Abstract

Atomic media are important in optics research since they can be conveniently manipulated and controlled due to easy selection of atomic levels, laser fields, and the active adjustment of many system parameters. In this paper, we investigate optical Bloch oscillation, Zener tunneling, and Bloch–Zener oscillation in atomic media both theoretically and numerically. We use two coupling fields to prepare the dynamical optical lattice through interference. To induce a transverse force, we make the frequency difference between the two coupling fields increase linearly along the longitudinal coordinate. These phenomena have potential application for beam splitters and optical interconnects, and are helpful for investigating quantum analogies.

© 2017 Optical Society of America

1. INTRODUCTION

Quantum mechanics posits that electrons in a periodic lattice perform Bloch oscillation if a constant transverse electric field is applied to the system [1]. Such a phenomenon was predicted by Felix Bloch in 1929 and experimentally observed some 60 years later, in a semiconductor superlattice [2]. Study of Bloch oscillation is still an active topic, and to date Bloch oscillations are reported in but not limited to cold atoms [36], optical waveguides [710], photonic lattices [1115], integrated photonic circuits [16,17], and non-Hermitian systems [18,19].

Optical Bloch oscillation (OBO), instead of considering the dynamics of electrons or atoms, focuses on the propagation of laser light in arrays of waveguides or in photonic crystals. Such a change in the focus of investigation and systems investigated is facilitated by the formal equivalence between the paraxial wave equation in photonics and the Schrödinger equation in quantum mechanics. If a sufficiently strong transverse force that acts on the beam is introduced [20], optical Zener tunneling (OZT) and the optical Bloch–Zener oscillation (OBZO) between Bloch bands can be observed [10,14,21]. In fact, OBO and OZT are the two typical examples of the so-called quantum-optical analogies [22]. The advantage of such analogies is that they map the temporal evolution of wavefunctions in coherent quantum phenomena onto the spatial propagation of optical fields in photonic devices. Investigations of OBO and OZT show applicative potential for producing beam combiners, splitters, and interferometers. For a thorough survey of research progress in OBO and OZT, we suggest review papers [2225]. It should be noted that, as pointed out in [21], BZO is different from Rabi oscillation [2630], where an ac field is introduced to resonantly induce transitions between Bloch states of different bands.

As mentioned above, previous investigations of OBO and OZT were mainly undertaken in photonic lattices or waveguide arrays that can be prepared by the femtosecond laser writing technique or the multi-wave interference method. These techniques are usually applied to solid materials, such as silicon-based materials or photorefractive crystals. We wonder if one can extend OBO and OZT to non-solid materials, e.g., to atomic systems, in which the lattices are prepared dynamically, by utilizing the multi-wave interference method [31]. Being convenient optical media for various applications, atomic systems have been intensely investigated in the past few decades. To overcome large absorption in atomic systems, electromagnetically induced transparency (EIT) [32] was introduced and played an important role in generating multi-wave mixing processes [33]. It is interesting to note that many phenomena of high current interest, such as optical condensates [34,35], photonic topological insulators [3638], (anti-)parity-time symmetric systems [3941], and other phenomena [42,43], have been observed in atomic systems.

Even though the Bloch oscillation of cold atoms was reported before, the questions addressed here will provide a different point of view on light propagation in atomic systems. In this paper, we report OBO, OZT, and OBZO in an atomic system. We believe that most of these effects have not been explored before in atomic media, except for the OBO effect [44]. In comparison with Ref. [44], where nonlinearity is elaborately designed to obtain the OBO effect, we adopt a completely different method. A periodic lattice with a transverse force is constructed using the interference between two coupling fields with a linearly increasing frequency difference. In comparison with the investigations in solid waveguide arrays and photonic crystals, the atomic medium has unique advantages, such as the tunability of beam confinement (the potential depth), the variable spatial extent between optically induced waveguides, and adjustable loss (or even gain), to name a few. Our results can be utilized in fabricating beam splitters and optical interconnects, and also in investigating novel quantum analogies.

