# Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

###### Abstract

We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically-unstable domains.

###### pacs:

03.75.Lm, 05.45.Mt, 03.65.Sq## I Introduction

Small-size bosonic lattices have attracted considerable attention in the last decade since they allow the exploration of a rich variety of dynamical behaviors in which macroscopic nonlinear effects are triggered through a few controllable parameters Nemoto -HDC . Such systems, formed by small arrays of coupled condensates, are governed by the discrete nonlinear Schrödinger equations

(1) |

where is the boson interaction, represents the tunneling amplitude, and is the population of the condensate at well in a -well lattice. Adopted almost thirty years ago to model small molecular systems and study energy-localization effects ELS1 -KC1 , equations (1) still represent the basic theoretic model for studying the dynamics of solitons and of low-energy excitations in lattices FW -flop where bosons can experience both attractive () and repulsive () interactions.

The appealing feature of small-size lattices is that equations (1) involve a number of dynamical variables which is small but sufficient to make the system nonintegrable. Thus, while preserving a character simple enough to allow a systematic analytic approach, mesoscopic lattices display strong dynamical instabilities and a variety of behaviors (including chaos) typically occurring in longer chains. This circumstance has stimulated a considerable interest in revisiting nonlinear behaviors CFFCSS , WEHMC first studied at the classical level within the fully quantum environment of bosonic lattices Persist -Alon . Quantum aspects become relevant for lattices involving low numbers of bosons per well. In this case, a realistic description of microscopic processes is provided by the second-quantized Bose-Hubbard (BH) model hald80 -jbcgz98

(2) |

where operators , obey commutators and are number operators. Equations (1), the semiclassical counterpart of model (2), are easily recovered within both the coherent-state apPrl80 ; ejc and the continuous-variable bpvPra84 picture showing how operators , are replaced by complex variables , .

Theoretical work on mesoscopic lattices has been mainly focused on the three-well array (trimer) this being the simplest nonintegrable model of this class of systems. Many interesting aspects of semiclassical trimer have been explored such as its unstable regimes with both repulsive FPpre67 and attractive joha1 interaction , the emergence of chaos in the presence of parabolic confinement BFPprl90 , external fields Chong or off-site interactions VF2 , and discrete breather-phonon collisions HDC .

Almost in parallel, the impressive development of laser trapping techniques, has made concrete the possibility to engineer small-size arrays whose dynamics is accessible to experiments. The most prominent example is the two-well system (dimer) Anker ; Albiez obtained by superposing a (sinusoidal) optical potential on the parabolic potential trapping the condensate. The same scheme should enable the realization of linear chains with an arbitrary number of wells by adjusting the laser wavelength (determining the interwell distance) and the parabolic amplitude.

In this scenario, the study of trimer dynamics provides a privileged standpoint to better understand the quantum counterpart of nonlinearity and instabilities. This issue has been discussed in a series of papers examining the spectral properties of quantum trimer Persist -Kol , its description within the phase-variable Mossman and the Husimi-distribution TWK pictures, and the inclusion of higher-order quantum correlations within the multiconfigurational Hartree method Alon . Quantum trimer has been also used to model coherent transport with weak interaction SGS1 and thermalization effects within the Fokker-Planck theory vardi . More recently, the single-depleted-well regime of the trimer has been studied in JJK to evidence the quantum signature of oscillatory instabilities.

In the same spirit, in this paper, we study the quantum aspects of the single-depleted-well regime for an open trimer trapped in a parabolic potential. Based on previous work BFPprl90 , jpa42 , we aim, in particular, to obtain 1) a satisfactory quantum description of stable macroscopic oscillations characterizing this special regime, and to detect 2) significant effects that distinguish the approach to unstable regimes.

The presence of the parabolic trap results in an effective local potential that favors the occupation of the central well. Owing the absence of a closed geometry, the translation-invariant single-depleted-well solution of ring trimers reduces, in the present contest, to a stationary solution where the central well is depleted and the lateral condensates exhibit twin populations and coherent (opposite) phases. For this reason the relevant regime will be called central-depleted-well (CDW) regime. Classically, this is represented by trajectories whose initial conditions are close to the CDW fixed point.

