Topological Quantum Fluctuations and Traveling Wave Amplifiers

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Topological Quantum Fluctuations and Traveling Wave Amplifiers

Vittorio Peano, Martin Houde, Florian Marquardt, and Aashish A. Clerk

Phys. Rev. X 6, 041026 – Published 1 November 2016

Abstract

It is now well established that photonic systems can exhibit topological energy bands. Similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against backscattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a nonzero topological invariant guarantees the presence of a parametrically unstable chiral edge mode in a system with boundaries, even though there are no bulk-mode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited traveling wave parametric amplifier. The device is naturally protected against both internal losses and backscattering; the latter feature is in stark contrast to standard traveling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.

Received 29 April 2016

DOI:https://doi.org/10.1103/PhysRevX.6.041026

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Band gap Edge states Topological materials Topological phases of matter

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Vittorio Peano^1,4, Martin Houde², Florian Marquardt^1,3, and Aashish A. Clerk²

¹Institute for Theoretical Physics, University of Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
²Department of Physics, McGill University, 3600 rue University, Montreal, Quebec H3A 2T8, Canada
³Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
⁴Department of Physics, University of Malta, Msida MSD 2080, Malta

Popular Summary

An essential step in processing quantum information is the amplification of quantum signals up to a classically accessible level. While the gain required for this signal boost is large, preserving the fragile coherence of the quantum source requires a suppression of any noise that might follow the time-reversed path from the amplifier output port to its input, which implies a strong violation of time-reversal symmetry (i.e., strong nonreciprocity). Moreover, an ideal quantum amplifier should be as precise as allowed by the laws of quantum mechanics, a so-called quantum-limited amplifier. Here, we present a new route toward nonreciprocal, quantum-limited amplification using a topological amplifier.

Our proposal combines two basic ingredients: parametric pumping and the presence of a synthetic gauge field (i.e., a mechanism imprinting a geometrical phase on the photons traveling along a closed path). The latter induces a topological phase transition in which a unidirectional propagation channel appears along the boundary of the system, which is a small array of photonic nanocavities. With an appropriately tuned pump, all down-converted photon pairs are injected into the edge channel. The edge state accordingly becomes amplifying, while no amplification occurs in the bulk and no backscattering is introduced. We complete the amplifier design by adding three ports (input, output, and a sink) in the form of waveguides. Like any topological system, our amplifier is extremely resilient against disorder. Moreover, in stark contrast to other proposed topological devices, it is also resilient against loss.

Our work may revolutionize the engineering of quantum amplifiers. From a theoretical perspective, it could also stimulate the investigation of the topology of unstable Hamiltonians.

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Vol. 6, Iss. 4 — October - December 2016

