Abstract

Quantum walks are powerful tools for quantum applications and for designing topological systems. Although they are simulated in a variety of platforms, genuine two-dimensional realizations are still challenging. Here we present an innovative approach to the photonic simulation of a quantum walk in two dimensions, where walker positions are encoded in the transverse-wavevector component of a single light beam. The desired dynamics is obtained by means of a sequence of liquid-crystal devices, which apply polarization-dependent transverse “kicks” to the photons in the beam. We engineer our quantum walk so that it realizes a periodically driven Chern insulator, and we probe its topological features by detecting the anomalous displacement of the photonic wavepacket under the effect of a constant force. Our compact, versatile platform offers exciting prospects for the photonic simulation of two-dimensional quantum dynamics and topological systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (5)

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

H. Chalabi, S. Barik, S. Mittal, T. E. Murphy, M. Hafezi, and E. Waks, “Synthetic gauge field for two-dimensional time-multiplexed quantum random walks,” Phys. Rev. Lett. 123, 150503 (2019).
[Crossref]

A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, “Q-plate technology: a progress review [Invited],” J. Opt. Soc. Am. B 36, D70–D87 (2019).
[Crossref]

K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, and P. Xue, “Simulating dynamic quantum phase transitions in photonic quantum walks,” Phys. Rev. Lett. 122, 020501 (2019).
[Crossref]

K. Yokomizo and S. Murakami, “Non-Bloch band theory of non-Hermitian systems,” Phys. Rev. Lett. 123, 066404 (2019).
[Crossref]

2018 (9)

S. Yao, F. Song, and Z. Wang, “Non-Hermitian Chern bands,” Phys. Rev. Lett. 121, 136802 (2018).
[Crossref]

Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, “Topological phases of non-Hermitian systems,” Phys. Rev. X 8, 031079 (2018).
[Crossref]

M. Heyl, “Dynamical quantum phase transitions: a review,” Rep. Prog. Phys. 81, 054001 (2018).
[Crossref]

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing Chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100501 (2018).
[Crossref]

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4, eaat3174 (2018).
[Crossref]

H. Tang, C. Di Franco, Z.-Y. Shi, T.-S. He, Z. Feng, J. Gao, K. Sun, Z.-M. Li, Z.-Q. Jiao, T.-Y. Wang, M. S. Kim, and X.-M. Jin, “Experimental quantum fast hitting on hexagonal graphs,” Nat. Photonics 12, 754–758 (2018).
[Crossref]

M. Aidelsburger, S. Nascimbene, and N. Goldman, “Artificial gauge fields in materials and engineered systems,” C. R. Physique 19, 394 (2018).
[Crossref]

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S. Summy, “Quantum walk in momentum space with a Bose-Einstein condensate,” Phys. Rev. Lett. 121, 070402 (2018).
[Crossref]

2017 (7)

E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi, “Observing topological invariants using quantum walks in superconducting circuits,” Phys. Rev. X 7, 031023 (2017).
[Crossref]

N. C. Harris, G. R. Steinbrecher, M. Prabhu, Y. Lahini, J. Mower, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13, 1117–1123 (2017).
[Crossref]

X. Zhan, L. Xiao, Z. Bian, K. Wang, X. Qiu, B. C. Sanders, W. Yi, and P. Xue, “Detecting topological invariants in nonunitary discrete-time quantum walks,” Phys. Rev. Lett. 119, 130501 (2017).
[Crossref]

A. B. Khanikaev and G. Shvets, “Two-dimensional topological photonics,” Nat. Photonics 11, 763–773 (2017).
[Crossref]

S. Longhi, “Parity-time symmetry meets photonics: a new twist in non-Hermitian optics,” Europhys. Lett. 120, 64001 (2017).
[Crossref]

2016 (2)

H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, “Measurement of Chern numbers through center-of-mass responses,” Phys. Rev. B 93, 245113 (2016).
[Crossref]

H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, “Two-photon quantum walk in a multimode fiber,” Sci. Adv. 2, e1501054 (2016).
[Crossref]

2015 (4)

P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner, “Strongly correlated quantum walks in optical lattices,” Science 347, 1229–1233 (2015).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Dynamical moments reveal a topological quantum transition in a photonic quantum walk,” Nat. Commun. 7, 11439 (2015).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

2014 (1)

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112, 143604 (2014).
[Crossref]

2013 (5)