2. THEORETICAL MODEL

We consider light propagation in a Λ-type rubidium atomic system [42,43], as shown in Fig. 1(a). For energy levels |0, |1, and |2, we choose the 5S1/2(F=3), 5P3/2, and 5S1/2(F=2) states of Rb85, respectively. The probe susceptibility in such a system can be written as [45] χ(1)=iNμ102[ϵ0(d10+|G12+G12|2/d20)]1, with N being the atomic density, μ10 the electric dipole moment, and d10=Γ10+iΔ10 and d20=Γ20+i(Δ10Δ12) the complex decay rates. In Fig. 1(a), the probe field E10 connects the transition |0|1, and the coupling fields E12 and E12 connect the transition |1|2. Γij are the decay rates between |i and |j states, and Δ10=Ω10ω10, Δ12=Ω12ω12 and Δ12=Ω12ω12 are the detunings. They are determined by the transition frequencies Ωij between |i and |j, and by the frequencies ω10 and ω12(ω12) of the probe and the coupling fields. G12 represents the Rabi frequency of the coupling fields, defined as G12=μ12E12/. In Fig. 1(b), the geometry of the probe and coupling fields is displayed. A small angle between the two coupling fields that are not exactly counterpropagating is assumed. It should be noted that the Doppler broadening effect in the EIT window is considered even in hot atomic systems [46].

 figure: Fig. 1.

Fig. 1. (a) Λ-type energy system in a rubidium atomic system. (b) Geometry of the beams in (a). (c) Rabi frequency of the two interfering coupling fields for (c1) δ=0, (c2) δ being a constant, and (c3) δ increasing linearly with t.

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One can put the coupling fields together, as Ec=E12exp[i(ω12tk12x)]+E12exp[i(ω12t+k12x)], which will form a standing wave [41,47]—the optically induced lattice (OIL)—as shown in Fig. 1(c1). If one deliberately prepares two coupling fields with a frequency difference δ=ω12ω12, the OIL will move with a constant speed along the transverse coordinate, as shown in Fig. 1(c2), and the Rabi frequency can be written as |G12+G12|2=(μ12/)2[E122+E122+2E12E12cos(δt+2kcx)]. The speed v can be written as v=δ/2kc with kc=(k12+k12)/2. We note that the speed of OIL can also be viewed as the slope of OIL along the t direction. If one fixes ω12 and changes ω12 to detune the direction of motion and speed, one obtains the effective period of the moving OIL as D=π/(k12+δ/2c), from which one can see that D is bigger for δ<0 than for δ>0, due to the Doppler effect. As a result, the reflectivity and the transmissivity of the same probe incidence will be different when launched into the OIL from directions which are nearly perpendicular to the case shown in Fig. 1(b); this property may be useful for producing an optical diode [42,43]. If δ is fixed, the OIL moves with a constant speed, while if δ increases with t linearly, i.e., δ=at, the OIL moves with a linearly increasing speed (δ is much smaller than ω12, and its influence on kc can be neglected). This means that there is a constant acceleration a [3]; such an OIL is displayed in Fig. 1(c3). In a potential experiment, the frequency difference can be realized by an acousto-optic modulator.

We consider evolution of the probe E10 in such a lattice, as shown in Fig. 1(b). The governing equation can be described as a Schrödinger-like paraxial equation:

iE10z+12k102E10x2+12k10χ(1)E10=0,
where z=vt, and for the parameters used in this paper, v is around 1m/s. For convenience, we set v=1m/s throughout. Since there is a frequency difference between the two coupling fields that results in the bending of OIL along the evolution direction, we introduce the transformation of coordinates τ=t and ξ=x+ζ(t) with ζ(t)=δt/2kc, to transform Eq. (1) in the bending frame of reference:
iE10τ+idζdτE10ξ+12k102E10ξ2+12k10χ(1)E10=0.

If one introduces the gauge transformation [48,49] E10(ξ,τ)=ψ(ξ,τ)exp{ik10[ξ(dζ/dτ)+120τ[dζ(α)/dα]2dα]}, Eq. (2) can be rewritten as

iψτ+12k102ψξ2+12k10χ(1)ψ+k10d2ζdτ2ξψ=0.

