Quantum Nonreciprocal Interactions via Dissipative Gauge Symmetry

One-way nonreciprocal interactions between two quantum systems are typically described by a cascaded quantum master equation, and rely on an effective breaking of time-reversal symmetry (TRS) as well as the balancing of coherent and dissipative interactions. Here, we present a new approach for obtaining nonreciprocal quantum interactions that is completely distinct from cascaded quantum systems, and that does not in general require broken TRS. Our method relies on a local gauge symmetry present in any Markovian Lindblad master equation. This new kind of quantum nonreciprocity has many implications, including a new mechanism for performing dissipative steady-state unitary gate operations on a target quantum system. We also introduce a new, extremely general quantum-information-based metric for quantifying quantum nonreciprocity.

Popular Summary

Nonreciprocal systems, where two parties interact with each other in an intrinsically asymmetric manner, play a key role in classical and quantum information processing. As their dynamics explicitly violate Newton's third law, such systems are also of fundamental interest and provide a route to realizing exotic phases of matter. In the quantum regime, a long-standing paradigm for describing nonreciprocity is given by the cascaded quantum setup, where the directional interaction is directly mediated by a one-way waveguide. However, recent advances in quantum technologies and quantum hardware call for the investigation of nonreciprocity beyond this standard paradigm.

In this work, we present a systematic framework for designing a new form of nonreciprocal interactions, which are completely distinct from standard cascaded schemes. Our work utilizes intrinsic gauge symmetries of Markovian dissipative processes, which describe degrees of freedom in the system-environment interaction that keep the system-only dynamics invariant. We show how this novel type of directional dynamics leads to a previously unexplored mechanism for dissipatively performing quantum gates. We also discuss strategies for experimental implementation; surprisingly, we find that the new nonreciprocal dissipator can be realized without invoking any explicitly broken time-reversal symmetry. This stands in stark contrast to the case with cascaded systems. In addition, we propose a general metric for characterizing nonreciprocity beyond scattering-based systems.

By proposing a recipe for designing novel classes of nonreciprocal quantum interactions, our work paves the way for investigating new dissipative phases of quantum matter, and it also has implications for quantum information applications including quantum control and autonomous quantum error correction.

Figure 1

(a) Schematic for quantum nonreciprocal interactions via gauge symmetry. This approach uses correlated dissipation, but is completely distinct from a cascaded quantum system. (b) Dissipative steady-state realization of a tunable unitary gate making use of the nonreciprocal interaction in (a), in a system where subsystem $A$ is a cavity mode $a$ that controls the gate, and subsystem $B$ is the target qubit. One can selectively apply a unitary gate $({\hat{U}}_B{)}^{\ell }$ on $B$ by initializing $A$ in the corresponding Fock state $|\ell ⟩$, and letting the system relax to the steady state [see Eq. (3) and Sec. 3c].

Figure 2

(a) Schematic of a setup that realizes a nonreciprocal interaction from a cavity mode $a$ ($A$) to a qubit ($B$). Both subsystems are coupled to an auxiliary damped bosonic mode $c$ that plays the role of a reservoir; see Eq. (28). (b) The effective dissipator describing the nonreciprocal interaction in the Markovian reservoir limit ${\kappa }_c\gg J$.

Figure 3

(a) State-averaged infidelity of the cavity-controlled steady-state qubit gate, as a function of the non-Markovianity parameter ${\kappa }_c/J$. We consider different choices of initial cavity state: either the Fock state $|{\varphi }_i{⟩}_A=|1⟩$ (solid) or $|2⟩$ (dashed), and take ${\theta }_{\mathrm{eff}}=\pi /6$. These different initial cavity states result in different gate operations. Infidelity tends to zero in the Markovian limit. (b) Isolation functions for both qubit and cavity as a function of the non-Markovianity parameter ${\kappa }_c/J$. We take a fixed evolution time $t=\pi /{\mathrm{\Gamma }}_{\mathrm{eff}}$ and ${\theta }_{\mathrm{eff}}=\pi /6$. Blue curves denote the cavity isolation $I^{(A)}(t)$; cf. Eqs. (7) and (33). Red curves denote the conditional qubit isolation function $I_{\{|0⟩,|\ell ⟩\}}^{(B)}(t)$ ($\ell =1,2$) that is an upper bound for the qubit isolation $I^{(B)}(t)$; see Eq. (34). We see strongly nonreciprocal behavior [i.e., $I^{(A)}(t)\simeq 1$, $I^{(B)}(t)<1$] even away from the Markovian limit. As discussed in the main text, when calculating $I^{(A)}(t)$, we truncate the cavity Fock space to $n_{\max }$ photons for simplicity; the solid (dashed) curve is for $n_{\max }=1$ ($n_{\max }=2$).

Figure 4

(a) Schematic illustrating a two-dissipator gauge symmetry nonreciprocal interaction. Two cavity modes $a_1$,$a_2$ are each coupled to waveguides, which are then sent to a beam splitter that depends on the state of a qubit. The dissipation is described by the Liouvillian $\mathrm{\Gamma }(\mathcal{D}[{\hat{z}}_1]+\mathcal{D}[{\hat{z}}_2])$ [see Eqs. (40) and (42a)]. (b) Numerically computed cavity isolation $I_{\{|{+}_x⟩,|{-}_x⟩\}}^{(A)}(t)$ for dynamics generated by the two-dissipator sum (solid red line) versus a single dissipator $2\mathrm{\Gamma }\mathcal{D}[{\hat{z}}_1]$ (dashed blue curve). The $A$ isolation stays unity in the former case, as expected for any dynamics that is fully nonreciprocal by design [see the discussion on Eq. (40)]. In contrast, individual dissipators from the setup in (a), e.g., $\mathcal{D}[{\hat{z}}_1]$, are not unidirectional. For illustrative purposes, the numerics plotted here is restricted to the subspace with at most two photons in $A$, but our result remains valid for $a_1,a_2$ modes with infinite levels. (c) Entanglement (logarithmic negativity $E_{\mathcal{N}}$) generation by the dissipations, assuming a product initial state $|11{⟩}_A\otimes {|{+}_y⟩}_B$. The fully nonreciprocal interactions generated by $\mathrm{\Gamma }(\mathcal{D}[{\hat{z}}_1]+\mathcal{D}[{\hat{z}}_2])$ creates entanglement (solid red curve), signaling nontrivial influence from the cavity modes to the qubit.