Sudden excitations of harmonic normal modes

Journal of Mathematical Chemistry (2023) 61:288–295 https://doi.org/10.1007/s10910-022-01343-w ORIGINAL PAPER Sudden excitations of harmonic normal modes I. Aldazabal1,2 ·I. Nagy2,3 Received: 25 January 2022 / Accepted: 2 March 2022 / Published online: 4 April 2022 © The Author(s) 2022 Abstract The N-harmonium boson system, i.e., a completely integrable model of Nparticles where both the external confinement and the two-particle interaction are harmonic, is investigated under the action of sudden time-dependent perturbation. This quenchlike external perturbation of confinement has a quadrupolar space-character. The time-independent transition probabilities, which characterize the impact of quench as average occupation numbers, form a complete distribution in the sense of probability theory. The quench-generated energy shift Ein the correlated many-body system, and a purity-type Rényi entropy Sα=2are calculated. Challenging reinterpretations of such an energy change in terms of variables of a classical thermodynamical system of N(N−1)/2 pairs are given as well. As in the case of the ground-state correlated system, an entropy could characterize a global link to energetically optimized independent-particle models. Keywords Correlation ·Entropy ·Excitations 1 Survey of the unperturbed model system Advances in optical trapping of cold atoms have allowed for an unprecedented manipulation over the size of these quantum systems such that the number Nof atoms being trapped can be [1] precisely specified. In general, operating on quantum many-body systems provides a way to understanding [2–5]. In particular, for precisely specified interacting quantum systems, time-dependent tuning (quench) of external harmonic This study on analogies between many-body systems is dedicated to the memory of János Pipek. BI. Aldazabal [email protected] 1Centro de Física de Materiales (CSIC-UPV/EHU)-MPC, P. Manuel de Lardizabal 5, 20018 San Sebastián, Spain 2Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 San Sebastián, Spain 3Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, 1521 Budapest, Hungary 123 Journal of Mathematical Chemistry (2023) 61:288–295 289 confinement seems to be an easily realizable experimental tool to generate dynamics. Remarkably, this external control can be a continuous one, in contrast to the discrete manipulation of many-body (target) systems with external projectile-impact. Motivated by the feasibility of experimental access to averages of driven many-body systems, here we apply a well-analyzed model system to an energetic study. Such a study on a model system can be considered as a clinical attempt to important energetic and statistical details, which might generate further efforts. Following earlier works [6,7], here we take a prototype one-dimensional system of Nidentical particles, bosons, with mass mand scalar coordinates xi, where i= 1,2, .....N. The Hamiltonian, introduced by Heisenberg as the simplest many-body form to his studies HN= N  i=1p2 i 2m+1 2mω2 0x2 i−1 2mω2 0 1≤i≤j≤N (xi−xj)2,(1) is separable (at = 0) and this fact results in independent normal modes. Thus the basic expectation value in quantum mechanics, the ground-state energy, becomes additive E=1 2ω1+(N−1) 2ω2(2) where, without loss of generality, we take units defined by m=1 and =1. The frequencies of harmonic normal modes are [6,7] given by ω1=ω0and ω2=ω0√1−N. Notice that the stability, for repulsive interparticle interaction ( > 0),ismarkedbytheN<1 condition. There is no such constraint for the attractive, like in nuclear physics, case. In the energetically-optimal (e), independent-particle modeling one gets for the energy Ee=N 2ωe=N 2ω01−(N−1), (3) which is based on the additive structure of Eq. (1) without xixjproduct-terms. For >0 the frequency-ordering becomes ω2<ω e<ω 0, and for <0 the ordering is ω2>ω e>ω 0, The difference Ec=(E−Ee)is, according to Wigner [8] pioneering definition, the correlation energy. It is instructive (c.f., next paragraph) to investigate the small-coupling (→0) limit of the correlation energy. By straightforward expansion one arrives at, in our units Ec( << 1)=− N(N−1) 2 2 8ω0.(4) The second derivative, in coupling, of the difference of two variational quantity is negative. This observation on sign is in accord with general statements in quantum chemistry [9]. 