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The ground state and the tunnelling dynamics of the Bose-Einstein condensate in a tilted shallow trap
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The ground state and the tunnelling dynamics of the Bose-Einstein condensate in a tilted shallow trap Yue Jian, Xin Qiao, Yan-Chao Zhang, Ai-Xia Zhang, Ju-Kui Xue ∗ College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070, China
a r t i c l e
i n f o
Article history: Received 7 July 2019 Received in revised form 24 October 2019 Accepted 4 November 2019 Available online xxxx Communicated by V.A. Markel Keywords: Bose-Einstein condensate Stability Tunnelling dynamics
a b s t r a c t The ground state and the tunnelling dynamics of the Bose-Einstein condensate (BEC) loaded in a tilted shallow trap is studied analytically and numerically. The stable bound state, the quasi-bound state and the diffusion state are predicted. The thresholds for transition between the different states are obtained and the stability diagram in parameter space is presented. The tunnelling dynamics of the system in different states is revealed. The shape of the potential well and the atomic interaction play important role and have coupled effect on the tunnelling dynamics of the system. Furthermore, the resonant tunnelling phenomenon in the parametrically modulated shallow trap is observed. The results show that when the modulating frequency approaches the dipolar mode of the system, resonant tunnelling occurs and the whole system is unstable. Our results provide a theoretical evidence for studying the tunnelling dynamics of the ultracold atomic system. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Macroscopic quantum tunnelling is the tunnelling of a manybody wavefunction through a potential barrier which is forbidden by classical analysis [1]. The escape of a particle due to tunnelling from a quasi-bound state, studied in α decay and nuclear fission and fusion, is one of the significant and earliest problems studied in quantum mechanics [2]. The shallow well is the most basic model to study the tunnelling dynamics which was originally applied to the study of chiral isomers [3] and the researches show that the probability to tunnel through an energy barrier is exponentially dependent on the barrier area. Since its inception, tunnelling plays an important role across physics, chemistry, and technology [4–6]. For example, as electronic devices reach the nanoscale, quantum tunnelling will play a more and more important role in developing nanoelectronics, such as tunnelling diodes [7]. Furthermore, with the advent of Josephson junctions, we can now measure voltage with unprecedented accuracy by the epitome of macroscopic quantum devices based on tunnelling [8]. At very low temperatures, Bose-Einstein condensate (BEC) is the ideal scenario to exhibit purely quantum phenomena [9–12]. A series of fundamental experiments on tunnelling of BEC have been carried out in recent years as the techniques for preparing and manipulating ultracold atoms become mature [13–16]. Such as
*
Corresponding author. E-mail address: [email protected] (J.-K. Xue).
https://doi.org/10.1016/j.physleta.2019.126126 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Josephson oscillations and self trapping [17], the dc and ac Josephson effect [18], the crossover from hydrodynamic and Josephson regimes [19]. Experimentally, the shallow well can be achieved by reducing the laser intensity on optical trapping and/or by reducing the electric current in magnetic trapping. It is shown that possible bound state can exist in the shallow well. However, experiments largely study Landau-Zener tunnelling out of an optical lattice, and studying tunnelling from a bound state into the continuum are fewer. In addition, the studies of tunnelling are usually concentrated in symmetric shallow wells [20–23]. Nevertheless, due to the influence of magnetic field gradient and gravity, tilted shallow trap is easier to generate experimentally. Therefore, it is of more general significance to study the tunnelling dynamics of BEC in tilted trap. In particular, a recent experiment about the macroscopic quantum tunnelling in a tilted shallow trap [24–26] reveals that the tunnelling rate is an experimentally measurable parameter and containing important information about the atomic dynamics inside the trap as well as the initial state of the quantum system. The results show that the nonlinearities due to the atomic interaction and the macroscopic nature of BEC create a highly nonexponential decay and open up the hitherto unexplored experimental regime of tunnelling from a single trapping well into the continuum. Motivated by the experimental observation [24], we study the ground state and its stability of BEC in the tilted shallow trap, and provide a deep quantitative understanding of the tunnelling dynamics of the system. By using the variational analysis and numerical stimulation, the stability of the ground state of BEC in the
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Fig. 1. The sketch of the model. V 0 = 140, a = 60,
σ0 = 0.4.
tilted shallow potential well is discussed theoretically. The stable bound state, the quasi-bound state and the diffusion state are predicted. The thresholds for transition between the different states and the stability diagram in parameter space are presented. The tunnelling dynamics of the system in different states is discussed. In addition, the resonant tunnelling phenomenon in the parametrically modulated shallow trap is further discussed. The results show that when the modulating frequency approaches the dipolar mode of the system, resonant tunnelling occurs and the whole system is unstable. This paper is organized as follows. In Section 2, we present the physical model and the numerically simulated tunnelling diagram. In Section 3, the ground state phase diagram and the stability of the ground state are discussed. In Section 4, the tunnelling and diffusion dynamics of the system are discussed. In Section 5, by applying the variational approximation and the numerical simulation, the tunnelling dynamics of the condensate in the parametrically modulated tilted shallow trap is investigated. Finally, in Section 6, a brief summary is given.
where ψ(x) is the mean-field wavefunction and the total number of atoms N satisfies the normalization |ψ(x)|2 dx = N. g is the nonlinearity which comes from the interaction between the particles and may be either positive (repulsive) or negative (attractive). Here length and time are expressed in units of l and (ω )−1 , respectively. For the shallow trap, it is particularly important to study the tunnelling properties of the condensate. The number of atoms in the potential well decreases with time according to
2. The model and the tunnelling diagram
dN
Fig. 2. The average tunnelling rate γ of the condensate as a function of g for different values of the potential-shape parameter given by numerical simulation of Eq. (2). The horizontal short dotted line at the bottom represents the boundary between the stable bound state and the quasi-bound state obtained by numerical simulation. Symbols represent the critical nonlinearities g crit and g max calculated by the variational method.
(−
dt We study the stability and tunnelling dynamics of the onedimensional 87 Rb BEC in a tilted shallow trap based on the experimental realization [24]. The trap is composed of a quadrupole magnetic field and a blue detuned light sheet. The magnetic trap provides harmonic confinement in the horizontal direction with trapping frequency ω . Due to a magnetic field gradient and gravity, there is a tilt in the vertical direction with a constant acceleration g eff . The one-dimensional tilted shallow trap in the vertical 2
− 2x
direction has the form V ( x) = mg eff x+ V 0 e σ0 , where m is the mass of an atom, g eff is the acceleration and represents the tilt of the potential well, V 0 is the peak height of the barrier and σ0 is the barrier waist. In units of h¯ ω , we reduce the potential well to the form 2
V (x) = ax + V 0 e
− 2x σ
0
,
(1)
where x = x/l, a = mg effl/¯hω , V 0 = V 0 /¯hω , and σ0 = σ0 /l with l = h¯ /mω . The tilted trap is shown in Fig. 1. In general, the GP equation is a sufficient theoretical tool for describing the ground state and dynamics of BEC at 0 K. So we begin with the time-dependent mean-field GP equation used to describe BEC. The one-dimensional GP equation for the BEC wavefunction ψ can be written in dimensionless form
1 ∂2 2 ∂2x
+ V (x) + g |ψ(x)|2 )ψ(x) = i
∂ ψ(x), ∂t
= −γ ( N ) N ,
(3)
where, the coefficient
γ (N ) = −
d ln N dt
(2)
.
