In the field of quantum simulation of condensed matter phenomena by artificially engineering the Hamiltonian of an atomic, molecular or optical system, the concept of synthetic dimensions has recently emerged as a powerful way to emulate phenomena such as topological phases of matter, which are now of great interest across many areas of physics. The main idea of a synthetic dimension is to couple together suitable degrees of freedom, such as a set of internal atomic states, in order to mimic the motion of a particle along an extra spatial dimension. This approach provides a way to engineer lattice Hamiltonians and enables the realization of higher-dimensional topological models in platforms with lower dimensionality. We give an overview of the recent progress in studying topological matter in synthetic dimensions. After reviewing proposals and realizations in various set-ups, we discuss future prospects in many-body physics, applications and topological effects in three or more spatial dimensions.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Communications Physics Open Access 18 October 2023
Scientific Reports Open Access 20 September 2023
Communications Physics Open Access 19 September 2023
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$99.00 per year
only $8.25 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Klitzing, Kv, Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).
Lu, L., Joannopoulos, J. D. & Soljačć, M. Topological photonics. Nat. Photonics 8, 821 (2014).
Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological states in photonic systems. Nat. Phys. 12, 626–629 (2016).
Khanikaev, A. B. & Shvets, G. Two-dimensional topological photonics. Nat. Photonics 11, 763 (2017).
Ozawa, T. et al. Topological photonics. Preprint at arXiv https://arxiv.org/abs/1802.04173 (2018).
Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Progress. Phys. 77, 126401 (2014).
Goldman, N., Budich, J. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016).
Cooper, N., Dalibard, J. & Spielman, I. Topological bands for ultracold atoms. Preprint at arXiv https://arxiv.org/abs/1803.00249 (2018).
Boada, O., Celi, A., Latorre, J. I. & Lewenstein, M. Quantum simulation of an extra dimension. Phys. Rev. Lett. 108, 133001 (2012).
Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).
Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008).
Tsomokos, D. I., Ashhab, S. & Nori, F. Using superconducting qubit circuits to engineer exotic lattice systems. Phys. Rev. A 82, 052311 (2010).
Jukić, D. & Buljan, H. Four-dimensional photonic lattices and discrete tesseract solitons. Phys. Rev. A 87, 013814 (2013).
Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).
Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).
Kraus, Y. E. & Zilberberg, O. Topological equivalence between the Fibonacci quasicrystal and the Harper model. Phys. Rev. Lett. 109, 116404 (2012).
Verbin, M., Zilberberg, O., Kraus, Y. E., Lahini, Y. & Silberberg, Y. Observation of topological phase transitions in photonic quasicrystals. Phys. Rev. Lett. 110, 076403 (2013).
Kraus, Y. E., Ringel, Z. & Zilberberg, O. Four-dimensional quantum Hall effect in a two-dimensional quasicrystal. Phys. Rev. Lett. 111, 226401 (2013).
Verbin, M., Zilberberg, O., Lahini, Y., Kraus, Y. E. & Silberberg, Y. Topological pumping over a photonic Fibonacci quasicrystal. Phys. Rev. B 91, 064201 (2015).
Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350 (2016).
Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296 (2016).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).
Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. Sect. A 68, 874 (1955).
Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).
Arkani-Hamed, N., Cohen, A. G. & Georgi, H. (De)Constructing dimensions. Phys. Rev. Lett. 86, 4757–4761 (2001).
Casati, G., Guarneri, I. & Shepelyansky, D. L. Anderson transition in a one-dimensional system with three incommensurate frequencies. Phys. Rev. Lett. 62, 345–348 (1989).
Edge, J. M., Tworzydlo, J. & Beenakker, C. W. J. Metallic phase of the quantum Hall effect in four-dimensional space. Phys. Rev. Lett. 109, 135701 (2012).
Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B. & Raizen, M. G. Atom optics realization of the quantum δ-kicked rotor. Phys. Rev. Lett. 75, 4598–4601 (1995).
