## Adiabatically induced coherent Josephson oscillations of ultracold atoms in an asymmetric two-dimensional magnetic lattice

Optics Express, Vol. 17, Issue 26, pp. 24358-24370 (2009)

http://dx.doi.org/10.1364/OE.17.024358

Acrobat PDF (4578 KB)

### Abstract

We propose a new method to create an asymmetric two-dimensional magnetic lattice which exhibits magnetic band gap structure similar to semiconductor devices. The quantum device is assumed to host bound states of collective excitations formed in a magnetically trapped quantum degenerate gas of ultracold atoms such as a Bose-Einstein condensate (BEC) or a degenerate Fermi gas. A theoretical framework is established to describe possible realization of the exciton-Mott to discharging Josephson states oscillations in which the adiabatically controlled oscillations induce *ac* and *dc* Josephson atomic currents where this effect can be used to transfer *n* Josephson qubits across the asymmetric two-dimensional magnetic lattice. We consider second-quantized Hamiltonians to describe the Mott insulator state and the coherence of multiple tunneling between adjacent magnetic lattice sites where we derive the self consistent non-linear Schrödinger equation with a proper field operator to describe the exciton Mott quantum phase transition via the induced Josephson atomic current across the *n* magnetic bands.

© 2009 Optical Society of America

## 1. Introduction

4. S. Ghanbari, T. D Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B **39**, 847 (2006). [CrossRef]

6. M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B: At. Mol. Opt. Phys. **41**, 065301 (2008). [CrossRef]

23. S. Whitlock, R. Gerritsma, T. Fernholz, and R. J. C. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. **11**, 023021 (2009). [CrossRef]

30. G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. **82**, 1060 (1999). [CrossRef]

27. S. Ghanbari, P. B. Blakie, P. Hannaford, and T. D. Kien, “Superfluid to Mott insulator quantum phase transition in a 2D permanent magnetic lattice,” Eur. Phys. J. B **70**305 (2009). [CrossRef]

30. G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. **82**, 1060 (1999). [CrossRef]

24. H. Zoubi and H. Ritsch, “Bright and dark excitons in an atom-pairfilled optical lattice within a cavity,” EPL **82**, 14001 (2008). [CrossRef]

25. H. Zoubi and H. Ritsch, “Excitons and cavity polaritons for cold-atoms in an optical lattice,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper QThE3.

31. B. D. Josephson, “Tunneling Into Superconductors,” Phys. Lett. **1**, 251 (1962). [CrossRef]

32. S. Shapiro, “Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations,” Phys. Rev. Lett. **11**, 80 (1963). [CrossRef]

29. S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A **59**, 620 (1999). [CrossRef]

33. S. Giovanazzi, A. Smerzi, and S. Fantoni, “Josephson effects in dilute Bose-Einstein condensates,” Phys. Rev. Lett. **84**, 4521 (2000). [CrossRef] [PubMed]

34. S. Ashhab and Carlos Lobo, “External Josephson effect in Bose-Einstein condensates with a spin degree of freedom,” Phys. Rev. A **66**, 013609 (2002). [CrossRef]

36. B. Juliá-Diaz, M. Guilleumas, M. Lewenstein, A. Polls, and A. Sanpera, “Josephson oscillations in binary mixtures of F=1 spinor Bose-Einstein condensates,” Phys. Rev. A **80**, 023616 (2009). [CrossRef]

35. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science **293**, 843 (2001). [CrossRef] [PubMed]

37. R. Gati, M. Albiez, J. Fölling, B. Hemmerling, and M. K. Oberthaler, “Realization of a single Josephson junction for Bose-Einstein condensates,” Appl. Phys. B **82**, 207 (2006). [CrossRef]

38. B. P. Anderson and M. A. Kasevich, “Macroscopic quantum interference from atomic tunnel arrays,” Science **282**, 1686 (1998). [CrossRef] [PubMed]

## 2. The Asymmetric Two-Dimensional Magnetic Lattice

*d*, from the surface. Depending on the fabricated patterns, the dimension of the periodicity can be selected, where one and two-dimensional magnetic lattices have been achieved and produced by trapping ultracold atoms using patterned permanent magnetic materials [6

_{min}6. M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B: At. Mol. Opt. Phys. **41**, 065301 (2008). [CrossRef]

23. S. Whitlock, R. Gerritsma, T. Fernholz, and R. J. C. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. **11**, 023021 (2009). [CrossRef]

*τ*by milling an

_{btm}*m*×

*m*array of blocks such that each block is an array of

*n*×

*n*square holes, where

*n*represents the number of holes of width

*α*and separated by

_{h}*α*within each block as shown in Figs. 1(a),1(b). The depths of all holes are equal and extend through the magnetic thin film down to the substrate surface level. The magnetic structure is magnetized in its remanently-magnetized state, with the magnetization direction perpendicular to the

_{s}*x-y*plane. The gaps between the blocks containing no holes are assumed to be greater than, or equal to

*α*. The thickness of the gaps is an important design feature since it introduces a control over an extra degree of confinement which is realized through the creation of magnetic field walls encircling the

_{s}*n*×

*n*matrices, and isolating them from one another. It also can control the magnetic bottom,

*B*, and the distance from the surface,

_{min}*d*, of the sites at the center of the magnetic lattice.

