Nonuniversality of quantum noise in optical amplifiers operating at exceptional points

The concept of exceptional points-based optical amplifiers (EPOAs) has been recently proposed as a new paradigm for miniaturizing optical amplifiers while simultaneously enhancing their gain-bandwidth product. While the operation of this new family of amplifiers in the classical domain provides a clear advantage, their performance in the quantum domain has not yet been evaluated. Particularly, it is not clear how the quantum noise introduced by vacuum fluctuations will affect their operation. Here, we investigate this problem by considering three archetypal EPOA structures that rely either on unidirectional coupling, parity-time symmetry, or particle-hole symmetry for implementing the exceptional point. By using the Heisenberg-Langevin formalism, we calculate the added quantum noise in each of these devices and compare it with that of a quantum-limited amplifier scheme that does not involve any exceptional points. Our analysis reveals several interesting results: most notably that while the quantum noise of certain EPOAs can be comparable to those associated with conventional amplifier systems, in general the noise does not follow a universal scaling as a function of the exceptional point but rather varies from one implementation to another.

Figure 1

Schematic of an optical amplification and detection system. The input signal $S_{\mathrm{in}}$ is boosted at the output $S_{\mathrm{out}}$ with the addition of extra noise (indicated by the blurring of the output arrow). The added noise value depends on the noise contribution from each independent open channel and their mutual interactions. To illustrate the main goal of this work, a hypothetical amplifier configuration with hypothetical open channels $f_{1-4}$ is depicted. Tuning the system to operate at an EP may introduce new open channels, say $f_{5,6}$, and/or feedback between some of these channels as illustrated by the dashed blue and red arrows for bidirectional and unidirectional feedback scenarios. However, the quantum noise of each individual channel is not affected since this is dictated by the Heisenberg uncertainty principle. These effects combined will consequently change the added noise at the output. Possible detection schemes include coherent or direct detection. In this work, we evaluate the added noise for some practical implementations of EPOAs for coherent detection, which is widely used in communication networks.

Figure 2

Optical amplifier made of a microring resonator (gain element) with evanescently coupled waveguides serving as input/output channels. In the ideal case of negligible back reflection, only one optical mode (here the CW wave) contributes to the amplification process. When ${\gamma }_W^{\prime }=0$, the output from port $P_1$ exhibits a minimum quantum noise of ${\overline{n}}_{\mathrm{add}}\approx \frac{1}{2}$. On the other hand, when ${\gamma }_W^{\prime }={\gamma }_W$, the output from port $P_2$ will feature a purely Lorentzian amplification but at the same time will have a larger quantum noise of ${\overline{n}}_{\mathrm{add}}\approx \frac{3}{2}$.

Figure 3

Optical amplifier configuration that implements a chiral EP via unidirectional coupling (for details, see [8]). The input/output channels as well as the noise sources are also depicted on the figure.

Figure 4

Series of two identical optical amplifiers with CW modes $a$ and $b$ coupled through a common waveguide. Both resonators have the same gain $g$ with identical coupling coefficients ${\gamma }_W$ and radiation loss ${\gamma }_R$.

Figure 5

PT-based optical amplifier. The two microrings are coupled to each other with a coupling coefficient $\kappa$ and also to identical waveguides with coupling coefficients ${\gamma }_W$ and ${\gamma }_W^{\prime }$. The top ring exhibits a gain $g_1$ while $-g_2$ represents a loss in the bottom ring. In our analysis, when ${\gamma }_W^{\prime }={\gamma }_W$, the output is collected from port $P_2$. On the other hand, when ${\gamma }_W^{\prime }=0$, $P_1$ is the output port.

Figure 6

Characteristics of PT-like optical amplifiers operated at an exceptional point. The dashed lines denote the corresponding results for a DP amplifier. (a) Added noise ${\overline{n}}_{\mathrm{add}}^{EP}$ and bandwidth $B_{\mathrm{EP}}$ for the PT-like architecture with two waveguides as a function of the factor $r=\sqrt{G_o}{\mathrm{\Gamma }}_1/{\gamma }_W$. The gray area denotes the regime where the PT-like architecture outperforms the DP amplifier in terms of reduced noise and enhanced bandwidth. Note that we only plot the interesting domain of $B_{\mathrm{EP}}/{\gamma }_W$ since this quantity vanishes at $r=0$. (b) Added noise ${\overline{n}}_{\mathrm{add}}^{EP}$ and bandwidth $B_{\mathrm{EP}}$ for the PT-like architecture with one waveguide as a function of the loss rate $g_2/{\mathrm{\Gamma }}_2$.

Figure 7

Cascaded optical amplifier configuration with an EP of order $M$. An input signal ${\widehat{S}}_{\mathrm{in}}$ from the leftmost waveguide enters amplifier ${\text{Amp}}_1$ that has quantum noise $\widehat{{\mathcal{N}}_o}$. The output of this amplifier is then fed into a second waveguide and then into a second amplifier that also has quantum noise $\widehat{{\mathcal{N}}_o}$, with this process then repeating for $M$ amplifiers. For large values of gain, the noise of ${\text{Amp}}_1$ will dominate the quantum noise in ${\widehat{S}}_{\mathrm{out}}$.