Resummation for Nonequilibrium Perturbation Theory and Application to Open Quantum Lattices

Lattice models of fermions, bosons, and spins have long served to elucidate the essential physics of quantum phase transitions in a variety of systems. Generalizing such models to incorporate driving and dissipation has opened new vistas to investigate nonequilibrium phenomena and dissipative phase transitions in interacting many-body systems. We present a framework for the treatment of such open quantum lattices based on a resummation scheme for the Lindblad perturbation series. Employing a convenient diagrammatic representation, we utilize this method to obtain relevant observables for the open Jaynes-Cummings lattice, a model of special interest for open-system quantum simulation. We demonstrate that the resummation framework allows us to reliably predict observables for both finite and infinite Jaynes-Cummings lattices with different lattice geometries. The resummation of the Lindblad perturbation series can thus serve as a valuable tool in validating open quantum simulators, such as circuit-QED lattices, currently being investigated experimentally.

Popular Summary

Photons are special particles: They are easily created by a light source, absorbed by matter, and may repel or attract each other when in contact with a suitable material. The idea of feeding photons into an engineered crystal-like structure in which they move between preferred positions and interact with one another has recently captured the attention of many physicists. Such photon lattices enable scientists to learn more about many-body physics in nonequilibrium. The first experiments with such systems are currently underway, but making theoretical predictions for their behavior poses a significant challenge. Here, we introduce a theoretical framework that allows us to make reliable predictions whenever a particular form of the interaction can be identified as “weak.”

We focus on open quantum lattices that can be driven by external fields and are affected by coupling to their environment. To validate experimental devices realizing such lattices, we formulate a perturbation theory and resummation scheme that allow us to investigate different lattice dimensionalities and sizes, including infinite ones. Since our perturbative formalism is directly built on the Lindblad master equation, it applies to a large class of open quantum systems. As a concrete example of interest, we study an array of resonators locally coupled to qubits and subject to photon decay and qubit relaxation. For such an open Jaynes-Cummings lattice, we show that accuracy of predictions is significantly improved by resummation of an infinite subset of perturbative corrections. We demonstrate that our findings are consistent with exact results obtained for small lattices.

We expect that open-system perturbation theory enhanced by resummation will be of use in many contexts, including the validation of experimental data from novel open-system quantum simulators and the study of dissipative phase transitions.

Figure 1

Open quantum lattices of different dimensionalities and geometries. The examples show Jaynes-Cummings lattices in which photons can hop between neighboring resonators (dark boxes) and experience an interaction mediated by the coupling to two-level systems (represented as spin systems). Lattice types of interest include (a) one-dimensional Jaynes-Cummings chains, (b) two-dimensional arrays such as the depicted square lattice, and (c) more artificial arrangements of theoretical interest, such as the global-coupling scenario where each site is connected to all other sites.

Figure 2

Constituents of the Jaynes-Cummings lattice: Two-level systems are driven coherently with strength $\epsilon$ and exchange excitations with local harmonic oscillators at a rate set by $g$. Together, a pair of two-level system and oscillator form one site of the Jaynes-Cummings system. Nearest-neighbor sites are coupled by propagation of oscillator excitations with rate $\sim \kappa$. Dissipation is included in the form of two-level relaxation (rate $\mathrm{\Gamma }$) and oscillator relaxation (rate $\gamma$). In the circuit-QED realization, two-level systems are implemented by superconducting qubits, and oscillators by photon modes of on-chip microwave resonators with typical frequencies in the range of a few GHz.

Figure 3

Diagrams for perturbative corrections. (a) The order-$j$ correction to the steady state [Eq. (15)] is depicted as a chain of $j$ lines, each representing one factor $\mathbb{U}$. The chain connects $j+1$ dots symbolizing ${\mathbb{L}}_0$ eigenstates. The leftmost state is the unperturbed steady state $|{\mathrm{\rho }}_j)=|{\mathsf{u}}_0^0)$, the rightmost one the unperturbed eigenstate $|{\mathsf{u}}_{{\mu }_j}^0)$, giving rise to one specific correction term. The full correction is obtained by summation over final and intermediate states ${\mu }_1,\dots ,{\mu }_j$. (b) The resummation superoperator $\mathbb{f}$ is the sum of all (reducible and irreducible) $\mathbb{S}$ superoperators. Each $\mathbb{S}$ diagram starts and ends with the same unperturbed eigenstate. (c) Resummation combines evaluation of $\mathbb{S}$ and $\mathbb{T}$ superoperators. Terms of rank $j$ are comprised of the fully off-diagonal chain specified by ${\mathbb{T}}_j$ and final application of the resummation superoperator.

