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Bound states of the spin-orbit coupled ultracold atom in a one-dimensional short-range potential
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10.1063/1.4807419
/content/aip/journal/jmp/54/5/10.1063/1.4807419
http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/5/10.1063/1.4807419

Figures

Image of FIG. 1.
FIG. 1.

Computed spectrum of operator (see Eq. (3.1) and Theorem 1) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = = 1 units). The eigenvalues divided by η > 0 are those of the one-dimensional Dirac-like operator for the particle of spin one-half and mass Ω/(2η) moving in the Fermi pseudopotential (3.7) . In the figure, red lines show the border of the essential spectrum of , which is ±Ω/2. The blue ɛ (green ɛ) line, showing the bound state as a function of the Raman coupling Ω > 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin = 0.

Image of FIG. 2.
FIG. 2.

Computed lower branch of dispersion in (4.4) for the spin-orbit-coupling strength η = 0.6 (in ℏ = = 1 units), for a range of Raman couplings Ω ⩾ 0. As Ω increases (Ω > η), the two dressed spin states ( ) are merged into a single minimum −Ω/2 at = 0. This is a regime when the spin-orbit coupling induced states σ(), Theorem 2, are observed below the continuous spectrum as well as above it. For Ω ⩽ η, the spin states have two minima −[η + (Ω/η)]/4 at , and the spin-orbit induced states are embedded into the essential spectrum of .

Image of FIG. 3.
FIG. 3.

Computed spin-orbit coupling induced states σ() ⊂ σ() (refer to Theorem 2) for the point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = = 1 units). In the figure, red line shows the border of the essential spectrum of (Lemma 2). The eigenvalues λ(ɛ) ∈ σ() (ɛ ∈ σ( )), as functions of the Raman coupling Ω > 0, are drawn by the blue (σ()) and green (σ()) lines. Resonant states of are drawn by yellow curves (R).

Image of FIG. 4.
FIG. 4.

The eigenvalues of associated with everywhere continuous eigenfunctions. The point-interaction strength γ = −1 and the spin-orbit-coupling strength η = 0.6 (in ℏ = = 1 units). In the figure, red line shows the border of the essential spectrum of (Lemma 2). The blue λ (green λ) line, showing the bound state as a function of the Raman coupling Ω ⩾ 0, corresponds to the eigenfunction with a zero-valued lower (upper) component at the origin = 0 (Theorem 3). The eigenvalue λ approaches at Ω = η + γ/4 and then disappears (for details, refer to Remark 4). Resonant states of are drawn by the yellow curve (R).

Tables

Generic image for table
Table I.

All possible solutions of (5.7) with respect to for , = ±1 for = 1, …, 4 given in (5.8) and (5.9) .

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/content/aip/journal/jmp/54/5/10.1063/1.4807419
2013-05-29
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Bound states of the spin-orbit coupled ultracold atom in a one-dimensional short-range potential
http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/5/10.1063/1.4807419
10.1063/1.4807419
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