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Adiabatic elimination, the rotating-wave approximation and two-photon transitions
Optics Communications 253 (2005) 125–137 www.elsevier.com/locate/optcom
Adiabatic elimination, the rotating-wave approximation and two-photon transitions M.P. Fewell
*
Physics and Electronics, University of New England, Armidale, NSW 2351, Australia Received 1 November 2004; received in revised form 11 April 2005; accepted 20 April 2005
Abstract The rotating-wave approximation (RWA) is a formalism of great utility in the description of the coherent excitation of atoms and molecules by laser light. Not only does it give results in agreement with experiment, it also provides a simple framework allowing the Hamiltonian of a system to be written down from inspection of the state-linkage diagram. Recent interest in systems with a two-photon coupling prompted an investigation of the structure of two-photon terms in RWA Hamiltonians. In carrying through the derivation, an interaction with adiabatic elimination was discovered. It is shown that adiabatic elimination must be performed before application of the RWA, else terms are dropped that ought to be retained. RWA Hamiltonians for three-state systems with one and two two-photon linkages are displayed. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Hz; 32.80.Wr Keywords: Rotating-wave approximation; Two-photon transitions; Adiabatic elimination
1. Introduction The rotating-wave approximation (RWA) is a key concept underlying the theory of the coherent excitation of atoms and molecules by laser light. Its success in accounting for experimentally observed features of energy and momentum transfer from photon to atom has long been known [1,2]. Also, the RWA provides a simple schema allowing the Hamiltonian array of a quantum system to be written down by inspection of the energy-level coupling diagram. For example, all the three-state systems shown in Fig. 1(a)–(c),
*
Present address: Defence Science and Technology Organisation, P.O. Box 1500, Edinburgh, SA 5111, Australia. Tel.: +61 8 8259 7608; fax: +61 8 8259 5139. E-mail address: [email protected]. 0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.04.049
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ωb, Ωb
ω a, Ωa
(a)
ωb, Ωb
ω a, Ωa
(c)
ωb, Ωb
ω a, Ωa
(b)
ωc, Ωc
ωb, Ωb ω a, Ωa (d)
Fig. 1. Linkage pattern in some simple STIRAP systems: (a) three-state K system; (b) 3-state V system; (c) three-state ladder system; (d) a 4-state K system. State j1æ is the initial state and the desired final state is (a)–(c) state j3æ, (d) state j4æ. The two or three laser fields are distinguished by letters and are shown next to the transition that they most impact upon, though in principle each interacts with every linkage. Quantities x are angular frequencies of the laser fields and X are Rabi frequencies.
when interacting with two laser fields with frequencies that approximate the Bohr frequencies of the transitions, are described by the Hamiltonian array 0 1 2D1 Xa 0 hB C H ¼ @ Xa 2D2 Xb A; ð1:1Þ 2 0 Xb 2D3 where the Xl are Rabi frequencies and the Dn are cumulative detunings at each level.1 The differences between the patterns of state energies enter in the definition of the Dn, as detailed in Chapter 13 of Shores book [2]. The two zeros in Eq. (1.1) are a consequence of the absence of a direct connection between states j1æ and j3æ; situations where such a connection exists (e.g. [3,4]) have non-zero entries for these elements. The extension to the 4-state system of Fig. 1(d) is plain. This paper is concerned with the description of two-photon processes in situations where the rotatingwave approximation may be expected to apply. The main result, the Hamiltonian for a two-photon transition, has been known for decades ([2,5] and references therein); the new feature presented here is its derivation from a Hamiltonian to which the RWA has been applied. That is, the aim of this paper is to bring two-photon transitions into the RWA purview, so to speak, so that one can write down by inspection the equivalent of Eq. (1.1) when one or more of the transitions is two-photon. The key to achieving this is the recognition that an interaction exists between adiabatic elimination, a natural step in the treatment of a two-photon transition, and the rotating-wave approximation, the consequence of which is the incorrect dropping of terms from the Hamiltonian when the RWA is applied prior to adiabatic elimination. To the best of the authors knowledge, this interaction has not been previously noted. Application of the two approximations in the opposite order (i.e., adiabatic elimination first) results in a Hamiltonian that agrees with expectations. As well as showing the consistency of RWA theory in the two-photon case, this work 1 In an attempt to reduce confusion, the Rabi frequencies are identified by letters. Full definitions and details of the notation are given in Section 4.
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also indicates that care must be taken in using an RWA Hamiltonian when one state is well off resonance; for this is equivalent to adiabatically eliminating the state. The work reported here was originally motivated by an issue in stimulated Raman scattering with adiabatic passage (STIRAP) [1,6–12]: the search for a STIRAP system in which the intermediate state has no spontaneous decay. (This is in addition to the initial and final states which, for practical reasons, must also not decay.) Such a system would be a useful laboratory for testing experimentally several predictions concerning coherent population transfer [13–15]. If one restricts oneself to single-photon transitions, then it is clear that such a system cannot exist; for the dipole couplings from the intermediate state to the initial and final states inevitably produce spontaneous decay. Thus, one is led to consider three-state systems in which both pump and Stokes couplings are two-photon. The M = 2, M = 0 and M = +2 projection sublevels of a J = 2 metastable state are an example. Such a system is denoted 2 + 2 STIRAP (e.g. [16]), to indicate that both transitions of the STIRAP process are two-photon. In the interests of simplicity, however, the theory is developed for 2 + 1 STIRAP, in which only the pump transition is two-photon. Also, the 2 + 1 system has been studied theoretically [16–18] and experimentally [19], whereas there has yet to be a successful experimental realisation of the 2 + 2 system. For definiteness, the discussion here deals with the K-type system shown in Fig. 1(d), since this arrangement of level energies matches the experimental case.2 The three levels of the STIRAP system are labelled j1æ, j3æ and j4æ; level j2æ represents the virtual level that provides the basis of the theoretical description of the two-photon process. The question is how to describe this system theoretically in the context of the RWA. The natural answer is first to consider level j2æ as a real level, giving a four-state problem. The Hamiltonian of such a system is clear. One then allows the detuning at level j2æ to become large and adiabatically eliminates the level, thereby effecting the transition to a Hamiltonian describing a 2 + 1 system. Section 2 presents this analysis and Section 3 indicates why the result is wrong. Section 4 shows the improvement obtained by performing the adiabatic elimination before applying the RWA, with the derivation of the two-photon RWA Hamiltonian being completed in Section 5. The results of the corresponding analysis of the 2 + 2 system are stated in Section 6. Section 7 contains a brief discussion of conditions required for the existence trapped states and Section 8 is a summary of the conclusions.
2. Working from the four-state RWA Hamiltonian If we take level j2æ in Fig. 1(d) to be a real level, then the system involves three single-photon couplings. Such a system was first treated in the context of STIRAP in 1992 [20] and its RWA Hamiltonian is well known (e.g. [2,20–22]). In the RWA, Schro¨dingers equation reads 0_ 1 0 10 1 C1 0 0 0 Xa C1 B C_ C 1 B X 2D C B Xb 0 CB C 2 C 2 B 2C B a C iB C ¼ B ð2:1Þ CB C; @ C_ 3 A 2 @ 0 Xb 2D3 Xc A@ C 3 A 0 0 Xc 2D4 C4 C_ 4 where the symbols have their usual meanings [2,20] – in the Hamiltonian matrix, the Dn are cumulative detunings at each level and the Xl are Rabi frequencies. 2 In most, perhaps all, experimental realisations of two-photon processes, the two photons come from the same laser field, i.e., xb = xa in the present notation. However, there is no inherent increase in complication in treating the more general case, so the distinction between the two photons is maintained throughout this analysis.
