Abstract

We study the tunneling dynamics and energy bands of three Bose-Einstein condensates which are coupled weakly with each other. The study is carried out with both the mean-filed model and the second-quantized model. The results from these two models are compared and found to agree with each other when the particle number is large. Without interaction, this system possesses a Dirac point in its energy band. This Dirac point is immediately destroyed and develops into a loop structure with arbitrary small interaction. This loop structure has a strong effect on the tunneling dynamics. We find that the tunneling dynamics in this system is very sensitive to the system parameter, e.g., the interaction strength. This sensitivity is found to be caused by the chaos in the mean-field model and the avoided energy crossings with tiny gaps in the second-quantized model. This result gives a certain indication on how the classical dynamics and quantum dynamics are connected in the semi-classical limit. Our mean-field results are also valid for three mutually coupled optical nonlinear waveguides.

© 2014 Optical Society of America

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  63. L. Dell Anna, G. Mazzarella, V. Penna, and L. Salasnich, “Entanglement entropy and macroscopic quantum states with dipolar bosons in a triple-well potential,” Phys. Rev. A 87, 053620 (2013).
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2014 (1)

X. B. Luo, L. P. Li, L. You, and B. Wu, “Coherent destruction of tunneling and dark Floquet state,” New J. Phys. 16, 013007 (2014)
[CrossRef]

2013 (2)

L. Dell Anna, G. Mazzarella, V. Penna, and L. Salasnich, “Entanglement entropy and macroscopic quantum states with dipolar bosons in a triple-well potential,” Phys. Rev. A 87, 053620 (2013).
[CrossRef]

V. Penna, “Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer,” Phys. Rev. E 87, 052909 (2013).
[CrossRef]

2012 (2)

C. J. Bradly, M. Rab, A. D. Greentree, and A. M. Martin, “Coherent tunneling via adiabatic passage in a three-well Bose-Hubbard system,” Phys. Rev. A. 85, 053609 (2012).
[CrossRef]

P. Jason, M. Johansson, and K. Kirr, “Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer,” Phys. Rev. E 86, 016214 (2012).
[CrossRef]

2011 (7)

T. F. Viscondi and K. Furuya, “Dynamics of a BoseEinstein condensate in a symmetric triple-well trap,” J. Phys. A. 44, 175301 (2011).
[CrossRef]

L. Cao, I. Brouzos, S. Zllner, and P. Schmelcher, “Interaction-driven interband tunneling of bosons in the triple well,” New J. Phys. 13, 033032 (2011).
[CrossRef]

Xin Jiang, Li-Bin Fu, Wen-shan Duan, and Jie Liu, “Phase transition of the ground state for two-component BoseEinstein condensates in a triple-well trap,” Journal of Phys. B,  44, 115301 (2011).
[CrossRef]

Z. Chen and B. Wu, “Bose-Einstein condensate in a honeycomb optical lattice: fingerprint of superfluidity at the dirac point,” Phys. Rev. Lett. 107, 065301 (2011).
[CrossRef] [PubMed]

J. v. Porto, “Optical lattices: More than a look,” Nature Phys. 7, 280–281 (2011).
[CrossRef]

Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, “Many-body LandauZener dynamics in coupled one-dimensional Bose liquid,” Nature Phys. 7, 61–67 (2011).
[CrossRef]

C. Kasztelan, S. Trotzky, Y.A. Chen, I. Bloch, I. P. McCulloch, U. Schollwock, and G. Orso, “Landau-Zener Sweeps and Sudden Quenches in Coupled Bose-Hubbard Chains,” Phys. Rev. Lett. 106, 155302 (2011).
[CrossRef] [PubMed]

2010 (4)

K. Maussang, G. E. Marti, T. Schneider, P. Treutlein, Y. Li, A. Sinatra, and R. Long, “Enhanced and Reduced Atom Number Fluctuations in a BEC Splitter,” J. Estéve and J. Reichel, Phys. Rev. Lett. 105, 080403 (2010).
[CrossRef]

A. Sinatra, Y. Castin, and Y. Li, “Particle number fluctuations in a cloven trapped Bose gas at finite temperature,” Phys. Rev. A 81, 053623 (2010).
[CrossRef]

T. Lahaye, T. Pfau, and L. Santos, “Mesoscopic ensembles of polar Bosons in triple-well potentials,” Phys. Rev. Lett. 104, 170404 (2010).
[CrossRef] [PubMed]

H. Hennig, J. Dorignac, and D. K. Campbell, “Transfer of Bose-Einstein condensates through discrete breathers in an optical lattice,” Phys. Rev. A. 82, 053604 (2010).
[CrossRef]

2009 (6)

