Matrix-product-state approach for qubits-waveguide systems in real space

TL;DR

The paper develops a matrix-product-state framework to simulate several qubits coupled to a common one-dimensional waveguide modeled in real space. By using a quantum discrete transmission-line representation and Bogoliubov modes, it captures non-Markovian dynamics and counter-rotating terms, while mitigating the large local bosonic Hilbert space with single-site TDVP/DMRG updates via controlled-bond expansion. The authors demonstrate key phenomena including single-qubit decay, two-qubit correlations, and superradiant scaling as the number of qubits grows, and they analyze vacuum quality and bond-dimension requirements. The approach provides a practical tool for ultrastrong-coupling waveguide QED and can be extended to driven or inhomogeneous waveguides, enabling optimization of control pulses and design of multi-qubit devices.

Abstract

We present a matrix-product-state-based numerical approach for simulating systems composed of several qubits and a common one-dimensional waveguide. In the presented approach, the one-dimensional waveguide is modeled in real space. Thus, one can use the advantage of matrix-product states that are suited for simulating low-entangled one-dimensional systems. The price to pay is that the vacuum of the waveguide in this modeling becomes the Bogoliubov vacuum, and one has to consider a not-so-small local Hilbert space for bosonic degrees of freedom. To manage the large local Hilbert space, we adopt the recently proposed single-site schemes. We demonstrate the potential of the presented approach by simulating superradiant phenomena within the Hamiltonian dynamics.

Figures (8)

• Figure 1: Circuit diagram for the discrete transmission line with the boundary condition $\phi_0 = \phi_{N+1} = 0$.
• Figure 2: (a) Excitation energy dependence of the occupancy of each Bogoliubov mode for different highest occupation numbers $N_\mathrm{max}$. We set the number of discretized points $N$ to $200$. (b) Highest occupation number dependence of the occupancy of the energetically lowest Bogoliubov mode $\braket{\hat{\alpha}^\dagger_1 \hat{\alpha}_1}$. The line is a guide for the eye.
• Figure 3: Excitation energy dependence of the occupancy of each Bogoliubov mode for different numbers of discretized points $N$. We set the maximum occupation number $N_{\mathrm{max}}$ to 10.
• Figure 4: Time evolution of (a) the population and (b) the decay rate of the single-qubit coupled to the waveguide. The waveguide model is composed of 200 discretized points and the qubit is coupled to the 100th point. We use three highest occupation numbers 12, 15, and 20 in the simulations.
• Figure 5: Time evolution of the correlation $\braket{\hat{\sigma}^\dagger_1(t)\hat{\sigma}_2(t)}$ between two distant qubits. The vertical lines represent the duration required for propagating emitted photons between the qubits. The waveguide model is composed of 1000 discretized points. The first qubit is coupled to the 410 th point. The second qubit is connected to the 570th (590th) point for the propagating phase $\phi = 8 \pi$ ($9\pi$).