Harris (1973). 6.2.2. Open Two-Level Atom The open two-level atom is shown schematically in Fig. 6.2.3. Here the upper and lower levels are allowed to exchange population with associated reservoir levels. These levels might, for example, be magnetic sublevels or hyperfine levels associated with states a and b. The system is called open because the population that leaves the upper l evel does not necessarily enter the lower level. This model is often encountered in connection with laser theory, in which case the upper level or both levels are assumed to acquire population at some controllable pump rates, which we take to be λ b and λ a for levels b and a, respectively. As previously, we assume that the induced dipole moment re- laxes in a characteristic time T 2 . In order to account for relaxation and pump- ing processes of the sort just described, the density matrix equations (6.2.10) 284 6 ♦ Nonlinear Optics in the Two-Level Approximation FIGURE 6.2.3 R elaxation processes for the open two-level atom. are modified to become ˙ρ ba =−  iω ba + 1 T 2  ρ ba + i ¯ h V ba (ρ bb −ρ aa ), (6.2.22a) ˙ρ bb = λ b − b (ρ bb −ρ bb ) (eq) − i ¯ h (V ba ρ ab −ρ ba V ab ), (6.2.22b) ˙ρ aa = λ a − a (ρ aa −ρ aa ) (eq) + i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.22c) Note that in this case the total population contained in the two levels a and b is not conserved and that in general all three equations must be considered. The relaxation rates are related to the collisional dephasing rate γ c and population rates  b and  a by 1 T 2 = 1 2 ( b + a ) +γ c . (6.2.23) 6.2.3. Two-Level Atom with a Non-Radiatively Coupled Third Level The energy level scheme shown in Fig. 6.3.1 is often used to model a sat- urable absorber. Population spontaneously leaves the optically excited level b at a rate  ba +  bc , where  ba is the rate of decay to the ground state a, and  bc is the rate of decay to level c.Levelc acts as a trap level; popula- tion decays from level c back to the ground state at a rate  ca . In addition, any dipole moment associated with the transition b etween levels a and b is damped at a rate γ ba . These relaxation processes are modeled by modifying 6.3. Steady-State Response of a Two-Level Atom 285 FIGURE 6.3.1 Relaxation processes for a two-level atom with a nonradiatively cou- pled third level. Eqs. (6.2.10) to ˙ρ ba =−(iω ba +γ ba )ρ ba + i ¯ h V ba (ρ bb −ρ aa ), (6.2.24a) ˙ρ bb =−( ba + bc )ρ bb − i ¯ h (V ba ρ ab −ρ ba V ab ), (6.2.24b) ˙ρ cc =  bc ρ bb − ca ρ cc , (6.2.24c) ˙ρ aa =  ba ρ bb + ca ρ cc + i ¯ h (V ba ρ ab −ρ ba V ab ). (6.2.24d) It can be seen by inspection that the population in the three levels is conserved, that is, that ˙ρ aa +˙ρ bb +˙ρ cc =0. 6.3. Steady-State Response of a Two-Level Atom to a Monochromatic Field We next examine the nature of the solution to the density matrix equations of motion for a two-level atom in the presence of a monochromatic, steady-state field. For definiteness, we treat the case of a closed two-level atom, although our results would be qualitatively similar for any of the models described above (see Problem 1 at the end of this chapter). For the closed two-level atomic system, the density matrix equations were shown above (Eqs. (6.2.13a) and (6.2.17)) to be of the form d dt ρ ba =−  iω ba + 1 T 2  ρ ba + i ¯ h V ba (ρ bb −ρ aa ), (6.3.1) d dt (ρ bb −ρ aa ) =− (ρ bb −ρ aa ) −(ρ bb −ρ aa ) (eq) T 1 − 2i ¯ h (V ba ρ ab −ρ ba V ab ). (6.3.2) 286 6 ♦ Nonlinear Optics in the Two-Level Approximation In the electric dipole approximation, the interaction Hamiltonian for an ap- plied field in the form of a monochromatic wave of frequency ω is given by ˆ V =−ˆμ ˜ E(t) =−ˆμ  Ee −iωt +E ∗ e iωt  , (6.3.3) and the matrix elements of the interaction Hamiltonian are then given by V ba =−μ ba  Ee −iωt +E ∗ e iωt  . (6.3.4) Equations (6.3.1) and (6.3.2) cannot be solved exactly for V ba given by Eq. (6.3.4). However, they can be solved in an approximation known as the rotating-wave approximation. We recall from the discussion of Eq. (6.2.20) that, in the absence of a driving field, ρ ba tends to evolve in time as exp(−iω ba t). For this reason, when ω is approximately equal to ω ba , the part of V ba that oscillates as e −iωt acts as a far more effective driving term for ρ ba than does the part that oscillates as e iωt . It is thus a good approximation to take V ba not as Eq. (6.3.4) but instead as V ba =−μ ba Ee −iωt . (6.3.5) This approximation is called the rotating-wave approximation. Within this approximation, the density matrix equations of motion (6.3.1) and (6.3.2) become d dt ρ ba =−  iω ba + 1 T 2  ρ ba − i ¯ h μ ba Ee −iωt (ρ bb −ρ aa ), (6.3.6) d dt (ρ bb −ρ aa ) =− (ρ bb −ρ aa ) −(ρ bb −ρ aa ) (eq) T 1 + 2i ¯ h  μ ba Ee −iωt ρ ab −μ ab E ∗ e iωt ρ ba  . (6.3.7) Note that (in the rotating-wave approximation) ρ ba is driven only at nearly its resonance frequency ω ba ,andρ bb −ρ aa is driven only at nearly zero fre- quency, which is its natural frequency. We next find the steady-state solution to Eqs. (6.3.6) and (6.3.7)—that is, the solution that is valid long after the transients associated with the turn-on of the driving field have died out. We do so by introducing the slowly varying quantity σ ba ,definedby ρ ba (t) = σ ba (t)e −iωt . (6.3.8) [...].. .6. 3 Steady-State Response of a Two-Level Atom 287 Equations (6. 3 .6) and (6. 3.7) then become d 1 i σba − μba E(ρbb − ρaa ), σba = i(ω − ωba ) − dt T2 h ¯ (6. 3.9) d (ρbb − ρaa )−(ρbb − ρaa )(eq) (ρbb − ρaa ) = − dt T1 2i μba Eσab − μab E ∗ σba + (6. 3.10) h ¯ The steady-state solution can now be obtained by setting the left-hand sides of Eqs (6. 3.9) and (6. 3.10) equal to zero We... e(1/2)i t (6. 5.24) The probability that the atom is in level a at time t is hence given by |Ca |2 = cos2 t + 1 2 2 2 sin2 1 2 t , (6. 5.25) while the probability of being in level b is given by |Cb |2 = Note that (since 2 = | |2 + | |2 2 sin2 1 2 t (6. 5. 26) 2) |Ca |2 + |Cb |2 = 1, which shows that probability is conserved (6. 5.27) 3 06 6 ♦ Nonlinear Optics in the Two-Level Approximation F IGURE 6. 5.1 Rabi... The coupled equations (6. 5.10) then reduce to the set ∗ μab E ˙ Ca = i Cb e i t , h ¯ μba E ˙ Ca e−i t Cb = i h ¯ ∗ See also the discussion preceding Eq (6. 3.5) (6. 5.12a) (6. 5.12b) 6 ♦ Nonlinear Optics in the Two-Level Approximation 304 This set of equations can be readily solved by adopting a trial solution of the form Ca = Ke−iλt (6. 5.13) This expression is introduced into Eq (6. 5.12a), which shows... 3ωba /c Is (6. 3.34b) Note also that the third-order susceptibility can be related to the linear susceptibility by χ (3) = −χ (1) −χ (1) = 0 3(1 + 2 T22 )|Es |2 3|Es |2 (6. 