γ is the tunnelling rate. Therefore (4)
In order to clearly show the tunnelling dynamics of the system, we solved the Eq. (2) numerically. Fig. 2 shows the change of the average tunnelling rate γ of the system with the nonlinearity g for different trap parameters. To facilitate numerical simulation, the tunnelling rate γ of the condensate is calculated by the average change rate of the logarithm of the normalized atomic number in a certain period of the evolution time. The evolution time of the condensate is t = 300, which is equivalent to about 1s in the actual experiment in the dimensional form. As shown in Fig. 2, the tunnelling of the condensate is closely related to the peak height of the barrier V 0 , i.e., γ decreases with the increase of V 0 . Because the narrower the potential well is, the more likely the condensate is to tunnelling, so γ increases much sharply with g and is much larger in the tight configuration (a = 90) than that in the weak configuration (a = 60). Since the influence of the barrier waist σ0 on tunnelling is nonlinearly, so the γ in the shallower potential well (σ0 = 0.8) is smaller than that in the wide potential well (σ0 = 0.4) in the case of a = 60. Interestingly, Fig. 2 indicates that the average tunnelling rate presents an obvious step structure as g changes, which clearly shows that the system has three
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states: the stable bound state, the quasi-bound state and the diffusion state. In the stable bound state, γ → 0, the condensate is all trapped in the potential well. In the quasi-bound state, there exits tunnelling phenomenon and γ increases quickly with the increase of g. In the diffused state, the condensate quickly spills out of the potential well and γ reaches its maximum. The transition from the stable bound state to the quasi-bound state occurs when γ → 0. Since the tunnelling rate of the system can be infinitely small, we can only qualitatively give a critical point from the numerical simulation, γc ≈ 10−4 , which is obtained by fitting the platform in the numerical simulation in Fig. 2, as shown by the horizontal short dotted line at the bottom of the figure. The average tunnelling rate below the critical line changes very slowly and basically approaches 0, so we regard it as the stable bound state. Above the critical line, the particles of the condensate can be considered to tunnelling. In the following sections, the existence and the dynamics of the three states will be discussed theoretically and numerically. 3. The ground state and its stability The stationary solution of the one-dimensional GP Eq. (2) has the form (x, t ) = ψ(x)e (−i μt ) , so that
(−
1 ∂2 2 ∂2x
+ V (x) + g |ψ(x)|2 )ψ(x) = μψ(x),
(5)
where μ is the chemical potential. To understand the stability of the ground state in the tilted shallow trap, we employ a variational method with the following Gaussian wave function for the solution of Eq. (5)
ψ(x) =
1
π
1 4
√ e
−
(x−x0 )2 2R 2
(6)
,
R
where R and x0 are the width and the central of the mass of the condensate, respectively. The ground state energy of the system is given by the functional
E=
1
[− ψ(x)∗ 2
∂2 1 ψ(x) + V (x)|ψ(x)|2 + g |ψ(x)|4 ]dx. 2 ∂ x2
(7)
Substituting the trial wave function (6) into the ground state energy (7), we obtain
E=
1 4R 2
+ √
g
2 2π R
+
V 0 σ0
e
−
2x2 0
σ02 +2R 2
σ02 + 2R 2
+ ax0 .
(8)
1 2R 3 a−
+ √
g
2 2π R 2
4V 0 σ0 x0 3
(σ02 + 2R 2 ) 2
+
e
2V 0 σ0 R (σ02 + 2R 2 − 4x20 )
2x2 − 2 0 2 σ0 +2R
5
(σ02 + 2R 2 ) 2
e
Figs. 3(a) and (b) show the energy of the system E versus the width of the wave packet R for different values of g. As can be seen from the figures, g max is an obvious boundary for the existence of minimum energy. In order to further reveal the diffusion mechanism of the system, we compare the relationship between the tilted shallow trap V (x) and the effective potential V eff ≡ V (x) + g |ψ(x)|2 of the system, as shown in Figs. 3 (c) and (d). When g > g max , V eff is obviously higher than that of the potential well, that is, the particles will overflow from the potential well. When g < g max , V eff is contained in the potential well, the system is in bound state. Fig. 2 indicates that, unlike an infinite well, the condensate can still exist tunnelling phenomenon in the bound state. That is, there is a critical nonlinearity termed g crit , when g < g crit , the condensate is trapped in potential well and the system is in the stable bound state; when g crit < g < g max , the condensate exists tunnelling phenomenon and the system is in the quasi-bound state. The g crit can be determined by the variational WKB approach [20]. The WKB tunnelling rate γ is given by the standard expressions:
x2
The ground state of the system can be obtained by minimizing the energy Eq. (8) with respect to R and x0 , 2x2 − 2 0 2 σ0 +2R
Fig. 3. (a) and (b) The energy of the system E versus the width of the wave packet R for different values of g. (c) and (d) The tilted shallow trap V (x) (Solid line) and the effective potential V eff (dash lines) versus x for different values of g. V 0 = 140, σ0 = 0.4.
γ = ν exp(−2 p (x) ≡
= 0 (9a)
ν −1 ≡ 2
(9b)
So, from Eqs. (9a)–(9b), we can find a critical nonlinearity called the maximum nonlinearity g max which divides the system into two states, one is the bound state (g < g max ) and another one is the diffusion state (g > g max ). The condensate has a minimum energy in the bound state and can be stabilized in the potential well, while the condensate does not have a minimum energy in the diffusion state and is pushed out over the top of the potential well. This is clearly illustrated in Fig. 3.
(10)
2(μ − V eff ) x3
(11)
dy
x2
= 0.
| p (x)|dx) x1
(12)
| p (x)|
where the limits of the integration x1 , x2 and x3 are found from setting the semiclassical momentum p (x) = 0. x2 − x1 is the barrier length through which the wavefuction must tunnel and
2
μ = [− 12 ψ(x)∗ ∂∂x2 ψ(x) + 2 4 V (x)|ψ(x)| + g |ψ(x)| ]dx is the chemical potential. ν −1 is the
x3 − x2 is the length between the trap.
oscillation frequency of the particles in the trap. The transition from the stable bound state to the quasi-bound state occurs when γ → 0. Since the tunnelling rate of the system can be infinitely small, here we take γc ≈ 10−4 as the boundary, which is obtained by fitting the platform in the numerical simulation in Fig. 2, as
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Fig. 4. The stability diagram in V 0 - g plane for the weak configuration (a) and the tight configuration (b) with different values of the potential-shape parameter.
shown by the horizontal short dotted line at the bottom of the figure. When γ > γc , it can be considered that the tunnelling takes place. According to Eqs. (9)–(12), the stability diagram of the system in V 0 - g plane for the weak configuration (a) and the tight configuration (b) is clearly depicted in Fig. 4. It is clear that given the values of the potential-shape parameter V 0 and σ0 , there exists two critical nonlinearities g crit (lower curves) and g max (upper curves), which divide the V 0 - g plane into three regions: the stable bound state region (g < g crit ), the quasi-bound state region (g crit < g < g max ) and the diffusion region (g > g max ). When g > g max , there is no minimum in the energy Eq. (8), in this case, the atomic interaction of the condensate is so strong that the system becomes highly repulsive and the condensate is completely diffused. When g < g max , there exists a minimum energy state and the system is in the bound state. While g crit divides the bound state into two states: the stable bound state (g < g crit ) in which the condensate is completely confined in the trap and the quasibound state (g crit < g < g max ) in which the tunnelling occurs. For both two configurations, the values of g max and g crit increase with the peak height of the barrier V 0 increasing. This means that the higher the V 0 is, the less likely the condensate is to diffuse and the more stable the system is. Different from the linear growth of g max with V 0 , the growth of g crit with V 0 is inhibited when V 0 increases to a certain extent. The reason is that, at a certain V 0 , strong nonlinearity makes the effective potential of the system form a barrier in the trap. This is equivalent to placing some particles in a shallower trap, making it easier for the particles to tunnelling and increasing the quasi-bound state region of the system. Furthermore, when V 0 increases to a certain extent, the quasi-bound state disappears, that is, excessive interaction causes the condensate to diffuse before it tunnelling. In this case, the peak height of the barrier V 0 is too high and its effect is equivalent to an infinite deep potential well, the condensate has only the stable bound state and the diffusion state, which is consistent with the dynamics of the condensate trapped in an infinite well. In addition, limited by the shape of the potential well, there is a minimum √ aσ
e
value of the peak height of the barrier V 0min (V 0min = 02 ). If V < V 0min , the potential well is too shallow to trap the condensate. As V 0 → V 0min , g max decreases rapidly and g max → g crit . In this case, the system is in diffusive state. Because the potential well gets shallower as σ0 increases, V 0min increases with the increase of the barrier waist σ0 . While, the effect of σ0 on g max and g crit is nonlinearly. Due to the potential well of the weak configuration (a = 60) is wider than that of the tight configuration (a = 90), the bound state region of the condensate is larger than that of the tight configuration, and the condensate is more stable in the weak configuration case. The stability diagram obtained by
Fig. 5. The normalized atom number N in the well as a function of time t under different g for different values of the potential-shape parameter obtained by numerical simulation of Eq. (2). V 0 = 140 and (a) σ0 = 0.4, a = 60; (b) σ0 = 0.4, a = 90; (c) σ0 = 0.8, a = 60; (d) σ0 = 0.8, a = 90.
