Phase-Dependent Chiral Transport and Effective Non-Hermitian Dynamics in a Bosonic Kitaev-Majorana Chain

We study a 1D chain of noninteracting bosonic cavities which are subject to nearest-neighbor parametric driving, thus realizing a bosonic Hamiltonian whose form is reminiscent of the celebrated Kitaev model of a 1D $p$-wave superconductor. For a suitable choice of drive phases, the model exhibits a number of remarkable properties. This includes phase-dependent chirality: Photons propagate and are amplified in a direction determined by the phase of the initial drive or excitation. It also exhibits a drastic sensitivity to boundary conditions: For a range of parameters, the boundaryless system has only delocalized, dynamically unstable modes, while a finite open chain is described by localized, dynamically stable modes. While our model is described by a Hermitian Hamiltonian, we show that it has a surprising connection to non-Hermitian asymmetric hopping models. In addition to being of fundamental interest as a new kind of topological bosonic system, our system also has potential practical utility as a quantum amplifier and a source of multimode entangled photons.

Popular Summary

One exciting route to building a quantum computer relies on exotic quantum states of matter that have built-in “topological” protection. The simplest model for realizing the necessary physics is the Kitaev chain model, which describes a kind of 1D superconducting state of electrons. But electrons represent just one of two fundamental particle classes—fermions—and so physicists would like to know if it is possible for the same physics to be realized also with bosons such as photons. Here, we show that by driving a photonic system in the correct manner, one can realize a bosonic analogue of the Kitaev chain.

The proposed system resembles the original Kitaev model, while exhibiting several properties that are distinctly bosonic. In particular, the propagation direction of a photonic wave packet is tied to its global phase. Moreover, propagation is associated with amplification. This lets our system serve as a powerful quantum amplifier and generator of entangled light.

Further, our system has properties that are topological, but which violate some of the most basic intuition one has for lattice systems. We find, for instance, that adding boundaries to our system does more than introduce localized edge states—it can cause all states to become localized and completely change the dynamical stability of the system.

Our work introduces a completely new kind of topological bosonic system. It will serve as a starting point for investigations of similar exotic phenomena in a range of related systems, and also serve as a new design principle for quantum amplifiers and sources of entangled photon.

Figure 1

(a) Schematic of the setup: An array of tunnel-coupled cavities (hopping $t$, on-site loss ${\kappa }_j$) is subject to resonant parametric driving (amplitude $\mathrm{\Delta }$) on each bond. We require a purely imaginary hopping matrix element, which is gauge equivalent to having staggered parametric drive phases. (b) When written in terms of local cavity quadratures ${\widehat{x}}_j$ and ${\widehat{p}}_j$, the model describes a spatially asymmetric pattern of couplings reminiscent of the asymmetric coupling of Majorana operators in the fermionic Kitaev-Majorana chain. The system exhibits phase-dependent chiral transport that is robust against arbitrary spatially varying loss.

Figure 2

Scattering properties of the bosonic Kitaev-Majorana chain. (a) Schematic of the setup. The leftmost, middle, and rightmost sites are attached to waveguides (coupling rates ${\kappa }_L$, ${\kappa }_M$, and ${\kappa }_R$, respectively). A signal with a frequency $\omega$ and global phase $\theta =0$ corresponding to an $x$ excitation is injected in the middle waveguide and is amplified (deamplified) as it propagates to the right (left). (b) Amplitude squared of the scattering matrix elements plotted as a function of the frequency of the input signal. As expected, signals propagating to the right (left) are amplified (deamplified). Note the reflection probability (black) is bounded by unity. (c) Same setup as in (a), except the phase of the signal is now $\theta =\pi /2$ corresponding to a $p$ excitation. (d) The signal is now amplified (deamplified) as it propagates to the left (right). For (b),(d), we take $N=13$ sites, $\mathrm{\Delta }=t/2$, uniform on-site internal loss rate $\kappa ={10}^{-2}t$, and waveguide couplings ${\kappa }_M=2{\kappa }_L=2{\kappa }_R=2t$.