The last term in Eq. (3) acts as a transverse force. If δ is a constant [as in Figs. 1(a) and 1(b)], then the transverse force is zero. If δ increases linearly with τ (or t), there will be a fixed transverse force, which leads to the OBO and OZT of light during evolution. The period of OBO is determined by the transverse force and the period D of the OIL [10], as T=2kc/(k10|d2ζ/dτ2|).

3. NUMERICAL SIMULATIONS

We first consider the case δ=0. As expected, the beam will undergo discrete diffraction during evolution, as shown in Fig. 2(a). If δ0, the OIL will be oblique in the (x,t) plane, as shown in Fig. 1(c2). Since the transverse force is still zero (d2ζ/dτ2=0), one can only observe a bent discrete diffraction, as displayed in Fig. 2(b). As the bending OIL can always be transformed into a straight lattice with a transverse force [Eq. (3)], we consider only the band structure of the straight lattice, which is exhibited in the inset of Fig. 2(a). One can see that there is a small bandgap between the bands.

 figure: Fig. 2.

Fig. 2. Discrete diffraction. (a) δ=0. (b) δ=10MHz. Inset shows the band structure. From top to bottom are the first, second, third, and fourth bands, respectively. Other parameters are: μ10=3×1029C·m, N=2×1013cm3, G12=30MHz, G12=25MHz, λ10=770.792nm, λ12=770.778nm, Γ10=1MHz, Γ20=2kHz, Δ12=0, and Δ10=1MHz.

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When the frequency difference changes linearly with evolution time, the OIL will bend along a parabolic curve, as in Fig. 1(c). In this case, we numerically determine the beam evolution, according to both Eqs. (1) and (3). To start with, we consider the case when only one lattice waveguide is excited; the results are displayed in Fig. 3. In Fig. 3(a), the beam bends during evolution, due to the guidance from the OIL. One can note at least two interesting phenomena: the beam oscillates periodically (exhibits breather-like behavior), and the energy of the beam leaks from oscillation—which are the indicators of OBO and OZT. The evolution according to the transformed equation is depicted in Fig. 3(b), in which the OBO and OZT are more easily observed. Analytically, the period of OBO is about T16.3μs, and the numerical result agrees with it very well. Since the OZT will dampen the OBO, it becomes weaker during evolution. In Fig. 3(c), we also present the evolution of Fig. 3(b) in the momentum space. One finds that the beam excites the whole Brillouin zone and indeed is not confined to it; the energy transfers to other bands over the whole OBO period, which by definition is the OZT.

 figure: Fig. 3.

Fig. 3. Optical Bloch oscillation and Zener tunneling when only one lattice waveguide is excited. (a) According to Eq. (1). (b) According to Eq. (3). (c) Same as (b), but in the momentum space, which is normalized with k12. We set a=1MHz2; other parameters are the same as those in Fig. 2. To see the evolution dynamics explicitly, we show the amplitudes rather than the intensity in (a) and (b).

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If one excites more than one lattice waveguide, the evolution dynamics of the input beam is shown in Fig. 4. Again, one observes OBO and OZT quite clearly. However, different from the case in Fig. 3, where OZT happened in the whole OBO period, the OZT here happens only at the edge of the Brillouin zone, as shown by the vertical dashed lines that connect Figs. 4(b) and 4(c). One can see that the beam moves across the Brillouin zones at the edges successively during evolution. In Fig. 4(b), one can also note that there is OBZO in addition to OBO. We classify the oscillations in Fig. 4(b) into four categories, which are labeled by symbols (I), (II), (III), and (IV) and simultaneously indicated by the solid, dashed, dashed–dotted, and dotted curves. Case (I) is the OBO, while cases (II)–(IV) are the OBZO progressing across different bands. Together with the inset in Fig. 2(a), one can conclude that the beam will escape from one band to the next one at the Brillouin zone edge, due to the large transverse force and mini-gaps between the adjacent bands, to form OZT. On the other hand, the escaped beam will oscillate in each band, to form OBO and OBZO. Finally, a net-like structure is formed in this continuum model during evolution, as shown in Fig. 4(b). From this point of view, the property may have potential applications for on-chip beam splitters and optical interconnect fabrication.

 figure: Fig. 4.