123 290 Journal of Mathematical Chemistry (2023) 61:288–295 In the second part of this survey, we derive a challenging correspondence for Ec( << 1)by using precise result on the one-particle reduced density matrix [6,7] of the interacting boson system, and its well-known [10,11]formal equivalence with the statistical density matrix of an ideal system of Noscillators with frequency ¯ω(ω0,,N)in thermal equilibrium at temperature T(ω0,,N). These oscillators do not interact with each other, but only with the heat bath. We note that in field-theoretic attempts to black-hole physics [10,11], the usual association is based on photons. There, the tracing out of high-energy degrees of freedom yields a low-energy effective field theory with an accompanying statistical measure of black-body-like entropy which may be considered as an information loss. For the equivalent thermodynamical system one has E=F+TS N, where F is Helmholtz’s free energy, and there is a heat-like product of the temperature T and the von Neumann entropy SN. These are the variables in the path based on an ideal canonical-ensemble to thermodynamics [12]. Employing precise [6] mappings between the formally equivalent one-matrices, we derive to a direct comparison at weak coupling TS N≡T(ω0,,N)SN(ω0,,N)=ω0N(N−1) 2 2 8,(5) where common logarithmic factors cancel out in the lhs product. Thus, based on formal equivalence of two density matrices, we get as partial correspondence TS N=−Ec. Notice that at N||<< 1 in the quantum-mechanical case, i.e., at (T/ω0)<<1in the thermal case, the thermal part of the Helmholtz free energy, i.e., the part beyond its zero-point energy (1/2)N¯ω, is exponentially small. To our best knowledge, the abovederived formal correspondence is novel on a prototype many-body system. Besides, the quadratic-in-and the linear-in-number-of-pair characters for the entropy-based part of the ordered-product in Eq. (5) suggests that the correspondence found might hold independently of the statistics in weakly correlated systems. We stress that a proportionality between −Ecand von Neumann entropy SNis known in quantum chemistry as conjecture [13]. 2 Time-dependent perturbation of the model system The selected details of the previous Section signal that the complete orthonormal sets for independent normal modes are oscillator wave functions φn(ω, u)=ω π1/41 √2nn!e−1 2ωu2Hn(√ωu). (6) As Eq. (2) shows, there is one mode with ω1=ω0, and (N−1)mode with ω2.In the ideal, i.e., noninteracting (=0) case all (N)modes are equivalent (ω1=ω2= ω0), and that system has zero information-theoretic entropy since its one-matrix is idempotent. Notice here, that below we will use, to simplify mathematics, a common 123 Journal of Mathematical Chemistry (2023) 61:288–295 291 notation ωfor these frequencies where this is possible and return to ω1and ω2channelnotation where it is needed to physics. The main goal below is to consider the energetic-impact of time-dependent, passing, perturbations with quadratic space-character in normal coordinates. In fact, the N=2 case, with such a perturbation of different sign, was already investigated recently in order to shed light on the sign-dependence of energy shift, a well-known problem in swift proton and antiproton close-impact on inert helium atom [14]. Besides, that study addressed few alarming problems inherent in the time-dependent density-functional method [15] where one works with auxiliary (density-optimal) orbitals instead of precise independent modes. In short, with perturbations of finite duration one can calculate the energy shift by using Dirac’s variation of constant method instead of following in time the evolving wave functions, since we know the Hamiltonian at the beginning and at the end of a passing perturbation. Therefore, the energy shift can be calculated in this case by considering the excitation probabilities as occupation numbers which characterize the transitions from a given (in our case: ground) state to other elements of the orthonormal complete sets. The sum of these probabilistic occupation numbers satisfy the normalization condition, as it should be. They weight the mode-energies in summation over quantum number nto get the total energy change. For a passing (vanishing at t→±∞) perturbation the expectation value of the Hamiltonian with evolving states results in the same [14] time-independent energy change. Which still remains to our enumeration of tools, is the concretization of the aboveoutlined occupation numbers. But an insightful method to that concretization is, fortunately, also well-documented due to established works [16,17]. In fact, a complete Chapter [18] written by experts is devoted to similar problems. Briefly, that insightful method rests on an asymptotic analysis via a clever variable-change to map time-dependence into a stationary scattering problem. In our comparative study we employ the expressions, deduced for a single oscillator [16–18], to our case with independent modes. We stress, however, that we consider N-mode systems with precise and energetically-optimized modes. In other words, we investigate the interplay of inherent correlation and external perturbation of quench-character. We believe that the such-obtained results could contribute to understanding. The required statistical weights to energy averaging, i.e., the occupation (transition) probabilities W2n,0are given by [16–18] the following expression W2n,0(R)=(2n)! 