_{min}*z*direction and creating magnetic field minima that are located at effective working distances,

*d*, above the plane of the thin film as shown in Figs. 1(d)–1(f). These minima are localized in confining volumes representing the magnetic potential wells that contain a certain number of quantized energy levels occupied by the ultracold atoms. In our design, we assumed that the width of the holes

_{min}*α*and the separation of the holes

_{h}*α*are equal,

_{s}*α*=

_{h}*α*≡a, to simplify the mathematical derivations and analyses which are similar to those reported in [4

_{s}4. S. Ghanbari, T. D Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B **39**, 847 (2006). [CrossRef]

7. S. Ghanbari, T. D. Kieu, and P. Hannaford, “A class of permanent magnetic lattices for ultracold atoms,” J. Phys. B: At. Mol. Opt. Phys. **40**, 1283 (2007). [CrossRef]

### 2.1. Detailed Analysis of the Distribution of the Magnetic Field Minima

*B*and

_{x}, B_{y}*B*can be written analytically as a combination of a field decaying away from the surface of the trap in the z-direction and a periodically distributed magnetic field in the

_{z}*x-y*plane produced by the magnetic induction,

*B**=*

_{o}*µ*=

_{o}M_{z}*π*at the surface of the permanently magnetized thin film. We define a surface reference magnetic field as

*=*

**B**_{ref}*(1-*

**B**_{o}*e*

^{-βτ}), where

*β*=

*π/α*, and

*τ*=

*τ*denotes the magnetic film thickness and a plane of symmetry is assumed at

_{btm}*z*=0. The analysis of the surface magnetic field includes components of external magnetic bias fields along the

*x;y*and

*z*directions,

*,*

**B**_{x-bias}*and*

**B**_{y-bias}*, respectively. Taking into account the surface effective field,*

**B**z_{-bias}*, and the characteristic periodicity interval*

**B**_{ref}*α*, analytical expressions can be derived to describe the periodically distributed local minima across the

*x-y*plane of the magnetic thin film for the case of an infinite magnetic lattice as follows

*d*, namely larger than

_{min}*α*/2

*π*, the cold atoms effectively interact with the local magnetic minima are loaded into the lattice sites. Thus the higher order terms in these equations can be neglected for

*d*>

_{min}*α*/2

*π*reducing Eqs. (1),(2),(3) to the following simplified set of expressions

*B*of the magnetic field at

*d*above the surface of the magnetic film can be written as

_{min}*α*, is reported elsewhere [2].

_{s}, α_{h}### 2.2. Magnetic Band Structure in the Asymmetric Two-Dimensional Magnetic Lattice

*B*and their magnetic bottoms are displaced in the gravitational field z-direction by a titling potential

*δB*. Both

*ΔB*and

*δB*can be controlled by applying an external magnetic bias field along the negative direction of the

*z*-axis, as shown in Fig. 2(i).

*B*, are distributed in space downward to the edges of the lattice, see Fig. 2(f). The distribution has a pyramid shape in the

_{min}*x-y*plane where each level has its

*B*value and the values of the

^{z}_{min}*B*are spaced along the

_{min}*z*-axis by a tilting potential δB. The potential tilt creates a gap between each two sets of magnetic minima distributed in two adjacent bands. This configuration exhibits a scenario similar to that of the energy band gap structure in semiconductor devices. We denote the pyramid-like distribution by a magnetic band gap structure in our proposed two-dimensional magnetic lattice. A schematic representation for the pyramid-like distribution is shown in Fig. 3.