The most part of fixed points of the trimer dynamics, which have the form of collective-mode stationary solutions, feature a strong dependence from interaction parameters. The change of the latter provides various mechanisms whereby one can control the coalescence (or the formation) of fixed-point pairs and thus the onset of dramatic macroscopic effects jpa42 . Different from the other fixed points, the CDW solution has the special feature to be always present it being independent from the model parameters. This property is advantageous at the experimental level in that the conditions for realizing configurations close to the CDW state appear to be weakly conditioned by the tuning of physical parameters. The semiclassical study of the CDW state has shown how its dynamics displays both stable and unstable regimes. In general, trajectories close to the CDW state exhibit oscillations both of the lateral macroscopically-occupied condensates and of the central condensate which typically involves small fractions of the whole population.

The interest for the CDW regime is motivated by its complex, manifold character which, in addition to a well-known, rich scenario of stable and unstable behaviors, features dynamical modes whose character is at the border between the classical and the quantum behavior. We will explore this regime showing that an almost exact description can be achieved.

If the small fraction of the central condensate indeed corresponds to a few-boson population, the semiclassical picture must be replaced by a purely quantum-mechanical approach. Apparently, the only variable that must be quantized is the order parameter of the central well while those pertaining to lateral wells (with many bosons) are expected to maintain their classical form. This possibility is only apparent. In this paper we develop a quantum picture based on replacing some of trimer space modes with collective, momentum-like modes. This alternative formulation of trimer dynamics and, in particular, of the CDW regime allows one to identify the dynamical variables that really feature a quantum behavior. The new description is particularly interesting in that the model we obtain can be diagonalized in an exact way by implementing the dynamical algebra method Zhang .

In Section II we review the semiclassical trimer model and its quantum counterpart, and propose our alternative description in terms of momentum-like modes. Section III is devoted to obtain the energy spectrum and the eigenstates for the two regimes characterizing the CDW dynamics. The diagonalization is made possible by recognizing that algebra sp(4) is the dynamical algebra of this model. In section IV we discuss the energy-spectrum properties for various limiting cases and highlight the link between the parameter-space regions where the spectrum has a discrete character and the regions corresponding to classically stable oscillations. The occurrence of a continuous spectrum is related to classical instability. Section V further illustrates this aspect by showing how various delocalization effects characterize the approach to classically unstable regions. Section VI is devoted to concluding remarks.

## Ii Trimer dynamics in the CDW regime

For a three-well array including parabolic confinement equations (1) take the form

(3) | |||||

(4) |

where on-site potentials are such that reflecting the form of the external potential. With no loss of generality one can assume and . The relevant semiclassical BH Hamiltonian

(5) |

where canonical variables obey the canonical Poisson brackets , exhibits a single constant of motion () representing the total boson number. Quantities and are interpreted, in fact, as the condensate order parameter and the boson population at the th well, respectively, defined by and , in the quantum-classical correspondence with the BH model jpa41 .

Among many interesting regimes, the CDW regime, formed by phase-space trajectories whose initial condition are close to the CDW solution

(6) |

indeed represents a special case owing to its evident independence from interaction parameters , and . The application of linear-stability analysis to the CDW solution jpa42 shows that the relevant dynamics features both stable and unstable subregimes depending on the value of and . In particular, numerical simulations jpa42 show that the stable regime displays a regular dynamics with periodic oscillations of the three populations possibly involving different time scales. For initial conditions close to the CDW configuration both the lateral and central populations exhibit small deviations from solution (6) thus confirming the fact that lateral condensates oscillate maintaining their macroscopic character whereas the central well remains almost empty. Similar to Hamiltonian (5), its quantum counterpart

(7) |

involves three space modes . Only mode , however, features a true quantum character, modes and being related to macroscopic boson populations for . This reasoning is only apparently correct. By introducing the new set of dynamical variables

(8) |

with nonzero canonical Poisson brackets , we obtain the trimer equations of motion in the alternative form

