Nonexponential tunneling decay of a single ultracold atom
NNonexponential tunneling decay of a single ultracold atom
Gast´on Garc´ıa-Calder´on ∗ and Roberto Romo † Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20 364, 01000 M´exico, Distrito Federal, Mexico Facultad de Ciencias, Universidad Aut´onoma de Baja California,Apartado Postal 1880, 22800 Ensenada, Baja California, Mexico (Dated: September 11, 2018)By using an exact analytical approach to the time evolution of decay we investigate the tunnelingdecay of ultracold single atoms, to discuss the conditions for deviations of the exponential decay law.We find that R , given by the ratio of the energy of the decaying fragment E r to its correspondingwidth Γ r , is the relevant quantity in this study. When R is less than 0 . − Γ r t/ (cid:126) ) t − / . We also find thatfor values of R ∼
1, the nonexponential behavior occurs in a post-exponential regime that goes as t − after around a dozen of lifetimes. The above conditions depend on suitable designed potentialparameters and suggest that for values R (cid:46)
1, the experimental verification of nonexponential decaymight be possible.
PACS numbers: 67.85.-d,74.50.+r,03.75.Lm,03.65.Ca
I. INTRODUCTION
The recent experimental work on tunable few-fermionsystems consisting of ultracold gases in optical traps, thatis characterized by the control of the quantum state ofthe system [1–3], has opened the way to investigate avariety of aspects of few and many-body physics [4–6].As described in these works, this can be achieved by ex-ploiting Pauli’s principle in a highly degenerate Fermigas in a trap so that it is possible to control the numberof particles by controlling the number of available low-est energy single-particle states. For a few particles, theconfining trapping potential consists of a one-dimensionaloptical potential created by a tight focus of a laser beamand a magnetic field gradient in the axial direction, insuch a way that the states above a well defined energybecome unbound. The resulting potential is formed byan impenetrable barrier on the left, a barrier of finiteheight on the right and a well in between with a con-trolled number of atoms which may decay out of the trapby tunneling through the barrier. The above setup hasbeen used to address experimentally the tunneling decayof two or more atoms [1], and has motivated theoreticalstudies on the dynamics of multiparticle decay [5, 7–10],in addition to studies on the decay of two interacting ornon-interacting particles [2, 11–16]. In Ref. [1] it is alsopointed out the possibility to prepare just one atom inthe lowest energy level of an optical trap.In contrast to the widespread view that tunneling de-cay of an isolated single particle into open space is amplyunderstood [5], here we call the attention to the old pre-diction of the deviations of the exponential decay lawat short and long times compared with the lifetime of ∗ gaston@fisica.unam.mx † [email protected] the decaying system. Here, by using an exact analyticalapproach for decay [17–19], we investigate the conditionsthat need to be fulfilled to be able to observe nonexponen-tial decay of single ultracold atoms tunneling out of po-tential profile using realistic parameters. We believe thatthese systems are the closest realization for tunneling de-cay in a highly isolated environment and hence might beappropriate to test nonexponential decay at long times.This is a relevant issue from a fundamental point of viewthat requires experimental verification.We first provide an overview of the subject of decayof particles by tunneling. It is well known that quantumdecay, a subject as old as quantum mechanics, was devel-oped to explain α -decay in radioactive nuclei. In 1928,Gamow derived the analytical expression for the expo-nential decay law exp( − Γ t/ (cid:126) ), with Γ the decay rate;an expression that has been widely used in the descrip-tion of particle decay. For single-particle decay, and itseems also in multi-particle decay [8, 10], one may usu-ally identify, in addition to the exponential regime, twononexponential regimes that, respectively, occur in gen-eral at short and long times compared with the lifetimeof the system. The short-time behavior, which is relatedto the existence of the energy moments of the Hamilto-nian [20], exhibits typically a t behavior (see however[21]) and has been the subject of a great deal of atten-tion, particularly in connection with the quantum Zenoeffect [22, 23]. The long-time, post-exponential, regime isa consequence of the fact that in most real systems, theenergy spectra E is bounded by below, i.e., E ∈ (0 , ∞ ),leading to integer inverse power in time behaviors, as dis-cussed by Khalfin [24]. In that work, Khalfin indicatedthe relevant role of the ratio of the energy of the decay-ing fragment E r to the decaying width Γ r , R = E r / Γ r , indetermining the time scale for the transition from expo-nential to nonexponential decay. Subsequent theoreticalwork investigated further the issue of the approximatenature of the exponential decay law [25, 26] and pro- a r X i v : . [ qu a n t - ph ] F e b vided also estimates of the above time scale for values R (cid:29) Mn up to 65lifetimes. Here R ∼ and Winter’s estimate in life-time units: t = 5 ln( R ) for the onset of nonexponentialdecay, yields t ∼ R , as the proposal for observing nonexponentialdecay in isolated autoionizing states located very close tothe energy threshold in atomic systems [32], that so farhas not been confirmed experimentally.The above results contributed to the widespread viewthat nonexponential decay contributions were beyond ex-perimental reach or even to the alternative explanationthat the interaction of the decaying system with the en-vironment would enforce exponential decay at all times[28, 33]. Some years ago, however, short-time deviationsfrom exponential decay [34] and the quantum Zeno effect[35] were finally observed, and more recently, in 2006,the measurement of post-exponential decay in a numberof organic molecules in solution, exhibiting distinct in-verse power in time behaviors was reported [36]. Theabove long-time deviations from the exponential decaylaw, however, were obtained due to the additional broad-ening of the excited state energy distributions producedby the solvent. That is, instead of looking for states withvery small E , these authors considered systems with largevalues of Γ. It is intriguing that the solvent, which maybe regarded as some sort of environment for these com-plex molecules, favors a nonexponential behavior. Tothe best of our knowledge, there is not at present time atheoretical approach that explains the above experimen-tal results. It is worthwhile to point out that these ex-periments refer to luminescence decays after laser pulseexcitation and hence do not refer to particle decay asconsidered in the present work.In 2005 Jittoh et. al. published a theoretical study onparticle decay using a single-pole approximation, wherethese authors estimate that for values of R < . s -wave spherical symmetric systems or one-dimensionalsystems, there exists a novel regime where the decay isnonexponential at all times [37]. These authors, how-ever, did not discuss that regime in actual physical sys-tems. In 2006, Garc´ıa-Calder´on and Villavicencio [38]suggested the possibility that this novel full time non-exponential regime could be observed in semiconductordouble-barrier resonant quantum structures.As pointed out at the beginning of this Introduction,in this work we investigate the conditions that need tobe fulfilled to be able to observe nonexponential decayin the deterministic preparation of a tunable single atomin an optical trap [1]. As pointed out above, we believethat these systems are the closest realization of decay by tunneling of a particle out of a single particle potential.We derive analytical expressions for the nonescape prob-ability as an expansion involving the full set of decayingstates of the system at all times and study the conditionsof validity for the single-pole approximation.It is worth to point out that the formulation consideredhere refers to the full hamiltonian H to the problem andhence it differs from approaches where the Hamiltonianis separated into a part H corresponding to a closedsystem and a part H responsible for the decay whichis usually treated to some sort of perturbation theory,as in the work by Weisskopf and Wigner to describe thedecay (also exponential) of an excited atom interactingwith a quantized radiation field [39]. These approximateapproaches have become a standard procedure for treat-ing a class of decay problems where perturbation theorycan be justified, as in studies of nonexponential decay inatomic spontaneous emission [40, 41].The manuscript is organized as follows. Section II pro-vides an overview of the theoretical formalism that weconsider here. Section III discusses some model calcula-tions and analyzes different post-exponential scenariosand finally, Section IV presents some Concluding Re-marks. II. FORMALISM
