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Quantum simulation of non-trivial topology

Boada, Octavi, Celi, Alessio, Rodríguez-Laguna, Javier, Latorre, José I, Lewenstein, Maciej·2015·New Journal of Physics
DOI 10.1088/1367-2630/17/4/045007
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Figures

FIG. 1 Idea of a synthetic lattice: a 1D chain of length L sites with M species is equivalent to a L × M synthetic lattice, once the chain is dressed with appropriate couplings between species.
FIG. 1 Idea of a synthetic lattice: a 1D chain of length L sites with M species is equivalent to a L × M synthetic lattice, once the chain is dressed with appropriate couplings between species.CC-BY · From the PDF
FIG. 2 Engineering a circumference of 2 L lattice sites from a L synthetic lattice that carries M = 2 species. The blue solid links indicate the hopping along the chain, e.g., the free hopping in a cold-atom implementation, while the two dashed red links are induced local interactions between the two species, e.g., a Raman coupling between hyperfine levels of atoms. The vertical black ellipses represent the physical sites in the real 1D-chain occupied by the two species.
FIG. 2 Engineering a circumference of 2 L lattice sites from a L synthetic lattice that carries M = 2 species. The blue solid links indicate the hopping along the chain, e.g., the free hopping in a cold-atom implementation, while the two dashed red links are induced local interactions between the two species, e.g., a Raman coupling between hyperfine levels of atoms. The vertical black ellipses represent the physical sites in the real 1D-chain occupied by the two species.CC-BY · From the PDF
FIG. 3 Engineering a cylinder of basis L x and height L y lattice sites from a L × M synthetic lattice with L = L x / 2 and M = 2 L y . (a) L y = 2 synthetic cylinder. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed red links are the 'sewing' hoppings C σσ ′ closing each circle of the cylinder, while the dashed green ones are the hoppings B σσ ′ along its height, connecting the different circles. (b) and (c) Spectroscopical arrangement of the four internal degrees of freedom in the one-manifold and two-manifold scheme, respectively. The former can be realized by using the groundstate of atoms F ≥ 3 2 like Li, K, Yb, Sr, Er, etc, and requires quadratic Zeeman splitting in order to have J ′ -coupling (in red) only between odd-even m F -states. The latter requires earth-alkali like atoms with F ≥ 1 2 as 171 Yb, 173 Yb, and 87 Sr and in this case the linear splitting is sufficient in order to make the spin accessible by Raman lasers or radio-frequency pulses.
FIG. 3 Engineering a cylinder of basis L x and height L y lattice sites from a L × M synthetic lattice with L = L x / 2 and M = 2 L y . (a) L y = 2 synthetic cylinder. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed red links are the 'sewing' hoppings C σσ ′ closing each circle of the cylinder, while the dashed green ones are the hoppings B σσ ′ along its height, connecting the different circles. (b) and (c) Spectroscopical arrangement of the four internal degrees of freedom in the one-manifold and two-manifold scheme, respectively. The former can be realized by using the groundstate of atoms F ≥ 3 2 like Li, K, Yb, Sr, Er, etc, and requires quadratic Zeeman splitting in order to have J ′ -coupling (in red) only between odd-even m F -states. The latter requires earth-alkali like atoms with F ≥ 1 2 as 171 Yb, 173 Yb, and 87 Sr and in this case the linear splitting is sufficient in order to make the spin accessible by Raman lasers or radio-frequency pulses.CC-BY · From the PDF
FIG. 4 Engineering a torus by converting each fiber into a circumference. In (a) we show the L y = 3 synthetic torus. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radiofrequency induced. The dashed red links are unchanged with respect to Fig. 3 (a), while additional green dashes ones are used to glue the basis of the cylinder connecting the top and bottom circles, and correspond to the additional terms in B σσ ′ . In (b) we show the spectroscopical arrangement of the six internal degrees of freedom in the two-manifold scheme requiring F ≥ 3 2 , for instance, 173 Yb.
FIG. 4 Engineering a torus by converting each fiber into a circumference. In (a) we show the L y = 3 synthetic torus. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radiofrequency induced. The dashed red links are unchanged with respect to Fig. 3 (a), while additional green dashes ones are used to glue the basis of the cylinder connecting the top and bottom circles, and correspond to the additional terms in B σσ ′ . In (b) we show the spectroscopical arrangement of the six internal degrees of freedom in the two-manifold scheme requiring F ≥ 3 2 , for instance, 173 Yb.CC-BY · From the PDF
