Received 10 September 2008, in final form 11 September 2008Published 3 February 2009Online at stacks.iop.org/JPhysB/42/044018
Abstract
The decay and tunnelling dynamics of repulsive few-boson systems through aone-dimensional potential barrier is studied from first principles. To this end, we solve thenumerically exact time-dependent few-boson Schrodinger equation by utilizing the successfulmulticonfiguration time-dependent Hartree method. Benchmark results for a wide range ofinteractions are reported. Deviations from the time-dependent Gross–Pitaevskii approach areidentified. Counterintuitively, the mean-field approach can overestimate the tunnelling timeseven for relatively weakly-interacting few-boson systems. Implications are discussed.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Metastable states and resonances have been a fundamentalquantum-mechanical problem for many years, spanningdifferent research fields in physics, chemistry and biology.Perhaps the most popular and simple-to-visualize version ofthis problem is that of a particle decaying by tunnelling througha potential barrier. In the context of atomic physics and therapidly growing field of Bose–Einstein condensates (BECs)[1–4], the interest in resonances and decay by tunnelling ofBECs [5–12] and few-boson systems [13] through a potentialbarrier has attracted much attention. Here, a principal goal is todecipher how boson–boson interactions modify the resonancesand tunnelling dynamics of the single-particle problem.
Moiseyev et al [5] have studied the transition ofresonances to bound states of trapped attractive BECs in one,two and three dimensions as the attraction between the bosonsincreases. For this, the popular stationary Gross–Pitaevskii(GP) equation has been analytically continued to the complexplane. In [6], Witthaut et al studied analytically resonances(positions and widths) of the stationary GP equation of aBEC held in a one-dimensional (1D) ‘box’ with a zero-ranged Dirac-δ(x)-function barrier. Carr et al [7] havestudied tunnelling rates of repulsive and attractive trappedBECs in one, two and three dimensions as a function of the
interaction strength, for both ground and excited soliton andvortex states, using a combined variational-WKB (Wentzel–Kramers–Brillouin) treatment of the stationary GP equation.In [8], Moiseyev and Cederbaum derived the complex-scaledstationary GP equation from the complex-scaled many-bosonSchrodinger equation and used the former to compute positionsand widths (inverse half-lifetimes) of metastable repulsiveBECs as a function of interaction strength in a 1D trappotential. Furthermore, they have realized that resonancesof interacting trapped repulsive BECs are associated with thefragmentation phenomenon for which a theoretical frameworkextending beyond the GP equation is a must. Schlagheck andPaul [9] have devised a real-time propagation scheme for thecomplex-scaled GP equation to compute resonance positionsand widths of BECs in a 1D trap, and found ‘reasonablygood’ agreement between the results based on complex scalingand two additional methods: absorbing boundary conditionsand complex absorbing potentials. They have furthermoreintegrated the time-dependent GP (TDGP) equation (withabsorbing boundary conditions) to follow the decay of the BECout of the trap, finding an ‘extremely good’ agreement withan adiabatic approximation approach to the decay process.Sub-exponential decay has been found in the beginning ofthe tunnelling process [9]. Wimberger et al [10] computedtunnelling rates for the 1D nonlinear Wannier–Stark problem
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
(a BEC in a tilted periodic potential) and showed that even aweak repulsive interaction can have a pronounced effect ontunnelling in this system. Schlagheck and Wimberger [11]have found on the basis of the complex-scaled GP equationdeviations from mono-exponential decay of repulsive BECsin 1D double-barrier and Wannier–Stark problems. Finally,Huhtamaki et al [12] have derived an analytic Thomas–Fermi-based WKB-like expression for the tunnelling rate of a BECthrough a 1D barrier, based on the GP equation.
At the other end of the interaction scale, where GP theorydoes not apply at all, del Campo et al [13] have studied theexact decay by tunnelling of a few-boson Tonks–Girardeaugas held in a 1D ‘box’ with a zero-ranged Dirac-δ(x)-functionbarrier of different opacities. Analytically exact treatment ofstatics [14, 16] and dynamics [15, 16] of the Tonks–Girardeaugas is, of course, possible. It has been shown in [13] thatthe average number of bosons remaining in this trap exhibitsa nonexponential concave-up decay at the beginning of thetunnelling process. This behaviour ‘builds-up’ in the decayingfew-boson Tonks–Girardeau gas as the number of bosons isincreased, see figure 4 in [13].
What is the exact decay dynamics of interacting Bosesystems with finite interaction strengths? Despite obviousrelevance and being of fundamental interest, an exact study oftunnelling through a 1D potential barrier of interacting few-boson systems for a wide range of interaction strengths has,to the best of our knowledge, yet to be conducted, whichis the subject of the present work. There is, in general,no analytically exact solution to this more general problem,and we therefore would attack it numerically exactly. Thisis a demanding task. Correspondingly, this work deals withthe exact decay by tunnelling of at present only few-bosonsystems.