the variational analysis (Fig. 4) is in good agreement with the full numerical simulation of Eq. (2) shown in Fig. 2. 4. Tunnelling and diffusion dynamics Now we discuss the tunnelling and diffusion dynamics of the condensate in the tilted shallow trap. To obtain an approximation for the time evolution of the condensate, the following Gaussian wavefunction is adopted:
ψ(x, t ) =
1
1 4
√
exp[−
(x − x0 (t ))2 i + β(t )(x − x0 (t ))2 2 2R (t )2
π R (t ) + i α (t ) + ik(t )(x − x0 (t ))],
(13) where R (t ), x0 (t ), β(t ), α (t ) and k(t ) are the width, the mass central, chirp, phase and momentum of the condensate, respectively. Fig. 5 displays the normalized atom number N in the well as a function of t under different g for different potential-shape obtained by numerical simulation of Eq. (2) with initial wavefunction of Eq. (13). Here we use the absorption boundary condition on the outside of the potential well, qualitatively, such boundary condition imply that the particles are directly pushed away when they escape from the trapping potential. The absorption boundary condition is implemented as an imaginary potential energy term with the strength linearly increasing with |x|. As shown in the Fig. 5, when g < g crit , the system is in the stable bound state and the condensate is trapped in the potential well, in this case, the normalized atomic number is conserved N ≈ 1; when g > g max , because of the strong interaction, the condensate diffuses and the normalized atom number decreases dramatically; when g crit < g < g max , the system is in the quasi-bound state and tunnelling occurs, in this region, the normalized atom number decreases gradually and is not conserved. It is worth noting that, in the weak configuration (see Figs. 5(a) and (c)), g crit and g max are relatively large, so the system is less likely to occur tunnelling (i.e. the tunnelling rate is low) and more likely to be stable. In the diffusion state, there are always some particles left in the well besides the ones that are diffused out. The number of particles left
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Ł=
i
5
1
2
˙ ∗ − ψ˙ ∗ ψ) − (− ψ ∗ (ψψ 2
∂2 1 ψ + V (x)|ψ|2 + g |ψ|4 ). (14) 2 ∂ x2
Substituting the trail wavefunction Eq. (13) into the Lagrangian density, the effective Lagrangian is calculated by integrating the Lagrangian density as L eff = Łdx. After some straightforward algebra, the effective Lagrangian takes the form
˙ (t ) − L eff = k(t )˙x0 (t ) − α 1
1
4
R 2 (t )
− (2k2 (t ) + − (ax0 (t ) +
1 4
˙ t) R 2 (t )β(
+ R 2 (t )β 2 (t )) − √
g (15)
2 2π R (t )
V 0σ
σ + 2R 2 (t ) 2
e
2x2 (t ) − 2 0 2 σ +2R (t )
).
In addition, in order to better quantify the dynamic characteristics of the system, we analyze the evolution of the energy and the tunnelling rate over time. The energy of the system can be written as:
E (t ) = Fig. 6. The time evolution of the wave packets. (a) and (b) are the stable bound state, (c) and (d) are the quasi-bound state, (e) and (f) are the diffusion state. The inserts are the enlargement of the wave packets. V 0 = 140, σ0 = 0.4.
in the well depends very closely on the shape of the well, in the weak configuration case, due to the well is wider, there are more particles left in the well. The tunnelling characters are also well illustrated in Fig. 2. The tunnelling and diffusion dynamics of the system can also be revealed by the wave packets dynamics of the system. The time evolution of the wave packets in the stable bound state, in the quasi-bound state and in the diffusion state are shown in Fig. 6. In the stable bound state, as shown in Figs. 6(a) and (b), the wave packet preserves its initial Gaussian shape for a limited time (t = 300), that is, the condensate is completely trapped in the potential well. In the quasi-bound state (Figs. 6(c) and (d)) and the diffusion state (Figs. 6(e) and (f)), it can be clearly seen that the particles escape from the potential well and the wave packet gradually decreases with time, and the decay rate in the diffusion state is more significant than that of the quasi-bound state. This is due to the fact that in the diffusion state, the decay of the condensate has a transition from the classical spill to quantum tunnelling and the classical spill time is very short [24]. Therefore, we only observed an obvious overflow of wave packets from the potential well at t = 1 in Figs. 6(e) and (f). With the evolution of time, the interaction between particles decreased, and the particles in the system gradually changed from the initial overflow barrier to the tunnelling. Interestingly, during the escaping process, the initial Gaussian-shape wave packet is deformed slightly in the quasi-bound state but seriously in the diffusion state, i.e., the wave packet front edge in the escaping direction is deformed slightly in the quasi-bound state but sharply in the diffusion state. Fig. 6 shows that the dynamics of the wave packets in different states behave significant different characters. 5. Parametrically modulated tunnelling dynamics Now we address the tunnelling dynamics in the parametrically modulated trap with the barrier waist varies periodically with time, i.e. σ = σ0 (1 + cos( t )). The barrier waist has a timeaveraged σ0 , which is parametrically modulated with the strength and the frequency . The Lagrangian density for Eq. (2) is given by
+
1 4R 2 (t )
+ √
g
2 2π R (t )
V 0σ
σ 2 + 2R 2 (t )
e
−
+ ax0 (t ) (16)
2x2 (t ) 0
σ 2 +2R 2 (t ) .
The Euler-Lagrange equations for this effective Lagrangian are given by
d ∂ L eff
∂ L eff , ∂ q(t )
=
dt ∂ q(˙t )
(17)
where q(t ) represents β(t ), α (t ), k(t ), R (t ) and x0 (t ). After some straightforward algebra, the variables β(t ), α (t ) and k(t ) from these equations can be eliminated and the following second-order differential equation for the evolution of the width R (t ) and the mass central x0 (t ) of the condensate are obtained:
d2 R (t ) dt 2
=
1 R 3 (t )
+√
g 2π R 2 (t )
(18a)
2x2 (t )
4V 0 σ R (t )(σ 2 + 2R 2 (t ) − 4x20 (t )) − 2 0 2 + e σ +2R (t ) , (σ 2 + 2R 2 (t ))5/2 d2 x0 (t )
=
dt 2
4V 0 σ x0 (t )
(σ 2 + 2R 2 (t ))3/2
e
−
2x2 (t ) 0
σ 2 +2R 2 (t )
− a.
(18b)
To show the influence of the external modulation on the tunnelling dynamics, we discuss the dipolar mode of Eq. (18a) and Eq. (18b). Assuming σ = σ0 , replacing R (t ) with R (t ) = R 0 + δ R (t ) and x0 (t ) with x0 (t ) = x0 + δ x0 (t ), where R 0 and x0 are the stable equilibrium states of Eq. (9a)–(9b), δ R (t ) and δ x0 (t ) are the perturbations of R 0 and x0 , and keeping the terms of the first order of δ R (t ) and δ x0 (t ), one may obtain
d2 (δ R (t )) dt 2 2 d (δ x0 (t )) dt 2
= −ω1 δ R (t ) − ω2 δ x0 (t ),
(19a)
= −ω3 δ R (t ) − ω4 δ x0 (t ),
(19b)
where
ω1 =
3 4
R0
+
2g
3√
R 0 2π
4V 0 σ0 s exp[−
−
2
2
2R 0 +σ02 9
(2R 0 + σ02 ) 2
2
ω2 =
2x20
16R 0 V 0 σ0 x0 (6R 0 + 3σ02 − 4x20 ) exp[− 2
7
(2R 0 + σ02 ) 2
] (20a)
, 2x20 2
2R 0 +σ02
] ,
(20b)
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Fig. 7. The dipolar mode 0 of the condensate as a function of g for different potential-shape parameter V 0 and σ0 .
2x20
2
ω3 =
8R 0 V 0 σ0 x0 (6R 0 + 3σ02 − 4x20 ) exp[− 2
7
(2R 0 + σ02 ) 2 2
ω4 =
2
2R 0 +σ02
4V 0 σ0 (4x20 − 2R 0 − σ02 ) exp[− 2
5
(2R 0 + σ02 ) 2
2x20 2
2R 0 +σ02
Fig. 8. The average tunnelling rate γ and the maximum Energy difference E max of the system as a function of the modulating frequency / 0 in the stable bound state (g = g crit − 0.5| g crit | < g crit ) under weak (a = 60) and tight (a = 90) configurations. Solid line represents the average tunnelling rate γ ; Solid dot line represents E max given by numerical solutions of Eq. (2); dash dot line represents E max of analytical results Eq. (16). V 0 = 140, = 0.1.
] (20c)
,
] (20d)
,
and 6
4
4
s = −16R 0 − 12R 0 σ02 + σ06 + 8R 0 x20 2
(21)
2
+ 32R 0 σ02 x20 − 4σ04 x20 − 32R 0 x40 .
After some algebra calculation, the lowest dipolar mode 0 is given by
0 =
(ω1 + ω4 ) −
(ω1 + ω4 )2 − 4(ω1 ω4 − ω2 ω3 ) 2
.