FIG. 5 Engineering a M¨ obius strip by twisting the cylinder. In (a) we show the L y = 2 synthetic strip. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed green links and half of the dashed red links are unchanged with respect to Fig. 3 (a), while the yellow dashed ones are the twisted closing hoppings connecting different circles, reflecting the change from C σσ ′ to C M σσ ′ . In (b) and (c) we see the spectroscopical arrangement of the four internal degrees of freedom in the one-manifold and two-manifold scheme, respectively. The former can be realized by using the ground state of atoms F ≥ 3 2 like Li, K, Yb, Sr, Er, etc., while the latter requires earth-alkali like atoms with F ≥ 1 2 as 171 Yb, 173 Yb, and 87 Sr. in both cases, the hyperfine levels are linearly split in order to make them accessible by multiple Raman lasers or radio-frequency pulses.
FIG. 5 Engineering a M¨ obius strip by twisting the cylinder. In (a) we show the L y = 2 synthetic strip. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed green links and half of the dashed red links are unchanged with respect to Fig. 3 (a), while the yellow dashed ones are the twisted closing hoppings connecting different circles, reflecting the change from C σσ ′ to C M σσ ′ . In (b) and (c) we see the spectroscopical arrangement of the four internal degrees of freedom in the one-manifold and two-manifold scheme, respectively. The former can be realized by using the ground state of atoms F ≥ 3 2 like Li, K, Yb, Sr, Er, etc., while the latter requires earth-alkali like atoms with F ≥ 1 2 as 171 Yb, 173 Yb, and 87 Sr. in both cases, the hyperfine levels are linearly split in order to make them accessible by multiple Raman lasers or radio-frequency pulses.CC-BY · From the PDF
FIG. 6 Engineering a twisted torus by wrapping the M¨ obius strip. Equivalently, a twisted torus can be visualized as a torus cut and glued with a twist. In (a) we see the L y = 3 synthetic twisted torus. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed red links are unchanged with respect to Fig. 5 (a), while the green dashed ones are used to glue the borders of the M¨ obius strip. In (b) we show the spectroscopical arrangement of the six internal degrees of freedom in the two-manifold scheme, which requires F ≥ 3 2 as, for instance, in 173 Yb.
FIG. 6 Engineering a twisted torus by wrapping the M¨ obius strip. Equivalently, a twisted torus can be visualized as a torus cut and glued with a twist. In (a) we see the L y = 3 synthetic twisted torus. In a cold atom implementation, the blue solid links indicate the free hopping while the dashed ones are laser or radio-frequency induced. The dashed red links are unchanged with respect to Fig. 5 (a), while the green dashed ones are used to glue the borders of the M¨ obius strip. In (b) we show the spectroscopical arrangement of the six internal degrees of freedom in the two-manifold scheme, which requires F ≥ 3 2 as, for instance, in 173 Yb.CC-BY · From the PDF
Image 7
CC-BY · From the PDF
FIG. 8 Hofstadter-like single-particle spectra of several quasi-1D systems under a continuous change of boundary conditions. In all cases the X -axis is labeled by φ ∈ [0 , 2 π ], and color corresponds to eigenvalue index. In (a) we present the spectrum for a PBC system such as that in Eq. (1) with L = 40, pierced with a flux J c = e iφ with φ ∈ [0 , 2 π ]. In (b) and (c) we show the single-body spectrum of a ladder of size 40 × 2, such as Eq. (3) in which the opposite extremes are joined with a unitary matrix of different types: cylindrical in (b) and M¨ obius-like in (c), as specified in the boundary conditions given in equations (12) and (13).
FIG. 8 Hofstadter-like single-particle spectra of several quasi-1D systems under a continuous change of boundary conditions. In all cases the X -axis is labeled by φ ∈ [0 , 2 π ], and color corresponds to eigenvalue index. In (a) we present the spectrum for a PBC system such as that in Eq. (1) with L = 40, pierced with a flux J c = e iφ with φ ∈ [0 , 2 π ]. In (b) and (c) we show the single-body spectrum of a ladder of size 40 × 2, such as Eq. (3) in which the opposite extremes are joined with a unitary matrix of different types: cylindrical in (b) and M¨ obius-like in (c), as specified in the boundary conditions given in equations (12) and (13).CC-BY · From the PDF
FIG. 9 Single-particle spectrum of the 40 × 2 ladder Hamiltonian represented in Eq. (3) undergoing a smooth transition between a cylinder and a M¨ obius strip, with boundary conditions specified in Eq. (14).