3.35) 6 ♦ Nonlinear Optics in the Two-Level Approximation 292 F IGURE 6. 3.3 Real and imaginary parts of the susceptibility χ (3) plotted as functions of the optical frequency ω Furthermore, through use of Eqs (6. 3.22b) and (6. 3.27),... P e−iωt + c.c., (6. 3.14) and we define the susceptibility χ as the constant of proportionality relating P and E according to P= 0 χ E (6. 3.15) We hence find from Eqs (6. 3.12) through (6. 3.15) that the susceptibility is given by χ= N|μba |2 (ρbb − ρaa ) , ¯ 0 h(ω − ωba + i/T2 ) (6. 3. 16) where ρbb − ρaa is given by Eq (6. 3.11) We introduce this expression for [ρbb − ρaa ] into Eq (6. 3. 16) and rationalize... σba μab e−iωt + σab μba eiωt ˜ (6. 4 .6) ˆ If we define the complex amplitude p of the dipole moment μ through the ˜ relation ˆ μ = pe−iωt + c.c., ˜ (6. 4.7) we find by comparison with Eq (6. 4 .6) that p = σba μab (6. 4.8) Equations (6. 4.5) can hence be rewritten in terms of the dipole amplitude p as 1 dp h ¯ = i − p − i|κ|2 Ew, (6. 4.9a) dt T2 4 w − w(eq) 4 dw =− − Im(Ep∗ ) (6. 4.9b) dt T1 h ¯ These equations... ωba ) 2 ¯ ba 1 2 (6. 3.17) 288 6 ♦ Nonlinear Optics in the Two-Level Approximation Note that this expression gives the total susceptibility, including both its linear and nonlinear contributions We next introduce new notation to simplify this expression We introduce the quantity = 2|μba | |E|/h, ¯ (6. 3.18) which is known as the on-resonance Rabi frequency, and the quantity = ω − ωba , (6. 3.19) which is... order to relate our present treatment of the nonlinear optical susceptibility to the perturbative treatment that we have used in the previous chapters, 6. 3 Steady-State Response of a Two-Level Atom 291 we next calculate the first- and third-order contributions to the susceptibility of a collection of two-level atoms By performing a power series expansion of 0 Eq (6. 3.28) in the quantity |E|2 /|Es |2 and... −α0 (0) ωba /c 1 + T2 − i 0 + |E|2 /|Es |2 2T 2 2 (6. 3.28) 0 We see from this expression that the significance of Es is that the absorption experienced by an optical wave tuned to line center (which is proportional to Im χ evaluated at = 0) drops to one-half its weak-field value when the 290 6 ♦ Nonlinear Optics in the Two-Level Approximation F IGURE 6. 3.2 Real and imaginary parts of the susceptibility... equations (6. 4.5) can be simplified in a different way We assume that the phase convention for describing the atomic energy eigenstates has been chosen such that μba and hence κ are real quantities It is then useful to express the density matrix element σ in terms of two real quantities u and v as σ = 1 (u − iv) 2 (6. 4.10) 2 96 6 ♦ Nonlinear Optics in the Two-Level Approximation The factor of one-half and . Chapter 6 Nonlinear Optics in the Two-Level Approximation 6. 1. Introduction Our treatment of nonlinear optics in the previous chapters has for the most part made. σ ba (t)e −iωt . (6. 3.8) 6. 3. Steady-State Response of a Two-Level Atom 287 Equations (6. 3 .6) and (6. 3.7) then become d dt σ ba =  i(ω −ω ba ) − 1 T 2  σ ba − i ¯ h μ ba E(ρ bb −ρ aa ), (6. 3.9) d dt (ρ bb −ρ aa ). 2γ ba . 282 6 ♦ Nonlinear Optics in the Two-Level Approximation Eq. (6. 2.13b) to find that d dt (ρ bb −ρ aa ) = −2ρ bb T 1 − 2i ¯ h (V ba ρ ab −ρ ba V ab ). (6. 2.15) The first term on the right-hand