(22)
The width of the trap varies periodically due to periodically modulating of the barrier waist σ . The condensate feels a periodic inertial force in the frame comoving with the trap. If the external drive frequency approaches the dipolar mode 0 , the resonant oscillation of the dipolar mode occurs, which makes the whole system unstable and the condensate quickly escape from the potential barrier, i.e., the resonant instability occurs. In Fig. 7, the dipolar mode 0 as a function of g for different values of the potential-shape parameter in the case of the weak configuration (a = 60) and the tight configuration (a = 90) respectively is shown. As can be seen from the Fig. 7, the dipole mode 0 decreases with the increase of g, which indicates that the stronger the interaction is, the weak stable the system is, the system is more prone to the resonant instability. In addition, 0 in the tight configuration is larger than that in the weak configuration and the tight configuration has a great influence on 0 . Figs. 7(a) and (b) show the effect of different peak heights of barrier V 0 on the dipole mode 0 with the fixed barrier waist σ0 = 0.4. As shown in the figures, in the case of fixed g, 0 increases with the increase of V 0 , which indicates that a deeper potential well requires a larger modulation frequency to excite the resonant instability. Furthermore, Figs. 7(c) and (d) show the effect of different barrier waist σ0 on 0 with fixed V 0 = 140. In these cases, when fix g,
Fig. 9. Same as in Fig. 8 but for the quasi-bound state (g = g crit + 0.5| g crit | > g crit ).
0 decreases with the increase of σ0 , due to the potential well becomes shallower as σ0 increases, and the system is more likely to excite the resonant instability. Figs. 8 and 9 show the average tunnelling rate γ and the maximum energy difference E max = max( E (t ) − E 0 ), i.e., the maximum difference between the energy E (t ) of the modulated system and the energy E 0 of the bound state in a period of time (t = 300), as a function of the modulating frequency / 0 in the stable bound state and quasi-bound state, respectively. Both the variational results given by Eq. (16) and the numerical simulations of Eq. (2) are shown. It is clear that the two graphs give a detailed comparison of γ and E max of the system under weak and tight configurations and the analytical results are confirmed by the direct numerical simulations. As shown in Fig. 8, in the stable bound state (g = g crit − 0.5| g crit | < g crit ), when / 0 = n, here n is an integer, the tunnelling rate is relatively small, but larger than that without modulation ( / 0 = 0), which is caused by the heating effect of modulation. However, when the modulating frequency approaches an integer multiple of the dipolar mode 0 , i.e., / 0 → n, the resonant escape occurs, the whole system is unstable. At this time, γ and E max suddenly increase,
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but the heating effect caused by the modulation will accelerate the escape of the particles. The time evolution diagram (Figs. 10(d) and 11(d)) of N clearly shows this acceleration effect. The results of Figs. 7–11 reveal that, the lowest dipole mode 0 plays important role for the stability and escapes dynamics of the modulated system. 6. Summary
Fig. 10. The time evolution of the wave packets (a)-(c) and the normalized atom number N (d) for different modulating frequency / 0 in the stable bound state. The inserts are the enlargement of the wave packets. V 0 = 140, a = 60, σ0 = 0.4, g = 6.4 < g crit .
In this work we have investigated the ground state and the dynamical properties of the one-dimensional BEC in the tilted shallow trap. By using the variational method, we calculated two critical values of the nonlinearity g crit and g max , which divide the system into three states: the stable bound state (g < g crit ) in which the condensate is completely trapped in the potential well, the quasi-bound state (g crit < g < g max ) in which the condensate exists tunnelling phenomenon, and the diffusion state (g > g max ) in which the condensate is pushed out over the top of the potential well. The stability diagram of the system, the evolution of particles with time and the average tunnelling rate γ of the system for different values of the potential-shape parameter are obtained. Due to the potential well in the tight configuration is narrower than that in the weak configuration, the condensate is more likely to undergo tunnelling in the tight case. The condensate dynamics in the parametrically modulated trap with the barrier waist varies periodically with time is also studied. Applying the variational approximation, the dipolar mode 0 of the system is obtained. By analyzing γ and E max at different modulation frequencies, we find that when → n 0 , the resonant tunnelling occurs and the whole system is unstable. Our results provide a theoretical basis for studying and controlling the tunnelling dynamics of ultracold atomic system. These predictions can be verified experimentally. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements
Fig. 11. Same as in Fig. 10 but for the quasi-bound state. V 0 = 140, a = 60, g crit < g = 19.1 < g max .
σ0 = 0.4,
This work is supported by the National Natural Science Foundation of China under Grants No. 11764039, 11847304, 11865014, 11475027, 11305132, and 11274255, by Natural Science Foundation of Gansu Province under Grant No. 17JR5RA076, and by Scientific research project of Gansu higher education under Grand No. 2016A-005. References
the system undergoes resonant heating, leading to the escape of the condensate. Fig. 9 shows the case in the quasi-bound state (g = g crit + 0.5| g crit | > g crit ). Similar to the results shown in Fig. 8, the resonant escape occurs when / 0 → n. For both Figs. 8 and 9, the system exhibits more complex escape dynamics after the modulation frequency is greater than 0 . In this case, the resonance point of / 0 is widened. In order to further demonstrate the resonance heating effect, we draw the time evolution diagrams of the wave packets and the normalized atom number N in the stable bound and the quasibound state respectively, as shown in the Figs. 10 and 11. When
/ 0 → 1 (Figs. 10(b) and 11(b)), the system undergoes resonant heating and the resonant oscillation of the dipolar mode occurs. The strong heating effect causes the wave packet to break, which makes the whole system unstable and the condensate quickly escapes from the potential barrier, i.e., the resonant instability occurs. Beyond that, the wave packet of the system does not break,
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A.J. Leggett, Rev. Mod. Phys. 73 (2001) 307. R.W. Gurney, E.U. Condon, Phys. Rev. 33 (1929) 127. E. Merzbacher, Phys. Today 55 (2002) 44. D. Gatteschi, R. Sessoli, Angew. Chem., Int. Ed. Engl. 42 (2003) 268. E. Collini, C.Y. Wong, K.E. Wilk, P.M.G. Curmi, P. Brumer, G.D. Scholes, Nature (London) 463 (2010) 644. H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 84 (2000) 2917. K. Choi, G. Ryu, F. Yesilkoy, A. Chryssis, N. Goldsman, M. Dagenais, M. Peckerar, J. Vac. Sci. Technol. B 28 (2010) C6050. P.D. Dresselhaus, M.M. Elsbury, D. Olaya, C.J. Burroughs, S.P. Benz, IEEE Trans. Appl. Supercond. 21 (2011) 693. Zi-Fa Yu, Ai-Xia Zhang, Rong-An Tang, Hong-Ping Xu, Ji-Ming Gao, Ju-Kui Xue, Phys. Rev. A 95 (2017) 033607. Yue Jian, Ai-Xia Zhang, Cai-Xia He, Xiu-Ying Qi, Ju-Kui Xue, Phys. Rev. E 87 (2013) 053201. E. Kengne, W.M. Liu, Phys. Rev. E 98 (2018) 012204. Xiao-Fei Zhang, Qin Yang, Jie-Fang Zhang, X.Z. Chen, W.M. Liu, Phys. Rev. A 77 (2008) 023613. T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, J. Schmiedmayer, Nature (London) 545 (2017) 323.
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[14] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80 (2008) 885. [15] Y. Shin, M. Saba, T.A. Pasquini, W. Ketterle, D.E. Pritchard, A.E. Leanhardt, Phys. Rev. Lett. 92 (2004) 050405. [16] D.S. Petrov, C. Salomon, G.V. Shlyapnikov, Phys. Rev. Lett. 93 (2004) 090404. [17] M. Albiez, R. Gati, J. Föling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Phys. Rev. Lett. 95 (2005) 010402. [18] S. Levy, E. Lahoud, I. Shomroni, J. Steinhauer, Nature (London) 449 (2007) 579. [19] L.J. LeBlanc, A.B. Bardon, J. McKeever, M.H.T. Extavour, D. Jervis, J.H. Thywissen, F. Piazza, A. Smerzi, Phys. Rev. Lett. 106 (2011) 025302. [20] L.D. Carr, M.J. Holland, B.A. Malomed, J. Phys. B 38 (2005) 3217.
[21] [22] [23] [24]
S.K. Adhikari, J. Phys. B, At. Mol. Opt. Phys. 38 (2005) 579. P. Schlagheck, T. Paul, Phys. Rev. A 73 (2006) 023619. N. Moiseyev, L.S. Cederbaum, Phys. Rev. A 72 (2005) 033605. S. Potnis, R. Ramos, K. Maeda, L.D. Carr, A.M. Steinberg, Phys. Rev. Lett. 118 (2017) 060402. [25] D.A. Alcala, G. Urban, M. Weidemüller, L.D. Carr, Phys. Rev. A 98 (2018) 023619. [26] X. Zhao, D.A. Alcala, M.A. McLain, K. Maeda, S. Potnis, R. Ramos, A.M. Steinberg, L.D. Carr, Phys. Rev. A 96 (2017) 063601.