*B*and

_{x-bias}*B*, all sites have magnetic minima close to zero. Once the external magnetic bias field is applied the values of the magnetic minima increase and differ from neighboring sites by the titling magnetic potential

_{y-bias}*δB*as shown in Fig. 2(d). The amount of tilt

*δB*can be calculated from

*i*=0,1,2; …,

*n*is the non-zero local minimum index at each band represented by

*l*=0,1,2,…,

*N*assuming there are

*N*magnetic bands starting from the center of the lattice. The magnetic band gap is given by the difference between the maximum of the tunneling barrier,

*B*, and the magnetic bottom of the lattice site,

_{max}*B*, and is denoted by the tunneling barrier height

_{min}*of the harmonic potential wells in which Λ*

_{depth}*of an individual potential well can be expressed as*

_{depth}**x**={

*x,y,z*}. The

*g*is the Landé

_{F}*g*-factor,

*µ*is Bohr magneton,

_{B}*F*is the atomic hyperfine state with the magnetic quantum number

*m*and

_{F}*k*is the Boltzmann constant. In Fig. 4, we show the measurement results for a fabricated 2×2 blocks of 9×9 asymmetric two-dimensional magnetic lattice. The quantum device is fabricated using the dual electron-focused ion beams technology and imaged with scanning electron microscope and the atomic force microscope, as shown in Figs. 4(a)–4(b). The structure is designed in such a way that allows in-situ magnetic field bias to be applied along the

_{B}*x*-axis. The measurement output and the simulation results are shown in Figs. 4(c)–4(d).

## 3. Tunneling Mechanisms in the Two-Dimensional Magnetic Lattice

9. Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, “Optical weak link between two spatially Separated Bose-Einstein Condensates,” Phys. Rev. Lett. **95**, 170402 (2005). [CrossRef] [PubMed]

10. 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]

11. M. Rigol, V. Rousseau, R. T. Scalettar, and R. R. P. Singh, “Collective Oscillations of Strongly Correlated One-Dimensional Bosons on a Lattice,” Phys. Rev. Lett. **95**, 110402 (2005). [CrossRef] [PubMed]

*z*-axis. Atoms tunnel through the magnetic barriers from the highest magnetic band to the neighboring lowest magnetic band following the pyramid-like distribution of energy levels. Figures 2(b)–2(c) shows simulation results of two adjacent edge lattice sites where the gap can also be regarded as the difference between the heights of the two sites in the earth’s gravitational fields.

*ϕ*and a vibrational excited state

^{g}_{i,j}*ϕ*in each single lattice site at the

^{e}_{i,j}*i*(or

^{th}*j*) magnetic band, where

^{th}*i*≠

*j*∈ {1,…,

*n*} is the potential well index starting from the center site. The sites are characterized by a configured magnetic bottom

*B*to trap alkali atoms which are magnetically prepared in a low magnetic field seeking state. Our system consists of

_{min}*n*quantum wells (QWs) that are indirectly coupled via the magnetic band gap. In the asymmetrical QWs, we consider the two lowest energy state, in each individual potential well, i.e., an individual lattice site, are closely spaced and well separated from the other higher levels within the lattice site. This is a picture of a tow-level quantum system with negligible interaction between the many bosons distributed in the two energy levels of the system, permitting the two-mode approximation of the many-body problem in our proposed magnetic lattice [9

9. Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, “Optical weak link between two spatially Separated Bose-Einstein Condensates,” Phys. Rev. Lett. **95**, 170402 (2005). [CrossRef] [PubMed]

10. 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]

13. B. J. Dalton, “Two-mode theory of BEC interferometry,” J. Mod. Opt. **54**, 615 (2007). [CrossRef]

**Φ**̂

^{†}(

**x**), and annihilation,

**Φ**̂F(

**x**), field operators, for a system of

*N*interacting boson of mass

*confined by an external magnetic potential*

**M****(**

*B**x*) at zero temperature is given by

*U*(

_{int}**x**-

**x**́)→

*g*=4

*πħ*

^{2}

*a*=

_{s}*M*. The coupling constant

*g*is determined from the scattering length

*a*. For an

_{s}*n*×

*n*asymmetric magnetic lattice with

*n*magnetic bands and no tunneling between sites, i.e., the case of uncoupled magnetic bands, the individual lattice site

*i*of the two energy levels

*k**{0,1} allows a localized single wave function of the condensate to be in the ground state, taking the form

*m*accounts for the quanta of angular momentum in the

*z*-direction, where the value of the index

*m*depends on the value of

*k*such that for

*k*=0 there is no angular momentum, i.e.,

*m*=0, and for

*k*=1,

*m*∈{-1,0,1} in the three dimensions [14,15

15. S. Fölling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera1, T. Müller, and I. Bloch, “Direct observation of second-order atom tunnelling,” Nature **448**, 1029 (2007). [CrossRef] [PubMed]

*ϕ*

^{[0,m]}

*, is essentially symmetric and the second mode,*

_{i}*ϕ*

^{[1,m]}

*i*, is antisymmetric. Due to the initially tilted

*n*potential wells, the tunneling of the condensate produces a superposition between the higher magnetic band ground state mode