M. Hiller, T. Kottos, and T. Geisel, “Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system,” Phys. Rev. A 79, 023621 (2009).
[CrossRef]

A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, “Nonlinearity-induced broadening of resonances in dynamically modulated couplers,” Opt. Lett. 34, 2700–2702 (2009).
[CrossRef] [PubMed]

P. Buonsante, R. Franzosi, and V. Penna, “Control of unstable macroscopic oscillations in the dynamics of three coupled Bose condensates,” J. Phys. A. 42, 285307 (2009).
[CrossRef]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for BoseEinstein condensates,” New J. Phys. 11, 043030 (2009).
[CrossRef]

F. Trimborn, D. Witthaut, and H. J. Korsch, “Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase-space approach,” Phys. Rev. A 79, 013608 (2009).
[CrossRef]

2008 (1)

J. Koch and K. L. Hur, “Discontinuous current-phase relations in small one-dimensional Josephson junction arrays,” Phy. Rev. L 101, 097007 (2008)
[CrossRef]

2007 (6)

C.L Pando L. and E. J. Doedel, “1/f noise in a thin stochastic layer described by the discrete nonlinear Schrdinger equation,” Phys. Rev. E 75, 016213 (2007).
[CrossRef]

A. R. Kolovsky, “Semiclassical Quantization of the Bogoliubov Spectrum,” Phys. Rev. Lett. 99, 020401 (2007).
[CrossRef] [PubMed]

B. Liu, L. B. Fu, S. P. Yang, and J. Liu, “Josephson oscillation and transition to self-trapping for Bose-Einstein condensates in a triple-well trap,” Phys. Rev. A. 75, 033601 (2007).
[CrossRef]

J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, “Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential,” Phys. Rev. A 75, 013608 (2007).
[CrossRef]

J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, “Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential,” Phy. Rev. A. 75, 013608 (2007).
[CrossRef]

X. B. Luo, Q. T. Xie, and B. Wu, “Nonlinear coherent destruction of tunneling,” Phys. Rev. A 76, 051802 (2007).
[CrossRef]

2006 (6)

B. Wu and J. Liu, “Commutability between the Semiclassical and Adiabatic Limits,” Phys. Rev. L 96, 020405 (2006).
[CrossRef]

R. Paredes, “Tunneling of ultracold Bose gases in multiple wells,” Phys. Rev. A 73, 033616 (2006).
[CrossRef]

S. Mossmann and C. Jung, “Semiclassical approach to Bose-Einstein condensates in a triple well potential,” Phys. Rev. A 74, 033601 (2006).
[CrossRef]

E. M. Graefe, H. J. Korsch, and D. Witthaut, “Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman adiabatic passage,” Phys. Rev. A 73, 013617 (2006).
[CrossRef]

M. Hiller, T. Kottos, and T. Geisel, “Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates,” Phys. Rev. A 73, 061604(R) (2006).
[CrossRef]

B. Wu and J. Liu, “Commutability between the Semiclassical and Adiabatic Limits,” Phys. Rev. Lett. 96, 020405 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nature Phys. 1, 23–30 (2005).
[CrossRef]

2004 (4)

Y. Shin, M. Saba, A. Schirotzek, T. A. Pasquini, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, “Distillation of Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 150401 (2004).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Persistence of mean-field features in the energy spectrum of small arrays of BoseEinstein condensates,” J. Phys. B. 37, 229–238 (2004).
[CrossRef]

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 050405 (2004).
[CrossRef]

M. Johansson, “Hamiltonian Hopf bifurcations in the discrete nonlinear Schrdinger trimer: oscillatory instabilities, quasi-periodic solutions and a ’new’ type of self-trapping transition,” J. Phys. A: Math. Gen. 37, 2201–2222 (2004).
[CrossRef]

2003 (6)

R. Franzosi and V. Penna, “Chaotic behavior, collective modes, and self-trapping in the dynamics of three coupled Bose-Einstein condensates,” Phys. Rev. E. 67, 046227 (2003).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates,” Phy. Rev. Lett. 90, 050404 (2003).
[CrossRef]

Q. Thommen, J. C. Garreau, and V. Zehnle, “Classical chaos with Bose-Einstein condensates in tilted optical lattices,” Phys. Rev. Lett. 91, 210405 (2003).
[CrossRef] [PubMed]

L.M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

J. Liu, B. Wu, and Q. Niu, “Nonlinear evolution of quantum states in the Adiabatic regime,” Phys. Rev. Lett. 90, 170404 (2003).
[CrossRef] [PubMed]

B. Wu and Q. Niu, “Superfluidity of BoseEinstein condensate in an optical lattice: LandauZener tunnelling and dynamical instability,” New J. Phys. 5, 1041–10424 (2003).
[CrossRef]

