Abstract

Quantum walks present novel tools for redesigning quantum algorithms, universal quantum computations, and quantum simulators. Hitherto, one- and two-dimensional quantum systems (lattices) have been simulated and studied with photonic systems. Here, we report the photonic simulation of cyclic quantum systems, such as hexagonal structures. We experimentally explore the wavefunction dynamics and probability distribution of a quantum particle located on a six-site system, along with three- and four-site systems while under different initial conditions. Various quantum walk systems employing Hadamard, C-NOT, and Pauli-Z gates are experimentally simulated, where we find configurations capable of simulating particle transport and probability density localization. Our technique can potentially be integrated into small-scale structures using microfabrication, and thus would open a venue towards simulating more complicated quantum systems comprised of cyclic structures.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]

2017 (2)

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]

T. Chen, B. Wang, and X. Zhang, “Controlling probability transfer in the discrete-time quantum walk by modulating the symmetries,” New J. Phys. 19, 113049 (2017).
[Crossref]

2016 (2)

S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi, “Measurement of topological invariants in a 2D photonic system,” Nat. Photonics 10, 180–183 (2016).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

2015 (3)

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]

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (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]

2014 (4)

T. Ozawa and I. Carusotto, “Anomalous and quantum Hall effects in lossy photonic lattices,” Phys. Rev. Lett. 112, 133902 (2014).
[Crossref]

M. Hafezi, “Measuring topological invariants in photonic systems,” Phys. Rev. Lett. 112, 210405 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljiačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
[Crossref]

I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014).
[Crossref]

2012 (6)

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (2012).
[Crossref]

R. Blatt and C. F. Roos, “Quantum simulations with trapped ions,” Nat. Phys. 8, 277–284 (2012).
[Crossref]

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

T. Kitagawa, “Topological phenomena in quantum walks: elementary introduction to the physics of topological phases,” Quantum Inf. Process. 11, 1107–1148 (2012).
[Crossref]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6(2012).
[Crossref]

2011 (3)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

O. Mülken and A. Blumen, “Continuous-time quantum walks: models for coherent transport on complex networks,” Phys. Rep. 502, 37–87 (2011).
[Crossref]

X. Qi and S. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

2010 (4)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (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]

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104, 100503 (2010).
[Crossref]

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

2009 (5)

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103, 090504 (2009).
[Crossref]

M. Karski, L. Förster, 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]

V. Potoček, A. Gábris, T. Kiss, and I. Jex, “Optimized quantum random-walk search algorithms on the hypercube,” Phys. Rev. A 79, 012325 (2009).
[Crossref]

I. Buluta and F. Nori, “Quantum simulators,” Science 326, 108–111 (2009).
[Crossref]

2006 (1)

D. Solenov and L. Fedichkin, “Continuous-time quantum walks on a cycle graph,” Phys. Rev. A 73, 012313 (2006).
[Crossref]

2005 (2)

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor,” Phys. Rev. A 72, 062317 (2005).
[Crossref]

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref]

2003 (2)

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

N. Shenvi, J. Kempe, and B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67, 052307 (2003).
[Crossref]

1990 (1)

D. J. Moore, “Berry phases and Hamiltonian time dependence,” J. Phys. A 23, 5523–5534 (1990).
[Crossref]

1989 (1)

J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747–2750 (1989).
[Crossref]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[Crossref]

1982 (2)

R. Feynman, “Simulating physics with computers,” Int. J. Theor. Phys. 21, 467–488 (1982).
[Crossref]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. 49, 405–408 (1982).
[Crossref]

1965 (1)

W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. 140, A1133 (1965).
[Crossref]

1964 (1)

P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev. 136, B864 (1964).
[Crossref]

Alt, W.

M. Karski, L. Förster, 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]

Ashhab, S.

I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod. Phys. 86, 153–185 (2014).
[Crossref]

Aspuru-Guzik, A.

A. Aspuru-Guzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8, 285–291 (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]

Berg, E.

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]

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

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[Crossref]

Blatt, R.

R. Blatt and C. F. Roos, “Quantum simulations with trapped ions,” Nat. Phys. 8, 277–284 (2012).
[Crossref]

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104, 100503 (2010).
[Crossref]

Blumen, A.

O. Mülken and A. Blumen, “Continuous-time quantum walks: models for coherent transport on complex networks,” Phys. Rep. 502, 37–87 (2011).
[Crossref]

Boileau, J. C.

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor,” Phys. Rev. A 72, 062317 (2005).
[Crossref]

Boyd, R. W.

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]

Broome, M. A.

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]

Buluta, I.

I. Buluta and F. Nori, “Quantum simulators,” Science 326, 108–111 (2009).
[Crossref]

Cardano, F.

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]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[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]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6(2012).
[Crossref]

Carusotto, I.

T. Ozawa and I. Carusotto, “Anomalous and quantum Hall effects in lossy photonic lattices,” Phys. Rev. Lett. 112, 133902 (2014).
[Crossref]

Cataudella, V.

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]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Chang, M.

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

Chen, T.

T. Chen, B. Wang, and X. Zhang, “Controlling probability transfer in the discrete-time quantum walk by modulating the symmetries,” New J. Phys. 19, 113049 (2017).
[Crossref]

Childs, A. M.

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

Choi, J. M.

M. Karski, L. Förster, 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]

Chong, Y. D.

W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, “Measurement of a topological edge invariant in a microwave network,” Phys. Rev. X 5, 011012 (2015).
[Crossref]

D’Errico, A.

