Vol.35 (Apr) 2025 | Article no.5 2025
We study the macroscopic quantum tunnelling of an interacting Bose-Einstein condensate in a cubic-plus-quadratic well that was approximately realized in recent experiment. We utilise the Gross-Pitaevskii equation and Wentzel-Kramers-Brillouin (WKB) method to investigate the effect of the inter-atomic interactions on the tunnelling rate of a quasi-bound condensate. We find that the existence of repulsive interaction enhances the quantum tunnelling of a trapped Bose-Einstein condensate.
As a quintessential quantum phenomenon, quantum tunnelling plays an essential role in various physical processes, ranging from the processes of \(\alpha\)-decay [1] to the quantum tunnelling in magnetic [2] and cold-atom systems [3, 4]. In particular, various aspects of quantum tunnelling can be studied using ultracold atoms, since different scenarios of quantum tunnelling can be readily realized in experiments. Furthermore, it is possible to engineer different Hamiltonians in cold atom systems and change the interaction strength via Feshbach resonances [5,6,7,8] or optical lattices [9, 10]. Using light-atom interaction, exotic interaction such as spin-orbit coupling [11], can be realized. Recently, the studies of the quantum tunnelling have attracted quite some interest in the Bose-Einstein condensates (BECs) [12,13,14,15,16,17,18], e.g., the macroscopic Klein tunnelling of a BEC with or without the spin-orbit coupling [19, 20], the Landau-Zener tunnelling out of an optical lattice [21], and the Landau-Zener tunnelling of a BEC with the spin-orbit coupling [22].
In most studies, it is assumed that the particle tunnels independently, leading to the expected exponential decay as in the case of \(\alpha\)-decay. The question of the effects of interaction in the tunnelling process is much less understood [23,24,25,26,27,28]. In ultracold atomic systems, the inter-particle interactions can be controlled via Feshbach resonances [5,6,7,8], thus providing an ideal platform for studying the effects of interaction in quantum tunnelling. We note that interaction effects have been explored in the ac and dc Josephson effects [3, 4] in double well potential, or the many-body quantum phase transitions of bosons in optical lattices [29, 30]. However, in those studies, the tunnelling amplitude is usually assumed to be density independent and arises from single particle processes. We are interested instead in the interaction modification of tunnelling amplitude itself.
In the above examples, furthermore, the quantum tunnelling was studied in systems that are bound in space. On the other hand, studies of effects of interaction on quantum tunnelling into un-bound state received relatively less attention. Recent experiment [31] along this line has shown that in the regime where quantum tunnelling was believed to occur, the tunnelling is non-exponential in time, revealing the crucial effects of inter-atomic interaction. In general, we note that the inclusion of inter-atomic interaction in the tunnelling processes is a formidable task compared with the single particle tunnelling because of the enormous configuration space in which the tunnelling occurs. For BEC, however, this problem might be alleviated because, to a first approximation, all bosons are occupying the same quantum state, and thus the possibility of using Gross-Pitaevskii ansatz which reduces the problem effectively to the single particle case. On the other hand, non-trivial interaction effects does enter through the non-linear interaction term and the chemical potential (see below). Our modest task in this paper is to investigate how these two effects affects the tunnelling of a condensate.
It should be mentioned that extensive theoretical analysis of the experiment were already carried out based on the numerical simulation of Gross-Pitaevskii equations and a modified WKB analysis that takes into account the interaction effects at the mean field level which was approximated by an simple ansatz (see Eq. (2) below) [27]. In this treatment, the tunnelling is essentially single-particle like, but the external potential is modified to have the following form
where \(V(\boldsymbol{r})\) is the external potential which was taken in Ref. [27] to be of a triangular shape. \(U_0={4\pi \hbar ^{2}a_{s}}/{m}\) is the interaction parameter and \(a_s\) the s-wave scattering length. \(\psi (\boldsymbol{r})\) is the order parameter (macroscopic wave function) for the Bose condensate. This still gives a non-linear equation in which \(\psi (\boldsymbol{r})\) must be determined self-consistently. A further approximation is made in Ref. [27] to replace the mean field term by
where \(V_s\) is an energy scale and a is a dimensionless fitting parameter. N(t) is the number of atoms in the condensate at time t. In this manner, the tunnelling was computed for the triangular potential \(V(\boldsymbol{r})\) and they found good fit with the experiment. In particular, they suggested that the tunnelling rate \(\Gamma\) can be written as
with \(\Gamma _{\text {bg}}\) describing the background loss and \(\alpha\) and \(\beta\) are fitting parameters. \(\mu\) is the chemical potential of the condensate. As the tunnelling proceeds, the chemical potential of the condensate decreases and this leads to the non-exponential decay found in experiment.