Manai, I. et al. Experimental observation of two-dimensional Anderson localization with the atomic kicked rotor. Phys. Rev. Lett. 115, 240603 (2015).
Chabé, J. et al. Experimental observation of the Anderson metal-insulator transition with atomic matter waves. Phys. Rev. Lett. 101, 255702 (2008).
Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).
Stuhl, B., Lu, H.-I., Aycock, L., Genkina, D. & Spielman, I. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).
Livi, L. F. et al. Synthetic dimensions and spin-orbit coupling with an optical clock transition. Phys. Rev. Lett. 117, 220401 (2016).
Kolkowitz, S. et al. Spin–orbit-coupled fermions in an optical lattice clock. Nature 542, 66 (2017).
Gadway, B. Atom-optics approach to studying transport phenomena. Phys. Rev. A 92, 043606 (2015).
Meier, E. J., An, F. A. & Gadway, B. Atom-optics simulator of lattice transport phenomena. Phys. Rev. A 93, 051602 (2016).
Meier, E. J., An, F. A. & Gadway, B. Observation of the topological soliton state in the Su–Schrieffer–Heeger model. Nat. Commun. 7, 13986 (2016).
An, F. A., Meier, E. J. & Gadway, B. Direct observation of chiral currents and magnetic reflection in atomic flux lattices. Sci. Adv. 3, e1602685 (2017).
An, F. A., Meier, E. J., Ang’ong’a, J. & Gadway, B. Correlated dynamics in a synthetic lattice of momentum states. Phys. Rev. Lett. 120, 040407 (2018).
Price, H. M., Ozawa, T. & Goldman, N. Synthetic dimensions for cold atoms from shaking a harmonic trap. Phys. Rev. A 95, 023607 (2017).
Sundar, B., Gadway, B. & Hazzard, K. R. Synthetic dimensions in ultracold polar molecules. Sci. Rep. 8, 3422 (2018).
Martin, I., Refael, G. & Halperin, B. Topological frequency conversion in strongly driven quantum systems. Phys. Rev. X 7, 041008 (2017).
Baum, Y. & Refael, G. Setting boundaries with memory: generation of topological boundary states in Floquet-induced synthetic crystals. Phys. Rev. Lett. 120, 106402 (2018).
Peng, Y. & Refael, G. Topological energy conversion through the bulk or the boundary of driven systems. Phys. Rev. B 97, 134303 (2018).
Andrijauskas, T., Spielman, I. B. & Juzeliūnas, G. Topological lattice using multi-frequency radiation. New J. Phys. 20, 055001 (2018).
Wall, M. L. et al. Synthetic spin-orbit coupling in an optical lattice clock. Phys. Rev. Lett. 116, 035301 (2016).
Genkina, D. et al. Imaging topology of Hofstadter ribbons. Preprint at arXiv https://arxiv.org/abs/1804.06345 (2018).
Mugel, S. et al. Measuring Chern numbers in Hofstadter strips. SciPost Phys. 3, 012 (2017).
Han, J. H., Kang, J. H. & Shin, Y.-i. Band gap closing in a synthetic Hall tube of neutral fermions. Preprint at arXiv https://arxiv.org/abs/1809.00444 (2018).
Li, C.-H. et al. A Bose-Einstein condensate on a synthetic Hall cylinder. Preprint at arXiv https://arxiv.org/abs/1809.02122 (2018).
Boada, O., Celi, A., Rodríguez-Laguna, J., Latorre, J. I. & Lewenstein, M. Quantum simulation of non-trivial topology. New J. Phys. 17, 045007 (2015).
Anisimovas, E. et al. Semisynthetic zigzag optical lattice for ultracold bosons. Phys. Rev. A 94, 063632 (2016).
Xu, J., Gu, Q. & Mueller, E. J. Realizing the Haldane phase with bosons in optical lattices. Phys. Rev. Lett. 120, 085301 (2018).
Suszalski, D. & Zakrzewski, J. Different lattice geometries with a synthetic dimension. Phys. Rev. A 94, 033602 (2016).
