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JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
doi:10.1088/0953-4075/42/9/091004
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004 (5pp)
FAST TRACK COMMUNICATION
Efficient generation and properties of mesoscopic quantum superposition states in an attractive Bose–Einstein condensate threaded by a potential barrier Alexej I Streltsov, Ofir E Alon and Lorenz S Cederbaum Theoretische Chemie, Physikalisch-Chemisches Institut, Universit¨at Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany E-mail: [email protected], [email protected] and [email protected]
Received 1 April 2009 Published 27 April 2009 Online at stacks.iop.org/JPhysB/42/091004 Abstract The dynamics of a low-dimensional ultracold attractive Bose gas threaded by a potential barrier is studied by solving the time-dependent many-boson Schr¨odinger equation. It is shown that quantum superposition states of two (or more) distinct mesoscopic Bose clouds can be formed. We demonstrate the efficiency and robustness of this proposed preparation scheme and characterize the properties of the generated quantum superposition states. (Some figures in this article are in colour only in the electronic version)
Quantum dynamics of ultracold low-dimensional attractive Bose gases is an active research field attracting both experimentalists and theorists [1–7]. Here, much work has been devoted to studying and analysing the evolution in time of attractive Bose systems with the popular Gross–Pitaevskii equation. In particular, the Gross–Pitaevskii equation predicts the famous bright-soliton solution for the ground state and soliton trains for excited states. We recall that the Gross– Pitaevskii theory [8] is a mean-field theory which assumes all bosons to reside in a single quantum state or an orbital, i.e. the Bose system has to remain condensed throughout its evolution in time. Recently, by computing the dynamics of low-dimensional attractive Bose gases beyond Gross– Pitaevskii, the arsenal of physical phenomena which attractive Bose gases exhibit has been enriched [9–12]. In particular, low-lying fragmented excited states can conspire to give dynamically stable fragmented objects in low-dimensional attractive Bose gases. These fragmented objects are distinct from and generally lie lower in energy than solitons, which are condensed objects. 0953-4075/09/091004+05$30.00
In the above context, it has been shown by two independent groups [11, 12] that mesoscopic quantum superposition states can be generated by scattering an attractive Bose gas from a potential barrier. In [11], a model for an elastic scattering process has been considered. In the model’s wavefunction, the internal and centre-of-mass degrees of freedom of the scattered Bose gas are separated from each other at all times during the collision with the barrier. In [12], using many-body theory a more general case has been treated, in which inelastic scattering and coupling between all degrees of freedom of the scattered Bose gas are allowed and taken explicitly into account. In both works [11, 12], it has been demonstrated that a mesoscopic quantum superposition state of two non-overlapping clouds can be formed as a result of the collision with a potential barrier. We have termed this object a caton in the low-dimensional attractive Bose gases [12]. We stress that several propositions have been made in the literature to creating mesoscopic quantum superposition states with ultracold Bose gases [14–20]. These propositions primarily involve quantum superposition states prepared in a trap. 1
© 2009 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
In this work, we propose an efficient and robust alternative preparation scheme for generating catons or mesoscopic quantum superposition states in low-dimensional attractive Bose gases. According to our time-dependent many-body calculations in [12], catons are generated due to coupling between the centre-of-mass and internal degrees of freedom of the scattered Bose gas while it interacts with the potential barrier. This suggests that the barrier itself plays a key role in generating catons. We suppose that extending the duration of the interaction between a Bose gas and a barrier will open up efficient ways for creating quantum superposition states. A natural way to achieve longer interaction times of a cloud with a barrier is to ramp up a barrier into a cloud at rest. In this communication, we explicitly show that quantum superposition states of two (or more) distinct mesoscopic Bose clouds can efficiently and in a controllable manner be formed by threading a low-dimensional attractive Bose gas with a potential barrier. We study N attractive bosons in one dimension, interacting To via the contact potential λ0 δ(x − x ), λ0 < 0. compute the dynamics of the bosonic systems, we integrate the time-dependent many-boson Schr¨odinger equation by the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method [13]. In the MCTDHB(M) method, the time-dependent many-boson wavefunction is expanded by all time-dependent permanents generated by distributing the N bosons over M time-dependent orbitals {φi (x, t)}. The MCTDHB(M) wavefunction reads as (x1 , x2 , . . . , xN , t) = ˆ C (t) S φ (x n 1 1 , t) · · · φ2 (xn1 +1 , t) · · · φ3 (xn1 +n2 +1 , t) · · · n φM (xN , t), where Sˆ is the symmetrization operator and n = (n1 , n2 , · · · nM ) collects the occupation numbers. The orbitals {φi (x, t)} as well as the expansion coefficients {Cn (t)} are time-dependent quantities and determined according to the Dirac–Frenkel time-dependent variational principle [13]. We will also contrast the results of the many-body dynamics with the respective Gross–Pitaevskii dynamics. We mention that the Gross–Pitaevskii mean-field theory can be seen as the limiting case of the MCTDHB many-body theory where only a single orbital, φ1 (x, t), is available for the dynamics (M = 1), i.e. when all bosons are forced to be coherent. The Gross–Pitaevskii wavefunction reads as (x1 , x2 , . . . , xN , t) = φ1 (x1 , t) · · · φ1 (xN , t). In this work, we consider N = 100 attractive bosons with λ0 = −0.04. We aim at simulating scenarios with a sudden ramp-up of a barrier, i.e. when the barrier is ramped up much faster than the cloud’s internal timescale (inverse of the first excitation energy). Thus, in practice, at t = 0 we switch on a potential barrier Vˆ (x) onto the Bose cloud which is located around the origin x = 0, and examine the evolution of the system in time. In this work we use dimensionless units for length, time and energy, which are h ¯2 readily arrived at by dividing the Hamiltonian by mL 2 , where m is the mass of a boson and L is a convenient length scale, say the size of the atomic cloud. The one-body Hamiltonian ∂2 ˆ ˆ A Gaussian barrier then reads as h(x) = − 12 ∂x 2 + V (x). 2 2 ˆ V (x) = V0 exp[(x + x) /2σ ] is chosen where V0 = 0.3 is the height of the barrier, σ is the width and x is a possible offset of the barrier with respect to the Bose cloud.
