Difference between revisions of "Superfluid to Mott Insulator Transition"
imported>Woochang m (commented out old descriptions on band structure) |
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matter systems, and can be explored deeply with BECs. | matter systems, and can be explored deeply with BECs. | ||
+ | The discussion of superfluid to Mott insulator (SF-MI) transition involves some basic knowledge of periodic potentials from solid state physics. We will review the concept briefly. To observe SF-MI transition, ultracold bosons are placed in an optical lattice, which is formed by a standing wave of light. A variety of optical lattice geometry can be generated by how the laser beams interfere with each other. In this article we only consider the simple (and most common) case of separable, sinusoidal potential, generated by a pair of retroreflected beams in each axis: | ||
+ | :<math> | ||
+ | V(x,y,z) = V_0 (\sin^2 (kx) + \sin^2 (ky) + \sin^2 (kz)) | ||
+ | </math> | ||
+ | The Hamiltonian <math> -\hbar^2 \nabla^2 / 2m + V </math> is periodic in space (ignoring interactions for now) with period <math> a </math>, so we can use Bloch's theorem, which says that the wavefunctions of the eigenstates <math> |\psi \rangle </math> of a periodic Hamiltonian can be written as | ||
+ | :<math> | ||
+ | \psi(x) = e^{ikx} u_{n,k} (x)\, ,\Big(-\frac{\pi}{a} \leq k \leq \frac{\pi}{a}, n = 1,2,\ldots \Big) | ||
+ | </math> | ||
+ | What this means is that if we translate the wavefunction by distance <math> a </math>, then the wavefunction can only change up to a complex phase. From a more formal point of view, because the continuous translation symmetry is lost and only discrete translation symmetry remains, momentum is only conserved modulo <math> \hbar K = \hbar \frac{2\pi}{a} </math>. Imagine | ||
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The ultracold atoms are trapped in a periodic potential, | The ultracold atoms are trapped in a periodic potential, |
Revision as of 03:00, 13 May 2017
A condenstate in a shallow standing wave potential is a BEC, well described by a Bogoliubov approximate solution. As the potential gets deeper, though, eventually the system transitions into a state of localized atoms, with no long-range coherence, known as a Mott insulator. These physics are important in a wide range of condensed matter systems, and can be explored deeply with BECs.
The discussion of superfluid to Mott insulator (SF-MI) transition involves some basic knowledge of periodic potentials from solid state physics. We will review the concept briefly. To observe SF-MI transition, ultracold bosons are placed in an optical lattice, which is formed by a standing wave of light. A variety of optical lattice geometry can be generated by how the laser beams interfere with each other. In this article we only consider the simple (and most common) case of separable, sinusoidal potential, generated by a pair of retroreflected beams in each axis:
The Hamiltonian is periodic in space (ignoring interactions for now) with period , so we can use Bloch's theorem, which says that the wavefunctions of the eigenstates of a periodic Hamiltonian can be written as
What this means is that if we translate the wavefunction by distance , then the wavefunction can only change up to a complex phase. From a more formal point of view, because the continuous translation symmetry is lost and only discrete translation symmetry remains, momentum is only conserved modulo . Imagine
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