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The ground state and the tunnelling dynamics of the Bose-Einstein condensate in a tilted shallow trap Yue Jian, Xin Qiao, Yan-Chao Zhang, Ai-Xia Zhang, Ju-Kui Xue ∗ College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070, China
a r t i c l e
i n f o
Article history: Received 7 July 2019 Received in revised form 24 October 2019 Accepted 4 November 2019 Available online xxxx Communicated by V.A. Markel Keywords: Bose-Einstein condensate Stability Tunnelling dynamics
a b s t r a c t The ground state and the tunnelling dynamics of the Bose-Einstein condensate (BEC) loaded in a tilted shallow trap is studied analytically and numerically. The stable bound state, the quasi-bound state and the diffusion state are predicted. The thresholds for transition between the different states are obtained and the stability diagram in parameter space is presented. The tunnelling dynamics of the system in different states is revealed. The shape of the potential well and the atomic interaction play important role and have coupled effect on the tunnelling dynamics of the system. Furthermore, the resonant tunnelling phenomenon in the parametrically modulated shallow trap is observed. The results show that when the modulating frequency approaches the dipolar mode of the system, resonant tunnelling occurs and the whole system is unstable. Our results provide a theoretical evidence for studying the tunnelling dynamics of the ultracold atomic system. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Macroscopic quantum tunnelling is the tunnelling of a manybody wavefunction through a potential barrier which is forbidden by classical analysis [1]. The escape of a particle due to tunnelling from a quasi-bound state, studied in α decay and nuclear fission and fusion, is one of the significant and earliest problems studied in quantum mechanics [2]. The shallow well is the most basic model to study the tunnelling dynamics which was originally applied to the study of chiral isomers [3] and the researches show that the probability to tunnel through an energy barrier is exponentially dependent on the barrier area. Since its inception, tunnelling plays an important role across physics, chemistry, and technology [4–6]. For example, as electronic devices reach the nanoscale, quantum tunnelling will play a more and more important role in developing nanoelectronics, such as tunnelling diodes [7]. Furthermore, with the advent of Josephson junctions, we can now measure voltage with unprecedented accuracy by the epitome of macroscopic quantum devices based on tunnelling [8]. At very low temperatures, Bose-Einstein condensate (BEC) is the ideal scenario to exhibit purely quantum phenomena [9–12]. A series of fundamental experiments on tunnelling of BEC have been carried out in recent years as the techniques for preparing and manipulating ultracold atoms become mature [13–16]. Such as
*
Corresponding author. E-mail address: [email protected] (J.-K. Xue).
https://doi.org/10.1016/j.physleta.2019.126126 0375-9601/© 2019 Elsevier B.V. All rights reserved.
Josephson oscillations and self trapping [17], the dc and ac Josephson effect [18], the crossover from hydrodynamic and Josephson regimes [19]. Experimentally, the shallow well can be achieved by reducing the laser intensity on optical trapping and/or by reducing the electric current in magnetic trapping. It is shown that possible bound state can exist in the shallow well. However, experiments largely study Landau-Zener tunnelling out of an optical lattice, and studying tunnelling from a bound state into the continuum are fewer. In addition, the studies of tunnelling are usually concentrated in symmetric shallow wells [20–23]. Nevertheless, due to the influence of magnetic field gradient and gravity, tilted shallow trap is easier to generate experimentally. Therefore, it is of more general significance to study the tunnelling dynamics of BEC in tilted trap. In particular, a recent experiment about the macroscopic quantum tunnelling in a tilted shallow trap [24–26] reveals that the tunnelling rate is an experimentally measurable parameter and containing important information about the atomic dynamics inside the trap as well as the initial state of the quantum system. The results show that the nonlinearities due to the atomic interaction and the macroscopic nature of BEC create a highly nonexponential decay and open up the hitherto unexplored experimental regime of tunnelling from a single trapping well into the continuum. Motivated by the experimental observation [24], we study the ground state and its stability of BEC in the tilted shallow trap, and provide a deep quantitative understanding of the tunnelling dynamics of the system. By using the variational analysis and numerical stimulation, the stability of the ground state of BEC in the
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Fig. 1. The sketch of the model. V 0 = 140, a = 60,
σ0 = 0.4.
tilted shallow potential well is discussed theoretically. The stable bound state, the quasi-bound state and the diffusion state are predicted. The thresholds for transition between the different states and the stability diagram in parameter space are presented. The tunnelling dynamics of the system in different states is discussed. In addition, the resonant tunnelling phenomenon in the parametrically modulated shallow trap is further discussed. The results show that when the modulating frequency approaches the dipolar mode of the system, resonant tunnelling occurs and the whole system is unstable. This paper is organized as follows. In Section 2, we present the physical model and the numerically simulated tunnelling diagram. In Section 3, the ground state phase diagram and the stability of the ground state are discussed. In Section 4, the tunnelling and diffusion dynamics of the system are discussed. In Section 5, by applying the variational approximation and the numerical simulation, the tunnelling dynamics of the condensate in the parametrically modulated tilted shallow trap is investigated. Finally, in Section 6, a brief summary is given.
where ψ(x) is the mean-field wavefunction and the total number of atoms N satisfies the normalization |ψ(x)|2 dx = N. g is the nonlinearity which comes from the interaction between the particles and may be either positive (repulsive) or negative (attractive). Here length and time are expressed in units of l and (ω )−1 , respectively. For the shallow trap, it is particularly important to study the tunnelling properties of the condensate. The number of atoms in the potential well decreases with time according to
2. The model and the tunnelling diagram
dN
Fig. 2. The average tunnelling rate γ of the condensate as a function of g for different values of the potential-shape parameter given by numerical simulation of Eq. (2). The horizontal short dotted line at the bottom represents the boundary between the stable bound state and the quasi-bound state obtained by numerical simulation. Symbols represent the critical nonlinearities g crit and g max calculated by the variational method.
(−
dt We study the stability and tunnelling dynamics of the onedimensional 87 Rb BEC in a tilted shallow trap based on the experimental realization [24]. The trap is composed of a quadrupole magnetic field and a blue detuned light sheet. The magnetic trap provides harmonic confinement in the horizontal direction with trapping frequency ω . Due to a magnetic field gradient and gravity, there is a tilt in the vertical direction with a constant acceleration g eff . The one-dimensional tilted shallow trap in the vertical 2
− 2x
direction has the form V ( x) = mg eff x+ V 0 e σ0 , where m is the mass of an atom, g eff is the acceleration and represents the tilt of the potential well, V 0 is the peak height of the barrier and σ0 is the barrier waist. In units of h¯ ω , we reduce the potential well to the form 2
V (x) = ax + V 0 e
− 2x σ
0
,
(1)
where x = x/l, a = mg effl/¯hω , V 0 = V 0 /¯hω , and σ0 = σ0 /l with l = h¯ /mω . The tilted trap is shown in Fig. 1. In general, the GP equation is a sufficient theoretical tool for describing the ground state and dynamics of BEC at 0 K. So we begin with the time-dependent mean-field GP equation used to describe BEC. The one-dimensional GP equation for the BEC wavefunction ψ can be written in dimensionless form
1 ∂2 2 ∂2x
+ V (x) + g |ψ(x)|2 )ψ(x) = i
∂ ψ(x), ∂t
= −γ ( N ) N ,
(3)
where, the coefficient
γ (N ) = −
d ln N dt
(2)
.
