Quantum non-Hermitian topological sensors

We investigate in the framework of quantum noise theory how the striking boundary sensitivity recently discovered in the context of non-Hermitian (NH) topological phases may be harnessed to devise novel quantum sensors. Specifically, we study a quantum-optical setting of coupled modes arranged in an array with broken ring geometry that would realize a NH topological phase in the classical limit. Using methods from quantum-information theory of Gaussian states, we show that a small coupling induced between the ends of the broken ring may be detected with a precision that increases exponentially in the number of coupled modes, e.g., by heterodyne detection of two output modes. While this robust effect only relies on reaching a NH topological regime, we identify a resonance phenomenon without direct classical counterpart that provides an experimental knob for drastically enhancing the aforementioned exponential growth. Our findings pave the way towards designing quantum NH topological sensors that may observe with high precision any physical observable that couples to the boundary conditions of the device.

Figure 1

Illustration of the quantum non-Hermitian topological sensor (QUANTOS) setup. The device consists of an odd number of bosonic modes (viewed as sites) arranged in an open ring geometry. The observed quantity (measurand) affects the weak coupling $\mathrm{\Gamma }$ between the ends. Observed input-output ports with coupling rates $\kappa$ are indicated in orange. Dissipative quantum channels inducing a staggered pattern of gain (coupling ${\eta }^g$) and loss (${\eta }^l$) are exemplified in black (see magnified inset). Formulas describe the underlying system-bath coupling Hamiltonians [30, 31] of all channels.

Figure 2

(a) Scaling with system size $N$ of the Fisher information $\mathcal{I}$ corresponding to a heterodyne detection scheme for several values of $\mathrm{\Gamma }$ (see plot legends). The exponential law corresponding to Eq. (1) with $2\alpha =0.43011$ is shown as a guide to the eye. (b) Topological phase diagram (defined by integer invariant $\nu$) of the model (4) as a function of parameters $t_1>0$ and $t_2>0$. Other parameter regimes are readily inferred using the symmetries $\nu (-t_1,t_2)=-\nu (t_1,t_2)$ and $\nu (t_1,-t_2)=\nu (t_1,t_2)$. Parameter points used in the three other panels as well as in Fig. 3 are indicated. (c) $\mathcal{I}\left(N\right)$ for another parameter set in the topological region ($\nu =1$) confirms qualitative similarity to (a). (d) $\mathcal{I}\left(N\right)$ for a parameter set in the topologically trivial phase ($\nu =0$) does not exhibit the exponential increase with $N$ that is characteristic of the QUANTOS. Other model parameters are $\gamma =0.7,\omega =0$ in all plots.

Figure 3

(a) Dependence of the exponential growth rate $\alpha$ [cf. Eq. (1)] as a function of $t_1$ at $t_2=0.5$ for several values of frequency $\omega$ (relative to the free mode frequency ${\omega }_0$). A sharp resonance around $\gamma =t_1$ is visible at $\omega =0$ that is pinched off at a finite maximum value of $\alpha$ for finite frequency shifts. Inset: Corresponding exponential growth rate ${\alpha }_c$ of the energy-level shift known from the classical NH Hamiltonian theory. (b) Steepening of the exponential growth of $\mathcal{I}\left(N\right)$ when approaching the resonance at $\gamma =t_1=0.7$, $\omega =0$ by varying $t_1$. (c) Frequency dependence of $\mathcal{I}$ at various system sizes $N$ at $t_1=0.69$, $t_2=0.5$, exhibiting a striking $N$-dependent frequency-shifted resonance. Other parameters are $\gamma =0.7$, $\mathrm{\Gamma }={10}^{-11}$ in all plots.

Figure 4

Scaling with the system size $N$ of the Fisher information $\mathcal{I}$ for several values of $\mathrm{\Gamma }$ of the unbalanced QUANTOS model [cf. Eq. (E1)]. In comparison to Fig. 2 we can see that the exponential growth rate $\alpha$ is smaller and thus more sites $N$ are needed until the Fisher information $\mathcal{I}$ reaches saturation. The following parameters were chosen: $t_1=t_2=1$, $\gamma =0.7$, and $\epsilon =0.1$.