2002 (2)

M. Greiner, O. Mandel, T. Esslinger, T W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature(London) 415, 39–44 (2002).
[CrossRef]

H. Pu, W. P. Zhang, and P. Meystre, “Macroscopic Spin Tunneling and Quantum Critical Behavior of a Condensate in a Double-Well Potential,” Phys. Rev. L 89090401 (2002).
[CrossRef]

2001 (3)

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

L. Pitaevskii and S. Stringari, “Thermal vs quantum decoherence in double well trapped Bose-Einstein condensates,” Phys. Rev. L 87180402 (2001).
[CrossRef]

R. Franzosi and V. Penna, “Self-trapping mechanisms in the dynamics of three coupled Bose-Einstein condensates,” Phys. Rev. A. 65, 013601 (2001).
[CrossRef]

2000 (3)

K. Nemoto, C. A. Holmes, G. J. Milburn, and W. J. Munro, “Quantum dynamics of three coupled atomic Bose-Einstein condensates,” Phys. Rev. A. 63, 013604 (2000).
[CrossRef]

R. Franzosi, V. Penna, and R. Zecchina, “Quantum dynamics of coupled bosonic wells within the Bose-Hubbard picture,” Mod. Phys. B 14, 943–961 (2000).
[CrossRef]

B. Wu and Q. Niu, “Nonlinear Landau-Zener tunneling,” Phys. Rev. A. 61, 023402 (2000).
[CrossRef]

1998 (3)

M. J. Steel and M. J. Collett, “Quantum state of two trapped Bose-Einstein condensates with a Josephson coupling,” Phys. Rev. A. 57, 2920–2930 (1998).
[CrossRef]

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices, ” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

L. Amico and V. Penna, “Dynamical Mean Field Theory of the Bose-Hubbard Model,” Phys. Rev. Lett. 80, 2189–2192 (1998).
[CrossRef]

1997 (2)

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 79, 4950–4953 (1997).
[CrossRef]

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 554318–4324 (1997).
[CrossRef]

1996 (1)

S. Aubry, S. Flach, K. Kladko, and E. Olbrich, “Manifestation of classical Bifurcation in the spectrum of the integrable quantum dimer,” Phys. Rev. Lett. 76, 1607–1610 (1996).
[CrossRef] [PubMed]

1995 (2)

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Albiez, M.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Altman, E.

Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, “Many-body LandauZener dynamics in coupled one-dimensional Bose liquid,” Nature Phys. 7, 61–67 (2011).
[CrossRef]

Amico, L.

L. Amico and V. Penna, “Dynamical Mean Field Theory of the Bose-Hubbard Model,” Phys. Rev. Lett. 80, 2189–2192 (1998).
[CrossRef]

Anderson, B. P.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

Anderson, D. Z.

J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, “Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential,” Phys. Rev. A 75, 013608 (2007).
[CrossRef]

J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, “Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential,” Phy. Rev. A. 75, 013608 (2007).
[CrossRef]

Anderson, M. H.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Aubry, S.

S. Aubry, S. Flach, K. Kladko, and E. Olbrich, “Manifestation of classical Bifurcation in the spectrum of the integrable quantum dimer,” Phys. Rev. Lett. 76, 1607–1610 (1996).
[CrossRef] [PubMed]

Bloch, I.

Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, “Many-body LandauZener dynamics in coupled one-dimensional Bose liquid,” Nature Phys. 7, 61–67 (2011).
[CrossRef]

C. Kasztelan, S. Trotzky, Y.A. Chen, I. Bloch, I. P. McCulloch, U. Schollwock, and G. Orso, “Landau-Zener Sweeps and Sudden Quenches in Coupled Bose-Hubbard Chains,” Phys. Rev. Lett. 106, 155302 (2011).
[CrossRef] [PubMed]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nature Phys. 1, 23–30 (2005).
[CrossRef]

M. Greiner, O. Mandel, T. Esslinger, T W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature(London) 415, 39–44 (2002).
[CrossRef]

Boshier, M. G.

K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for BoseEinstein condensates,” New J. Phys. 11, 043030 (2009).
[CrossRef]

Bradley, C. C.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Bradly, C. J.

C. J. Bradly, M. Rab, A. D. Greentree, and A. M. Martin, “Coherent tunneling via adiabatic passage in a three-well Bose-Hubbard system,” Phys. Rev. A. 85, 053609 (2012).
[CrossRef]

Brouzos, I.