γ is the tunnelling rate. Therefore (4)
In order to clearly show the tunnelling dynamics of the system, we solved the Eq. (2) numerically. Fig. 2 shows the change of the average tunnelling rate γ of the system with the nonlinearity g for different trap parameters. To facilitate numerical simulation, the tunnelling rate γ of the condensate is calculated by the average change rate of the logarithm of the normalized atomic number in a certain period of the evolution time. The evolution time of the condensate is t = 300, which is equivalent to about 1s in the actual experiment in the dimensional form. As shown in Fig. 2, the tunnelling of the condensate is closely related to the peak height of the barrier V 0 , i.e., γ decreases with the increase of V 0 . Because the narrower the potential well is, the more likely the condensate is to tunnelling, so γ increases much sharply with g and is much larger in the tight configuration (a = 90) than that in the weak configuration (a = 60). Since the influence of the barrier waist σ0 on tunnelling is nonlinearly, so the γ in the shallower potential well (σ0 = 0.8) is smaller than that in the wide potential well (σ0 = 0.4) in the case of a = 60. Interestingly, Fig. 2 indicates that the average tunnelling rate presents an obvious step structure as g changes, which clearly shows that the system has three
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states: the stable bound state, the quasi-bound state and the diffusion state. In the stable bound state, γ → 0, the condensate is all trapped in the potential well. In the quasi-bound state, there exits tunnelling phenomenon and γ increases quickly with the increase of g. In the diffused state, the condensate quickly spills out of the potential well and γ reaches its maximum. The transition from the stable bound state to the quasi-bound state occurs when γ → 0. Since the tunnelling rate of the system can be infinitely small, we can only qualitatively give a critical point from the numerical simulation, γc ≈ 10−4 , which is obtained by fitting the platform in the numerical simulation in Fig. 2, as shown by the horizontal short dotted line at the bottom of the figure. The average tunnelling rate below the critical line changes very slowly and basically approaches 0, so we regard it as the stable bound state. Above the critical line, the particles of the condensate can be considered to tunnelling. In the following sections, the existence and the dynamics of the three states will be discussed theoretically and numerically. 3. The ground state and its stability The stationary solution of the one-dimensional GP Eq. (2) has the form (x, t ) = ψ(x)e (−i μt ) , so that
(−
1 ∂2 2 ∂2x
+ V (x) + g |ψ(x)|2 )ψ(x) = μψ(x),
(5)
where μ is the chemical potential. To understand the stability of the ground state in the tilted shallow trap, we employ a variational method with the following Gaussian wave function for the solution of Eq. (5)
ψ(x) =
1
π
1 4
√ e
−
(x−x0 )2 2R 2
(6)
,
R
where R and x0 are the width and the central of the mass of the condensate, respectively. The ground state energy of the system is given by the functional
E=
1
[− ψ(x)∗ 2
∂2 1 ψ(x) + V (x)|ψ(x)|2 + g |ψ(x)|4 ]dx. 2 ∂ x2
(7)
Substituting the trial wave function (6) into the ground state energy (7), we obtain
E=
1 4R 2
+ √
g
2 2π R
+
V 0 σ0
e
−
2x2 0
σ02 +2R 2
σ02 + 2R 2
+ ax0 .
(8)
1 2R 3 a−
+ √
g
2 2π R 2
4V 0 σ0 x0 3
(σ02 + 2R 2 ) 2
+
e
2V 0 σ0 R (σ02 + 2R 2 − 4x20 )
2x2 − 2 0 2 σ0 +2R
5
(σ02 + 2R 2 ) 2
e
Figs. 3(a) and (b) show the energy of the system E versus the width of the wave packet R for different values of g. As can be seen from the figures, g max is an obvious boundary for the existence of minimum energy. In order to further reveal the diffusion mechanism of the system, we compare the relationship between the tilted shallow trap V (x) and the effective potential V eff ≡ V (x) + g |ψ(x)|2 of the system, as shown in Figs. 3 (c) and (d). When g > g max , V eff is obviously higher than that of the potential well, that is, the particles will overflow from the potential well. When g < g max , V eff is contained in the potential well, the system is in bound state. Fig. 2 indicates that, unlike an infinite well, the condensate can still exist tunnelling phenomenon in the bound state. That is, there is a critical nonlinearity termed g crit , when g < g crit , the condensate is trapped in potential well and the system is in the stable bound state; when g crit < g < g max , the condensate exists tunnelling phenomenon and the system is in the quasi-bound state. The g crit can be determined by the variational WKB approach [20]. The WKB tunnelling rate γ is given by the standard expressions:
x2
The ground state of the system can be obtained by minimizing the energy Eq. (8) with respect to R and x0 , 2x2 − 2 0 2 σ0 +2R
Fig. 3. (a) and (b) The energy of the system E versus the width of the wave packet R for different values of g. (c) and (d) The tilted shallow trap V (x) (Solid line) and the effective potential V eff (dash lines) versus x for different values of g. V 0 = 140, σ0 = 0.4.
γ = ν exp(−2 p (x) ≡
= 0 (9a)
ν −1 ≡ 2
(9b)
So, from Eqs. (9a)–(9b), we can find a critical nonlinearity called the maximum nonlinearity g max which divides the system into two states, one is the bound state (g < g max ) and another one is the diffusion state (g > g max ). The condensate has a minimum energy in the bound state and can be stabilized in the potential well, while the condensate does not have a minimum energy in the diffusion state and is pushed out over the top of the potential well. This is clearly illustrated in Fig. 3.
(10)
2(μ − V eff ) x3
(11)
dy
x2
= 0.
| p (x)|dx) x1
(12)
| p (x)|
where the limits of the integration x1 , x2 and x3 are found from setting the semiclassical momentum p (x) = 0. x2 − x1 is the barrier length through which the wavefuction must tunnel and
2
μ = [− 12 ψ(x)∗ ∂∂x2 ψ(x) + 2 4 V (x)|ψ(x)| + g |ψ(x)| ]dx is the chemical potential. ν −1 is the
x3 − x2 is the length between the trap.
oscillation frequency of the particles in the trap. The transition from the stable bound state to the quasi-bound state occurs when γ → 0. Since the tunnelling rate of the system can be infinitely small, here we take γc ≈ 10−4 as the boundary, which is obtained by fitting the platform in the numerical simulation in Fig. 2, as
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Fig. 4. The stability diagram in V 0 - g plane for the weak configuration (a) and the tight configuration (b) with different values of the potential-shape parameter.
shown by the horizontal short dotted line at the bottom of the figure. When γ > γc , it can be considered that the tunnelling takes place. According to Eqs. (9)–(12), the stability diagram of the system in V 0 - g plane for the weak configuration (a) and the tight configuration (b) is clearly depicted in Fig. 4. It is clear that given the values of the potential-shape parameter V 0 and σ0 , there exists two critical nonlinearities g crit (lower curves) and g max (upper curves), which divide the V 0 - g plane into three regions: the stable bound state region (g < g crit ), the quasi-bound state region (g crit < g < g max ) and the diffusion region (g > g max ). When g > g max , there is no minimum in the energy Eq. (8), in this case, the atomic interaction of the condensate is so strong that the system becomes highly repulsive and the condensate is completely diffused. When g < g max , there exists a minimum energy state and the system is in the bound state. While g crit divides the bound state into two states: the stable bound state (g < g crit ) in which the condensate is completely confined in the trap and the quasibound state (g crit < g < g max ) in which the tunnelling occurs. For both two configurations, the values of g max and g crit increase with the peak height of the barrier V 0 increasing. This means that the higher the V 0 is, the less likely the condensate is to diffuse and the more stable the system is. Different from the linear growth of g max with V 0 , the growth of g crit with V 0 is inhibited when V 0 increases to a certain extent. The reason is that, at a certain V 0 , strong nonlinearity makes the effective potential of the system form a barrier in the trap. This is equivalent to placing some particles in a shallower trap, making it easier for the particles to tunnelling and increasing the quasi-bound state region of the system. Furthermore, when V 0 increases to a certain extent, the quasi-bound state disappears, that is, excessive interaction causes the condensate to diffuse before it tunnelling. In this case, the peak height of the barrier V 0 is too high and its effect is equivalent to an infinite deep potential well, the condensate has only the stable bound state and the diffusion state, which is consistent with the dynamics of the condensate trapped in an infinite well. In addition, limited by the shape of the potential well, there is a minimum √ aσ
e
value of the peak height of the barrier V 0min (V 0min = 02 ). If V < V 0min , the potential well is too shallow to trap the condensate. As V 0 → V 0min , g max decreases rapidly and g max → g crit . In this case, the system is in diffusive state. Because the potential well gets shallower as σ0 increases, V 0min increases with the increase of the barrier waist σ0 . While, the effect of σ0 on g max and g crit is nonlinearly. Due to the potential well of the weak configuration (a = 60) is wider than that of the tight configuration (a = 90), the bound state region of the condensate is larger than that of the tight configuration, and the condensate is more stable in the weak configuration case. The stability diagram obtained by
Fig. 5. The normalized atom number N in the well as a function of time t under different g for different values of the potential-shape parameter obtained by numerical simulation of Eq. (2). V 0 = 140 and (a) σ0 = 0.4, a = 60; (b) σ0 = 0.4, a = 90; (c) σ0 = 0.8, a = 60; (d) σ0 = 0.8, a = 90.
