A unified treatment of polynomial solutions and constraint polynomials of the Rabi models
General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, , where , are polynomials, is a one-dimensional coordinate, and . It is not required that ODE is either (i) Fuchsian or (ii) leads to a usual Sturm-Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of th degree to exist forces energy to a th baseline. Once the constraint relations on the th baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the th energy level of a displaced harmonic oscillator with varying coupling ). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual and driven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.
The Rabi model Rb () describes the simplest interaction between a cavity mode with a bare frequency and a two-level system with the levels separated by a frequency difference , where is the Planck constant and is a bare resonance frequency. The model is characterized by the Hamiltonian Rb (); Schw ()
where is the unit matrix, and are the conventional boson annihilation and creation operators of a boson mode with frequency , which satisfy commutation relation , and is a coupling constant. Here and elsewhere the standard representation of the Pauli matrices , , with diagonal is assumed. The Rabi model applies to a great variety of physical systems, including cavity and circuit quantum electrodynamics, quantum dots, polaronic physics and trapped ions KGK (); BGA (); NDH (); TAP (); TPG (); AMdn (). The Rabi model is not exactly solvable. Yet the model has been known for a long time to possess polynomial solutions, the so called Juddian isolated exact solutions Jd (); Ks (), at energy levels corresponding to those of a displaced harmonic oscillator, which is the limit of the Rabi model Schw (). The latter has been known as the baseline condition for the Rabi model Schw (); Jd (); Ks () [e.g. Eqs. (38), (48), (57), (60), (67) below].
In what follows we shall consider the (driven) Rabi model Zh2 (); Zh6 (); Li15 (); Wa (); KRW (), together with its nonlinear two-photon AMdn (); Zh2 (); Zh6 (); NLL (); EB1 () and nonlinear two-mode AMdn (); Zh2 (); Zh6 () versions, and the generalized Rabi model of Refs. TAP (); TPG (); AMdn (). A typical nd order linear ordinary differential equation (ODE) for the Rabi models turns out to be of the form , where , are polynomials, is a one-dimensional coordinate, and . What sets those equations apart from other common equations is that comprises energy dependent terms , , Schw (); KKT (); Zh2 (); Zh6 () and, therefore, does not reduce to a standard eigenvalue problem. The physical problem is rather to find zero modes of . Differential equations for the Rabi models are not even Fuchsian, having an irregular singular point at infinity. Obviously in analyzing a given 2nd order ODE with polynomial coefficients one can always switch from the above form into a Schrödinger equation (SE) (also known as normal) form, where the first derivative term has been eliminated and the coefficient of has been set to one KO (); GKO (). Such a transformation leads to an energy dependent potential and to a non-Sturm-Liouville problem.
Inspired by nd order ODE which occur when trying to solve the Rabi models Rb (); Schw (); TAP (); TPG (); AMdn (); Zh2 (); Zh6 (); NLL (); EB1 (), we developed general necessary and sufficient conditions for the existence of a polynomial solution of th degree of such equations. In recent years, the corresponding differential operator for a (driven) quantum Rabi model KKT (); Zh2 (); Zh6 (), the two-photon and two-mode quantum Rabi models Zh2 (); Zh6 (), and the generalized Rabi model TAP () was shown to be expressible as a bilinear combination of algebra generators, and hence to be an element of the enveloping algebra for a certain choice of model parameters and of energy. For a typical eigenvalue problem, , one can always find polynomial solutions corresponding to different eigenvalues of in a corresponding module characterized by the spin . The constant term of is a free parameter that can always be absorbed to an eigenvalue . However, in the case of the Rabi models, the value of is fixed and only zero modes of , which satisfy , are physical solutions. One can easily show that the latter problem can have at most a single polynomial solution. We show that with , the ODE need not have in general any polynomial solution. In particular, algebra does not explain why the Juddian isolated exact solutions can be analytically computed Jd (); Ks (); Li15 (); Wa (); KRW (), whereas the remaining part of the spectrum not. The Rabi models are thus an unusual example of quasi-exactly solvable (QES) models KO (); GKO (); TU (); Trb (); ShT (); Shfm (); Trb1 (); Tams (); FGR (); KKT (). The QES models are distinguished by the fact that a finite number of their eigenvalues and corresponding eigenfunctions can be determined algebraically KO (); GKO (); TU (); Trb (); ShT (); Shfm (); Trb1 (); Tams (); FGR (). Initially, QES was essentially a synonym for algebraization in one dimensional quantum mechanical problems KO (); GKO (); TU (); Trb (); ShT (); Shfm (); Trb1 (); Tams (); FGR (); KKT (), up to the point that no difference was made between the terms Lie-algebraic and quasi-exactly solvable in the literature. The reason behind this was that is the only algebra of first order differential operators with finite dimensional invariant modules. Burnside’s classical theorem ensures that every differential operator which leaves the space invariant belongs to the enveloping algebra , since is an irreducible module for the action.