Figure 1. Threading an attractive Bose cloud by a Gaussian-shaped potential barrier. Both the cloud and the barrier are located at the origin x = 0. The barrier height is fixed at V0 = 0.3; the barrier width σ is varied. The wavepacket density is shown as a function of time. Left and middle panels: many-body dynamics for σ = 0.2 and σ = 0.4. Catons—mesoscopic quantum superposition states of two distinct wavepackets—are formed. Right panel: Gross–Pitaevskii dynamics for σ = 0.4 reveals only small amplitude breathing (note the different scale), signifying that the formation of mesoscopic quantum superposition states is a many-body phenomenon not accessed within the Gross–Pitaevskii theory. All quantities are dimensionless.
In figure 1, we present the results of threading an attractive Bose cloud by a barrier exactly in the middle, i.e. the ‘ideal’ symmetric case where there is no offset of the barrier with respect to the cloud, x = 0. The cloud is prepared in its ground state. The time-dependent density ρ(x, t) = N dx2 · · · dxN |(x, x2 , . . . , xN , t)|2 is shown. We first employ a barrier of width σ = 0.2 (see the left panel of figure 1). For comparison, the width of the cloud is about 0.5. The barrier excites the Bose cloud and initiates the splitting. The splitting process takes some time at the end of which the cloud is divided into two symmetric parts which propagate in opposite directions, one with respect to the other. To show that the split object shown in figure 1 (and throughout this work) is a fragmented object, we resort to the reduced one-body density matrix of the many-boson wavefunction and diagonalize it: ρ(x|x ; t) = N dx2 · · · dxN ∗ (x , x2 , . . . , xN , t)(x, x2 , . . . , xN , t) = i ρi (t) φi∗ (x , t)φi (x, t). Analysis of the natural occupation numbers ρi (t) will tell us whether the system is condensed [21] or fragmented [22], i.e. whether there is only one or several natural orbitals which are macroscopically occupied. Thus, the initial wavepacket is condensed because ρ1 (0) = 99.1% and ρ2 (0) = 0.9%. Once the split object is formed, the natural occupation numbers indeed become macroscopic and saturate, slightly oscillating around the symmetric values of ρ1 (t) = 50% and ρ2 (t) = 50%. Hence, the system has become fragmented. For purposes of presentation, the explicit proof that the fragmented split object shown in figure 1 (and throughout this work) corresponds to a quantum superposition state of two distinct Bose clouds is presented below, when we discuss a more general, asymmetric scenario of threading an attractive Bose cloud by a potential barrier (see figure 3). 2
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
The main purpose of this work, after demonstrating that quantum superposition states can be generated by ramping up a barrier onto an attractive Bose cloud, is to establish that such a mode of preparation of catons is efficient and robust. For this, our next ‘experiment’ involves a wider barrier of width σ = 0.4, twice as much as the original barrier. As can be seen in the middle panel of figure 1, a split object is formed as well. Analysis of the many-body wavefunction again shows that the split object is fragmented and corresponds to a mesoscopic quantum superposition state. Comparison between the left and middle panels of figure 1 suggests two generic trends. The first is that the velocity of the split clouds is larger for wider threading barriers (to guide the eye, see the distance each split cloud has travelled at t = 30). The second is that the splitting process takes (slightly) longer times for wider barriers. These trends can be anticipated and understood from the geometry of the barrier and based on energetical considerations. First, imagine the limit of an infinitely wide barrier: the ramp-up process does not perturb the system and splitting cannot occur. The flatter (wider) the barrier, the slower the external-potential energy is transformed to the internal energy of the cloud by the sudden ramp-up process and the longer is the time needed to form a caton. Moreover, the wider the barrier, the larger is the (one-body) potential energy added to the system, and thus more kinetic energy can be transformed to the split clouds. Can the phenomenon of splitting an attractive cloud by ramping up a barrier be found on the Gross–Pitaevskii meanfield level? We take the Gross–Pitaevskii ground state—the famous bright-soliton solution, sech[γ x] with γ = |λ0 |(N − 1)/2—and study the dynamics following the sudden ramp-up of a Gaussian barrier of width σ = 0.4. The corresponding density is depicted in the right panel of figure 1 as a function of time. Comparing it with the respective MCTDHB manybody calculation depicted in the middle panel of figure 1, we conclude the following: the Gross–Pitaevskii theory cannot describe the splitting phenomenon. On the Gross–Pitaevskii level, we see that the barrier excites small-amplitude breathinglike oscillations of the cloud; however, the cloud does not split. Note the difference in the coordinate scale between the panels. Interestingly, small asymmetries or offsets x = 0 between the cloud and the barrier (see below) would cause the cloud to slide down the barrier, of course without being split. There are two main reasons why splitting does not occur within the Gross–Pitaevskii theory. The first reason is that the split object—the caton—is a fragmented many-body state, whereas the Gross–Pitaevskii theory is based on the assumption that the Bose cloud remains always fully condensed, i.e. not fragmented. The second reason is energetics—the lowest energy, coherent two-hump soliton appears at much higher energies than all split scenarios studied here. Consequently, in all the scenarios studied in this work the Gross–Pitaevskii theory cannot show the splitting of the attractive cloud. We conclude that the formation of quantum superposition states is a pure many-body phenomenon originating to fragmented excited states. The caton-formation scenarios demonstrated so far in figure 1 are ‘ideal’ cases, where the barrier threads the Bose cloud in a symmetrical manner. Of course, no ‘ideal’ or
