Experimental observation of non-Markovian quantum exceptional points

Abstract

One of the most remarkable features that distinguish open systems from closed ones is the presence of exceptional points (EPs), where two or more eigenvectors of a non-Hermitian operator coalesce, accompanying the convergence of the correcponding eigenvalues. So far, EPs have been demonstrated on a number of platforms, ranging from classical optical systems to fully quantum-mechanical spin-boson models. In these demonstrations, the reservoir that induced the non-Hermiticity was treated as a Markovian one, without considering its memory effect. We here present the first experimental demonstration of non-Markovian quantum EPs, engineered by coupling a Josephson-junction-based qubit to a leaky electromagnetic resonator, which acts as a non-Markovian reservoir. We map out the spectrum of the extended Liouvillian superoperator by observing the quantum state evolution of the qubit and the pseudomode, in which the memory of the reservoir is encoded. We identify a two-fold second-order EP and a third-order EP in the Liouvillian spectrum, which cannot be realized with a Markovian reservoir. Our results pave the way for experimental exploration of exotic phenomena associated with non-Markovian quantum EPs.

Figures (4)

• Figure 1: (a) Spectral density $J(\omega )$ of the reservoir. The reservoir is a continuum of bosonic modes, having a Lorentzian shape with the spectral width, centered at the qubit frequency $\omega _{0}$. (b) Effective qubit-reservoir interaction model. The memory effect of the reservoir can be captured by a pseudomode (PM), which coherently swaps excitations with the qubit, and undergoes a continuous energy decay with a rate $\kappa$. (c) Real and imaginary parts of the spectrum of the extended Liouvillian superoperator. LEP2 and LEP3 are represented by red and blue stars, respectively. The LEP2 features simultaneous coalescence of the eigenvectors ${\bf V}_{1}$ with ${\bf V}_{3}$, and ${\bf V}_{2}$ with ${\bf V}_{4}$, while the LEP3 corresponds to coalescence of three eigenvectors ${\bf V}_{5}$, ${\bf V}_{6}$, and ${\bf V}_{7}$.
• Figure 2: (a) Sketch of the experimental system. The LEPs are realized with a circuit quantum electrodynamics architecture, where a bus resonator (R$_{b}$) connects five frequency-tunable Xmon qubits, one of which (Q) is used to test the non-Markovian dynamics. The readout resonator of Q (R) acts as a reservoir. (b) Pulse sequence. The experiment starts with the application of a $\pi /2$ pulse to Q, preparing it in the superposition state $(\left\vert l\right\rangle +i\left\vert u\right\rangle )/\sqrt{2}$. Then a parametric modulation with the frequency $\nu$ and amplitude $\varepsilon$ is applied to Q, coupling it to R at a sideband. The effective coupling strength is controlled by $\varepsilon$. After a preset interaction time, the Q-R coupling is switched off. The output state of Q and the PM is read out by subsequently performing the state mappings: Q$\rightarrow$R$_{b}\rightarrow$Q$_{a}$ and the PM$\rightarrow$Q.
• Figure 3: Evolutions of amplitudes of the extended Liouvillian eigenvectors measured for different Q-PM coupling. The top row is Re$(A_j)$ and bottom row is Im$(A_j)$ (for $j=0,1,..,7$). The amplitudes for each time are obtained by expressing the reconstructed Q-PM density matrix in terms of the Liouvillian eigenvectors. The initial state does not include the last eigenvector, whose amplitude ($A_8$) remains zero and is not shown.
• Figure 4: Reconstructed Liouvillian spectrum. The eigenvalues associated with the first eight eigenvectors of the extended Liouvillian superoperator are inferred by the exponential fitting of the time evolving amplitude $A_j(t) = A_j(0)e^{\lambda_jt}$.