The article is organized as follows. On defining the grade of a term as integer , Sec. II introduces other necessary definitions to perform a gradation slicing of a given ordinary linear differential equation (ODE) with polynomial coefficients,
In Sec. III basic theorems are formulated that yield necessary and sufficient conditions for the existence of a polynomial solution of th degree. The necessary condition in general constraints energy in a parameter space for each given to a different baseline [e.g. Eqs. (38), (48), (57), (60), (67) below]. Provided that the highest grade of the terms of ODE (2) is , there are recursively defined constraints to be satisfied for a polynomial solution on the th baseline to exist. It turns out that, with the exception of the generalized Rabi model TAP (), each of the Rabi models considered in this article is characterized by an ordinary linear differential equation comprising terms with highest grade , the lowest grade , gradation width , the highest grade slice , and its induced multiplicator as follows:
For the generalized Rabi model one finds, depending on parameters, either or . The corresponding constraints are shown to (i) reproduce known constraint polynomials for the usual and driven Rabi models Jd (); Ks (); Li15 (); Wa (); KRW () (cf. Figs. 1, 2) and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and the generalized Rabi models. Usual road to constraint polynomials required to reveal an ingenious Ansatz for the polynomial solutions. For example, the original Kus construction Ks () consisted in an insightful observation that an exact polynomial solution of the Rabi model on the th baseline can be constructed as a finite linear combination of the solutions of a displaced harmonic oscillator ( limit of the Rabi model) from all baselines and of the same parity. An analogous approach was attempted later by Emary and Bishop EB1 () in the case of two-photon Rabi model, and the others in the case of the driven Rabi model Li15 (); Wa (); KRW (). Yet the origin of constraint polynomials remained mysterious. It was not a priori clear if they at all exist. In this regard Theorem 3 of Sec. III yields a recipe for determining constraint polynomials by a downward recurrence (11), (14) with well defined coefficients for any problem, and in particular for any conceivable Rabi model generalization. In Theorem 4 of Sec. III we have succeeded to generalize an important result of Zhang (cf. Eqs. (1.8-10) of Ref. Zh ()) obtained for nd order ODE’s to the case of arbitrary . A algebraization with spin is shown in Sec. III.2 to be equivalent merely to the necessary condition for the existence of a polynomial solution of th degree. A lemma is formulated which yields necessary condition for a spectral problem , where is an eigenvalue, to have degenerate energy levels in an invariant module of spin .
In Sec. IV our approach is illustrated in detail on the example of the usual quantum Rabi model. A driven Rabi model is considered in section V. Nonlinear two-photon and two-mode Rabi models are dealt with in section VI, and the generalized Rabi model is the subject of section VII. In each of the above cases explicit expressions of the recurrence coefficients for the constraint polynomials are presented. Our results open a number of different avenues of further research which are discussed in Sec. VIII. We then conclude in Sec. IX. For the sake or presentation, Appendix A provides an overview of the basics of algebraization. Relevant features of nd order linear differential equation with all solutions being polynomials are summarized in Appendix B. Singular points at spatial infinity are dealt with in Appendix C. Some other alternative forms of nd order ODE of Refs. Schw (); KKT () for the Rabi model are examined in Appendix D.