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We now suppose that D2 is large and adiabatically eliminate state j2æ. A widely used prescription for this is to set C_ 2 to zero and solve the resulting equations for C2, so eliminating C2 from the remaining three equations of Eq. (2.1).3 The result is 0 X2 10 1 0 1 Xa Xb a 0 C1 C_ 1 D2 D2 CB C B _ C iB 2 B C X ¼ ð2:2Þ X X @ C3 A @ C 3 A. a b b 4 @ D2 D2 4D3 2Xc A C4 C_ 4 0 2X 4D c
4
The diagonal elements in Eq. (2.2) give the detunings, including the dynamic Stark shifts caused by the pulses, and the off diagonal elements give the two-photon couplings. The following section argues that the diagonal elements must be wrong. In fact, the off-diagonal elements are also incorrect, as shown at the end of Section 4. This is related to the accuracy of the above method of adiabatic elimination, a point dealt with in Section 5. 3. Dynamic stark shifts According to Eq. (2.2), the dynamic (or ac) Stark shift DE1(x) of the initial level is X2a . ð3:1Þ 4D2 Now consider what happens when the system encounters a set of pulses in the so-called counterintuitive order characteristic of STIRAP. In this case, pulse c must lead. That is, at first Xa and Xb are zero, only Xc is present. Eq. (3.1) says that the initial state experiences no dynamic Stark shift during this time, no matter how strong Xc may be. Physical intuition tells us that this cannot be right. However, intuition is sometimes misleading, so it is as well to exhibit the errors in detail. The remainder of this section undertakes this task. Any textbook on the subject (e.g. [2], p. 282) gives the dynamic Stark shift DEn(x) of level n as E2 X ðdnk e Þðdkn eÞ ðdnk eÞðdkn e Þ þ DEn ðxÞ ¼ . ð3:2Þ E k En Ek En þ hx 4 k hx DE1 ðxÞ ¼
Here, dnk is a dipole matrix-element vector and e is the polarisation vector of the electric field, which is characterised in addition by its amplitude E and angular frequency x, and k enumerates all states of the system other than state n. Comparison of Eqs. (3.1) and (3.2) raises the following points: (i) The most obvious feature of Eq. (3.2) is the sum over all states. The absence of such a sum in Eq. (3.1) is not a serious problem, however; for level j2æ can be interpreted as representing all other levels of the system. Thus, it is easy to rationalise the rewriting of Eq. (3.1) to include a sum over all states. (ii) On the other hand, it is impossible to rationalise rewriting Eq. (3.1) to include all fields. In Eq. (3.2), E refers to the entire electric field – everything that the system experiences – whereas Eq. (2.2), and therefore Eq. (3.1), clearly state that certain fields act only on certain couplings. This is an important consequence of the RWA. To introduce a sum over fields arbitrarily in Eq. (3.1) would, it seems, be to threaten the self-consistency of the theory. (iii) The denominator D2 of Eq. (3.1) is exactly the denominator of the first term in Eq. (3.2).4 The second term of Eq. (3.2), with the + hx in its denominator, is often called the anti resonant term. It is nowhere to be seen in Eq. (3.1) and there seems to be no rational ground for arguing for its insertion. 3 Recent examples of the use of this procedure occur in [23] (sentences before Eq. (6)) [24] (Section IV.C), [25] (Section VI.A), [26] (Section V). 4 Setting k = 2 and with an appropriate interpretation of x.
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The following section shows that the problem with Eq. (2.2) lies in part in the application of adiabatic elimination to a system of equations to which the RWA has already been applied. 4. Adiabatic elimination prior to application of the RWA As is shown in detail below, performing adiabatic elimination prior to applying the rotating-wave approximation results in the retention of terms that are dropped if the two steps are carried out in the opposite order. In fact, reversing the order is not sufficient to fully resolve the problems with Eq. (2.2); a more accurate method of adiabatic elimination is also required. However, in the interests of reducing the complexity of the equations so as to make the exposition as clear as possible, we retain in this section the method of adiabatic elimination used in Section 2. This section follows the method and notation of Shore ([2], Chapter 13). 4.1. Setting the theory up The total electric field E comprises the sum of the three fields shown schematically in Fig. 1: ( ) c X ixl t E ¼ Re El e el ;
ð4:1Þ
l¼a
where El , xl and el are, respectively, the (real) amplitude, angular frequency and polarisation vector of each field. Transition strengths between levels are quantified by dipole matrix-element vectors dnk. It is assumed that levels are coupled only to their neighbours; that is, dnk ¼ 0
if k 6¼ n 1.
ð4:2Þ
The polarisations of the fields can further reduce the couplings in that dnk Æ el may be zero for particular values of n, k and l. However, we do not make explicit use of this in the following. The state vector W of the system is expanded on the basis of the unperturbed states jnæ of the system 4 X WðtÞ ¼ C n ðtÞeifn ðtÞ jni; ð4:3Þ n¼1
where Cn(t) are the expansion coefficients and fn(t) are arbitrary phases to be chosen so as to facilitate the purpose at hand. As always with RWA theory, the choice of these is crucial and is considered in detail in Section 4.2. Substitution of Eqs. (4.1)–(4.3) into Schro¨dingers equation yields a set of four coupled differential equations, the first of which is X X21l eixl t þ X eixl t 12l ð4:4Þ iC_ 1 ¼ D1 C 1 þ C 2 eiðf1 f2 Þ 2 l and the second and third are X Xknl eixl t þ X eixl t X nkl C k eiðfn fk Þ iC_ n ¼ Dn C n þ 2 l k¼n1;nþ1
ðn ¼ 2; 3Þ.
ð4:5Þ
In these equations, En is the unperturbed eigenenergy of state jnæ, the Dn are the usual portmanteau variables for the coefficients of Cn: hDn ¼ En hf_ n ; ð4:6Þ and the Xs are the usual Rabi frequencies, but with an extension of notation in the interests of clarity5 5
Note the order of the subscripts on the right-hand side of Eq. (4.7).
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hXnkl ¼ dkn el El .
ð4:7Þ
4.2. Choosing phases with the RWA in mind A suitable choice of phases simplifies Eqs. (4.4) and (4.5). Since we are considering the case in which state j2æ lies between states j1æ and j3æ, the choices: f_ 2 ¼ f_ 1 þ xa ;
f_ 3 ¼ f_ 2 þ xb
are appropriate (Section 13.1 of [2]). Eqs. (4.4) and (4.5) then become " # X C 2 X21a e2ixa t þ X12a þ X21l eiðxa þxl Þt þ X12l eiðxa xl Þt iC_ 1 ¼ D1 C 1 þ 2 l¼b;c and
" # X C 1 2ix t iðx x Þt iðx þx Þt a a a l l iC_ 2 ¼ D2 C 2 þ X12a þ X21a e þ X21l e þ X12l e 2 l¼b;c " # X C3 X32b e2ixb t þ X23b þ X32l eiðxb þxl Þt þ X23l eiðxb xl Þt . þ 2 l¼a;c
ð4:8Þ
ð4:9Þ
ð4:10Þ
The Dn are related because of Eq. (4.8) and the corresponding equation for f_ 4 . For reference, their relationships are: hD2 ¼ E2 E1 hxa þ hD 1 ; D3 ¼ E3 E1 h hxa hx b þ hD1 ; hD4 ¼ E4 E1 hxa hx b þ hxc þ hD1 .
ð4:11Þ
4.3. The elimination step It is usual at this stage to apply the rotating-wave approximation; that is, to drop all terms containing complex exponential factors. However, the point of the present work is to adiabatically eliminate state j2æ first. In this Section, this is done by the same method as used in Section 2, that is, by setting derivatives to zero: we set the left-hand side of Eq. (4.10) to zero and solve for C2. The result is then used to eliminate C2 from all other equations. To demonstrate explicitly how this gives rise to new terms, an intermediate step is shown in detail: insertion of the expression for C2 into Eq. (4.9) gives, after a slight rearrangement, ( c c X iðxa x Þt C1 X iðxa þxl Þt l _ iC 1 ¼ D1 C 1 X12l e þ X21l e X12l0 eiðxa xl Þt þ X21l0 eiðxa þ xl Þt 4D2 l0 ¼a l¼a ) c C3 X X32l0 eiðxb þxl Þt þ X23l0 eiðxb xl Þt . þ ð4:12Þ 4D2 l0 ¼a Now consider the first two sums. Had the rotating-wave approximation already been applied, only the first term of each sum (that with l = a or l 0 = a) would be present, and the product of the two sums would be exactly the first term of the first line of Eq. (2.2). However, it is now apparent that every term in the first sum of Eq. (4.12) has a partner in the second sum such that the complex exponential factors cancel when the two sums are multiplied out. This means that five additional terms survive the application of the RWA.