Few-boson systems and, in particular in the contextof the present study, exact many-body treatments of few-boson problems—analytically or numerically—have attractedthroughout the years enormous attention [13–35]. Theseinclude exact solutions and studies of ground-state properties[14, 16, 17, 19–21, 23–28, 30], excited-state properties[14, 16, 18–22, 29] and dynamics [13, 15, 16, 31–35] offew-boson systems. In this context, the present work aims atadding to this substantial wealth, by studying the numericallyexact dynamics of tunnelling through a barrier of interactingfew-boson systems.
To achieve this goal, we employ the multi-configurationtime-dependent Hartree (MCTDH) method [36–38]. TheMCTDH method is considered at present the mostefficient wave-packet propagation approach for in generaldistinguishable coupled degrees of freedom, and hassuccessfully and routinely been used for multidimensionaldynamical systems like molecular vibrations [39–43]. Ofcourse, the system under investigation by MCTDH canconsist of identical particles such as bosons. In the pasttwo years, MCTDH has been extremely successful andunique in obtaining fundamental and influential ground-state,excited-state and dynamical properties of weakly to stronglyinteracting trapped few-boson systems, on the numericallyexact many-body level [26, 27, 29, 31, 33, 34]. In particular,
Zollner et al have studied the exact dynamics of tunnellingof few-boson systems in a closed system—a 1D double-wellpotential [33, 34]. Covering the full crossover from weakinteractions to the fermionization limit, Zollner et al havediscovered that the tunnelling dynamics in the double wellevolves from Rabi oscillations in the weak interaction regimeto correlated pair tunnelling as the interaction strength isincreased.
The structure of the paper is as follows. In section 2 wepresent a theoretical background including a brief expositionof the MCTDH method and package and their usage for ourproblem. In section 3 we present the results and discuss them.Finally, in section 4, we present a summary and outlook.
2. Theoretical background
2.1. Physical framework
In this work we solve numerically the time-dependentSchrodinger equation (TDSE) in one spatial dimension:
i∂�
∂t= H�, (1)
where the many-body Hamiltonian is given by
H =N∑
j=1
h(xj ) +N∑
j<k
W (xj − xk). (2)
Here, h(x) = − 12
∂2
∂x2 + V (x) is the one-body Hamiltoniancontaining kinetic and potential V (x) terms. We work indimensionless units which are readily arrived at when dividingthe original Hamiltonian by the energy unit h2
mL2 , where m isthe mass of a boson and L is a convenient length scale, say thesize of the trap V (x) in which the bosons are initially prepared.The two-body interaction of the bosons is represented by thesecond term in the Hamiltonian (2). The interaction betweenultracold bosons in a quasi-1D setup can be modelled by aDirac-δ(x)-function potential, W (x − x ′) = λ0δ(x − x ′),where the strength of the interaction, λ0, is proportional tothe (tunable) s-wave-scattering length [44, 45], also see [3, 4].Transverse confinement is implicitly assumed.
2.2. The MCTDH method
In this section we briefly outline the theoretical frameworkof the calculations to be presented. The MCTDH method[36, 37, 40] and package [38] constitute a numericallyefficient algorithm for solving the TDSE (1), given someinitial conditions �(x1, . . . , xN ; t = 0) ≡ �(0) for themany-body wavefunction. The way the MCTDH ansatzis constructed, namely as a weighted expansion of Hartreeproducts φ1
j1(x1, t) · · · φN
jN(xN, t) of the so-called single-
particle functions (SPFs){φκ
jκ(x, t), κ = 1, . . . , N
},
�(x1, . . . , xN ; t) =n1∑
j1=1
. . .
nN∑jN=1
Aj1...jN(t)
N∏κ=1
φκjκ
(xκ, t), (3)
is responsible for the efficiency of the MCTDH algorithm.Since both the expansion coefficients
{Aj1···jN
(t)}
of thewavefunction and the SPFs themselves are time-dependent
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quantities, it is possible to reach numerically exact resultswith a comparatively small number {nκ, κ = 1, . . . , N} ofSPFs. By employing the Dirac–Frenkel variational principle,one arrives at coupled equations of motion for the expansioncoefficients
{Aj1...jN
(t)}
and SPFs{φκ
jκ(x, t), κ = 1, . . . , N
}.
By solving, i.e., integrating these equations of motion one getsa solution of the TDSE which is variationally optimal. Fora detailed discussion of the underlying theory and numericalimplementation of MCTDH the reader shall be referred to theliterature [36, 37, 40, 42].
The ansatz for the wavefunction (3) does not a priorihave the bosonic permutational symmetry, so it has to beproperly adjusted and symmetrized. Explicitly, the SPFsfor each degree of freedom are taken to be identical toone another, φκ
l (x, t) ≡ φl(x, t); the number of SPFs foreach degree of freedom is taken to be the same, nκ ≡ n;and the coefficients are forced to have the correct bosonicsymmetry, Aj1...jk ...jl ...jN
(t) = Aj1...jl ...jk ...jN(t). The fact
that the Hamiltonian (2) is symmetric to permutations ofany two particles and the use of the MCTDH equationsof motion ensure two relevant properties: (i) given initialconditions �(0) that are symmetric to permutations of any twoparticles, the time-dependent wavefunction �(t) preserves thebosonic permutational symmetry as well, and (ii) the ground-state of the many-body Hamiltonian (2) satisfies the bosonicpermutational symmetry.
