Stability, complex modes and non-separability in rotating quadratic potentials

We examine the dynamics of a particle in a general rotating quadratic potential, not necessarily stable or isotropic, using a general complex mode formalism. The problem is equivalent to that of a charged particle in a quadratic potential in the presence of a uniform magnetic field. It is shown that the unstable system exhibits a rich structure, with complex normal modes as well as non-standard modes of evolution characterized by equations of motion which cannot be decoupled (non-separable cases). It is also shown that in some unstable cases the dynamics can be stabilized by increasing the magnetic field or tuning the rotational frequency, giving rise to dynamical stability or instability windows. The evolution in general non-diagonalizable cases is as well discussed.


I. INTRODUCTION
Quadratic forms in boson operators or generalized coordinates and momenta are an ubiquitous presence in the theoretical description of diverse physical systems. They often arise through the linearization of the equations of motion around a stationary point, as in the case of the random phase approximation (RPA) [1,2], providing a basic tractable scenario. They play an important role in the description of Bose-Einstein condensates (BEC) [3][4][5][6][7][8] as well as in other areas like quantum optics [9], disordered systems [10] and dynamical systems [11][12][13]. Nonetheless, while positive forms, characteristic of stable systems, are well known to be diagonalizable, i.e., they can be written in terms of normal coordinates and viewed as a set of independent bosons or separate oscillators (plus eventually free particle terms in the presence of standard zero frequency modes [1,2]), non-positive ones may not admit such a diagonal representation [14]. Non-positive forms can arise in the description of BEC instabilities [6][7][8] and fast rotating condensates [15][16][17][18][19][20], as well as in generalized RPA treatments [21,22].
In [14] we have extended the standard methodology for diagonalizing quadratic bosonic forms by using generalized quasiparticle operators fulfilling boson-like commutation relations, associated with non-hermitian coordinates and momenta. This allows to characterize the operators exhibiting purely exponential evolutions (complex modes) in non-positive forms, enabling a precise description of the dynamics and quadratic invariants. We have also pointed out that non-positive forms can in some cases be dynamically stable, as the evolution can remain quasiperiodic, irrespective of the initial conditions. Moreover, we have noticed the existence of nondiagonalizable cases where the equations of motion cannot be fully decoupled and which may arise even if all eigenfrequencies are non-zero. The method of ref. [14] has been found useful in the context of BEC, being employed to study the emergence of instabilities in trapped BEC with a highly quantized vortex [6][7][8] through the Bogoliubov-de Gennes equations.
Here we will apply this methodology to the basic prob-lem of a particle in a rotating anisotropic quadratic potential, not necessarily stable. This system is formally equivalent to that of a charged particle in a uniform magnetic field in a general quadratic potential [15,16,23]. The problem is therefore relevant for many fields. In particular, the rotating case has recently become relevant in the context of BEC in rotating anisotropic traps [15][16][17][18][19][20], which in the Landau level approach are basically described by a cranked quadratic potential of the type here considered. The stable system is well known [1,2,23,24] and diverse aspects of the stable anisotropic rotating case in the context of rotating condensates have recently been investigated [17][18][19][20].
We will here examine the general unstable case, which is of interest for fast rotating condensates as the Hamiltonian ceases to be positive definite at high frequency due to the centrifugal force. As we shall see, the unstable system exhibits a rich structure, with several different dynamical regimes as well as some quite remarkable features, including: a) The possibility of becoming nonseparable at the boundaries of regions with distinct dynamics, in the sense that the Hamiltonian can no longer be written as a sum of two independent standard or complex modes. In such cases the system will exhibit anomalous evolutions characterized by a set of linear equations which cannot be decoupled and which may lead to coordinates and/or momenta evolving with terms ∝ te λt or even ∝ t 3 ; b) The possibility of achieving dynamical stability in some unstable cases by increasing the magnetic field or tuning the rotational frequency. In particular, a stable anisotropic rotating potential becomes dynamically unstable just in a finite frequency window, recovering dynamical stability at high rotational frequency, whereas an unstable saddle type potential can become dynamically stable in a certain frequency window.
In sec. II we briefly revisit the main features of the formalism, discussing the concept of separability in generalized coordinates and momenta and the evolution for general non-separable cases. The application is discussed in III while conclusions are drawn in IV.