*ϕ*

^{[0,m]}

*and the adjacent lower band excited mode,*

_{i}*ϕ*

^{[1,m]}

*. The superposition amplitude strongly depends on the spatial separation of the wells determined by*

_{j}*α*where Δ

_{s}*=*

^{Bi}*B*-

^{i}_{max}*B*

^{i}_{min}*≈*α

^{2}=2 [19]. Dynamically, interacting bosonic cold atoms in n potential wells with tight

**(**

*B***) magnetic field confinement can be described by generalizing, for**

*x**n*sites, the quantized Josephson or a two-mode Bose-Hubbard Hamiltonian, assuming the site number distribution starts from the center of the magnetic lattice

**b**̂

^{†}

*and*

_{i,j}**b**̂

*are the bosonic creation and annihilation operators obeying the canonical commutation rules, and*

_{i,j}**n**

*=*

_{i,j}**b**̂

^{†}

_{i,j}**b**̂

*is the bosons number operator and*

_{i,j}*δB*is the tilting magnetic potential. The

*J*

^{[k,m]}

_{i,j}and

*U*

^{[k,m]}

*are tunneling and inter-lattice site boson-boson interaction parameters, respectively, defined as*

_{i,j}*δB*=0, the Bose-Hubbard Hamiltonian describes the noninteracting magnetic bands where the condensate is assumed to occupy the ground state

*ϕ*

^{[0,0]}

*of an individual lattice site in each magnetic band with a single-mode one-level configuration regardless of the existence of the excited mode. This is because the no-tunneling condition results in inter-site many boson interaction which creates a lattice structure described in the Fock regime as will be explained in the following section. A typical characteristic that can be encountered in such a situation is the Mott insulator state [13*

_{i,j}13. B. J. Dalton, “Two-mode theory of BEC interferometry,” J. Mod. Opt. **54**, 615 (2007). [CrossRef]

27. S. Ghanbari, P. B. Blakie, P. Hannaford, and T. D. Kien, “Superfluid to Mott insulator quantum phase transition in a 2D permanent magnetic lattice,” Eur. Phys. J. B **70**305 (2009). [CrossRef]

*ϕ*

^{[0,m]}

*in the QW*

_{i}*and the vibrational excited state*

_{i}*ϕ*

^{[1,m]}j in the QW

*. This is a simultaneous bi-directional Josephson transition between each pair of adjacent magnetic bands propagating from the center towards the edges of the lattice, i.e. along*

_{j}*x,y*and -

*x,-y*directions, as schematically represented in Fig. 5(b).

*n*creates interesting configurations at the center of the lattice. When

*n*takes odd values, i.e.,

*n*=5,7,9,…, there is only one center site screened by the surrounding four sites. In this case of a single center site, the induced tunneling interaction is dominated by the equivalent Coulomb potentials between the four surrounding sites distributed along the

*x-y*magnetic bands and taking a molecule-like configuration which can be created only in bound states assuming that there are symmetrical tunneling amplitude in the four directions. When

*n*takes even values, i.e.,

*n*=4,6,8, …, there are four symmetrical center sites which exhibit a first Brillouin zone dimensionality and hence have Bloch interacting wave functions and a field operator expanded in localized single-particle wave functions of the form

*ϕ*

^{[k,m]}

_{1,γ}represents the Bloch wave function with

*k*=0, and g is the wave vector in the first Brillouin zone.

*ϕ*

^{[k,m]}

_{1},

*γ*via the standard eigenvalue problem

*n*is even [8

8. V. S. Shchesnovich and V. V. Konotop, “Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival,” Phys. Rev. A **75**, 063628 (2007). [CrossRef]

*δB*=0. When the tunneling is allowed between the four sites only the excited states,

*ϕ*

^{[1,m]}

_{1}contribute to the local tunneling amplitude between the four sites where the superposition state is described for the four center sites {

*a*;

*b*;

*c*;

*d*} via the coupling amplitude such that

*J*

^{[1,m]}

_{1}[

*ϕ*

^{[1,m]}

_{1},a

*ϕ**

^{[1,m]}

_{1},

*b+ϕ*

^{[1,m]}1,

*bϕ**

^{[1,m]}

_{1},

*+*

_{c}*ϕ*

^{[1,m]}

_{1,c}ϕ*

^{[1,m]}

_{1,d}+

*ϕ*

^{[1,m]}

_{1,d}ϕ*

^{[1,m]}

_{1,a}].