Figure 2. Formation of quantum superposition states for a displaced barrier. The same as figure 1 but for a barrier displaced by x = 0.05 to the left. Left and middle panels: formation of a slightly asymmetric caton for a barrier’s width σ = 0.2 and σ = 0.4. Right panel: the Gross–Pitaevskii evolution for σ = 0.4, predicting that the cloud can only slide down without visible distortions. All quantities are dimensionless.
perfectly symmetric ramp-up scenarios are possible in an experiment, and the question we would like to raise is whether catons can be formed and what is the stability of their formation in an asymmetric ramp-up scenario. In figure 2, we repeat the previous three ‘experiments’ in the presence of an offset x = 0 between the barrier and the atomic cloud. We fix an offset of x = 0.05. Let us concentrate on the left panel. With this offset, which is of the order of 25% of the barrier width (σ = 0.2), and of the order of 10% of the cloud size (∼1/γ ≈ 0.5), a slightly asymmetric caton is generated. Once the caton is formed, the occupation numbers of the reduced one-body density matrix saturate, slightly oscillating around ρ1 (t) ≈ 56.7% and ρ2 (t) ≈ 43.3%. We have found numerically the following behaviour. For an offset of the barrier to the left, the ‘smaller’ cloud propagates to the left, in the direction of the offset, whereas the ‘larger’ cloud propagates to the right, opposite to the barrier offset. Furthermore accompanying this asymmetry, the velocity of the ‘smaller’ cloud travelling to the left is slightly faster than that of the ‘larger’ cloud travelling to the right. We see that an asymmetric ramp-up scenario generates a caton which exhibits interesting properties. This demonstrates the robustness of the caton-formation process when ramping up a barrier. The same offset of x = 0.05 applied to the broader barrier (σ = 0.4) also generates an asymmetric caton (see the middle panel of figure 2) but the split parts are less asymmetric. As the offset is now only about 12% of the barrier width, the generation of a less asymmetric caton can be anticipated. After the split, the occupation numbers of the reduced onebody density matrix saturate, slightly oscillating now around ρ1 (t) ≈ 54.2% and ρ2 (t) ≈ 45.8%. For comparison, the Gross–Pitaevskii ‘experiment’ for the same setup is presented in the right panel of figure 2. The Gross–Pitaevskii mean-field theory predicts sliding of the cloud, without any signature of a split. 3
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
Sˆφ1 (x1 , t) · · · φ1 (xn , t)φ2 (xn+1 , t) · · · φ2 (xN , t). In the middle panel of figure 3, we plot the evolution of the expansion coefficients (probabilities) |Cn (t)|2 in the Fock space spanned by the |N, 0, |N −1, 1, . . . , |1, N −1, |0, N configurations. The initial wavepacket is a slightly depleted condensed state (ρ1 (0) = 99.1% and ρ2 (0) = 0.9%). Its pattern of the expansion coefficients in the Fock space is mainly formed by the |N, 0 configuration augmented by few neighbouring excited configurations responsible for the depletion of the cloud. The respective natural orbitals at t = 0 are plotted in the right bottom panel of figure 3. In time, the expansion coefficients’ pattern evolves from the above mainly condensed initial pattern at t = 0 to two groups of coefficients, well localized in opposite sides of the Fock space, after the split has occurred. Within each group, the major contribution is provided by either the |N, 0 or the |0, N configuration. Therefore, the final state is a superposition state of mainly the |N, 0 and |0, N configurations. Side by side, the natural orbitals after the split, depicted in the right upper panel of figure 3 for t = 30, describe two distinct objects of almost identical shapes and localized in different parts of the coordinate space. All in all, we conclude that the split object obtained corresponds to a quantum superposition state of two distinct attractive Bose clouds. We mention that a similar analysis has been performed (not displayed) for all split objects (cases) described above, identifying them as catons as well. In conclusion, we have studied in this work the formation of quantum superposition states in low-dimensional ultracold attractive Bose gases by threading the Bose cloud with a potential barrier. The proposed scheme with a Gaussianshaped barrier, of width comparable to or smaller than the cloud size, has been shown to be efficient and promising for generating catons. In particular, it has been found to be robust with respect to moderate variations of the width of the barrier, the position (offset) of the barrier with respect to the cloud and even to moderate internal excitations of the cloud. Generally, the barriers of other shapes and parameters may lead to other different split scenarios. In this context, the formation of a larger number of split clouds than two can be foreseen and has indeed been found in some of our calculations with broader barriers. On the theoretical side, we have demonstrated that low-dimensional attractive Bose gases exhibit intriguing many-body phenomena not accessible within the popular and amply employed Gross– Pitaevskii theory. Threading an attractive Bose cloud by a potential barrier suggests a simple and controllable way to generating mesoscopic quantum superposition states (catons). The proposed half-collision scheme allows in principle to create in the ‘ideal’ case two exact copies of the original manyboson quantum state. We hope that our work will stimulate experiments.