M.P. Fewell / Optics Communications 253 (2005) 125–137
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A similar effect occurs when sum 1 of Eq. (4.12) is multiplied by sum 3, this time generating one additional term. 4.4. The Hamiltonian Multiplying Eq. (4.12) out and dropping the rapidly oscillating terms gives c C1 X C3 2 iC_ 1 ¼ D1 C 1 X12a X32b þ X21b X23a . jX12l j 2D2 l¼a 4D2
ð4:13Þ
The full Hamiltonian can now be written down by analogy. The result for Schro¨dingers equation is: 1 0 c P X12a X23b þX12b X23a 2 2 0 1 0 1 jX j 0 D2 C C1 B D2 l¼a 12l C_ 1 C B CB C B _ C iB c ð4:14Þ C@ C 3 A; @ C 3 A ¼ B X12a X23b þX12b X23a 2 P 2 jX j 4D 2X 4B 23l 3 34c C D2 D2 A C4 @ _C 4 l¼a 0 2X43c 4D4 where f1, the last remaining arbitrary phase, has been chosen to make D1 zero, as usual for this ordering of energies. In comparing Eq. (4.14) to Eq. (2.2), we see that the additional terms on the diagonal go some way toward answering point (ii) in Section 3. However: The dynamic Stark shifts are now twice the magnitude expected from Eq. (3.2). The anti resonant terms still have not been generated. As might be suspected, these two effects are related. Also, the extra off-diagonal terms do not have the right form – Eq. (14.9–16a) in [2] suggests that these should have anti resonant denominators. The problem lies in the manner in which the adiabatic elimination was performed: simply setting derivatives to zero is not sufficiently accurate when dealing with two-photon effects. The method in Section 14.9 of [2] provides the model for how to proceed; this is adapted to the present case in the following section. 5. A more accurate adiabatic elimination We seek to solve Eq. (4.5) for the states to be eliminated by a technique that mimics direct integration as closely as possible. That is, we write Z t C n ðtÞ ¼ ð5:1Þ C_ n ðt0 Þ dt0 ; 0
where we use the fact that, for states being eliminated, Cn(0) = 0. Eq. (5.1) is, of course, formally tautologous; the issue is to evaluate the integral. Choice of phase is important for this; as discussed below, the standard RWA choice is not appropriate. 5.1. Choice of phase – states being eliminated If state n is to be eliminated, an expression for Cn(t) is required. This is ultimately obtained by inserting Eq. (4.5) into Eq. (5.1), but the difficulty caused by the DnCn term is clear. This term owes its form to the choice of phase for state n, and so may be removed by appropriate adjustment of this choice. Hence, for any state n that is to be eliminated, we must choose
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hf_ n ¼ En ;
ð5:2Þ
so that Dn = 0. With this choice of phase, and following the method of Shore (Section 14.9 of [2]), we write Eq. (5.1) for any state being eliminated as c i X X ½J ðtÞ þ Jnkl ðtÞ; ð5:3Þ C n ðtÞ ¼ 2 k¼n1;nþ1 l¼a nkl where Jnkl ðtÞ ¼
Z
t
0
0
C k ðt0 ÞGknl ðt0 Þei½fk ðt ÞEn t =h dt0
ð5:4Þ
0
with Gknl ðtÞ ¼ Xknl ðtÞeixl t .
ð5:5Þ
The Jnkl ðtÞ differ from the Jnkl ðtÞ only in containing the complex conjugates of the field quantities:6 Z t 0 0 Jnkl ðtÞ ¼ C k ðt0 ÞGknl ðt0 Þei½fk ðt ÞEn t =h dt0 . ð5:6Þ 0
The phases fk have deliberately not been inserted in Eqs. (5.4) and (5.6) because, for a two-photon transition, states k cannot be candidates for elimination if state n (= k ± 1) is being eliminated, and so Eq. (5.2) does not apply to states k. This is a consequence of the assumed pattern of linkages between the levels. 5.2. Phases of states not being eliminated The definition of Gknl given in Eq. (5.4) has been chosen so as to cast Eqs. (5.4) and (5.6) into a form suitable for application of the method of Shore [2], which has the advantage of making explicit the approximations that required to arrive at a solution. This Section details the evaluation of the integral in Eq. (5.4); the method for Eq. (5.6) is identical. Following Shore, Cn and Gknl in Eq. (5.4) are replaced by their representations as Fourier integrals Z 1 ~ k ðxÞeixt dx C k ðtÞ ¼ ð5:7Þ C 1
and similarly for Gknl. A rearrangement of the order of integration in Eq. (5.4) gives7 Z 1 Z 1 Z t 0 0 0 0 0 ~ knl ðx0 Þ ~ k ðxÞG Jnkl ðtÞ ¼ ei½fk ðt Þþxt þx t En t =h dt0 dx0 dx. C 1
1
ð5:8Þ
0
To allow evaluation of the innermost integral in Eq. (5.8), fk is chosen to be a linear function of time. The choice in Eq. (5.2) has this property, but so does the conventional choice of the RWA (Eq. (4.8)). To maintain generality, we write fk ðtÞ ¼ f_ k t; ð5:9Þ _ where the fk are understood to be independent of time. Their values are specified further on in the derivation. This choice of phase in Eq. (5.9) allows the evaluation of the innermost integral of Eq. (5.8). Application of the approximations detailed by Shore (p. 917 of [2]) gives
6 7
Note the order of the subscripts on the right-hand side of Eq. (5.6). The quantities x,x 0 are, of course, Fourier conjugate variables quite distinct from the externally applied optical frequencies xl.
M.P. Fewell / Optics Communications 253 (2005) 125–137
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_
C k ðtÞXknl ðtÞeiðfk þxl En =hÞt C k ð0ÞXknl ð0Þ Jnkl ðtÞ ¼ i . f_ k þ xl En = h
ð5:10Þ
The second term in Eq. (5.10) represents initial conditions. Since the population resides in state j1æ initially, Ck(0) = dk,1. To avoid having to retain the initial condition when k = 1, we place the origin of the time coordinate at a time when all fields coupling to state j1æ are zero. That is, we choose t = 0 so as to ensure that X1nl(0) = 0 for all n,l. The Jnkl ðtÞ contain complex conjugates of the field quantities only, so the expression corresponding to Eq. (5.10) is not obtained by simply replacing the Xknl by their complex conjugates. Not only must the order of subscripts k and n be reversed in recognition of the fact that the matrix elements are not conjugated, we must also allow for the fact that conjugated field quantities refer to the negative-frequency part of Fourier space. Hence, _ C k ðtÞXknl ðtÞeiðfk xl En =hÞt Jnkl ðtÞ ¼ i . ð5:11Þ f_ k xl En = h The different signs in the denominators of Eqs. (5.10) and (5.11) are the source of the two sorts of terms in Eq. (3.2). 5.3. The elimination step and values of the phase rates We are now in a position to assemble an expression for C2(t), by inserting Eqs. (5.10) and (5.11) into Eq. (5.3) and setting n = 2. The result is used to eliminate C2(t) from the other lines of Schro¨dingers equation. For example, the expression for C_ 1 is h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X e X X þ X e 0 0 X 12l 21l 21l 12l 21l 12l C1 þ iC_ 1 ¼ D1 C 1 þ ; 4 ll0 : f_ 1 þ xl E2 =h f_ 1 xl E2 =h h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X e X X þ X e 0 0 X 32l 21l 21l 12l 23l 12l C3 _ _ þ þ eiðf1 f3 Þt . : ; 4 f_ 3 þ xl E2 =h f_ 3 xl E2 =h ll0 ð5:12Þ
This time, application of the RWA leaves 12 terms remaining in the first sum, 6 with each denominator. Before applying the RWA to the second sum, the remaining phase rates must be chosen. To give a result that matches as closely as possible the conventional expressions, it seems best to adopt the usual RWA convention, f_ 1 f_ 3 ¼ xb xa ;
ð5:13Þ
and to make the standard assignment of E1t/h for f1. For convenience, we define a frequency w2 by hw2 ¼ E2 E1
ð5:14Þ
(remembering that E2 > E1 in our system). All this reduces Eq. (5.12) to h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X X X þ X 0e 0e X 12l 21l 21l 12l 21l 12l C1 iC_ 1 ¼ þ ; 4 ll0 : w2 xl w2 þ xl h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X X X þ X 0e 0e X 32l 21l 21l 12l 23l 12l C3 þ eiðx2 x1 Þt . : ; 4 w2 x1 þ x2 xl w2 x1 þ x2 þ xl ll0 ð5:15Þ
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The terms with zero exponential phase are: in the first sum, all terms with l = l 0 ; in the second sum, term 2 with l = 2, l 0 = 1 and term 3 with l = 1, l 0 = 2. For completeness, the values of f and D resulting from the present choices of phase are listed below: h; f1 ¼ E1 t=
hD1 ¼ 0;
f2 ¼ E2 t= h; hD2 ¼ 0; _f3 ¼ f_ 1 þ xa þ xb ; hD3 ¼ E3 E1 hxa hxb ; f_ 4 ¼ f_ 3 xc ; hD 4 ¼ E 4 E 1 hxa hxb þ hxc .
ð5:16Þ
The phases and D values of the retained levels are the same as in the standard RWA analysis, but that of the eliminated level is different. That is, the values of D3 and D4 still give the cumulative detunings at the respective levels; that of D2 does not. It is interesting to observe that the value of D2 required by the derivation of Eq. (5.12) is inadmissible in the derivation leading to Eq. (4.12). 5.4. Construction of the Hamiltonian We now drop the rapidly oscillating terms in Eq. (5.15) to give one line of the RWA Hamiltonian array. The result is C 1 X jX12l j2 jX12l j2 iC_ 1 ¼ þ 4 l w2 xl w2 þ xl
!