Fig. 4. Same as Fig. 3, but with many lattice waveguides excited.

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Another evidence for the existence of OZT comes from the projection of the beam during evolution onto an orthogonalized Floquet–Bloch base [50]: An(κ)=D2πdκnexp(iβκnt)ϕκn*(x)E10(x,t)dx, where κ is the Bloch momentum, n is the band number, ϕκn(x)=ϕκn(x)/cκn, cκn=Dϕκn*(x)ϕκn(x)dx, dκn=±1, and ϕκn(x) is the Floquet–Bloch mode, with βκn being the corresponding eigenvalue. We exhibit the projection coefficients An of the beam during evolution in Fig. 5. Clearly, the beam excites the modes of other bands during evolution; in the beginning it excites the modes of mainly the first band and, as it evolves, it starts exciting the other modes. It is interesting to note that the projection coefficients at t=25μs are quite small almost over the whole Brillouin zone, except at the boundaries. The reason is that this distance is almost 1.5 times the OBO period, which is close to the turning point, where the beam starts to escape from the OBO. From Fig. 4(c), one can infer that the beam is indeed quite close to the boundaries of different Brillouin zones (t=25μs is close to the vertical dashed line).

 figure: Fig. 5.

Fig. 5. Projection coefficients of the beam during evolution. Amplitude is normalized by max{|A14|}.

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We should note that the detunings and the intensity of coupling fields can be adjusted momentarily in atomic systems, which is more advantageous than in the lattices fabricated by femtosecond lasers in silicon-based materials. As a result, the value of χ(1) can be made to increase, which will improve the confinement of the lattice, and the OZT will decrease to some extent. Therefore, the quality of OBO will become better. Admittedly, this system is also lossy, even in the EIT window. As a result, the beam intensity will decrease during evolution. According to previous literature [41,46], if one introduces an additional pump field into the system (changing it from a Λ-type to an N-type), the pumping gain will be obtained, which can compensate for the absorption losses.

4. CONCLUSION

In conclusion, we have shown that OBO, OZT, and OBZT can be observed in an atomic system. Two coupling beams with a linearly increasing frequency difference are used to form a bending OIL via interference. Due to the multi-parameter controllable properties of an atomic system, the phenomena obtained in this paper can be easily adjusted. Also, this investigation can be conveniently transplanted to atomic-like solid media, such as the praseodymium-doped yttrium orthosilicate crystal. We would like to note that, in an experiment, the diffraction patterns can be recorded using a method similar to [51]—that is, one puts a CCD camera at the output plane of the atomic cell to monitor the diffraction patterns in time. Our study not only shows the potential for fabricating beams splitters and optical interconnects, but also provides a new platform for observing optical quantum analogies.

Funding

China Postdoctoral Science Foundation (2016M600777, 2016M590935, 2016M600776); Key Scientific and Technological Innovation Team of Shaanxi Province (2014KCT-10); National Natural Science Foundation of China (NSFC) (11474228, 11534008, 61605154); Ministry of Science and Technology of the People’s Republic of China (MOST) (2016YFA0301404); Qatar National Research Fund (QNRF) (NPRP 6-021-1-005).

