Classical and quantum analysis of the parametric oscillator with Morse-like time-dependent frequency
CClassical and quantum analysis of the parametric oscillator with
Morse-like time-dependent frequency
Mariagiovanna Gianfreda a ∗ and Giulio Landolfi b † a Department of Physics, Washington University, St. Louis, MO 63130, USA b Dipartimento di Matematica e Fisica ”Ennio De Giorgi” Universit`adel Salento and INFN Sezione di Lecce, 73100 Lecce, Italy (Dated: July 7, 2020)We consider the problem of understanding the basic features displayed by quantum systemsdescribed by parametric oscillators whose time-dependent frequency parameter ω ( t ) varies duringevolution so to display either a non harmonic hole or barrier. To this scope we focus on the casewhere ω ( t ) behaves like a Morse potential, up to possible sign reversion and translations in the( t, ω ) plane. We derive closed form solution for the time-dependent amplitude of quasi-normalmodes, that is known to be the very fundamental dynamical object entering the description ofboth classical and quantum dynamics of time-dependent quadratic systems. Once such quantity isdetermined and its significant characteristics highlighted, we provide a more refined insight on theway quantum states evolve by paying attention on the position-momentum Heisenberg uncertaintyprinciple and the statistical aspects implied by second-order correlation functions over number-typestates. PACS numbers:Keywords: Parametric oscillator; Ermakov equation; time-dependent Schr¨odinger equation; Lewis-Riesenfeld invariant method; quantum correlations.
I. INTRODUCTION
Time-dependent quadratic hamiltonians play a funda-mental role in physics, and have been therefore inten-sively studied both at the classical and the quantum level.They have widespread applications in modern physics in-cluding, among the others, quantum optics and quantuminformation [1–6], quantum cosmology [7–11], plasmaphysics [12], disordered systems [13], and Bose-Einsteincondensates [14–19]. Studies in the Bose-Einstein con-densation context, in particular, are currently attractinga very major interest, and in fact they are pushing to-wards the broadening of analytical results. An instancein this respect can be found in [18] where three prototyp-ical examples have been considered for squared instan-taneous time-dependent frequency ω ( t ), consisting in astep-like, a Woods-Saxon, and a modified P¨oschl-Tellermodel. The discussion of these examples has provided aview of how oscillating phenomena may characterize thephysics of quantum condensates.While dealing with time-dependent quadratic Hamil-tonian systems, a convenient guideline is the known factthat their quantum dynamics is in fact ruled by the clas-sical one: expectation values follow the classical motion exactly , and the spreading of solutions to Schr¨odinger canbe understood in classical terms; see e.g. [20–22]. Forthis reason one should be well acquainted with featuresof the classical dynamics, and particularly of amplitudes ∗ Electronic address: [email protected] † Electronic address: Giulio.Landolfi@unisalento.it,Giulio.Landolfi@le.infn.it of solution to the classical equation of motion. Indeed, itis the amplitude function σ , which obeys by the Ermakovdifferential equation [23]¨ σ ( t ) = − ω ( t ) σ ( t ) + cσ ( t ) (1)where c is a real positive constant parameter [50], thatis later recognized to play the pivotal role in the quan-tum description of parametric oscillator dynamics. So-lutions to the Ermakov amplitude equation and theirtime-differential specialize the classical system orbits inphase-space, and in turns this determines the elements ofthe position-momentum correlation matrix among quan-tum wave-packets solving the Schr¨odinger equation. Thisis an instance of what is arguably the most profoundconnection of quantum mechanics with the classical one:the uncertainty principle, in its Robertson-Schr¨odingerstrong form, relies on the existence of Poincar´e invari-ants on the classical side [24–28], the Planck’s constantsimply representing a scale requirement. For the onedegree of freedom systems ruled by the time-dependenthamiltonian, the basic invariant is the area of the in-variant ellipses in phase space determined for assignedvalues of the so-called Ermakov invariant, a quadraticN¨other invariant for the system. It is then clear thatmanifestation of oscillations in expectation values forobservables governed by a one-degree of freedom time-dependent quadratic hamiltonian operator are an im-age of the time-dependent deformation of classical Er-makov ellipses in phase-space pertinent to the motion ofa classical particle. The deformation is ruled by evolu-tion of amplitude σ , whose oscillatory features intertwinewith those of classical Newton equation of motion forthe parametric oscillator, a Sturm-Liouville problem in- a r X i v : . [ qu a n t - ph ] J u l deed. Initial condition and adiabaticity can be expectedto play a key role in this context, possibly giving rise tohighly oscillating amplitudes determining strong squeez-ing phenomena on classical orbits, and consequently oncognate Wigner ellipses. Peculiar oscillating phenomenacan therefore enter quantum correlation functions andmode-counting statistics.In this paper we fix some specific time dependent fre-quencies ω ( t ) and address the study of one-degree of free-dom quantum systems described by a parametric oscilla-tor hamiltonian operator. Our motivation stands in theobvious remark that it is worth investigating new casesstudy, possibly exhibiting features that differ not onlyquantitatively but also qualitatively from examples dis-cussed by other authors. We focus, on general grounds,on the oscillation mechanism that arise already at theclassical level, the way it is influenced by deviation fromadiabaticity, how it is inherited at the quantum level,and the influence it has on quantum correlations. Hav-ing this in mind, we shall perform therefore a study con-cerned with square parametric frequency functions ω assuming a form connected with that of Morse potentialfunction. Acting on a Morse potential through possi-ble time-translations and sign-reversions will enable us toprovide another analytically solvable example of a time-dependent quadratic problem. This allows us to ana-lyze, within a single analytical problem, the role playedby changes in the energy subtraction/pumping scheme,such as whenever parameter are chosen in a way thatasymmetrical barrier or well show up and a monotonic-ity is lost for ω ( t ) , or giving rise to changes that performnonadiabatically on a finite time-scale. We discuss exten-sively the derivation of solution to the classical equationof motion and the features of their amplitudes σ . For thereasons expounded above, the step put in the position tobuild a solid bridge leading to the quantum dynamics.Examples will be given of highly oscillating amplitudesdetermining strong squeezing phenomena on phase-spaceorbits and on mode-counting statistics pertinent nonclas-sical states of the number type.The outline of the paper is as follows. To start with,in next Section we provide a recap of the treatment oftime-dependent quadratic quantum systems through theLewis-Riesenfeld invariant approach [29], thence high-lighting how solution to the Schrodinger equation are es-sentially determined by classical dynamics via solutionsto an Ermakov equation. After recalling this linkage,in Section III we make some general comment on theconstruction and features of solutions to parametric os-cillator and Ermakov equations. In Section IV, we shalldiscuss to a wide extend the analytical construction ofsolutions to the Ermakov equation when the paramet-ric frequency is assumed to vary by the Morse potentialbased frequency parameters that we have chosen to focuson in this communication. In Section V, we will providefigures showing solution to Ermakov equation based onsuitable initial conditions and representative sets of pa-rameters entering the time-frequency function ω ( t ), and will analyze thoroughly the features displayed. In Sec-tion VI we pay attention on implied squeezing of clas-sical orbits for the parametric oscillator, which reallymeans to comprehend the Heisenberg uncertainties forposition-momentum operators at the quantum level. InSections VII and VIII our attention will be devoted tomode-counting statistics. There, we consider normalizedsecond order correlation functions and argue on the typi-calities that are visible from their plots. Last Section willbe for conclusions. All along the paper all necessary de-tails will be given to make the exposition self-contained. II. THEORETICAL BACKGROUND
The quantization of time-dependent quadratic systemsis a well established topic, see e.g. [26]. The most effec-tive way to proceed has been recognised in the approachproposed by Lewis and Riesenfeld in [29], a method in-trinsically relying on the identification of symmetries andinvariants. In this Section we recall the basic steps con-necting the solutions to the Schr¨odinger equation perti-nent to a parametric oscillator hamiltonian operator tothe spectral problem for the quantum Ermakov invariantoperator. After doing so, the Bogolubov map relating theinvariant description to Sch¨rodinger picture is presented.This will enable us to delineate the squeezing dynamicsexperienced by the system.
A. The Lewis-Riesenfeld approach to quantumparametric oscillator
Consider the time-dependent one-dimensional Hamil-tonian H ( t ) = p m + m ω ( t )2 q , (2)where the variables q and p satisfy the canonical com-mutation relation [ q, p ] = i (cid:126) . The mass parameter m isnot time dependent. For such oscillator systems, a con-venient representation can be given in terms of a timedependent hermitian operator I ( t ) which is an invariant.That is, I ≡ I ( t ) satisfies the conditions I † = I and˙ I = ∂ t I + 1 i (cid:126) [ I, H ] = 0 , (3)where the dot stands for the total time derivative. Adynamical invariant I corresponds to a symmetry of thequantum system and comply with obvious properties: ex-pectation values and eigenvalues are constant (while theeigenstates are time dependent). A state vector | ψ ( t ) (cid:105) that satisfies the Schrodinger equation for (2), i (cid:126) ∂ t | ψ ( t ) (cid:105) = H ( t ) | ψ ( t ) (cid:105) , (4)can be written in terms of the eigenstates | φ n ( t ) (cid:105) associ-ated to the invariant I ( t ) with eigenvalues λ ( I ) n , I | φ n ( t ) (cid:105) = λ ( I ) n | φ n ( t ) (cid:105) , (5)via the general formula | ψ ( t ) (cid:105) = ∞ (cid:88) n =0 c n e i α n ( t ) | φ n ( t ) (cid:105) , (6)where the phase α n ( t ) satisfies the equation (cid:126) ˙ α n = (cid:104) φ n | i (cid:126) ∂ t − H | φ n (cid:105) . (7)By assuming that I ( t ) can be written as a quadratic ex-pression in the dynamical variables q and p , one can findthat I = cσ ( t ) q + (cid:104) σ ( t ) pm − ˙ σ ( t ) q (cid:105) , (8)where σ ( t ) is a real function of time, solution to theErmakov differential equation (1). Operator (8) is es-sentially the quantised version of the classical invariantdiscussed originally by Ermakov in his prime analysis in[23]. The invariant operator I ( t ) in Equation (8) can bewritten as I ( t ) = 2 √ c (cid:126) m (cid:20) a † ( t ) a ( t ) + 12 (cid:21) (9)where a ( t ) = 1 √ (cid:126) (cid:20) c / √ mσ q + i (cid:18) σ √ m c / p − √ m ˙ σc / q (cid:19)(cid:21) .. (10)Operators a ( t ) and a † ( t ) are time-dependent operators ofthe annihilation and creation type, satisfying the canon-ical commutation rule [ a ( t ) , a † ( t )] = 1. Accordingly, N = a † ( t ) a ( t ) is an operator of the number type, fromwhich it follows that the spectrum of I ( t ) consists inthe set of stationary eigenvalues λ ( I ) n = √ c m (cid:126) ( n + ),for n = 0 , , . . . , and the eigenstates are the same ofthe number operator N , | φ n ( t ) (cid:105) ≡ | n (cid:105) being N | n (cid:105) = n | n (cid:105) . Otherwise said, the invariant operator I ( t ) can bemapped into a time-independent harmonic oscillator viaa unitary transformation. Precisely, a unitary transfor-mation in the form [30] U ( t ) = e − i m ˙ σ ( t )2 σ ( t ) q e i log √ ω σ ( t ) c / ( pq + qp ) (11)is such that U I U † = (2 √ c/mω ) H , where H is thehamiltonian of an harmonic oscillator with constant mass m and frequency ω . In this way, denoting as | n (cid:105) theeigenstates of the time independent harmonic oscillator H , the eigestates | n (cid:105) of I ( t ) can be written as | n (cid:105) = U † | n (cid:105) , that in position representation provides (cid:104) q | U † | ψ n (cid:105) = (cid:115) c m / n n ! ω / π / (cid:126) / × e m (cid:126) (cid:104) i ˙ σ ( t ) σ ( t ) − √ cσ t ) (cid:105) q H n (cid:20) c / (cid:114) m (cid:126) qσ ( t ) (cid:21) . (12) Moreover, by expressing the Hamiltonian H in eq.(2) interms of a † ( t ) and a ( t ) through inversion of (10), we canthen evaluate explicitly the phase factor (7), which re-sults to be α n = − (cid:18) n + 12 (cid:19) (cid:90) t dτ √ cσ ( τ ) . (13)Finally, the solution | ψ ( t ) (cid:105) to the Schr¨odinger equationfor the Hamiltonian operator in (2) can be written as | ψ ( t ) (cid:105) = ∞ (cid:88) n =0 c n e − i ( n + ) (cid:82) t dτ √ cσ τ ) U † | n (cid:105) . (14)As in the standard harmonic oscillator case, exact ana-lytical solutions in the form of gaussian wave-packets canbe therefore found for the Schr¨odinger equation associ-ated to a parametric oscillator hamiltonian. A strongconnection is consequently established between classicaland quantum dynamics. In fact, the time-evolution ofthe maximum and the width of a wave-packet (the twoquantities that uniquely determine the packets) are gov-erned by classical dynamics: the maximum behaves likea particle following just the classical trajectory and thewidth is proportional to the amplitude σ of solution toclassical Newton-Lagrange equation for the parametricoscillator; see e.g. [20, 22]. B. Unitary Bogolubov transformation forannihilation-creation operators