Having the bosonic wavefunction �(t) at hand, one canquantify its properties. For our needs, the reduced one-bodydensity matrix is used,
ρ(x|x ′; t) =∫
dx2 · · · dxN�∗(x ′, x2, . . . , xN ; t)
× �(x, x2, . . . , xN ; t)
=∑
j
ρj (t){φNO
j (x ′, t)}∗
φNOj (x, t), (4)
where ρj (t) are referred to as natural occupation numbers andφNO
j (x, t) natural orbitals. The diagonal part of the reducedone-body density matrix ρ(x, t) ≡ ρ(x|x ′ = x; t) is simplyreferred to as the density. Throughout this work the density andsum of natural occupation numbers are normalized at t = 0to 1.
Finally, when one uses only one SPF, i.e., n = 1,the MCTDH ansatz for the wavefunction (3) boils down tothe TDGP which is identical to the Hartree ansatz for thewavefunction, φ1(x1, t) · · · φ1(xN, t). The TDGP theory is amean-field approximation to the TDSE (1) [3, 4]. We thereforenot only get a tool with MCTDH to solve the numerically exactTDSE, but also a quantitative measure where the widely (andsuccessfully) used TDGP approximation is suitable.
2.3. Numerical matters
2.3.1. Complex absorbing potentials. As the subject ofthis work is open or, in other words, decaying few-bosonsystems, this subsection is to briefly introduce the numericaltool implemented in the MCTDH package [38]—monomialcomplex absorbing potentials (CAPs)—to emulate the factthat at one end of the system there should be no ‘border’.
The underlying concept is to add a monomial complex part−iηQ(x) to the one-body Hamiltonian h(x),
which simply absorbs the wavefunction far away from the trap.Here, η is referred to as the strength of the CAP, xc is the CAPstarting point and p is the order of the CAP. θ(x) denotes theHeaviside step function. To represent an open system, one hasalso to take care of that the CAP added to the total HamiltonianH is practically reflection free. Therefore one has to make acareful trade-off between the length, i.e., the proper choice ofxc and the grid size, on the one hand, and the strength η andorder p of the CAP, on the other hand. If η is chosen to be toolarge, the CAP becomes more reflecting. If, on the other hand,η is too small, there would be no unwanted reflections but thesize of the grid required for perfect absorption (by the CAP)will become larger. For a detailed discussion on reflection andabsorption properties as well as of implementation of differentCAPs see, e.g., Refs. [46–50]. We will only briefly state herethat the strength η and p are optimized for the chosen energyof the system whereas the xc and the grid size have been fixedthroughout the work.
2.3.2. Convergence. Applying the MCTDH methodnumerically to solve a physical problem, one of course has touse a truncated Hilbert space in the representation of the many-body wavefunction (3). Thus, it has to be clearly defined whatthe terms numerically exact or converged calculations meanthroughout this work. Whenever these terms are in use, thefollowing points apply to the numerical simulations of theinitial conditions �(0) and the time evolution of the few-boson systems. (i) The calculation is converged with respectto the grid density—adding more primitive basis functions tothe discrete variable representation (DVR) does not changethe results. When using a grid density which is lower by25%, the energy as well as the occupation numbers of theinitial conditions differ by less than 10−5 respectively. (ii)The calculation is converged with respect to the CAP, i.e., ithas been checked that the CAP is essentially nonreflectingand its position xc does not interfere with the dynamics of thewavepacket. This is achieved already when using xc = 16.75with the same grid density. For convenience xc = 20 was used(see subsection 3.1). The length of the CAP was chosen to be30 in order to reach values which are at least as low as 10−10
for the sum of the squares of the reflection and transmissioncoefficients. (iii) The calculation is converged with respectto the number of SPFs used. Adding more orbitals does notchange the energy, density, and natural occupation numbers ofthe initial conditions by more than 10−5, 5 × 10−4 and 10−5,respectively. (iv) The usual checks for the convergence of thedynamics have been performed according to [40]. Specifically,the difference between the time-dependent autocorrelationfunctions with n and n − 1 SPFs is less than 10−5 − 10−4
for all times t. In practice, the lowest occupation number iskept below 10−5 − 5 × 10−4 for all times t.
