Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering

We discuss a general method for constructing nonreciprocal, cavity-based photonic devices, based on matching a given coherent interaction with its corresponding dissipative counterpart; our method generalizes the basic structure used in the theory of cascaded quantum systems and can render an extremely wide class of interactions directional. In contrast to standard interference-based schemes, our approach allows directional behavior over a wide bandwidth. We show how it can be used to devise isolators and directional, quantum-limited amplifiers. We discuss in detail how this general method allows the construction of a directional, noise-free phase-sensitive amplifier that is not limited by any fundamental gain-bandwidth constraint. Our approach is particularly well suited to implementations using superconducting microwave circuits and optomechanical systems.

Popular Summary

Engineering photonic systems that artificially break time-reversal symmetry and reciprocity is an open area of research. The transmission amplitude of photons in one direction in such devices can differ in both magnitude and phase from the amplitude for transmission in the other direction. Breaking these symmetries is interesting on a fundamental level, as it can enable truly new quantum many-body phases of light. Additionally, nonreciprocal devices are crucial ingredients in optical networks both in the classical and quantum domain. Nonreciprocal devices are particularly important in quantum information processing with superconducting circuits since quantum bits must be protected from noise generated in the amplifiers used to detect them.

Here, we theoretically formulate a new principle that allows for the construction of nonreciprocal interactions in photonic devices without the use of traditional magneto-optic materials. Our general method is as simple as it is powerful and is based on the concept of “reservoir engineering” in which controlled dissipation is used to generate useful behavior. In our approach, we balance the “dissipative interaction” generated by controlled dissipation with a standard coherent interaction to generate a net nonreciprocal interaction. This idea can be used to make an extremely wide class of interactions (both linear and nonlinear) unidirectional. We demonstrate theoretically how this simple principle can be used to construct both isolators and amplifiers (both phase sensitive and phase preserving), using only simple interactions that are easily realized in a variety of systems. Crucially, these devices can be quantum limited: They add the absolute minimum of noise required by quantum mechanics. In contrast to alternate approaches, our method allows for the realization of directional behavior over relatively wide ranges of bandwidth.

Our ideas can be easily implemented using microwave-frequency superconducting circuits or optomechanical systems. Our findings may also have applications in cavity lattice structures, thereby opening the door to new kinds of photonic quantum many-body states.

Figure 1

(a) Basic recipe for generating directionality: Two cavities are directly coupled to one another via a coherent Hamiltonian ${\widehat{\mathcal{H}}}_{\text{coh}}$, and are also each coupled to the same (nondirectional) dissipative environment. (b) The dissipative environment in (a) mediates a reciprocal dissipative interaction between the two cavities. This interaction can be modeled using a Lindblad master equation and dissipative superoperator $\mathcal{L}[\widehat{z}]$ [cf. Eq. (2)]. By balancing the strength of coherent and bath-induced dissipative interactions between the cavities, one can break reciprocity. (c) Schematic cascaded quantum system, where one cavity drives another via a waveguide supporting only a right-propagating mode. The effective theory used to describe such systems corresponds to (b).

Figure 2

(a) Realization of the engineered reservoir via an auxiliary cavity mode that is damped at a rate ${\kappa }^{\prime }$. For strong damping ${\kappa }^{\prime }\gg \kappa$, this setup corresponds to a Markovian reservoir. (b) Implementation based on a transmission line, which supports propagation of photons in both directions. (c) Scattering matrix elements for the dissipative isolator setup at zero frequency, as a function of the coherent hopping $J$; the phase of $J$ is fixed so that $\arg (J/\mathrm{\Gamma })=\pi /2$. When $J$ is tuned as per Eq. (11), the system only allows directional transmission between cavities 1 and 2. We have fixed the dissipative coupling strength $\mathrm{\Gamma }$ to be equal to the cavity damping rate $\kappa$ and have taken the Markovian limit for the engineered reservoir (${\kappa }^{\prime }\gg \kappa$). (d) Scattering matrix elements as a function of frequency, when the directionality condition of Eq. (11) is fulfilled. In the Markovian limit, directionality holds over all frequencies.

Figure 3

$s$-matrix elements of the directional, phase-preserving amplifier, as a function of scaled frequency; the cooperativity $\mathcal{C}\equiv \mathrm{\Gamma }/\kappa =0.95$, where $\mathrm{\Gamma }$ is the dissipative interaction strength and $\kappa$ is the damping rate of cavities 1 and 2. (a) Auxiliary mode damping ${\kappa }^{\prime }=\kappa$, indicating a strong deviation from the Markovian limit; while perfect isolation exists at $\omega =0$, it is rapidly lost for nonzero frequencies. (b) Auxiliary mode damping ${\kappa }^{\prime }=100\kappa$, closer to the Markovian limit. The directionality is much better at finite frequencies, while the gain is unchanged.

Figure 4

Schematic illustrating the directional phase-sensitive amplifier. The coherent QND Hamiltonian of Eq. (25) causes the cavity-1 $P$ quadrature to drive the cavity-2 $P$ quadrature, and the cavity-2 $X$ quadrature to drive the cavity-1 $X$ quadrature (blue arrows); there is gain associated with each of these drivings, cf. Eq. (27). The engineered reservoir [jump operator described by Eq. (26)] also mediates the same drivings (green and magenta arrows), again with gain. By balancing these interactions, one can cancel the $X_2\to X_1$ driving, resulting in directional amplification.

Figure 5

Gain ${\mathcal{G}}_{\varphi }[\omega ]$ and reverse gain ${\overline{\mathcal{G}}}_{\varphi }[\omega ]$ of the directional phase-sensitive amplifier, plotted as a function of signal frequency $\omega$ and coherent coupling strength ${\lambda }_{\mathrm{QND}}$ [cf. Eq. (25)], assuming that the dissipative coupling $\mathrm{\Gamma }$ always satisfies the matching condition $\mathrm{\Gamma }={\lambda }_{\mathrm{QND}}$ [cf. Eq. (28)]. ${\mathcal{G}}_{\varphi }[\omega ]$ describes the amplification in transmission of signals incident on the cavity-1 $P$ quadrature, while ${\overline{\mathcal{G}}}_{\varphi }[\omega ]$ describes the amplification in transmission of signals incident on the cavity-2 $X$ quadrature. We have taken the engineered reservoir to be an auxiliary cavity mode with damping rate ${\kappa }^{\prime }/\kappa =100$ (see Sec. 3c). In this limit, deviations from the Markovian-reservoir approximation are small.

Figure 6

Properties of the directional phase-sensitive amplifier. (a) Sketch of couplings and drivings used to impedance match the amplifier. (b) Reverse gain ${\overline{\mathcal{G}}}_{\varphi }[\omega ]$ for $\omega$ set to half of the amplification bandwidth $\mathrm{\Delta }\omega$, for various choices of the auxiliary-mode damping rate ${\kappa }^{\prime }$. On resonance, we always have perfect directionality: ${\overline{\mathcal{G}}}_{\varphi }[0]=0$. (c) Amplification bandwidth $\mathrm{\Delta }\omega$ as a function of zero-frequency forward gain ${\mathcal{G}}_{\varphi }$ for various values of ${\kappa }^{\prime }$. The amplifier does not suffer from a standard gain-bandwidth constraint.