C 3 X12a X23b X12b X23a þ . 4 w2 xa w2 þ xb
ð5:17Þ
The chief difference from Eq. (4.13) lies in the denominators: both resonant and anti resonant denominators are now present. The resonant denominators are not identical to the denominator in Eq. (4.13); for, with the choice of phase applying there, D2 = w2 xa. In the sum in Eq. (5.17), the resonant denominator contains the frequency of the field concerned rather than using xa for all terms. In attempting to generate the full Hamiltonian by analogy from Eq. (5.17), care is needed to track correctly the manner in which the denominators vary. The details are not reproduced here, but the resulting Schro¨dingers equation is 0
1
0
c P jX12l j2
2
j þ wjX221l w2 xl þxl
B l¼a C_ 1 B B _ C iB @ C 3 A ¼ B X12a X23b X12b X23a 4 B w2 xa þ w2 xb @ C_ 4 0
X12a X23b w2 xa
þ
X12b X23a w2 xb
c P jX32l j2 l¼a
jX23l j2 þ 4D3 w3 xl w3 þxl 2X43c
1
0 1 C C1 C CB C C@ C 3 A; 2X34c C A C4 4D4 0
ð5:18Þ
where the additional variable hw3 ¼ E2 E1 hxa hx b
ð5:19Þ
is used. Written this way, one can begin to see a pattern in the denominators of the various terms, and also in the order of subscripts in the Rabi frequencies. This becomes clearer on comparison with the results for the 2 + 2 system presented in the following section. Eq. (5.18) has all the properties mentioned in Section 3 and at the end of Section 4.4 as required in a correct expression for a two-photon linkage.
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135
6. Results for a 2 + 2 system The 2 + 2 system with the energy-level relationship shown in Fig. 2 is of the double K or M type, of interest as a potential laboratory for testing certain predictions concerning the STIRAP process. The method of the preceding section applied to this system gives the Hamiltonian. 0 1
d P X12a X32b X21b X23a jX12l j2 jX21l j2 þ þ 0 B C w2 xa w2 þxb 0 1 B l¼a w2 xl w2 þxl C C_ 1 B C
d B C P jX32l j2 jX23l j2 jX34l j2 jX43l j2 X34c X54d X43d X45c B _ C i B X12a X32b X21b X23a C ¼ þ þ þ þ þ 4D @ C3 A 3 w2 xa w2 þxb w3 xl w3 þxl w4 xl w4 þxl w4 xc w4 þxd B C 4B l¼a C B C C_ 5
d P @ A X34c X54d X43d X45c jX54l j2 jX45l j2 0 þ þ 4D 5 w4 xc w4 þxd w5 xl w5 þxl l¼a 0 1 C1 B C ð6:1Þ @ C 3 A. C5 To assist in capturing the regularities that allow such Hamiltonians to be written down from an inspection of the energy-level diagram, the phases, D values and w values applying in Eq. (6.1) are listed below: f1 ¼ E1 t= h;
hD1 ¼ 0;
f2 ¼ E2 t= h; hD2 ¼ 0; D3 ¼ E3 E1 hxa þ hxb ; h f_ 3 ¼ f_ 1 þ xa xb ; f4 ¼ E4 t= h; hD4 ¼ 0; hD5 ¼ E5 E1 hxa þ hxb hxc þ hxd f_ 5 ¼ f_ 3 þ xc xd ;
ð6:2Þ
and hw2 ¼ E2 E1 ; hw3 ¼ E2 E1 hxa þ hxb ;
ð6:3Þ
w4 ¼ E4 E1 h hxa þ hxb ; hw5 ¼ E4 E1 hxa þ hxb hx c þ hxd .
7. Conditions for
STIRAP
It is well known that, for STIRAP in the strict sense to be possible, the Hamiltonian must have an eigenstate with zero eigenvalue [1,11,17]. This state can then act as a trapped state, in which population can be
ωa, Ωa
ωb, Ωb
ωc , Ωc
ωd, Ωd
Fig. 2. As in Fig. 1, but for double-K or M type 2 + 2
STIRAP.
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M.P. Fewell / Optics Communications 253 (2005) 125–137
retained by suitable manipulation of the laser pulse ordering. In the case of the 2 + 1 system, the Hamiltonian of Eq. (5.18) can be rewritten as 0 1 ~p S1 2X 0 hB ~ S 3 4D3 2Xc C ð7:1Þ H ¼ @ 2X A; p 4 0 2Xc 4D4 ~ p is the effective pump Rabi frequency.8 Recall where the Sn are the dynamic Stark shifts of level n and X that D3 is the two-photon detuning at the intermediate level and D4 is the cumulative three-photon detuning at the final level. The condition for this Hamiltonian to have a zero eigenvalue is 2
~ D4 þ X2 S 1 4D3 D4 S 1 þ D4 S 1 S 3 ¼ 0. 4X c p
ð7:2Þ
Since the Stark shifts and Rabi frequencies all vary with laser intensity, it is not possible for this equality to be maintained at all times during a pulse sequence. This question has been explored in detail by Yatsenko et al. [17] and Gue´rin et al. [18]. They find that high transfer probabilities can be obtained if the detunings Dn are carefully chosen, a prediction that has experimental confirmation [19]. In the 2 + 2 system (Section 6), the final state also has a dynamic Stark shift. A corresponding analysis applied to the Hamiltonian of Eq. (6.1) yields the condition for a zero eigenstate as ~ 2 D5 þ X ~ 2 S1 X ~ 2 S 5 4D3 D5 S 1 þ D3 S 1 S 5 þ D5 S 1 S 3 1S 1 S 3 S 5 ¼ 0; 4X p S p 4
ð7:3Þ
~ S is the effective Stokes Rabi frequency. Presumably the analysis of Yatsenko and Gue´rin et al. where X could be extended to cover this case. 8. Conclusions The formalism presented above shows how to derive Hamiltonians with two-photon linkages in situations where the rotating-wave approximation can be expected to apply. The key new result is the identification of an interaction between the rotating-wave approximation and adiabatic elimination: to obtain correct results, adiabatic elimination must be applied before the rotating-wave approximation. Application of the two approximations in the opposite order results in terms being dropped that ought to be retained. Comparison of Eqs. (5.18) and (6.1) suggests some patterns from which rules describing the structure two-photon RWA Hamiltonians may be postulated. Thus, the goal of developing a formalism that allows Hamiltonians of systems with two-photon linkages to be written down by inspection is in sight. But perhaps a few rather more elaborate cases should be worked out from first principles to give confidence in the heuristic procedures. Acknowledgements The author thanks Dr. B.W. Shore for drawing his attention to the question of the form of the RWA Hamiltonian with two-photon transitions, Professor K. Bergmann for his warm hospitality at the Universita¨t Kaiserslautern, where this work was begun, and members of the Arbeitsgruppe Bergmann for much stimulating discussion. This work was supported by the Australian Government Department of Industry, Science and Technology under the Bilateral Science and Technology Program with Germany, and by the Universita¨t Kaiserslautern Graduierten Kolleg Laser und Teilchen Spektroskopie. 8
These quantities are defined by simple comparison with Eq. (5.18).
M.P. Fewell / Optics Communications 253 (2005) 125–137
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
K. Bergmann, H. Theuer, B.W. Shore, Rev. Mod. Phys. 70 (1998) 1003. B.W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, 1990. R.G. Unanyan, L.P. Yatsenko, K. Bergmann, B.. Shore, Opt. Commun. 139 (1997) 48. M. Fleischhauer, R. Unanyan, B.W. Shore, K. Bergmann, Phys. Rev. A59 (1999) 3751. M. Shapiro, Phys. Rev. A54 (1996) 1504. U. Gaubatz, P. Rudecki, S. Schiemann, K. Bergmann, J. Chem. Phys. 92 (1990) 5363. B.W. Shore, J. Martin, M.P. Fewell, K. Bergmann, Phys. Rev. A52 (1995) 566. J. Martin, B.W. Shore, K. Bergmann, Phys. Rev. A52 (1995) 583. T. Halfmann, K. Bergmann, J. Chem. Phys. 104 (1996) 7068. J. Martin, B.W. Shore, K. Bergmann, Phys. Rev. A54 (1996) 1556. M.P. Fewell, B.W. Shore, K. Bergmann, Aust. J. Phys. (1997) 281. A. Kuhn, S. Steuerwald, K. Bergmann, Eur. Phys. J. D1 (1998) 57. N.V. Vitanov, S. Stenholm, Opt. Commun. 127 (1996) 215. N.V. Vitanov, S. Stenholm, Opt. Commun. 135 (1997) 394. N.V. Vitanov, S. Stenholm, Phys. Rev. A56 (1997) 1463. S. Gue´rin, H.R. Jauslin, Eur. Phys. J. D2 (1998) 99. L.P. Yatsenko, S. Gue´rin, T. Halfmann, K. Bo¨hmer, B.W. Shore, K. Bergmann, Phys. Rev. A58 (1998) 4683. S. Gue´rin, L.P. Yatsenko, T. Halfmann, B.W. Shore, K. Bergmann, Phys. Rev. A58 (1998) 4691. K. Bo¨hmer, T. Halfmann, L.P. Yatsenko, B.W. Shore, K. Bergmann, Phys. Rev. A64 (2001), paper 023404. J. Oreg, K. Bergmann, B.W. Shore, S. Rosenwaks, Phys. Rev. A45 (1992) 4888. V.S. Malinovsky, D.J. Turner, Phys. Rev. A56 (1997) 4929. T. Nakajima, Phys. Rev. A59 (1999) 559. L.P. Yatsenko, R.G. Unanyan, K. Bergmann, T. Halfmann, B.W. Shore, Opt. Commun. 135 (1997) 406. N.V. Vitanov, Phys. Rev. A58 (1998) 2295. N.V. Vitanov, S. Stenholm, Phys. Rev. A60 (1999) 3820. I.R. Sola, V.S. Malinovsky, Phys. Rev. A68 (2003), paper 013412.