Figure 3

Spectrum of the system with periodic boundary condition (PBC) $E_{k,\pm }^{\mathrm{PBC}}=t\sin (k)\pm i\mathrm{\Delta }\cos (k)$ versus open-boundary conditions (OBC) $E_k^{\mathrm{OBC}}=\sqrt{t^2-{\mathrm{\Delta }}^2}\cos (k)$ with $\mathrm{\Delta }=t/2$. The spectrum with PBC is complex for any nonzero $\mathrm{\Delta }$, indicating parametric instability. In contrast, the system with OBC is stable as long as $t>\mathrm{\Delta }$, regardless of system size.

Figure 4

(a) An $x$ excitation is amplified as it propagates to the right. After being reflected off the boundary, it remains an $x$ excitation as it propagates to the left while being deamplified. The scattering mechanism is the same if we consider reflecting off of on-site loss. (b) An $x$ excitation can turn into a $p$ excitation after scattering off an impurity (dark blue site). In combination with the chiral nature of amplification, multiple reflections between impurities can lead to instability.

Figure 5

Disorder-averaged transmission coefficient $⟨|s_{\mathrm{RM}}^x[\omega ]{|}^2{⟩}_{\mathrm{dis}}$ for the same setup as is Fig. 2 but with on-site disordered detuning perturbations (disorder strength $W={10}^{-3}t$). The shaded region corresponds to the variance, and the dashed orange line corresponds to the clean system. Although stability is no longer guaranteed after introducing the detuning perturbations, with the chosen parameters, instability occurs only in less than 0.01% of realizations. For smaller values of $r$ and/or $N$, one can tolerate even larger amounts of disorder. Parameters: $N=13$ sites, $\mathrm{\Delta }=t/2$, uniform on-site internal loss $\kappa ={10}^{-2}t$, waveguide coupling rates ${\kappa }_M=2{\kappa }_L=2{\kappa }_R=2t$.

Figure 6

Schematic showing an equivalent depiction of scattering off our lattice with $M$ input-output waveguides coupled to $M\le N$ arbitrary sites. Input states in each waveguide are first locally squeezed (squeeze parameters $R_j$). They then pass through a beam-splitter network and are then finally locally antisqueezed at the outputs. Crucially, the squeeze parameters $R_j$ and beam-splitter unitary matrix $K$ have a direct and simple relation to the system Hamiltonian. In particular, $K$ is the unitary of a simple tight-binding model [see Eqs. (e5) and (e6)].

Figure 7

A possible realization of the bosonic Kitaev-Majorana chain. On each bond, adjacent cavity modes ${\widehat{a}}_j$ and ${\widehat{a}}_{j+1}$ are tunnel coupled with hopping amplitude $t$. In addition, they are both coupled to a coherently driven auxiliary mode ${\widehat{b}}_j$ via a three-wave mixing interaction [Eq. (30)]. At the mean-field level, this interaction yields the parametric two-photon drive in our model. Staggering the drive phases results in a Hamiltonian that is gauge equivalent to a purely imaginary hopping phase, a key ingredient in our setup.

Figure 8

(a) Schematic of a realization of our model which uses purely intracavity nonlinearities. There are two lattice sites per unit cell, consisting of even and odd lattice sites which have different resonant frequency ${\omega }_E$ and ${\omega }_O$, respectively. Each site is tunnel coupled to four auxiliary resonators. (b) Single unit cell of our 1D lattice. One of the auxiliary cavities has a Kerr-type nonlinearity and is driven nonresonantly at some carefully chosen frequency ${\omega }_P$, which induces the nearest-neighbor parametric drive $\mathrm{\Delta }$ on the main lattice sites. Dynamically modulating the dielectric constant of the second auxiliary resonator $\epsilon (t)\propto \cos [({\tilde{\omega }}_E-{\tilde{\omega }}_O)t]$ gives rise to an effective nearest-neighbor hopping $t$ [35, 36, 37]. Here, ${\tilde{\omega }}_E$ and ${\tilde{\omega }}_O$ are detuned cavity frequencies; see Appendix pp7 for details.