The discussion presented above implicitly states thatFock-type spaces are defined at different times in termsof the common eigenstates | n (cid:105) of the number opera-tor N and Ermakov invariant I , and that these spacescan be then mapped one into the other through a time-dependent transformation. Similarly, coherent-like basesat different times may be defined by means of the eigen-states | α (cid:105) associated with the spectral problem for thetime-dependent operator ˆ a , i.e. ˆ a | α (cid:105) = α | α (cid:105) . These arethe so-called Lewis-Riesenfeld coherent states (thoughtthey are actually squeezed states), and are expressiblein terms of the states | n (cid:105) through the usual relation-ship. The number-type states | n (cid:105) , and consequently thecoherent-type states a l´a Lewis-Riesenfeld | α (cid:105) , do notsolve the Schrodinger equation for the Hamiltonian op-erator (2). For this to happen the action of a unitaryoperator is needed that merely equips each component | n (cid:105) with the proper time-dependent phase (13), an oper-ation that do not affect means and let these states be ofinterest for practical purposes.General formulae for position-momentum correlationsover generalized number and coherent states of time-dependent quadratic Hamiltonian (2) are well known.They are very straightforwardly obtained by employingthe inverse of transformation (10) to express position andmomentum quadratures in terms of the time-dependentladder operators a and a † according to [31]: q = √ (cid:126) σ √ c / √ m ( a + a † ) (15) p = √ (cid:126) m √ c / σ ( t ) (cid:2)(cid:0) σ ˙ σ − i √ c (cid:1) a + (cid:0) σ ˙ σ + i √ c (cid:1) a † (cid:3) . (16)Formulae (10), (15)-(16) enable us to address the iden-tification of features associated with the quantum-mechanical dynamics for the quantum parametric hamil-tonian by resorting to a squeezing formalism [5]. Thiscan be done by writing operators a ( t ) and a † ( t ) as lin-ear combination of annihilation and creation operators a and a † identified by diagonalization of the harmonic os-cillator hamiltonian structure at an initial instant. Thatis: a ( t ) = µ ( t ) a + ν ( t ) a † , | µ | − | ν | = 1 (17)where a = 1 √ (cid:126) (cid:18) √ m ω q + i p √ m ω (cid:19) . (18)and its hermitian conjugate a † are the the ladder-typeoperators applicable to the fixed-time operator H ≡ H (0) = p m + 12 m ω q = (cid:126) ω (cid:18) a † a + 12 (cid:19) ,ω ≡ ω (0) . (19)The constraint µ | − | ν | = 1 on Bogolubov coefficients µ, ν conseques from the unitarity of transformation. Wehave [51] µ ( t ) = 12 c / √ ω (cid:18) √ cσ + ω σ − i ˙ σ (cid:19) ,ν ( t ) = 12 c / √ ω (cid:18) √ cσ − ω σ − i ˙ σ (cid:19) . (20)Any element of the Fock space determined by the in-variant spectral problem at time t, i.e. | m (cid:105) the stateobtained by application of (ˆ a † ) m on the ˆ a -vacuum | (cid:105) ,can be expressed as an infinite superposition of the num-ber states | n (cid:105) belonging to the Fock space associatedwith the operators ˆ a and ˆ a † , for which ˆ a † ˆ a | n (cid:105) = n | n (cid:105) .The fundamental overlaps (cid:104) n | m (cid:105) are established via Eqs.(17) and (20), and so the evolution of arbitrary wave-functions can be obtained accordingly in either of thebasis.Notice that at this stage there is a freedom in respectto the choice of initial condition for the Ermakov dif-ferential problem (1), and this also affect the explicitform of Bogolubov mapping. Unitarily inequivalent time-dependent vacua can be thus introduced via a ( t ) | (cid:105) = 0by the general form of the invariant factorization, Eqs. (9)-(10). This is not surprising, as it generalizes to thenon-autonomous case the well understood unitary actionof squeezing of orbits for the stationary case. To removethe vacuum ambiguity, in this paper we shall proceed inthe most economical way, by demanding that at initialtime t = 0 the operator a (0) coincides with a in (18): µ (0) = 1 and ν (0) = 0. This requirement fixes initialconditions to the Ermakov differential equation (1) asfollows [32]: σ ≡ σ (0) = c / √ ω , ˙ σ ≡ ˙ σ (0) = 0 . (21)Evidently, the Hamiltonian and the invariant operatorsdo not generally commute, [ H, I ] = µ ∗ ν (ˆ a † ) − hc . Butonce initial conditions (21) are superimposed I (0) = m √ cω H (0), and Lewis-Riesenfeld states at initial time co-incide with standard states for the initial-time harmonicoscillator. Of course other choices can be made for µ (0)and ν (0) that more conveniently allow to follow the evo-lution of initial states possessing a squeezed character. C. Position-momentum variances and Heisenberguncertainty
Under the time-dependent quadratic Hamiltonian dy-namics, physical states generally loose coherence dur-ing their time evolution as a consequence of the time-dependent Bogolubov (squeezing) transformation. Thisaffects position-momentum correlations. In particular,from the expressions Eqs. (15)-(16), it follows that posi-tion and momentum operators means and variances overLewis-Riesenfeld coherent states | α (cid:105) take the form (cid:104) α | q | α (cid:105) = (cid:114) (cid:126) m σc / ( α + c.c ) , (22) (cid:104) α | p | α (cid:105) = (cid:114) (cid:126) m c / σ (cid:20)(cid:18) σ ˙ σ √ c − i (cid:19) α + c.c. (cid:21) , (23)∆ α q = (cid:114) (cid:126) m σc / , (24)∆ α p = (cid:114) (cid:126) m c / σ (cid:114) σ ˙ σ c , (25)yielding to the Heisenberg uncertainty∆ α q ∆ α p = (cid:126) (cid:114) σ ˙ σ c . (26)The quantity (cid:126) σ ˙ σ/ √ c can be thus recognized as twice theposition-momentum correlation among Lewis-Riesenfeldcoherent states | α (cid:105) , (cid:104) α | qp + pq | α (cid:105) − (cid:104) α | q | α (cid:105)(cid:104) α | p | α (cid:105) = (cid:126) σ ˙ σ/ √ c . Obviously, means and variances among thenumber-type states | n (cid:105) can be desumed as well immedi-ately to find (cid:104) n | q | n (cid:105) = (cid:104) n | p | n (cid:105) = 0, ∆ n q = √ n ∆ α q and ∆ n p = √ n ∆ α p . III. SOLUTIONS TO ERMAKOV EQUATION
Ermakov systems have widespread occurrence mathe-matics and physics [33–41]. In particular, the Ermakovequation (1) naturally enters the description of time-dependent quadratic systems. It basically represents thedifferential constraint for the amplitude of solutions tothe corresponding equation of motion once, in the spiritof the pivotal Ermakov and B¨ohl discussions [23, 42],they are expressed via the sine and the cosine functions.In the next two Subsections we shall argue on to whatextend the nonlinear superposition principle applicablevia Ermakov-B¨ohl-type transformation may help to un-derstand on some general grounds the origin and fea-tures of fluctuations for the amplitude variable σ . Thishelps in shedding light on oscillatory effects for phase-space orbits in the classical dynamics of time-dependentquadratic systems, and thence for wave-packets motionat the quantum level. Later, we highlight the conditionfor adiabatic dynamics. A. Ermakov-B¨ohl type transformations andsolutions to Ermakov equation
The Ermakov differential equation follows from theparametric oscillator equation by treating it through anErmakov-B¨ohl strategy, or, equivalently said, by meansof an amplitude-phase decomposition of solutions. Inpractice, Equation (1) is nothing but one of the two dif-ferential constraints into which the parametric oscillatorequation ¨ x ( t ) + ω ( t ) x ( t ) = 0 (27)splits once solutions are sought in the quasi-normal modeform x = √ I √ c σ ( t ) cos[ θ ( t ) + θ ] (28)with real constants θ and I , c >
0, the other differentialconstraint to be satisfied for consistency determining thephase function via ˙ θ = √ cσ (29)( I in Eq. (28) corresponds to the Noether invariant ofthe theory obtained by taking q and p in (8) as the clas-sical position and momentum phase-space coordinate).Relying on this connection, it is possible to employ solu-tions to the linear parametric equation to the purpose ofdesuming solutions to the nonlinear Ermakov amplitudeequation (1). In exact terms, the general solution to Eq.(1) can be expressed in terms of two linearly independentsolutions x and x of (27) via [43] σ = (cid:113) A x + 2 B x x + C x , (30) with the condition A C − B = cW ( x , x ) ≥ , (31)where W is the Wronskian of x and x . The Ermakov-Bohl-type formulae (30)-(31) generalize those with B = 0considered at early stages [23, 42], and can also be derivedbrightly using projective geometry [35]. As a matter offact, they establish a nonlinear superposition principlefor Ermakov systems. The constants A, B, C can be com-pletely fixed by imposing the initial conditions on σ and˙ σ along with (31). Real solutions of Ermakov equationare thus positively defined in terms of quadratic forms,and do not vanish. Furthermore, one infers that oscil-latory behavior characterizes the solutions to Eq. (27)iff (cid:82) ∞ σ − dt = ∞ , a long dated observation in fact [44].This feature transfers to the amplitude σ through trans-formation (30). The resulting oscillatory effect for ampli-tude σ can be marked even when extremely small vari-ations of ω ( t ) take place. Variation in the initial condi-tions for the nonlinear Ermakov differential problem areclearly influential in this respect. Retaining arbitrary ini-tial condition for σ and ˙ σ at a given time t = 0, fromEquations (30)-(31) we get: A = c x (0) + σ [ ˙ x (0) σ − x (0) ˙ σ ] σ W ( x , x ) ,B = − σ W ( x , x ) { c x (0) x (0)++[ ˙ x (0) σ − ˙ σ x (0)][ ˙ x (0) σ − ˙ σ x (0)] σ (cid:9) ,C = c x (0) + σ [2 ˙ x (0) σ − x (0) ˙ σ ] σ W ( x , x ) (32)with σ = σ (0) and σ = ˙ σ (0). Effects on function σ of deviation from the set of initial conditions (21) canbe then promptly evidenced via Eqs. (30) and (32), forinstance. Especially, adjustments appear for σ on thescales of ˙ σ and ˙ σ . B. Oscillatory second-order differential equationsand solutions to the Ermakov equation
The possible presence of an oscillating pattern in thebehavior of solutions to the Ermakov equation is obvi-ously a reflex of features that are typical of solutionsto the linear second order differential equation for theparametric oscillator. For this reason, it is helpful to re-member that the understanding and the identification ofoscillation criteria for the second order differential equa-tion ¨ x + f ( t ) x = 0 (33)and for its generalizations (including damping and forc-ing, or with functions f allowed to assume negative val-ues) has been a fundamental issue for several mathemati-cians, and has led to well-recognized fundamental results;see e.g. [45, 46] and references therein. The well-knownFite-Leighton criterion can be resorted, for instance. Itstates that if f ( t ) is a nonnegative real function in theinterval t ∈ [ t , ∞ ) such thatlim T →∞ (cid:90) Tt f ( t ) dt = ∞ (34)then equation (33) with x = x ( t ) and t ∈ [ t , ∞ ) is oscil-latory , i.e. has arbitrarily large zeroes on the independentvariable domain [ t , ∞ ). We also recall that, when f ( t )is allowed to assume negative values for arbitrarily largevalues of t, the generalization of the criterium in the formof Wintner theorem can be resorted, according to whichif lim T →∞ T (cid:90) Tt dt (cid:90) tt f ( t (cid:48) ) dt (cid:48) = ∞ (35)then (33) is oscillatory. Furthermore, theorems have beenformulated that provide an insight in respect to the lo-cation of successive zeros [46]. A famous example is, inparticular, the Sturm comparison theorem, which sup-ports the conclusion that the larger is the parametricfrequency the more rapidly solutions to the parametricoscillator equation will oscillate. Besides, the Sturm sep-aration theorem guarantees that given two linear inde-pendent solutions of the parametric oscillator equationthe zeros of the two solutions are alternating.While the theorems cannot be likewise stated for theErmakov equation, it is however apprehensible that theyprovide a convenient platform for appraising implicationsfrom the latter owing to the connection established via(30)-(31). Independent solutions for the parametric os-cillator equation that alternate zeros introduces indeeda mechanism where terms in (30) alternate minor andmajor contribution, and the quasi-normal amplitude σ consequently displays at least some kind of undulations(contrary to the harmonic oscillator case ˙ ω = 0 where theamplitude is constant). Depending on the shape of para-metric frequency curves and the time interval where it isdefined, the fluctuation mechanism can be sustained todifferent degrees, possibly giving rise to oscillations aboutan asymptotic value. What we will see later dealing withMorse type square frequencies is indeed consistent withthe Sturm theorem, curves σ resulting for majorant ω possessing (in a sufficiently large time interval) more ex-trema (see Figs. 5-8 below).We conclude the Subsection with a final remark aboutan analytical result known for the parametric oscillatorthat can be similarly adopted for solutions to Ermakovequation. It concerns the boundness of solutions. It isknown indeed that the inequality | x | ≤ ( | x ( t ) | + | ˙ x ( t ) | ) exp (cid:90) tt | c ( s ) | ds (36)generally holds for solutions to the differential equation(33) with f ( t ) = 1+ c ( t ) (in proper units). The polar form of Ermakov-B¨ohl decomposition, Eq. (28), then allowsto transfer the inequality (36) directly to an inequalityfor the amplitude σ comprising initial conditions for theimplied Ermakov equation. C. Adiabatic dynamics