The formalism that we shall consider here has its rootsin the old work by Gamow which imposed outgoingboundary conditions on the solutions to the Schr¨odingerequation to describe the process of decay [42, 43]. Asis well known, these boundary conditions lead to com-plex energy eigenvalues, its imaginary part being twicethe decay rate that appears in the expression of the ex-ponential decay law. Outside the interaction region, theamplitude of such solutions, known as decaying, resonantor quasi-normal states, grows exponentially with distanceand hence the usual rules of normalization and complete-ness do not apply. The approach initiated by Gamow,however, evolved over the years. In particular, significantdevelopments in the 1970s on the analytical properties ofthe outgoing Green’s function to the problem provided asuitable framework to study distinct approaches to theissues of normalization and eigenfunction expansions in-volving these states [44, 45]. In particular some of thesedevelopments have led to an exact analytical descriptionof decay by tunneling [17, 18].The effective trap potential that results after applica-tion of the magnetic field to the initially confining trapand of the spilling process that guarantees that the de-caying atom remains in the lowest decaying state, corre-sponds to a one-dimensional system where the transmis-sion channel is closed [1, 14]. This potential is analogousto a spherical potential of zero angular momentum.The solution to the time-dependent Schr¨odinger equa-tion as an initial value problem, may be written at time t in terms of the retarded Green’s function g ( x, x (cid:48) ; t ) ofthe problem as [17, 18]Ψ( x, t ) = (cid:90) L g ( x, x (cid:48) , t )Ψ( x (cid:48) ,
0) d x (cid:48) , (1)where Ψ( x,
0) stands for a state initially confined withinthe internal interaction region (0 , L ). Here, for simplic-ity of the discussion and without loss of generality it isassumed that ψ ( x,
0) is a real function. A convenientform of the retarded time-dependent Green’s functionis expressed in terms of the outgoing Green’s function G + ( x, x (cid:48) ; k ) of the problem. Both quantities are relatedby a Laplace transformation [17, 18]. In the present ap-proach, instead of the common practice of assuming theanalytical properties of G + ( x, x (cid:48) ; k ), we impose the con-dition, justified on physical grounds, that the potentialvanishes after a distance, i.e. V ( x ) = 0 , x > L . As a con-sequence, it can be rigorously proved that G + ( x, x (cid:48) ; k )may be extended analytically to the whole complex k plane where it has an infinite number of poles distributedin a well known manner [46].The relevant point here is that the residue of G + ( x, x (cid:48) ; k ) at a pole κ n is proportional to the functions u n ( x ) and u n ( x (cid:48) ) and provides its normalization condi-tion [17, 18, 45]. The decaying or resonant states u n ( x )satisfy the Schr¨odinger equation of the problem[ E n − H ] u n ( x ) = 0 , (2)where H is the full Hamiltonian H = − ( (cid:126) / m ) d /dx + V ( x ), with m the mass of the decaying particle. Equa-tion (1) satisfies outgoing boundary conditions at x = L ,namely, u n (0) = 0 , (cid:20) du n ( x ) dx (cid:21) x = L = iκ n u n ( L ) , (3)with κ n = α n − iβ n . The quantity E n in (2) refersto the complex energy eigenvalue E n = ( (cid:126) / m ) κ n = E n − i Γ n /