Cooper, N. R. & Moessner, R. Designing topological bands in reciprocal space. Phys. Rev. Lett. 109, 215302 (2012).
Ozawa, T., Price, H. M. & Carusotto, I. Momentum-space Harper-Hofstadter model. Phys. Rev. A 92, 023609 (2015).
Price, H. M., Ozawa, T. & Carusotto, I. Quantum mechanics with a momentum-space artificial magnetic field. Phys. Rev. Lett. 113, 190403 (2014).
Berceanu, A. C., Price, H. M., Ozawa, T. & Carusotto, I. Momentum-space Landau levels in driven-dissipative cavity arrays. Phys. Rev. A 93, 013827 (2016).
Ozawa, T., Price, H. M. & Carusotto, I. Quantum Hall effect in momentum space. Phys. Rev. B 93, 195201 (2016).
Claassen, M., Lee, C. H., Thomale, R., Qi, X.-L. & Devereaux, T. P. Position-momentum duality and fractional quantum Hall effect in Chern insulators. Phys. Rev. Lett. 114, 236802 (2015).
An, F. A., Meier, E. J. & Gadway, B. Diffusive and arrested transport of atoms under tailored disorder. Nat. Commun. 8, 325 (2017).
Meier, E. J. et al. Observation of the topological Anderson insulator in disordered atomic wires. Science 362, 929–933 (2018).
Cai, H. et al. Experimental observation of momentum-space chiral edge currents in room-temperature atoms. Phys. Rev. Lett. 122, 023601 (2018).
Wang, D.-W., Liu, R.-B., Zhu, S.-Y. & Scully, M. O. Superradiance lattice. Phys. Rev. Lett. 114, 043602 (2015).
Wang, D.-W., Cai, H., Yuan, L., Zhu, S.-Y. & Liu, R.-B. Topological phase transitions in superradiance lattices. Optica 2, 712–715 (2015).
Chen, L. et al. Experimental observation of one-dimensional superradiance lattices in ultracold atoms. Phys. Rev. Lett. 120, 193601 (2018).
Salerno, G. et al. The quantized Hall conductance of a single atomic wire: a proposal based on synthetic dimensions. Preprint at arXiv https://arxiv.org/abs/1811.00963 (2018).
Sundar, B., Thibodeau, M., Wang, Z., Gadway, B. & Hazzard, K. Strings of ultracold molecules in a synthetic dimension. Phys. Rev. A 99, 013624 (2018).
Signoles, A. et al. Coherent transfer between low-angular-momentum and circular rydberg states. Phys. Rev. Lett. 118, 253603 (2017).
Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).
Luo, X.-W. et al. Quantum simulation of 2D topological physics in a 1D array of optical cavities. Nat. Commun. 6, 7704 (2015).
Zhou, X.-F. et al. Dynamically manipulating topological physics and edge modes in a single degenerate optical cavity. Phys. Rev. Lett. 118, 083603 (2017).
Cardano, F. et al. Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons. Nat. Commun. 8, 15516 (2017).
Wang, B., Chen, T. & Zhang, X. Experimental observation of topologically protected bound states with vanishing Chern numbers in a two-dimensional quantum walk. Phys. Rev. Lett. 121, 100501 (2018).
Schwartz, A. & Fischer, B. Laser mode hyper-combs. Opt. Express 21, 6196–6204 (2013).
Ozawa, T., Price, H. M., Goldman, N., Zilberberg, O. & Carusotto, I. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics. Phys. Rev. A 93, 043827 (2016).
Yuan, L., Shi, Y. & Fan, S. Photonic gauge potential in a system with a synthetic frequency dimension. Opt. Lett. 41, 741–744 (2016).
Yuan, L., Xiao, M., Lin, Q. & Fan, S. Synthetic space with arbitrary dimensions in a few rings undergoing dynamic modulation. Phys. Rev. B 97, 104105 (2018).
Bell, B. A. et al. Spectral photonic lattices with complex long-range coupling. Optica 4, 1433–1436 (2017).
Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Preprint at arXiv https://arxiv.org/abs/1807.01983(2018).
Boyd, R. W. In Handbook of Laser Technology and Applications (Three-Volume Set) (eds Webb, C. E. & Jones, J. D. C.) 161–183 (Taylor & Francis, 2003).
Schreiber, A. et al. Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010).
Regensburger, A. et al. Photon propagation in a discrete fiber network: an interplay of coherence and losses. Phys. Rev. Lett. 107, 233902 (2011).
Schreiber, A. et al. Decoherence and disorder in quantum walks: from ballistic spread to localization. Phys. Rev. Lett. 106, 180403 (2011).
Vatnik, I. D., Tikan, A., Onishchukov, G., Churkin, D. V. & Sukhorukov, A. A. Anderson localization in synthetic photonic lattices. Sci. Rep. 7, 4301 (2017).
Regensburger, A. et al. Parity–time synthetic photonic lattices. Nature 488, 167 (2012).
Regensburger, A. et al. Observation of defect states in ð’« ð’ˉ-symmetric optical lattices. Phys. Rev. Lett. 110, 223902 (2013).
Wimmer, M. et al. Observation of optical solitons in PT-symmetric lattices. Nat. Commun. 6, 7782 (2015).
Wimmer, M., Price, H. M., Carusotto, I. & Peschel, U. Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys. 13, 545 (2017).
Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012).
Chen, C. et al. Observation of topologically protected edge states in a photonic two-dimensional quantum walk. Phys. Rev. Lett. 121, 100502 (2018).
Schmidt, M., Kessler, S., Peano, V., Painter, O. & Marquardt, F. Optomechanical creation of magnetic fields for photons on a lattice. Optica 2, 635–641 (2015).
Poshakinskiy, A. V. & Poddubny, A. N. Phonoritonic crystals with a synthetic magnetic field for an acoustic diode. Phys. Rev. Lett. 118, 156801 (2017).
Ozawa, T. & Carusotto, I. Synthetic dimensions with magnetic fields and local interactions in photonic lattices. Phys. Rev. Lett. 118, 013601 (2017).
Chang, M.-S., Qin, Q., Zhang, W., You, L. & Chapman, M. S. Coherent spinor dynamics in a spin-1 bose condensate. Nat. Phys. 1, 111 (2005).
Barbarino, S., Taddia, L., Rossini, D., Mazza, L. & Fazio, R. Magnetic crystals and helical liquids in alkaline-earth fermionic gases. Nat. Commun. 6, 8134 (2015).
Yan, Z., Wan, S. & Wang, Z. Topological superfluid and Majorana zero modes in synthetic dimension. Sci. Rep. 5, 15927 (2015).
Cornfeld, E. & Sela, E. Chiral currents in one-dimensional fractional quantum hall states. Phys. Rev. B 92, 115446 (2015).
Barbarino, S., Taddia, L., Rossini, D., Mazza, L. & Fazio, R. Synthetic gauge fields in synthetic dimensions: interactions and chiral edge modes. New J. Phys. 18, 035010 (2016).
Zeng, T.-S., Wang, C. & Zhai, H. Charge pumping of interacting fermion atoms in the synthetic dimension. Phys. Rev. Lett. 115, 095302 (2015).
Taddia, L. et al. Topological fractional pumping with alkaline-earth-like atoms in synthetic lattices. Phys. Rev. Lett. 118, 230402 (2017).
Jünemann, J. et al. Exploring interacting topological insulators with ultracold atoms: the synthetic Creutz-Hubbard model. Phys. Rev. X 7, 031057 (2017).
Ghosh, S. K., Yadav, U. K. & Shenoy, V. B. Baryon squishing in synthetic dimensions by effective SU(m) gauge fields. Phys. Rev. A 92, 051602 (2015).
Ghosh, S. K. et al. Unconventional phases of attractive fermi gases in synthetic Hall ribbons. Phys. Rev. A 95, 063612 (2017).