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, “Experimental realization of a delayed-choice quantum walk,” Nat. Commun. 4, 2471 (2013).
[Crossref]

M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
[Crossref]

A. Dauphin and N. Goldman, “Extracting the Chern number from the dynamics of a Fermi gas: Implementing a quantum Hall bar for cold atoms,” Phys. Rev. Lett. 111, 135302 (2013).
[Crossref]

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3, 031005 (2013).
[Crossref]

2012 (4)

S. E. Venegas-Andraca, “Quantum walks: a comprehensive review,” Quantum Inf. Process. 11, 1015–1106 (2012).
[Crossref]

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle Bosonic-Fermionic quantum walk via integrated photonics,” Phys. Rev. Lett. 108, 010502 (2012).
[Crossref]

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3, 882 (2012).
[Crossref]

2011 (3)

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

C. Di Franco, M. Mc Gettrick, and T. Busch, “Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin,” Phys. Rev. Lett. 106, 080502 (2011).
[Crossref]

C. Di Franco, M. Mc Gettrick, T. Machida, and T. Busch, “Alternate two-dimensional quantum walk with a single-qubit coin,” Phys. Rev. A 84, 042337 (2011).
[Crossref]

2010 (5)

D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Rev. Mod. Phys. 82, 1959–2007 (2010).
[Crossref]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversion via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104, 153602 (2010).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

2009 (2)

M. Karski, L. Forster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174–177 (2009).
[Crossref]

A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102, 180501 (2009).
[Crossref]

2008 (2)

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129, 174106 (2008).
[Crossref]

X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B 78, 195424 (2008).
[Crossref]

2006 (1)