(9) | |||||

equipped with the constant of motion . The ensuing version of the CDW solution is

showing that the entire boson population is attributed to mode . In this scenario, trajectories representing small deviations from the previous CDW state involve two “microscopic” modes, namely and , whose populations are small with respect to the total boson number . The only macroscopic quantity is thus the population of mode . Equations (9) makes it evident that the rapid phase oscillations of mode can be easily removed from the trimer dynamics. By setting

(10) |

with and , , where , and describe small deviations from the CDW state, the motion equations reduce to the linear form

(11) |

if quadratic and cubic terms involving microscopic variables , and are neglected. The effective dynamics of the CDW regime is thus driven by the first two equations involving modes and , since must be imposed to ensure condition in the time evolution of trimer. Simple calculations show that the Hamiltonian corresponding to equations (11) reads

(12) |

can be derived as well from Hamiltonian (5) by substituting variables (10). The time-dependent canonical transformations (10) giving , and in terms of , and , also show that the canonical structure is preserved, namely that with .

### ii.1 The quantum model and its dynamical algebra

The discussion of appendix A shows how, similar to its classical counterpart , quantum Hamiltonian (7) can be reduced to the quadratic form

This is achieved by introducing quantum collective modes , and and implementing a suitable time-dependent unitary transformation able to eliminate the macroscopic dynamics of mode described, at the classical level, by factor in equations (10). The crucial point is that in the new scenario quantum modes , and play the same role of classical modes , , in equations (11). Operators , and satisfy the standard commutation relation , and . The interesting part of is

which should characterize the quantum dynamical behavior of CDW-like states. exhibits an unusual, composite form where, in addition to the standard coupling between bosonic modes, mode involves a pair of two-boson creation/destruction terms, and , typically causing squeezing effects in atom-photon interaction models of quantum optics. These are also responsible for considerably increasing the complexity of the dynamical algebra of .

One should recall that, given a Hamiltonian , its dynamical algebra is the set of operators forming an algebraic structure, with definite commutators , based on which can be expressed as a hermitian linear combination . In the absence of and the dynamical algebra of is the spin algebra su(2) formed by generators

with and , in the well-known two-boson Schwinger realization. In this case, the diagonalization of with , would reduce to perform a simple rotation SU(2) such that where is, by definition, the diagonal generator of su(2).

Owing to and the dynamical algebra is the more complex symplectic algebra sp(4), reviewed in appendix B, involving ten independent generators. To simplify the diagonalization of it is advantageous to rewrite it in terms of canonical operators , , and such that , defined by

(13) |

The final form of is

(14) |

where , and . Even if we have assumed , reflecting the form of the parabolic-potential profile, we shall consider as well the possibility to realize negative , representing the presence of a repulsive central potential. In the latter case the three local potentials mimic the mexican-hat profile. The critical value where significant changes are expected is , below which becomes positive.

## Iii Energy spectrum

Hamiltonian can be diagonalized by a many-step process where one exploits the knowledge of the transformations of group Sp(4) and, in particular, of the effects of their action on momentum and coordinate operators. Diagonalization is achieved by combining in a suitable way squeezing transformations , standard rotations and hyperbolic transformation . The latter belong to group Sp(4) they being generated, through the usual Lie-group exponential map, by generators , and , respectively, of sp(4) (see Appendix B).

### iii.1 Diagonalization of case

In this case can be diagonalized by a three-step process where the diagonal Hamiltonian can be shown to be given by with .

First step.

(15) |

The action of on Hamiltonian allows one to get the same coefficient (up to a factor ) for and

provided the condition defining is imposed.

Second step.

(16) |

where and . Transformation generates the new Hamiltonian introducing an undefined parameter whereby terms depending on can be suppressed. This condition is achieved by imposing which provides a complete definition of through . As a consequence, the new Hamiltonian takes the form

Third step.

