Journal ArticleA computation with unprecedented precision is presented for quasienergies of a two-level atom in a monochromatic  radiation on the basis of a recently obtained exact expression [D.-S. Guo et al., Phys. Rev. A 73,023419 (2006)]. We start with the proof of an expression theorem. With this theorem the quasienergies for any  two-level atom can be expressed in terms of the quasienergies for only those with the original energy spacing (per field photon energy) being an integer (preresonances). Then we carry out a numerical evaluation of the quasienergies at the two-photon preresonance, which involves computing an infinite determinant, up to the 18th power of the coupling strength. The theoretical prediction presents an experimental challenge for highprecision tests of quantum mechanics and could be exploited for precise calibration of high laser intensities

Guo, Dong-Sheng

Wu, Yong-Shi

English

The University of Utah: J. Willard Marriott Digital Library

PHYSICAL REVIEW A 77, 025401 (2008)High-precision quasienergies for a driven two-level atom at the two-photon preresonanceDoiig-Sheng Guo,' J. T. Wang,' and Yong-Shi Wu21 Department o f  Physics, Southern University and A& M  College, Baton Rouge, Louisiana 70813, USA ‘Department o f  Physics, University o f  Utah, Salt Lake City, Utah 84112, USA (Received 14 March 2007; published 4 February 2008)A computation with unprecedented precision is presented for quasienergies of a two-level atom in a mono­chromatic radiation on the basis of a recently obtained exact expression [D.-S. Guo et al., Phys. Rev. A 73,023419 (2006)]. We start with the proof of an expression theorem. With this theorem the quasienergies for any two-level atom can be expressed in terms of the quasienergies for only those with the original energy spacing (per field photon energy) being an integer (preresonances). Then we carry out a numerical evaluation of the quasienergies at the two-photon preresonance, which involves computing an infinite determinant, up to the 18th power of the coupling strength. The theoretical prediction presents an experimental challenge for high- precision tests of quantum mechanics and could be exploited for precise calibration of high laser intensities,DOI: 10,1103/PhysRevA.77.025401 PACS number(s): 32.80.RmIn a recent paper [I], a closed expression for quasiener­gies exhibiting explicit Floquet periodicity, together with the corresponding wave functions, is derived for a two-level atom in a monochromatic radiation field without any restric­tion on the range of the parameters. The study of the two- level atom [2 ] in a radiation field has occupied a central position in quantum optics (see, e.g., [3-6]). The ultimate goal of quantum optics is to achieve a complete understand­ing of atoms interacting with radiation fields. To reach this goal, the most important thing is to obtain accurate wave functions with corresponding accurate quasienergies for an arbitrary atom (or ion) interacting with a radiation field. A real atom (or ion) has an infinite number of energy levels, which may include a finite number of bound-state energy levels (occupied or nonoccupied ground states), an enumer­able infinite number of excited states (occupied or nonoccu­pied Rydberg states), and a nonenumerable infinite number of states in its continuous spectrum. With present computing techniques, all continuous quantities are discretized to sets of finite numbers. Thus the iV-level atom model with an arbi­trarily large N  number is an appropriate way to describe a real atom.The importance of the model of a two-level atom in a radiation field is in the following two aspects, (i) The model can apply to many physical systems directly—e.g., atoms in a laser field when the laser frequency is near a transition frequency between two levels of a real atom and a two-level system of magnetic resonances, (ii) The model is the first step in the development of an accurate theory for the iV-level atom in radiation fields: Analytic or numerical techniques for solving a driven two-level system may be generalized to the driven iV-level system; and as a necessary condition, the ana­lytical and numerical solutions for a driven two-level atom will provide strong