Enhanced sensitivity via non-Hermitian topology

Abstract

Sensors are indispensable tools of modern life that are ubiquitously used in diverse settings ranging from smartphones and autonomous vehicles to the healthcare industry and space technology. By interfacing multiple sensors that collectively interact with the signal to be measured, one can go beyond the signal-to-noise ratios (SNR) attainable by the individual constituting elements. Such techniques have also been implemented in the quantum regime, where a linear increase in the SNR has been achieved via using entangled states. Along similar lines, coupled non-Hermitian systems have provided yet additional degrees of freedom to obtain better sensors via higher-order exceptional points. Quite recently, a new class of non-Hermitian systems, known as non-Hermitian topological sensors (NTOS) has been theoretically proposed. Remarkably, the synergistic interplay between non-Hermiticity and topology is expected to bestow such sensors with an enhanced sensitivity that grows exponentially with the size of the sensor network. Here, we experimentally demonstrate NTOS using a network of photonic time-multiplexed resonators in the synthetic dimension represented by optical pulses. By judiciously programming the delay lines in such a network, we realize the archetypal Hatano-Nelson model for our non-Hermitian topological sensing scheme. Our experimentally measured sensitivities for different lattice sizes confirm the characteristic exponential enhancement of NTOS. We show that this peculiar response arises due to the combined synergy between non-Hermiticity and topology, something that is absent in Hermitian topological lattices. Our demonstration of NTOS paves the way for realizing sensors with unprecedented sensitivities.

Fig. 1

Non-Hermitian topological sensors (NTOS). Schematic diagram of the NTOS demonstrated here based on the Hatano-Nelson model which features nonreciprocal couplings between the adjacent elements of the array. Depending on the boundary conditions, this lattice exhibits different eigenvalue spectra, as shown in the top part of the figure. This can be represented by the strength Γ of the coupling between the first and last resonators in the system. When Γ is equal to the other couplings in the array (the rightmost part of the scale), the structure follows periodic boundary conditions (PBC), where the eigenvalues form an ellipse around the origin in the complex plane. In this case, a nonzero winding number $\mathcal{W}$ can be defined. On the other hand, when Γ = 0, i.e., under open boundary conditions (OBC), all the eigenvalues reside on the real axis, with one eigenvalue exactly equal to zero E = 0 (for odd values of N). This eigenvalue tends to shift from its original value by ΔE which is proportional to the strength of the boundary coupling Γ, as long as the coupling is sufficiently small. This mechanism can be effectively harnessed for sensing any perturbation that modifies Γ

Fig. 2

Schematic of the network of time-multiplexed resonators used to demonstrate NTOS. Synthetic resonators are defined by femtosecond pulses emitted by a mode-locked laser with a repetition rate of T_R passing through an electro-optic modulator (EOM) before injection into the optical fiber-based cavity (yellow fibers). An Erbium-doped fiber amplifier (EDFA) is used in the main cavity to compensate for the losses and increase the number of measurement roundtrips. Two delay lines with smaller and larger lengths than the main cavity (corresponding to delays of −T_R and +T_R, respectively) are utilized to provide nonreciprocal couplings between the nearest-neighbor resonators, necessary to implement the non-Hermitian topological model of Eq. (1). In addition, a third delay line with a length that corresponds to an optical delay of +(N − 1)T_R associated with the perturbation $\mathrm{\Delta }\widehat{H}$ is also included. In practice, the strength of such a perturbation, i.e., Γ, can be modified by an absorptive gas inside a cell (bottom). In such a scenario, NTOS can be used to accurately measure the concentration of the target gas molecules

Fig. 3

Measurement procedure for the time-multiplexed NTOS. Experimental time trace showing the pulse patterns at the output of the time-multiplexed resonator network for N = 23. At the beginning (t < 10.5 μs) optical pulses representing the zero eigenstate ${\left.∣{\psi }_0\right)>}_{\mathrm{R}}$ of the unperturbed Hamiltonian in Eq. (1) (bottom green inset) are repeatedly injected into the closed cavity (power build-up regime). After this, the input path to the cavity is blocked while the delay lines are opened, allowing for the pulses to circulate inside the cavity and the delay lines. This results in a temporal decay of the input eigenstate for t > 10.5 μs. By measuring these pulses and projecting them onto the left eigenstate of the unperturbed Hamiltonian ${\left.∣{\psi }_0\right)>}_{\mathrm{L}}$, we experimentally estimate the shift in the zero eigenvalue ΔE associated with the Hatano-Nelson model resulting from the nonzero perturbation in the system. In addition to the zero eigenstate, we also inject two control pulses (shown in the red boxes in the top plot) into the cavity. We use the first control pulse to accurately measure the intrinsic cavity decay, while the second one is intended to characterize the ±1T_R delay lines (Methods)

Fig. 4

Experimental demonstration of NTOS. Experimentally measured shifts in the eigenvalue ΔE as the boundary coupling strength Γ is perturbed from zero value (OBC conditions), for different lattice sizes N = 7, 13, 17 and 23. As evident in the figure, as long as Γ is small enough, our NTOS responds linearly to the induced perturbations. However, as Γ passes a threshold which depends on the size of the non-Hermitian topological lattice N, the change in the eigenvalue is no longer linear. The transition to this nonlinear regime is marked for each case in the figure by vertical dashed lines. Theoretically expected values are shown as solid curves. Here, T_RT represents the round-trip time of the optical cavity

Fig. 5

Exponential enhancement in the sensitivity of the NTOS. Experimentally obtained sensitivities S of the NTOS for different lattice sizes N are shown as green circles on the left plot. The corresponding theoretically predicted values are also depicted as orange squares. The data shows an exponential enhancement in the sensitivity S as the NTOS lattice size grows (green dashed line). For comparison, we performed similar analysis for other types of lattices including a trivial lattice with uniform couplings as well as the Hermitian topological lattice represented by the SSH Hamiltonian (depicted on the right side of the figure). As shown in the plot, in sharp contrast to NTOS, such lattices tend to become less sensitive to their boundary conditions as the structure grows. The plot also displays the effective enhancement in the sensitivity resulting from the noise suppression via averaging N different measurements