137
Adiabatic elimination, the rotating-wave approximation and two-photon transitions M.P. Fewell
*
Physics and Electronics, University of New England, Armidale, NSW 2351, Australia Received 1 November 2004; received in revised form 11 April 2005; accepted 20 April 2005
Abstract The rotating-wave approximation (RWA) is a formalism of great utility in the description of the coherent excitation of atoms and molecules by laser light. Not only does it give results in agreement with experiment, it also provides a simple framework allowing the Hamiltonian of a system to be written down from inspection of the state-linkage diagram. Recent interest in systems with a two-photon coupling prompted an investigation of the structure of two-photon terms in RWA Hamiltonians. In carrying through the derivation, an interaction with adiabatic elimination was discovered. It is shown that adiabatic elimination must be performed before application of the RWA, else terms are dropped that ought to be retained. RWA Hamiltonians for three-state systems with one and two two-photon linkages are displayed. 2005 Elsevier B.V. All rights reserved. PACS: 42.50.Hz; 32.80.Wr Keywords: Rotating-wave approximation; Two-photon transitions; Adiabatic elimination
1. Introduction The rotating-wave approximation (RWA) is a key concept underlying the theory of the coherent excitation of atoms and molecules by laser light. Its success in accounting for experimentally observed features of energy and momentum transfer from photon to atom has long been known [1,2]. Also, the RWA provides a simple schema allowing the Hamiltonian array of a quantum system to be written down by inspection of the energy-level coupling diagram. For example, all the three-state systems shown in Fig. 1(a)–(c),
*
Present address: Defence Science and Technology Organisation, P.O. Box 1500, Edinburgh, SA 5111, Australia. Tel.: +61 8 8259 7608; fax: +61 8 8259 5139. E-mail address: [email protected]. 0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.04.049
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ωb, Ωb
ω a, Ωa
(a)
ωb, Ωb
ω a, Ωa
(c)
ωb, Ωb
ω a, Ωa
(b)
ωc, Ωc
ωb, Ωb ω a, Ωa (d)
Fig. 1. Linkage pattern in some simple STIRAP systems: (a) three-state K system; (b) 3-state V system; (c) three-state ladder system; (d) a 4-state K system. State j1æ is the initial state and the desired final state is (a)–(c) state j3æ, (d) state j4æ. The two or three laser fields are distinguished by letters and are shown next to the transition that they most impact upon, though in principle each interacts with every linkage. Quantities x are angular frequencies of the laser fields and X are Rabi frequencies.
when interacting with two laser fields with frequencies that approximate the Bohr frequencies of the transitions, are described by the Hamiltonian array 0 1 2D1 Xa 0 hB C H ¼ @ Xa 2D2 Xb A; ð1:1Þ 2 0 Xb 2D3 where the Xl are Rabi frequencies and the Dn are cumulative detunings at each level.1 The differences between the patterns of state energies enter in the definition of the Dn, as detailed in Chapter 13 of Shores book [2]. The two zeros in Eq. (1.1) are a consequence of the absence of a direct connection between states j1æ and j3æ; situations where such a connection exists (e.g. [3,4]) have non-zero entries for these elements. The extension to the 4-state system of Fig. 1(d) is plain. This paper is concerned with the description of two-photon processes in situations where the rotatingwave approximation may be expected to apply. The main result, the Hamiltonian for a two-photon transition, has been known for decades ([2,5] and references therein); the new feature presented here is its derivation from a Hamiltonian to which the RWA has been applied. That is, the aim of this paper is to bring two-photon transitions into the RWA purview, so to speak, so that one can write down by inspection the equivalent of Eq. (1.1) when one or more of the transitions is two-photon. The key to achieving this is the recognition that an interaction exists between adiabatic elimination, a natural step in the treatment of a two-photon transition, and the rotating-wave approximation, the consequence of which is the incorrect dropping of terms from the Hamiltonian when the RWA is applied prior to adiabatic elimination. To the best of the authors knowledge, this interaction has not been previously noted. Application of the two approximations in the opposite order (i.e., adiabatic elimination first) results in a Hamiltonian that agrees with expectations. As well as showing the consistency of RWA theory in the two-photon case, this work 1 In an attempt to reduce confusion, the Rabi frequencies are identified by letters. Full definitions and details of the notation are given in Section 4.
M.P. Fewell / Optics Communications 253 (2005) 125–137
127
also indicates that care must be taken in using an RWA Hamiltonian when one state is well off resonance; for this is equivalent to adiabatically eliminating the state. The work reported here was originally motivated by an issue in stimulated Raman scattering with adiabatic passage (STIRAP) [1,6–12]: the search for a STIRAP system in which the intermediate state has no spontaneous decay. (This is in addition to the initial and final states which, for practical reasons, must also not decay.) Such a system would be a useful laboratory for testing experimentally several predictions concerning coherent population transfer [13–15]. If one restricts oneself to single-photon transitions, then it is clear that such a system cannot exist; for the dipole couplings from the intermediate state to the initial and final states inevitably produce spontaneous decay. Thus, one is led to consider three-state systems in which both pump and Stokes couplings are two-photon. The M = 2, M = 0 and M = +2 projection sublevels of a J = 2 metastable state are an example. Such a system is denoted 2 + 2 STIRAP (e.g. [16]), to indicate that both transitions of the STIRAP process are two-photon. In the interests of simplicity, however, the theory is developed for 2 + 1 STIRAP, in which only the pump transition is two-photon. Also, the 2 + 1 system has been studied theoretically [16–18] and experimentally [19], whereas there has yet to be a successful experimental realisation of the 2 + 2 system. For definiteness, the discussion here deals with the K-type system shown in Fig. 1(d), since this arrangement of level energies matches the experimental case.2 The three levels of the STIRAP system are labelled j1æ, j3æ and j4æ; level j2æ represents the virtual level that provides the basis of the theoretical description of the two-photon process. The question is how to describe this system theoretically in the context of the RWA. The natural answer is first to consider level j2æ as a real level, giving a four-state problem. The Hamiltonian of such a system is clear. One then allows the detuning at level j2æ to become large and adiabatically eliminates the level, thereby effecting the transition to a Hamiltonian describing a 2 + 1 system. Section 2 presents this analysis and Section 3 indicates why the result is wrong. Section 4 shows the improvement obtained by performing the adiabatic elimination before applying the RWA, with the derivation of the two-photon RWA Hamiltonian being completed in Section 5. The results of the corresponding analysis of the 2 + 2 system are stated in Section 6. Section 7 contains a brief discussion of conditions required for the existence trapped states and Section 8 is a summary of the conclusions.
2. Working from the four-state RWA Hamiltonian If we take level j2æ in Fig. 1(d) to be a real level, then the system involves three single-photon couplings. Such a system was first treated in the context of STIRAP in 1992 [20] and its RWA Hamiltonian is well known (e.g. [2,20–22]). In the RWA, Schro¨dingers equation reads 0_ 1 0 10 1 C1 0 0 0 Xa C1 B C_ C 1 B X 2D C B Xb 0 CB C 2 C 2 B 2C B a C iB C ¼ B ð2:1Þ CB C; @ C_ 3 A 2 @ 0 Xb 2D3 Xc A@ C 3 A 0 0 Xc 2D4 C4 C_ 4 where the symbols have their usual meanings [2,20] – in the Hamiltonian matrix, the Dn are cumulative detunings at each level and the Xl are Rabi frequencies. 2 In most, perhaps all, experimental realisations of two-photon processes, the two photons come from the same laser field, i.e., xb = xa in the present notation. However, there is no inherent increase in complication in treating the more general case, so the distinction between the two photons is maintained throughout this analysis.