Figure 4

Diagrams for perturbative treatment and resummation of a JC lattice. In each panel, the top part shows the general diagram labeled by constituents deviating from the steady-state configuration. The bottom parts are JC-lattice specific diagrams, with the upper branch denoting photon modes and the lower one qubit degrees of freedom. (a) Rank-1 corrections. There are two classes of terms with either a photon mode or a cluster of a photon mode $\mathbf{k}$ and a qubit at $\mathbf{r}$ deviating from the unperturbed steady-state configuration. Each interaction vertex $g$ must switch the photon mode configuration but may leave that of the qubit unchanged. Terminating symbols on the right signal application of the resummation superoperator $(\mathbb{1}-\mathbb{S}{)}^{-1}$. (b) Two examples of rank-2 corrections, which differ in the number of involved photon modes and qubits and in the number of constituents deviating from the steady-state configuration.

Figure 5

Evaluation of the $\mathbb{S}$ superoperator. $\mathbb{S}$ is needed for resummation, and it is composed of irreducible diagrams starting and terminating in the same state. The two diagrams show the leading-order contributions, ${\mathbb{S}}_2$.

Figure 6

Comparison between perturbative results and exact solution. (a) Within the Bloch-sphere picture, the steady-state expectation value $|⟨{\mathrm{\sigma }}^{-}⟩|$ is directly proportional to the distance of the Bloch vector (representing ${\mathrm{\rho }}_{\mathrm{s}}^{\text{qubit}}$) from the $z$ axis. Panel (b) shows $|⟨{\mathrm{\sigma }}^{-}⟩|$ as a function of detuning $\delta \mathrm{\Omega }$ between drive and bare qubit frequency, in units of the qubit relaxation rate $\mathrm{\Gamma }$. Qubits are held in resonance with the bare resonator frequency, $\mathrm{\Omega }=\omega$. Exact and perturbative results for chain sizes $N=2$ and 4 are in good agreement. (See text for explanation of deviations close to $\delta \mathrm{\Omega }=0$ in the $N=4$ case.) (c) Curves here depict the absolute deviations between exact and perturbative results for calculations with (solid lines) and without resummation (dashed lines). Perturbation theory is not sufficient to describe the region close to $\delta \mathrm{\Omega }=0$ for $N=4$ (see text). Outside this region, resummation consistently improves agreement with the exact solution. (Parameters: $g/\mathrm{\Gamma }=3$, $\epsilon /\mathrm{\Gamma }=20$, $\kappa /\mathrm{\Gamma }=10$, and $\gamma /\mathrm{\Gamma }=4$.)

Figure 7

Comparison between perturbative results with resummation (solid line) and without (dashed line) for the three-site chain. The result without resummation shows three regions with significant deviations from the exact solution (dots) for $\delta \mathrm{\Omega }/\mathrm{\Gamma }\approx -24$, $-14$ and 1. These deviations (spurious and wrongly shifted resonances) are cured by including corrections obtained by resummation. (All system parameters are identical to those in Fig. 6.)

Figure 8

Perturbative results and spectral functions for periodic Jaynes-Cummings chains and global-coupling model. (a) shows the qubit expectation value $|⟨{\mathrm{\sigma }}^{-}⟩|$ for periodic chains with different site numbers $N$. We observe three resonance dips (labeled A–C). A and B are located roughly at the driven photon mode and qubit frequencies, respectively. (b) Spectral functions resonator modes differ for small site numbers. Presence and position of resonances near the qubit frequency explain the $N$-dependence of strength and shift of the B resonance. (c,d) show analogous plots for infinite lattices with periodic chain or global-coupling geometry. Resonances A and B are visible, but resonance C is (nearly) absent. [All system parameters are identical to those in Fig. 6.]

Figure 9

Dependence of the anomalous resonance C on site number $N$ and drive strength $\epsilon$. (a) The anomalous dip becomes less prominent with increasing $N$ (even) and nearly vanishes for the infinite chain. (The same trend applies to odd site numbers.) (b) For the dimer case, $N=2$, the position of the anomalous resonance shifts monotonically with increasing drive strength $\epsilon$. The same trend is observed for longer chains. (Parameters are chosen the same as in Fig. 6.)

Figure 10

Effective four-level model explaining the anomalous resonance. The transitions $|g⟩\leftrightarrow |q⟩$ and $|r⟩\leftrightarrow |\psi ⟩$ strongly hybridize two pairs of states. The anomalous resonance results from transitions between the two hybridized doublets. Since the energy separation between hybridized doublets depends on drive strength $\epsilon$, so does the position of the anomalous resonance.