The differential equation for the position-type coordi-nate x of a parametric oscillator system is linear, Eq.(27), and if the frequency parameter ω is slowly varyingthe effects on x advance in a like manner. By contrast,the evolution for the amplitude σ entering the quasi-normal mode representation (28) is governed by a dif-ferential equation which is nonlinear, and adiabaticity ofvariations for the frequency parameter ω do not sufficeto guarantee that even changes for mode amplitude σ arerealized slowly. The matter of adiabaticity of amplitudechanges is immediately dealt with though because of thevery simple normal form structure of the Ermakov differ-ential equation. As far as quasi-normal mode amplitudesand Ermakov equation are concerned, in fact attentionhas to be paid on the quantity c σ − − ω σ , the right-hand side of Eq. (1). Demanding its vanishing functionsas retaining the leading order in the WKBJ expansiontreatment of the parametric equation. The condition ω σ (cid:39) c (37)then maintains ˙ σ slowly-varying, while ( ω + 3 c σ − ) ˙ σ +2 ω ˙ ωσ (cid:39) σ be negligible. So, if ω σ /c takes values that are held approximately aboutthe unity, changes in square amplitude σ and paramet-ric frequency ω are comparable as expected, their loga-rithmic differentials being linked according to ˙ ωω (cid:39) − ˙ σσ ,variations in ¨ σ being negligible. The adiabaticity of timeevolution of quasi-normal mode amplitude σ is thus mea-sured by the deviation from unity of quantity ω σ /c . Aslong as ω σ significantly deviates from attaining valuesabout c , the dynamical changes in amplitude σ are lesssmooth. Comparison between ˙ ω/ω and 4 ω σ will beshown in the forthcoming section for the Morse-type fre-quency problem under consideration in this communica-tion. The oscillating behavior for the amplitude σ we willobserve is not captured by first lowest terms in expansionof the generalized Bohl transformation (30), as it can betested by resorting to the standard formula √ cσ (cid:39) ω + 3 ˙ ω ω − ¨ ω ω (38)produced by application of WKBJ expansion method upto second order. Also notice that if condition ω σ = c holds at a given time, then one of the needed ingredientsfor ensuring the minimal classical squeezing of trajecto-ries in phase space is guaranteed, see Eqs. (20)-(21). Thesimultaneous vanishing of ˙ σ at settles a strong instanta-neous adiabatic condition securing no-squeezing at thatgiven time. IV. ANALITYCAL TREATMENT FOR AMORSE-LIKE TIME DEPENDENT FREQUENCY
In this communication we consider a time-dependentquadratic system (2) with square parametric frequencyof the form ω ( t ) = D e (cid:104) e − t + ts ) b − e − t + tsb (cid:105) + V ∞ , (39)where D e (cid:54) = 0, b > V ∞ are real parameters. Theform of Equation (39) basically represents a generaliza-tion of the well-known Morse potential, which is indeedrecovered for V ∞ = 0, where the parameters D e and b , both positive, determine the depth and the effectivewidth of the Morse potential hole, see Figure 1.The Morse potential is known to provide a solvableexample of the Schr¨odinger equation, and we can ex-ploit the formal analogy with the equation governing thevariations of the quasi-normal mode amplitude σ [52].Adopting (39) we also consider a possible upside-downflipping caused by the change of sign of D e , as well ashorizontal translation of the standard Morse curves via t → t + t s . This enables us to argue on the problem witha wider generality. Obviously, we shall be concerned onlywith the cases for which ω ( t ) ≥
0, and it shall be under-stood that ω ≡ ω (0) is strictly positive. Four dynamicalcases can be then distinguished, the qualitative behav-ior thence implied for ω being summarized in Figure1.b). Precisely, when V ∞ < t ≥ t − = − t s + b log D e − (cid:112) D e ( D e − V ∞ ) V ∞ (40)of equation ω = 0.In the following Subsections the derivation of closedform solution to the Ermakov equation (1) for the fourcases will be discussed. We will perform a step-by-stepanalysis drawing special attention on factors affecting theway in which independent solutions for the parametric os-cillator equation with frequency (39) are permitted. Af-ter that we compute their Wronskians, whose role is ofcentral importance in explicitly determining solutions tothe Ermakov equation via formulae (30)-(31). Featuresof the solutions to Ermakov equation in each of the fourcases will be argued separately in Section V, where theplots summarize the behavior of σ for different values ofthe parameters involved and suitable initial conditionswe shall be adopting. A. Case 1: D e , V ∞ < The first case that we consider is a square paramet-ric frequency defined through an inverse Morse potential
FIG. 1: a) Morse potential V ( t ) = D ( e − tb − e − tb ) com-pared to its harmonic approximation (dotted). b) Examplesof admissible ω ( t ) once all the possible actions on the Morsepotential ( D e sign reversion, potential shift V ∞ , and finitetime-translation) are put forward. with a negative nonvanishing asymptotic value, say ω ( t ) = d (cid:16) e − t + tsb − e − t + ts ) b (cid:17) − v , (41)where the positive parameters v = (cid:112) − V ∞ , d = (cid:112) − D e (42)have been defined for convenience. Since we want theHamiltonian (2) to collapse into a standard harmonic os-cillator at t = 0, we consider the case ω = ω (0) = V ∞ − D e = d − v >
0. The homogeneous second orderlinear differential equation (27), with a change of variable ξ = 2 d b e − t + tsb , (43)reads¨ x ( ξ ) + ˙ x ( ξ ) ξ + (cid:18) d bξ −
14 + v b ξ (cid:19) x ( ξ ) = 0 . (44)If we let x ( ξ ) to be x ( ξ ) = e − ξ/ ξ b v z ( ξ ) , (45)the differential equation (44) becomes, upon dividing by e − ξ/ ξ b v , the confluent hypergeometric equation ξ ¨ z ( ξ ) + ( β − ξ ) ˙ z ( ξ ) − α z ( ξ ) = 0 , (46)with parameters α = 12 + b ( v − d ) , β = 1 + 2 b v . (47)A solution to the confluent hypergeometric equation isthe Kummer’s function of the first kind F [ α ; β ; ξ ] de-fined as F [ α ; β ; ξ ] = ∞ (cid:88) n =0 ( α ) n ξ n ( β ) n , (48)where ( α ) n = α ( α + 1) ... ( α + n −
1) and ( α ) = 1.Note that the Kummer’s function F [ α ; − n ; ξ ] is notdefined for n non-negative integer, except for the case α = − m with m non-negative integer and m < n . Inthese cases, other representations of the solutions to (44)should be used (for more details here, see Refs. [47]-[48]).It is easy to show that ξ − β F [1 + α − β ; 2 − β ; ξ ] isanother solution to (44). In fact, by letting z ( ξ ) be z ( ξ ) = ξ − β v ( ξ ) , (49)upon dividing by ξ − β , equation (44) becomes ξ v ( ξ ) + (2 − β − ξ ) ˙ v ( ξ ) − (1 + α − β ) v ( ξ ) = 0 . (50)We can use a combination of the above solutions to definethe Tricomi confluent hypergeometric function U [ α ; β ; ξ ][47], U [ α, β, ξ ] = π sin( π β ) (cid:18) F [ α ; β ; ξ ]Γ[ β ] Γ[1 + α − β ] −− ξ − β F [ α + 1 − β ; 2 − β ; ξ ]Γ[ α ] Γ[2 − β ] (cid:19) . (51)The advantage of the function U [ α, β, ξ ] is that, even ifit does not exist for β integer, it can be extended analyt-ically for all values of β .For most combinations of real or complex α and β ,the two solutions (48) and (51) are independent. How-ever, if α is a non-positive integer, then (48) and (51) areproportional. This is due to the relation [47, 48], U [ − n ; α ; x ] = ( α − n )!( α − − n F [ − n ; α ; x ] , (52)(with n = 0 , , , ... ). In these cases we should use F [ α ; β ; ξ ] and ξ − β F [1 + α − β ; 2 − β ; ξ ] as indepen-dent solutions, when they exist.In the light of all above considerations, we can finallywrite the independent solutions to Eq.(44) as x ( ξ ) = e − ξ/ ξ b v U (cid:20)
12 + b ( v − d ); 1 + 2 b v, ξ (cid:21) ,x ( ξ ) = e − ξ/ ξ b v F (cid:20)
12 + b ( v − d ); 1 + 2 b v ; ξ (cid:21) , (53)that in terms of the time variable t becomes x ( t ) = e − d b e − ( t + ts ) /b e − t v × U (cid:20)
12 + b ( v − d ); 1 + 2 b v, d b e − ( t + t s ) /b (cid:21) ,x ( t ) = e − d b e − ( t + ts ) /b e − t v × F (cid:20)
12 + b ( v − d ); 1 + 2 b v ; 2 d b e − ( t + t s ) /b (cid:21) , (54)for + b ( v − d ) (cid:54) = − m , with m non negative integer(wehave omitted the factor e − t s v (2 d b ) b v arising after the change of variable), and˜ x ( ξ ) = e − ξ/ ξ − b v F (cid:20) − b ( v + d ); 1 − b v ; ξ (cid:21) , ˜ x ( ξ ) = e − ξ/ ξ b v F (cid:20)
12 + b ( v − d ); 1 + 2 b v ; ξ (cid:21) , (55)that in terms of the time variable t becomes˜ x ( t ) = e − d b e − ( t + ts ) /b e t v × F (cid:20) − b ( v + d ); 1 − b v ; 2 d b e − ( t + t s ) /b (cid:21) , ˜ x ( t ) = e − d b e − ( t + ts ) /b e − t v × F (cid:20)
12 + b ( v − d ); 1 + 2 b v ; 2 d b e − ( t + t s ) /b (cid:21) , (56)for + b ( v − d ) = − m , with m non negative integer (wehave omitted the multiplying factor e t s v (2 b d ) − b v in x and e − t s v (2 b d ) b v in x that arises after the change ofvariable).The Wronskian of the two solutions (53) can be easilycomputed by resorting to the known forms for expan-sions pertaining small arguments ξ of special functionsinvolved [53]. So we can take W ( x ( ξ ) , x ( ξ )) = cos[ π b ( d − v )]Γ( + b ( d + v )) π ξ . (57)The Wronskian of (54) in the variable t is related to theWronskian in the variable ξ by means of the relation W ( x ( t ) , x ( t )) = − W ( x ( ξ ) , x ( ξ )) ξb , (58)giving explicitly W ( x ( t ) , x ( t )) = − cos[ π b ( d − v )] Γ[ + b ( d + v )] π b . (59)Analogously, the Wronskian of the two solutions (55) isfound in the form W (˜ x ( ξ ) , ˜ x ( ξ )) = − b vξ . (60)and the Wronskian of (56) in the variable t , simply re-lated to (60), W (˜ x ( t ) , ˜ x ( t )) = − W ( x ( ξ ) , x ( ξ )) ξb , (61)explicitly becomes W ( x ( t ) , x ( t )) = 2 v. (62)This completes our derivation of the ingredients forexpressing in closed form the solutions to the Ermakovequation with the parametric frequency (41) with D e < V ∞ <
0. Note that when D e < V ∞ = 0, two independent solutions to (27) can onlybe obtained for b d (cid:54) = m + , with m non negativeinteger [47]. Their analytical expression, together withthe expression for their Wronskian, is simply given bysubstituting v = 0 into Eqs. (54) and (59). In particular,for b d = + m, ( m = 0 , , , . . . ) all the solutions to theconfluent hypergeometric equation (46) are proportionalto the Laguerre polynomials L m ( ξ ) [47]. B. Case 2: D e < and V ∞ > We now consider an inverse a Morse potential with apositive nonvanishing asymptotic value, that is V ∞ > D e < t = 0 is ω = V ∞ + | D e | , which is always positive.The treatment of the equation (27) in the present casefollows a route similar to that of Subsection IV A, andsolutions in the ξ variable (43) will be obtained after per-forming the replacement V ∞ → − V ∞ , i.e. v → i v into(44). Compared to Case 1 of Subsection IV A, the analy-sis is evidently complicated by the presence of complex-valued objects. Since x ( ξ ) and x ( ξ ) are now complexfunctions, for the evaluation of σ we use their real part, y ( ξ ) = x ( ξ ) + x ∗ ( ξ ) , y ( ξ ) = x ( ξ ) + x ∗ ( ξ ) . (63)In this way the Wronskian becomes, in the original vari-able t, W ( y ( t ) , y ( t )) == − π b (cid:26) cos[ π b ( d − i v )] Γ (cid:20)
12 + b ( d + i v ) (cid:21) ++ cos[ π b ( d + i v )] Γ (cid:20)