2, where E n yields the resonance energy of thedecaying fragment and Γ n stands for the correspondingdecaying width. Using Cauchy’s Integral Theorem allowsto obtain a discrete expansion of G + ( x, x (cid:48) ; k ) in termsof the functions { u n ( x ) } and the poles { κ n } along theinternal potential region. This expansion may be usedto obtain a representation of g ( x, x (cid:48) , t ) that may be in-serted into Eq. (1) to obtain the time-dependent solution[17, 18]Ψ( x, t ) = ∞ (cid:88) n = −∞ (cid:40) C n u n ( x ) M ( y ◦ n ) , x ≤ LC n u n ( L ) M ( y n ) , x ≥ L, (4)where the coefficients C n are given by C n = (cid:90) L Ψ( x, u n ( x ) dx, (5)and the functions M ( y n ) are defined as [17] M ( y n ) = i π (cid:90) ∞−∞ e ik ( x − L ) e − i (cid:126) k t/ m k − κ n dk = 12 e ( imr / (cid:126) t ) w ( iy n ) , (6)where y n = e − iπ/ ( m/ (cid:126) t ) / [( x − L ) − ( (cid:126) κ n /m ) t ], andthe function w ( z ) = exp( − z )erfc( − iz) stands for theFaddeyeva or complex error function [47] for which thereexist efficient computational tools [48]. The argument y ◦ n of the functions M ( y n ) in (4) is that of y n given abovewith x = L .Notice that the sums in (4) run, respectively, over thepoles κ − n = − α n − iβ n , located on the third quadrant ofthe k plane, and the poles κ n = α n − iβ n , located on thefourth quadrant. It follows from time reversal invariancethat κ − n = − κ ∗ n [46].The functions { u n ( x ) } are normalized according to thecondition (cid:90) L u n ( x ) dx + i u n ( L )2 κ n = 1 , (7)and satisfy a closure relationship along the internal re-gion of the potential which, provided the initial state isnormalized to unity, leads to the expression [17, 18],Re (cid:40) ∞ (cid:88) n =1 C n (cid:41) = 1 , (8)Equation (8) indicates that the terms Re { C n } cannotbe interpreted as a probability, since in general they arenot positive definite quantities, however, each of themrepresents the ‘strength’ or ‘weight’ of the initial statein the corresponding decaying state. One might see thecoefficients Re { C n } as some sort of quasi-probabilities[49].The equivalence between the non-Hermitian formula-tion that leads to the time-dependent solution given byEq. (4) and the Hermitian formulation based on contin-uum wave functions is discussed in Ref. [19]. There, theadvantage of using the analytical expressions for the dis-tinct decaying regimes that follow from the former formu-lation is contrasted with the ‘black box’ numerical treat-ment that characterizes the latter formulation.It is worth mentioning that the formalism outlinedabove differs from the so called rigged Hilbert space for-mulation in many respects, as discussed in Refs. [17, 50].For example, since in that approach the poles located onthe third quadrant of the k plane are not taken explicitlyinto consideration, there is no analytical description ofthe nonexponential contributions to decay, as given bythe Eq. (14) discussed below. It might also be worth-while to mention here that decaying states, in spite of itsnon-Hermitian nature, have been used in a large varietyof topics yet with different names: resonant states, quasi-normal modes or Siegert states, as for example in quan-tum transients [51, 52], gravitational waves and blackholes [53] and nonadiabatic processes involving molecules[54]. Nonescape probability
Two quantities of interest in decaying problems are thesurvival probability S ( t ), that yields the probability thatat time t the system remains in the initial state, and thenonescape probability P ( t ), that provides the probabilitythat at time t the particle remains within the confiningregion of the potential. When the initial state overlapsstrongly with the lowest decaying state, both quantitiesexhibit a very similar behavior with time [55]. It seemsthat such is the case for atom decay in ultracold traps,where the spilling process leaves just one atom in thelowest decaying state [1].Here we consider the nonescape probability which isdefined as P ( t ) = (cid:90) L Ψ ∗ ( x, t )Ψ( x, t ) dx. (9)In order to calculate the above quantity one requires thetime-dependent solution along the internal region of thepotential. Hence one may insert the top expression ofEq. (4) into Eq. (9), to obtain the expansion of thenonescape probability in terms of decaying states, P ( t ) = ∞ (cid:88) m,n = −∞ C m C ∗ n I mn M ( y m ) M ∗ ( y n ) . (10)where I mn = (cid:90) L u m ( x ) u ∗ n ( x ) dx. (11)Equations (4) and (10) are given in terms of M func-tions, and consequently their exponential and nonexpo-nential behavior is not exhibited explicitly. This may beobtained by using the symmetry relations κ − n = − κ ∗ n and u − n ( x ) = u ∗ n ( x ), to write the sums over the poleslocated on the fourth quadrant. Here one may use therelation M ( y ◦ n ) = exp( − i E n t/ (cid:126) ) exp( − Γ n t/ (cid:126) ) − M ( − y ◦ n )[17, 47], to write the time-dependent wave function alongthe internal region as,Ψ( x, t ) = ∞ (cid:88) n =1 C n u n ( x )e − i E n t (cid:126) e − Γ n t/ (cid:126) − I n ( x, t ) , x ≤ L, (12)where the nonexponential contribution I n ( x, t ) is givenby I n ( x, t ) = ∞ (cid:88) n =1 C n u n ( x ) M ( − y ◦ n ) − C ∗ n u ∗ n ( x ) M ( y ◦− n ) . (13)In this last expression the argument y ◦− n is equal to y ◦ n with κ n substituted by κ − n = − κ ∗ n .Substitution of Eqs. (12) and (13) into Eq. (9) pro-vides, therefore, an expression for the nonescape proba-bility that exhibits explicitly the exponential and nonex-ponential contributions to decay. Notice, that assuming an initial state that overlaps strongly with the longestlifetime state, say n = r , it may be seen in view of (8),that Re { C r } ≈
1, and also that I rr ≈
1, and ignoringthe nonexponential contributions, one obtains the wellknown exponential decay law P ( t ) = exp( − Γ t/ (cid:126) ).Equations (4) and (10) are exact and may be used tofind out the validity of different approximations.The functions M ( − y ◦ n ) and M ( y ◦− n ) that appear in thenonexponential contribution given by Eq. (13), exhibitat long times a t − / behavior with time [17]. As a con-sequence, the time-dependent solution may be writtenalong the exponential and long-time regimes as [17],Ψ( x, t ) ≈ ∞ (cid:88) n =1 C n u n ( x )e − i E n t/ (cid:126) e − Γ n t/ (cid:126) − ib Im (cid:40) ∞ (cid:88) n =1 C n u n ( x ) κ n (cid:41) t / ; x ≤ L, (14)with b = e − iπ/ √ π (cid:18) m (cid:126) (cid:19) / . (15) Decay of a single level
In what follows we restrict the discussion to the sit-uation that corresponds to an atom located in the low-est decaying state, n = 1, of the effective ultracold trappotential. On physical grounds one expects that the ini-tial state overlaps strongly with that state, and thereforeRe { C } may provide the main contribution to Eq. (8).This justifies to consider the single pole approximation, n = 1, in the expansion of the decaying wave functiongiven by Eq. (14). This is a good approximation exceptnear the time origin where more poles are needed [21].Hence we may write,Ψ( x, t ) ≈ C u ( x )e − i E t/ (cid:126) e − Γ t/ (cid:126) − ib Im (cid:26) C u ( x ) κ (cid:27) t / ; x ≤ L. (16)Inserting Eq. (16) into Eq. (9) allows to write thenonescape probability as P ( t ) ≈ P e ( t ) + P e,ne ( t ) + P ne ( t ) , (17)where P e ( t ) stands for the purely exponential decay con-tribution, P e ( t ) = | C | I e − Γ t/ (cid:126) , (18) P e,ne ( t ) refers to the interference term that involves ex-ponential and nonexponential contributions, P e,ne ( t ) = − Re (cid:26)(cid:20) | C | I κ ∗ − C Y κ (cid:21) b ∗ e − i E t/ (cid:126) (cid:27) × e − Γ t/ (cid:126) t / (19)and P ne ( t ) stands for the long-time post-exponential con-tribution, P ne ( t ) = | b | (cid:26)(cid:20) | C | I | κ | − C Y ( κ ) (cid:21)(cid:27) t . (20)In the above expressions I y Y are defined, respectively,as I = (cid:90) L | u ( x ) | dx. (21)and Y = (cid:90) L u ( x ) dx. (22)We end this Section by referring to an exact single-level resonance expression for the survival probability[56], that allows to derive an approximate expression forthe time t , in lifetime units, for the transition from ex-ponential to post-exponential decay, t ≈ .
41 ln( R ) + 12 . , (23)where we recall that R = E / Γ . Equation (23) yieldsa good estimate of t for values of R (cid:38) III. MODELS
As pointed out in the Introduction, a relevant aspectof the recent developments on the preparation of tunablefew-body quantum systems is the control of the quan-tum states in these systems [1–3]. For tunneling decay,the resulting potential corresponds to an impenetrablebarrier on the left, a barrier of finite height on the rightand a well in between that, in particular, it may be usedto study tunneling decay of a single atom located in thelowest energy level of the system [1].Here we consider two potential profiles to study thedecaying regimes corresponding to different values of R for single Li atom decay. The first potential profile isthe bathtub potential, which has been considered in Ref.[10] to study multiparticle tunneling decay, and the sec-ond potential profile refers to the tunneling decay po-tential considered by the Heidelberg group on tunablefew-fermion systems, which consists of the summation ofthe optical one-dimensional confinement potential plus alinear magnetic term [1].