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2.3.3. Interaction potential. Following [26, 27, 29, 31, 33,34] and without loss of generality, we emulate in this work theDirac-δ(x)-function zero-ranged inter-particle interaction bya narrow Gaussian-shaped function, as the short-ranged two-body interaction potential W entering the Hamiltonian (2),
δσ (xi − xj ) = 1
σ√
2πe− 1
2 (xi−xj
σ)2. (6)
By using the smoothed δσ (x) instead of the Dirac-δ(x)-function one makes a trade-off. The smaller σ is, the closeris the interaction potential to the true Dirac-δ(x)-functioninteraction (obviously: limσ→0 δσ (x) = δ(x)), but thesampling of δσ (x) by the DVR for a given grid density becomesless accurate. Checks of the dependence of quantities and theirconvergence with the width σ have been made. Throughoutthis work σ = 0.05 is used (as in [26, 27, 29, 31, 33, 34]),because it turned out to be suitable by means of keeping anefficient (not too high) grid density while reproducing well thephysical properties of the static interacting two-boson systemin the harmonic trap for all interaction strengths. See the insetof figure 5 and compare to figure 2 in [20].
3. Decay and tunnelling: setup, results anddiscussion
3.1. Setup
The one-body potential used in this work to investigatethe tunnelling process of the few-boson systems through apotential barrier reads
V (x) ={
12x2 t < 0
θ(2 − x) 12x2 + θ(x − 2)A e−B(x−C)2
t � 0,(7)
where the parameters A = 2.2663, B = 2 and C = 2.25have been chosen such that the connection at x = 2 is smooth(zeroth, first and second derivatives are equal). For a plot ofthe potential V (x) see figure 1.
The wavefunctions used as initial conditions �(0) toinvestigate the tunnelling dynamics are the numerically exactground states of the interacting few-boson systems in theparabolic trap 1
2x2, obtained by improved relaxation withinMCTDH [42]. The occupation numbers of the initialconditions of the two-boson systems are depicted in figure 2,covering a whole range of weak to strong interactions.
To reach convergence we employed as much as n = 9SPFs in the strongly interacting, fermionized limit for thetwo-boson dynamics and up to n = 10 SPFs for the dynamicsof the four-boson systems. The grid was represented by512 sine DVR functions in a box of size 60, x ∈ [−10, 50].The CAP starts at xc = 20 and thus half the box sizeis employed to absorb the wavefunction tunnelling throughthe barrier, see subsection 2.3. The other CAP parameterswere chosen appropriately for the different energies of thepropagated wavefunctions. The CAP strengths η were of theorder of 10−5, and the CAP order was p = 2 to p = 4.The parameters were optimally chosen for the individualpropagation runs according to Riss and Meyer [47].
0
0.5
1
1.5
2
2.5
3
3.5
4
-5 0 5 10 15 20 25
V(x
)
x
xc
CAP
.Figure 1. The trap potential V (x) (solid black curve), see equation(7), from which the interacting bosons leak out by tunnelling. Thebosons are initially prepared as the ground state of an harmonicpotential (dotted curve). As an illustrative example, the density ρ(x)of the ground state of two interacting bosons with interactionstrength λ0 = 0.5 is shown (solid blue line; scaled by a factor of 4).The complex absorbing potential (CAP) starts at xc = 20. Thequantities shown are dimensionless.
0.001
0.01
0.1
1
0 10 20 30 40 50 60
ρ j
λ0
j=1
j=2
j=3
j=4
j=5
j=6
Figure 2. The six largest natural occupation numbers ρj for theground state of N = 2 bosons in an harmonic trap as a function ofthe interaction strength λ0. Note the logarithmic scale. Theseoccupation numbers characterize the initial conditions �(0) used forthe tunnelling of the two-boson systems. As λ0 increases, theoccupation numbers saturate, signifying that the system is fullyfermionized. The quantities shown are dimensionless.
3.2. Tunnelling dynamics of two-boson systems
In this subsection we investigate the exact decay dynamicsof two-boson systems tunnelling through the potential barrier(7) with the MCTDH approach. It is instructive to comparethe exact results with those obtained by the TDGP method.Obviously, the norm of the wavefunction �(t) within the gridborders [−10, 50] is not conserved and decays in time due tothe absorption of outgoing waves tunnelling out of the potentialV (x) by the CAP. To investigate how it decays we first look atthe time evolution of the density ρ(x, t).
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
0
0.1
0.2
0.3
0.4
0.5
-10 -5 0 5 10 15 20
ρ(x
)
x
λ0=0.5
t= 0 t= 8 t=12t=60
0
0.005
0.01
5 10 15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-10 -5 0 5 10 15 20
ρ(x
)
x
λ0=60
t= 0 t= 6 t= 8 t=30
0
0.02
0.04
5 10 15
Figure 3. Density of two interacting bosons decaying by tunnelling.Top: snapshots of the density ρ(x, t) as a function of time forinteraction strength λ0 = 0.5. The largest occupation number of theinitially prepared two-boson state is ρ1 = 99.6%, where the systemis referred to as condensed. The inset shows enlarged snapshots ofthe density within and outside the barrier maximum atx = C = 2.25. Bottom: the same but for interaction strengthλ0 = 60. The largest occupation number of the initially preparedtwo-boson state is ρ1 = 74.0%, where the system is fermionized.See text for discussion. The quantities shown are dimensionless.