the variational analysis (Fig. 4) is in good agreement with the full numerical simulation of Eq. (2) shown in Fig. 2. 4. Tunnelling and diffusion dynamics Now we discuss the tunnelling and diffusion dynamics of the condensate in the tilted shallow trap. To obtain an approximation for the time evolution of the condensate, the following Gaussian wavefunction is adopted:
ψ(x, t ) =
1
1 4
√
exp[−
(x − x0 (t ))2 i + β(t )(x − x0 (t ))2 2 2R (t )2
π R (t ) + i α (t ) + ik(t )(x − x0 (t ))],
(13) where R (t ), x0 (t ), β(t ), α (t ) and k(t ) are the width, the mass central, chirp, phase and momentum of the condensate, respectively. Fig. 5 displays the normalized atom number N in the well as a function of t under different g for different potential-shape obtained by numerical simulation of Eq. (2) with initial wavefunction of Eq. (13). Here we use the absorption boundary condition on the outside of the potential well, qualitatively, such boundary condition imply that the particles are directly pushed away when they escape from the trapping potential. The absorption boundary condition is implemented as an imaginary potential energy term with the strength linearly increasing with |x|. As shown in the Fig. 5, when g < g crit , the system is in the stable bound state and the condensate is trapped in the potential well, in this case, the normalized atomic number is conserved N ≈ 1; when g > g max , because of the strong interaction, the condensate diffuses and the normalized atom number decreases dramatically; when g crit < g < g max , the system is in the quasi-bound state and tunnelling occurs, in this region, the normalized atom number decreases gradually and is not conserved. It is worth noting that, in the weak configuration (see Figs. 5(a) and (c)), g crit and g max are relatively large, so the system is less likely to occur tunnelling (i.e. the tunnelling rate is low) and more likely to be stable. In the diffusion state, there are always some particles left in the well besides the ones that are diffused out. The number of particles left
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Ł=
i
5
1
2
˙ ∗ − ψ˙ ∗ ψ) − (− ψ ∗ (ψψ 2
∂2 1 ψ + V (x)|ψ|2 + g |ψ|4 ). (14) 2 ∂ x2
Substituting the trail wavefunction Eq. (13) into the Lagrangian density, the effective Lagrangian is calculated by integrating the Lagrangian density as L eff = Łdx. After some straightforward algebra, the effective Lagrangian takes the form
˙ (t ) − L eff = k(t )˙x0 (t ) − α 1
1
4
R 2 (t )
− (2k2 (t ) + − (ax0 (t ) +
1 4
˙ t) R 2 (t )β(
+ R 2 (t )β 2 (t )) − √
g (15)
2 2π R (t )
V 0σ
σ + 2R 2 (t ) 2
e
2x2 (t ) − 2 0 2 σ +2R (t )
).
In addition, in order to better quantify the dynamic characteristics of the system, we analyze the evolution of the energy and the tunnelling rate over time. The energy of the system can be written as:
E (t ) = Fig. 6. The time evolution of the wave packets. (a) and (b) are the stable bound state, (c) and (d) are the quasi-bound state, (e) and (f) are the diffusion state. The inserts are the enlargement of the wave packets. V 0 = 140, σ0 = 0.4.
in the well depends very closely on the shape of the well, in the weak configuration case, due to the well is wider, there are more particles left in the well. The tunnelling characters are also well illustrated in Fig. 2. The tunnelling and diffusion dynamics of the system can also be revealed by the wave packets dynamics of the system. The time evolution of the wave packets in the stable bound state, in the quasi-bound state and in the diffusion state are shown in Fig. 6. In the stable bound state, as shown in Figs. 6(a) and (b), the wave packet preserves its initial Gaussian shape for a limited time (t = 300), that is, the condensate is completely trapped in the potential well. In the quasi-bound state (Figs. 6(c) and (d)) and the diffusion state (Figs. 6(e) and (f)), it can be clearly seen that the particles escape from the potential well and the wave packet gradually decreases with time, and the decay rate in the diffusion state is more significant than that of the quasi-bound state. This is due to the fact that in the diffusion state, the decay of the condensate has a transition from the classical spill to quantum tunnelling and the classical spill time is very short [24]. Therefore, we only observed an obvious overflow of wave packets from the potential well at t = 1 in Figs. 6(e) and (f). With the evolution of time, the interaction between particles decreased, and the particles in the system gradually changed from the initial overflow barrier to the tunnelling. Interestingly, during the escaping process, the initial Gaussian-shape wave packet is deformed slightly in the quasi-bound state but seriously in the diffusion state, i.e., the wave packet front edge in the escaping direction is deformed slightly in the quasi-bound state but sharply in the diffusion state. Fig. 6 shows that the dynamics of the wave packets in different states behave significant different characters. 5. Parametrically modulated tunnelling dynamics Now we address the tunnelling dynamics in the parametrically modulated trap with the barrier waist varies periodically with time, i.e. σ = σ0 (1 + cos( t )). The barrier waist has a timeaveraged σ0 , which is parametrically modulated with the strength and the frequency . The Lagrangian density for Eq. (2) is given by
+
1 4R 2 (t )
+ √
g
2 2π R (t )
V 0σ
σ 2 + 2R 2 (t )
e
−
+ ax0 (t ) (16)
2x2 (t ) 0
σ 2 +2R 2 (t ) .
The Euler-Lagrange equations for this effective Lagrangian are given by
d ∂ L eff
∂ L eff , ∂ q(t )
=
dt ∂ q(˙t )
(17)
where q(t ) represents β(t ), α (t ), k(t ), R (t ) and x0 (t ). After some straightforward algebra, the variables β(t ), α (t ) and k(t ) from these equations can be eliminated and the following second-order differential equation for the evolution of the width R (t ) and the mass central x0 (t ) of the condensate are obtained:
d2 R (t ) dt 2
=
1 R 3 (t )
+√
g 2π R 2 (t )
(18a)
2x2 (t )
4V 0 σ R (t )(σ 2 + 2R 2 (t ) − 4x20 (t )) − 2 0 2 + e σ +2R (t ) , (σ 2 + 2R 2 (t ))5/2 d2 x0 (t )
=
dt 2
4V 0 σ x0 (t )
(σ 2 + 2R 2 (t ))3/2
e
−
2x2 (t ) 0
σ 2 +2R 2 (t )
− a.
(18b)
To show the influence of the external modulation on the tunnelling dynamics, we discuss the dipolar mode of Eq. (18a) and Eq. (18b). Assuming σ = σ0 , replacing R (t ) with R (t ) = R 0 + δ R (t ) and x0 (t ) with x0 (t ) = x0 + δ x0 (t ), where R 0 and x0 are the stable equilibrium states of Eq. (9a)–(9b), δ R (t ) and δ x0 (t ) are the perturbations of R 0 and x0 , and keeping the terms of the first order of δ R (t ) and δ x0 (t ), one may obtain
d2 (δ R (t )) dt 2 2 d (δ x0 (t )) dt 2
= −ω1 δ R (t ) − ω2 δ x0 (t ),
(19a)
= −ω3 δ R (t ) − ω4 δ x0 (t ),
(19b)
where
ω1 =
3 4
R0
+
2g
3√
R 0 2π
4V 0 σ0 s exp[−
−
2
2
2R 0 +σ02 9
(2R 0 + σ02 ) 2
2
ω2 =
2x20
16R 0 V 0 σ0 x0 (6R 0 + 3σ02 − 4x20 ) exp[− 2
7
(2R 0 + σ02 ) 2
] (20a)
, 2x20 2
2R 0 +σ02
] ,
(20b)
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Fig. 7. The dipolar mode 0 of the condensate as a function of g for different potential-shape parameter V 0 and σ0 .
2x20
2
ω3 =
8R 0 V 0 σ0 x0 (6R 0 + 3σ02 − 4x20 ) exp[− 2
7
(2R 0 + σ02 ) 2 2
ω4 =
2
2R 0 +σ02
4V 0 σ0 (4x20 − 2R 0 − σ02 ) exp[− 2
5
(2R 0 + σ02 ) 2
2x20 2
2R 0 +σ02
Fig. 8. The average tunnelling rate γ and the maximum Energy difference E max of the system as a function of the modulating frequency / 0 in the stable bound state (g = g crit − 0.5| g crit | < g crit ) under weak (a = 60) and tight (a = 90) configurations. Solid line represents the average tunnelling rate γ ; Solid dot line represents E max given by numerical solutions of Eq. (2); dash dot line represents E max of analytical results Eq. (16). V 0 = 140, = 0.1.
] (20c)
,
] (20d)
,
and 6
4
4
s = −16R 0 − 12R 0 σ02 + σ06 + 8R 0 x20 2
(21)
2
+ 32R 0 σ02 x20 − 4σ04 x20 − 32R 0 x40 .
After some algebra calculation, the lowest dipolar mode 0 is given by
0 =
(ω1 + ω4 ) −
(ω1 + ω4 )2 − 4(ω1 ω4 − ω2 ω3 ) 2
.