In this paper, we carry out an analytic calculation of \(\Gamma\) and determine its dependences on \(\mu\). We found that there is a logarithmic correction to the formula (3) above and that the \(\beta \mu\) term will be replaced by \((c_1+c_2\ln \mu )\mu\) at least in the low chemical potential regime. Here \(c_1\) and \(c_2\) are constants that can be computed explicitly. To model the external potential, we shall adopt the standard cubic-plus-quadratic well that is well-known in the literature on quantum tunnelling.
The rest of this paper is organized as follows. In Sect. 2, we analyze the time-independent Gross-Pitaevskii equation and obtain the wave functions of the condensate in the whole regions by applying the Thomas-Fermi approximation and adapted WKB method. We then calculate the decay rate, i.e., the WKB integral \(\gamma\) (see below), of the quasi-bound BEC in presence of the inter-particle interactions. In Sect. 3, we present the explicit formulas relating the renormalized WKB integral, i.e. \(\Gamma =\gamma /\gamma _{0}\), where \(\gamma _{0}\) is the WKB integral of the single-particle quantum tunnelling in the absence of inter-particle interactions. In Sect. 4, we give a brief conclusion.
Let us consider an interacting Bose-Einstein condensate described by the s-wave scattering length \(a_s\). At low energies and low density, the effective interaction between two particles can be modelled as a contact interaction,
Here \(\hbar\) is the Planck constant and m is the mass of the boson. For N bosons, the effective Hamiltonian can be written as
where \(V(\boldsymbol{r})\) is the external potential, and we consider the repulsive interactions, i.e., \(U_0>0\).
At zero-temperature, we adopt the Gross-Pitaevskii ansatz and assume that all bosons are condensed in the same single-particle state, \(\phi (\boldsymbol{r})\), and the wave function of the N-boson gas can be written as
The single-particle wave function \(\phi (\boldsymbol{r}_{i})\) is normalized as usual, \(\int d\boldsymbol{r}|\phi (\boldsymbol{r})|^{2}=1\).
Taking the expectation value of the Hamiltonian Eq. (5) in the state Eq. (6), one can obtain the energy functional of the Bose condensate
where the condensate wave function is denoted by \(\psi (\boldsymbol{r})=\sqrt{N}\phi (\boldsymbol{r})\) (we assume \(N\gg 1\)). One wants to find out the minimal of E subjected to the constraint that the total number of bosons is conserved. This leads to the celebrated time-independent Gross-Pitaevskii (GP) equation [32]
where \(\mu\) is the chemical potential. Our analysis of the quantum tunnelling will be based on GP equation above.
For orientation, let us first consider the simple situation when the condensate is trapped in a harmonic potential \(V(\boldsymbol r)=\frac{1}{2}m\omega_x^2x^2+\frac{1}{2}m\omega_\perp^2\left(y^2+z^2\right)\) with a tight confinement \(\omega _{\perp }\) in the transverse direction and a weak confinement \(\omega _{x}\) along the axial \(\hat{x}\)-axis. For a sufficient large cloud and away from the cloud boundary, we can neglect the kinetic energy in the Gross-Pitaevskii equation and make use of the so-called the Thomas-Fermi approximation, whereupon the condensate wave function can be written in the form
The chemical potential is determined by requiring the total number of bosons to be N. This gives
with \(\tilde{a}={Na_{s}}/{\bar{a}_{ho}}\) a dimensionless measure of the interaction strength. Here the characteristic length \(\bar{a}_{ho}=\sqrt{{\hbar }/{m\bar{\omega }}}\) and \(\bar{\omega }=(\omega _{x}\omega _{y}\omega _{z})^{1/3}=\omega _{\perp }^{2/3}\omega _{x}^{1/3}\). Also, the spatial extent of the cloud along the weak confinement direction is given by the following:
where \(\bar{R}=(R_{x}R_{y}R_{z})^{1/3}\). Similar expressions can be written down for \(R_y\) and \(R_z\).