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References

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  1. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
    [Crossref]
  2. J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, “Optical investigation of Bloch oscillations in a semiconductor superlattice,” Phys. Rev. B 46, 7252–7255 (1992).
    [Crossref]
  3. M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
    [Crossref]
  4. R. Battesti, P. Cladé, S. Guellati-Khélifa, C. Schwob, B. Grémaud, F. Nez, L. Julien, and F. Biraben, “Bloch oscillations of ultracold atoms: a tool for a metrological determination of h/mrb,” Phys. Rev. Lett. 92, 253001 (2004).
    [Crossref]
  5. P. Cladé, E. de Mirandes, M. Cadoret, S. Guellati-Khélifa, C. Schwob, F. M. C. Nez, L. Julien, and F. M. C. Biraben, “Determination of the fine structure constant based on Bloch oscillations of ultracold atoms in a vertical optical lattice,” Phys. Rev. Lett. 96, 033001 (2006).
    [Crossref]
  6. G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, “Long-lived Bloch oscillations with bosonic Sr atoms and application to gravity measurement at the micrometer scale,” Phys. Rev. Lett. 97, 060402 (2006).
    [Crossref]
  7. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
    [Crossref]
  8. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [Crossref]
  9. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
    [Crossref]
  10. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
    [Crossref]
  11. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
    [Crossref]
  12. V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch oscillations in porous silicon optical superlattices,” Phys. Rev. Lett. 92, 097401 (2004).
    [Crossref]
  13. M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
    [Crossref]
  14. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
    [Crossref]
  15. Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
    [Crossref]
  16. Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
    [Crossref]
  17. M. Lebugle, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6, 8273 (2015).
    [Crossref]
  18. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
    [Crossref]
  19. Y.-L. Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu, Z.-Y. Li, A. Scherer, and Y.-F. Chen, “Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice,” Nat. Commun. 7, 11319 (2016).
    [Crossref]
  20. C. Zener, “A theory of the electrical breakdown of solid dielectrics,” Proc. R. Soc. London Ser. A 145, 523–529 (1934).
    [Crossref]
  21. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch–Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
    [Crossref]
  22. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
    [Crossref]
  23. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
    [Crossref]
  24. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
    [Crossref]
  25. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
    [Crossref]
  26. X.-G. Zhao, G. A. Georgakis, and Q. Niu, “Rabi oscillations between Bloch bands,” Phys. Rev. B 54, R5235–R5238 (1996).
    [Crossref]
  27. M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of Rabi oscillations between Bloch bands in an optical potential,” Phys. Rev. A 58, R2648–R2651 (1998).
    [Crossref]
  28. Y. V. Bludov, V. V. Konotop, and M. Salerno, “Rabi oscillations of matter-wave solitons in optical lattices,” Phys. Rev. A 80, 023623 (2009).
    [Crossref]
  29. Y. V. Bludov, V. V. Konotop, and M. Salerno, “Matter waves and quantum tunneling engineered by time-dependent interactions,” Phys. Rev. A 81, 053614 (2010).
    [Crossref]
  30. X. Zhang, F. Ye, Y. V. Kartashov, and X. Chen, “Rabi oscillations and stimulated mode conversion on the subwavelength scale,” Opt. Express 23, 6731–6737 (2015).
    [Crossref]
  31. P. Windpassinger and K. Sengstock, “Engineering novel optical lattices,” Rep. Prog. Phys. 76, 086401 (2013).
    [Crossref]
  32. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  33. Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
    [Crossref]
  34. H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96, 023903 (2006).
    [Crossref]
  35. Z. K. Wu, Y. Q. Zhang, C. Z. Yuan, F. Wen, H. B. Zheng, Y. P. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2013).
    [Crossref]
  36. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model,” Nature 515, 237–240 (2014).
    [Crossref]
  37. Y. Q. Zhang, Z. K. Wu, M. R. Belić, H. B. Zheng, Z. G. Wang, M. Xiao, and Y. P. Zhang, “Photonic Floquet topological insulators in atomic ensembles,” Laser Photon. Rev. 9, 331–338 (2015).
    [Crossref]
  38. D.-W. Wang, H. Cai, L. Yuan, S.-Y. Zhu, and R.-B. Liu, “Topological phase transitions in superradiance lattices,” Optica 2, 712–715 (2015).
    [Crossref]
  39. C. Hang, G. Huang, and V. V. Konotop, “PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  40. P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
    [Crossref]
  41. Z. Y. Zhang, Y. Q. Zhang, J. T. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. P. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
    [Crossref]
  42. D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).
    [Crossref]
  43. J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
    [Crossref]
  44. C. Hang and V. V. Konotop, “All-optical steering of light via spatial Bloch oscillations in a gas of three-level atoms,” Phys. Rev. A 81, 053849 (2010).
    [Crossref]
  45. J.-H. Wu, M. Artoni, and G. C. L. Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
    [Crossref]
  46. J. Sheng, X. Yang, H. Wu, and M. Xiao, “Modified self-Kerr-nonlinearity in a four-level N-type atomic system,” Phys. Rev. A 84, 053820 (2011).
    [Crossref]
  47. M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
    [Crossref]
  48. S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. 30, 2137–2139 (2005).
    [Crossref]
  49. F. Dreisow, M. Heinrich, A. Szameit, S. Döring, S. Nolte, A. Tünnermann, S. Fahr, and F. Lederer, “Spectral resolved dynamic localization in curved fs laser written waveguide arrays,” Opt. Express 16, 3474–3483 (2008).
    [Crossref]
  50. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  51. Z. Y. Zhang, X. Liu, D. Zhang, J. T. Sheng, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Observation of electromagnetically induced Talbot effect in an atomic system,” arXiv:1608.00298 (2016).

