Topological Pumping of Photons in Nonlinear Resonator Arrays

Jirawat Tangpanitanon, Victor M. Bastidas, Sarah Al-Assam, Pedram Roushan, Dieter Jaksch, and Dimitris G. Angelakis

Phys. Rev. Lett. 117, 213603 – Published 17 November 2016

Abstract

We show how to implement topological or Thouless pumping of interacting photons in one-dimensional nonlinear resonator arrays by simply modulating the frequency of the resonators periodically in space and time. The interplay between the interactions and the adiabatic modulations enables robust transport of Fock states with few photons per site. We analyze the transport mechanism via an effective analytic model and study its topological properties and its protection to noise. We conclude by a detailed study of an implementation with existing circuit-QED architectures.

Received 15 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.213603

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Geometric & topological phases Quantum simulation Topological effects in photonic systems Topological phase transition

General Physics

Authors & Affiliations

Jirawat Tangpanitanon^1,*, Victor M. Bastidas¹, Sarah Al-Assam², Pedram Roushan³, Dieter Jaksch^2,1, and Dimitris G. Angelakis^1,4,†

¹Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
²Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
³Google Inc., Santa Barbara, California 93117, USA
⁴School of Electrical and Computer Engineering, Technical University of Crete, Chania, Crete 73100, Greece

^*a0122902@u.nus.edu
^†dimitris.angelakis@qubit.org

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Vol. 117, Iss. 21 — 18 November 2016

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Images

Figure 1
(a) Depictions of the sublattices $A$ , $B$ , and $C$ at the sites $3 l$ , $3 l + 1$ , and $3 l + 2$ , respectively. (b) Pump cycle in the 2D parameter space spanned by ( $ω_{A} - ω_{B}$ ) and ( $ω_{A} - ω_{C}$ ) for $U = - J$ . It encircles the critical point at $ω_{A} = ω_{B} = ω_{C}$ labeled as a red dot. A gray-dashed path is displayed as an example of a topologically trivial path. (c1) On-site energies $E_{0}^{μ} (t) = ⟨ μ | H_{0}^{l} (t) | μ ⟩_{l}$ as a function of the modulation phase $ϕ (t)$ . Different bands $μ = 300$ , 030, 003 are labeled as blue, green, and orange, respectively. Crossing points between two bands are labeled as gray dots. (c2) Eigenenergies emerging in the presence of a small photon hopping $J ≪ Δ$ . As discussed in the text, near every crossing point in (c1), an effective three-photon hopping can be derived, which converts these points into the anticrossing points shown in (c2) with the gap $2 J = \sqrt{2} J^{3} / U^{2}$ . As a result, the quantized transport of the Fock states can then be understood by adiabatically following one of the bands in (c2). (d) Illustration of the quantized transport. $T_{p} = 2 π / Ω$ is the pumping period. In the path $I$ , the state $| 300 ⟩_{l}$ is initialized at the highest band with $ϕ (0) = 0$ . The three photons hop from one site to another when passing through each anticrossing point. Since in the upper band there are three anticrossing points for $\forall ϕ (t) \in [0, 2 π)$ , after one pump cycle, the three photons are pumped from $| 300 ⟩_{l}$ to $| 300 ⟩_{l - 1}$ . The transport corresponds to the effective Chern number $C = 1$ . For path II, the transport has different topology with $C = - 2$ . The lowest band has the same topology as the highest one.
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Figure 2
Density plot $⟨ {\hat{n}}_{m} ⟩$ as a function of time, illustrating a quantized transport of a three-photon state. In (a) a three-photon Fock state $| 3 ⟩$ is prepared at the sublattice $A$ located at the middle of an array of size $L = 120$ ( $Δ = 10 J$ , $ϕ_{0} = 0$ , $U = - J$ , $T_{p} = 2 π / Ω$ , and $Ω = 0.01 J$ ). In (b) each sublattice $A$ is filled with the three-photon Fock state. We left five trimers near the edges empty to avoid boundary effects during the evolution. The density plot shows a clear steplike behavior in both cases. In (c) the initial modulation phase is set at $ϕ_{0} = π / 2$ , and the ramping speed is $Ω_{p} = 0.002 J$ . As discussed in the text, this results in a quantized transport in the reversed direction and twice the speed of the pump. In (b) and (c) the local Hilbert space in the numerics is truncated at the five-photon Fock state.
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Figure 3
(a1) Illustration of a topologically trivial pumping scheme in the 2D parameter space. We fix $ω_{A} = ω_{0}$ and vary $ω_{B}$ and $ω_{C}$ as a square loop with the starting point $(x_{0}, y_{0}) = [- Δ \cos (2 π / 3), - Δ \cos (4 π / 3)]$ . The pumping period $T_{p}$ and the initial state are the same as those in Fig. 2. (a2) Density plot showing the corresponding motion. (b) C.m. displacement, $Δ x$ , of a three-photon state as a function of time with the nontrivial pumping topology in the presence of random noise. A black solid line corresponds to the perfect case with no noise $η = 0$ . The parameters of the Hamiltonian and the initial state are the same as those in Fig. 2. The plot shows that the quantized motion is robust against weak perturbations, such that the amplitude of the noise $η$ is smaller than the smallest energy gap $2 J$ .
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Figure 4
(a) Circuit-QED diagram showing an implementation of the Hamiltonian $H (t)$ . We introduce the flux variable, which is defined as $ϕ_{m} = - \int V_{m} d t$ , where $V_{m}$ is a voltage at the corresponding position. As shown in Ref. [39], this quantity can be quantized to the form $ϕ_{m} = α (a_{m} + a_{m}^{†})$ , where $α$ is a constant depending on the circuit’s elements. The Josephson junctions $E_{J 1}$ , $E_{J 2}$ and the shunting capacitor $C_{J}$ act as a nonlinear resonator, whose frequency can be tuned via the flux bias $Φ_{g}$ . Each resonator is coupled by the capacitor $C$ . (b1),(b2) Quantum trajectory simulations of a nine-site lossy resonator array. (b1) Density plot $⟨ {\hat{n}}_{m} ⟩$ as a function of time in a lossy case. (b2) Center of mass of the three photons as a function of time. A clear steplike behavior is observed in both plots.
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