Ii Gradation slicing
The subset of ODE (2) where are polynomial of degree at most , , , respectively, covers (i) all QES models within the context of [cf. Eqs. (24), (76)] KO (); GKO (); TU (); Trb (); ShT (); Shfm (); Trb1 (); Tams (); FGR (), (ii) all Fuchsian 2nd order ODE In1 () and (iii) non-Fuchsian 2nd order ODE of the present article [such as Eq. (37) below, which has an irregular singular point at infinity (see Appendix C)].
For the purpose of looking for (monic) polynomial solutions,
of the ODE (2), or in general of
where are also polynomials, it is expedient to rearrange it into a more convenient form. Obviously for any polynomial the image is also a polynomial. The basic idea is to characterize the terms of the operator which contribute to the same polynomial degree in the image . In what follows we call (a positive or negative) integer the grade of the term . The grade describes a change of the degree of a monomial under the action . This is similar to the grading (73) of generators (72) employed by Turbiner Trb1 (); Tams ().
We introduce the concept of a gradation slicing of an ODE (6), which comprises the following steps:
Consider a given differential equation as a linear combination of terms and determine the grade of each term.
Rearrange all the terms of the ODE according to their grade. The subset of the ODE with an identical grade will be called a slice. Hence the differential equation can be recast as
where the sum runs over all grades . In what follows we will use an abbreviation for the highest grade and for the lowest grade.
Definition: A decomposition of original ordinary linear differential equation (6) into (7) will be called gradation slicing. We call the grade of an ordinary linear differential equation the highest grade . A width of the gradation slicing will be called the integer . Define a function by
We shall call the function an induced multiplicator corresponding to the slice .
The width counts the number of possible slices with the grade between the minimal and maximal grades, and , respectively. Unless is identically zero, one has always . In what follows, we shall assume that . The case can always be reduced to the case by factorizing out of the polynomial coefficients of the differential equation (6).
Remark 1: A hypergeometric equation is characterized by , , and . A typical Heine-Stieltjes problem Heine1878 (); Stl (); Sz (); Schhs (); Shp (); AMu (); AGM (), where are polynomials of exact degree , , , respectively, is grade , , problem.
It turns out that each of the Rabi models considered in this article is described by an ordinary linear differential equation characterized by , , , , and induced multiplicator as summarized by (4). The lowest grade and width are not absolute invariants of an ODE, because they may depend on the origin of coordinates.
Obviously the condition that the slice with the highest grade annihilates a monomial of degree , , provides a necessary condition for the existence of a polynomial solution of degree ,
Provided that depends on energy one can consider the condition (8) as equation for . (For the Rabi models considered here this will be typically a linear equation.) Its solution constraints energy in the parameter space [e.g. Eqs. (38), (48), (57), (60), (67) below]. Therefore, we will refer to the condition also as the baseline condition, although it defines a line only in the case of the original Rabi model, where it depends on a single parameter Jd (); Ks ().
For the ensuing analysis of Rabi models we need both necessary and sufficient conditions for the existence of a polynomial solution of th degree. In what follows we shall distinguish two main alternative types of differential equations (6):
An example of the alternative (A1) are the Fuchsian equations, which include a hypergeometric one, and the Heine-Stieltjes problem Heine1878 (); Stl (); Sz (); Schhs (); Shp (); AMu (); AGM (). The alternative (A2) is usually omitted in the analysis of polynomial solutions. Yet for all Rabi model examples which follow, the alternative (A2) will be the only relevant one. Anomalous alternative (A2) can be encountered also in other problems (cf. Eq. (5) of Ref. Tr94 (); Eq. (31) of Ref. CH () for relative motion of two electrons in an external oscillator potential). Obviously, one has automatically for the alternative (A2). Therefore, the necessary condition (81) for a 2nd order ODE (2) with fixed polynomial coefficients to have two linearly independent (and hence to possess only) polynomial solutions is always violated. Consequently if Eq. (2) has a polynomial solution, such a polynomial solution is necessarily unique.