12 + b ( d − i v ) (cid:21)(cid:27) . (64) C. Case 3: D e , V ∞ > . We now consider the standard Morse potential with apositive shift, that is (39) with both D e and V ∞ positiveconstants. We assume V ∞ > D e so to have at t = 0 apositive instantaneous frequency ω = √ V ∞ − D e . Thesolution to the equation (27) in the ξ variable will beobtained with the replacement ξ → i ξ , v → i v and d → i d into formulae of Subsection IV A, see e.g. Eqs. (43)-(44). Since x ( ξ ) and x ( ξ ) are now complex functions,for the evaluation of σ we proceed as in the case with D e < V ∞ > W ( y ( t ) , y ( t )) == − π b (cid:26) cos[ b π ( d − v )] Γ (cid:20) − i b ( d + v ) (cid:21) ++ cosh[ b π d ] cosh[ b π v ] Γ (cid:20)
12 + i b ( d + v ) (cid:21)(cid:27) . (65)
1. The particular case of D e > and V ∞ = 0 For the sake of completeness, in this Subsection weshall provide analytical details concerning the case ofvanishing asymptotic limit, though we shall not discussexplicitly its dynamical implications in forthcoming sec-tions. In the case V ∞ = 0, we have x ( ξ ) = e − i ξ/ U (cid:20) − i b d, , i ξ (cid:21) ,x ( ξ ) = e − ξ/ F (cid:20) − i b d, , i ξ (cid:21) . (66)where now x ( ξ ) is a real valued function while x ( ξ ) isa complex function. For finding solutions to (44), thistime we can take the two functions y ( ξ ) = x ( ξ ) + x ∗ ( ξ ) , y ( ξ ) = x ( ξ ) . (67)In this way we have W ( y ( t ) , y ( t )) = − b d π ) b π (cid:90) ∞ dt e − t √ t cos( b d log t ) , (68)where we have used the integral representation of theGamma functionΓ[ z ] = (cid:90) ∞ e − t t z − dt, Re ( z ) > . (69) D. Case 4: D e > and V ∞ < . The Morse-type structure (39) can be finally consid-ered with D e >
0, and with a negative vertical shift V ∞ <
0. In order to avoid ω <
0, the case has tobe concerned with suitable negative shifts of time vari-able t s < ξ → i ξ and d → i d are performed intoformulae of Subsection IV A from Eqs. (43) -(44) on. V. ANALYSIS OF AMPLITUDES σ ANDPHASES θ . Here we analyze solutions σ to Ermakov equation (1)when ω takes the Morse-like form (39) and the parame-ters involved are varied to modify width and height of0barriers/holes as well as the position of the curve inthe ( t, ω ) plane. Attention will be also paid on phases θ = √ c (cid:82) t σ − dt . The examination can be conducted be-ing conscious that the Fite-Leighton criterion applies ina natural manner to Cases 2 and 3, for which V ∞ , b > t →∞ , that yields indeed to lim T →∞ (cid:82) Tt ω ( t (cid:48) ) dt (cid:48) = ∞ . InCases 1 and 4, application of Wintner theorem interdictsthe independent solutions to the equation ¨ x + ω x = 0from possessing arbitrarily large zeroes. Nevertheless,in our study this differential problem is actually posedin a finite interval for the time independent variable.Zeroes in this interval are not excluded and, depend-ing on the manner initial conditions and parameters arefixed, visible fluctuations may be produced. Also no-tice that the outcome of integration in (36) (with para-metric frequency cast in adimensional units) is alwaysfinite, even as t → ∞ (in cases where the limit makessense). Hence solutions to parametric and Ermakovequations are bounded, but nontrivial initial conditionsfor t-differentials determine higher upper bounds for themagnitude of position/amplitude oscillations. We willassume the mild initial conditions of Equation (21). A. Case 1: D e , V ∞ < The case under consideration in this Section is realizedwith negative parameters D e and V ∞ , and such that thatcurve (39) intercepts the positive time axis once . The fullset of parameters involved in (39) has also to guaranteethat ω > ω = 0 have to be real, noncoincident and opposite insign.Curves in Figure 2 summarize the typical behavior for σ in this case upon superimposition of the initial con-ditions (21) [54]. For t s = 0, solutions to the Ermakovequation tend to increase their value during the course ofevolution, in a way similar to the black curve in Fig. 2.a)and dashed curves in Fig. 2.b). Although they may bealmost imperceptible, footprints of fluctuations for quasi-normal amplitudes σ are actually present. These can bebetter enucleated looking at differentials ˙ σ , as in Figure2.b). There undulations in the profile of ˙ σ are revealed,never attaining negative values though. Function σ growsmonotonically, with small rate fluctuations. What hap-pens by translating the ω curve via t → t + t s is that thoseundulations are deformed and part of them is bent in theside determined by negative value for ˙ σ , thus being as-sociated to visible oscillations for the amplitude function σ . When t s (cid:54) = 0 the resulting horizontal translations ofthe parametric frequency shape deform solutions σ inways that may lead to significantly different curves evenif the changes in the shift parameter t s appear commen-surable. Besides the obvious remark in respect to thequantitative change in the initial value of quasi-mode FIG. 2: Amplitude σ and ˙ σ when ω ( t ) > D e , V ∞ >
0. Dotted curves refer to scaled ω ;dashed curves in b) refer to σ/
4. Turning on at the initialtime faster varying parametric drivings gives rise to greaterfluctuations for the amplitude function.FIG. 3: Functions ˙ ω/ω , ω σ /c and ¨ σ/ω σ . a) D e , b and V ∞ as in Figure 2.a) and t s = 1 .
5. Dashed curves refer to thesame quantities of continuous curves when t s = 0. b) Same D e , b and V ∞ as in previous plot, but t s = −
2. c) ω σ /c for fixed b and V ∞ and different couples ( D e , t s ). ω ∗ denotesthe parametric frequency evaluated at initial time with thesame potential parameters as the red curve in Fig. 2.c) and t s = − amplitude σ , from the analytical point of view it is topoint out that evolution now would start no longer at aninstant when the time-differential of the frequency pa-rameter vanishes. This circumstance plays a role in themanner the system dynamics deviates from comportmentit would display if t s = 0. Whenever initial data areweakly altered, smooth deformations of σ are expectedthat progressively unmask the latent fluctuations. How-1ever sudden changes in the energy pumping mechanismmay take place that lead to more notable mutations ofthe system dynamical responses. Varying the parameter t s to negative values, the system is progressively drivento a dynamics whose initial course acquires a higher andhigher nonadiabatic character indeed. A major ampli-fication of magnitude of fluctuation can be foretold inconsequences.The predicted effects are confirmed once plots for am-plitude σ are implemented, see Figure 2. It is also helpfulto complement their discussion by the exemplification ofadiabaticity estimators as in Figure 3. As it can be seenfrom Figure 3.a), when t s > ω decreasesmonotonically in a way that maintains ¨ σ scaled-downwith respect to ω σ (by about a factor 5 − × − in thefigure) and the quantity 4 ω σ relatively close to unityfor most of the evolution (final divergences and collapseto zero of curves in plots is evidently to be ascribed to ω approaching the trivial value). With the increasing of t s the time interval where ω > ω = ω (0) (i.e. the overall jump experienced by ω inthis case). There is no inclination to make more intenseoscillations.Moving to the case t s <
0, the curve ω ( t ) grows untilit reaches its maximum value at t = − t s to later de-crease until its vanishing at t − , Eq. (40). If the timeinterval [0 , − t s ) is sufficiently wide, the initial value of ω cuts down meaningfully. As Figures 2.a) and 2.c) show,the effect is quite a gain for the initial value for func-tion σ and for the magnitude of fluctuations occurringthereafter. Remark that in the plot 2.c) the ω ’s at ini-tial time are very small, but never vanishing: resultinginitial conditions for the corresponding σ are relatively large, but in fact finite. The point is that when t s < ω . The system is thus required toadapt itself first to a continuous positive frequency jump∆ ω, = ω max − ω = | D e − V ∞ | − ω , and later to a con-tinuous negative jump ∆ ω, = ω max = | D e − V ∞ | . Sucha driving dynamics calls for a bigger and bigger adapta-tion effort as long as t s takes negative and negative val-ues, and the system communicate this happening throughenhanced oscillations for the quasi-normal mode ampli-tude. Figures 3 provide examples of functions that canbe taken as adiabaticity indicators for ω and σ , showingthe net differences originated t s from positive to negativevalues. In particular, Figure 3.c) compares the quantities4 ω σ that corresponds to a t s < t s = 0 casewith a different parameter D e chosen so to equal the sum∆ ω, + ∆ ω, of the two jumps identifiable in the formercase, thus showing the higher degree of adiabaticity ofthe second case. Finally note that when t s < σ tends to increase its value during the course ofevolution after a transient time where it decreases. Suchdynamics is related to the occurrence of a sign changefor ˙ ω . Indeed, there is a basic process by which varia-tions of amplitude σ are brought about: the raising offrequency parameter ω acts to reduce σ and viceversa.If the frequency parameter varies adiabatically the cor-respondence is stronger, in the sense that a net 1-to-1map is entailed between parametric frequency and quasinormal-mode amplitude, the relationship ˙ θ = √ c/σ be-tween amplitude and phase of quasi-normal modes effec-tively turning into ˙ θ (cid:39) ω (the lowest order term in Eq.(38)). If meaningful deviations from adiabatic regime oc-cur, non-negligible fluctuations start to be generated for σ . FIG. 4: Comparison between solutions to the Ermakov equation σ and solutions to the parametric oscillator equation. Whenits oscillations increase due to non-adiabaticity of frequency changes and of dynamics, amplitude σ progressively acquiresfeatures of solutions to the parametric oscillator equation, approaching the positive portions of x = σ cos θ and the reflectionsacross the time axes of negative ones. Accordingly, phases θ develop continuous jumps. It is instructive at this stage also to show explic-itly how the process of generation of amplitude fluc-tuations relates to the features of the other importantoutcomes and of the governing pumping ω . When thedynamical regime is such that weakly oscillating quasi-normal mode amplitudes are generated, the phase func- tion θ ( t ) = √ c (cid:82) t σ − ( t (cid:48) ) dt (cid:48) deviates from the linearbehavior to a minor extend. If its oscillations are ofmore considerable importance, the function σ tends tomimic more extensively the quasi-normal mode function y = σ ( t ) cos θ ( t ) solving the parametric oscillator equa-tion with the same initial conditions, and, especially,2its absolute value | y ( t ) | = σ ( t ) | cos θ ( t ) | . The shape ofphase function is affected accordingly, and a staircasetype shape ensues with raisers developing in correspon-dence of minima for σ . Owing to the development ofcontinuous phase jumps of the order of π , the functioncos θ modifies to a sequence of square-wave like elementswith minimum and maximum about the values -1 and 1. B. Case 2: D e < and V ∞ > The case we dwell upon in this Subsection differs fromthe previous one in that the parametric frequency tendsto an asymptotic value V ∞ >
0, allowing for a potentiallyinfinite time evolution. The system is driven exponen-tially fast to a dynamical regime that basically act as anharmonic oscillator that is perturbed extremely weaklyby a time-dependent correction to the frequency. Unlessthe initial and the asymptotic value of the parametric fre-quency ω = ω (0) and ω ∞ ≡ ω ( t → ∞ ) ≡ √ V ∞ coincide,initial and asymptotic amplitudes σ and σ ∞ ≡ σ ( t →∞ ) are affected unequally by a variation of the verticaltranslation parameter V ∞ → V ∞ = V ∞ + δV ∞ . In partic-ular, if σ and σ ∞ denote the initial and the asymptoticvalues for amplitude σ after the shift V ∞ → V ∞ in ω ,the asymmetry in the initial and asymptotic responses ofamplitudes can be measured by the ratio R = σ ∞ − σ ∞ σ − σ = (cid:18) ω V ∞ (cid:19) / (cid:16) V ∞ V ∞ + δV ∞ (cid:17) / − (cid:16) ω ω + δV ∞ (cid:17) / − ω = (cid:112) D e ( e − t s /b − e − t s /b ) + V ∞ the initial fre-quency pertinent the case of vertical translation V ∞ . Theratio R is just the unity when t s = − b ln 2 so that ω = ω ∞ = √ V ∞ , being R < − < δV ∞ ω < δV ∞ V ∞ < < δV ∞ V ∞ < δV ∞ ω , and R > σ after a square frequency constant shift is greater thanthat at the initial time if ω > ω ∞ = V ∞ .Let us analyze the solutions to the Ermakov equationin the present case once we set t s = 0. Solution σ ini-tially grows until it stabilizes to an oscillating regime.For fixed height of the frequency parameter shape (i.e.for fixed D e ), the magnitude of oscillation is stronglyaffected by the width of the time interval where ˙ ω issignificant and the jump for ω from maximum to asymp-totic value is essentially realized. As long as the positivetime-scaling parameter b increases, the amplitude of os-cillations becomes smaller and smaller, the fluctuationbeing basically negligible for sufficiently large values of b . Magnitude of oscillations varies significantly providedthat the different b are taken so to exclude at a too sud-den initial jump for ω . This is evident from Figure 5.a)(there the b = 0 . b = 0 .