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We now suppose that D2 is large and adiabatically eliminate state j2æ. A widely used prescription for this is to set C_ 2 to zero and solve the resulting equations for C2, so eliminating C2 from the remaining three equations of Eq. (2.1).3 The result is 0 X2 10 1 0 1 Xa Xb a 0 C1 C_ 1 D2 D2 CB C B _ C iB 2 B C X ¼ ð2:2Þ X X @ C3 A @ C 3 A. a b b 4 @ D2 D2 4D3 2Xc A C4 C_ 4 0 2X 4D c
4
The diagonal elements in Eq. (2.2) give the detunings, including the dynamic Stark shifts caused by the pulses, and the off diagonal elements give the two-photon couplings. The following section argues that the diagonal elements must be wrong. In fact, the off-diagonal elements are also incorrect, as shown at the end of Section 4. This is related to the accuracy of the above method of adiabatic elimination, a point dealt with in Section 5. 3. Dynamic stark shifts According to Eq. (2.2), the dynamic (or ac) Stark shift DE1(x) of the initial level is X2a . ð3:1Þ 4D2 Now consider what happens when the system encounters a set of pulses in the so-called counterintuitive order characteristic of STIRAP. In this case, pulse c must lead. That is, at first Xa and Xb are zero, only Xc is present. Eq. (3.1) says that the initial state experiences no dynamic Stark shift during this time, no matter how strong Xc may be. Physical intuition tells us that this cannot be right. However, intuition is sometimes misleading, so it is as well to exhibit the errors in detail. The remainder of this section undertakes this task. Any textbook on the subject (e.g. [2], p. 282) gives the dynamic Stark shift DEn(x) of level n as E2 X ðdnk e Þðdkn eÞ ðdnk eÞðdkn e Þ þ DEn ðxÞ ¼ . ð3:2Þ E k En Ek En þ hx 4 k hx DE1 ðxÞ ¼
Here, dnk is a dipole matrix-element vector and e is the polarisation vector of the electric field, which is characterised in addition by its amplitude E and angular frequency x, and k enumerates all states of the system other than state n. Comparison of Eqs. (3.1) and (3.2) raises the following points: (i) The most obvious feature of Eq. (3.2) is the sum over all states. The absence of such a sum in Eq. (3.1) is not a serious problem, however; for level j2æ can be interpreted as representing all other levels of the system. Thus, it is easy to rationalise the rewriting of Eq. (3.1) to include a sum over all states. (ii) On the other hand, it is impossible to rationalise rewriting Eq. (3.1) to include all fields. In Eq. (3.2), E refers to the entire electric field – everything that the system experiences – whereas Eq. (2.2), and therefore Eq. (3.1), clearly state that certain fields act only on certain couplings. This is an important consequence of the RWA. To introduce a sum over fields arbitrarily in Eq. (3.1) would, it seems, be to threaten the self-consistency of the theory. (iii) The denominator D2 of Eq. (3.1) is exactly the denominator of the first term in Eq. (3.2).4 The second term of Eq. (3.2), with the + hx in its denominator, is often called the anti resonant term. It is nowhere to be seen in Eq. (3.1) and there seems to be no rational ground for arguing for its insertion. 3 Recent examples of the use of this procedure occur in [23] (sentences before Eq. (6)) [24] (Section IV.C), [25] (Section VI.A), [26] (Section V). 4 Setting k = 2 and with an appropriate interpretation of x.
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The following section shows that the problem with Eq. (2.2) lies in part in the application of adiabatic elimination to a system of equations to which the RWA has already been applied. 4. Adiabatic elimination prior to application of the RWA As is shown in detail below, performing adiabatic elimination prior to applying the rotating-wave approximation results in the retention of terms that are dropped if the two steps are carried out in the opposite order. In fact, reversing the order is not sufficient to fully resolve the problems with Eq. (2.2); a more accurate method of adiabatic elimination is also required. However, in the interests of reducing the complexity of the equations so as to make the exposition as clear as possible, we retain in this section the method of adiabatic elimination used in Section 2. This section follows the method and notation of Shore ([2], Chapter 13). 4.1. Setting the theory up The total electric field E comprises the sum of the three fields shown schematically in Fig. 1: ( ) c X ixl t E ¼ Re El e el ;
ð4:1Þ
l¼a
where El , xl and el are, respectively, the (real) amplitude, angular frequency and polarisation vector of each field. Transition strengths between levels are quantified by dipole matrix-element vectors dnk. It is assumed that levels are coupled only to their neighbours; that is, dnk ¼ 0
if k 6¼ n 1.
ð4:2Þ
The polarisations of the fields can further reduce the couplings in that dnk Æ el may be zero for particular values of n, k and l. However, we do not make explicit use of this in the following. The state vector W of the system is expanded on the basis of the unperturbed states jnæ of the system 4 X WðtÞ ¼ C n ðtÞeifn ðtÞ jni; ð4:3Þ n¼1
where Cn(t) are the expansion coefficients and fn(t) are arbitrary phases to be chosen so as to facilitate the purpose at hand. As always with RWA theory, the choice of these is crucial and is considered in detail in Section 4.2. Substitution of Eqs. (4.1)–(4.3) into Schro¨dingers equation yields a set of four coupled differential equations, the first of which is X X21l eixl t þ X eixl t 12l ð4:4Þ iC_ 1 ¼ D1 C 1 þ C 2 eiðf1 f2 Þ 2 l and the second and third are X Xknl eixl t þ X eixl t X nkl C k eiðfn fk Þ iC_ n ¼ Dn C n þ 2 l k¼n1;nþ1
ðn ¼ 2; 3Þ.
ð4:5Þ
In these equations, En is the unperturbed eigenenergy of state jnæ, the Dn are the usual portmanteau variables for the coefficients of Cn: hDn ¼ En hf_ n ; ð4:6Þ and the Xs are the usual Rabi frequencies, but with an extension of notation in the interests of clarity5 5
Note the order of the subscripts on the right-hand side of Eq. (4.7).
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hXnkl ¼ dkn el El .
ð4:7Þ
4.2. Choosing phases with the RWA in mind A suitable choice of phases simplifies Eqs. (4.4) and (4.5). Since we are considering the case in which state j2æ lies between states j1æ and j3æ, the choices: f_ 2 ¼ f_ 1 þ xa ;
f_ 3 ¼ f_ 2 þ xb
are appropriate (Section 13.1 of [2]). Eqs. (4.4) and (4.5) then become " # X C 2 X21a e2ixa t þ X12a þ X21l eiðxa þxl Þt þ X12l eiðxa xl Þt iC_ 1 ¼ D1 C 1 þ 2 l¼b;c and
" # X C 1 2ix t iðx x Þt iðx þx Þt a a a l l iC_ 2 ¼ D2 C 2 þ X12a þ X21a e þ X21l e þ X12l e 2 l¼b;c " # X C3 X32b e2ixb t þ X23b þ X32l eiðxb þxl Þt þ X23l eiðxb xl Þt . þ 2 l¼a;c
ð4:8Þ
ð4:9Þ
ð4:10Þ
The Dn are related because of Eq. (4.8) and the corresponding equation for f_ 4 . For reference, their relationships are: hD2 ¼ E2 E1 hxa þ hD 1 ; D3 ¼ E3 E1 h hxa hx b þ hD1 ; hD4 ¼ E4 E1 hxa hx b þ hxc þ hD1 .
ð4:11Þ
4.3. The elimination step It is usual at this stage to apply the rotating-wave approximation; that is, to drop all terms containing complex exponential factors. However, the point of the present work is to adiabatically eliminate state j2æ first. In this Section, this is done by the same method as used in Section 2, that is, by setting derivatives to zero: we set the left-hand side of Eq. (4.10) to zero and solve for C2. The result is then used to eliminate C2 from all other equations. To demonstrate explicitly how this gives rise to new terms, an intermediate step is shown in detail: insertion of the expression for C2 into Eq. (4.9) gives, after a slight rearrangement, ( c c X iðxa x Þt C1 X iðxa þxl Þt l _ iC 1 ¼ D1 C 1 X12l e þ X21l e X12l0 eiðxa xl Þt þ X21l0 eiðxa þ xl Þt 4D2 l0 ¼a l¼a ) c C3 X X32l0 eiðxb þxl Þt þ X23l0 eiðxb xl Þt . þ ð4:12Þ 4D2 l0 ¼a Now consider the first two sums. Had the rotating-wave approximation already been applied, only the first term of each sum (that with l = a or l 0 = a) would be present, and the product of the two sums would be exactly the first term of the first line of Eq. (2.2). However, it is now apparent that every term in the first sum of Eq. (4.12) has a partner in the second sum such that the complex exponential factors cancel when the two sums are multiplied out. This means that five additional terms survive the application of the RWA.