II. FORMALISM
We consider a general quadratic hamiltonian where t denotes transpose, T , V are symmetric matrices and p, q are hermitian coordinates and momenta satisfy- The ensuing Heisenberg equations of motion lead to a closed set of linear equations which can be written as whereH c represents the RPA matrix in coordinate representation [14]. It completely determines the system dynamics. Its eigenvalues come in pairs of opposite sign and can be complex in unstable systems. Moreover,H c can also be non-diagonalizable (as in the case of free particles U = V = 0, T ij = t i δ ij , although other cases can also arise, as discussed later). A positive definite H c (R t H c R > 0 ∀ real R = 0) ensures a diagonalizableH c together with a real spectrum [1,2,14] (standard stable case) but the converse is not true. Under a general linear canonical transformation R = UR ′ , with the matrix U satisfying UM c U t = M c in order to preserve Eq.
It is obviously equivalent to use a representation of h in The ensuing RPA matrix, defined by idZ/dt =HZ, is justH = SH c S † and has obviously the same eigenvalues (and Jordan canonical form) asH c .

A. General evolution and dynamical stability
For a time independent h, the solution of system (3) is where R ≡ R(0). Eq. (7) is itself a linear canonical transformation. A system which is dynamically stable, i.e., leading to a bounded quasiperiodic evolution of all operators p i , q i , corresponds to a matrixH c which i) is diagonalizable and ii) has only real eigenvalues: a) IfH c is diagonalizable, such thatH c = WH ′ c W −1 with (H ′ c ) µν = λ ν δ µν , we may expand Eq. (7) as where W ν is the ν th column of the eigenvector matrix W and Z ′ = W −1 R is a set of normal operators satisfying and evolving then as Z ′ ν (t) = e −iλν t Z ′ ν . For complex eigenvalues λ ν , these generalized normal operators represent exponentially increasing or decreasing modes (complex modes) and the dynamics is unbounded (±λ ν are both eigenvalues). They can always be ordered and normalized such that those associated with [14]. b) IfH c is non-diagonalizable, the system (3) cannot be fully decoupled, but we may use its Jordan canonical formH c = WH ′ c W −1 , withH ′ c having blocks of the form We may then expand Eq. (7) as where again Z ′ = W −1 R, d ν is the dimension of the block and ν k , k = 1, . . . , d ν labels elements within each block, with W ν k the ν k column of the generalized eigenvector matrix W. The generalized normal operators Z ′ satisfy the "minimally coupled" evolution equations allowed by the Jordan form, i.e., idZ The dynamics is then unbounded even for real λ ν . The free particle case corresponds to λ ν = 0 and d ν = 2. Other cases are discussed in sec. III.

B. Separability
We will denote the Hamiltonian (1) as separable if there is a linear canonical transformation In this case we may then rewrite h as a sum of independent elementary quadratic systems, ), but the products α ν β ν = λ 2 ν determine the eigenvalues ofH c and are hence unique.
A diagonalizableH c ensures separability since in this case we may rewrite h in terms of the generalized normal operators In case iv) we may still rewrite the ensuing term in (13) However, the separable case also includes the free particle case iii) whereH c is non-diagonalizable, asH ′ c will contain a Jordan block c having a Jordan block of dimension d ν > 2 or d ν = 2 and λ ν = 0, are non-separable. They may arise even in simple unstable cases (sec. III) and their evolution can be determined through the general solution (10).
Dynamically stable quadratic systems correspond to a separable h with terms just of the form i), ii) or 0), and have then a discrete spectrum where λ ν > 0, < 0 or 0 in cases i), ii) or 0). An example of a non-positive dynamically stable form is an angular which is the sum of a positive plus an inverted oscillator. HereL c is diagonalizable with eigenvalues ±1, ∓1.