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Figure 1
Setup figure. (a) Scheme of the basic interactions. (i) Photons hopping anticlockwise around a plaquette pick up a phase $Φ$ that can be interpreted as a synthetic gauge field flux. (ii) Pump photons with frequency $ω_{p}$ and quasimomentum $k_{p}$ are down-converted into a photon pair with frequency $ω_{p} / 2$ and quasimomentum $k_{p} / 2$ . (b),(c) Combining the two interactions in a finite geometry allows one to engineer a topologically protected quantum-limited amplifier. A signal injected into the device via a tapered fiber propagates unidirectionally along the edge. (b) With the appropriate choice of pump frequency $ω_{p}$ and quasimomentum $k_{p}$ , the signal is amplified while it travels along the upper edge. A second tapered fiber detects the amplified signal. (c) When the input and the output fiber are exchanged, the signal propagates along a different path where it decays due to the lack of phase matching. This leads to nonreciprocal amplification.
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Figure 2
Topological band structure. (a) Band structure of a semi-infinite strip for the Hofstadter model with a flux of $π / 2$ (when the parametric driving is switched off). The edge states are plotted in dark blue. The energy is counted off from half of the pump photon energy. The solid green circle indicates the tuning of the pump photon quasimomentum $k_{p}$ required to resonantly excite pairs of down-converted edge state photons with quasimomentum $k_{p} / 2$ (at the quasimomentum $k_{p} / 2$ the edge state should have energy $E = 0$ ). (b) Zoom of the band structure for a finite laser power. In the unstable energy interval highlighted in green, pairs of Bogoliubov excitations having quasimomenta $k_{p} / 2 \pm δ k$ are also excited. The corresponding amplification amplitude $λ$ is shown in (c). Parameters are $ω_{0} = 2.15 J$ , $Φ = π / 2$ , $k_{p} = 2.2$ . In (b) and (c), $ν = 0.08 J$ .
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Figure 3
Linear response of the topological amplifier. (a),(b) The topological amplifier is formed by a $30 \times 12$ array of photonic nanocavities. Three of the cavities at the edge of the sample are attached to waveguides, the input and output ports of the amplifier, and an additional sink port. (c) The amplifier has the geometry of a circulator. (a),(b) The red ellipses represent the linear response of the field inside the photonic array as a function of the incoming signal phase. The signal is injected at the port marked by an inward arrow. (a) A signal injected at the input port propagates unidirectionally toward the output port. The response is strongly phase sensitive; a signal with the right phase is amplified along the way. (d) Transmission power gain for the amplified quadrature as a function of the frequency of the input signal (counted off from half of the pump frequency) for a disordered (light thick line) and a clean sample (dark thin line). The reflection coefficient at the input is shown in (e) for the disordered sample. (b) A signal propagating from the output port toward the input port follows a different path and it is not amplified. Moreover, an appropriate matching of the impedances ensures that it leaks out at the sink port. The resulting (small) reverse transmission from the output to the input of the amplifier is the blue curve in (e). Parameters are $ω_{0} = 2.14 J$ , $Φ = π / 2$ , $ν = 0.08 J$ , $k_{p} = 2.2$ , $κ = 0.001 J$ , $κ_{in} = 2.6 J$ , $κ_{out} = 3 J$ , $κ_{sink} = 4.2 J$ . In the disordered simulations, the offset energies $δ ω_{j}$ , represented by the gray scale in (a) and (b), are random numbers in the interval $- 0.1 J < δ ω_{j} < 0.1 J$ .
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Figure 4
Quantum stationary state and noise properties of the topological amplifier. (a) Noise spectral densities for both the amplified (red) and squeezed (blue) quadratures of the field leaving the output waveguide as a function of frequency, cf. Eq. (c9); the noise is plotted in units of quanta (left-hand axis), and also in decibels relative to the vacuum noise level (right-hand axis). The field leaking out of the output waveguide [top left corner in (c)] is strongly squeezed. We plot results for both a disordered system (light thick lines) and a clean sample (dark thin lines). The noise in the amplified quadrature is only slightly larger than the quantum-limit value for a phase-preserving amplifier (i.e., the amplified vacuum noise from the input port). The added noise in the amplified quadrature [in units of quanta; cf. Eq. (c10))] for both a disordered and a clean sample is shown in (b). (c) Ellipses representing the Gaussian Wigner function of state of each site inside the cavity array [cf. Eq. (c11)]. In the bulk, the ellipses have a circular shape and their area is as small as allowed by the Heisenberg principle, representing a standard vacuum state. In contrast, the ellipses at the edge are anisotropic and have areas larger than the minimum required by the uncertainty principle, implying that one has a thermal squeezed state. This excess noise does not come from a finite temperature of the environment, but rather by the amplification of the zero-point fluctuations (quantum heating). Plotted as a color code is also the average number of photons on each site. The gray bars indicate the sites that are attached to coupling waveguides. Parameters are $40 \times 12$ sites, $ω_{0} = 2.14 J$ , $Φ = π / 2$ , $ν = 0.08 J$ , $k_{p} = 2.2$ , $κ = 0.001 J$ , $κ_{in} = 2.6 J$ , $κ_{out} = 3 J$ , $κ_{sink} = 4.2 J$ . For the disordered simulations the offset energies $δ ω_{j}$ are random numbers in the interval $- 0.1 J < δ ω_{j} < 0.1 J$ . For all plots, there is only vacuum noise entering from each waveguide.
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Figure 5
Bogoliubov–de Gennes band structures. (a) BdG band structure for a system without boundaries. Here and in (b) and (c), the particle (hole) bands are plotted in blue (gray). The stability of the bulk is protected by the complete band gaps separating neighboring particle and hole bands. (b),(c) BdG band structure of a finite strip of width $M = 40$ magnetic unit cells. The upper- (lower-)edge states are represented as thick solid (dashed) lines. (d) Corresponding amplification amplitudes $λ_{n}$ . The parameters are the same as for Fig. 2 ( $ω_{0} = 2.15 J$ , $Φ = π / 2$ , $k_{p} = 2.2$ , $ν = 0.08 J$ ).
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Figure 6
Gain profile obtained in the effective 1D model with impedance-matched (black dashed line) and impedance-mismatched (red solid line) waveguides. Small (almost invisible) ripples appear in the gain for the impedance-mismatched case. Parameters are total length 80 sites, length of the amplification region 30 sites, $ω_{0} = 2.14 J$ , $Φ = π / 2$ , $ν = 0.08 J$ , $κ = 0.001 J$ , impedance mistmatch $ε = - 0.1$ ).
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