*ϕ*

^{[0,m]}

_{1}contribute to the interaction mechanisms between the four center sites and the sites of the surrounding first magnetic band where the local interaction of the excited mode,

*ϕ*

^{[1,m]}

_{1}, between the four sites is assumed to be negligible. As will be described in the following section, this picture can be generalized to describe, regardless of the asymmetrical feature, the Mott insulator quantum phase transition across the

*n*×

*n*two-dimensional magnetic lattice via the discharging Josephson state. The quantum phase transition can be oscillating adiabatically, a significant feature for quantum information processing in such type of magnetic lattices.

## 4. Josephson Oscillations and the Excitons Mott Phase Transition

16. 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 (1997). [CrossRef]

*ϕ*

^{[0,m]}

*is regarded as the ground state and as a solution for the Schrödinger equation with negligible inter-site interactions regardless of the existence of the excited mode*

_{i, j}*ϕ*

^{[1,m]}

*, the condensate is said to have an individual mode solution of the form*

_{i,j}*ϕ*=

^{g}_{i,j}*ϕ*

^{[0,m]}

*(*

_{i,j}**x**-

**x**́) [19]. Initially, there is an approximately equal number of atoms in each site

*N*≈

_{s}*N=n*, where this number is fixed until tunneling is allowed. The atoms are completely localized at the lattice site where we assume the coupling strength between the two levels is very weak such that

*E*

^{[k,m]}

*is the energy difference. Localization in our scenario means that the first symmetric mode,*

_{i,j}*ϕ*, is dominating over all lattice sites and the expansion of the excited mode,

^{g}_{i,j}*ϕ*, between each two adjacent magnetic bands is negligible. This condition can be thought of as the Fock regime where the on-site interaction is greater than the hopping strength,

^{e}_{i,j}*U*

^{[0,m]}

*↔*

_{i}*U*

^{[1,m]}

*≫*

_{i}*J*

^{[0→1,m]}

*. It is a signature of the Mott insulator state where all particles are localized in the ground state, neglecting the excited level, with a defined number of atoms at each site exhibiting no coherence nor a macroscopic wave function and permitting a one-level approximation having a one-level Bose-Hubbard Hamiltonian for the*

_{i↔j}*n*×

*n*asymmetric magnetic lattice of the form

*δB=0*, and

*H*̂

_{1HB}describes a finite number of sites. In the case of an infinite number of sites, Eq. (18) describes the single-band Bose-Hubbard Hamiltonian for weakly interacting bosons in a symmetric magnetic lattice, i.e.,

*H*̂

_{1HB}→

*H*̂

*. Note that the operators*

_{HB}**b**̂

^{†}

*,*

_{i,j}**b**̂

*and*

_{i,j}**n**̂

*are for the case of*

_{i,j}*k*=

*m*=0. The Fock space state vector takes the form

*N*+1) and

*c*

^{[0,0]}

*represents the ground state amplitude.*

_{i,j}17. J. Javanainen, “Oscillatory exchange of atoms between traps containing Bose condensates,” Phys. Rev. Lett. **57**, 3164 (1986). [CrossRef] [PubMed]

18. A. Smerzi, A. Trombettoni, T. Lopez-Arias, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Macroscopic oscillations between two weakly coupled Bose-Einstein condensates,” Eur. Phys. J. B **31**, 457 (2003). [CrossRef]

28. J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, “Nonlinear Josephson-type oscillations of a driven, two-component Bose-Einstein condensate,” Phys. Rev. A **59**, R31 (1999). [CrossRef]

*n*asymmetric magnetic bands where the field operator is expanded in terms of the static ground state solution,

*ϕ*, of the GPE Eq. (21) for each individual uncoupled lattice site, and in terms of the amplitude of the relative population

^{g}_{i}*N*≡

_{s}*N*expressed as

^{s}_{i}*θ*is the corresponding phase [29

_{i}29. S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A **59**, 620 (1999). [CrossRef]

*N*(

^{s}_{i}*t*) of the

*n*sites can be obtained by integrating the spatial distributions

*φ*(

_{i}*x,t*) to obtain the time dependent equation [16

16. 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 (1997). [CrossRef]

18. A. Smerzi, A. Trombettoni, T. Lopez-Arias, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Macroscopic oscillations between two weakly coupled Bose-Einstein condensates,” Eur. Phys. J. B **31**, 457 (2003). [CrossRef]

29. S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A **59**, 620 (1999). [CrossRef]

*n*sites. A similar case of a double well is discussed in [19]. The transition from self-trapping to Josephson oscillating states [10

10. 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]

12. M. Holthaus and S. Stenholm, “Coherent control of the self-trapping transition,” Eur. Phys. J. B **20**, 451 (2001). [CrossRef]

*B-*, the number of atoms in each site

_{z-bias}*N*varies in time to a critical number of atoms that is periodically oscillating and predicted to produce the n magnetic inter-band Josephson oscillations. Schematic representations are shown in Fig. 5. The quantum state of the asymmetrical magnetic lattice at this point of the transition belongs to what is known as the Josephson regime, where the in-site (inter-well) many boson interaction of the two energy states

