The purpose of this paper is the discussion of a pair of coupled linear
oscillators that has recently been proposed as a model of a system of two
optical resonators. By means of an algebraic approach we show that the
frequencies of the classical and quantum-mechanical interpretations of the
optical phenomenon are exactly the same. Consequently, if the classical
frequencies are real, then the quantum-mechanical eigenvalues are also real

CM Bender

D Chruściński

E Celechini

FA Cotton

FM Fernández

Francisco M. Fernández

H Dekker

M Tinkham

S Klainman

International Journal of Theoretical Physics

English

arXiv

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the quantum-mechanical interpretation of the optical phenomenon are linear combinations of the classical frequencies. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.Fil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico la Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentina. Universidad Nacional de La Plata; Argentin

Fernández, Francisco Marcelo

CONICET Digital

Int J Theor PhysDOI 10.1007/s10773-014-2201-yAlgebraic Treatment of PT -Symmetric CoupledOscillatorsFrancisco M. FernándezReceived: 1 May 2014 / Accepted: 21 May 2014© Springer Science+Business Media New York 2014Abstract The purpose of this paper is the discussion of a pair of coupled linear oscillatorsthat has recently been proposed as a model of a system of two optical resonators. By meansof an algebraic approach we show that the frequencies of the quantum-mechanical inter-pretation of the optical phenomenon are linear combinations of the classical frequencies.Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvaluesare also real.Keywords PT symmetry · Coupled oscillators · Gain and loss · PT transitions · Algebraictreatment · Adjoint matrix representation1 IntroductionIn a recent paper Bender et al. [1] discussed the classical and quantum-mechanical versionsof a system of two coupled linear oscillators one with gain and the other with loss. When thegain and loss parameters are equal the Hamiltonian derived from the classical equations ofmotion is PT -symmetric and exhibits two PT -transitions in terms of the coupling param-eter. In the unbroken-PT region the classical frequencies are real otherwise they appearas pairs of complex conjugate numbers. This analysis is straightforward because one canderive analytical expressions for such frequencies from the equations of motion. The authorsalso obtained analytical expressions for the eigenvalues and eigenfunctions of the quantum-mechanical Hamiltonian and showed that both the eigenvalues of the quantum-mechanicalHamiltonian and the classical frequencies are real in the same region of model parameters.However, at first sight the analytical expressions of the quantum-mechanical eigenvalueson the one hand and the classical frequencies on the other do not resemble each other. Thistheoretical investigation was motivated by recent experiments on a PT -symmetric systemof two coupled optical resonators [2].F. M. Fernández ()INIFTA (UNLP, CCT La Plata-CONICET), División Quı́mica Teórica, Blvd. 113 y 64 S/N, Sucursal 4,Casilla de Correo 16, 1900 La Plata, Argentinae-mail: fernande@quimica.unlp.edu.arInt J Theor PhysThe purpose of the present paper is to derive a clearer connection between the clas-sical frequencies and the quantum-mechanical energies. In Section 2 we apply a wellknown algebraic method and show that the quantum-mechanical frequencies (spacingbetween eigenvalues) are linear combinations of the classical frequencies. We explicitlywrite the quantum-mechanical energies in terms of the classical frequencies and analysethe spectrum. Finally, in Section 3 we review the main results of the paper and drawconclusions.2 Quantum-Mechanical ModelAs indicated above, the analysis of the classical model is straightforward and here we focuson the quantal version in the case of equal gain and loss. The Hamiltonian operator derivedfrom the classical equations of motion is given by [1]H = pxpy + γ (ypy − xpx) +(ω2 − γ 2)xy + ε2(x2 + y2), (1)where px and py are the momenta conjugate to coordinates x and y: [x, px] = [y, py] = i.It is PT symmetric PT HPT = H since it is invariant under the combined effect of par-ity P : {x, y, px, py} → {−y,−x,−py,−px} and time reversal T : {x, y, px, py} →{x, y,−px,−py} [1]. Note that we can also choose P : {x, y, px, py} → {y, x, py, px}for exactly the same purpose. In the language of point-group symmetry such parity trans-formations are given by reflection planes σv and σ ′v perpendicular to the x − y plane[3, 4]. Since the Hamiltonian H is also