(22)
The width of the trap varies periodically due to periodically modulating of the barrier waist σ . The condensate feels a periodic inertial force in the frame comoving with the trap. If the external drive frequency approaches the dipolar mode 0 , the resonant oscillation of the dipolar mode occurs, which makes the whole system unstable and the condensate quickly escape from the potential barrier, i.e., the resonant instability occurs. In Fig. 7, the dipolar mode 0 as a function of g for different values of the potential-shape parameter in the case of the weak configuration (a = 60) and the tight configuration (a = 90) respectively is shown. As can be seen from the Fig. 7, the dipole mode 0 decreases with the increase of g, which indicates that the stronger the interaction is, the weak stable the system is, the system is more prone to the resonant instability. In addition, 0 in the tight configuration is larger than that in the weak configuration and the tight configuration has a great influence on 0 . Figs. 7(a) and (b) show the effect of different peak heights of barrier V 0 on the dipole mode 0 with the fixed barrier waist σ0 = 0.4. As shown in the figures, in the case of fixed g, 0 increases with the increase of V 0 , which indicates that a deeper potential well requires a larger modulation frequency to excite the resonant instability. Furthermore, Figs. 7(c) and (d) show the effect of different barrier waist σ0 on 0 with fixed V 0 = 140. In these cases, when fix g,
Fig. 9. Same as in Fig. 8 but for the quasi-bound state (g = g crit + 0.5| g crit | > g crit ).
0 decreases with the increase of σ0 , due to the potential well becomes shallower as σ0 increases, and the system is more likely to excite the resonant instability. Figs. 8 and 9 show the average tunnelling rate γ and the maximum energy difference E max = max( E (t ) − E 0 ), i.e., the maximum difference between the energy E (t ) of the modulated system and the energy E 0 of the bound state in a period of time (t = 300), as a function of the modulating frequency / 0 in the stable bound state and quasi-bound state, respectively. Both the variational results given by Eq. (16) and the numerical simulations of Eq. (2) are shown. It is clear that the two graphs give a detailed comparison of γ and E max of the system under weak and tight configurations and the analytical results are confirmed by the direct numerical simulations. As shown in Fig. 8, in the stable bound state (g = g crit − 0.5| g crit | < g crit ), when / 0 = n, here n is an integer, the tunnelling rate is relatively small, but larger than that without modulation ( / 0 = 0), which is caused by the heating effect of modulation. However, when the modulating frequency approaches an integer multiple of the dipolar mode 0 , i.e., / 0 → n, the resonant escape occurs, the whole system is unstable. At this time, γ and E max suddenly increase,
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but the heating effect caused by the modulation will accelerate the escape of the particles. The time evolution diagram (Figs. 10(d) and 11(d)) of N clearly shows this acceleration effect. The results of Figs. 7–11 reveal that, the lowest dipole mode 0 plays important role for the stability and escapes dynamics of the modulated system. 6. Summary
Fig. 10. The time evolution of the wave packets (a)-(c) and the normalized atom number N (d) for different modulating frequency / 0 in the stable bound state. The inserts are the enlargement of the wave packets. V 0 = 140, a = 60, σ0 = 0.4, g = 6.4 < g crit .
In this work we have investigated the ground state and the dynamical properties of the one-dimensional BEC in the tilted shallow trap. By using the variational method, we calculated two critical values of the nonlinearity g crit and g max , which divide the system into three states: the stable bound state (g < g crit ) in which the condensate is completely trapped in the potential well, the quasi-bound state (g crit < g < g max ) in which the condensate exists tunnelling phenomenon, and the diffusion state (g > g max ) in which the condensate is pushed out over the top of the potential well. The stability diagram of the system, the evolution of particles with time and the average tunnelling rate γ of the system for different values of the potential-shape parameter are obtained. Due to the potential well in the tight configuration is narrower than that in the weak configuration, the condensate is more likely to undergo tunnelling in the tight case. The condensate dynamics in the parametrically modulated trap with the barrier waist varies periodically with time is also studied. Applying the variational approximation, the dipolar mode 0 of the system is obtained. By analyzing γ and E max at different modulation frequencies, we find that when → n 0 , the resonant tunnelling occurs and the whole system is unstable. Our results provide a theoretical basis for studying and controlling the tunnelling dynamics of ultracold atomic system. These predictions can be verified experimentally. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements
Fig. 11. Same as in Fig. 10 but for the quasi-bound state. V 0 = 140, a = 60, g crit < g = 19.1 < g max .
σ0 = 0.4,
This work is supported by the National Natural Science Foundation of China under Grants No. 11764039, 11847304, 11865014, 11475027, 11305132, and 11274255, by Natural Science Foundation of Gansu Province under Grant No. 17JR5RA076, and by Scientific research project of Gansu higher education under Grand No. 2016A-005. References
the system undergoes resonant heating, leading to the escape of the condensate. Fig. 9 shows the case in the quasi-bound state (g = g crit + 0.5| g crit | > g crit ). Similar to the results shown in Fig. 8, the resonant escape occurs when / 0 → n. For both Figs. 8 and 9, the system exhibits more complex escape dynamics after the modulation frequency is greater than 0 . In this case, the resonance point of / 0 is widened. In order to further demonstrate the resonance heating effect, we draw the time evolution diagrams of the wave packets and the normalized atom number N in the stable bound and the quasibound state respectively, as shown in the Figs. 10 and 11. When
/ 0 → 1 (Figs. 10(b) and 11(b)), the system undergoes resonant heating and the resonant oscillation of the dipolar mode occurs. The strong heating effect causes the wave packet to break, which makes the whole system unstable and the condensate quickly escapes from the potential barrier, i.e., the resonant instability occurs. Beyond that, the wave packet of the system does not break,
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A.J. Leggett, Rev. Mod. Phys. 73 (2001) 307. R.W. Gurney, E.U. Condon, Phys. Rev. 33 (1929) 127. E. Merzbacher, Phys. Today 55 (2002) 44. D. Gatteschi, R. Sessoli, Angew. Chem., Int. Ed. Engl. 42 (2003) 268. E. Collini, C.Y. Wong, K.E. Wilk, P.M.G. Curmi, P. Brumer, G.D. Scholes, Nature (London) 463 (2010) 644. H.J. Choi, J. Ihm, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 84 (2000) 2917. K. Choi, G. Ryu, F. Yesilkoy, A. Chryssis, N. Goldsman, M. Dagenais, M. Peckerar, J. Vac. Sci. Technol. B 28 (2010) C6050. P.D. Dresselhaus, M.M. Elsbury, D. Olaya, C.J. Burroughs, S.P. Benz, IEEE Trans. Appl. Supercond. 21 (2011) 693. Zi-Fa Yu, Ai-Xia Zhang, Rong-An Tang, Hong-Ping Xu, Ji-Ming Gao, Ju-Kui Xue, Phys. Rev. A 95 (2017) 033607. Yue Jian, Ai-Xia Zhang, Cai-Xia He, Xiu-Ying Qi, Ju-Kui Xue, Phys. Rev. E 87 (2013) 053201. E. Kengne, W.M. Liu, Phys. Rev. E 98 (2018) 012204. Xiao-Fei Zhang, Qin Yang, Jie-Fang Zhang, X.Z. Chen, W.M. Liu, Phys. Rev. A 77 (2008) 023613. T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, J. Schmiedmayer, Nature (London) 545 (2017) 323.
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[14] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80 (2008) 885. [15] Y. Shin, M. Saba, T.A. Pasquini, W. Ketterle, D.E. Pritchard, A.E. Leanhardt, Phys. Rev. Lett. 92 (2004) 050405. [16] D.S. Petrov, C. Salomon, G.V. Shlyapnikov, Phys. Rev. Lett. 93 (2004) 090404. [17] M. Albiez, R. Gati, J. Föling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Phys. Rev. Lett. 95 (2005) 010402. [18] S. Levy, E. Lahoud, I. Shomroni, J. Steinhauer, Nature (London) 449 (2007) 579. [19] L.J. LeBlanc, A.B. Bardon, J. McKeever, M.H.T. Extavour, D. Jervis, J.H. Thywissen, F. Piazza, A. Smerzi, Phys. Rev. Lett. 106 (2011) 025302. [20] L.D. Carr, M.J. Holland, B.A. Malomed, J. Phys. B 38 (2005) 3217.
[21] [22] [23] [24]
S.K. Adhikari, J. Phys. B, At. Mol. Opt. Phys. 38 (2005) 579. P. Schlagheck, T. Paul, Phys. Rev. A 73 (2006) 023619. N. Moiseyev, L.S. Cederbaum, Phys. Rev. A 72 (2005) 033605. S. Potnis, R. Ramos, K. Maeda, L.D. Carr, A.M. Steinberg, Phys. Rev. Lett. 118 (2017) 060402. [25] D.A. Alcala, G. Urban, M. Weidemüller, L.D. Carr, Phys. Rev. A 98 (2018) 023619. [26] X. Zhao, D.A. Alcala, M.A. McLain, K. Maeda, S. Potnis, R. Ramos, A.M. Steinberg, L.D. Carr, Phys. Rev. A 96 (2017) 063601.