_{s}*k*=0 and

*k*=1 is very small compared to the off-site many boson interaction, namely the condition

*n*magnetic band one-directional Josephson transition,

*ϕ*

^{[0,m]}

*→*

_{i}*ϕ*

^{[1,m]}

*. Oscillating bound states will be created in both situations, on and off-sites leading to exciton Mott multi-transitions. The on-site bound state is created between the ground state and the excited state, e.g., two different components of a spinor BEC trapped in a single site. This is due to the dipole interaction which occurs when the condensate in the excited state periodically couples to the oscillating ground state at each individual site. The number of atoms may play an important role in such a scenario where it might be required to have an equal number of atoms in both modes [20*

_{j}20. M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, William D. Phillips, and J. V. Porto, “Controlled exchange interaction between pairs of neutral atoms in an optical lattice,” Nature **448**, 452 (2007). [CrossRef] [PubMed]

22. W. Zhang, S. Yi, and L. You, “Bose-Einstein condensation of trapped interacting spin-1 atoms,” Phys. Rev. A **70**, 043611 (2004). [CrossRef]

**95**, 010402 (2005). [CrossRef] [PubMed]

## 5. Conclusion

*dc*and

*ac*Josephson current, a significant feature that can be used to encode and transfer, across the 2D asymmetrical lattice with

*n*×

*n*Josephson qubits for quantum information processing. We have established a theoretical framework that can be used to calculate the relevant parameters required to describe the fundamental concepts.

## Acknowledgment

## References and links

1. | L. Perakis, “Condensed-matter physics: Exciton developments,” Nature |

2. | A. Abdelrahman, P. Hannaford, M. Vasiliev, and K. Alameh, “Asymmetric Two-dimensional Magnetic Lattices for Ultracold Atoms Trapping and Confinement,” in progress, arXiv:0910.5032v1 [quant-ph] (2009). |

3. | L. V. Butov, A. C. Gossard, and D. S. Chemla, “Towards Bose-Einstein condensation of excitons in potential traps,” Nature |

4. | S. Ghanbari, T. D Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B |

5. | B.V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford, and A. Sidorov, “A permanent magnetic film atom chip for Bose-Einstein condensation,” J. Phys. B: At. Mol. Opt. Phys. |

6. | M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B: At. Mol. Opt. Phys. |

7. | S. Ghanbari, T. D. Kieu, and P. Hannaford, “A class of permanent magnetic lattices for ultracold atoms,” J. Phys. B: At. Mol. Opt. Phys. |

8. | V. S. Shchesnovich and V. V. Konotop, “Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival,” Phys. Rev. A |

9. | Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, “Optical weak link between two spatially Separated Bose-Einstein Condensates,” Phys. Rev. Lett. |

10. | 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. |

11. | M. Rigol, V. Rousseau, R. T. Scalettar, and R. R. P. Singh, “Collective Oscillations of Strongly Correlated One-Dimensional Bosons on a Lattice,” Phys. Rev. Lett. |

12. | M. Holthaus and S. Stenholm, “Coherent control of the self-trapping transition,” Eur. Phys. J. B |

13. | B. J. Dalton, “Two-mode theory of BEC interferometry,” J. Mod. Opt. |

14. | D. R. Dounas-Frazer and L. D. Carr, “Tunneling resonances and entanglement dynamics of cold bosons in the double well,” quant-ph/0610166 (2006). |

15. | S. Fölling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera1, T. Müller, and I. Bloch, “Direct observation of second-order atom tunnelling,” Nature |

16. | A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum coherent atomic tunneling between two Trapped Bose-Einstein condensates,” Phys. Rev. Lett. |

17. | J. Javanainen, “Oscillatory exchange of atoms between traps containing Bose condensates,” Phys. Rev. Lett. |

18. | A. Smerzi, A. Trombettoni, T. Lopez-Arias, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, “Macroscopic oscillations between two weakly coupled Bose-Einstein condensates,” Eur. Phys. J. B |

19. | G. J. Milburn and J. Corney, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. Lett. |

20. | M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, William D. Phillips, and J. V. Porto, “Controlled exchange interaction between pairs of neutral atoms in an optical lattice,” Nature |

21. | J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, “Spin domains in ground state spinor Bose-Einstein condensates,” Nature |

22. | W. Zhang, S. Yi, and L. You, “Bose-Einstein condensation of trapped interacting spin-1 atoms,” Phys. Rev. A |

23. | S. Whitlock, R. Gerritsma, T. Fernholz, and R. J. C. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. |