Figure 3. Formation of quantum superposition states by ramping up a displaced barrier into a ‘hotter’ cloud. The parameters are as in figure 2 except for a slightly wider, excited initial cloud. Left panel: the density as a function of time is shown. Middle panel: representation of the many-body state and proof that the split object corresponds to a superposition state of two distinct wavepackets. Evolution of the expansion coefficients in the Fock space spanned by the |N, 0, |N − 1, 1, . . . , |1, N − 1, |0, N configurations. The probabilities |Cn (t)|2 are plotted as a function of time. Right upper and lower panels: natural orbitals |φi (x, t)|2 (upper two curves of each of the right panels, in green and blue; normalized to 1) and densities (lower curve of each of the right panels, in red) before, at t = 0, and after the split, at t = 30. The barrier is indicated in black. The initial wavepacket is described essentially by |N, 0 configuration while mainly the |N, 0 and |0, N configurations contribute to the split object. All quantities are dimensionless.
To provide an ultimate motivation to pursuing the formation of catons in ramp-up scenarios, we consider the case where the attractive atomic cloud is not in its ground state, i.e. it is a slightly excited or ‘hotter’ cloud. We model an excited cloud by starting from a slightly broader cloud (technically, this is achieved by preparing the cloud as the ground state of an attractive system with smaller attraction, here with λ0 = −0.035, and then switching back to λ0 = −0.04). We consider the case of a Gaussian barrier of width σ = 0.2 and the same offset x = 0.05 as done in the panels of figure 2. The results are plotted in figure 3. As can be seen, starting from the left panel, a clear generation of a caton is observed. After the splitting process, two non-exact replicas of the original ‘hot’ cloud are produced. The occupation numbers of the reduced one-body density matrix saturate, slightly oscillating now around ρ1 (t) ≈ 55.2% and ρ2 (t) ≈ 44.8%, meaning that with asymmetry and internal excitations a rather symmetric caton is still generated. Now is the time to prescribe the proof that the split object is a caton, i.e. a mesoscopic quantum superposition state of two distinct attractive clouds. In the middle and right panels of figure 3, we depict the detailed analysis of the many-boson wavefunction in the asymmetric and ‘hot’ case studied here. We remind the reader that the many-boson wavefunction is written as |(t) = N C (t)|n, N − n; t where the configurations are defined n n=0 with respect to the natural orbitals {φi (x, t)} and expressed in the coordinate space as follows: x1 , . . . , xN |n, N − n; t =
Acknowledgment Financial support by DFG is acknowledged. 4
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
[11] Weiss C and Castin Y 2009 Phys. Rev. Lett. 102 010403 [12] Streltsov A I et al 2008 arXiv:0812.3573v1 [13] Streltsov A I et al 2007 Phys. Rev. Lett. 99 030402 Alon O E et al 2008 Phys. Rev. A 77 033613 [14] Ruostekoski J et al 1998 Phys. Rev. A 57 511 [15] Cirac J I et al 1998 Phys. Rev. A 57 1208 [16] Steel M J and Collett M J 1998 Phys. Rev. A 57 2920 [17] Gordon D and Savage C M 1999 Phys. Rev. A 59 4623 [18] Mahmud K W et al 2002 Phys. Rev. A 66 063607 [19] Ho T-L and Ciobanu C V 2004 J. Low Temp. Phys. 135 257 [20] Huang Y P and Moore M G 2006 Phys. Rev. A 73 023606 [21] Penrose O and Onsager L 1956 Phys. Rev. 104 576 [22] Nozi`eres P 1996 Bose–Einstein Condensation ed A Griffin, D W Snoke and S Stringari (Cambridge: Cambridge University Press)
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Ruprecht P A et al 1995 Phys. Rev. A 51 4704 P´erez-Garc´ıa V M et al 1998 Phys. Rev. A 57 3837 Strecker K E et al 2002 Nature 417 150 Khaykovich L et al 2002 Science 296 1290 Salasnich L et al 2003 Phys. Rev. Lett. 91 080405 Carr L D and Brand J 2004 Phys. Rev. Lett. 92 040401 Martin A D et al 2007 Phys. Rev. Lett. 98 020402 Gross E P 1961 Nuovo Cimento 20 454 Pitaevskii L P 1961 Zh. Eksp. Teor. Fiz. 40 646 Pitaevskii L P 1961 Sov. Phys.—JETP 13 451 [9] Buljan H et al 2005 Phys. Rev. Lett. 95 180401 [10] Streltsov A I et al 2008 Phys. Rev. Lett. 100 130401
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JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
doi:10.1088/0953-4075/42/9/091004
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004 (5pp)
FAST TRACK COMMUNICATION
Efficient generation and properties of mesoscopic quantum superposition states in an attractive Bose–Einstein condensate threaded by a potential barrier Alexej I Streltsov, Ofir E Alon and Lorenz S Cederbaum Theoretische Chemie, Physikalisch-Chemisches Institut, Universit¨at Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany E-mail: [email protected], [email protected] and [email protected]
Received 1 April 2009 Published 27 April 2009 Online at stacks.iop.org/JPhysB/42/091004 Abstract The dynamics of a low-dimensional ultracold attractive Bose gas threaded by a potential barrier is studied by solving the time-dependent many-boson Schr¨odinger equation. It is shown that quantum superposition states of two (or more) distinct mesoscopic Bose clouds can be formed. We demonstrate the efficiency and robustness of this proposed preparation scheme and characterize the properties of the generated quantum superposition states. (Some figures in this article are in colour only in the electronic version)