Iii Basic theorems
In this section general necessary and sufficient conditions for the existence of a polynomial solution of th degree are formulated.
iii.1 General theory
The condition that solves (7) is equivalent to that all the coefficients of respective powers of of the image of vanish. The latter brings us to the linear system of equations
where each line summarizes all the terms contributing to the same power of , beginning from of the first equation down to of the last equation. We recall that is the gradation slicing width. If one tries to determine the coefficients of in the expansion (5) by direct substitution into underlying differential equation, the width thus yields the length of a downward recurrence. For both and its image are polynomials of the same degree. Moreover, one has necessarily . We have the following Theorem.
Theorem 1: A necessary and sufficient conditions for the ODE (6) with the grade to have a unique polynomial solution of th degree is that
where the second condition applies for .
Proof: The condition is nothing but the baseline condition (8) in the special case , and is obviously necessary. In order to demonstrate sufficiency, note that the second condition ensures that each subsequent line in the system (11) of equations, when progressing from the very top down, enables one to uniquely determine newly appearing coefficient (i.e. in the th line) and thus to determine at the end a unique set of coefficients , . The initial condition is used here to simply fix an arbitrary irrelevant multiplication factor. The point of crucial importance is that for (and only for) the image and are polynomials of the same degree. Hence on summing up the lines of the above system one recovers the original system of equations for unknown coefficients , . Indeed, because by definition, and on substituting ,
Thereby the theorem is proven.
Corollary 1: If we drop “unique” in Theorem 1, then the condition (8) is both necessary and sufficient condition for the existence of a polynomial solution of an ODE of grade zero.
Proof: Apply Theorem 1 to the smallest nonnegative zero of .
Corollary 2: If is a linear function of , there is always at most a single unique polynomial solution, because a linear function can have at most a single root.
Remark 2: For the hypergeometric equation characterized by and , the system of equations (11) reduces to the three-term recurrence relation (TTRR) studied exhaustively by Lesky Lsk (). is a quadratic function of and there are, in principle, possible two linearly independent polynomial solutions, because quadratic function has in general two roots (cf. Appendix B).
Remark 3: In the case of the Heine-Stieltjes problem Heine1878 (); Stl (); Sz (); Schhs (); Shp (); AMu (); AGM (), the usual condition for ODE (2) to have only polynomial solutions (see Appendix B for more detail) requires that for some
Remark 4: Theorem 1 does not rely on, and is independent of, the Frobenius analysis of a regular singular point of nd order ODE (see for instance Chap. 10.3 of Whittaker and Watson WW (), or Chap. 5.3 of Hille Hi ()). Yet there are many parallels between the two approaches. In Theorem 1 the condition gives an entry point to a downward recurrence. In the Frobenius analysis, one needs instead an entry point for an upward recurrence. Such an entry point is provided by the solutions of the so-called indicial equation, which can be viewed as , . Hence in the Frobenius analysis one is instead of the highest grade slice concerned with the lowest grade slice . If is a solution, i.e. , then the condition , , guarantees that a unique, in general infinite, set of coefficients of a solution at a regular singular point can be determined by the relevant recurrence.
As explained above, we assume that , which can always be achieved by a suitable factorization of the polynomial coefficients of differential equation (6).
Theorem 2: A necessary and sufficient conditions for the ODE (6) with the grade to have a unique polynomial solution is that, in addition to the conditions (12) which determine the unique set of coefficients by the recurrence (11) of Theorem 1, the subset of the coefficients satisfies additional constraints:
Proof: According to the definition, . Therefore, whenever , the recurrence (11) does not take into account the terms of degree of the image . There are exactly of such polynomial terms with . One can verify that the vanishing of the coefficients of , amounts to solving the system (14). The vanishing of the coefficients of , thus imposes constraints on the (up to a multiplication by a constant) unique set of coefficients .
Remark 5: Grade problem has always a polynomial solution, because it leads to a system of equations for unknowns. Obviously, whenever , the differential operator is not exactly solvable. The image of is a polynomial of th degree. The vanishing of all the polynomial coefficients of the image then imposes different conditions.