24. | H. Zoubi and H. Ritsch, “Bright and dark excitons in an atom-pairfilled optical lattice within a cavity,” EPL |

25. | H. Zoubi and H. Ritsch, “Excitons and cavity polaritons for cold-atoms in an optical lattice,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper QThE3. |

26. | A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannaford, Yong-Tak Lee, and Byoung S. Ham, “Towards Bose-Einstein condensation of excitons in an asymmetric multi-quantum state magnetic lattice,” Numerical Simulation of Optoelectronic Devices (NUSOD) (2009). |

27. | S. Ghanbari, P. B. Blakie, P. Hannaford, and T. D. Kien, “Superfluid to Mott insulator quantum phase transition in a 2D permanent magnetic lattice,” Eur. Phys. J. B |

28. | J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, “Nonlinear Josephson-type oscillations of a driven, two-component Bose-Einstein condensate,” Phys. Rev. A |

29. | S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A |

30. | G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. |

31. | B. D. Josephson, “Tunneling Into Superconductors,” Phys. Lett. |

32. | S. Shapiro, “Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations,” Phys. Rev. Lett. |

33. | S. Giovanazzi, A. Smerzi, and S. Fantoni, “Josephson effects in dilute Bose-Einstein condensates,” Phys. Rev. Lett. |

34. | S. Ashhab and Carlos Lobo, “External Josephson effect in Bose-Einstein condensates with a spin degree of freedom,” Phys. Rev. A |

35. | F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science |

36. | B. Juliá-Diaz, M. Guilleumas, M. Lewenstein, A. Polls, and A. Sanpera, “Josephson oscillations in binary mixtures of F=1 spinor Bose-Einstein condensates,” Phys. Rev. A |

37. | R. Gati, M. Albiez, J. Fölling, B. Hemmerling, and M. K. Oberthaler, “Realization of a single Josephson junction for Bose-Einstein condensates,” Appl. Phys. B |

38. | B. P. Anderson and M. A. Kasevich, “Macroscopic quantum interference from atomic tunnel arrays,” Science |

**OCIS Codes**

(020.1475) Atomic and molecular physics : Bose-Einstein condensates

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: November 2, 2009

Manuscript Accepted: December 5, 2009

Published: December 18, 2009

**Citation**

A. Abdelrahman, P. Hannaford, and K. Alameh, "Adiabatically induced coherent Josephson oscillations of ultracold atoms in an asymmetric two-dimensional magnetic lattice," Opt. Express **17**, 24358-24370 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24358