III. APPLICATION
We will consider the quantum problem of a particle in an anisotropic quadratic potential, not necessarily stable, rotating around one of its principal axes (z). It is formally equivalent to that of a particle of charge e in a uniform magnetic field H parallel to this axis in a quadratic potential. The Hamiltonian of the latter reads where A = 1 2 H ×R is the vector potential, Ω = e|H|/mc the cyclotron frequency, For Ω → 2Ω, Eq. (16) is just the cranked Hamiltonian describing the intrinsic motion of a particle in a rotating quadratic potential with constants K ′ Since H z is fully decoupled from the rest and its treatment is trivial, it will be omitted in what follows and all considerations will refer to the motion in the x, y plane. Defining dimensionless operators q = R mΩ 0 / , where Ω 0 is a reference frequency, we can rewrite h ≡ H xy / Ω 0 as with From the form of Eq. (15), it is apparent that for fixed k µ , the field cannot change the number of positive or negative eigenvalues of the Hamiltonian matrix H c (the number of positive and negative diagonal elements is the same in any real diagonal representation of a quadratic form). For m > 0, H c will then have none, one or two negative eigenvalues if and only if none, one or both of the constants k x , k y are, respectively, negative. The positive definite case corresponds then to k x > 0, k y > 0 ∀ ω, i.e., k ′ µ > ω 2 [17,18], although we will now see that dynamical stability is not restricted to this case.

A. Dynamical stability
The RPA matrixH c = M c H becomes and its eigenvalues, which are the system eigenfrequencies (in units of Ω 0 ), are ±λ + , ±λ − with They satisfy λ 2 We first note that for k x k y = 0 and ∆ = 0,H c is diagonalizable since it will have four different eigenvalues. We will then show that for ∆ = 0,H c is non-diagonalizable and, moreover, h is non-separable. At fixed k µ , ∆ = 0 if k x,y < 0 and It also follows that λ ± can be both real only if k x k y ≥ 0. For a charged particle in a magnetic field, this opens up the possibility of full dynamical stability around a quadratic maximum (k x,y < 0) but dismisses it for a saddle point (k x k y < 0). It is indeed verified that λ ± are both real for k x,y > 0 as well as for k x,y < 0 and At fixed k µ , the dynamics in the vicinity of a quadratic maximum can then be stabilized by increasing the field. The behavior of λ ± for increasing |ω| at fixed k µ is depicted on the left panels of Fig. 1. As the scaled cyclotron frequency 2|ω| is increased, at an anisotropic maximum λ ± evolve from imaginary (|ω| ≤ ω − c ) to full complex (ω − c < |ω| < ω + c ) and finally to real (|ω| ≥ ω + c ) values, reaching the system dynamical stability for |ω| > ω + c , whereas for k x k y < 0, λ + is real but λ − is imaginary ∀ω.
Stability in rotating potential. At fixed k ′ µ , the previous picture is seen quite differently and leads to dynamical stability and instability windows in the anisotropic case k ′ x = k ′ y (right panels in Fig. 1), i.e., when the rotation has a non-trivial effect. Owing to the centrifugal force, H c is here positive definite just for ω 2 < k ′ x,y . However, λ ± are real also for ω 2 > k ′ x,y > 0, implying that motion in a rotating stable potential becomes dynamically stable at high rotational frequencies, and dynamically unstable just in the finite interval where λ − becomes imaginary or 0 (see III B). In contrast, for a saddle point with k ′ x > 0 > k ′ y (or viceversa) the system becomes dynamically stable in the windows  (20) for selected fixed values of kx,y (left panels, corresponding to a particle in a magnetic field) and k ′ x,y (right panels, corresponding to a particle in a rotating potential), in terms of the scaled cyclotron or rotational frequency. Vertical dotted lines separate dynamically stable and unstable regions. We have set ω0 = |kx| the single value of |ω| where ∆ = 0 at fixed k ′ µ (requires k ′ x + k ′ y < 0). Thus, l z can turn unstable the dynamics of a stable anisotropic oscillator and stabilize that around a saddle point within the limits (25). It can never stabilize a quadratic maximum.
In the isotropic case k ′ x = k ′ y = k ′ , [h, l z ] = 0 and Eq. (20) leads then to |λ ± | = | √ k ′ ± ω|. The rotation has no effect except for the shift ±ω. The instability window collapses into a single point ω 2 = k ′ where the dynamics remains stable (see below), since it corresponds to the Landau case k x = k y = 0 [23,25].