checks for the correctness of the solu­tions for a driven AMevel atom. Now, obtaining accurate ana­lytic and numerical solutions for an arbitrary AMevel atom is in our short-term plan. Thus an accurate numerical solution for a two-level atom plays a key role in our current and future researches.There is a vast literature on the two-level atom in a mono­chromatic classical field described by the Bloch equation [7].Analytic methods with various approximations have been used (see, e.g., [3-6]), while numerically accurate, direct in­tegration of the Bloch equation in strong radiation fields still remains elusive, because the pertinent time-dependent differ­ential equations involve discrete quasienergies and discrete boundary conditions to be determined.The closed expression for quasienergies obtained in Ref. [1] involves infinite determinants. Numerical evaluations of these infinite determinants involve some complicated alge­braic manipulations, such as multiple infinite series and infi­nite products, which cannot be found in any mathematics textbook. One of the main purposes of the present paper is to demonstrate, with physically interesting examples, how to evaluate these infinite determinants in a numerically satisfac­tory fashion. Moreover, the precise numerical values of quasienergies provide theoretical predictions which can be put to precise experimental tests and exploited to calibrate laser intensities. The techniques developed here are useful for evaluating quasienergies and wave functions at any pre­resonance. The special case we are presenting here is when the level spacing 2A is twice the photon energy w of the radiation. We call this the two-photon preresonance, which has been proven [1] to be not a true resonance. (The true resonances occur only when the radiation-s/i(/te</ level spac­ing is an integral multiple of the laser-photon energy, such as Freeman resonances in above-threshold photoelectron spec­tra [8 ].) However, as shown below, the preresonance cases play a very important role in calculating quasienergies for a generic value of 2 A /w¥=n, since those quasienergies can be expressed in terms of all quasienergies only at the prereso­nances.Theorem 1. The quasienergies of a driven two-level atom whose original spacing (before interacting with the driving radiation field) is a nonintegral multiple of the photon energy of the driving field can be expressed in terms of the quasien­ergies of all two-level atoms with an integral original spacing interacting with the same radiation field.Proof. In Ref. [1], the quasienergy when the original spac­ing 2A=£ integer (we set w = l, as in Ref. [1]), the quasiener­gies can be expressed as any one of the following two ex­pressions:1050-2947/2008/77(2)/025401 (4) 025401-1 ©2008 The American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 77, 025401 (2008)E  = —cos \/cos‘(irA) + t t R + sin(2irA) (2A #  n),7T1E = — sin vsin‘(7rA)-  7r/?+ sin(27rA) (2A #  «), (1)fl*A-l4A_4A -  ( 2 k -  1 ) jCOS2(77£2A-l)4A4A2 -  (2k):;&m2(TrE2k) (5)where, for notation simplification, we do not write the addi­tive arbitrary integer explicitly. In the square root of above equations, R+ has an expression4AA4- — t4A2 -  n(2)where the residues r„ relate to the quasienergies of the two- level atoms with the original spacing 2A = integer asE 2k-1  = -c o s  1 (ttV- r2JM) (2 A = 2k -  1),77E 2k = ~sin 1 (-tt-V— r 2A.) (2A = 2k) (k=  1 ,2 ,3 ,...), (3)TTwhich can be converted to'M- 1 = ^ c o s 2(77£2A'-i) (2A = 2k -  1),r2k = - — mr(7TE2k) (2A = 2k) (k=  1 ,2 ,3 ,...). (4)Thus we haveNow, the quasienergies for any two-level atom with an arbi­trary original spacing are expressed in terms of the quasien­ergies of all preresonance cases through Eqs. (1), (3), and (5). Q.E.D.According to this theorem, the entire calculations of the quasienergies of two-level atoms, where the original spacing 2A is an arbitrary number, greatly reduce to the calculations in the preresonance cases, where 2A = integer. The applica­tion of this theorem is not limited to the exact solutions obtained in [1]. For example, one may develop other solution techniques to get the quasienergies from other methods with­out calculating the residues /•„, such as solving the