invariant under inversion ı̂ : {x, y, px, py} ={−x,−y,−px,−py} its eigenvectors |ψ〉 satisfy ı̂ |ψ〉 = ± |ψ〉. This result is consistentwith the solutions of the form Pmn(x, y)e−(2axy+bx2+cy2) obtained by Bender et al. [1] inthe coordinate representation, where the polynomials Pmn(x, y) satisfy Pmn(−x,−y) =(−1)mPmn(x, y). Point-group symmetry proved to be useful for the discussion of broken-and unbroken-PT symmetry in some anharmonic oscillators [5, 6]. As far as we knowKlaiman and Cederbaum [5] were the first to construct a non-Hermitian Hamiltonian withconsecutive PT phase transitions in terms of the coupling parameter. Those authors coinedthe term space-time symmetry when P is different from ı̂ as in the present case.It is worth noting that the Hamiltonian (1) is also a symmetric operator 〈Hψ |ϕ〉 =〈ψ |Hϕ〉, which for brevity we formally express by means of the usual notation for Hermi-tian operators as H † = H . In particular, note that (ypy − xpx)† = (pyy − pxx) = (ypy −xpx). Therefore, one expects real eigenvalues at least for some range of the model parame-ters ω, γ and ε where the above turnover relation holds. In fact, if ψ is an eigenfunction ofH with eigenvalue E then it follows from 〈Hψ |ψ〉 = 〈ψ |Hψ〉 that E∗ = E.In order to solve the eigenvalue equation H |ψ〉 = E |ψ〉 in a way that clearly revealsthe connection with the classical interpretation we resort to a well known algebraic method[7] (see also http://en.wikipedia.org/wiki/Ladder operator). It is suitable when there existsa set of symmetric operators {O1,O2, . . . , ON } that satisfy the commutation relations[H, Oi] =N∑j=1HjiOj . (2)Int J Theor PhysWe look for an operator of the formZ =N∑i=1ciOi, (3)such that[H, Z] = λZ. (4)The operator Z is important for our purposes becauseHZ |ψ〉 = (E + λ)Z |ψ〉 . (5)It follows from equations (2), (3) and (4 ) that(H − λI)C = 0, (6)where H is an N ×N matrix with elements Hij , I is the N ×N identity matrix, and C is anN × 1 column matrix with elements ci . H is called the adjoint or regular matrix representa-tion of H in the operator basis {O1,O2, . . . , ON } [7]. In the case of an Hermitian operatorwe expect all the roots λi , i = 1, 2, . . . , N to be real. These roots are obviously the nat-ural frequencies of the quantum-mechanical system (the quantum-mechanical frequenciesare linear combinations of them). Here we apply the same approach to symmetric Hamil-tonians because all the relevant equations are formally identical. If λ is real then it followsfrom equation (4) that[H, Z†] = −λZ†, (7)where Z† is a linear combination like (3) with coefficients c∗i . This equation tells us that ifλ is a real root of det(H − λI) = 0, then −λ is also a root. Obviously, Z and Z† are a pairof annihilation-creation or ladder operators because, in addition to (5), we also haveHZ† |ψ〉 = (E − λ)Z† |ψ〉 . (8)Other authors have already applied Lie-algebraic methods to the Hamiltonian (1) with-out coupling (ε = 0) [8–10] but they were not interested in the quantum-mechanicalfrequencies.In the present case, the obvious choice {O1, O2,O3,O4} = {x, y, px, py} leads to thematrix representationH = i⎛⎜⎜⎝γ 0 ε ω2 − γ 20 −γ ω2 − γ 2 ε0 −1 −γ 0−1 0 0 γ⎞⎟⎟⎠ , (9)with characteristic polynomialλ4 + λ2(4γ 2 − 2ω2)− ε2 + ω4 = 0, (10)that is exactly the one that yields the classical frequencies [1]. Two of its roots areλ1 =√√ε2 + 4γ 4 − 4γ 2ω2 − 2γ 2 + ω2λ2 =√−√ε2 + 4γ 4 − 4γ 2ω2 − 2γ 2 + ω2, (11)Int J Theor Physand the other two ones are λ3 = −λ1 and λ4 = −λ2 in agreement with the more generalequations (4) and (7). The operators Z1 and Z2 associated to λ1 and λ2 are creation or rising,while Z3 = Z†1 and Z4 = Z†2 are annihilation or lowering. The classical and quantal naturalfrequencies are exactly the same because the relevant Poisson brackets and commutatorsare similar: i{H, Oi} → [H, Oi]. This result reveals why the condition for real classicalfrequencies2γ√ω2 − γ 2 < ε < ω2 (12)is also the condition for real spectrum (unbroken-PT region) in the quantum-mechanicalcounterpart [1].It is obvious that the algebraic method described above only applies to some exactlysolvable problems. In particular, the form of the Poisson brackets and quantum-mechanicalcommutators is similar because the Hamiltonian operator (1) is a quadratic function of thecoordinates and their conjugate momenta.If we write the polynomial equation (10) as(λ2 − λ21) (λ2 − λ22) = 0 then we realize thatλ21 + λ22 = 2ω2 − 4γ 2λ21λ22 = ω4 − ε2. (13)Throughout this paper we keep the parameter ω in order to facilitate the discussion of theresults of Bender et al. However, it is worth noting that we can choose ω = 1 without lossof generality as