Quantum dynamics of ultracold low-dimensional attractive Bose gases is an active research field attracting both experimentalists and theorists [1–7]. Here, much work has been devoted to studying and analysing the evolution in time of attractive Bose systems with the popular Gross–Pitaevskii equation. In particular, the Gross–Pitaevskii equation predicts the famous bright-soliton solution for the ground state and soliton trains for excited states. We recall that the Gross– Pitaevskii theory [8] is a mean-field theory which assumes all bosons to reside in a single quantum state or an orbital, i.e. the Bose system has to remain condensed throughout its evolution in time. Recently, by computing the dynamics of low-dimensional attractive Bose gases beyond Gross– Pitaevskii, the arsenal of physical phenomena which attractive Bose gases exhibit has been enriched [9–12]. In particular, low-lying fragmented excited states can conspire to give dynamically stable fragmented objects in low-dimensional attractive Bose gases. These fragmented objects are distinct from and generally lie lower in energy than solitons, which are condensed objects. 0953-4075/09/091004+05$30.00
In the above context, it has been shown by two independent groups [11, 12] that mesoscopic quantum superposition states can be generated by scattering an attractive Bose gas from a potential barrier. In [11], a model for an elastic scattering process has been considered. In the model’s wavefunction, the internal and centre-of-mass degrees of freedom of the scattered Bose gas are separated from each other at all times during the collision with the barrier. In [12], using many-body theory a more general case has been treated, in which inelastic scattering and coupling between all degrees of freedom of the scattered Bose gas are allowed and taken explicitly into account. In both works [11, 12], it has been demonstrated that a mesoscopic quantum superposition state of two non-overlapping clouds can be formed as a result of the collision with a potential barrier. We have termed this object a caton in the low-dimensional attractive Bose gases [12]. We stress that several propositions have been made in the literature to creating mesoscopic quantum superposition states with ultracold Bose gases [14–20]. These propositions primarily involve quantum superposition states prepared in a trap. 1
© 2009 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
In this work, we propose an efficient and robust alternative preparation scheme for generating catons or mesoscopic quantum superposition states in low-dimensional attractive Bose gases. According to our time-dependent many-body calculations in [12], catons are generated due to coupling between the centre-of-mass and internal degrees of freedom of the scattered Bose gas while it interacts with the potential barrier. This suggests that the barrier itself plays a key role in generating catons. We suppose that extending the duration of the interaction between a Bose gas and a barrier will open up efficient ways for creating quantum superposition states. A natural way to achieve longer interaction times of a cloud with a barrier is to ramp up a barrier into a cloud at rest. In this communication, we explicitly show that quantum superposition states of two (or more) distinct mesoscopic Bose clouds can efficiently and in a controllable manner be formed by threading a low-dimensional attractive Bose gas with a potential barrier. We study N attractive bosons in one dimension, interacting To via the contact potential λ0 δ(x − x ), λ0 < 0. compute the dynamics of the bosonic systems, we integrate the time-dependent many-boson Schr¨odinger equation by the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method [13]. In the MCTDHB(M) method, the time-dependent many-boson wavefunction is expanded by all time-dependent permanents generated by distributing the N bosons over M time-dependent orbitals {φi (x, t)}. The MCTDHB(M) wavefunction reads as (x1 , x2 , . . . , xN , t) = ˆ C (t) S φ (x n 1 1 , t) · · · φ2 (xn1 +1 , t) · · · φ3 (xn1 +n2 +1 , t) · · · n φM (xN , t), where Sˆ is the symmetrization operator and n = (n1 , n2 , · · · nM ) collects the occupation numbers. The orbitals {φi (x, t)} as well as the expansion coefficients {Cn (t)} are time-dependent quantities and determined according to the Dirac–Frenkel time-dependent variational principle [13]. We will also contrast the results of the many-body dynamics with the respective Gross–Pitaevskii dynamics. We mention that the Gross–Pitaevskii mean-field theory can be seen as the limiting case of the MCTDHB many-body theory where only a single orbital, φ1 (x, t), is available for the dynamics (M = 1), i.e. when all bosons are forced to be coherent. The Gross–Pitaevskii wavefunction reads as (x1 , x2 , . . . , xN , t) = φ1 (x1 , t) · · · φ1 (xN , t). In this work, we consider N = 100 attractive bosons with λ0 = −0.04. We aim at simulating scenarios with a sudden ramp-up of a barrier, i.e. when the barrier is ramped up much faster than the cloud’s internal timescale (inverse of the first excitation energy). Thus, in practice, at t = 0 we switch on a potential barrier Vˆ (x) onto the Bose cloud which is located around the origin x = 0, and examine the evolution of the system in time. In this work we use dimensionless units for length, time and energy, which are h ¯2 readily arrived at by dividing the Hamiltonian by mL 2 , where m is the mass of a boson and L is a convenient length scale, say the size of the atomic cloud. The one-body Hamiltonian ∂2 ˆ ˆ A Gaussian barrier then reads as h(x) = − 12 ∂x 2 + V (x). 