015 curves almost overlaps).Although at the initial stage σ grows monotonically, os-cillations underlay the growth rate. They are very little visible and are evidenced via ˙ σ ; see Figure 5.b) where thet-differentials for the smoothest curves in Figure 5.a) areplotted. At fixed b and D e , the greater is the asymptoticvalue of frequency parameter | V ∞ | > σ are realized, as wellas their amplitude and their characteristic period; see thecomparison between the black curve and the (essentiallyoverlapping) yellow and magenta ones in figure 5.a) aswell as the blue curve in figure 5.c). For assigned V ∞ and b , the progressive augment of | D e | results into moreand more substantial departures of ω from the ”opti-mal” constant value V ∞ , and wider oscillation regimesarise, Fig. 5.c). It is also worth to notice that oscilla-tions do not develop symmetrically with respect to thevalue σ HO = (4 V ∞ ) / , and mean values of σ tend toattain a higher value -see Figure 5.a). FIG. 5: σ and ˙ σ when V ∞ > D e > t s = 0 in Eq.(39). Color dotted curves refer to corresponding potentialsup to scaling factors. Oscillations of different magnitude aregenerated for σ depending on the properties of the actingparametric driving. Having an overall insight on the case where dynamicsis assumed to start at the maximum of ω , we can moveto the other case of our current interest by consideringa nonvanishing time-translation parameter t s and lettingthe evolution start prior or later that point.Let us start first with negative time translation pa-rameters, t s < σ and ˙ σ we adopted sofar.By considering negative t s , the driving frequency turnsfrom monotonic to unimodal in a manner that is, evi-dently, similar to the Case 1 argued in previous Subsec-tion. By progressively diminishing the parameter t s < ω enters more and more effectively into action in a waythat is in in agreement with findings from Subsection3IV A. From Figures 6 one infers indeed that larger fluc-tuations are generated for lower negative values of t s .Explicit examples are also provided concerning therepercussion of diversifying the time-scaling parameter b , as in plots 6.a) and 6.b) that make definite the result-ing amplification of σ -oscillations (compare with Figs.5.a) and 5.b)). Once again, it is found that that mereadiabaticity of frequency changes is not enough to en-sure a smooth outcome for amplitudes σ . In contrast, ifvery gentle modifications of parametric frequency beginto rule the dynamics at a certain time, then oscillationsof large magnitude keeps being sustained. This is not FIG. 6: σ when time translations are performed with t s (cid:54) = 0.Colored dotted curves refer to scaled ω . Oscillations of dif-ferent magnitude are generated depending on the propertiesof the acting parametric driving. surprising. It simply reflects the fact that the systemsenters into that dynamical regime with a largely oscillat-ing amplitude σ owing the features of the initial transientregime, and the very adiabatic changes of hamiltoniantaking place thereinafter are so extremely slow that theyare not much effective in altering the evolution trend thatthe amplitude σ is already experiencing. That is, thesystem is led in a relatively short time and exponentiallyfast to a dynamic regime governed by a hamiltonian thatis essentially harmonic, with wide amplitudes surviving.If one looks at the envelope profile surrounding oscilla-tions its modifications are just as adiabatic as expectedby comparison with the parametric frequency .The case t s > t s begins to increase the oscillations tendto become more extensive, but for sufficiently large t s the opposite trend is engaged owing to the progressivediminishing of ω (which gets closer to V ∞ ) and of ˙ ω (since the faster decreasing component e − t + t s ) /b doesnot play an effective role anymore).We conclude the Subsection by providing plots mak-ing evident the dynamics of amplitude and phase of solu- tions to the Ermakov equation for the current case if fre-quency parameters are changed so that a greater degreeof nonadiabaticity characterizes the ω pumping at earlyevolution times. They confirm the expectations aboutdynamics of amplitude σ oscillations and of its extrema,and consequently of phases and their sudden changes, seeFigure 7. FIG. 7: Comparison of quantities σ , θ , σ cos θ when ω isgiven by (39) with V ∞ > D e <
0. When the magnitudeof oscillations increases, solutions to the Ermakov equationapproach solutions to the parametric equation and tend toreproduce their absolute values.
C. Case 3: D e , V ∞ > The case we consider now is a dual counterpart to Case2, to which it is linked by a reflection of the ω shapeabout the horizontal line identified in terms of the asymp-totic value V ∞ . By virtue of this, the case gives rise todynamical mechanisms rather predictable in the light ofarguments expounded in previous two subsection. Thereis however a difference that has to be accounted for: atfixed V ∞ and D e , quite fast initial variations of para-metric frequency may happen because of the arbitrarilygreat differences can be achieved between the maximumand the minimum values attained by the curve ω when t s < ω and ω min = V ∞ − D e ).Having defined D e as a positive quantity, in the sim-plest case t s = 0 the request ω ( t ) > V ∞ > D e . The increasing of ω from its initial valuemeans that some energy is pumped into the system, andthe evolution of quantities of dynamical interest reflectsthe presence of such basic mechanism. When t s = 0 thefunction σ exhibits an asymptotic oscillating shape thatis more and more suppressed as long as the difference V ∞ − D e increases, the average value also lowering; seeFig. 8.a). Fluctuations are of greater magnitude when-ever V ∞ and D e are comparable. This happens because4of the higher value of the initial value for σ along with thewider gap between the initial minimum and the asymp-totic value for ω , the latter giving rise to a less adiabaticdynamics. For sufficiently great values of V ∞ , the curve obtained for σ essentially flattens, as consequence of theprogressive disappearance of the initial hole and of valu-able rate changes for ω . FIG. 8: Examples of amplitude σ for D e > V ∞ >
0. Colored dotted curves refer to corresponding scaled ω . Oscillationsof different magnitude are generated by variation of the parameters determining the time-dependent frequency driving. The picture clearly modifies if the ω shape previouslyruling the dynamics is shifted either to the right or tothe left in the ( ω , t ) plane, Figs. 8.b) and 8.c). Considerfirst the case t s <
0. With the increasing of | t s | keeping D e , b, V ∞ fixed, the initial value of function σ diminishes,lessening the amplitude of oscillations maintained in thecourse of evolution. It is pretty clear, however, that dis-tinguished situations can be expected depending on themagnitude of t s . Indeed, for sufficiently small t s <
0, theoverall qualitative dynamics tailors pretty much to the t s = 0 case: the initial decreasing trait introduced forthe parametric frequency only acts on quite a short timewindow and the difference between curves’ fluctuations atlater times (i.e. when ω approaches its asymptotic value V ∞ ) is slightly reduced by comparison with the initial gap σ ( t s ) − σ (0), see Fig. 8. While lowering further the valueof t s , the first part of the amplitude curve goes down andlarger fluctuations appear. So, for sufficiently small nega-tive t s the magnitude of σ fluctuations reduces comparedto the t s = 0 case, and it increases as expected only laterfor smaller negative t s . Besides, we checked that, by di-minishing t s further, fluctuations invert few times againthe tendency to vary magnitude monotonically, until forsufficiently low t s the part of σ with regular oscillationsappear to freeze and simply translates with the almostconstant part of the parametric frequency. This unpre-dicted dynamics seems to be very peculiar of the caseunder consideration in this Subsection that, along withthe opposite way to introduce the alternation betweengrowing and decreasing regimes for the parametric fre-quency, differs from cases in Subsections V A and V Balso by not having a lower bound for t s .Vertical translation of the ω curve towards the timeaxis augments the magnitude of σ fluctuations. Notwith-standing it may result in a minor variation for ω (andhence σ ), a change in the asymptotic value V ∞ impor- tantly alters the evolution of solutions to the Ermakovequation for quasi normal amplitude σ . Figures 8 showthat the effect is really worthy of attention. For instance,in cases shown in graphics with t s < V ∞ = 7 is translated by a factor that is only either1 /
14 or 1 / ω experi-ences a percentage variation that are only about 0 . . . − .