Bathtub potential
Let us first refer to the bathtub potential. Figure 1 (a)exhibits a profile for this potential (blue solid line), givenby the formula V ( x ) = 12 V [tanh X − tanh X ] Θ (cid:16) w − x (cid:17) + FIG. 1. (Color online) (a) Trapping potential V ( x ) (bluesolid line) obtained by smoothing the squared barriers (greendashed line) using Eq. (24). (b) Two potential profiles ap-propriate for the analysis of the transition from exponentialto post exponential decay. See parameters in the Table I.FIG. 2. (Color online) Example of the trajectories followed bya few poles located on the complex energy plane as the tun-neling barrier is gradually reduced. We consider neV units,where 1neV = 241 . R = 13 . R = 0 . FIG. 3. (Color online) Plot of the natural logarithm ofthe nonescape probability as a function of time (in units ofthe lifetime τ ) for different values of the ratio R = E / Γ .The transition from the exponential to the post-exponentialregime is clearly appreciated in each curve. V [tanh X − tanh X ] Θ (cid:16) x − w (cid:17) , (24)where Θ( u ) is the Heaviside step function, X = (cid:0) | x − w | − w (cid:1) /σ , X = ( x − L ) /σ , and L = w + b be-ing the total width of the potential depicted in Fig. 1(a)(green dashed line). The parameter σ determines thesmoothness of the potential. As required for single atomdecay, the potential supports only one decaying state( n = 1), with the higher states ( n ≥
2) lying above thetunneling barrier height, i.e. , E n > V for n = 2 , , , ... .The above may be accomplished by choosing appropri-ately the parameters of the potential shown in Fig. 1(a)and the value of σ . Here we fix for all calculations alarge value for V , namely, V /h = 241 . V . Figure 1(b) displaystwo potential profiles which may be appropriate for theanalysis of the distinct decaying regimes.Using an appropriate combination of the system pa-rameters may allow to select the transition time atwhich the decay changes from exponential to the post-exponential regime. According to Eq. (23), this tran-sition time is tunable through variations of the ratio R = E / Γ , which may be manipulated by realizing thatthe location of the poles on the complex wave number orenergy planes depends on the system parameters. Theequation for the complex poles follows by imposing theoutgoing boundary condition to Eq. (2). The polesmay be easily calculated for potentials with rectangularshapes by well known procedures [51, 57] that may beimplemented to potentials of arbitrary shape by noticingthat a given potential profile can be described as a se-quence of rectangles of appropriate high and width. Asan example, Fig. 2 shows that the main effect of dimin- TABLE I. Potential parameters for the distinct values of R shown in Fig. 3. V e /h (kHz) is the effective potential heightthat results from the right hand side of Eq. (24), where V /h (kHz) is the potential height, w ( µm ) is the well width, σ the smoothness of the potential, b ( µm ) is the barrier width, R = E / Γ , τ ( ms ) the lifetime, and Re { C } the expan-sion the coefficient for n = 1. The mass of Li is taken as10 , . m e , with m e the electron mass. See text. V e /h V /h w σ b R τ Re { C } ishing the barrier width is to increase the values of theimaginary energies of the poles. One appreciates in Fig.2 the trajectories followed by some poles on the fourthquadrant of the energy plane (dashed line).We model the initial state Ψ( x,
0) as the lowest energystate of a potential with infinite walls, namely,Ψ( x,
0) = (cid:18) w (cid:19) / sin (cid:104) πw x (cid:105) . (25)We choose the initial state (25), in addition to its math-ematical simplicity, because it has the essential physicalingredient that initially there must a large probabilityto find the particle within the interaction region. Theabove initial state guarantees that Re { C } is the largestcontribution to Eq. (8) that is the condition that jus-tifies the single pole approximation in the expansion ofthe decaying wave function, as discussed in the previousSection.Using the initial state given by Eq. (25) and the setof poles { κ n } and decaying states { u n ( r ) } correspondingto a given set of potential parameters, allows to calcu-late the nonescape probability given by Eq. (10). Figure3 exhibits a plot of the time evolution of the nonescapeprobability as a function of time for different values of R in units of the corresponding lifetime. One may clearlyappreciate the transitions from exponential to nonexpo-nential behavior for distinct values of R . The potentialparameters that correspond to the above systems for dis-tinct values of R are given in the Table I, which in addi-tion, displays the corresponding values of the lifetimes τ and the expansion coefficients Re { C } . Notice that thebarrier width b is a relevant parameter to diminish thevalues of R .It is worth emphasizing a number of features exhibitedby Fig. 3. A first one is that the transition time is re-duced as the values of R decrease; a second one, is thatthe frequency of oscillations that arise from the interfer-ence between the purely exponential and the long-time FIG. 4. (Color online) Comparison of ln P ( t ) vs t/τ with R = 0 .
