An insulating optical lattice with double-well sites is considered. In the case of the unity filling factor, an effective Hamiltonian in the pseudospin representation is derived. A method is suggested for manipulating the properties of the system by varying the shape of the double-well potential. In particular, it is shown that the atomic imbalance can be varied at will and a kind of the Morse-alphabet sequences can be created.
Deep Dive into Regulating atomic imbalance in double-well lattices.
An insulating optical lattice with double-well sites is considered. In the case of the unity filling factor, an effective Hamiltonian in the pseudospin representation is derived. A method is suggested for manipulating the properties of the system by varying the shape of the double-well potential. In particular, it is shown that the atomic imbalance can be varied at will and a kind of the Morse-alphabet sequences can be created.
Physics of cold atomic gases, Bose [1][2][3][4][5][6][7][8] and Fermi [9,10], is a fastly developing field of research, both theoretically and experimentally. Cold atoms in optical lattices provide a versatile tool for various applications [11][12][13][14]. Recently, a novel type of optical lattices has been realized in experiments [15][16][17], the so-called double-well lattices. Each site of such a lattice is formed by a double-well potential.
Generally, the theoretical description of the double-well lattices is essentially more complicated than that of the usual lattices. Therefore, in order to describe the properties of such lattices, it is useful to consider some particular types of them.
In the present paper, we consider a particular kind of the double-well lattice, which is characterized by the following main features: atoms in the lattice are in an insulating state, and the atomic filling factor is equal to unity. Such a type of a lattice is of principal importance providing a convenient setup for realizing quantum information processing and quantum computing [11]. Another important property of the lattices, we shall be considering, is the existence of atomic interactions between different lattice sites. The principal goal of the present paper is to demonstrate that the atomic imbalance between the wells of each double well can be manipulated and that arbitrary sequences of the order-disorder transition can be generated.
We start with the standard form of the energy Hamiltonian, expressed through the field operators in the Heisenberg representation. Since the system is assumed to be in the insulating state, the field operators can be expanded over localized orbitals, which yields the Hamiltonian
in which c nj is a field operator labelled by the quantum index n and the site number j, related to a lattice vector a j . The atoms can be either bosons or fermions, with the operator commutation relations
where the commutator is assumed for bosons and anticommutator for fermions. The value E n represents the energy levels of an atom in a double-well located at a site of the lattice. The quantity Φ n 1 n 2 n 3 n 4 j 1 j 2 j 3 j 4 is a matrix element of the interaction potential with respect to the localized orbitals labelled by the indices n and j. We assume that the interaction potential is sufficiently strong, so that the interactions of atoms between at least nearest-neighbor sites cannot be neglected. This can be easily achieved, for instance, with the long-range potentials, such that exist between polar molecules and between Rydberg atoms [18] or between the atoms with large magnetic moments [19]. A known example of the latter atoms is Cr that can be cooled to ultracold temperatures [20]. In the case of ions, long-range forces would be due to the Coulomb interaction.
The assumption that each lattice site contains just a single atom can be formalized by means of the unipolarity condition
This is the known condition, used earlier by Bogolubov [21] for treating ferromagnets. Keeping in mind low temperatures, we can consider only two lowest energy levels of a double-well potential, enumerated by n = 1, 2, so that E 1 < E 2 . As is known [22], the ground-state orbital, of an atom in a double-well potential, is symmetric with respect to spatial inversion, while that for the first excited state is antisymmetric.
Retaining two lowest levels allows us to invoke the pseudospin representation by introducing the pseudospin operators
To understand the meaning of these operators, one can introduce the left and right location operators, respectively, as
Then operators (3) become
Note that this representation is valid for both statistics, so that atoms can be either bosons or fermions. The physical meaning of these operators is as follows: S x j characterizes the tunneling intensity between the wells of a double-well potentials, S y j describes the Josephson current between the wells, and S z j defines the atomic imbalance between the wells. It is worth emphasizing that for a double well it is necessary to take into account at least two lowest energy levels, but not just one ground state, since only with two levels there exists tunneling between the wells. Actually, it is exactly because of the tunneling that the energy levels split into pairs [22].
Let us use the notation
and
where the sign plus or minus is for bosons or fermions, respectively. Also, let us denote the matrix elements
An important quantity is the tunneling frequency
C ij (7) characterizing the atomic tunneling between the wells of a double-well potential.
Using the above conditions and notations, we reduce Hamiltonian (1) to the pseudospin form
The first two terms do not contain operators, so do not play role in what follows. The magnitude of B ij can be comparable with Ω, hence, it cannot be neglected. The value of the tunneling frequency Ω can be varied in a wide range, depending on the shape of the double-well potential. To illust
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