differen­tial equation set directly. This theorem tells us that we can just concentrate on the preresonance cases in quasienergy calculations with any kind of solution techniques.In the following, we show the detailed evaluation process for r2. The mathematical technique developed here should be useful for future calculations for __We proceed to present details of evaluating r2. The quasienergy at the two-photon preresonance is given by E2 = ± ^sii'T1 (7r \ —r2), with r2 being the following determinant infinite in the two directions along the main diagonal:J>'2 :3 D  2 -44 D3 X 52D1 X 33D2 x 4D92D1 x  300D21D9002D3 x  1D93D4 X 22D3 X 14  D5 X 33D4 X 21(6)025401-2BRIEF REPORTS PHYSICAL REVIEW A 77, 025401 (2008)We simplify this determinant by multiplying and dividing the 1 / 77 2 \row and the column crossing each main-diagonal element 1 E2 = — ~ s n^ I "3 ^  r2 ) ' ®by the same factor, respectively, and then break it into theproduct of two subdeterminants with equal value, resulting in We haye developed techniques t0 calculate exactly the coef-r2-~~9 r2 > where >2 is a half-infinite determinant (infinite in ficients of the expansion of r ' in the down-right direction):1D2-  1 --  1 -D71DD51DThe quasienergies are now given byD61(7)r'2= 1 -  K xD2 + K2D4 -  K3D 6 + K4D s + ■■■.in terms of the partial sums1 1 <A(«) = 2 “— 7, = k= 1 *  +  1Define ;*;(«) = l /« (n  + 3); then,=  2  x M ^ — , 2 36(9)(10)(ID(12)^3= 2  x(n)x(n+j)x(n+j + i)i f 4 1 m 1 5 1 2 1 4 1 1 = — I 2  “ 2  ~*Kn) -  2  ~*Kn) + 2  + 3) -  0(3)] + 2  — [*Kn + 3) -  3)] f2 7  [  m=2 m  n=2 «  „=4 «  «=1 «  «=2 n  J1271 7T2-  + —  -  0(4) 4 61 1  1 2  <A(3) + 2  — “ [*/<» + 3) -  vH3)] f -  —  m=l w  + 2 „=1 « I 277T2 1—  vH3) - 2  -<^(« + 4)6  n=2 «= 3.278683997349973... X 10“3,= 2  x(n)x(n + y)/^ + 7 + + ,/ + / + £) = 1.402452866666796 ... X 10 4.n , j , l ,k >  2(13)(14)The above exact expression of K 3 gives the reader a flavor of what is going on. For K4, to save space we did not present its exact expression, but its numerical value to high preci­sion. Similarly the higher-order coefficients are calculated to be [9]andK* = 4.089694836636745 ... X 10“K*= 1.761826370939839... X 10‘i-ii(15)K6= 8.661234370774032... X 10-8, (16)K7 = 1.393524486588925... X 10-9, (17)(18)With these coefficients, the quasienergies can be calculated up to order Z)18. Our method is applicable to any desired higher order.We have carried out a numerical evaluation using the above coefficients. We set D -  0.6 in a sample calculation. (For a hydrogen atom in a laser field of wavelength 243.2 nm, the transition between a 2p  state and the Is state is at two-photon preresonance; when the intensity is 2 X 1014 W /cm 2, the corresponding dipole coupling has ap­proximately the value D - 0.6.)A quasienergy calculated by using above coefficients up to K7 is -0.107 131 400 930 337.... The precision of this evaluation is estimated to be better than 15 figures: From Eq. (8) the error due to neglecting Ks and higher-order terms satisfies, 1 \SE2\ -13 C0s(7T'£,2)K*p 18 . 1 X IO-15, (19)025401-3BRIEF REPORTS PHYSICAL REVIEW A 77. 025401 (2008)2.5•2.5------------1------------ 1------------ 1------------1------------ 1------------ 1------------1------------0 0.5 1 1.5 2 2.5 3 3.5 4dipole moment square D2FIG. 1. The two thick lines are the main quasienergy levels, the upper one (solid line) and the lower one (dot-dashed line), which reduce to the original two levels, respectively, when the dipole mo­ment vanishes. The thin lines, in each respective group, are quasien­ergy levels derived by the Floquet periodicity.which means the 15 figures for this quasienergy are all ac­curate.To compare, the quasienergy from the usual rotating- wave approximation (RWA) calculation is -1.083 095 189 4 (or -0.083 095 189 4 by the Floquet theorem). Its error is about 22%. The quasienergy from an improved RWA result [1] is -1.120 483 682 2 (o r-0.120 483 682 2 by the Floquet theorem), still with an error of 12%. High precision of the predicted quasienergy is clearly guaranteed by the analyti­cally exact expression.In Fig. 1, we present the quasienergy spectrum, calculated at the two-photon preresonance, versus the dimensionless di­pole coupling squared, D 2. Recall that the