L. Cao, I. Brouzos, S. Zllner, and P. Schmelcher, “Interaction-driven interband tunneling of bosons in the triple well,” New J. Phys. 13, 033032 (2011).
[CrossRef]

Bruder, C.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices, ” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Buonsante, P.

P. Buonsante, R. Franzosi, and V. Penna, “Control of unstable macroscopic oscillations in the dynamics of three coupled Bose condensates,” J. Phys. A. 42, 285307 (2009).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Persistence of mean-field features in the energy spectrum of small arrays of BoseEinstein condensates,” J. Phys. B. 37, 229–238 (2004).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates,” Phy. Rev. Lett. 90, 050404 (2003).
[CrossRef]

Campbell, D. K.

H. Hennig, J. Dorignac, and D. K. Campbell, “Transfer of Bose-Einstein condensates through discrete breathers in an optical lattice,” Phys. Rev. A. 82, 053604 (2010).
[CrossRef]

Cao, L.

L. Cao, I. Brouzos, S. Zllner, and P. Schmelcher, “Interaction-driven interband tunneling of bosons in the triple well,” New J. Phys. 13, 033032 (2011).
[CrossRef]

Castin, Y.

A. Sinatra, Y. Castin, and Y. Li, “Particle number fluctuations in a cloven trapped Bose gas at finite temperature,” Phys. Rev. A 81, 053623 (2010).
[CrossRef]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

Chen, Y. A.

Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, “Many-body LandauZener dynamics in coupled one-dimensional Bose liquid,” Nature Phys. 7, 61–67 (2011).
[CrossRef]

Chen, Y.A.

C. Kasztelan, S. Trotzky, Y.A. Chen, I. Bloch, I. P. McCulloch, U. Schollwock, and G. Orso, “Landau-Zener Sweeps and Sudden Quenches in Coupled Bose-Hubbard Chains,” Phys. Rev. Lett. 106, 155302 (2011).
[CrossRef] [PubMed]

Chen, Z.

Z. Chen and B. Wu, “Bose-Einstein condensate in a honeycomb optical lattice: fingerprint of superfluidity at the dirac point,” Phys. Rev. Lett. 107, 065301 (2011).
[CrossRef] [PubMed]

Cirac, J. I.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices, ” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Clark, C. W.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Steel and M. J. Collett, “Quantum state of two trapped Bose-Einstein condensates with a Josephson coupling,” Phys. Rev. A. 57, 2920–2930 (1998).
[CrossRef]

Collins, L. A.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

Cornell, E. A.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Corney, J.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A 554318–4324 (1997).
[CrossRef]

Cristiani, M.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Dell Anna, L.

L. Dell Anna, G. Mazzarella, V. Penna, and L. Salasnich, “Entanglement entropy and macroscopic quantum states with dipolar bosons in a triple-well potential,” Phys. Rev. A 87, 053620 (2013).
[CrossRef]

Demler, E.

L.M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Doedel, E. J.

C.L Pando L. and E. J. Doedel, “1/f noise in a thin stochastic layer described by the discrete nonlinear Schrdinger equation,” Phys. Rev. E 75, 016213 (2007).
[CrossRef]

Dorignac, J.

H. Hennig, J. Dorignac, and D. K. Campbell, “Transfer of Bose-Einstein condensates through discrete breathers in an optical lattice,” Phys. Rev. A. 82, 053604 (2010).
[CrossRef]

Dreisow, F.

Duan, L.M.

L.M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Duan, Wen-shan

Xin Jiang, Li-Bin Fu, Wen-shan Duan, and Jie Liu, “Phase transition of the ground state for two-component BoseEinstein condensates in a triple-well trap,” Journal of Phys. B,  44, 115301 (2011).
[CrossRef]

Ensher, J. R.

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[CrossRef] [PubMed]

Esslinger, T.

M. Greiner, O. Mandel, T. Esslinger, T W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature(London) 415, 39–44 (2002).
[CrossRef]

Fantoni, S.

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 79, 4950–4953 (1997).
[CrossRef]

Feder, D. L.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

Flach, S.

S. Aubry, S. Flach, K. Kladko, and E. Olbrich, “Manifestation of classical Bifurcation in the spectrum of the integrable quantum dimer,” Phys. Rev. Lett. 76, 1607–1610 (1996).
[CrossRef] [PubMed]

Fölling, J.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Franzosi, R.