Snapshots of the density of the tunnelling two-bosonwavefunction are depicted in figure 3 for the representativecase of a condensed (λ0 = 0.5, first natural orbital occupationis ρ1 = 99.6%) and a fermionized (λ0 = 60, first naturalorbital occupation is ρ1 = 74.0%) system. We first examinethe part of the density that has tunnelled out of the trap. Thelarger the interaction strength λ0 is, the larger is the value ofρ(x, t) just outside the trap region, i.e. for x � C = 2.25, forthe same time t, see insets in figure 3. This behaviour of thedensity can be anticipated. With increasing repulsion betweenthe bosons, the energy of the system increases and therefore itbecomes more likely for the particles to be in the free regionof the potential V (x). This also holds for the fact that thetunnelling rate is enhanced by stronger repulsive interactions,as we shall see below.
Let us discuss the density inside the trap. An interestingbehaviour is found in the fermionized case λ0 = 60, seebottom part of figure 3. As time proceeds and more of the
density of the repulsive two-boson system is running away,the part of the density which remains inside the trappedregion evolves from having two density maxima (which isa characteristic of a fermionized two-boson system) to havingonly one (which resembles the density of a condensed system).This time-dependent behaviour of the density can be intuitivelyexplained: as boson density is leaving the parabolic trap,less particles and therefore less repulsion is left inside thetrap. Consequently, the system remaining in the trap becomeseffectively less fermionized and hence more condensed. Thisanalysis is supported by the fact that, for a given interactionstrength λ0, the largest occupation number ρ1 of a few-bosonsystem in the harmonic trap increases when the size of thefew-boson system decreases, see figure 7(a) in [26].
To get a quantitative picture of how and how fast thesystem is decaying for different interaction strengths, it isinstructive to define a quantity that measures the fraction ofthe density remaining localized in the trap, i.e., a measure ofthe total probability to find the system inside the trap region.Building on del Campo et al [13], we define the non-escapeprobability Pnot(t) as follows:
Pnot(t) = 1
N
∫ C
−∞dx ρ(x, t), (8)
which is simply the integral of the density up to the maximumof the barrier x = C in (7). At t = 0, Pnot(0) is essentially1. For a plot depicting the time-evolution of Pnot(t) for non-interacting λ0 = 0 bosons to strong λ0 = 205 interactions seefigure 4. It can be seen that Pnot(t) saturates with the interactionstrength, since the wavefunction and density saturate in thislimit.
It is natural to define the half-life τ1/2 of the non-escapeprobability as the time t for which Pnot(t) = 1
2 . The inset offigure 4 displays τ1/2 for non-interacting to strongly interactingtwo-boson systems. It is seen that the half-life saturates. Itsvalue at the fermionized regime for λ0 = 205 is τ1/2 = 37.5.For comparison, the half-life value for the non-interactingsystem is τ1/2 = 589.5.
Having at hand benchmark exact results for the tunnellingof two-boson systems through a barrier, it would prove to beinstructive to compare the exact Pnot(t) computed by MCTDHto the mean-field Pnot(t) computed by the TDGP. For weakinteraction strengths, λ0 = 0.3 (ρ1 = 99.8(6)%) and λ0 = 0.5(ρ1 = 99.6%), the TDGP non-escape probability Pnot(t) isslightly smaller (tunnelling is slightly faster) than the exactPnot(t) for all times t, see figure 4. This is in line with thesituation that the system is in the condensed regime, and thatthe energy of the exact solution is slightly lower than theenergy of the mean-field solution, see figure 5. For a slightlylarger interaction strength, λ0 = 1 (ρ1 = 98.5%), however,the situation reverses. Counterintuitively, Pnot(t) is smaller(tunnelling is faster) in the exact MCTDH calculation thanwithin the TDGP one, see figure 4, although the mean-fieldenergy is higher than the exact energy, see figure 5. Thisbehaviour persists for larger interaction strengths as well.
What is the explanation? In the TDGP picture the bosonssit in the one and the same time-dependent orbital φ1(x, t) at alltimes. When tunnelling sets in, the bosons are both inside and
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
0.125
0.25
0.5
1
0 200 400 600 800 1000 1200
Pn
ot(
t)
t
τ1/2
λ0
λ0=0
205
10 40
160 640
0 200
Figure 4. Non-escape probability of two interacting bosonsdecaying by tunnelling. Shown is, on a logarithmic scale, thenon-escape probability Pnot(t) as a function of time for weak tostrong interaction strengths λ0 (curves from right to left): 0 (black);0.3 (grey); 0.5 (turquoise); 1 (red); 2 (green); 3 (blue); and4 (magenta). For these interaction strengths a comparison is givenbetween the exact MCTDH solution (solid curves) and thecorresponding TDGP solution (solid curves with crosses). The sameinteraction strength in the MCTDH and TDGP solutions has thesame color. Counterintuitively, the decay of the mean-field solutionfor λ0 � 1 is slower. See text for more details. Finally, the fivedashed black curves are the exact solution computed by MCTDHfor interaction strengths: λ0 = 5, 10, 35, 135, 205. Saturation ofPnot(t) with increasing interaction strength is seen. The inset plotsthe non-escape probability half-life τ1/2 defined as the time whenPnot = 1
2 , which saturates to a value of τ1/2 = 37.5. For comparison,τ1/2 = 589.5 for the non-interacting system. The quantities shownare dimensionless.