energy-level spac­ing is 2 (twice the photon energy) at D 2 = 0 and we have chosen the zero energy to be at the midpoint of the two levels in the absence of a radiation field (i.e., at D = 0 ), so the quasienergy spectrum is symmetric with energy zero. The quasienergy shift, corresponding to the well-known Bloch- Siegert (BS) shift [7], increasing with D 2, is shown by the two thick lines. The thin lines are the quasienergies due to the Floquet periodicity: There are two sets of quasienergies[ 1 0 ], corresponding to the quasi stationary states derived from the original two levels. For any two quasienergies, if their difference (or sum) is always an integral multiple ofphoton energy, then they belong to the same (or the distinct) quasistationary states. A weak probe laser beam with appro­priate frequency (allowing small detuning) can induce a tran­sition between two quasienergies belonging to distinct qua­sistationary states. This may be used to test the predicted quasienergies with high precision. To test the nonperturba- tive nature of our results, we note that a transition induced by a probe beam may happen even between quasienergies with spacing smaller than the original spacing.In the driven two-level atom model, various atomic ef­fects that affect the atomic levels and wave functions show up only through two parameters: the level spacing 2A and the coupling strength D. As shown in Eq. (8 ), the quasien­ergy at the preresonance depends only on the coupling D. With this advantage, the measurements of the quasienergy at the two-photon preresonance may provide a new possibility to test quantum mechanics to high precision.Moreover, since the coupling strength D 2 is proportional to the driving laser intensity, measurements of quasienergy at the two-photon preresonance can also be used for calibration of laser intensities with high precision. For example, for the 2 /7 -1 s transition of a hydrogen atom in two-photon prereso­nance, we haveD 2 = [Dp [2 = 2 n X 3”“8cv—— m„. (20)2 TTUfjHere k  = 2iTlco is the photon wavelength, a = \ / m ea() the fine-structure constant, and up=2ire2l l m (,(jy' is the pondero- motive parameter—i.e., the ponderomotive energy per field- photon energy, with I  being the light field intensity. We see that D 2 is proportional to up or to /, numerically, when the field intensity /= 2 .0 0 x 1 0 14 W /cm 2 (or 1.00X1015 W / cm2), «;)=0.216 (or «;,= 1.08), and /)2=0.360 (or /)2=1.80). The corresponding quasienergies can be easily read from Fig. 1.The demonstration here for evaluating the infinite deter­minant r2 for two-photon preresonance up to high precision gives us confidence that the other preresonance cases can be dealt with in a similar way. According to the theorem proved in this paper, quasienergies for a generic driven two-level atom could be calculated with high precision after the calcu­lation of r,, r4, etc.D.S.G. and J.T.W. were supported in part by the U.S. Navy through Grant No. N00014-05-1-0706. The work of Y.S.W. was supported in part by the NSF through Grant No. PHY-0457018. We thank Y. Wang for his help in these cal­culations.[1] D.-S. Guo, Y.-S. Wu, and L. Van Woerkom, Phys. Rev. A 73, 023419 (2006).[2] A. Einstein, Phys. Z. 18, 121 (1917).[3] J. H. Eberly and P. W. Milonni, in Encyclopedia o f  Physical Science and Technology, edited by R. Myers (Academic Press, New York, 1987), Vol. II, pp. 471-502.[4] C. Cohen-Tannoudji, J. Dupont-Roc, and C. Fabre, J. Phys. B 6, L214 (1973).[5] R. Loudon, The Quantum Theory o f  Light, 2nd ed. (OxfordUniversity Press, New York, 1983).[6] M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Phys­ics (Addison-Wesley, London, 1973).[7] F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).[8] R. R. Freeman, P. H. Bucksbaum, H. Milchberg, S. Darack, D. Schumacher, and M. E. Geusic, Phys. Rev. Lett. 59, 1092 (1987).[9] Y. Wang (private communication).[10] Ya. B. Zel'dovich, Sov. Phys. JETP 24, 1006 (1967).025401-4

High-precision quasienergies for a driven two-level atom at the two-photon preresonance

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High-precision quasienergies for a driven two-level atom at the two-photon preresonance

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