Resolving the topology of encircling multiple exceptional points

Abstract

Non-Hermiticity has emerged as a new paradigm for controlling coupled-mode systems in ways that cannot be achieved with conventional techniques. One aspect of this control that has received considerable attention recently is the encircling of exceptional points (EPs). To date, most work has focused on systems consisting of two modes that are tuned by two control parameters and have isolated EPs. While these systems exhibit exotic features related to EP encircling, it has been shown that richer behavior occurs in systems with more than two modes. Such systems can be tuned by more than two control parameters, and contain EPs that form a knot-like structure. Control loops that encircle this structure cause the system's eigenvalues to trace out non-commutative braids. Here we consider a hybrid scenario: a three-mode system with just two control parameters. We describe the relationship between control loops and their topology in the full and two-dimensional parameter space. We demonstrate this relationship experimentally using a three-mode mechanical system in which the control parameters are provided by optomechanical interaction with a high-finesse optical cavity.

Figures (6)

• Figure 1: Controlling the spectrum of a three-mode system.a The full space of control parameters. The EPs are shown in yellow, and form a trefoil knot $\mathcal{K}$. The coordinates $X,Y,Z$ are defined in Methods. The grey plane shows an example of a 2D subspace ($\mathcal{B}$). The yellow discs are the five intersections of $\mathcal{B}$ with $\mathcal{K}$. The red, green, and blue loops all lie in $\mathcal{B}$ and have a common basepoint (black circle). b The control plane $\mathcal{B}$, showing the five EPs it contains (yellow circles) and the three control loops. The coordinate along each loop is $\rho$. For each loop, the basepoint (black circle) corresponds to $\rho = 0$ and $\rho = 1$. c The eigenvalue spectrum $\boldsymbol{\lambda}$ calculated as a function of $\rho$ along each of the loops. A detailed description of these plots is in Methods.
• Figure 2: Experimental setup.a A Si$_3$N$_4$ membrane (red) is placed between the mirrors of a Fabry-Pérot cavity (white) in a cryostat (blue). Three of the membrane's modes (pink box) are tuned using three tones generated from the "control" laser via an AOM (cAOM). The membrane is driven by modulating the "probe" laser's intensity (via the amplitude modulation port (AM-in) of the source that drives a second AOM (pAOM)). The probe laser also provides a local oscillator (LO) which generates a signal $\tilde{V}$ that is proportional to the membrane's displacement and is monitored via a lock-in amplifier (LIA). b The detunings $\Delta$ of the three control tones (with respect to the cavity’s resonance). Dark blue: the magnitude of the cavity's optical susceptibility. The parameter $\eta = -2 \pi \times 100$ Hz is chosen to provide an optimal rotating frame Patil2022. c A measurement of the membrane’s mechanical susceptibility for $\Psi = (2\pi\times46\;\mathrm{kHz},109.4\uW,376.8\uW,77.0\uW)$. For each frequency range, the left panel shows $|\tilde{V}(\tilde{\omega}_{\mathrm{AM}})|$ and the right panel shows a parametric plot of $\tilde{V}$. Each data point is colored according to the value of $\tilde{\omega}_{\mathrm{AM}}$. The black lines are a global fit to all the data shown. This fit returns the system's eigenvalues $\boldsymbol{\lambda} = 2\pi\times\{49.670 - i\;{84.977},57.636 -i\;{29.834},112.222-i\;{26.325}\}\mathrm{Hz}$ in the rotating frame $\mathcal{R}$. The magnitude of each mode’s contribution (as determined from the fit) is shown as the orange, green, and light blue curves in the left-hand column.
• Figure 3: Equivalent eigenvalue braids from different loops.a The 2D control space $\mathcal{B}^{(1)}$. Color scale: $\mathrm{arg}(D)$. Yellow circles: vortices in $\mathrm{arg}(D)$, which correspond to EPs. Green and blue curves: control loops. Black circle: the loops' basepoint. A view of $\mathcal{B}^{(1)}$ in $\mathcal{L}_3$ is in the Methods. b Stereographic projection of the hypersurface $\mathcal{S}$. The axes $X,Y,Z$ are defined in Methods. Yellow curve: the measured EPs. The black circle, green curve, and yellow circles are as in (a). The control loops consist of straight segments in (a) but appear curved in (b) owing to the stereographic projection. Inset: a simplified cartoon of the relationship between the EPs and the control loop. c The same as in b, but for the blue control loop. d The eigenvalue spectrum $\boldsymbol{\lambda}$ as a function of position along the green control loop. $\rho$ indexes the measurements of $\boldsymbol{\lambda}$ along the loop. e As in (d), but for the blue control loop. Although the loops are not homotopic in $\bar{\mathcal{B}}^{(1)}$, they are homotopic in $\mathcal{G}_3$, and so produce isotopic braids.
• Figure 4: Different braids by encircling the same EPs.a The 2D control space $\mathcal{B}^{(2)}$. Color scale: $\mathrm{arg}(D)$. Yellow circles: vortices in $\mathrm{arg}(D)$ corresponding to EPs. Blue and red curves: control loops. Black circle: the loops' basepoint. A view of $\mathcal{B}^{(2)}$ in $\mathcal{L}_3$ is in Methods. b Stereographic projection of $\mathcal{S}$. Yellow curve: the measured EPs. The black circle, blue curve, and yellow circles are as in a. Inset: a simplified cartoon of the relationship between the EPs and the control loop. c The same as in (b), but for the red control loop. d The eigenvalue spectrum $\boldsymbol{\lambda}$ as a function of position along the blue control loop. $\rho$ indexes the measurements of $\boldsymbol{\lambda}$ along the loop. e As in (d), but for the red control loop. The braids produced by the two loops are not isotopic (and do not produce the same permutation) even though they encircle the same EPs in $\mathcal{B}^{(2)}$.
• Figure 5: Measuring the basepoint dependence of homotopy equivalence.a The 2D control space $\mathcal{B}^{(3)}$. Color scale: $\mathrm{arg}(D)$. Yellow circles: vortices in $\mathrm{arg}(D)$ corresponding to EPs. Blue and red curves: control loops. Black and white circles: two choices for the loops' basepoint. A view of $\mathcal{B}^{(3)}$ in $\mathcal{L}_3$ is in Methods. b Stereographic projection of $\mathcal{S}$. Yellow curve: the measured EPs. The black circle, white circle, red curve, and yellow circles are as in (a). Inset: a simplified cartoon of the relationship between the EPs, the control loop, and the basepoints. c As in (b), but for the blue control loop. d The eigenvalue spectrum $\boldsymbol{\lambda}$ as a function of position along the red control loop, using the white basepoint. $\rho$ indexes the measurements of $\boldsymbol{\lambda}$ along the loop. e As in (d), but for the blue control loop. f As in (d), but for the black basepoint. g As in (e), but for the black basepoint. Thus, when the two loops shown in (a) are based at the white point, they are homotopic and so produce isotopic braids; however, the same loops based at the black point are not homotopic and produce non-isotopic braids.