3% interval. This appear to sug-gest that if jumps ω − ω min and ω min − ω ∞ are fixed,then wider oscillation are necessary for the amplitude σ if the system is driven closer to a free dynamics. Fig-ures 8.b)-8.d) clearly indicate also that oscillations are,once again, not to be meant as centered about the value σ HO = (4 V ∞ ) − / naturally associated with an harmonicoscillator whose frequency is determined by the asymp-totic value ω ( t → ∞ ) = √ V ∞ of parametric frequency.What happens instead when the time shift parame-ter attains positive values t s > ω tends to progressively disappear and theinitial value ω assumes higher values, thereby implyingfor σ that oscillations of smaller magnitude are realizedabout the value σ HO = (4 V ∞ ) − / .Figures 8, as well as Figs. 5 provided dealing withCase 2, show an evident consistency with the Sturm the-orem, in that the curves σ resulting for majorant ω possess (in a sufficiently large time interval) more ex-trema. Even though the concrete examples we explicitlyreported by specifying numerical values for the parame-ters involved are such that the resulting parametric fre-quency ω are relatively similar and the effect may notbe marked, they still reveal that the frequency of occur-rence for the maxima/minima exhibited by the quasi-normal mode amplitude σ does respond to the introduc-tion of majoring/minoring the pumping mechanism for5the parametric oscillator (27), curves σ resulting for ma-jorant ω possessing (in a sufficiently large time interval)more extrema. In figure 6.a), for instance, effects of thedifferent number of maxima of Ax + Cx + 2 Bx x arealready visible in few time units through an increasing de-phasing that is realized. The dynamics of the harmonicoscillator with the minimum value attained by paramet-ric frequency introduces some limits on the oscillationmechanism for the solution to the parametric oscillatorequation with the Milne-type frequency (39). From theSturm comparison theorem follows indeed that a zero forthe parametric equation with the Milne-type frequency(39) occurs between two successive zeros of solutions tothe harmonic oscillator with the asymptotic frequency ω ∞ = ω ( t → ∞ ) = √ V ∞ (Case 2) or ω min = √ V ∞ − D e (Case 3).We conclude this Subsection by calling attention of thefact that variations of b , and the dynamics of progressivedrawing near of functions σ and σ | cos θ | , have not explic-itly argued here since both issue have been understoodpreviously. D. Case 4: D e > and V ∞ < The final case regards a parametric function startingfrom a positive value at initial time, and next monoton-ically decreasing until it vanishes at a certain time. Thedifference with Case 1 with t s > ω ( t ) because the two cases are associated todifferent parts of the basic Morse curve (recall Figure 1).The frequency curve now decreases much more rapidly,and the time window of interest is smaller accordingly.Inspection of the independent solution to the paramet-ric oscillator equation now shows a possible generationof initial zeroes and a rapid transition to the exponentialgrowth regime. The analysis of the case develops thuseasily. In Figure 9 we plot the quasi-normal amplitudefunction σ ( t ) of the form (30) with proper x , x andWronskian (see Sec. IV D), and coefficients A, B, C inEq. (32), for some parameters D e and b . Once one looksat function σ ( t ) an initial fluctuating shape it is thereforeimplied that later tends to bend itself and rectifies, reach-ing its maximum at the end of evolution. The modestundulations reflect that relatively small variations takeplace for the independent solutions to the parametric os-cillator equation. VI. QUANTUM VARIANCES ANDUNCERTAINTIESA. Classical and quantum squeezing of orbits
Solutions to equation of motion for the parametric os-cillator define curves in the extended phase space ( q, p, t ),the collection of all possible orbits in such space spanning
FIG. 9: Examples of σ when Case 4 holds for ω . Dottedcurves refer to scaled ω . a surface that has the topology of a cylinder. This sur-face is essentially determined by the classical Ermakovinvariant I = Q + P = c q σ + (cid:16) σ pm − ˙ σq (cid:17) (71)(here q and p are classical variables) which, at any giventime, defines an elliptic curve in the position-momentumspace that collects all the possible pairs ( q, p ) obtained byvariation of initial conditions for the Ermakov amplitudeequation. During the course of evolution the invariant I generates different ellipses about the origin in the plane( q, p ) because the functions σ and ˙ σ vary. These ellipsesare of fixed area ( √ c π m = πmI √ c ), but with foci placement,eccentricity e ∗ , major axes a t = 2 a ∗ , minor axes b t = 2 b ∗ and length L ∗ changing in time. The ellipses motion canbe schematically be ascribed to variations of quantities σ and δ = σ ˙ σ √ c , (72)since it results: e ∗ = (cid:115) √ T − c m σ T + √ T − c m σ (73) a ∗ = √ I √ c σ (cid:113) T + (cid:112) T − c m σ (74) b ∗ = √ I √ c σ (cid:113) T − (cid:112) T − c m σ (75)with T = c m (1 + δ ) + σ (76)( T > cm σ ) and L ∗ = a ∗ E ( e ∗ ), being E the ellipticintegral of the second kind. The role of quantities σ and˙ σ is also tangible while examining the dynamics of inter-sections of ellipses with axis in phase space ( q, p ), thatare identified through q ∗± = ± (cid:114) I c σ √ δ , p ∗± = ± m √ I σ . (77)6If ˙ σ = 0 foci turn out to be located on the q -axis or on p -axis, and eccentricity simply reads e ∗ = (cid:112) − σ /cm if σ < √ cm (in which case foci coordinates read q f, = q f, = 0, p f, = − p f, = − √ cm − σ mσ , so that a ∗ = m √ I σ ), or e ∗ = (cid:112) − cm /σ if σ > √ cm (inwhich case q f, = − q f, = √ σ − cm mσ , p f, = − p f, = 0,so a ∗ = √ I σ/ √ c ). A circular profile is thence gener-ated only for ˙ σ = 0 and σ = c / √ m , the radius being R ∗ = a ∗ = b ∗ = c − / √ mI .The above identification of the classically accessibleregion in phase space put a basis for the connectionbetween classical and quantum aspects of the paramet-ric oscillator dynamics, which is rather naturally vali-dated within a coherent-states approach, where the mo-tion (73)-(75) of the elliptic sets collecting all couplesof position-momentum coordinates that can be realizedclassically at each time is conveyed at the quantum levelto the Wigner ellipses [55], e.g. for the vacuum | (cid:105) . Bear-ing in mind Subsection II C, it is evident that coefficientsof the quadratic invariant I can be rephrased as the sec-ond statistical momenta for position and operators, and,in turn, that the latter directly connects to the dynamicalreshaping of ellipses in phase space, see Eqs. (24)-(25)and (77).Having already scrutinized the behavior of the ampli-tude σ , to complete gaining insight on the transfer atquantum level of the squeezing dynamics for classical Er-makov ellipses it is therefore in order to pay attention onthe quantity δ . Aimed at this, in the next Subsectionswe shall provide examples that outcome by taking theinitial conditions (21) and by considering the function∆ UP = (cid:126) (cid:16)(cid:112) δ − (cid:17) (78)measuring the deviation of Heisenberg uncertainty princi-ple over the vacuum | (cid:105) (or any other Lewis-Reiesenfeldcoherent state, Eq. (26)) from the minimal value (cid:126) / UP also gives straight indication on what re-sults over arbitrarily excited states | n (cid:105) due to ∆ n ˆ q ∆ n ˆ p =(1 + 2 n ) ∆ ˆ q ∆ ˆ p . Results of previous Section for quasi-normal mode amplitudes σ provide a first glimpse in re-spect to possible dynamics for δ and ∆ UP . Besides, wecan notice that the Ermakov equation can be managedto generally write dδdt = (cid:0) δ (cid:1) √ c σ − ω σ √ c = (cid:0) c − ω σ (cid:1) + c δ √ c σ , (79)a formula that gives an indication about the way nona-diabatic dynamics can act in squeezing Ermakov-Wignerellipses and build up nontrivial correlations in the courseof system’s evolution. We thus expect that in some casesdeviation of function (cid:126) √ δ that cannot be neglectedcompared to the familiar minimum value (cid:126) /
2, and mayappear even astonishing if one would not have shed lighton the features of solution to the Ermakov equation andthe trajectories implied for the parametric oscillator clas-sical dynamics.
B. Position-momentum uncertainties. Case 1: D e , V ∞ < Assuming the initial conditions (21), the quantity (78)is vanishing at initial time. Main aspects concerningwhat happens after that time are shown in Figure 10.They can be can be subsumed as follows.When t s = 0 the growth of ∆ UP is fairly smooth andslow, before to markedly raising in the final course ofdynamics as the time (40) is going to be approached. InFigure 10, for instance, the maximum gain for ∆ ˆ q ∆ ˆ p isa bit below 20% of its initial value. Lowering the param-eter V ∞ , the pumping mechanism operates on a shortertime interval and ∆ UP varies faster.If t s is varied from zero, forewarning of oscillations beginto arise. A generation of a succession of peaks is distinctwhen t s <
0, which is the case where a sign change is ex-perienced by ˙ ω . The lowering of t s makes the oscillatingpattern more uniform, but also of greater magnitude asthe shape of the frequency parameter ω varies more be-fore the maximum is reached at t = − t s . When ω getsvery close to zero, minor variations of t s leading to in-appreciable variation of the frequency parameter shapeobviously provoke impressive alteration for the magni-tude of oscillations; in Figure 10.c), this is shown withvalues for the maxima of ∆ UP that jump at great ratefrom about (cid:126) to more that fifteen times that value. FIG. 10: ∆ UP in (cid:126) = 1 units when the parametric frequencyis given by means of Eq. (41). a) Examples with fixed D e , b and t = 0 and varying V ∞ . The plot in the upper cornergives a close-up of initial trait of the curves. b) Initial traitsof curves ∆ UP obtained with positive, vanishing and negativevalues of t s . c) Generation of large fluctuations when t s < C. Position-momentum uncertainties. Case 2: D e < and V ∞ > . Once again, the quantity (cid:126) (cid:112) σ ˙ σ /c may de-part pretty much from the canonical value (cid:126) / c = 1 / b and t s . Figure 11.a) shows examples where b issmall and peaks of ∆ UP range from about 10% to 80%of the initial uncertainty value (cid:126) /
2. In Figure 11.b) theparameter b that scales the independent variable t is in-creased and magnitude of ∆ UP falls off consistently. Atransient initial dynamics is clearly seen, especially for t s (cid:54) = 0, where the largest contributions to the position-momentum uncertainty are recorded (not exceeding 2%of (cid:126) / FIG. 11: Examples of ∆ UP curves in (cid:126) = 1 units when V ∞ > D e < D e and V ∞ are hold fixed while thereare two different varying sets for b and t s . Smaller values of b amplify the deviation from minimal Heisenberg uncertainty (cid:126) / D. Position-momentum uncertainties. Case 3
Even when both D e and V ∞ in (39) are positive con-stants, amplitude fluctuations may generate oscillatingenhancements of the Heisenberg uncertainty for certainvalues of parameters determining the shape of frequencyparameter. Examples are given in Figure 12 with param-eters that make the effect standing out. E. Position-momentum uncertainties. Case 4: D e > and V ∞ < We do not present figures for this case. We foundcurves for ∆ UP that resemble those of Fig. 10.a) of Case1, but with a marginal indication of an underlying fluc-tuation mechanism before the final fast growth. FIG. 12: Oscillations generated for ∆ UP in (cid:126) = 1 units whenboth D e and V ∞ in (39) are positive constants a) The increas-ing of V ∞ at fixed D e suppresses responses. When t s decreasesfrom the null value, a damping of the ∆ UP -oscillations first re-sults that is next replaced by the foreseen amplification trend.b) Scaled ω for originating curves in Fig. a). Relatively lim-ited reshaping of the parametric frequency driving can resultin evident changes for magnitude of oscillations; see red andblue curves in both panels. VII. ZERO-DELAY SECOND ORDERCORRELATION FUNCTIONS
For investigating the evolution of a quantum field gov-erned by a time dependent quadratic hamiltonian, it isconvenient to construct either the eigenstates of the time-dependent annihilation operator or those of the time-dependent number operator, and to connect them to thestandard coherent and number states for the harmonicoscillator identified by factorization of the hamiltonianat an initial time. This allows to realize a descriptionof the dynamic based on Lewis-Riesenfeld coherent- ornumber-type states, and to interpret the results in termsof the stationary coherent states or modal excitation con-structed through a diagonalization of the Hamiltonian atsome initial time. Features of the eigenstates of I are inti-mately connected to those of eigenstates of the harmonicoscillator with mass m and frequency ω because of theextended canonical transformation linking each to theother (Section II), but they present specific peculiaritiesinduced by the concrete form of the transformation, i.e.the vacuum state actually considered. The probability offinding an element from the modal Fock basis {| n (cid:105) } ina Lewis-Riesenfield squeezed coherent state | α (cid:105) does notfollow a Poisson distribution. Related to the mechanismof time-dependent squeezing of quadrature components,modifications of modes counting statistics are manifestedduring the evolution of the one-mode quantum systemgoverned by a parametric oscillator hamiltonian opera-tor. They can be highlighted by determining the valuesof the the second-order degree of coherence [5]. In thisSection we shall therefore analyze the zero-delay correla-tion functions g = (cid:104) n | ˆ a † ˆ a | n (cid:105) (cid:104) n | ˆ a † ˆ a | n (cid:105) = (cid:104) n | N | n (cid:105) − (cid:104) n | N | n (cid:105) (cid:104) n | N | n (cid:105) (80)pertinent to states resulting from evolution of elements | n (cid:105) from the Fock basis identified by diagonalization atinitial time of Hamiltonian. The function g can be de-8termined on purely algebraic grounds on account of theBogolubov mapping (17) with (20), yielding to (cid:104) n | N | n (cid:105) = n + (1 + 2 n ) | ν | , (81) (cid:104) n | N | n (cid:105) = n + 2(3 n + 2 n + 1) | ν | ++ 3 | ν | (2 n + 2 n + 1) , (82)so that g = n ( n −