follows from the transformation {λ, a, ω, γ, ε} → { λω, aω, 1, γω, εω2}.Bender et al. [1] derived the energiesEmn = (m + 1)a + (2n − m)	, (14)where m = 0, 1, . . . and n = 0, 1, . . . , m. In this equation a is a root of4a4 + 4a2(2γ 2 − ω2)+ ε2 + 4γ 2(γ 2 − ω2)= 0, (15)and	 =√bc − γ 2,b = c∗ = ε2(a + iγ ) . (16)The expressions for a, b and c come from solving the eigenvalue equation H |ψ00〉 =E00 |ψ00〉 in the coordinate representation with the ansatz ψ00(x, y) = e−(bx2+cy2+2axy),procedure that also yields E00 = a. If we write ξ = a2 then we obtain the rootsξ1 = ω2 − 2γ 2 −√ω4 − ε22,ξ2 = ω2 − 2γ 2 +√ω4 − ε22. (17)Following Bender et al we write a1 = −√ξ1, a2 = √ξ1, a3 = −√ξ2, a4 = √ξ2.For concreteness, from now on we choosea = a2 = 12√2ω2 − 4γ 2 − 2√ω4 − ε2. (18)Int J Theor PhysSo far, we have shown that the quantum-mechanical frequencies can be expressed aslinear combinations of the classical frequencies; it only remains to rewrite the eigenvalues(14) in terms of the latter. One can easily verify thatλ1 = 	 + a, λ2 = 	 − a, (19)is consistent with (13). Thus, the expression for the energies becomesEm,n = λ1 (2n + 1)2− λ2 (2m − 2n + 1)2== nλ1 + (n − m)λ2 + a. (20)Bender et al. [1] showed numerically that a2 and 	 are real and positive in the unbroken-PT region and concluded that the eigenvalues are also real and positive. However, this isnot the case because a − 	 < 0 and for every value of n Emn → −∞ as m → ∞. Asan example, consider the model parameters ω = 1, γ = 0.05 and ε = 0.5 that lie theunbroken-PT region. In this case 	 ≈ 0.964630863 and a ≈ 0.2539434939 that confirmswhat we have just said.We can analyse those results by means of the algebraic method. If E00 were the lowesteigenvalue then both Z3ψ00 and Z4ψ00 would be expected to vanish. However, for theparameters chosen above we found that Z2ψ00 and Z3ψ00 vanish while Z1ψ00 and Z4ψ00do not. Therefore, the eigenfunctions are given byψnk = Zn1Zk4ψ00 (21)with eigenvalues Enk = nλ1 − kλ2 + a which agree with (20) if k = m − n. The algebraicmethod clearly shows that the spectrum is unbounded from below.3 ConclusionsThe main purpose of this paper is to show that part of the mathematical analysis of someclassical systems also applies to their quantum-mechanical counterparts. The underlyingconnection is that the frequencies of both interpretations are closely related because of thesimilarity between the Poisson brackets and commutators. Therefore, if the frequencies ofthe motion of the classical system are real then the quantum-mechanical eigenvalues arealso real. The quantum-mechanical natural frequencies are the eigenvalues of the regular oradjoint matrix representation of the Hamiltonian operator in a suitable basis set of operators,whereas the corresponding eigenvectors provide the ladder operators. This well known alge-braic approach applies to many exactly solvable problems [7] (see also http://en.wikipedia.org/wiki/Ladder operator) and in particular for Hamiltonians that are quadratic functions ofthe coordinates and their conjugate momenta. Such Hamiltonians are suitable models formany physical problems like the one that motivated the paper by Bender et al. [1], amongothers [8–10].In closing it is worth mentioning that the fact that the spectrum of the Hamiltonian (1) isnot strictly positive, contrary to what Bender et al assumed, does not appear to be relevantto the interpretation of the physical data which is fitted by the classical (and also naturalquantal) frequencies [2].Acknowledgements This report has been financially supported by PIP No. 11420110100062 (ConsejoNacional de Investigaciones Cientificas y Tecnicas, República Argentina)Int J Theor PhysReferences1. Bender, C.M., Gianfreda, M., Özdemir, Ş.K., Peng, B., Yang, L.: Phys. Rev. A 88, 062111 (2013)2. Peng, B., K. Özdemir, Ş., Lei, F., Monifi, F., Gianfreda, M., Long, G.L., Fan, S., Nori, F., Bender, C.M.,Yang, L.: Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities,arXiv:physics.optics/1308.45643. Cotton, F.A.: Chemical Applications of Group Theory, 3rd edn. Wiley, New York (1990)4. Tinkham, M.: Group Theory and Quantum Mechanics. McGraw-Hill Book Company, New York (1964)5. Klainman, S., Cederbaum, L.S.: Phys. Rev. A 78, 062113 (2008)6. Fernández, F.M., Garcia, J.: Ann. Phys. 342, 195 (2014)7. Fernández, F.M., Castro, E.A.: Algebraic Methods in Quantum Chemistry and Physics (CRC, BocaRaton, New York, London, Tokyo) (1996)8. Dekker, H.: Phys. Rep. 80, 1 (1981)9. Celechini, E., Rasetti, M., Vitiello, G.: Ann. Phys. 215, 156 (1992)10. Chruściński, D., Jurkowski, J.: Ann. Phys. 321, 854 (2006)

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