(17) |

with and . This hyperbolic transformation has the property to leave and unchanged. By acting on the last two terms of one can exploit parameter to eliminate term in the final Hamiltonian. This is given by which reduces to a linear combination of , , and if condition

is satisfied. The latter condition gives the formula

(18) |

whereby and can be expressed in terms of the interaction parameters. The final Hamiltonian reads

where

In order to ensure that assumes real values, the latter formula requires that . This condition, combined with gives

(19) |

which identifies a well defined portion of the parameter space. Curve , denoted by , is represented by the red curve of Fig. 1. Hence, among the four domains visible in Fig. 1, corresponds to the upper region bounded by from below. For the present diagonalization scheme assigns a well-defined discrete spectrum to Hamiltonian . Note that, for , no upper limit constraints the range of . A second domain exhibiting a discrete spectrum can be found for which will be identified in the sequel.

The energy spectrum is easily worked out by observing that Hamiltonian can be rewritten in terms of two harmonic-oscillator Hamiltonians

with

(recall that ) and

(20) |

The crucial condition that can be easily verified. The eigenstates are defined as product states such that, for each oscillator,

and being standard harmonic-oscillator eigenfunctions. The resulting spectrum reads

(21) |

with , and the initial Hamiltonian satisfies the eigenvalue equation where, in view of the preceding diagonalzation process, one has

(22) |

### iii.2 Diagonalization of case

In this regime, involving a large central barrier, the diagonalization process can be performed by means of four subsequent steps. These correspond to the four unitary transformations forming whose action allows one to obtain the new diagonal Hamiltonian . The progressive action of such transformations is discussed in appendix C. One obtains

with , ,

(23) |

and

(24) |

Similar to the preceding case where , the eigenstates are product states formed by harmonic-oscillator eigenfunctions such that

The resulting spectrum reads

(25) |

where . The original Hamiltonian satisfies the eigenvalue equation where

(26) |

The range of validity of transformations and in terms of parameter defines the region of parameter space in which, for , the diagonalization process succeeds. The actions of is well defined if (this ensures that ), while , for a given , requires that , namely,

(27) |

(see appendix C). We will denote curve and the straight line with and , respectively. Thus domain corresponds to the region bounded by from below and by from above (see the caption of Fig. 1). This result confirms that the regions of plane where the current diagonalization scheme is effective correspond to the regions where classical oscillations are stable.

## Iv structure of the energy spectrum and classical instability

### iv.1 Spectrum of case

Domain described by inequalities (19) reproduces in Fig. 1 the first of the two regions of the stability diagram relevant to the CDW regime jpa42 in which classical trajectories with initial condition close to the CDW solution are dynamically stable. The property that CDW classical states are stable thus corresponds, quantum-mechanically, to the fact that the diagonalization process reduces the system to a simple double oscillator.

The dependence of energy spectrum (21) on parameters and allows one to identify two regimes characterized by the inequalities

respectively, in which the spectrum manifests significant changes. Region of Fig. 1 shows that, for a given value of in the interval , parameter ranges in , where defines boundary of . While the first regime corresponds to values of close to the vertical axis (interwell tunneling inhibited), the second regime, where , corresponds to approaching boundary from the left, namely, . Thus drives the approach to the region where the classical instability occurs.

Regime . In the weak-tunneling case, by exploiting the Taylor expansion of to the second order in , one obtains the expression

with

showing that quantum number describes the large-scale energy changes while quantum number describes the fine structure of the spectrum. The reference case, of course, is represented by , the energy of the quantum states corresponding to the minimum deviation from the pure CDW configuration.

The presence of the parabola, represented by , diminishes the fine-structure level separation with respect to the case involving no confinement. A more apparent effect occurs for where the attractive potential of the central well becomes a potential barrier. In this case the fine-structure level separation obtained with small is contrasted by factor . Decreasing by maintaining constant shows that such an effect is maximum when one approaches boundary from above.

Regime . The two oscillators tend to become identical () and the relevant energy levels are almost indistinguishable. This determines a macroscopic change of the energy spectrum of which becomes visible by effecting, with , the Taylor expansion of in terms of variable . The resulting spectrum has the form

with for . Eigenvalues feature a band structure described by and a fine structure controlled by parameter and the composite quantum number . The choice describes the band exhibiting the minimal deviation from the pure CDW configuration.