P. Buonsante, R. Franzosi, and V. Penna, “Control of unstable macroscopic oscillations in the dynamics of three coupled Bose condensates,” J. Phys. A. 42, 285307 (2009).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Persistence of mean-field features in the energy spectrum of small arrays of BoseEinstein condensates,” J. Phys. B. 37, 229–238 (2004).
[CrossRef]

P. Buonsante, R. Franzosi, and V. Penna, “Dynamical Instability in a Trimeric Chain of Interacting Bose-Einstein Condensates,” Phy. Rev. Lett. 90, 050404 (2003).
[CrossRef]

R. Franzosi and V. Penna, “Chaotic behavior, collective modes, and self-trapping in the dynamics of three coupled Bose-Einstein condensates,” Phys. Rev. E. 67, 046227 (2003).
[CrossRef]

R. Franzosi and V. Penna, “Self-trapping mechanisms in the dynamics of three coupled Bose-Einstein condensates,” Phys. Rev. A. 65, 013601 (2001).
[CrossRef]

R. Franzosi, V. Penna, and R. Zecchina, “Quantum dynamics of coupled bosonic wells within the Bose-Hubbard picture,” Mod. Phys. B 14, 943–961 (2000).
[CrossRef]

Fu, L. B.

B. Liu, L. B. Fu, S. P. Yang, and J. Liu, “Josephson oscillation and transition to self-trapping for Bose-Einstein condensates in a triple-well trap,” Phys. Rev. A. 75, 033601 (2007).
[CrossRef]

Fu, Li-Bin

Xin Jiang, Li-Bin Fu, Wen-shan Duan, and Jie Liu, “Phase transition of the ground state for two-component BoseEinstein condensates in a triple-well trap,” Journal of Phys. B,  44, 115301 (2011).
[CrossRef]

Furuya, K.

T. F. Viscondi and K. Furuya, “Dynamics of a BoseEinstein condensate in a symmetric triple-well trap,” J. Phys. A. 44, 175301 (2011).
[CrossRef]

Gardiner, C. W.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices, ” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Garreau, J. C.

Q. Thommen, J. C. Garreau, and V. Zehnle, “Classical chaos with Bose-Einstein condensates in tilted optical lattices,” Phys. Rev. Lett. 91, 210405 (2003).
[CrossRef] [PubMed]

Gati, R.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

Geisel, T.

M. Hiller, T. Kottos, and T. Geisel, “Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system,” Phys. Rev. A 79, 023621 (2009).
[CrossRef]

M. Hiller, T. Kottos, and T. Geisel, “Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates,” Phys. Rev. A 73, 061604(R) (2006).
[CrossRef]

Giovanazzi, S.

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 79, 4950–4953 (1997).
[CrossRef]

Graefe, E. M.

E. M. Graefe, H. J. Korsch, and D. Witthaut, “Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman adiabatic passage,” Phys. Rev. A 73, 013617 (2006).
[CrossRef]

Greentree, A. D.

C. J. Bradly, M. Rab, A. D. Greentree, and A. M. Martin, “Coherent tunneling via adiabatic passage in a three-well Bose-Hubbard system,” Phys. Rev. A. 85, 053609 (2012).
[CrossRef]

Greiner, M.

M. Greiner, O. Mandel, T. Esslinger, T W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature(London) 415, 39–44 (2002).
[CrossRef]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

Haljan, P. C.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein Condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[CrossRef] [PubMed]

Hänsch, T W.

M. Greiner, O. Mandel, T. Esslinger, T W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature(London) 415, 39–44 (2002).
[CrossRef]

Heinrich, M.

Henderson, K.

K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, “Experimental demonstration of painting arbitrary and dynamic potentials for BoseEinstein condensates,” New J. Phys. 11, 043030 (2009).
[CrossRef]

Hennig, H.

H. Hennig, J. Dorignac, and D. K. Campbell, “Transfer of Bose-Einstein condensates through discrete breathers in an optical lattice,” Phys. Rev. A. 82, 053604 (2010).
[CrossRef]

Hiller, M.

M. Hiller, T. Kottos, and T. Geisel, “Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system,” Phys. Rev. A 79, 023621 (2009).
[CrossRef]

M. Hiller, T. Kottos, and T. Geisel, “Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates,” Phys. Rev. A 73, 061604(R) (2006).
[CrossRef]

Holmes, C. A.

K. Nemoto, C. A. Holmes, G. J. Milburn, and W. J. Munro, “Quantum dynamics of three coupled atomic Bose-Einstein condensates,” Phys. Rev. A. 63, 013604 (2000).
[CrossRef]

Huber, S. D.

Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman, “Many-body LandauZener dynamics in coupled one-dimensional Bose liquid,” Nature Phys. 7, 61–67 (2011).
[CrossRef]

Hulet, R. G.

C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, “Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions,” Phys. Rev. Lett. 75, 1687–1690 (1995).
[CrossRef] [PubMed]

Hunsmann, S.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction,” Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Hur, K. L.