FIG. 9 Single-particle spectrum of the 40 × 2 ladder Hamiltonian represented in Eq. (3) undergoing a smooth transition between a cylinder and a M¨ obius strip, with boundary conditions specified in Eq. (14).CC-BY · From the PDF
FIG. 10 Ground state particle-number fluctuations for the Bose-Hubbard model of Eq. 15 at half filling, defined on a 2D lattice with periodic boundary conditions in both directions a torus- and on a 2D lattice with the boundary conditions of a Klein bottle, computed via exact diagonalization. The results are for a 4x4 lattice. Similarly to the results for the M¨ obius band, the ground state in twisted boundary conditions seems to favor larger particle-number fluctuations at intermediate values of J/U . The computation has been done with small disorder in µ to remove the degeneracy at J = 0, as the two complementary checkerboard coverings of the lattice are ground states.
FIG. 10 Ground state particle-number fluctuations for the Bose-Hubbard model of Eq. 15 at half filling, defined on a 2D lattice with periodic boundary conditions in both directions a torus- and on a 2D lattice with the boundary conditions of a Klein bottle, computed via exact diagonalization. The results are for a 4x4 lattice. Similarly to the results for the M¨ obius band, the ground state in twisted boundary conditions seems to favor larger particle-number fluctuations at intermediate values of J/U . The computation has been done with small disorder in µ to remove the degeneracy at J = 0, as the two complementary checkerboard coverings of the lattice are ground states.CC-BY · From the PDF
FIG. 11 Particle-number fluctuations in the ground state of the Bose-Hubbard model of Eq. (15) at half filling defined on a strip with periodic and M¨ obius boundary conditions computed with exact diagonalization. The results are for strips 4, 6 and 8 sites long. For large interactions compared to the hopping parameter the ground state presents larger particlenumber fluctuations for twisted boundary conditions than for regular ones. This behavior does not disappear as the system size increases, for the range of sizes analyzed. The computation has been done with small disorder in µ to remove spurious degeneracy.
FIG. 11 Particle-number fluctuations in the ground state of the Bose-Hubbard model of Eq. (15) at half filling defined on a strip with periodic and M¨ obius boundary conditions computed with exact diagonalization. The results are for strips 4, 6 and 8 sites long. For large interactions compared to the hopping parameter the ground state presents larger particlenumber fluctuations for twisted boundary conditions than for regular ones. This behavior does not disappear as the system size increases, for the range of sizes analyzed. The computation has been done with small disorder in µ to remove spurious degeneracy.CC-BY · From the PDF
FIG. 12 Representation of the ground state and first excited state of the fermionic Hubbard model (16) on a cylinder and a M¨ obius strip of size L x = 6 and L y = 2. The repulsion only takes place along the same rung. The color of each node represents the expected value of 〈 n i 〉 . The color of the bent line represents the correlator 〈 n i n j 〉 - 〈 n i 〉 〈 n j 〉 . The dashed lines are the correlators 〈 a † i a j 〉 .
FIG. 12 Representation of the ground state and first excited state of the fermionic Hubbard model (16) on a cylinder and a M¨ obius strip of size L x = 6 and L y = 2. The repulsion only takes place along the same rung. The color of each node represents the expected value of 〈 n i 〉 . The color of the bent line represents the correlator 〈 n i n j 〉 - 〈 n i 〉 〈 n j 〉 . The dashed lines are the correlators 〈 a † i a j 〉 .CC-BY · From the PDF

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Metadata

Authors
Boada, O., Celi, A., Rodríguez-Laguna, J., Latorre, J. I., & Lewenstein, M.
Venue
New Journal of Physics
Publisher
IOP Publishing
Published
2015-04-15
Volume
17
Issue
4
Pages
045007
ISSN
1367-2630
DOI
10.1088/1367-2630/17/4/045007
Cited by
64
References
7
Licence
Funders
  • Ministerio de Economia y Competitividad: (FIS2012-33642, FIS2013-41757-P)
  • Instituto de Telecomunicações: (IT/QuSim)
  • European Research Council: (AdG OSYRIS)
  • EU FP7: (GA 318287, GA 323901)
  • Fundação para a Ciência e a Tecnologia: (PEst-OE/EGE/UI0491/2013,)

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Boada, O., Celi, A., Rodríguez-Laguna, J., Latorre, J. I., & Lewenstein, M. (2015). Quantum simulation of non-trivial topology. New Journal of Physics. https://doi.org/10.1088/1367-2630/17/4/045007