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 1 2 3 4 5 6 7
ε
λ
ε
λ
εGP
ε4B
ε2B
0.5
1
0 100
Figure 5. Exact energy per particle ε of the initially-preparedsystems with N = 2 and N = 4 bosons as a function of themean-field factor λ = λ0(N − 1). For comparison and reference,the Gross–Pitaevskii energy is also shown. The inset displays thesaturation of the energy for N = 2 bosons with λ, corroborating theuse of a smoothed δσ=0.05(x) function, see equation (6), for theinter-particle interaction. See text for more details. The quantitiesshown are dimensionless. The exact ε saturates with λ whereas theGross–Pitaevskii one does not.
outside the trap because only one orbital is used to describethe tunnelling process. In the exact MCTDH picture moreorbitals can get involved, making it possible for the bosonsthat have tunnelled out to occupy them, and thus tunnel faster.Of course, differences in the energy of the exact and mean-fieldinitial conditions �(0) play a role as well. In combination,we obtain the different behaviour of Pnot(t) for the exact andmean-field dynamics as a function of the interaction strength.
Let us pause for a moment and briefly analyse theimplications. Although we are looking at the simplest few-body system, namely an interacting two-boson system whichis decaying by tunnelling through a barrier, we note a verydistinct difference between numerically exact results of theMCTDH calculation and the mean-field results of the TDGPcalculation even for regimes of interaction strengths whereone would expect the TDGP to provide a good or at least a fairdescription of the reality. This is found for a comparativelysimple property of the system —the non-escape probability—which is an integral of a certain region of the diagonal partof the reduced one-body density matrix. This questions thepredictions of the TDGP equation when considering open few-boson quantum systems which decay by tunnelling.
3.3. Tunnelling dynamics of four-boson systems
In this section the tunnelling of four-boson systems throughthe same potential barrier (7) used for two-boson systemsin the previous subsection is to be investigated. Thethree foci of the study are: (i) to obtain benchmarkexact results on the decay by tunnelling of four interactingbosons; (ii) to investigate and learn from the differencesbetween the exact and mean-field dynamics of four-bosontunnelling; and (iii) to compare the exact dynamics offour- and two-boson systems with the same mean-fieldfactor λ = λ0(N − 1). For the same value of λ theTDGP equation produces identical dynamics for all valuesof N.
The studies of the dynamics of four-boson systemsin the same potential V (x) employed for the two-bosonsystems concentrate on the regime of weak up to intermediateinteraction strengths. In turn, one has to deal with thefollowing issues. (a) The number of SPFs needed to getconverged results in the sense defined in subsection 2.3 islarger already for weak interactions which renders longercomputation times. (b) In general, the computational time(effort) of MCTDH needed to propagate a four-boson systemwith the same number of SPFs but with a larger interactionstrength is longer, because the wavefunction and density aremore structured. (c) One can only compare the systemswith weak and intermediate repulsions to TDGP calculations,because the energy of the latter exceeds the barrier height ofthe potential (7), and therefore one can no longer speak of atunnelling through a barrier process.
We computed the exact and TDGP dynamics of severalfour-boson systems. The interaction strengths chosen are:λ0 = 0, 0.1, 0.3, 0.5, 0.7, 1, 1.5, 2.5, where the respectivelargest occupation numbers of the initial conditions �(0) areρ1 = 100%, 99.9(5)%, 99.6%, 99.0%, 98.2%, 96.7%, 94.0%,
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
0.125
0.25
0.5
1
0 50 100 150 200 250 300 350 400
Pnot(
t)
t
λ =0
7.5
0.5
1
0 50 100 150 200 250
Pno
t(t)
t
λ=0
7.50.375
Figure 6. Non-escape probability of four interacting bosonsdecaying by tunnelling plotted on a logarithmic scale. Top:comparison of the exact non-escape probability Pnot(t) computed bythe MCTDH method (solid curves) and the correspondingprobability computed by the TDGP approximation (solid curveswith crosses) for the same mean-field factors λ = λ0(N − 1). TheMCTDH and TDGP solutions have the same color for the same λ.Curves from right to left are for λ values of: 0 (thick black line);0.3 (grey); 0.9 (red); 1.5 (green); 2.1 (blue); 3.0 (magenta);4.5 (turquoise); and 7.5 (black). Bottom: comparison of the exactPnot(t) of four-boson (solid curves) and two-boson (solid curveswith crosses) systems having the same values of λ. The four- andtwo-boson solutions have the same color for the same λ. λ valuesand color code are as for the top panel. The quantities shown aredimensionless.