1) + (6 n + 2 n + 1) | ν | + 3 | ν | (2 n + 2 n + 1)[ n + (1 + 2 n ) | ν | ] . (83)The creation of modes out of a state (e.g. the vac-uum) is determined by the square modulus of the Bo-golubov coefficient ν . At early stages of dynamics theprocess follows a power-law with scale exponent de-termined on account of initial conditions superimposedto σ and ˙ σ , and of initial values for parametric fre-quency and its t-differentials. In general, | ν (0) | =(4 √ c ω σ ) − (cid:2) σ ˙ σ + ( √ c − ω σ ) (cid:3) and one can distin-guish the following main subcases:i) ˙ σ ˙ ω (cid:54) = 0, so that | ν ( t ) | (cid:39) | ν (0) | − σ ˙ σ ˙ ω √ c t plushigher order terms;ii) ˙ ω = 0 and ˙ σ ¨ ω (cid:54) = 0, so that | ν ( t ) | (cid:39) | ν (0) | − σ ˙ σ ¨ ω √ c t +h.o.t.;iii) ˙ σ = 0, so that | ν ( t ) | (cid:39) | ν (0) | +( ω σ − c ) √ cσ ˙ ω t +h.o.t.;iv) ˙ ω = ˙ σ = 0 and ω σ (cid:54) = c , so that | ν ( t ) | (cid:39) | ν (0) | +( ω σ − c ) √ cσ ¨ ω t +h.o.t.;v) ˙ σ = 0 and ω σ = c , so that | ν ( t ) | (cid:39) √ c ω σ ˙ ω t +h.o.t;vi) ˙ σ = ˙ ω = 0 and ω σ = c , so that | ν ( t ) | = √ c ¨ ω ω σ t +h.o.t.The approximate form for the leading correction to zerodelay correlation function g follows accordingly via g (cid:39) − n + n +4 n +3 n | ν | .Working with Lewis-Riesenfeld invariant numberstates based on the minimal uncertainty conditions (21),the last two cases are thus implied. For the problem un-der consideration in this communication, in particular,the distinction relies on the value for time-translationparameter t s in (39) since ˙ ω = 0 for t s = 0. Once itis assumed that minimal uncertainty is realized at initialtime for elements of the invariant Fock space owing to(21), the variance of the invariant number operator(∆ n N ) = 2 ( n + n + 1) | ν | (1 + | ν | ) (84)so vanishes at initial time t = 0, and initial value forcorrelation function g is the standard 1 − /n (with n (cid:54) =0). A. Case 1: D e , V ∞ < n = 1 , ,
100 harmonic oscillator eigenstates that resultfrom the instantaneous Hamiltonian spectral problem at t .Left panel: t s = − . − . t s = 0 (solid), 0 . t s < g curves originate and switchesbetween sub- and super-poissonian regimes are realized. In general, for fixed frequency parameters the curvesfor g assume higher values by increasing the occupa-tion number n . Figures 13 subsume the essential fea-tures displayed when one is concerned with a paramet-ric frequency associated with the case, Eq. (39) with D e , V ∞ <
0. If t s = 0, there are no signs of oscilla-tions. The system starts to evolve from an adiabaticcondition and initial variations of frequency parameter ω do not affect much the 2-point correlation function.But after a certain time g begins to increase its val-ues almost linearly until reaching its maximum at thefinal time t − , Eq. (40). So there is mechanism acted bythe driving parametric frequency that steadily directs thesub-poissonian initial statistics to a super-poissonian oneeventually. Needless to say, for the occupation number n = 1 state the process is longer, but when the memoryof adiabatic initial condition is lost the rate of growthis noticeably higher than for other number states. Suchrate decreases with the increasing of the occupation num-ber, albeit very soon the differences become insignificant.At the final time values for g group closely. Figures 13also show what happens whenever a shift of the frequencycurve is performed by means of translations of time vari-able. For positive shifts t s the time interval where dy-namics takes place is shorter, and it turns out that thegrowing manifests itself with a greater rate but the fi-nal maximum value slightly decreases. It goes withoutsaying that for negative time shift parameter dynamicalresponses for g change. For small negative t s there aremarks of fluctuations forming. For larger negative t s awell defined bell-like profile characterizes g , whose max-imum develops about the time t = | t s | when the para-metric frequency displays its extremum. The top of thebell is in the g > t s = − . n > t s < D e result in the enlargement of the bell, and inthe replacement of the maximum peak with a bigger andbigger plateau. In contrast, lower values of parameter D e introduce more oscillation of lower magnitude, and hencemore changes in the statistics. For t s >
0, one would seeinstead that the decrease of D e merely tends to loweringcurves 13.b) and spreading them in a wider time interval. FIG. 14: Case 1: second-order correlation functions eval-uated over harmonic oscillator eigenstates. a) n = 1 and D e = − , − , − n = 10. Byadjusting the frequency parameters, essential changes affectthe particle creation mechanism and remould g , with moresquared shapes or more peaks emerging, or with suppres-sion/amplification of fluctuations. For low quantum numbers,inverted spikes can be seen due to the sudden changes in thedynamics. It is now important to see what happens by refining theparameter b which fixes the scale for the time-evolutionparameter t . The operation really has a repercussion onthe determination of ω , and of adiabaticity of ω and σ .Indeed, we can see that the dynamics of correlation func-tions g contrasts dramatically with that comprehendedpreviously for b = 1. For higher values of b , the adiabatic- ity of the driving improves: ω approaches the maximumvalue of ω ( t ); in addition, the parametric frequency curvestretches out on a wider time domain, Eq. (40). The ini-tial prominence previously seen for the correlation func-tion g is appreciably dampened, and the successive goingup again is evidently done much less quickly. In contrast,for lower values of b , the previously observed initial bell-like shape turns into a more squared profile, with theformation of a second plateau. The role b has in suppres-sion or enhancement of sudden changes is obviously leadto situations based on different choices for the other pa-rameters defining the frequency parameter. In particular,we have already noted through Figs. 14.a) and 14.b) thatlowering D e introduces more fluctuations for the correla-tion function, though with lessened magnitudes. Figures14.c) and 14.d) present what results if a variation for b is performed in combination the assumption of largevalues for D e . In these figures, D e = − g are shown that aregenerated in the first part time evolution for b = 1, andwider ones, with higher mean values, are sustained once b diminishes. In the remaining part of evolution, thatpattern breaks, to eventually conduct (with or withoutfluctuations, depending on the case) the function g inthe super-poissonian regime. For higher values of b , theinitial oscillations are strongly suppressed and values for g in the first part of dynamics also are much contained.For the second part of evolution a stretching of the b = 1curve over the enlarged time interval appears. B. Case 2: D e < and V ∞ > . Bearing in mind the discussion in Section V concerningthe amplitude σ dynamics, one is aware beforehand of theincidence oscillatory features for the parametric oscilla-tor problem when evolution is governed by the pumpingmechanism of Cases 2 or 3 defined in the infinite time in-terval t ∈ [0 , ∞ ). Specifically, translations of the Morse-type curve for ω are expected to give rise to a gradualdeveloping of oscillation in the g spectrum.In Figure 15 we have plotted typical curves for thecorrelation function g that elucidate the various influ-ences of the shift parameter t s entering the parametricfrequency.If t s = 0, an initial augment for g that reminds of thatseen in case 1 is followed by a sort of asymptotic satu-ration with extremely weak signs of fluctuations. Whenone is concerned with high occupation numbers, a super-poissonian regime is obviously realized in relatively shorttime. The major differences one has in the initial magni-tude of g for the very low occupation numbers vanishesin the course of dynamics, as curves tend asymptoticallyto attain values in a small domain. Being more atten-tive to the first trait of the curves, it is seen that thesuccessive minor fluctuations are in fact anticipated byrather unperceivable variations in the rate of the func-tion’s growth. So it is intuitive that the case possesses0 in potentia the characteristics for sustaining larger oscil-lation in the g spectra whenever changes are introducedvia nontrivial time shifts t s (cid:54) = 0. If t s <
0, pronouncedoscillations indeed arise owing to the major degree ofnonadiabaticity of frequency variations to which statesare subjected in their initial dynamics. A regime of reg-ularity in the oscillator of g appears after a transienttime. In the first course of dynamics, the frequency in- creases to reach its maximum, and a first local maxima isin fact exhibited by g through a peak above the t s = 0curve. After this peak, the g ’s profile stays below the t s = 0 curve, but goes towards it oscillating. When suffi-cient time is passed, the curve for g with t s < t s = 0 curve. The increasing of the occupation numberlet the magnitude of shape variations diminish. FIG. 15: Case 2: g over number states. a) t s = − . , , . n, D e , V ∞ are as in a), but with smaller b and different t s ; c) n = 50 and b, D e , V ∞ as in figure b). d) n = 20 and b, V ∞ as in previous two plots. Figures show oscillationpatterns forming after an initial time transient. Oscillation may be as enlarged to reach the initial values and to provoke abouncing of curves along with reduced minima/maxima excursions, see b)-c). For greater values of | D e | an inverted spike isseen to develop for the lowest allowed values of t s . If t s >
0, obviously departures from the t curves arerealized to a minor extend. By comparison with plotswhen t s = 0, responses differ to a little extend: a bithigher values can be detected in the initial growth of g ,and lower ones later on.It is now interesting to comprehend what happens byrefining the time scale where the exponential terms inthe frequency act more importantly. In view of findingsin Subsection VII A, we expect appreciable alterations ofthe previous big picture. Figure 15.b) shows an exampleon the consequences a scaling b → b/ n = 1 (the case experiencing the greater excursions ofvalues assumed), we can see that large oscillations showup even when t s = 0, and ˙ ω vanishes at initial time.Alternation of sub/super poissonian changes occurs. In-creasing positive t s both the maxima and the magnitudeof oscillations progressively reduce, and a damping mech-anism acts on the curves. For t s < t s oscillations actually increase, but later they areinhibited: when t s is decreased further, the curve is grad-ually lifted up to a reference stationary value, reachedin very short time. For low negative t s the minima at-tained in the course of oscillation approaches the initialvalue for g . When t s is decreased further, curves g arebounced back. The initial value for g sets a referencebound value also quantum numbers n are greater than 1,an instance being given in Figure 15.c). As for countingstatistics, the parametric driving is capable to imprint an essentially super-poissonian behavior, unless very lowquantum occupation numbers n and monotonically vary-ing frequency ω are concerned. C. Case 3: D e , V ∞ > We can proceed in discussing this case as we did beforefor the other examples resulting from different choices forparameters’ signs in the time-dependent frequency. Plot-ting the zero time delay correlation function g versustime we see the expected generation of fluctuating pat-terns. Figures 16 give the gist of variation of occupationnumber n and time-shifts t s . The oscillating pattern isflattened down by higher values of positive t s , as upshotof the declined initial variation and nonadiabaticity expe-rienced by the parametric frequency. For small negativevalues of t s , one whiteness to a right and down trans-lation of the profile. This mechanism is accompaniedby the birth of a new initial hump that, once lower val-ues for t s are assumed, it is lifted up with all the otherspresent about the portion of the t s = 0 curve that attainsmaximal values. Diminishing further the parameter t s , asaturation character emerges: plateau form that are in-terrupted by the small holes generated by the minimum-maximum excursions traits being suppressed and pushedup.Remark that both in this case and the one dealt within Subsection VII B, in the limit t → ∞ it is possible to1 FIG. 16: Case 3: g over number states. a) t s = 0 and n =1 , , , t s and n = 1; c) positive t s and n = 1.Inverted spikes are introduced in the oscillating pattern thatare smoothed out with the increase of the quantum number n and the tuning of t s . resort to formulae µ ∞ = ω ∞ + ω √ ω ∞ ω , ν ∞ = ω ∞ − ω √ ω ∞ ω , (85)with ω ∞ = √ V ∞ , for the Bogolubov coefficients. If theinitial and the asymptotic value of frequency parametercoincide, in the asymptotic future t → ∞ the Lewis-Riensenfield number states go back to the harmonic oscil-lator number states | n (cid:105) and the standard Poisson statis-tics is recovered. If ω ∞ (cid:54) = ω , ν ∞ quantifies the degreeof squeezing surviving in the asymptotic future limit anddetermine the asymptotic modification of statistics via(80)-(83). D. Case 4: D e > and V ∞ < The driving mechanism exerted by the parametric fre-quency runs for a short time interval (in b units) andselecting the most rapidly varying portion of the Morsecurve. The duration of time interval where the drivingmechanism is active does not get through t − , Eq. (40),creasing limitedly for higher values of D e and t s . Vari-ations of D e influence more effectively the quantity ω .However, most influential alteration for the frequency pa-rameter, and thereby for correlation functions, are thoseof parameter b, whose reduction gives a notable boostto ω , affecting the rate of ω . Functions g are sub-jected to a basic growth mechanism, and move into the g > t s ,the case shares some similarities with Case 1 with t s ≥ FIG. 17: Examples of g over number states in Case 4 fordifferent parametric frequencies. a)-b) b = 1, V ∞ = − t s = − , − . , − D e = 5 ,