2016 (4)

Y. V. Kartashov, V. V. Konotop, D. A. Zezyulin, and L. Torner, “Bloch oscillations in optical and Zeeman lattices in the presence of spin-orbit coupling,” Phys. Rev. Lett. 117, 215301 (2016).
[Crossref]

Y.-L. Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu, Z.-Y. Li, A. Scherer, and Y.-F. Chen, “Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice,” Nat. Commun. 7, 11319 (2016).
[Crossref]

P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity-time symmetry with flying atoms,” Nat. Phys. 12, 1139–1145 (2016).
[Crossref]

Z. Y. Zhang, Y. Q. Zhang, J. T. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. P. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117, 123601 (2016).
[Crossref]

2015 (4)

Y. Q. Zhang, Z. K. Wu, M. R. Belić, H. B. Zheng, Z. G. Wang, M. Xiao, and Y. P. Zhang, “Photonic Floquet topological insulators in atomic ensembles,” Laser Photon. Rev. 9, 331–338 (2015).
[Crossref]

D.-W. Wang, H. Cai, L. Yuan, S.-Y. Zhu, and R.-B. Liu, “Topological phase transitions in superradiance lattices,” Optica 2, 712–715 (2015).
[Crossref]

X. Zhang, F. Ye, Y. V. Kartashov, and X. Chen, “Rabi oscillations and stimulated mode conversion on the subwavelength scale,” Opt. Express 23, 6731–6737 (2015).
[Crossref]

M. Lebugle, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6, 8273 (2015).
[Crossref]

2014 (2)

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref]

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model,” Nature 515, 237–240 (2014).
[Crossref]

2013 (4)

C. Hang, G. Huang, and V. V. Konotop, “PT symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).
[Crossref]

P. Windpassinger and K. Sengstock, “Engineering novel optical lattices,” Rep. Prog. Phys. 76, 086401 (2013).
[Crossref]

Z. K. Wu, Y. Q. Zhang, C. Z. Yuan, F. Wen, H. B. Zheng, Y. P. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A 88, 063828 (2013).
[Crossref]

2012 (1)

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[Crossref]

2011 (2)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[Crossref]

J. Sheng, X. Yang, H. Wu, and M. Xiao, “Modified self-Kerr-nonlinearity in a four-level N-type atomic system,” Phys. Rev. A 84, 053820 (2011).
[Crossref]

2010 (3)

C. Hang and V. V. Konotop, “All-optical steering of light via spatial Bloch oscillations in a gas of three-level atoms,” Phys. Rev. A 81, 053849 (2010).
[Crossref]

Y. V. Bludov, V. V. Konotop, and M. Salerno, “Matter waves and quantum tunneling engineered by time-dependent interactions,” Phys. Rev. A 81, 053614 (2010).
[Crossref]

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

2009 (4)

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Y. V. Bludov, V. V. Konotop, and M. Salerno, “Rabi oscillations of matter-wave solitons in optical lattices,” Phys. Rev. A 80, 023623 (2009).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Bloch–Zener oscillations in binary superlattices,” Phys. Rev. Lett. 102, 076802 (2009).
[Crossref]