Now, let us consider turning on a cubic potential along the x-direction, i.e., \(V_{c}=-\beta x^{3}\). As shown in Fig. 1, this allows the condensate to tunnel through the finite barrier along \(\hat{x}\)-direction. For simplicity, we assume that the condensate dynamics in the \(\hat{y}\) and \(\hat{z}\)-directions are frozen and focus on the one-dimensional Gross-Pitaevskii equation:
where the trapping potential is a now given by the cubic-plus-quadratic well (denoting \(\omega _x\) as \(\omega\) below):
where \(\omega\) denotes the harmonic frequency close to the origin \(x=0\), and \(\beta\) characterizes the strength of the cubic potential. In general, Eq. (12) is a nonlinear differential equation that takes into account of the mean-field interaction. However, it is clear that when the density becomes very dilute close to the edge of the cloud where tunnelling occurs, the nonlinear term in Eq. (12) is correspondingly small and can be neglected. On the other hand, the single particle states at the edge of the cloud is certainly modified by the mean field interaction inside the bulk of the condensate.
MQT of a quasi-bound BEC in a Cubic-Plus-Quadratic Well. The condensate is initially trapped in a harmonic potential. \({{\mu }}\) is the chemical potential of the condensate. The dashed line denotes the value of \(\mu\), which is much less than the height of the potential barrier. After the cubic potential is turned on, the quasi-bound condensate will be mainly trapped in a classical region, i.e., \(X_{0}
We assume that the cubic potential is much less than the quadratic one \(\frac{1}{2}m\omega ^{2}x^{2}\) within the spatial extent of the trapped condensate. As shown in Fig. 1, there are three turning points, i.e., \(X_{0}\), \(X_{1}\), \(X_{2}\), determined by \(\mu =V(X_{i})\). In the trapping region where \(X_{0}
Therefore, for a sufficient large cloud, the previous estimate of the relation between \(\mu\) and \(\tilde{a}\), i.e. Eq. (10), is still valid. We then apply the Thomas-Fermi approximation to Eq. (12):
In the interior region, the condensate wave function is thus accurately given by
Close to the turning point \(X_1\), however, \(V(x)\approx \mu\) and the kinetic energy becomes important, and the Thomas-Fermi approximation fails. Therefore, it is crucial to find the condensate wave function close to the boundary. To achieve this goal, we adopt the WKB approximation.
In the vicinity of the turning point \(X_{1}\), i.e., \(\left| x-X_{1}\right| \ll X_{1}\), we approximate the potential:
For such a region, we rewrite the Gross-Pitaevskii equation as
Let us introduce the dimensionless variable
where
denotes the typical thickness of the boundary, in which the Thomas-Fermi approximation fails. For simplicity, we define \(\phi\) by
In terms of \(\phi _{1}\), the Gross-Pitaevskii equation takes the form
where the nonlinear term \(\phi _{1}^{3}\) arises from the inter-particle interactions. In the limit of large positive \(\xi _{1}\), where the condensate is sufficiently dilute, the nonlinear term can be ignored. Then Eq. (12) reduces to the Airy’s equation \(\phi ^{\prime \prime }_{1}=\xi _{1}\phi _{1}\), and the solution takes the form of Airy function. The asymptotic behavior is given by
where \(A_{1}\) and \(B_{1}\) are coefficients of the two exponential functions. In the limit of large negative \(\xi _{1}\), where the kinetic term is less important, the second derivative \(\phi ^{\prime \prime }_{1}\) can be safely neglected. Correspondingly, the asymptotic behavior reads
In the tunnelling region under the barrier, i.e., \(X_{1}
where we have defined
Let the WKB integral be
and define \(A'\equiv A_{1}e^{-\gamma }\), \(B'\equiv B_{1}e^{\gamma }\), we can then rewrite Eq. (25) as
As is clear, the effects of the inter-particle interactions on the condensate wave function are mainly contained via the magnitude of the chemical potential. The coefficient of proportionality in Eq. (28) is fixed to match the solution Eq. (23) in the patching region around \(X_1\).
Now, near the outer turning point \(X_{2}\), we need to find another patching wave function via the WKB method. Similarly, we approximate the potential:
and solve the Gross-Pitaevskii equation as
Again, we introduce a dimensionless variable
where
and the Gross-Pitaevskii equation reduces to
where the nonlinear term \(\phi _{2}^{3}\) has been neglected because the condensate is sufficiently dilute around \(X_2\). Solving this equation, we obtain its asymptotic forms in two opposite limits:
and
where \(\Delta (\xi _{2})=\frac{2}{3}\left( -\xi _{2}\right) ^{3/2}+\frac{\pi }{4}\), and \(A_{2}\) and \(B_{2}\) are coefficients determined by Eq. (33).
For the region where \(x>X_{2}\), i.e. \(\xi_{2}<0\), we find \(\mu>V(x)\) and the WKB approximation gives
where the constant C is the coefficient of the outgoing wave.