Remark 6: After imposing on the energy one of the baseline conditions , the energy is expressed as a function of model parameters [e.g. Eqs. (38), (48), (57), (60), (67) below]. Therefore after imposing the baseline condition, the coefficients of the recurrence (11) and of (14) cease to depend on . When solving for the expansion coefficients by the -term downward recurrence (11), each subsequent , beginning from in the first equation down to in the last equation, is obtained by dividing the corresponding equation line by . The necessary and sufficient conditions for the ODE (6) with the grade to have a unique polynomial solution ensures that the product . Provided that the coefficients are polynomials in model parameters [e.g. examples (39), (50), (59), (62), (68) below], each , defined by Eq. (14), when multiplied by the product , is necessarily a polynomial in model parameters. The resulting constraint polynomials are defined by the -term recurrence (11), which yields the unique solution for , , , that is substituted into the constraints (14) and each of the constraints (14) is multiplied by ,
We have thus proven the following fundamental result:
Theorem 3: Provided that each in (11) and (14) is a polynomial in model parameter(s) and the hypotheses of Theorem 2 are satisfied, each recursively determined , of Eq. (14) is proportional to a polynomial in model parameter(s).
Remark 7: The Rabi models in this article are all characterized by the same grading parameters summarized by (4). Given that , the recurrence system (11), (14) reduces to a downward four-term recurrence relation (FTRR) for the coefficients , , of . The necessary condition (8) for the existence of a polynomial solution becomes in view of (4)
For there is a single constraint (14) to be satisfied,
to guarantee the existence of a unique polynomial solution. For the Rabi models considered here all the coefficients are polynomials in physical parameters such as the coupling strength , detuning , and frequency in Eq. (1) [e.g. examples (39), (50), (59), (62), (67) below].
Corollary 3: For the Rabi model (1) the constraint (16) for each baseline is equivalent to the corresponding Kus polynomial Ks (). For the driven Rabi model, the constraint (16) on a given baseline is equivalent to the corresponding generalized Kus polynomial Li15 (); Wa (); KRW ().
The polynomial has to be equivalent to either the Kus polynomial Ks (), or generalized Kus polynomial Li15 (); Wa (); KRW (), respectively, because they express the necessary and sufficient conditions for the existence of a unique polynomial solution. The equivalence will be illustrated on examples and numerically in Secs. IV-V.
Remark 8: For the grade there would be two constraint polynomials. Common zeros of two different polynomials are the zeros of the so-called resultant MS (). This is the case of the generalized Rabi model discussed in Sec. VII below.
Note in passing that if a zero of were not simple, then the Wronskian of with any other (not necessary polynomial) nonsingular function would be zero at any multiple root of (see Appendix B for more detail). Here and below the coefficients with a negative subscript are assumed to be identically zero.
Proof: According to the hypothesis we have
The first of the recurrences of the system (11) requires
On substituting for from (17) and solving for yields
On taking into account that , this leads immediately to (18).
We now continue with the second of the recurrences of the system (11),
Solving for yields
To this end one makes use of
Thereby one recovers (19).
Remark 9: In the special case of , the conditions (17)-(19) reduce to those of Theorem 1.1 of Zhang (cf. Eqs. (1.8-10) of Ref. Zh ()), where they were derived by means of a functional Bethe Ansatz. Yet in the latter case the level of complexity increases significantly with and each -case has to be treated separately. In contrast to that, the gradation slicing approach enables one to prove the formulas of Theorem 4 for the coefficients , , in one go. Theorem 4 applies to both alternatives (9) and (10). Note in passing that if , the condition (18) for reduces to the necessary condition (38) for the existence of a polynomial solution of degree . Similarly for the condition (19) for , provided that additionally .
Remark 10: Provided that the simple roots are required to satisfy the set of the Bethe Ansatz algebraic equations (cf. Eqs. (1.11) and (2.5) of Ref. Zh ()),
then the conditions together with those of Theorem 4 provide necessary and sufficient conditions for the coefficients of of the ODE (6) with grade to have a polynomial solution of th degree with zeros . Yet this does not answer the question under which conditions has the system of the Bethe Ansatz algebraic equations a solution. Theorem 1.1 of Zhang Zh () is rather a set of general compatibility conditions between the polynomial zeros that satisfy the Bethe Ansatz algebraic equations for a given and on one hand, and the coefficients of of the ODE (6) on the other hand. Similarly to the Kus recipe Ks (), the Bethe Ansatz equations for a polynomial of th degree have a solution only for a discrete set of model parameters, which corresponds to zeros of a certain polynomial in model parameters TAP ().