For the fine structure of each band becomes infinitely dense since the interlevel distance tends to zero. In this regime the behaviors of the two oscillators are strongly correlated in that small energy changes requires the simultaneous change of and in order to preserve . All the effects described so far disappear at (and, more in general, for ) since parameter no longer vanishes. Also, the range of becomes unlimited consistent with the fact that does not cross the unstable-regime boundary .

### iv.2 Spectrum of case

The present regime features as well two significant limiting cases occurring in the proximity of boundaries and . These are

(see definition (23)) that correspond to approaching 1) the curve from the right, at a given , and 2) the straight line from below, at a given , respectively. Hence, the approach to the regions where the classical instability crops up is driven by either or .

Regime . One easily checks that amounts to effecting the limit . By substituting in , and considering the Taylor expansion in , one obtains

with entailing that

Quantum number represents the band index, while the fine structure of the spectrum is controlled by number . For , similar to the case of subsection IV.1, the spectrum acquires an almost continuous character.

Regime . In this case, effecting the limit in equation (25) implies that

where the approximation to the first order in

has been used. Spectrum (25) thus undergoes a macroscopic change with energy levels forming a band structure described by index , and a -dependent fine structure described by index . The separation between two subsequent levels tends to zero for . Transitions in which , are thus favoured in that small energy changes are involved. Similar to the case of section IV.1, the two oscillators appear to be strongly correlated in that .

### iv.3 Transition to a continuous spectrum

Transformations and (see formulas (22) and (26), respectively) no longer work at the boundaries , and of the relevant stability domains. In particular, in the case , the action of in is defined for arbitrarily large values of as shown by equation (18). The latter shows that for meaning that is approached (but not reached) from below. Likewise, for , the range of parameters and (see appendix C) relevant to and , respectively, allows one to get closer and closer to boundaries and without reaching them.

At such boundaries the spectrum displays a structural change corresponding to the fact that the diagonalization of yields an operator pertaining to a sector of the dynamical algebra disjoint from the one where reduces to a simple two-oscillator model. In the new sector the discrete character of the spectrum is lost. A paradigmatic example of such an effect is supplied by the harmonic oscillator with time-dependent parameters AP96 .

Hamiltonian at the boundary where well exemplifies this situation. In this case, reduces to . The action of transformation on giving followed by the unitary transformation , takes into the form

which exhibits the two-dimensional Laplacian together with the generator of planar rotations. After noting that such operators commute with each other, one easily observes that, despite has a discrete spectrum, the energy spectrum features a continuous character due to the presence of Laplacian. Since the same result can be shown to occur at boundaries and , then we conclude that at the border separating (classically) stable from unstable regimes the energy spectrum acquires a continuous character.

The correct interpretation of such an effect is twice. At the boundaries , and of the approximation inherent in model (14), involving microscopic modes and , is no longer valid. This is obvious noting that the spectrum of quantum trimer is discrete by definition. The onset of a continuous spectrum is thus an artifact of the momentum-mode picture when this is used outside its domains of validity . On the other hand, the emergence of a continuous spectrum just amplifies the effect observed in the proximity of , and which consists in the vanishing of energy-level separation (see subsections IV.1 and IV.2). In this sense, the continuous spectrum is a dramatic manifestation of the transition to the classically-unstable regions.

## V Delocalization effects

The discussion of sections IV.1 and IV.2 shows that both regimes and feature limiting cases where a suitable choice of parameters and allows one to approach the boundaries of the domains classically involving stable oscillations. The significant change of the spectrum structure observed in such cases is accompanied by a dramatic change of the localization properties of the system. To see this one must consider deviations where represents one of canonical operators , and , and

with . The action of propagator on energy eigenstates allows one to considerably simplify this calculation. Formulas (22) and (26) show that and in which

In the two cases and , one has , , and , , respectively. As a consequence the calculation of expectation values reduces to

being in propagator . One easily shows that so that the determination of amounts to calculating

with . The derivation of such formulas is discussed in Appendix D.