J. Koch and K. L. Hur, “Discontinuous current-phase relations in small one-dimensional Josephson junction arrays,” Phy. Rev. L 101, 097007 (2008)
[CrossRef]

Jaksch, D.

D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices, ” Phys. Rev. Lett. 81, 3108–3111 (1998).
[CrossRef]

Jason, P.

P. Jason, M. Johansson, and K. Kirr, “Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer,” Phys. Rev. E 86, 016214 (2012).
[CrossRef]

Jiang, Xin

Xin Jiang, Li-Bin Fu, Wen-shan Duan, and Jie Liu, “Phase transition of the ground state for two-component BoseEinstein condensates in a triple-well trap,” Journal of Phys. B,  44, 115301 (2011).
[CrossRef]

Johansson, M.

P. Jason, M. Johansson, and K. Kirr, “Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer,” Phys. Rev. E 86, 016214 (2012).
[CrossRef]

M. Johansson, “Hamiltonian Hopf bifurcations in the discrete nonlinear Schrdinger trimer: oscillatory instabilities, quasi-periodic solutions and a ’new’ type of self-trapping transition,” J. Phys. A: Math. Gen. 37, 2201–2222 (2004).
[CrossRef]

Jung, C.

S. Mossmann and C. Jung, “Semiclassical approach to Bose-Einstein condensates in a triple well potential,” Phys. Rev. A 74, 033601 (2006).
[CrossRef]

Kartashov, Y. V.

Kasztelan, C.

C. Kasztelan, S. Trotzky, Y.A. Chen, I. Bloch, I. P. McCulloch, U. Schollwock, and G. Orso, “Landau-Zener Sweeps and Sudden Quenches in Coupled Bose-Hubbard Chains,” Phys. Rev. Lett. 106, 155302 (2011).
[CrossRef] [PubMed]

Keil, R.

Ketterle, W.

Y. Shin, M. Saba, A. Schirotzek, T. A. Pasquini, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, “Distillation of Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 150401 (2004).
[CrossRef]

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 050405 (2004).
[CrossRef]

Kirr, K.

P. Jason, M. Johansson, and K. Kirr, “Quantum signatures of an oscillatory instability in the Bose-Hubbard trimer,” Phys. Rev. E 86, 016214 (2012).
[CrossRef]

Kladko, K.

S. Aubry, S. Flach, K. Kladko, and E. Olbrich, “Manifestation of classical Bifurcation in the spectrum of the integrable quantum dimer,” Phys. Rev. Lett. 76, 1607–1610 (1996).
[CrossRef] [PubMed]

Koch, J.

J. Koch and K. L. Hur, “Discontinuous current-phase relations in small one-dimensional Josephson junction arrays,” Phy. Rev. L 101, 097007 (2008)
[CrossRef]

Kolovsky, A. R.

A. R. Kolovsky, “Semiclassical Quantization of the Bogoliubov Spectrum,” Phys. Rev. Lett. 99, 020401 (2007).
[CrossRef] [PubMed]

Korsch, H. J.

F. Trimborn, D. Witthaut, and H. J. Korsch, “Beyond mean-field dynamics of small Bose-Hubbard systems based on the number-conserving phase-space approach,” Phys. Rev. A 79, 013608 (2009).
[CrossRef]

E. M. Graefe, H. J. Korsch, and D. Witthaut, “Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman adiabatic passage,” Phys. Rev. A 73, 013617 (2006).
[CrossRef]

Kottos, T.

M. Hiller, T. Kottos, and T. Geisel, “Wave-packet dynamics in energy space of a chaotic trimeric Bose-Hubbard system,” Phys. Rev. A 79, 023621 (2009).
[CrossRef]

M. Hiller, T. Kottos, and T. Geisel, “Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates,” Phys. Rev. A 73, 061604(R) (2006).
[CrossRef]

Lahaye, T.

T. Lahaye, T. Pfau, and L. Santos, “Mesoscopic ensembles of polar Bosons in triple-well potentials,” Phys. Rev. Lett. 104, 170404 (2010).
[CrossRef] [PubMed]

Leanhardt, A. E.

Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, “Atom interferometry with Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 050405 (2004).
[CrossRef]

Y. Shin, M. Saba, A. Schirotzek, T. A. Pasquini, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, “Distillation of Bose-Einstein condensates in a double-well potential,” Phys. Rev. L 92, 150401 (2004).
[CrossRef]

Lederer, F.

Li, L. P.

X. B. Luo, L. P. Li, L. You, and B. Wu, “Coherent destruction of tunneling and dark Floquet state,” New J. Phys. 16, 013007 (2014)
[CrossRef]

Li, Y.