2 2 ˆ V (x) = V0 exp[(x + x) /2σ ] is chosen where V0 = 0.3 is the height of the barrier, σ is the width and x is a possible offset of the barrier with respect to the Bose cloud.
Figure 1. Threading an attractive Bose cloud by a Gaussian-shaped potential barrier. Both the cloud and the barrier are located at the origin x = 0. The barrier height is fixed at V0 = 0.3; the barrier width σ is varied. The wavepacket density is shown as a function of time. Left and middle panels: many-body dynamics for σ = 0.2 and σ = 0.4. Catons—mesoscopic quantum superposition states of two distinct wavepackets—are formed. Right panel: Gross–Pitaevskii dynamics for σ = 0.4 reveals only small amplitude breathing (note the different scale), signifying that the formation of mesoscopic quantum superposition states is a many-body phenomenon not accessed within the Gross–Pitaevskii theory. All quantities are dimensionless.
In figure 1, we present the results of threading an attractive Bose cloud by a barrier exactly in the middle, i.e. the ‘ideal’ symmetric case where there is no offset of the barrier with respect to the cloud, x = 0. The cloud is prepared in its ground state. The time-dependent density ρ(x, t) = N dx2 · · · dxN |(x, x2 , . . . , xN , t)|2 is shown. We first employ a barrier of width σ = 0.2 (see the left panel of figure 1). For comparison, the width of the cloud is about 0.5. The barrier excites the Bose cloud and initiates the splitting. The splitting process takes some time at the end of which the cloud is divided into two symmetric parts which propagate in opposite directions, one with respect to the other. To show that the split object shown in figure 1 (and throughout this work) is a fragmented object, we resort to the reduced one-body density matrix of the many-boson wavefunction and diagonalize it: ρ(x|x ; t) = N dx2 · · · dxN ∗ (x , x2 , . . . , xN , t)(x, x2 , . . . , xN , t) = i ρi (t) φi∗ (x , t)φi (x, t). Analysis of the natural occupation numbers ρi (t) will tell us whether the system is condensed [21] or fragmented [22], i.e. whether there is only one or several natural orbitals which are macroscopically occupied. Thus, the initial wavepacket is condensed because ρ1 (0) = 99.1% and ρ2 (0) = 0.9%. Once the split object is formed, the natural occupation numbers indeed become macroscopic and saturate, slightly oscillating around the symmetric values of ρ1 (t) = 50% and ρ2 (t) = 50%. Hence, the system has become fragmented. For purposes of presentation, the explicit proof that the fragmented split object shown in figure 1 (and throughout this work) corresponds to a quantum superposition state of two distinct Bose clouds is presented below, when we discuss a more general, asymmetric scenario of threading an attractive Bose cloud by a potential barrier (see figure 3). 2
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
The main purpose of this work, after demonstrating that quantum superposition states can be generated by ramping up a barrier onto an attractive Bose cloud, is to establish that such a mode of preparation of catons is efficient and robust. For this, our next ‘experiment’ involves a wider barrier of width σ = 0.4, twice as much as the original barrier. As can be seen in the middle panel of figure 1, a split object is formed as well. Analysis of the many-body wavefunction again shows that the split object is fragmented and corresponds to a mesoscopic quantum superposition state. Comparison between the left and middle panels of figure 1 suggests two generic trends. The first is that the velocity of the split clouds is larger for wider threading barriers (to guide the eye, see the distance each split cloud has travelled at t = 30). The second is that the splitting process takes (slightly) longer times for wider barriers. These trends can be anticipated and understood from the geometry of the barrier and based on energetical considerations. First, imagine the limit of an infinitely wide barrier: the ramp-up process does not perturb the system and splitting cannot occur. The flatter (wider) the barrier, the slower the external-potential energy is transformed to the internal energy of the cloud by the sudden ramp-up process and the longer is the time needed to form a caton. Moreover, the wider the barrier, the larger is the (one-body) potential energy added to the system, and thus more kinetic energy can be transformed to the split clouds. Can the phenomenon of