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### References

- L. Perakis, "Condensed-matter physics: Exciton developments," Nature 33, 417 (2002).
- A. Abdelrahman, P. Hannaford, M. Vasiliev, and K. Alameh, "Asymmetric Two-dimensional Magnetic Lattices for Ultracold Atoms Trapping and Confinement," in progress, arXiv:0910.5032v1 [quant-ph] (2009).
- L. V. Butov, A. C. Gossard, and D. S. Chemla, "Towards Bose-Einstein condensation of excitons in potential traps," Nature 47, 417 (2002).
- S. Ghanbari, T. D Kieu, A. Sidorov, and P. Hannaford, "Permanent magnetic lattices for ultracold atoms and quantum degenerate gases," J. Phys. B 39, 847 (2006). [CrossRef]
- B.V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford, and A. Sidorov, "A permanent magnetic film atom chip for Bose-Einstein condensation," J. Phys. B: At. Mol. Opt. Phys. 39, 27 (2006). [CrossRef]
- M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, "One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip," J. Phys. B: At. Mol. Opt. Phys. 41, 065301 (2008). [CrossRef]
- S. Ghanbari, T. D. Kieu, and P. Hannaford, "A class of permanent magnetic lattices for ultracold atoms," J. Phys. B: At. Mol. Opt. Phys. 40, 1283 (2007). [CrossRef]
- V. S. Shchesnovich and V. V. Konotop, "Nonlinear tunneling of Bose-Einstein condensates in an optical lattice: Signatures of quantum collapse and revival," Phys. Rev. A 75, 063628 (2007). [CrossRef]
- Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, "Optical weak link between two spatially Separated Bose-Einstein Condensates," Phys. Rev. Lett. 95, 170402 (2005). [CrossRef] [PubMed]
- M. Albiez, R. Gati, J. Folling, 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]
- M. Rigol, V. Rousseau, R. T. Scalettar, and R. R. P. Singh, "Collective Oscillations of Strongly Correlated One- Dimensional Bosons on a Lattice," Phys. Rev. Lett. 95, 110402 (2005). [CrossRef] [PubMed]
- M. Holthaus and S. Stenholm, "Coherent control of the self-trapping transition," Eur. Phys. J. B 20, 451 (2001). [CrossRef]
- B. J. Dalton, "Two-mode theory of BEC interferometry," J. Mod. Opt. 54, 615 (2007). [CrossRef]
- D. R. Dounas-Frazer and L. D. Carr, "Tunneling resonances and entanglement dynamics of cold bosons in the double well," quant-ph/0610166 (2006).
- S. Folling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera1, T.Muller, and I. Bloch, "Direct observation of second-order atom tunnelling," Nature 448, 1029 (2007). [CrossRef] [PubMed]
- 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 (1997). [CrossRef]
- J. Javanainen, "Oscillatory exchange of atoms between traps containing Bose condensates," Phys. Rev. Lett. 57, 3164 (1986). [CrossRef] [PubMed]
- A. Smerzi, A. Trombettoni, T. Lopez-Arias, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, "Macroscopic oscillations between two weakly coupled Bose-Einstein condensates," Eur. Phys. J. B 31, 457 (2003). [CrossRef]
- G. J. Milburn and J. Corney, "Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential," Phys. Rev. Lett. 55, 4318 (1997).
- M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, William D. Phillips, and J. V. Porto, "Controlled exchange interaction between pairs of neutral atoms in an optical lattice," Nature 448, 452 (2007). [CrossRef] [PubMed]
- J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, "Spin domains in ground state spinor Bose-Einstein condensates," Nature 396, 345 (1998). [CrossRef]
- W. Zhang, S. Yi, and L. You, "Bose-Einstein condensation of trapped interacting spin-1 atoms," Phys. Rev. A 70, 043611 (2004). [CrossRef]
- S. Whitlock, R. Gerritsma, T. Fernholz, and R. J. C. Spreeuw, "Two-dimensional array of microtraps with atomic shift register on a chip," New J. Phys. 11, 023021 (2009). [CrossRef]
- H. Zoubi and H. Ritsch, "Bright and dark excitons in an atom-pairfilled optical lattice within a cavity," EPL 82, 14001 (2008). [CrossRef]
- H. Zoubi and H. Ritsch, "Excitons and cavity polaritons for cold-atoms in an optical lattice," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper QThE3.
- A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannaford, Yong-Tak Lee, and Byoung S. Ham, "Towards Bose- Einstein condensation of excitons in an asymmetric multi-quantum state magnetic lattice," Numerical Simulation of Optoelectronic Devices (NUSOD) (2009).
- S. Ghanbari, P. B. Blakie, P. Hannaford, and T. D. Kien, "Superfluid to Mott insulator quantum phase transition in a 2D permanent magnetic lattice," Eur. Phys. J. B 70305 (2009). [CrossRef]
- J. Williams, R. Walser, J. Cooper, E. Cornell, and M. Holland, "Nonlinear Josephson-type oscillations of a driven, two-component Bose-Einstein condensate," Phys. Rev. A 59, R31 (1999). [CrossRef]
- S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, "Coherent oscillations between two weakly coupled Bose- Einstein condensates: Josephson effects, p oscillations, and macroscopic quantum self-trapping," Phys. Rev. A 59, 620 (1999). [CrossRef]
- G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, "Quantum logic gates in optical lattices," Phys. Rev. Lett. 82, 1060 (1999). [CrossRef]
- B. D. Josephson, "Tunneling Into Superconductors," Phys. Lett. 1, 251 (1962). [CrossRef]
- S. Shapiro, "Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations," Phys. Rev. Lett. 11, 80 (1963). [CrossRef]
- S. Giovanazzi, A. Smerzi, and S. Fantoni, "Josephson effects in dilute Bose-Einstein condensates," Phys. Rev. Lett. 84, 4521 (2000). [CrossRef] [PubMed]
- S. Ashhab and Carlos Lobo, "External Josephson effect in Bose-Einstein condensates with a spin degree of freedom," Phys. Rev. A 66, 013609 (2002). [CrossRef]
- F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, "Josephson junction arrays with Bose-Einstein condensates," Science 293, 843 (2001). [CrossRef] [PubMed]
- B. Julia-Diaz, M. Guilleumas, M. Lewenstein, A. Polls, and A. Sanpera, "Josephson oscillations in binary mixtures of F=1 spinor Bose-Einstein condensates," Phys. Rev. A 80, 023616 (2009). [CrossRef]
- R. Gati, M. Albiez, J. Folling, B. Hemmerling, and M. K. Oberthaler, "Realization of a single Josephson junction for Bose-Einstein condensates," Appl. Phys. B 82, 207 (2006). [CrossRef]
- B. P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686 (1998). [CrossRef] [PubMed]

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