M.P. Fewell / Optics Communications 253 (2005) 125–137
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A similar effect occurs when sum 1 of Eq. (4.12) is multiplied by sum 3, this time generating one additional term. 4.4. The Hamiltonian Multiplying Eq. (4.12) out and dropping the rapidly oscillating terms gives c C1 X C3 2 iC_ 1 ¼ D1 C 1 X12a X32b þ X21b X23a . jX12l j 2D2 l¼a 4D2
ð4:13Þ
The full Hamiltonian can now be written down by analogy. The result for Schro¨dingers equation is: 1 0 c P X12a X23b þX12b X23a 2 2 0 1 0 1 jX j 0 D2 C C1 B D2 l¼a 12l C_ 1 C B CB C B _ C iB c ð4:14Þ C@ C 3 A; @ C 3 A ¼ B X12a X23b þX12b X23a 2 P 2 jX j 4D 2X 4B 23l 3 34c C D2 D2 A C4 @ _C 4 l¼a 0 2X43c 4D4 where f1, the last remaining arbitrary phase, has been chosen to make D1 zero, as usual for this ordering of energies. In comparing Eq. (4.14) to Eq. (2.2), we see that the additional terms on the diagonal go some way toward answering point (ii) in Section 3. However: The dynamic Stark shifts are now twice the magnitude expected from Eq. (3.2). The anti resonant terms still have not been generated. As might be suspected, these two effects are related. Also, the extra off-diagonal terms do not have the right form – Eq. (14.9–16a) in [2] suggests that these should have anti resonant denominators. The problem lies in the manner in which the adiabatic elimination was performed: simply setting derivatives to zero is not sufficiently accurate when dealing with two-photon effects. The method in Section 14.9 of [2] provides the model for how to proceed; this is adapted to the present case in the following section. 5. A more accurate adiabatic elimination We seek to solve Eq. (4.5) for the states to be eliminated by a technique that mimics direct integration as closely as possible. That is, we write Z t C n ðtÞ ¼ ð5:1Þ C_ n ðt0 Þ dt0 ; 0
where we use the fact that, for states being eliminated, Cn(0) = 0. Eq. (5.1) is, of course, formally tautologous; the issue is to evaluate the integral. Choice of phase is important for this; as discussed below, the standard RWA choice is not appropriate. 5.1. Choice of phase – states being eliminated If state n is to be eliminated, an expression for Cn(t) is required. This is ultimately obtained by inserting Eq. (4.5) into Eq. (5.1), but the difficulty caused by the DnCn term is clear. This term owes its form to the choice of phase for state n, and so may be removed by appropriate adjustment of this choice. Hence, for any state n that is to be eliminated, we must choose
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hf_ n ¼ En ;
ð5:2Þ
so that Dn = 0. With this choice of phase, and following the method of Shore (Section 14.9 of [2]), we write Eq. (5.1) for any state being eliminated as c i X X ½J ðtÞ þ Jnkl ðtÞ; ð5:3Þ C n ðtÞ ¼ 2 k¼n1;nþ1 l¼a nkl where Jnkl ðtÞ ¼
Z
t
0
0
C k ðt0 ÞGknl ðt0 Þei½fk ðt ÞEn t =h dt0
ð5:4Þ
0
with Gknl ðtÞ ¼ Xknl ðtÞeixl t .
ð5:5Þ
The Jnkl ðtÞ differ from the Jnkl ðtÞ only in containing the complex conjugates of the field quantities:6 Z t 0 0 Jnkl ðtÞ ¼ C k ðt0 ÞGknl ðt0 Þei½fk ðt ÞEn t =h dt0 . ð5:6Þ 0
The phases fk have deliberately not been inserted in Eqs. (5.4) and (5.6) because, for a two-photon transition, states k cannot be candidates for elimination if state n (= k ± 1) is being eliminated, and so Eq. (5.2) does not apply to states k. This is a consequence of the assumed pattern of linkages between the levels. 5.2. Phases of states not being eliminated The definition of Gknl given in Eq. (5.4) has been chosen so as to cast Eqs. (5.4) and (5.6) into a form suitable for application of the method of Shore [2], which has the advantage of making explicit the approximations that required to arrive at a solution. This Section details the evaluation of the integral in Eq. (5.4); the method for Eq. (5.6) is identical. Following Shore, Cn and Gknl in Eq. (5.4) are replaced by their representations as Fourier integrals Z 1 ~ k ðxÞeixt dx C k ðtÞ ¼ ð5:7Þ C 1
and similarly for Gknl. A rearrangement of the order of integration in Eq. (5.4) gives7 Z 1 Z 1 Z t 0 0 0 0 0 ~ knl ðx0 Þ ~ k ðxÞG Jnkl ðtÞ ¼ ei½fk ðt Þþxt þx t En t =h dt0 dx0 dx. C 1
1
ð5:8Þ
0
To allow evaluation of the innermost integral in Eq. (5.8), fk is chosen to be a linear function of time. The choice in Eq. (5.2) has this property, but so does the conventional choice of the RWA (Eq. (4.8)). To maintain generality, we write fk ðtÞ ¼ f_ k t; ð5:9Þ _ where the fk are understood to be independent of time. Their values are specified further on in the derivation. This choice of phase in Eq. (5.9) allows the evaluation of the innermost integral of Eq. (5.8). Application of the approximations detailed by Shore (p. 917 of [2]) gives
6 7
Note the order of the subscripts on the right-hand side of Eq. (5.6). The quantities x,x 0 are, of course, Fourier conjugate variables quite distinct from the externally applied optical frequencies xl.
M.P. Fewell / Optics Communications 253 (2005) 125–137
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_
C k ðtÞXknl ðtÞeiðfk þxl En =hÞt C k ð0ÞXknl ð0Þ Jnkl ðtÞ ¼ i . f_ k þ xl En = h
ð5:10Þ
The second term in Eq. (5.10) represents initial conditions. Since the population resides in state j1æ initially, Ck(0) = dk,1. To avoid having to retain the initial condition when k = 1, we place the origin of the time coordinate at a time when all fields coupling to state j1æ are zero. That is, we choose t = 0 so as to ensure that X1nl(0) = 0 for all n,l. The Jnkl ðtÞ contain complex conjugates of the field quantities only, so the expression corresponding to Eq. (5.10) is not obtained by simply replacing the Xknl by their complex conjugates. Not only must the order of subscripts k and n be reversed in recognition of the fact that the matrix elements are not conjugated, we must also allow for the fact that conjugated field quantities refer to the negative-frequency part of Fourier space. Hence, _ C k ðtÞXknl ðtÞeiðfk xl En =hÞt Jnkl ðtÞ ¼ i . ð5:11Þ f_ k xl En = h The different signs in the denominators of Eqs. (5.10) and (5.11) are the source of the two sorts of terms in Eq. (3.2). 5.3. The elimination step and values of the phase rates We are now in a position to assemble an expression for C2(t), by inserting Eqs. (5.10) and (5.11) into Eq. (5.3) and setting n = 2. The result is used to eliminate C2(t) from the other lines of Schro¨dingers equation. For example, the expression for C_ 1 is h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X e X X þ X e 0 0 X 12l 21l 21l 12l 21l 12l C1 þ iC_ 1 ¼ D1 C 1 þ ; 4 ll0 : f_ 1 þ xl E2 =h f_ 1 xl E2 =h h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X e X X þ X e 0 0 X 32l 21l 21l 12l 23l 12l C3 _ _ þ þ eiðf1 f3 Þt . : ; 4 f_ 3 þ xl E2 =h f_ 3 xl E2 =h ll0 ð5:12Þ
This time, application of the RWA leaves 12 terms remaining in the first sum, 6 with each denominator. Before applying the RWA to the second sum, the remaining phase rates must be chosen. To give a result that matches as closely as possible the conventional expressions, it seems best to adopt the usual RWA convention, f_ 1 f_ 3 ¼ xb xa ;
ð5:13Þ
and to make the standard assignment of E1t/h for f1. For convenience, we define a frequency w2 by hw2 ¼ E2 E1
ð5:14Þ
(remembering that E2 > E1 in our system). All this reduces Eq. (5.12) to h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X X X þ X 0e 0e X 12l 21l 21l 12l 21l 12l C1 iC_ 1 ¼ þ ; 4 ll0 : w2 xl w2 þ xl h i h i9 8 iðxl þxl0 Þt iðxl0 xl Þt iðxl xl0 Þt iðxl þxl0 Þt < = 0e 0e X X þ X X X þ X 0e 0e X 32l 21l 21l 12l 23l 12l C3 þ eiðx2 x1 Þt . : ; 4 w2 x1 þ x2 xl w2 x1 þ x2 þ xl ll0 ð5:15Þ
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The terms with zero exponential phase are: in the first sum, all terms with l = l 0 ; in the second sum, term 2 with l = 2, l 0 = 1 and term 3 with l = 1, l 0 = 2. For completeness, the values of f and D resulting from the present choices of phase are listed below: h; f1 ¼ E1 t=
hD1 ¼ 0;
f2 ¼ E2 t= h; hD2 ¼ 0; _f3 ¼ f_ 1 þ xa þ xb ; hD3 ¼ E3 E1 hxa hxb ; f_ 4 ¼ f_ 3 xc ; hD 4 ¼ E 4 E 1 hxa hxb þ hxc .