Greschner, S. et al. Symmetry-broken states in a system of interacting bosons on a two-leg ladder with a uniform abelian gauge field. Phys. Rev. A 94, 063628 (2016).
Greschner, S. & Vekua, T. Vortex-hole duality: a unified picture of weak- and strong-coupling regimes of bosonic ladders with flux. Phys. Rev. Lett. 119, 073401 (2017).
Bilitewski, T. & Cooper, N. R. Synthetic dimensions in the strong-coupling limit: supersolids and pair superfluids. Phys. Rev. A 94, 023630 (2016).
Calvanese Strinati, M. et al. Laughlin-like states in bosonic and fermionic atomic synthetic ladders. Phys. Rev. X 7, 021033 (2017).
Łącki, M. et al. Quantum Hall physics with cold atoms in cylindrical optical lattices. Phys. Rev. A 93, 013604 (2016).
Saito, T. Y. & Furukawa, S. Devil’s staircases in synthetic dimensions and gauge fields. Phys. Rev. A 95, 043613 (2017).
An, F. A., Meier, E. J. & Gadway, B. Engineering a flux-dependent mobility edge in disordered zigzag chains. Phys. Rev. X 8, 031045 (2018).
Luo, X.-W. et al. Synthetic-lattice enabled all-optical devices based on orbital angular momentum of light. Nat. Commun. 8, 16097 (2017).
Yuan, L. & Fan, S. Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator. Optica 3, 1014–1018 (2016).
Sun, B. Y., Luo, X. W., Gong, M., Guo, G. C. & Zhou, Z. W. Weyl semimetal phases and implementation in degenerate optical cavities. Phys. Rev. A 96, 013857 (2017).
Lin, Q., Xiao, M., Yuan, L. & Fan, S. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension. Nat. Commun. 7, 13731 (2016).
Lin, Q., Sun, X.-Q., Xiao, M., Zhang, S.-C. & Fan, S. A three-dimensional photonic topological insulator using a two-dimensional ring resonator lattice with a synthetic frequency dimension. Sci. Adv. 4, eaat2774 (2018).
Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009).
Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).
Fröhlich, J. & Pedrini, B. In Mathematical Physics 2000 (eds Fokas, A., Grigoryan, A., Kibble, T. & Zegarlinski, B.) 9–47 (Imperial College Press, 2000).
Zhang, S.-C. & Hu, J. A four-dimensional generalization of the quantum Hall effect. Science 294, 823–828 (2001).
Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008).
Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Four-dimensional quantum Hall effect with ultracold atoms. Phys. Rev. Lett. 115, 195303 (2015).
Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55 (2018).
Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59 (2018).
Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Measurement of Chern numbers through center-of-mass responses. Phys. Rev. B 93, 245113 (2016).
Lian, B. & Zhang, S.-C. Five-dimensional generalization of the topological Weyl semimetal. Phys. Rev. B 94, 041105 (2016).
Lee, C. H., Wang, Y., Chen, Y. & Zhang, X. Electromagnetic response of quantum Hall systems in dimensions five and six and beyond. Phys. Rev. B 98, 094434 (2018).
Petrides, I., Price, H. M. & Zilberberg, O. Six-dimensional quantum hall effect and three-dimensional topological pumps. Phys. Rev. B 98, 125431 (2018).
Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009).
Viebahn, K., Sbroscia, M., Carter, E., Yu, J.-C. & Schneider, U. Matter-wave diffraction from a quasicrystalline optical lattice. Phys. Rev. Lett. 122, 110404 (2019).
T.O. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant number JP18H05857, the RIKEN Incentive Research Project and the Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) program at RIKEN. H.M.P. was supported by funding from the Royal Society.
The authors declare no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Ozawa, T., Price, H.M. Topological quantum matter in synthetic dimensions. Nat Rev Phys 1, 349–357 (2019). https://doi.org/10.1038/s42254-019-0045-3
This article is cited by
Communications Physics (2023)
Light: Science & Applications (2023)
Nature Physics (2023)
Nature Communications (2023)