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

2008 (4)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. L. Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[Crossref]

F. Dreisow, M. Heinrich, A. Szameit, S. Döring, S. Nolte, A. Tünnermann, S. Fahr, and F. Lederer, “Spectral resolved dynamic localization in curved fs laser written waveguide arrays,” Opt. Express 16, 3474–3483 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

2007 (1)

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[Crossref]

2006 (6)

H. Michinel, M. J. Paz-Alonso, and V. M. Pérez-García, “Turning light into a liquid via atomic coherence,” Phys. Rev. Lett. 96, 023903 (2006).
[Crossref]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[Crossref]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[Crossref]

P. Cladé, E. de Mirandes, M. Cadoret, S. Guellati-Khélifa, C. Schwob, F. M. C. Nez, L. Julien, and F. M. C. Biraben, “Determination of the fine structure constant based on Bloch oscillations of ultracold atoms in a vertical optical lattice,” Phys. Rev. Lett. 96, 033001 (2006).
[Crossref]

G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, “Long-lived Bloch oscillations with bosonic Sr atoms and application to gravity measurement at the micrometer scale,” Phys. Rev. Lett. 97, 060402 (2006).
[Crossref]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[Crossref]

2005 (3)

S. Longhi, “Self-imaging and modulational instability in an array of periodically curved waveguides,” Opt. Lett. 30, 2137–2139 (2005).
[Crossref]

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

2004 (2)

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch oscillations in porous silicon optical superlattices,” Phys. Rev. Lett. 92, 097401 (2004).
[Crossref]

R. Battesti, P. Cladé, S. Guellati-Khélifa, C. Schwob, B. Grémaud, F. Nez, L. Julien, and F. Biraben, “Bloch oscillations of ultracold atoms: a tool for a metrological determination of h/mrb,” Phys. Rev. Lett. 92, 253001 (2004).
[Crossref]

2003 (1)

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

1999 (2)

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

1998 (2)

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
[Crossref]

M. C. Fischer, K. W. Madison, Q. Niu, and M. G. Raizen, “Observation of Rabi oscillations between Bloch bands in an optical potential,” Phys. Rev. A 58, R2648–R2651 (1998).
[Crossref]

1996 (2)

X.-G. Zhao, G. A. Georgakis, and Q. Niu, “Rabi oscillations between Bloch bands,” Phys. Rev. B 54, R5235–R5238 (1996).
[Crossref]

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

1992 (1)

J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, “Optical investigation of Bloch oscillations in a semiconductor superlattice,” Phys. Rev. B 46, 7252–7255 (1992).
[Crossref]

1934 (1)

C. Zener, “A theory of the electrical breakdown of solid dielectrics,” Proc. R. Soc. London Ser. A 145, 523–529 (1934).
[Crossref]

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Agarwal, V.

V. Agarwal, J. A. del Río, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch oscillations in porous silicon optical superlattices,” Phys. Rev. Lett. 92, 097401 (2004).
[Crossref]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[Crossref]

Artoni, M.

J.-H. Wu, M. Artoni, and G. C. La Rocca, “Non-Hermitian degeneracies and unidirectional reflectionless atomic lattices,” Phys. Rev. Lett. 113, 123004 (2014).
[Crossref]

J.-H. Wu, M. Artoni, and G. C. L. Rocca, “Controlling the photonic band structure of optically driven cold atoms,” J. Opt. Soc. Am. B 25, 1840–1849 (2008).
[Crossref]

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96, 073905 (2006).
[Crossref]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[Crossref]

Battesti, R.

R. Battesti, P. Cladé, S. Guellati-Khélifa, C. Schwob, B. Grémaud, F. Nez, L. Julien, and F. Biraben, “Bloch oscillations of ultracold atoms: a tool for a metrological determination of h/mrb,” Phys. Rev. Lett. 92, 253001 (2004).
[Crossref]

Belic, M. R.