splitting an attractive cloud by ramping up a barrier be found on the Gross–Pitaevskii meanfield level? We take the Gross–Pitaevskii ground state—the famous bright-soliton solution, sech[γ x] with γ = |λ0 |(N − 1)/2—and study the dynamics following the sudden ramp-up of a Gaussian barrier of width σ = 0.4. The corresponding density is depicted in the right panel of figure 1 as a function of time. Comparing it with the respective MCTDHB manybody calculation depicted in the middle panel of figure 1, we conclude the following: the Gross–Pitaevskii theory cannot describe the splitting phenomenon. On the Gross–Pitaevskii level, we see that the barrier excites small-amplitude breathinglike oscillations of the cloud; however, the cloud does not split. Note the difference in the coordinate scale between the panels. Interestingly, small asymmetries or offsets x = 0 between the cloud and the barrier (see below) would cause the cloud to slide down the barrier, of course without being split. There are two main reasons why splitting does not occur within the Gross–Pitaevskii theory. The first reason is that the split object—the caton—is a fragmented many-body state, whereas the Gross–Pitaevskii theory is based on the assumption that the Bose cloud remains always fully condensed, i.e. not fragmented. The second reason is energetics—the lowest energy, coherent two-hump soliton appears at much higher energies than all split scenarios studied here. Consequently, in all the scenarios studied in this work the Gross–Pitaevskii theory cannot show the splitting of the attractive cloud. We conclude that the formation of quantum superposition states is a pure many-body phenomenon originating to fragmented excited states. The caton-formation scenarios demonstrated so far in figure 1 are ‘ideal’ cases, where the barrier threads the Bose cloud in a symmetrical manner. Of course, no ‘ideal’ or
Figure 2. Formation of quantum superposition states for a displaced barrier. The same as figure 1 but for a barrier displaced by x = 0.05 to the left. Left and middle panels: formation of a slightly asymmetric caton for a barrier’s width σ = 0.2 and σ = 0.4. Right panel: the Gross–Pitaevskii evolution for σ = 0.4, predicting that the cloud can only slide down without visible distortions. All quantities are dimensionless.
perfectly symmetric ramp-up scenarios are possible in an experiment, and the question we would like to raise is whether catons can be formed and what is the stability of their formation in an asymmetric ramp-up scenario. In figure 2, we repeat the previous three ‘experiments’ in the presence of an offset x = 0 between the barrier and the atomic cloud. We fix an offset of x = 0.05. Let us concentrate on the left panel. With this offset, which is of the order of 25% of the barrier width (σ = 0.2), and of the order of 10% of the cloud size (∼1/γ ≈ 0.5), a slightly asymmetric caton is generated. Once the caton is formed, the occupation numbers of the reduced one-body density matrix saturate, slightly oscillating around ρ1 (t) ≈ 56.7% and ρ2 (t) ≈ 43.3%. We have found numerically the following behaviour. For an offset of the barrier to the left, the ‘smaller’ cloud propagates to the left, in the direction of the offset, whereas the ‘larger’ cloud propagates to the right, opposite to the barrier offset. Furthermore accompanying this asymmetry, the velocity of the ‘smaller’ cloud travelling to the left is slightly faster than that of the ‘larger’ cloud travelling to the right. We see that an asymmetric ramp-up scenario generates a caton which exhibits interesting properties. This demonstrates the robustness of the caton-formation process when ramping up a barrier. The same offset of x = 0.05 applied to the broader barrier (σ = 0.4) also generates an asymmetric caton (see the middle panel of figure 2) but the split parts are less asymmetric. As the offset is now only about 12% of the barrier width, the generation of a less asymmetric caton can be anticipated. After the split, the occupation numbers of the reduced onebody density matrix saturate, slightly oscillating now around ρ1 (t) ≈ 54.2% and ρ2 (t) ≈ 45.8%. For comparison, the Gross–Pitaevskii ‘experiment’ for the same setup is presented in the right panel of figure 2. The Gross–Pitaevskii mean-field theory predicts sliding of the cloud, without any signature of a split. 3
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