ð5:16Þ
The phases and D values of the retained levels are the same as in the standard RWA analysis, but that of the eliminated level is different. That is, the values of D3 and D4 still give the cumulative detunings at the respective levels; that of D2 does not. It is interesting to observe that the value of D2 required by the derivation of Eq. (5.12) is inadmissible in the derivation leading to Eq. (4.12). 5.4. Construction of the Hamiltonian We now drop the rapidly oscillating terms in Eq. (5.15) to give one line of the RWA Hamiltonian array. The result is C 1 X jX12l j2 jX12l j2 iC_ 1 ¼ þ 4 l w2 xl w2 þ xl
!
C 3 X12a X23b X12b X23a þ . 4 w2 xa w2 þ xb
ð5:17Þ
The chief difference from Eq. (4.13) lies in the denominators: both resonant and anti resonant denominators are now present. The resonant denominators are not identical to the denominator in Eq. (4.13); for, with the choice of phase applying there, D2 = w2 xa. In the sum in Eq. (5.17), the resonant denominator contains the frequency of the field concerned rather than using xa for all terms. In attempting to generate the full Hamiltonian by analogy from Eq. (5.17), care is needed to track correctly the manner in which the denominators vary. The details are not reproduced here, but the resulting Schro¨dingers equation is 0
1
0
c P jX12l j2
2
j þ wjX221l w2 xl þxl
B l¼a C_ 1 B B _ C iB @ C 3 A ¼ B X12a X23b X12b X23a 4 B w2 xa þ w2 xb @ C_ 4 0
X12a X23b w2 xa
þ
X12b X23a w2 xb
c P jX32l j2 l¼a
jX23l j2 þ 4D3 w3 xl w3 þxl 2X43c
1
0 1 C C1 C CB C C@ C 3 A; 2X34c C A C4 4D4 0
ð5:18Þ
where the additional variable hw3 ¼ E2 E1 hxa hx b
ð5:19Þ
is used. Written this way, one can begin to see a pattern in the denominators of the various terms, and also in the order of subscripts in the Rabi frequencies. This becomes clearer on comparison with the results for the 2 + 2 system presented in the following section. Eq. (5.18) has all the properties mentioned in Section 3 and at the end of Section 4.4 as required in a correct expression for a two-photon linkage.
M.P. Fewell / Optics Communications 253 (2005) 125–137
135
6. Results for a 2 + 2 system The 2 + 2 system with the energy-level relationship shown in Fig. 2 is of the double K or M type, of interest as a potential laboratory for testing certain predictions concerning the STIRAP process. The method of the preceding section applied to this system gives the Hamiltonian. 0 1
d P X12a X32b X21b X23a jX12l j2 jX21l j2 þ þ 0 B C w2 xa w2 þxb 0 1 B l¼a w2 xl w2 þxl C C_ 1 B C
d B C P jX32l j2 jX23l j2 jX34l j2 jX43l j2 X34c X54d X43d X45c B _ C i B X12a X32b X21b X23a C ¼ þ þ þ þ þ 4D @ C3 A 3 w2 xa w2 þxb w3 xl w3 þxl w4 xl w4 þxl w4 xc w4 þxd B C 4B l¼a C B C C_ 5
d P @ A X34c X54d X43d X45c jX54l j2 jX45l j2 0 þ þ 4D 5 w4 xc w4 þxd w5 xl w5 þxl l¼a 0 1 C1 B C ð6:1Þ @ C 3 A. C5 To assist in capturing the regularities that allow such Hamiltonians to be written down from an inspection of the energy-level diagram, the phases, D values and w values applying in Eq. (6.1) are listed below: f1 ¼ E1 t= h;
hD1 ¼ 0;
f2 ¼ E2 t= h; hD2 ¼ 0; D3 ¼ E3 E1 hxa þ hxb ; h f_ 3 ¼ f_ 1 þ xa xb ; f4 ¼ E4 t= h; hD4 ¼ 0; hD5 ¼ E5 E1 hxa þ hxb hxc þ hxd f_ 5 ¼ f_ 3 þ xc xd ;
ð6:2Þ
and hw2 ¼ E2 E1 ; hw3 ¼ E2 E1 hxa þ hxb ;
ð6:3Þ
w4 ¼ E4 E1 h hxa þ hxb ; hw5 ¼ E4 E1 hxa þ hxb hx c þ hxd .
7. Conditions for
STIRAP
It is well known that, for STIRAP in the strict sense to be possible, the Hamiltonian must have an eigenstate with zero eigenvalue [1,11,17]. This state can then act as a trapped state, in which population can be
ωa, Ωa
ωb, Ωb
ωc , Ωc
ωd, Ωd
Fig. 2. As in Fig. 1, but for double-K or M type 2 + 2
STIRAP.
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retained by suitable manipulation of the laser pulse ordering. In the case of the 2 + 1 system, the Hamiltonian of Eq. (5.18) can be rewritten as 0 1 ~p S1 2X 0 hB ~ S 3 4D3 2Xc C ð7:1Þ H ¼ @ 2X A; p 4 0 2Xc 4D4 ~ p is the effective pump Rabi frequency.8 Recall where the Sn are the dynamic Stark shifts of level n and X that D3 is the two-photon detuning at the intermediate level and D4 is the cumulative three-photon detuning at the final level. The condition for this Hamiltonian to have a zero eigenvalue is 2
~ D4 þ X2 S 1 4D3 D4 S 1 þ D4 S 1 S 3 ¼ 0. 4X c p
ð7:2Þ
Since the Stark shifts and Rabi frequencies all vary with laser intensity, it is not possible for this equality to be maintained at all times during a pulse sequence. This question has been explored in detail by Yatsenko et al. [17] and Gue´rin et al. [18]. They find that high transfer probabilities can be obtained if the detunings Dn are carefully chosen, a prediction that has experimental confirmation [19]. In the 2 + 2 system (Section 6), the final state also has a dynamic Stark shift. A corresponding analysis applied to the Hamiltonian of Eq. (6.1) yields the condition for a zero eigenstate as ~ 2 D5 þ X ~ 2 S1 X ~ 2 S 5 4D3 D5 S 1 þ D3 S 1 S 5 þ D5 S 1 S 3 1S 1 S 3 S 5 ¼ 0; 4X p S p 4
ð7:3Þ
~ S is the effective Stokes Rabi frequency. Presumably the analysis of Yatsenko and Gue´rin et al. where X could be extended to cover this case. 8. Conclusions The formalism presented above shows how to derive Hamiltonians with two-photon linkages in situations where the rotating-wave approximation can be expected to apply. The key new result is the identification of an interaction between the rotating-wave approximation and adiabatic elimination: to obtain correct results, adiabatic elimination must be applied before the rotating-wave approximation. Application of the two approximations in the opposite order results in terms being dropped that ought to be retained. Comparison of Eqs. (5.18) and (6.1) suggests some patterns from which rules describing the structure two-photon RWA Hamiltonians may be postulated. Thus, the goal of developing a formalism that allows Hamiltonians of systems with two-photon linkages to be written down by inspection is in sight. But perhaps a few rather more elaborate cases should be worked out from first principles to give confidence in the heuristic procedures. Acknowledgements The author thanks Dr. B.W. Shore for drawing his attention to the question of the form of the RWA Hamiltonian with two-photon transitions, Professor K. Bergmann for his warm hospitality at the Universita¨t Kaiserslautern, where this work was begun, and members of the Arbeitsgruppe Bergmann for much stimulating discussion. This work was supported by the Australian Government Department of Industry, Science and Technology under the Bilateral Science and Technology Program with Germany, and by the Universita¨t Kaiserslautern Graduierten Kolleg Laser und Teilchen Spektroskopie. 8
These quantities are defined by simple comparison with Eq. (5.18).
M.P. Fewell / Optics Communications 253 (2005) 125–137
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