The condition of a algebraization is that a corresponding differential operator can be expressed as a normally ordered bilinear combination of the generators ,
where KO (); GKO (); TU (); Trb (); ShT (); Shfm (); Trb1 (); Tams (); FGR (). Strictly speaking belongs to the central extension of (cf. Theorem 2 of Ref. GKO ()). With a slight abuse of notation we continue writing . The properties of the generators are summarized in Appendix A.
Let us consider an anomalous characterized by the grading parameters as in Eq. (4). In order that such a reproduces in Eq. (75), one has to have and in Eq. (76). The latter immediately requires in Eqs. (24), (76), whereas there has to satisfy
The above two conditions on require that the coefficients and of the terms of the highest grade satisfy
The necessary condition (26) for algebraization reproduces the necessary condition (15) for the existence of a polynomial solution of th degree if and only if the spin of an irreducible representation satisfies .
There are at most different polynomial solutions of th degree on each th baseline Ks (). The latter would not be surprising if we had an identical along whole given th baseline. Yet each of those polynomial solutions corresponds to a different (e.g. because depends on and the polynomial solutions are nondegenerate on a given (base)line in the -plane Schw (); Ks (); AMops (); AMtb (); AMef ()).
The remaining conditions for the algebraization in the anomalous and case are
From the general necessary condition (18) for the existence of a polynomial solution of th degree of Theorem 4 we know that for an anomalous characterized by the grading parameters as in Eq. (4), i.e. with , we have to have
The general necessary condition (28) for the existence of a polynomial solution of th degree of Theorem 4 for can be then recast in terms of ,
iii.2.1 Nondegenerate energy levels
Let us examine the conditions under which cannot have degenerate energy levels in an invariant module. A degeneracy can only occur if the necessary condition (81), , is satisfied. The latter requires
which is impossible to satisfy for . Hence
If , the condition (81) requires
or the constraint
If , the condition (81) requires
or the constraint
We have thus proven the following result:
Usual algebraization means that the corresponding spectral problem possesses eigenfunctions in the form of polynomials of degree for any given irreducible representation of of spin . The foregoing analysis of anomalous grade problems implies for the Rabi model problems that any given irreducible representation of of spin can in the most optimal case add only at most a single new polynomial eigenfunction of degree relative to a lower dimensional irreducible representation of of spin - a kind of onion algebraization. The forthcoming examples will demonstrate that more often than not no new polynomial eigenfunction will be added to the spectrum.
Iv The Rabi model
For a theoretical investigation of the Rabi model it is expedient to work in an equivalent single-mode spin-boson picture, which amounts to interchanging and in (1). The latter is realized by unitary transformation , where . Assuming , one arrives at
where , and becomes diagonal. The matrix possesses the parity symmetry , where unitary induces reflections of the annihilation and creation operators: , , and leaves the boson number operator invariant Schw (); AMdn (); Ks (); AMep (). is immediately recognized to be of the Fulton-Gouterman form AMdn (); FG (), where the Fulton-Gouterman symmetry operation is realized by . The projected parity eigenstates have generically two independent components. The advantage of the Fulton and Gouterman form is that the projected parity eigenstates are characterized by a single independent component. In the Bargmann realization Brg (): , and the Hamiltonian of Eq. (34) becomes a matrix differential operator. After is diagonalized in the spin subspace, the corresponding one-dimensional differential operators are found to be of Dunkl type AMdn (). The Fulton-Gouterman form and the one-dimensional differential operators of Dunkl type can also be determined for all the remaining Rabi models discussed here AMdn ().
In terms of the two-component wave function , the time-independent Schrödinger equation gives rise to a coupled system of two first-order differential equations (cf. Eqs. (2.4a-b) of Kus Ks ())
If these two equations decouple and reduce to the differential equations of two uncoupled displaced harmonic oscillators which can be exactly solved separately Schw (); Zh2 (). For this reason we will concentrate on the case in the following.