50. c) D e = 5, V ∞ = − t s = − b = 0 . , n = 1 ,
5. d) D e = 5, V ∞ = − t s = − b = 0 . , , n = 1 , g is subjected to a fast growingbehavior turning into a plateau if the parametric driving isactive for sufficiently long time. trait resembling a saturation mechanism, with possibleprior vague signs of undulations. It is however the lower-ing the value of parameter b that allows the saturation totake a more concrete form, by simultaneously smoothingthe modest undulations. Under these circumstances, themode creation mechanism thus can approach a quality ofstationarity. VIII. DELAYED SECOND ORDERCORRELATION FUNCTIONS
To complete the understanding of statistical aspectswe earned so far through the zero-delay correlation func-tions, it is helpful to considering also the two-time corre-lation function over states | n (cid:105) , i.e. the function g ( τ ) = (cid:104) n | a † ( t ) a † ( t + τ ) a ( t + τ ) a ( t ) | n (cid:105) (cid:104) n | a † ( t ) a ( t ) | n (cid:105) (86)expressing the probability of detecting a second modeafter a time τ once a first one has been detected at time t [5]. This enables one to get an insight into the fulfillmentof anti-bunching/bunching requisites and the possibilityto measuring modes in a time-delayed fashion with higherprobability than being measured at the same time. Weshall limit ourselves to the cases where D e and V ∞ areboth either positive or negative (Cases 1 and 3) becausethey allow to epitomize the basic features displayed alsofor different choices of the driving frequency parameters.Yet, we shall focus on correlations over the lowest excitednumber states | n (cid:105) as they present wider domains whereanti-bunching is accomplished.2 FIG. 18: Case 1: parametric frequency (39) with D e , V ∞ <
0. Level plots for the delayed correlation function g ( τ ) when D e = −
10 and V ∞ = −
2. Thick solid curves refer to g ( τ ) = 1 / g ( τ ) = 1 (black), g ( τ ) = 2 (gray). a)-c) Dynamicsunderlying the reduction of anti-bunching and the surfacing of strong bunching when n = 1 and the parametric frequencyinitial shape is modified by assuming t s = 0 , − . , − . n = 3 and t s = − .
6; by comparison with figure b) the effect ofchanging the reference state is noted.FIG. 19: Case 3: parametric frequency (39) with D e , V ∞ >
0. Level plots for the Glauber function g ( τ ) over number states | n (cid:105) when D e = 3 and V ∞ = 3 .
2. Thick solid curves refer to g ( τ ) = 1 / g ( τ ) = 1 (black), g ( τ ) = 2 (gray). a)-c)Revival and collapse of bunching and antibunching when n = 1 and the parametric frequency initial shape is modified byassuming t s = 0 , − . , − .
8. d) n = 3 and t s = − .
8; by comparison with figure c) the effect of changing the reference state isnoted.
Plots in Figure 18 provide instances of the bunchingand anti-bunching mechanisms possibly realized by con-sidering different time-translation parameters t s ≤ D e , V ∞ <
0. Hav-ing a finite time interval over which the dynamics takesplace clearly introduces a finite domain for the delay vari-able τ . If t s = 0 and the very low occupation numbers n are concerned, there is a mostly anti-bunched behav-ior that is displayed by varying the time τ determin-ing the second counting. For n = 1 a relatively vastearea denoting strong anti-bunching ( g ( τ ) < /
2) obvi-ously arises, a strong anti-bunching being sustained forall the allowed τ for lower t ’s -Fig. 18.a). Diminishing t s , the moderate extension of the time domain allowedfor evolution causes an enlargement of the region in the( t, τ ) plane that identifies anti-bunched responses; thisalso holds for the strongly anti-bunched domain appear-ing for n = 1. At this stage, contour plots for g ( τ )thus qualitatively resemble those obtained for t s = 0. Bydiminishing further the values of t s so to have less adi-abatic initial changes for the parametric frequency and more pronounced fluctuation effects for physical quan-tities, the picture for second-order correlation functionsmodifies: islands reflecting bunching phenomena are in-troduced, possibly tending to connect each to the otherwhile introducing higher values for g ( τ ). Remark thatthe condition t s < ω at early times) does not exclude the possibility to sup-port the detection of strong anti-bunching g ( τ ) < / n = 1, even though an overall inclination to buildbunched super-poissonian responses may be displayed.Figure 18.b) shows for instance that for n = 1 a stronganti-bunched path can be identified in plane ( t, τ ) thatoriginates when t s is about 1 . τ is varied in itswhole allowed domain. For the closely successive val-ues of the quantum number n , the behavior of correla-tions g ( τ ) is affected accordingly, see Figure 18.d) deter-mined for n = 3 and with the same frequency parameters D e , b, V ∞ and t s of Figure 18).b). The zero-delayed sec-ond order correlation function does not present stronglysub-poissonian traits as in the n = 1 case, but delayedcorrelation can attain low values. In Fig. 18.d) it is seen3that strong bunching or strong anti-bunching can be de-tected.When D e , V ∞ > g ( τ ), and the tendency to developrepeating patterns is recognized. Figures 19.a)-c) show,in particular, examples referring to the two-time correla-tions over the states | (cid:105) . The alternation of domains as-sociated with strong manifestations of anti-bunching andbunching is visible in Figure 19.a), concerned with a casefor which t s = 0. Upon decreasing the parameter t s pat-terns modify, a suppression on responses takes place andlarger islands associated with anti-bunching show up for τ >
0, Fig. 19.b). By further decreasing t s the foreseenreinforcement for the mechanism of bunching follow, Fig.19.c), with higher values for g ( τ ) attained by increasingthe quantum number n , Fig. 19.d). IX. CONCLUSIONS
We have discussed the dynamics of one-degree of free-dom Hamiltonian of the parametric oscillator type witha time-dependent frequency based on a Morse potentialshape, up to possible translation and sign reversion. Thisprovided us an analytical framework to examine bothcases where there is a monotonic increasing or decreasingenergy pumping mechanism, and cases where there is analternance of them realized continuously. Depending onchoices of the parameters, the pumping mechanism canbe active unlimited or cease after a certain time interval.For time-dependent quadratic systems, solution to theSch¨odinger equation and quantum dynamics are mani-festly based on classical dynamics as the centroid andthe spreads of a wave-packet follow exactly the evolutionof a classical particle. We thus detailed the derivation ofsolution to classical equation of motion for the parametric oscillator with the time-dependent frequency we set out,and next we analyzed the features of their amplitudes.The generation of oscillation for solution amplitude, aconsequence of the Sturm-Liouville problem determiningthe classical dynamics, has been argued as in connec-tion with adiabaticity of initial condition and dynamics.Enhancement and suppression of oscillations by virtue offrequency parameter shape changes have been thoroughlyinspected.We later argued on the squeezing mechanism impliedat the classical behavior and how quantum dynamics mir-rors it within the standard treatment of parametric oscil-lators by means of the squeezing formalism, and in thisrespect presented plots related to the Heisenberg uncer-tainty relation on number-type and coherent-type states.Finally, to refine the insight into the statistical contentsimplied, the generation during the course of evolutionof nonclassical features for number-type states has beendiscussed by investigating deviation from Poisson statis-tics of two-point correlation function with zero time de-lay owing to the squeezing dynamics taking place. De-layed two-point correlations have also been considered toattest the phenomena of time-depending bunching andanti-bunching showing up. Results obtained clearly showtrends of collapse and revival phenomena.The analysis we performed in this communication canbe continued by taking into account more complicatedstates or by implementation of quasi-probability tools. Inparticular, linear combination of states can be consideredto analyze the competition between interference effectsand parametric frequency variations both on sufficientlylong-time behavior and at early stage of the evolution(where they are more consistent), and the consequenteffects on correlations. It definitively appears of interestalso to proceed with multi-mode generalization and toexamine dynamics of entanglement. We plan to addressthese issues in the future. [1] V.V. Dodonov and V.I. Man’ko Eds,
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Ermakov equation:a commentary , Appl. Anal. Discrete Math. , 146-157(2008), and refererences therein.[42] P. Bohl, ¨Uber eine Differentialgleichung derSt¨orungstheorie , J. Reine Angew. Math. , 268-321 (1906).[43] C.J. Eliezer and A. Gray, A Note on the Time-DependentHarmonic Oscillator , SIAM J. Appl. Math. , 463-468(1976).[44] M. R´ab, Kriterien fiir die Oszillation der Losungen derDifferentialgleichung [ p ( x ) y (cid:48) ] (cid:48) + q ( x ) y = 0, Casopis Pesf.Mat. , 335-370 (1959); Erratum: ibid. , 91 (1960).[45] For a recent recollection of main results achieved in thevarious cases of interest, see e.g. : Q Kong and M.Paˇsi´c, Second-order differential equations: some signifi-cant results due to James S.W. Wong , Differential Equa-tions and Applications , 99-163, (2014), and referencestherein.[46] A. Zettl, Sturm-Liouville theory , Mathematical surveysand monographs, 121, AMS, Providence, 2005.[47] Abramowitz, M. and Stegun, I. A. (Eds.).
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Wave packets and statis-tics concerned woth SUSY-QM partners of Paul-traphamiltonians , Theor. Math. Phys. , 924-938 (2011).[50] The parameter c acts as a scale factor for σ having noinfluence on actual dynamics. The two most commonchoices are c = 1 or c = 1 /
4. When needed for the prac-tical purpose of plots generation, we will adopt the latterconvention.[51] Close formulae can be given also when the mass-typeparameter is time-dependent [32]. Adaptation to SUSY-QM formalism can be found in [49].[52] Analogy with a quantum diffusion problem may workpartially owing to the very different nature of the vari-ables and supplementary conditions involved.[53] This implies working with very large values for the timevariable t , in principle outside the domain where ourproblem (Eqs. (1) and (27) with (39)) is formulated. Thisaspect is obviously irrelevant if we are interested in eval-uating the Wronskian. [54] For the advantage of physical interpretation of results,hereinafter a reference to the frequency parameter willoften be present in plots through of dotted curves ofthe same color of the quantities that will be evaluatedwith the same assignment of parameters. The way thefrequency parameter reference will be given will be estab-lished case by case depending on individual plot’s conve- nience. Moreover, from now on all parameters enteringits definition will be considered in adimensional units.[55] Conventional Wigner ellipses [21, 26] result upon replace-ments q → q − (cid:104) q (cid:105) and p → p − (cid:104) p (cid:105) in the Ermakovinvariant, which is set to the value (cid:126) / (2 mm