22n(n!)2√1−R(R)n=1 √π (n+1/2) (n+1)√1−R(R)n,(7) which reflects the selection rule for allowed (upward) transition with a quadratic perturbation. This complete distribution function is normalized since in general 1 (1−x)η=∞  n=0 (n+η) n!(η) xn. 123 292 Journal of Mathematical Chemistry (2023) 61:288–295 The reflection coefficient Rof the mentioned (auxiliary) scattering problem (see, above) can be calculated in the knowledge of time-details on a quadrupolar perturbation [14,16]. In our work on a confined boson system we restrict ourselves to the quench-like situation. In this abrupt case at t=0, where ω2⇒ω2 f=(ω2+λω2 0), one gets [17] for the reflection R(ω, λ) =ω−ωf ω+ωf2 in terms of initial (ω) and final (ωf) frequencies which characterize the normal modes before and after the quench, respectively. It should be noted that the occupation numbers in Eq. (7) are now simply the squares of expansion coefficients obtained by expanding a given stationary ground-state φ0(ω, u)in terms of a complete set of stationary orthonormal φn(ω f,u)functions. The selection rule mentioned is based on parity-consideration in expansion. Thus, for one mode we have ω=ω1and for the other (N−1)modes we have ω=ω2. The parameter λmeasures the strength of a sudden-change in external confinement. It can have both sign, within the stability range [ω2>−|λ|ω2 0]of the system. Furthermore, in the stability range, there is a duality in R(λ) under the mathematical constraint of ω1ω2=ω2 f. Under such a special constraint, the magnitude of reflection Rcan not distinguish between physical cases with corresponding λ>0 or λ<0, i.e., upor down-tuning. This duality clearly signals, similarly to earlier observations [19,20] with eigenvalues of one-matrices of the unperturbed system, that simple probabilistic measures alone can not characterize completely the physics. We add here based on Eq. (7) (mode i, with Ri, where i=1,2,e) the so-called purity (R), a frequently [19] applied information measure (R)=∞  n=0[W2n,0(R)]2=(1−R)2 πK(R2)≤1,(8) where K(x)is the complete elliptic integral of the first kind. An other measure, the socalled Rényi’s min-entropy [21], is given by Sα=∞(R)=ln[1/(1−R)].HisSα=2(R) is related to a purity via =exp(−S2)in mode i. Such a connection was considered earlier [22] as a promising path to S2via an experimental estimation for . In Rényi’s classification Sαis a measure of order αof the amount of information. We consider [23] such mathematical measures as potentially useful ones even to not-scale-less problems, and return to physics. Despite the above-mentioned duality in occupation numbers (statistical weights), the total final energy (Ef) of the system after quench, and thus the E=(Ef−E) total energy shift, reflect the informations (energy scales) encoded in the Hamiltonian. Keeping in mind the remark at Eq. (6) on simplification in notations, we continue with the determination of channel-contributions, denoted by Eiwhere i=1,2. To 123 Journal of Mathematical Chemistry (2023) 61:288–295 293 E=(E1+E2)we obtain E1(ω1,R1)=ω1 ∞  n=0 (2n)W2n,0(R1)=1 2ω1 2R1 1−Ri ,(9) E2(ω2,R2)=(N−1)ω 2 ∞  n=0 (2n)W2n,0(R2)=(N−1) 2ω2 2R2 1−R2 .(10) In the knowledge of this precise result, we add the one, denoted by Ee, which is based on an energetically (e)pre-optimized independent-particle modeling outlined in Section I, under the impact of the same change (∼λω2 0)in external confinement. We arrive at Ee(ωe,Re)=Nωe ∞  n=0 (2n)W2n,0(Re)=N 2ωe 2Re 1−Re .