30 calculated from the formal solution, Eq (10) with N = 10 resonant terms (solid line) and the values the sum P e,ne + P pne (long dashed line) calculated from Eqs. (19)and (20) respectively. The purely exponential contribution(dashed line) is included to help the eye. inverse power contributions in (14) also diminishes as R decreases; the third one is that for values of R (cid:46)
1, thenonescape probability starts to exhibit a departure frompurely exponential decay before the transition to the t − long-time behavior occurs, and finally, as R diminishesfurther, as exemplified by R = 0 .
30, the decay becomesnonexponential at all times.Previous studies involving distinct systems have shownthat the single-pole approximation for the nonescapeprobability, given by Eq. (17), is an excellent approx-imation for
R > R much larger than unity the exponential decay lawholds for many lifetimes and hence the long-time nonex-ponential contribution is very small.On the other hand, it follows also from inspection ofFig. 3, that for values of R ∼
1, as R = 0 .
96, or evenwith R = 0 .
57, the nonescape probability exhibits a cleardeparture from exponential decay just after a few life-times to then follow a nonexponential behavior as t − ,according to Eq. (20). This suggests that in systemsaround these values of R nonexponential decay could beamenable to experimental verification.We have found that for values of R <
1, the single-poleapproximation is still a good approximation. As pointedout above, the case R = 0 .
30 is particularly interestingbecause it exhibits nonexponential decay in the full timeinterval. Figure 4 provides a comparison of a calculationof the nonescape probability (solid line) using Eq. (10),where 10 poles are sufficient to get convergence of theexpansion, with the single-pole approximate calculation P e,ne + P ne (long-dash), using, respectively, Eqs. (19)and (20). One sees that the interference term P e,ne , thatgoes as exp( − Γ t/ (cid:126) ) t − / , describes the decay for the FIG. 5. Color online) Potential profiles V ( x ), given by Eq.(26), for three distinct values of the parameter B (cid:48) , as indicatedin each curve. Each curve is shifted so that V /h = 0 at thetrap bottom. The dashed lines indicate the energies of thelowest decaying levels of each potential. The level with thelargest energy corresponds to the potential with the highestbarrier, and thus successively in descending order. See text.FIG. 6. Color online) Plot of the natural logarithm of thenonescape probability P ( t ) as a function of time in lifetimeunits τ for three different values of the R = E / Γ , as indi-cated in each curve. See text. first few lifetimes whereas the last term P ne , that goesas t − , becomes the dominant contribution from approx-imately fourteen lifetimes onwards. The above suggeststhat systems with R (cid:46) . Heidelberg potential
Let us now refer to the second potential profile. Thisconsists of a cigar-shaped cylindrically symmetric opticalpotential created by the sum of two terms, the first onea tightly focused laser beam which accounts by for anoptical one-dimensional confinement of atoms and thesecond one, a linear magnetic potential term, that allowsfor tunneling decay. The potential is given by the formula[1, 3, 14], V ( x ) = pV (cid:20) −
11 + ( x/x R ) (cid:21) − µ m B (cid:48) x, (26)where V = (3 . µK ) k B is the initial depth at thecenter of the optical dipole trap, with k B the Boltz-mann constant; p = 0 . x R = πω /λ stands forthe Rayleigh range, with λ = 1064 nm , the wavelengthof the the trapping light; µ m is the Bohr magneton;and B (cid:48) = 18 . G/cm , is the magnetic field gradient[1, 3, 14]. The above parameters determine a value for R . The analysis by Rontani on single atom decay [14],which is based on the experiments reported in Ref. [3],gives a value of R ≈
70, where E = (316 . h andΓ W KB = ( γ s / π ) = (4 .
516 Hz) h , with h the Planckconstant and γ s = (1 / . ms [14]. The above valueof R ≈
70 yields using (23), an onset for nonexponentialdecay around t ≈
35 lifetimes, which lies well beyondthe range of 6 lifetimes that were considered in these ex-periments. That analysis involves a trap parametriza-tion involving a WKB analysis. In recent theoreticalwork, however, it is argued that the trap calibration viaa WKB analysis leads to an inaccurate trap parametriza-tion [16]. Our own analysis, involving the complex polefor the above potential parameters, gives R = 27 .
