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arXiv:2307.06900 · Physical Review A

Solomon equations for qubit and two-level systems: Insights into non-Poissonian quantum jumps

Martin Spiecker, Andrei I. Pavlov, Alexander Shnirman, Ioan M. Pop

Published May 16, 2024• 9 citations
arXivHTMLPDF

Abstract

We measure and model the combined relaxation of a qubit coupled to a discrete two-level system (TLS) environment, also known as the central spin model. If the TLSs are much longer-lived than the qubit, nonexponential relaxation and non-Poissonian quantum jumps can be observed. In the limit of large numbers of TLSs, the relaxation is likely to follow a power law, which we confirm with measurements on a superconducting fluxonium qubit. Moreover, the observed relaxation and quantum jump statistics are described by the Solomon equations, for which we present a derivation starting from the general Lindblad equation for an arbitrary number of TLSs. We also show how to reproduce the non-Poissonian quantum jump statistics using a diffusive stochastic Schr\"odinger equation. The fact that the measured quantum jump statistics can be reproduced by the Solomon equations, which ignore the quantum measurement back action, hints at a quantum-to-classical transition.

DOI
10.1103/physreva.109.052218

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We measure and model the combined relaxation of a qubit coupled to a discrete two-level system (TLS) environment, also known as the central spin model. If the TLSs are much longer-lived than the qubit, nonexponential relaxation and non-Poissonian quantum jumps can be observed. In the limit of large numbers of TLSs, the relaxation is likely to follow a power law, which we confirm with measurements on a superconducting fluxonium qubit. Moreover, the observed relaxation and quantum jump statistics are described by the Solomon equations, for which we present a derivation starting from the general Lindblad equation for an arbitrary number of TLSs. We also show how to reproduce the non-Poissonian quantum jump statistics using a diffusive stochastic Schr"odinger equation. The fact that the measured quantum jump statistics can be reproduced by the Solomon equations, which ignore the quantum measurement back action, hints at a quantum-to-classical transition.
Achieving fault tolerance with superconducting quantum processors requires qubits to operate within the regime of threshold theorems based on the Born-Markov approximation. This approximation, which models dissipation as constant energy decay into a memoryless environment, breaks down when qubits couple to long-lived two-level systems (TLSs) that become polarized during operation and retain memory of past qubit states. Here, we show that non-Poissonian quantum jump traces carry the information required to distinguish long-lived TLSs from the standard Markovian bath. By fitting the Solomon equations to measured quantum jumps dynamics arising naturally due to thermal fluctuations, we can disentangle the coupling of the qubit to the two environments. Sweeping the qubit frequency reveals distinct peaks, each associated with a TLS that outlives the qubit, providing a handle to understand their microscopic origin.
Abstract Recent studies have shown that parasitic two-level systems (TLS) in superconducting qubits, which are a leading source of decoherence, can have relaxation times longer than the qubits themselves. However, the standard techniques used to characterize qubit relaxation is only valid for measuring T 1 under the Born-Markov approximation and could mask environmental memory effects in practice. Here, we introduce two-timescale relaxometry, a technique to probe the qubit and environment relaxation simultaneously and efficiently. We apply it to high-coherence fluxonium qubits over a frequency range of 0.1-0.4 GHz, and reveal a discrete spectrum of TLS with millisecond lifetimes. Our analysis of the spectrum is consistent with a random distribution of TLS in the aluminum oxide tunnel barrier of the Josephson junction chain of the fluxonium, with a spectral and volumetric density and average electric dipole similar to previous TLS studies at much higher frequencies. Our study suggests that investigating and mitigating TLS in the junction chain is crucial to the development of various types of noise-protected qubits in circuit QED.
As the energy relaxation time of superconducting qubits steadily improves, nonequilibrium quasiparticle excitations above the superconducting gap emerge as an increasingly relevant limit for qubit coherence. We measure fluctuations in the number of quasiparticle excitations by continuously monitoring the spontaneous quantum jumps between the states of a fluxonium qubit, in conditions where relaxation is dominated by quasiparticle loss. Resolution on the scale of a single quasiparticle is obtained by performing quantum nondemolition projective measurements within a time interval much shorter than T₁, using a quantum-limited amplifier (Josephson parametric converter). The quantum jump statistics switches between the expected Poisson distribution and a non-Poissonian one, indicating large relative fluctuations in the quasiparticle population, on time scales varying from seconds to hours. This dynamics can be modified controllably by injecting quasiparticles or by seeding quasiparticle-trapping vortices by cooling down in a magnetic field.
The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models.
Abstract Material defects fundamentally limit the coherence times of superconducting qubits, and manufacturing completely defect-free devices is not yet possible. Therefore, understanding the interactions between defects and a qubit in a real quantum processor design is essential. We build a model that incorporates the standard tunneling model, the electric field distributions in the qubit, and open quantum system dynamics, and draws from the current understanding of two-level system (TLS) theory. Specifically, we start with one million TLSs distributed on the surface of a qubit and pick the 200 systems that are most strongly coupled to the qubit. We then perform a full Lindbladian simulation that explicitly includes the coherent coupling between the qubit and the TLS bath to model the time dependent density matrix of resonant TLS defects and the qubit. We find that the 200 most strongly coupled TLSs can accurately describe the qubit energy relaxation time. This work confirms that resonant TLSs located in areas where the electric field is strong can significantly affect the qubit relaxation time, even if they are located far from the Josephson junction (JJ). Similarly, a strongly-coupled resonant TLS located in the JJ does not guarantee a reduced qubit relaxation time if a more strongly coupled TLS is far from the JJ. In addition to the coupling strengths between TLSs and the qubit, the model predicts that the geometry of the device and the TLS relaxation time play a significant role in qubit dynamics. Our work can provide guidance for future quantum processor designs with improved qubit coherence times.