A. Sinatra, Y. Castin, and Y. Li, “Particle number fluctuations in a cloven trapped Bose gas at finite temperature,” Phys. Rev. A 81, 053623 (2010).
[CrossRef]

K. Maussang, G. E. Marti, T. Schneider, P. Treutlein, Y. Li, A. Sinatra, and R. Long, “Enhanced and Reduced Atom Number Fluctuations in a BEC Splitter,” J. Estéve and J. Reichel, Phys. Rev. Lett. 105, 080403 (2010).
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Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Lieberman, Regular and stochastic Motion (ASpringer-Verlag, New York) p. 1983B.

Lieberman, M. A.

A. J. Lichtenberg and M. A. Lieberman, Regular and stochastic Motion (ASpringer-Verlag, New York) p. 1983B.

Liu, B.

B. Liu, L. B. Fu, S. P. Yang, and J. Liu, “Josephson oscillation and transition to self-trapping for Bose-Einstein condensates in a triple-well trap,” Phys. Rev. A. 75, 033601 (2007).
[CrossRef]

Liu, J.

B. Liu, L. B. Fu, S. P. Yang, and J. Liu, “Josephson oscillation and transition to self-trapping for Bose-Einstein condensates in a triple-well trap,” Phys. Rev. A. 75, 033601 (2007).
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Figures (11)

Fig. 1
Fig. 1

Illustration of a potential with three mutually coupled wells.

Fig. 2
Fig. 2

Mean-field energy levels at v = 0.1 for different c. (a) c = 0; (b) c = 0.06; (c) c = 0.1; (d) c = 0.2.

Fig. 3
Fig. 3

Mean particle numbers of three wells as a function of c at v = 0.1. The sweeping rate is α = 0.0001. Pi denotes the mean particle number in the ith well. The insets are the enlargement of the sections indicated in dashed boxes.

Fig. 4
Fig. 4

The Poincaré sections at θ3 = 0 for γ = 0.008, c = 0.2, v = 0.1 with different energies E. (a) E = −0.1215, (b) E = −0.1144, (c) E = −0.1103, (d) E = −0.1019, (e) E = −0.0975, (f) E = −0.0939, (g) E = −0.0780, (h) E = 0.0628. The green stars are for energy level D in Fig. 2, the red triangles (left) are for C, the black triangles (right) for B, and the black diamonds for A.

Fig. 5
Fig. 5

The dynamic evolution of the particle numbers Pi = |ψ(t)i|2 in each well. Blue solid line indicates P1, green dashed line is P2, red dash dotted line corresponds to P3, we use c = 0.14, v = 0.1. (a) The initial state is the lowest eigenstate at γ = −3; (b) The initial state is a small perturbation to the initial state in (a). The final mean particle numbers in the three wells are very different, indicating that a small change in the initial condition can cause a big change in the final state.

Fig. 6
Fig. 6

Mean particle numbers of three wells during the adiabatic evolution of γ. The adiabatic rate is α = 0.0001. The initial condition is the lowest eigenstate at γ = −3. (a) v = 0.1, c = 0.12; (b) v = 0.1, c = 0.14. The final probabilities in the three wells are very different, indicating that a small change in the system parameters can also cause a large change in the final state.

Fig. 7
Fig. 7

Mean particle numbers as a function of c in a chain-shaped three-well system [39]. v = 0.1, α = 0.0001. The critical point is labeled by red point F, the critical value of c is around c = 0.076. The insets are the enlargement of the sections indicated in dashed boxes.

Fig. 8
Fig. 8

Energy levels from the second-quantized model for different c: (a) c = 0, (b) c = 0.06, (c) c = 0.1, (d) c = 0.2. The red circles are for the mean-field energy levels; the blue lines stand for the quantized levels. N = 8, v = 0.1

Fig. 9
Fig. 9

Energy levels from the second-quantized model (blue solid lines) and mean-field model (red open circles) for a double-well model and the ring-shaped triple-well model. (a) The energy levels for two-mode model for N = 20, c = 0.4, v = 0.2, the quantized hamiltonian of this model is H ^ = γ ( a ^ 1 a ^ 1 a ^ 2 a 2 ) / 2 + v ( a ^ 1 a ^ 2 + a ^ 1 a ^ 2 ) 2 c ( a ^ 1 a ^ 1 a ^ 2 a ^ 2 ) 2 / ( 4 N ) ) [13]. (b) The energy levels for the ring-shaped triple-well energy for N = 10, c = 0.1, v = 0.1. (c) The enlarged rectangle part in (b). In (a), all the avoided-crossing points are enveloped in the mean-field energy levels. In (b,c), this is not the case; many avoided-crossing points lie outside of the envelope of the mean-field energy levels.