Y. Q. Zhang, Z. K. Wu, M. R. Belić, H. B. Zheng, Z. G. Wang, M. Xiao, and Y. P. Zhang, “Photonic Floquet topological insulators in atomic ensembles,” Laser Photon. Rev. 9, 331–338 (2015).
[Crossref]

Ben Dahan, M.

M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, “Bloch oscillations of atoms in an optical potential,” Phys. Rev. Lett. 76, 4508–4511 (1996).
[Crossref]

Biraben, F.

R. Battesti, P. Cladé, S. Guellati-Khélifa, C. Schwob, B. Grémaud, F. Nez, L. Julien, and F. Biraben, “Bloch oscillations of ultracold atoms: a tool for a metrological determination of h/mrb,” Phys. Rev. Lett. 92, 253001 (2004).
[Crossref]

Biraben, F. M. C.

P. Cladé, E. de Mirandes, M. Cadoret, S. Guellati-Khélifa, C. Schwob, F. M. C. Nez, L. Julien, and F. M. C. Biraben, “Determination of the fine structure constant based on Bloch oscillations of ultracold atoms in a vertical optical lattice,” Phys. Rev. Lett. 96, 033001 (2006).
[Crossref]

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[Crossref]

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and M. Salerno, “Matter waves and quantum tunneling engineered by time-dependent interactions,” Phys. Rev. A 81, 053614 (2010).
[Crossref]

Y. V. Bludov, V. V. Konotop, and M. Salerno, “Rabi oscillations of matter-wave solitons in optical lattices,” Phys. Rev. A 80, 023623 (2009).
[Crossref]

Bräuer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[Crossref]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[Crossref]

Bromberg, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch oscillations of path-entangled photons,” Phys. Rev. Lett. 105, 263604 (2010).
[Crossref]

Brown, A. W.

Y. P. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
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Z. Y. Zhang, X. Liu, D. Zhang, J. T. Sheng, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Observation of electromagnetically induced Talbot effect in an atomic system,” arXiv:1608.00298 (2016).

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Figures (5)

Fig. 1.
Fig. 1. (a)  Λ -type energy system in a rubidium atomic system. (b) Geometry of the beams in (a). (c) Rabi frequency of the two interfering coupling fields for (c1)  δ = 0 , (c2)  δ being a constant, and (c3)  δ increasing linearly with t .
Fig. 2.
Fig. 2. Discrete diffraction. (a)  δ = 0 . (b)  δ = 10 MHz . Inset shows the band structure. From top to bottom are the first, second, third, and fourth bands, respectively. Other parameters are: μ 10 = 3 × 10 29 C · m , N = 2 × 10 13 cm 3 , G 12 = 30 MHz , G 12 = 25 MHz , λ 10 = 770.792 nm , λ 12 = 770.778 nm , Γ 10 = 1 MHz , Γ 20 = 2 kHz , Δ 12 = 0 , and Δ 10 = 1 MHz .
Fig. 3.
Fig. 3. Optical Bloch oscillation and Zener tunneling when only one lattice waveguide is excited. (a) According to Eq. (1). (b) According to Eq. (3). (c) Same as (b), but in the momentum space, which is normalized with k 12 . We set a = 1 MHz 2 ; other parameters are the same as those in Fig. 2. To see the evolution dynamics explicitly, we show the amplitudes rather than the intensity in (a) and (b).
Fig. 4.
Fig. 4. Same as Fig. 3, but with many lattice waveguides excited.
Fig. 5.
Fig. 5. Projection coefficients of the beam during evolution. Amplitude is normalized by max { | A 1 4 | } .

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

i E 10 z + 1 2 k 10 2 E 10 x 2 + 1 2 k 10 χ ( 1 ) E 10 = 0 ,
i E 10 τ + i d ζ d τ E 10 ξ + 1 2 k 10 2 E 10 ξ 2 + 1 2 k 10 χ ( 1 ) E 10 = 0 .
i ψ τ + 1 2 k 10 2 ψ ξ 2 + 1 2 k 10 χ ( 1 ) ψ + k 10 d 2 ζ d τ 2 ξ ψ = 0 .

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