88.5%. For more data on the occupation numbers of thefour interacting bosons in an harmonic potential the reader isreferred to figure 3(a) of Zollner et al [27]. We concentrate onthe non-escape probability Pnot(t), see equation (8), to get aquantitative measure of the tunnelling dynamics. The resultsof the exact and mean-field calculations are plotted for com-parison in figure 6. Note the logarithmic scale. Clearly, thefour-boson exact Pnot(t) are closer to the mean-field Pnot(t),see top part of figure 6, in comparison to the observations ofthe preceding subsection 3.2 for two bosons, see figure 4. Thisresult could be anticipated from the energy per particle curvesdepicted in figure 5. The exact four-boson energy is closerthan the exact two-boson energy to the mean-field energy forthe same factor λ = λ0(N − 1), see figure 5. This difference
in the properties of exact non-escape probabilities of two- andfour-boson systems is the first explicit N-dependence of thedynamics of decay by tunnelling of few-boson systems thatwe encounter.
The comparison of the non-escape probabilities of thefour-boson systems and those of the two-boson systems revealadditional differences, namely N-dependent properties. First,we can see from figure 6 that the four-boson Pnot(t) exhibitprimarily two interaction-dependent decay rates, a slowerdecay rate at shorter times and a faster decay rate at longertimes. This observation stands for both exact and mean-fieldfour-boson dynamics. In contrast, the two-boson non-escapeprobability Pnot(t) exhibits essentially a single interaction-dependent decay rate, see figure 4. Thus, we may speak ofan equilibration of the four-boson initial conditions �(0) inthe trap potential V (x). Here, the larger λ is, the larger is thefraction of the density that has tunnelled out of the trap beforethe second, slower decay rate sets in, see figure 6.
As has been found for the two-boson decay, there aretwo interaction regimes. For weakly interacting four-bosonsystems, the mean-field decay is somewhat faster than the exactdecay. Here, the interaction strengths are about λ0 � 0.5, thecorresponding largest occupation numbers are ρ1 � 99.0%,and the systems can be referred to as condensed. For λ0 = 0.7the situation changes. The short-time mean-field decay isfaster than the short-time exact decay, whereas the long-timemean-field decay is slower than the long-time exact decay, seetop part of figure 6. The explanation of this intricate behaviourcan, as for the two-boson case, be associated with much moredegrees-of-freedom the exact dynamics has in comparison tothe mean-field dynamics. The higher energy of the mean-field solution ‘wins’ at short times. At longer times it ‘loses’,however, as the mean-field dynamics forces the particles whichhave already tunnelled out of the trap to be connected withthose remaining in the trap. It is evident from these resultsthat, especially for the dynamics of decay by tunnelling of few-boson systems, the TDGP approximation is not an adequatetool, even for such a simple global measure of the system asthe non-escape probability Pnot.
To get another perspective on the N-dependence ofthe decay by tunnelling process of few-boson systems, wecompare the exact dynamics of four- and two-boson systemscomputed with the same mean-field factor λ = λ0(N−1). Thebottom part of figure 6 presents the results. For any nonzeroλ, the exact four- and two-boson dynamics are different fromone another. We recall that the respective mean-field dynamicsare, of course, identical. Moreover, even for the lowestcorresponding interactions taken, for λ = 0.3, where thesystems can be referred to as condensed (the largest occupationnumbers are ρ1 = 99.9(5)% and ρ1 = 99.8(6)% for thefour- and two-boson systems, respectively), differences in therespective exact dynamics emerge after a relatively short time,see figure 6.
The exact four-boson decay is always faster than the two-boson decay, for the same value of λ. This corroborates theenergetics, where the exact four-boson energy per particle isalways higher than the two-boson energy per particle for thesame value of λ, see figure 5. It also suggests that the exact
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
0
0.05
0.1
0.15
0.2
0.25
-10 -5 0 5 10 15 20
ρ(x
)
x
t= 0 t= 1 t= 4 t=11
0.125
0.25
0.5
1
0 10 20P
no
t(t)
t
Figure 7. Snapshots of the density ρ(x, t) of four interactingbosons as a function of time for interaction strength λ0 = 20. Thelargest occupation number of the initially prepared four-boson stateis ρ1 = 55.6% (the lowest is ρ9 = 0.1(0)%), where the system isseen to be fermionized. Observe the N = 4 humps of the density att = 0. The inset shows the non-escape probability Pnot(t) as afunction of time. See text for discussion. The quantities shown aredimensionless.
energy per particle of an interacting few-boson system is anadequate predictor to the tunnelling pace.
Finally, as mentioned above the four-boson decaydynamics exhibits primarily two decay rates, at shorter andlonger times. Side by side, we can observe that the larger λ
is, the shorter is the time which the four and two-boson non-escape probabilities Pnot(t) are relatively close to one another,see bottom part of figure 6.
Summarizing, the above-described intricate N-particleproperties of the non-escape probability Pnot(t) demonstratethat the TDGP approximation is not an adequate tool todescribe the physics of even weakly interacting few-bosonsystems decaying by tunnelling out of a trap.