B. Separability
Let us now examine the separable representation of Eq. (17), feasible for ∆ = 0, and the ensuing distinct dynamical regimes. Defining, as in the stable case [1,2], , we may rewrite (17) for ∆ = 0 as a sum of two independent elementary quadratic forms , with α ± β ± = λ 2 ± , For real ∆ = 0, p ± , q ± are hermitian and λ ± is real or imaginary. Eqs. (27-28) are, however, also applicable for imaginary ∆ = 0, where p ± , q ± Dashed lines indicate separable cases with a free particle term (standard in g,i and inverted in h) and a stable (g,h) or unstable (i) oscillator. Bottom: Same details in the k ′ x,y plane, corresponding to the rotating system: All regions and curves are just shifted by ω 2 . Plots in Fig. 1 depict the behavior of λ± along straight lines running from ∞ (ω = 0) to the origin (|ω| → ∞).
are non-hermitian and λ ± complex. From the previously mentioned property of diagonal quadratic forms, it follows that all four coefficients α ± , β ± must be positive for k x,y > 0, just one (two) of them will be negative for k x k y < 0 (k x,y < 0 and ∆ real), one of them will vanish for k y = 0, k x = 0 (or vice-versa) whereas two of them will vanish in the Landau case k x = k y = 0.
In cases A, B and F, p ′ ν , q ′ ν are hermitian, withb ′ ν = b ′ † ν , and h possesses then a discrete spectrum with λ + > 0, while λ − > 0 in A, λ − < 0 in B and λ − = 0 in F. These are the dynamically stable cases. At fixed k ′ µ (rotating potential) all regions are just shifted by +ω 2 (lower panel in fig. 2). This shift leads to the different behavior of λ ± with ω depicted in fig. 1.
Separable non-diagonalizable cases. They arise for k y = 0 and k x = 0 or viceversa (sectors g,h,i in Fig.  2). For k y = 0, ∆ = 2ω 2 + k x /2 and we obtain λ − = 0 but λ + = √ 4ω 2 + k x = 0 for ∆ = 0, with In g, k x > 0 and Eq. (31) corresponds to a stable oscillator (λ + > 0) plus a free particle (cases i+iii). In h, −4ω 2 < k x < 0 and λ + is still real, but the second term in (31) becomes negative: h becomes a stable oscillator plus an "inverted" free particle term. The latter "absorbs" here the instability, allowing dynamical stability in the coordinates p + , q + . Finally, in sector i (k x < −4ω 2 ) λ + becomes imaginary: Eq. (31) corresponds here to an unstable oscillator plus a standard free particle (iv+iii). Eq. (31) leads to an evolutionb ′ We now examine the very peculiar case ∆ = 0 and ω = 0, where the eigenfrequencies become degenerate, = λ, and bothH c andH 2 c are non-diagonalizable. It occurs at the threshold for full complex solutions, i.e., |ω| = ω ± c (ω ′ c ) at fixed k µ (k ′ µ ) and corresponds at fixed ω to a parabola rotated π/4 with respect to the k x,y axes, with vertex at k x,y = −ω 2 , i.e., k ′ x,y = 0 (curves j, k, and point L in fig. 2). j) Here |ω| = ω + c and λ = 4 k x k y = ω + H c can be reduced to two non-trivial Jordan blocks which indicates non-separability. The transformation allows to express h in this case as which is the cranked hamiltonian for a rotating free particle and is a basic non-separable form: The equations of motion dp ± /dt = ±λp ∓ , dq ± /dt = ±λq ∓ + p ± , cannot be fully decoupled even though there is no vanishing eigenfrequency. More explicitly, defining the operators The ensuing equations of motion, i which gives rise to unbounded spiral-like trajectories in the variables p ± , q ± . This is apparent from Eq. (36): Since [l, h] = 0, where l = q + p − − q − p + , the evolution operator e −iht = e iλlt e −i(h+λl)t represents a rotation of frequency λ applied to a free motion. Such motion cannot arise from a separable h: Although h is the sum of two commuting quadratic forms and the equations of motion can be reduced to two separate blocks, the operators in each block do not commute with those of the other.
where p ± , q ± are hermitian. With a complex scaling p − → ip − , q − → −iq − Eq. (38) becomes identical to Eq. (36) with an imaginary λ. The remaining equations remain then unchanged but lead to exponentially increasing or decreasing evolutions for the operators b ′ ± , b ′ ± , corresponding to a "boost" applied to a free motion.
L) This is an exceptional critical point where cases B,C,D,E,h,i,j and k merge. Here k y = 0, k x = −4ω 2 (i.e., k ′ y = ω 2 , k ′ x = −3ω 2 ) or vice-versa, implying |ω| = ω + c = ω − c and λ ± = 0: All four eigenvalues of H c vanish. Nonetheless,H c is still of rank 3, implying that it becomes similar to a full Jordan block (d ν = 4): instead of two free particle blocks, as would appear from Eq. (33) for λ → 0. In this case the equations of motion cannot be even partially decoupled. The transformation with δ + = 1, δ − = 2, allows to rewrite h at this point as which is again a basic non-separable form: The ensuing equations of motion, dp− dt = 0, dp+ dt = ωp − , dq+ dt = p + , dq− dt = −ωq + , exhibit the structure of the Jordan form (39) for ω = 0 and lead to a polynomial evolution of third degree in t for q − : Coordinates q + , q − experience then a constant and linearly increasing acceleration respectively. In terms of the Their equations of motion, db ′ − /dt = 0, idb ′ follow exactly the Jordan form (39). Note that the disappearance of one of the kinetic terms in (41) is not exceptional for a non-separable form: Eq. (36) (and hence (38)) can also be rewritten as p 2 + − λl if q ± → q ± + 1 2λ p ∓ . We may also rewrite (41) with two kinetic terms with a similar transformation.