Fig. 10
Fig. 10

Mean particle numbers as a function of c for v = 0.1 in the ring-shaped triple-well model. v = 0.1, α = 0.0001. The solid lines are for the mean-field model; the green, red and black stars are the quantized results for N = 8, 10, 15, respectively.

Fig. 11
Fig. 11

The probability Ai of the ith level (Ai = 〈ϕi|ϕt〉). In (a,b), the first, second, and third largest Ai during the evolution are traced with black, green, and red stars, respectively, on the energy levels. (a) c = 0.12, N = 8, v = 0.1; (b) c = 0.14, N = 8, v = 0.1. (c) The time evolution of A11 (black), A7(green), A6 (red) for the case of (a). (d) The time evolution of A7 (black), A8(green), A11 (red) for the case of (b). The sharp peaks in (c) and (d) are caused by the avoided-crossings.

Tables (2)

Tables Icon

Table 1 The mean particle number Pi in each well at different energy levels. The parameters are γ = 0.4, N = 8, c = 0.12, v = 0.1

Tables Icon

Table 2 The mean particle number Pi in each well at different energy levels. The parameters are γ = 0.4, N = 8, c = 0.14, v = 0.1

Equations (19)

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i t ( a 1 a 2 a 3 ) = H ( γ ) ( a 1 a 2 a 3 ) .
H ( γ ) = ( c | a 1 | 2 + γ v / 2 v / 2 v / 2 c | a 2 | 2 v / 2 v / 2 v / 2 c | a 3 | 2 γ ) .
( a 1 , a 2 , a 3 ) A = ( 1 3 , 1 3 , 1 3 ) ,
μ A = v c 3 ;
( a 1 , a 2 , a 3 ) B = ( 1 3 , 1 2 3 ± 1 2 i , 1 2 3 1 2 i ) ,
μ B = c 3 v 2 .
( a 1 , a 2 , a 3 ) C = ( 0 , b + f i , b f i ) ,
μ C = c + v 2 .
( a 1 , a 2 , a 3 ) D = ( a , a , 1 2 a 2 ) ,
μ D = c a 2 + v ( a + 1 2 a 2 ) / ( 2 a ) ,
H = c / 2 [ n 1 2 + n 3 2 + n 2 2 ] γ ( n 1 n 3 ) + v n 1 n 2 cos θ 1 + v n 2 n 3 cos θ 3 + v n 1 n 3 cos ( θ 3 θ 1 ) .
H = ( c | a 1 | 2 + γ v / 2 0 v / 2 c | a 2 | 2 v / 2 0 v / 2 c | a 3 | 2 γ ) .
H ^ = γ ( a ^ 1 a ^ 1 a ^ 3 a ^ 3 ) + v / 2 ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 + a ^ 2 a ^ 3 + a ^ 3 a 2 + a ^ 3 a 1 + a 1 a 3 ) c / ( 2 N ) ( a ^ 1 a ^ 1 a ^ 1 a ^ 1 + a ^ 2 a ^ 2 a ^ 2 a ^ 2 + a ^ 3 a ^ 3 a ^ 3 a ^ 3 ) .
H = c / 2 [ n 1 2 + n 3 2 + n 2 2 ] γ ( n 1 n 3 ) + v n 1 n 2 cos θ 1 + v n 2 n 3 cos θ 3 + v n 1 n 3 cos ( θ 3 θ 1 ) .
H = c / 2 [ 1 + 2 n 1 2 + 2 n 3 2 + 2 n 1 n 3 2 ( n 1 + n 3 ) ] γ ( n 1 n 3 ) + v n 1 ( 1 n 1 n 3 ) cos θ 1 + v n 3 ( 1 n 1 n 3 ) cos θ 3 + v n 1 n 3 cos ( θ 3 θ 1 ) .
d n 3 d t = v n 3 n 2 sin θ 3 + v n 1 n 3 sin ( θ 3 θ 1 ) ,
d θ 3 d t = c ( n 3 n 2 ) + γ + v n 2 2 n 3 cos θ 3 v 2 n 2 ( n 1 cos θ 1 + n 3 cos θ 3 ) + n 1 2 n 1 n 3 v cos ( θ 3 θ 1 ) ,
d n 1 d t = v n 1 n 2 sin θ 1 + v n 1 n 3 sin ( θ 3 θ 1 ) ,
d θ 1 d t = c ( n 1 n 2 ) γ + v n 2 2 n 3 cos θ 1 v 2 n 2 ( n 1 cos θ 1 + n 3 cos θ 3 ) + n 3 2 n 1 n 3 v cos ( θ 3 θ 1 ) .

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