We wish to conclude the investigations on the tunnellingdynamics of four-boson systems by looking at an exampleof strongly interacting four-boson system with the interactionstrength λ0 = 20 in the same trap V (x) of figure 1. Forthis interaction strength the fully-converged time propagationis more demanding. Therefore, in order to make thecomputation tractable, the strict convergence conditionsdefined in subsection 2.3 are slightly ‘softened’. Specifically,we used n = 9 SPFs. The largest natural orbital occupationnumber is ρ1 = 55.6% whereas the smallest one is ρ9 =0.1(0)% for the initial conditions �(0). These values areconverged to within 5 × 10−4. The energy per particle isfound to be 1.97(3) and is converged to within 4 × 10−3. Notethat the energy per particle is slightly below the maximumvalue A = 2.2663 of the trap at V (C = 2.25), seeequation (7). Finally, the difference between the time-dependent autocorrelation functions with n and n − 1 SPFsis less than 5 × 10−3 for all times t, and the lowest occupationnumber is kept below 1.1 × 10−2 for all times t.
Snapshots of the density ρ(x, t) for the times t =0, 1, 4, 11 are plotted in figure 7. Intriguing behaviour
develops in time. Initially, at t = 0 the density exhibitsfour maxima as one would expect from a fermionized system[25, 26, 30, 51]. For t = 1 there are three maxima remainingin the trap region. The right-most fourth density maximum att = 0 is seen to evolve to a ‘shoulder’ just outside the trapregion. Thereafter, until 90% of the density has been absorbedby the CAP (the propagation in time is stopped then), thedensity remaining inside the trap has only two maxima.
To interpret the decay dynamics in the fermionizedregime, we compute the non-escape probability Pnot(t), seeinset of figure 7. The non-escape probability shows a differentstructure and is no longer decaying exponentially in the wayobserved for two- and four-boson systems throughout thiswork, compare figures 4 and 6 to figure 7. Moreover, the decayis much faster with τ1/2 ≈ 7.5, which one would expect for anescape rather than for a tunnelling through a barrier. Indeed,figure 7 indicates that one boson leaves the well region directlyand eventually a second one follows, whereas the remainingtwo bosons undergo a complex decay mechanism.
Concluding, the decay dynamics of the fermionized four-boson system seen in figure 7 is intricate. In order tounderstand the differences between this decay dynamics andthe other studies performed in this work for interacting two-and four-boson systems, more delicate many-body analysistools than the density ρ(x, t) and non-escape probabilityPnot(t) are required. These include momentum distributions,reduced density matrices and correlation functions.
4. Summary and outlook
In the present work we have studied from first principles,by integrating the time-dependent Schrodinger equation withthe help of the multiconfiguration time-dependent Hartree(MCTDH) package, the decay by tunnelling dynamics ofinteracting few-boson systems through a one-dimensionalpotential barrier. Particular attention has been paid toobtaining numerically exact benchmarks, i.e., converged,results. The exact results have been contrasted with those of thetime-dependent Gross–Pitaevskii equation. It has been foundthat for few-boson systems the exact decay dynamics dependsexplicitly on the number of bosons N. In particular, for two-boson systems the decay dynamics is governed by essentiallyone exponential decay parameter, whereas for four-bosonsystems it is governed primarily by two exponential decayparameters, at least for the times and interaction strengthsconsidered here.
Another valuable result is the deviation in tunnellingdynamics recorded between the exact and mean-field non-escape probabilities Pnot(t), even for relatively weaklyinteracting systems. This finding motivates one to examinecorrelation functions of few-boson systems decaying bytunnelling, a study that goes beyond the scope of the presentwork.
What to do next? There are many relevant directionsto follow. We have studied here the tunnelling dynamics ofrepulsive few-boson systems, and it is certainly interestingto enquire whether and how attractive interactions would
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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 044018 A U J Lode et al
influence the exact decay process. A natural extensionof the present work is to study the tunnelling through abarrier of larger than the interacting few-boson systemsconsidered here. For this, it would become essential to takeexplicitly into account the bosonic permutational symmetryof the wavefunction, namely, to truncate the large amountof redundancies of coefficients in the distinguishable-particlemulticonfigurational expansion of the MCTDH bosonicwavefunction. The multiconfigurational time-dependentHartree for bosons (MCTDHB) [52–54] exactly takes careof that. In the MCTDHB the many-boson wavefunctionis expanded as a linear combination of time-dependentpermanents. Consequently, much larger systems of interactingbosons can be treated with MCTDHB than with MCTDH.
Another natural direction is to compute resonances—positions and widths—of decaying interacting few- and many-boson systems beyond the mean field. In this context, it wouldbe valuable to extend the stationary multiconfigurationalHartree for bosons [55] used to compute bound statesto resonance states. This will allow one in the nextstage to compute reduce density matrices and coherenceproperties [56] of interacting few- and many-boson resonancestates. Here, the complex-coordinate method [57–60] andthe Non-Hermitian Quantum Mechanics, which Moiseyev hascontributed so greatly to, would prove to be valuable.
Acknowledgments
The work is dedicated to Professor Nimrod Moiseyev, acolleague and dear friend, on the occasion of his 60th birthday.Financial support by the Deutsche Forschungsgemeinschaft isgratefully acknowledged.
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