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F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Dynamical moments reveal a topological quantum transition in a photonic quantum walk,” Nat. Commun. 7, 11439 (2015).
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F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
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H. Tang, C. Di Franco, Z.-Y. Shi, T.-S. He, Z. Feng, J. Gao, K. Sun, Z.-M. Li, Z.-Q. Jiao, T.-Y. Wang, M. S. Kim, and X.-M. Jin, “Experimental quantum fast hitting on hexagonal graphs,” Nat. Photonics 12, 754–758 (2018).
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C. Di Franco, M. Mc Gettrick, T. Machida, and T. Busch, “Alternate two-dimensional quantum walk with a single-qubit coin,” Phys. Rev. A 84, 042337 (2011).
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C. Di Franco, M. Mc Gettrick, and T. Busch, “Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin,” Phys. Rev. Lett. 106, 080502 (2011).
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A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
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H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4, eaat3174 (2018).
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M. Genske, W. Alt, A. Steffen, A. H. Werner, R. F. Werner, D. Meschede, and A. Alberti, “Electric quantum walks with individual atoms,” Phys. Rev. Lett. 110, 190601 (2013).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Experimental concept and apparatus. (a) A collimated beam crosses a sequence of liquid-crystal (LC) devices. Different LC patterns implement coin rotations ($ W $) and spin-dependent walker discrete translations ($ {T_x} $ and $ {T_y} $). Each evolution step $ U = {T_y}{T_x}W $ is realized with three LC devices. The walker position is encoded in the transverse momentum of photons, so that walker steps physically correspond to transverse kicks that tilt slightly the photon propagation direction. The transverse diffraction of light remains negligible across the whole setup, and the entire evolution effectively occurs in a single beam. At the exit of the walk, a lens (with focal distance equal to 50 cm) Fourier-transforms transverse momentum into position, allowing us to resolve and measure individual modes. (b) The recorded intensity pattern is a regular grid of small Gaussian spots, whose intensities are proportional to the walker’s spatial probability distribution. We set the modes beam radius to $ {w_0} = 5 \,\, {\rm mm} $, which corresponds to a spot size of $ \simeq 20 \,\, \unicode{x00B5}{\rm m} $ (radius) on the camera plane. (c) LC optic-axis pattern for a $ g $-plate that realizes a $ {T_x} $ operator. The spatial period $ \Lambda $ fixes the transverse momentum lattice spacing $ \Delta {k_ \bot } = 2\pi /\Lambda $. We use $ \Lambda = 5 \,\,{\rm mm} $, so that $ \Delta {k_ \bot } = 1.26 \,\,{{\rm mm}^{ - 1}} $, corresponding to a spacing between spots of $ \simeq 63 \,\, \unicode{x00B5}{\rm m} $ on the camera. (d) Action of a single $ g $-plate $ {T_x} $ on a linearly polarized beam $ |{\Psi _0}\rangle = |0,0,H\rangle $, where $ |H\rangle = (|L\rangle + |R\rangle )/\sqrt 2 $, for three different values of $ \delta $.
Fig. 2.
Fig. 2. 2D quantum walk on a square lattice. Spatial probability distributions for a quantum walk with initial condition $ |0,0,H\rangle $ and optical retardation $ \delta = \pi /2 $. From top to bottom, we display results after 0, 3, and 5 evolution steps. Datapoints are averages of four independent measures.
Fig. 3.
Fig. 3. Detection of the group velocity at $ \delta = \pi /2 $. (a) Quasi-energy spectrum of the effective Hamiltonian $ {H_{{\rm eff}}} $. (b) Light intensity distribution measured for a wavepacket with $ {\textbf{q}_0} = (\pi /2,\pi ) $ in the upper band, where the expected group velocity is $ {\textbf{v}^{( + )}} = (0, - 0.5) $. The white marker indicates the center of mass of the wavepacket. The radius $ {w_g} $ of the input beam is $ (0.62 \pm 0.02) \,\,{\rm mm} $. In the camera plane, we measure a beam diameter of $ (0.32 \pm 0.01) \,\,{\rm mm} $, corresponding to $ \approx 5 $ lattice sites. (c) Displacement of the wavepacket center of mass, extracted from images as in panel (b). Experimental results (datapoints) are compared to semiclassical predictions of uniform motion (straight continuous lines) and to complete numerical simulations (dashed lines). Statistical uncertainties include estimated misalignment effects, as discussed in the main text. (d) Experimental mapping of the upper band’s group velocity $ {\textbf{v}^{( + )}} $ along $ x $ across the whole Brillouin zone, compared to a complete numerical simulation. Each datapoint is obtained from a linear fit of the center-of-mass displacement of a Gaussian wavepacket in five steps.
Fig. 4.
Fig. 4. Anomalous displacement for trivial and nontrivial Chern bands. (a) Quasi-energy spectra computed on a cylinder open along $ x $ for $ \delta = \pi /2 $ (top) and $ \delta = 7\pi /8 $ (bottom), showing the Chern numbers $ \nu $ of the various bands. In our Floquet system, edge states (shown as darker lines inside the gaps) may be present even if all bands have vanishing Chern numbers. (b) Center-of-mass average displacement $ {\langle \Delta {m_x}\rangle ^{( - )}}(t) $ and $ {\langle \Delta {m_y}\rangle ^{( - )}}(t) $ measured for $ \delta = \pi /2 $ in the lower band. Empty markers show results from the simple protocol $ U $, while solid symbols show the improved results obtained by combining protocol $ U $ with its inverse $ {U^{ - 1}} $. Straight lines correspond to the theoretical results dictated by the semiclassical equations of motion, predicting an anomalous displacement proportional to the band Chern number. (c) Center of mass displacements measured with $ \delta = 7\, \pi /8 $. Meaning of all symbols and lines as in panel (b). Statistical uncertainties include estimated misalignment effects, as discussed in the main text.

Equations (11)

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| m , ϕ = A ( x , y , z ) e i [ Δ k ( m x x + m y y ) + k z z ] | ϕ ,
L δ ( x , y ) ( cos ( δ / 2 ) i sin ( δ / 2 ) e 2 i α ( x , y ) i sin ( δ / 2 ) e 2 i α ( x , y ) cos ( δ / 2 ) ) .
α ( x , y ) = π x / Λ + α 0 ,
T x ( cos ( δ / 2 ) i sin ( δ / 2 ) e 2 i α 0 t ^ x i sin ( δ / 2 ) e 2 i α 0 t ^ x cos ( δ / 2 ) ) ,
W = 1 2 ( 1 i i 1 ) .
U = T y T x W ,
cos ε = 1 2 ( cos 2 ( δ / 2 ) cos ( δ / 2 ) sin ( δ / 2 ) × ( cos ( q x ) + cos ( q y ) ) sin 2 ( δ / 2 ) cos ( q x q y ) ) .
Ω xy ( ± ) ( q ) = ± 1 2 n(q) [ n q x × n q y ] .
ν ( ± ) = B Z d 2 q 2 π Ω xy ( ± ) ( q ) .
Δ m y = 0 t d τ [ v y ( ± ) ( q τ ) + F x Ω xy ( ± ) ( q τ ) ] ,
Δ m y ( ± ) F x ν ( ± ) 2 π t .

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