Now matching the solutions at the boundaries \(X_{1}\) and \(X_{2}\), we obtain the following connection formulas
Solving the above equations, we find
To summarize, we have obtained the condensate wave function in the entire regions:
and the current density in the tunnelling region can be computed explicitly
As one can see, the current depends exponentially on the the WKB integral \(\gamma\) in Eq. (27), as expected.
We define a normalized WKB integral, which is the ratio between the WKB integral of interacting condensate \(\gamma (\mu )\) and that of non-interacting atom \(\gamma_0\equiv\gamma(\mu=0)\):
where the elliptic modulus are given by \(k^{2}=1-k'^{2}\) with
and the elliptic function K(k) and E(k) are given by
It is important to notice that this normalized WKB integral \(\Gamma\) is a function of the parameter k, which is defined by the classical turning points (\(X_{0}\), \(X_{1}\), \(X_{2}\)). And these turning points are determined by the chemical potential \(\mu =V(x)\). Therefore, we can obtain the normalized WKB integral \(\Gamma \left( {\mu }\right)\) as a function of chemical potential \({\mu }\), or as a function of the s-wave scattering length \(a_s\) by using the Thomas-Fermi approximation Eq. (10).
Here we consider the case in which the chemical potential is much lower than the potential barrier and the cloud is sufficiently dilute, so that the decay mechanism of the condensate is supposed to be caused by macroscopic quantum tunnelling, rather than classical spilling or three-body losses [31]. Thus, the decay rate of the condensate is essentially the tunnelling rate:
In order to describe the tunnelling behavior of a Condensate in the limit of low chemical potential, we expand the (normalized) WKB integral \(\Gamma ({\mu })\), Eq. (45) in powers of the chemical potential \({\mu }\) up to the second order. This gives the following expression
where \(\tilde{\mu }=\mu /(\hbar \bar{\omega })\) is a dimensionless measure of chemical potential, \(c_{1}=-\frac{1}{240}(1+2\ln 120)\) and \(c_{2}=\frac{1}{240}\) are constants.
Figure 2 shows that the normalized WKB integral \(\Gamma\) decreases as the chemical potential \(\mu\) increases, which leads to a larger tunnelling rate T in Eq. (49). In other words, it is easier for the condensate to tunnel as the chemical potential is increased. On the contrary, the tunnelling of the condensate diminishes as the chemical potential is decreased. When \(\mu \rightarrow 0\), the tunnelling of the condensate is essentially like that of a single particle.
The normalized WKB integral in the limit of low chemical potential. The solid line is given by the Eq. (45) and the dashed line is given by the expansion Eq. (50). The parameters are chosen as \(\beta =m\omega ^{2}/(30\bar{a}_{ho})\), \(\omega _{x}=\omega =2\bar{\omega }\), \(\tilde{\mu }=\mu /(\hbar \bar{\omega })\), \(\tilde{a}=Na_{s}/\bar{a}_{ho}\)
By using the Thomas-Fermi approximation, i.e., Eq. (10), one can equivalently expand \(\Gamma\) in powers of \(\tilde{a}\):
where we recall that \(\tilde{a}={Na_{s}}/{\bar{a}_{ho}}\) and \(f_{1}\) and \(f_{2}\) are constants.
Figure 3 shows that the normalized WKB integral \(\Gamma\) decreases as the interaction \(\tilde{a}\) increases, which leads to a larger tunnelling rate T in Eq. (49). Namely, the repulsive interactions between atoms enhances the macroscopic quantum tunnelling of a metastable BEC in the cubic-plus-quadratic well. When \(\tilde{a}\rightarrow 0\), the system enters almost the non-interacting regime where \(\Gamma =1\), as expected.
In this paper, we discuss the interaction effects on the MQT of a BEC in a cubic-plus-quadratic well. With the Gross-Pitaevskii equation and the WKB method, we present the explicit expression relating the chemical potential and tunnelling rate of the condensate, i.e. \(\Gamma (\mu )\) in Eq. (50), and estimate the effect of atomic interactions on the tunnelling rate, i.e., \(\Gamma (a)\) in Eq. (51). We find that the repulsive interaction leads to an enhancement of quantum tunnelling of a trapped BEC in a cubic-plus-quadratic potential well. Our analytical results supplement the earlier theoretical calculation and reveal extra dependences on the chemical potential, and could be readily confirmed in current cold atom experiments.
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It is a great pleasure for one of the authors (SZZ) to dedicate this small note to the memory of Professor Chang Lee.
This work is supported by HK GRF Grants No 17313122, CRF Grants No. C7012-21G, and a RGC Fellowship Award No. HKU RFS2223-7S03.
All authors contributed to the research reported in the note, and all are involved in the writing of the final draft.
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The authors declare that they have no competing interests.
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