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]

Dauphin, A.

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]

De Filippis, G.

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]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

de Lisio, C.

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]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[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]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6(2012).
[Crossref]

Demler, E.

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Supplementary Material (1)

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» Supplement 1       Supplementary Information

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Figures (4)

Fig. 1.
Fig. 1. Geometry and physical properties of a six-site cyclic quantum system. (a) Schematic representation of a single quantum particle wavefunction propagating on the periphery of a ring with identical sites. The quantum particle hops to its neighboring sites, 2 or 6, depending on its spin state, i.e., |H or |V. (b) Energy band structure E(k) (top) and group velocity v(k) profile (bottom) of the six-site cyclic quantum walk. The bold points on the curves indicate the quasi-momentum eigenvalues of the system and are positioned along two bands over k space as the Brillouin zone is traversed. The red and blue bands are related to j=1 and j=2, respectively. (c) Path of the indicative unit vector on the Bloch sphere indicates spin eigenstates of the coin operator for the six-site ring. These points on the Bloch sphere correspond to the allowed values of the quasi-momenta. This great circle for the Hadamard walk can, indeed, be obtained by applying a π/4 rotation on the equator circle about the |L/|R-axes.
Fig. 2.
Fig. 2. Illustration of the experimental apparatus used to perform the cyclic quantum walk. Photon pairs are generated through SPDC, where one of the photons is used to trigger an ICCD camera for the detection of its partner that performs the CQW. The initial spatial state of the photon |ϕin is set by a SLM (lower inset), and its polarization state is set by a HWP and QWP. Each step of the CQW is performed by means of a HWP, which acts as the coin operator, and a polarizing Sagnac interferometer with an imbedded Dove prism (DP). The DP performs the conditional shift operator. Whenever the photon is incident on the 50:50 beam splitter (BS), it will either be sent to the ICCD camera (to be detected) or fed into the interferometers where it will perform two steps of the CQW. Imaging lenses (L) are used to image the plane of the SLM into the interferometers and onto the ICCD camera, indicated by the colored planes. To preserve polarization (especially circular polarization), all mirrors used in the experimental setup are silver mirrors. Due to the 50% loss of photons at the BS after every two steps and 2% loss for every reflection off a silver mirror, the photons can be fed back into the interferometers only twice before their signal is washed out by background noise on the camera. Thus, with two interferometers, only steps 2, 4, and 6 can be recorded, and by removing the PBS and DP in the second interferometer, steps 1 and 3 are recorded. The final polarization of the photons can be measured (via Stokes measurements) by placing a polarizer in front of the ICCD camera. Figure legends: ppKTP, periodically polled KTP; L, lens; HM, half-mirror; SMF, single-mode fiber; SLM, spatial light modulator; SPAD, single-photon avalanche diode; ICCD, intensified CCD; BS, 50:50 beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; DP, Dove prism; and LP, long-pass filter.
Fig. 3.
Fig. 3. Probability distribution of a cyclic quantum walk with localized initial states. Experimental data and theoretical calculation for the probability distribution of the walker over the course of a six-step evolution for a state initially prepared as (a) |ψ(0)=|0|V and (b) |ψ(0)=|0|L. The right columns in (a) and (b) represent the experimental data recorded on the ICCD camera for the initial state (0th step) and consecutive steps of the CQW up to six steps (step 5 is excluded). Each step of the walk is shown as the probability distribution of the photons, displayed as circles split into six sectors, which represent the six sites of a hexagonal-like structure. In both cases, the initial position of the walker is localized on one site. At the first step of the walk, the particle’s wavefunction is split into two equal portions on the neighboring sites (this is expected for the Hadamard coin). In the second step, the wavefunction favors localization in its initial position, but two nonzero probabilities for the walker are realized on the next nearest neighbors of the initial site. The probability distributions for the third, fourth, and sixth steps differ for the two cases as a result of interference effects discussed in the main text. The average quantum fidelities (defined in Supplement 1) over six steps for the initial conditions |ψ(0)=|0|V and |ψ(0)=|0|L are 0.8815±0.003 and 0.8882±0.001, respectively. Due to the exponential accumulation of loss in our setup, the signal for higher steps is washed out by background noise, and thus no acceptable data could be presented here.
Fig. 4.
Fig. 4. Probability distribution of a cyclic quantum walk with a stationary eigenstate. The experimental and theoretical results for the time evolution of a stationary eigenstate of the hexagonal ring indicating standing waves. The initial state is prepared as |ψ(0)=16(|0|1+|2|3+|4|5)|L. (a) Theoretical (left side) and experimental (right side) results indicating a stable and uniform distribution over each step of the walk with an average fidelity of 0.9982±0.0002 over the time evolution. (b) Polarization measurements over four steps indicating a conserved state of left-hand circular polarization, the initial state |L, and the absence of right-hand circular polarization state upon time evolution. (c) Bloch vectors for each site over four steps are experimentally inferred using Stokes measurements, i.e., polarization tomography [45]. The polarization state for all six sites remains unchanged, and the measured left-hand circular polarization |L is described via the red arrows on the Bloch vectors.

Equations (3)

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Ej(k,π4)=(1)jcos1[cos(k)2],
Vg,j(k,π4)=Ej(k,π4)k=(1)jsin(k)2cos2(k).
nπ4x(k)=sin(k)2cos2(k),nπ4y(k)=cos(k)2cos2(k),nπ4z(k)=sin(k)2cos2(k).

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