Sˆφ1 (x1 , t) · · · φ1 (xn , t)φ2 (xn+1 , t) · · · φ2 (xN , t). In the middle panel of figure 3, we plot the evolution of the expansion coefficients (probabilities) |Cn (t)|2 in the Fock space spanned by the |N, 0, |N −1, 1, . . . , |1, N −1, |0, N configurations. The initial wavepacket is a slightly depleted condensed state (ρ1 (0) = 99.1% and ρ2 (0) = 0.9%). Its pattern of the expansion coefficients in the Fock space is mainly formed by the |N, 0 configuration augmented by few neighbouring excited configurations responsible for the depletion of the cloud. The respective natural orbitals at t = 0 are plotted in the right bottom panel of figure 3. In time, the expansion coefficients’ pattern evolves from the above mainly condensed initial pattern at t = 0 to two groups of coefficients, well localized in opposite sides of the Fock space, after the split has occurred. Within each group, the major contribution is provided by either the |N, 0 or the |0, N configuration. Therefore, the final state is a superposition state of mainly the |N, 0 and |0, N configurations. Side by side, the natural orbitals after the split, depicted in the right upper panel of figure 3 for t = 30, describe two distinct objects of almost identical shapes and localized in different parts of the coordinate space. All in all, we conclude that the split object obtained corresponds to a quantum superposition state of two distinct attractive Bose clouds. We mention that a similar analysis has been performed (not displayed) for all split objects (cases) described above, identifying them as catons as well. In conclusion, we have studied in this work the formation of quantum superposition states in low-dimensional ultracold attractive Bose gases by threading the Bose cloud with a potential barrier. The proposed scheme with a Gaussianshaped barrier, of width comparable to or smaller than the cloud size, has been shown to be efficient and promising for generating catons. In particular, it has been found to be robust with respect to moderate variations of the width of the barrier, the position (offset) of the barrier with respect to the cloud and even to moderate internal excitations of the cloud. Generally, the barriers of other shapes and parameters may lead to other different split scenarios. In this context, the formation of a larger number of split clouds than two can be foreseen and has indeed been found in some of our calculations with broader barriers. On the theoretical side, we have demonstrated that low-dimensional attractive Bose gases exhibit intriguing many-body phenomena not accessible within the popular and amply employed Gross– Pitaevskii theory. Threading an attractive Bose cloud by a potential barrier suggests a simple and controllable way to generating mesoscopic quantum superposition states (catons). The proposed half-collision scheme allows in principle to create in the ‘ideal’ case two exact copies of the original manyboson quantum state. We hope that our work will stimulate experiments.
Figure 3. Formation of quantum superposition states by ramping up a displaced barrier into a ‘hotter’ cloud. The parameters are as in figure 2 except for a slightly wider, excited initial cloud. Left panel: the density as a function of time is shown. Middle panel: representation of the many-body state and proof that the split object corresponds to a superposition state of two distinct wavepackets. Evolution of the expansion coefficients in the Fock space spanned by the |N, 0, |N − 1, 1, . . . , |1, N − 1, |0, N configurations. The probabilities |Cn (t)|2 are plotted as a function of time. Right upper and lower panels: natural orbitals |φi (x, t)|2 (upper two curves of each of the right panels, in green and blue; normalized to 1) and densities (lower curve of each of the right panels, in red) before, at t = 0, and after the split, at t = 30. The barrier is indicated in black. The initial wavepacket is described essentially by |N, 0 configuration while mainly the |N, 0 and |0, N configurations contribute to the split object. All quantities are dimensionless.
To provide an ultimate motivation to pursuing the formation of catons in ramp-up scenarios, we consider the case where the attractive atomic cloud is not in its ground state, i.e. it is a slightly excited or ‘hotter’ cloud. We model an excited cloud by starting from a slightly broader cloud (technically, this is achieved by preparing the cloud as the ground state of an attractive system with smaller attraction, here with λ0 = −0.035, and then switching back to λ0 = −0.04). We consider the case of a Gaussian barrier of width σ = 0.2 and the same offset x = 0.05 as done in the panels of figure 2. The results are plotted in figure 3. As can be seen, starting from the left panel, a clear generation of a caton is observed. After the splitting process, two non-exact replicas of the original ‘hot’ cloud are produced. The occupation numbers of the reduced one-body density matrix saturate, slightly oscillating now around ρ1 (t) ≈ 55.2% and ρ2 (t) ≈ 44.8%, meaning that with asymmetry and internal excitations a rather symmetric caton is still generated. Now is the time to prescribe the proof that the split object is a caton, i.e. a mesoscopic quantum superposition state of two distinct attractive clouds. In the middle and right panels of figure 3, we depict the detailed analysis of the many-boson wavefunction in the asymmetric and ‘hot’ case studied here. We remind the reader that the many-boson wavefunction is written as |(t) = N C (t)|n, N − n; t where the configurations are defined n n=0 with respect to the natural orbitals {φi (x, t)} and expressed in the coordinate space as follows: x1 , . . . , xN |n, N − n; t =
Acknowledgment Financial support by DFG is acknowledged. 4
J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 091004
Fast Track Communication
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