IV. CONCLUSIONS
We have first analyzed, within the formalism of ref. [14], the dynamics in general unstable quadratic bosonic forms, discussing the treatment of the general nondiagonalizable case and determining the conditions for dynamical stability and separability. We have then applied the formalism to the basic problem of a particle in a general rotating quadratic potential, relevant in the context of fast rotating condensates in harmonic traps and formally equivalent to that of a charged particle in a uniform magnetic field in a quadratic potential. The present analysis unveils the rich variety of behaviors that can be exhibited by the unstable system, summarized in fig. 2, together with some quite remarkable features, which could lead to observable effects in fast rotating condensates. In particular, we have determined: a) The regions of dynamical stability. Intrinsic motion in a rotating stable potential remains dynamically stable at high frequencies, becoming unstable just in a finite frequency window in the anisotropic case, whereas in a rotating saddle potential it can become dynamically stable in a certain window (Eqs. (24), (25) and fig. 1); b) The regions in parameter space where H xy can written as a sum of two independent quadratic systems (separability), employing non-hermitian normal coordinates and momenta if necessary (sector E), and those where such a representation is not feasible (non-separability); c) The explicit transformations and final forms for all cases, including the energy spectrum in the dynamically stable cases and the "minimally coupled" standard forms and equations of motion in the non-separable cases; d) The existence of an exceptional non-separable zero mode case (point L in fig. 2) where all eigenfrequencies vanish and the Jordan Block has dimension 4. It is not equivalent to a standard zero frequency mode and leads to coordinates evolving as a third degree polynomial in time. These results indicate that similar peculiar effects can arise in more complex unstable quadratic systems, which can be analyzed with the same general formalism and techniques of sec. II.
RR and AMK are supported by CIC of Argentina.