(11) Motivated by Wigner’s definition of correlation energy Ec() =[E() −Ee()] in the ground-state situation, we are tempted to introduce a quench-related term defined as Ec(, λ) =[E(, λ) −Ee(, λ)] which also reflects the difference between exact and energetically pre-optimized independent-particle descriptions. Now we take the perturbative limit where λ→0 at fixed N,ω0and (small) , thus all Ri<< 1. To a useful comparison with Eq. (4) we obtain Ec(, λ) =+N(N−1) 2 2λ2 64 ω0(12) The positivity is expected on physical grounds since the quench acts as an external agent which tries to diminish rigid individual behaviors (i.e., difference in modes) reflected in Eq. (4) into the direction of a common behavior (i.e., similar, energetically optimized modes). Precisely, it is this observation which suggests us to make finally a somewhat cavalier conjecture via Ec(, λ) =[TS N]. By such a conjecture, which is motivated by the first law of macroscopic thermodynamics as well, we are tempted to view Ecas the result of certain heat-transfer (Q) to a classical system of pairs. 3 Summary, remarks and outlook In this work the N-harmonium boson system, i.e., a completely integrable model of Nparticles where both the external confinement and the two-particle interaction are harmonic, is investigated under the action of a quench-like-in-time perturbation. This external perturbation of confinement has a quadrupolar space-character. The 123 294 Journal of Mathematical Chemistry (2023) 61:288–295 time-independent transition probabilities, which characterize the impact of quench as average occupation numbers, are used to calculate analytically the energy shift E in the many-body system, and the purity-related Rényi’s entropy Sα=2. Challenging reinterpretations, Eqs. (4) and (12), of characteristic energy differences, in terms of variables of a classical thermodynamical system of N(N−1)/2 pairs, are given as well for both the groundand exited-state situations. We stress, as first remark, that our controllable quench is in the external [14] confinement and not in the particle-particle interaction. Their coupling (∼) is not changed. However, controllable quench in that coupling could also be interesting to a general, detailed understanding. Indeed, such a change is in the focus of efforts in [2–5]at fixed confinement. Our previous experience [23] with such a quench in the simpler two-particle (N=2) harmonic model suggests that a similar connection as the one in Eq. (12) for the important difference of stationary energies, i.e., quantum mechanical expectation values, can be found as well. Details due to different quenches, and their possible interplay, require a dedicated study. We add for completeness, that the time-dependence (after sudden quenches at t=0) of the evolving wave functions and associated time-dependent one-matrices [23] contain, via their time-dependent eigenvalues, useful probabilistic information on inherent dynamics in isolated interacting systems. Comparison of the such-obtained time-dependent entropic measures, say a time-dependent system purity [23], with stationary quantities characterized in this study based on independent modes, could allow important [14] conclusions on observable quantities. As a final remark, we note that it would be interesting to extend the present approach with abrupt confinement-tuning, to cases with other particle-particle interaction. Say, for the contact interaction which seems to be realistic and externally tunable (via Feshbach resonances) in Bose systems of harmonically confined atoms [24,25]. As an outlook we turn to a really theoretical challenge. According to earlier insights on the black-hole aspect of matter [26], there one also has a large number of unobservable internal configurations which may reflect, via an entropy, the end of certain processes. Our quench-mediated changes, generated in a given closed entangled system, in its energy and an associated entropy, already suggest that one may identify realistic processes, and their modulating role, in that fascinating field as well. For instance, a process in which there is a suddenly captured cloud of matter which changes the internal energy of a black-hole. In our modeling this would correspond to situation with total (t)energy E(t)(ω0,,N1+N2)≡[E1(ω0,,N1)+E2(ω0,,N2)]≡ F(t)+T(t)S(t) N. A future analysis of this situation, along the second law of thermodynamics, is desirable. Indeed, the "thermalization process" from subsystem’s T1and T2to a common Tis quite challenging. Acknowledgements One of us (I.N.) is indebted to the deceased Professor János Pipek for his earlier advices in many details of underlying mathematics behind this study. This work was supported in part by Grant PID2019-105488GB-I00 funded by MCIN/AEI/10.13039/501100011033. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give 123 Journal of Mathematical Chemistry (2023) 61:288–295 295 appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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