36 (seebelow) which gives an onset for nonexponential decayaround t ≈
30 lifetimes, that still lies beyond experi-mental verification.It turns out that varying slightly the value of the mag-netic field gradient B (cid:48) modifies the potential profile. Thisis exemplified in Fig. 5 which exhibits three poten-tial profiles corresponding respectively, to B (cid:48) = 18 . G/cm , the case considered above, B (cid:48) = 19 . G/cm and B (cid:48) = 19 . G/cm , as indicated for each curve in thatfigure. Notice that we have shifted the origin of energyto the bottom of the potential for each case. The lowestenergy decaying levels for each potential profile are indi-cated by dashed lines in Fig. 5. The first level from abovecorresponds to the potential profile with B (cid:48) = 18 .
92, andthus successively in descending order, as shown in TableII.Figure 6 displays the natural logarithm of thenonescape probability as a function of time in lifetimeunits for the above potential profiles corresponding, asindicated in the figure, to values of R = 1 .
09, 2 .
85 and27 .
36. In these calculations the box model initial stategiven by (25) is chosen to yield, as expected on physi-cal grounds, values of Re { C } around unity. The largestvalue of R , namely, R = 27 .
36, corresponds to the poten-tial profile with the smallest value of B (cid:48) in Fig. 5, that is, B (cid:48) = 18 .
92, and thus respectively for the other cases asindicated in Table II. One sees, therefore, that by varying
TABLE II. Values of the magnetic field gradient B (cid:48) ( G/cm ), w ( µm ), R = E / Γ , the energy of the decaying state E (kHz), the lifetime τ ( ms ), and the expansion coefficient for n = 1, Re { C } , corresponding to Figs. 5 and 6. See text. B (cid:48) w R E τ Re { C } slightly the values of the magnetic field gradient allowsfor the design of potential profiles with distinct values of R including values R <
1. Presumably one might alsovary some other parameters of the potential, as the op-tical trap depth p , in order to look for values of R whichmight be adequate for the experimental verification ofnonexponential decay. Table II groups also some otherrelevant parameters for the calculations shown in Figs. 5and 6.There is a feature that is worth pointing out here thatresults from our treatment concerning the potential givenby Eq. (26). It occurs for values of R (cid:46) . R = 1 .
09 appearing in Fig.6 and refers to the fact that the lowest energy decayinglevel has an energy that lies above the top of the corre-sponding potential barrier ( B (cid:48) = 19 . R = 0 .
96 in Fig. 3, follows from thefact that both cases have a large value of the coefficientRe { C } , as shown in Tables I and II. This follows fromthe notion that the initial state overlaps strongly withthe lowest decaying level. In these calculations the val-ues of w for the initial states are given in Table II andthe maxima of the corresponding probability densities arecentered at the maxima of the corresponding probabilitydecaying densities. Hence, it does not seem to matter ifthe lowest decaying level is located above or below thepotential barrier height. The higher energy decaying lev-els, in addition that decay much faster, have very smallvalues of the coefficients Re { C n } , with n = 2 , , ... , as fol-lows from Eq. (8), and hence do not play a relevant rolein the decay process except at very small times. Clearlythe above considerations lie beyond the WKB framework.It is worth stressing that the behavior with time ofthe nonescape probability for the potentials consideredhere is quite similar, as follows from a comparison be-tween Figs. 3 and 6. This suggests that what matters isthe value of R independently of the specific shape of thepotential profile. IV. CONCLUDING REMARKS
The approach discussed in this work provides a con-sistent analytical framework to discuss exponential andnonexponential contributions to quantum decay. Wehave exemplified the above for two model calculation forthe decay of ultracold atoms out of a trap having a bar-rier, with realistic parameters. We have pointed out therelevance of the ratio of the energy of the decaying frag-ment to the decaying width, R = E / Γ to determine thedecaying regime as a function of time, in particular valuesof R < R ∼ R . It is not crucial to know the precise analytical formof the potential. Essentially, the barrier height controlsthe number of decaying states within the well; the wellwidth controls the energy value of the decaying state, andthe barrier width, the value of decaying width. From anexperimental point of view in order to fix a value of R requires to acquire control over these parameters.We hope that the analysis presented here will stimulateexperimentalists interested in fundamental issues to lookfor the verification of the nonexponential contributionsto quantum decay of ultracold atoms in these systems. ACKNOWLEDGMENTS
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