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arXiv:1811.05126 · New Journal of Physics

Propagation and ramification of a solitary pulse through an environmentally coupled qubit

Yibo Gao, Shijie Jin, Hou Ian

Published Sep 29, 2020• 3 citations
arXivHTMLPDF

Abstract

Abstract We compute the relaxations experienced by a superconducting qubit and the simultaneous variation induced on the shape of a microwave pulse during the propagation of the pulse through the qubit. The environmentally affected propagation and the dressed relaxations are accounted by a microscopic-master-Maxwell equation pair. It is shown that the qubit longitudinal relaxation vanishes when the pulse envelope adopts a solitonic shape of nπ area whereas its transverse relaxation vanishes when the pulse phase has a periodic variation that is orthogonal to the spectral density of the environment. The pulse would propagate absorption-free when its area matches 2nπ. Otherwise, the environmental feedback decelerates the velocity of the soliton envelope and induces an monotonic increase of phase in the microwave. A pulse of non-2nπ area thus ramifies into a transparent part that travels absorption-free at incident velocity and a slowing part that decays through space. The ramification explains the environmental origin of pulse splitting observed in self-induced transparency.

DOI
10.1088/1367-2630/abbca4

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Abstract We compute the relaxations experienced by a superconducting qubit and the simultaneous variation induced on the shape of a microwave pulse during the propagation of the pulse through the qubit. The environmentally affected propagation and the dressed relaxations are accounted by a microscopic-master-Maxwell equation pair. It is shown that the qubit longitudinal relaxation vanishes when the pulse envelope adopts a solitonic shape of nπ area whereas its transverse relaxation vanishes when the pulse phase has a periodic variation that is orthogonal to the spectral density of the environment. The pulse would propagate absorption-free when its area matches 2nπ. Otherwise, the environmental feedback decelerates the velocity of the soliton envelope and induces an monotonic increase of phase in the microwave. A pulse of non-2nπ area thus ramifies into a transparent part that travels absorption-free at incident velocity and a slowing part that decays through space. The ramification explains the environmental origin of pulse splitting observed in self-induced transparency.
Microwave pulses are used ubiquitously to control and measure qubits fabricated on superconducting circuits. Due to continual environmental coupling, the qubits undergo decoherence both when it is free and during its interaction with the microwave pulse. As quantum logic gates are executed through pulse-qubit interaction, we study theoretically the decoherence-induced effects during the interaction, especially the variations of the pulse, under a dissipative environment with linear spectral distribution. We find that a transmissible pulse of finite width adopts an asymmetric multi-hump shape, due to the imbalanced pumping and emitting rates of the qubit during inversion when the environment is present. The pulse shape reduces to a solitonic pulse at vanishing dissipation and a pulse train at strong dissipation. We give detailed analysis of the environmental origin from both the perspectives of envelope and phase of the propagating pulse.
The waveguide propagation of a microwave magnetic field pulse at electron paramagnetic resonance (EPR) with a free radical sample of electron spins distributed along the guide displays the phenomenon of self-induced transparency (SIT). Predictions of plane-wave optical SIT are modified by summation of the magnetic-moment-microwave-field coupling over the transverse waveguide mode for specific sample geometries. Mode modified pulse area theorems are presented. Observed output pulse areas and shapes are compared with computer predictions. After pulse propagation over several absorption lengths a spontaneous zero-area pulse propagation effect is observed for a short weak input pulse with spectral width comparable to or larger than the spin-resonance linewidth. Alternating 180\ifmmode^\circ\else\textdegree\fi{} phase shifts in the pulse carrier wave, accounting for zero area, signify magnetic dipole transitions which alternate between coherent absorption and emission to some extent, and therefore anomalous pulse propagation is observed. For pulses applied off-resonance, linear and nonlinear shifts of the average carrier frequency occur toward the center of the spin resonance.
We study decoherence effects in qubits coupled to environments that exhibit resonant frequencies in their spectral function. We model the coupling of the qubit to its environment via the Caldeira-Leggett formulation of quantum dissipation/coherence, and study the simplest example of decoherence effects in circuits with resonances such as a dc superconducting quantum interference device (SQUID) phase qubit in the presence of an isolation circuit. Using the dc SQUID phase qubit as an example, we obtain two quite general results. First, when the frequency of the qubit is at least two times larger than the resonance frequency of the environmental spectral density (the isolation circuit in the case of a dc SQUID phase qubit), we find that the decoherence time of the qubit is a few orders of magnitude larger than that of the typical Ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the spectral density. Second, we show that, when the qubit frequency is nearly the same as the resonant frequency of the environmental spectral density, an oscillatory non-Markovian decay emerges, as the qubit and its environment self-generate Rabi oscillations of characteristic time scales shorter than the decoherence time.
Formation of optical solitons in self-induced transparency (SIT) regime, where light pulses propagate virtually without loss in nominally strongly absorbing medium, is one of the most striking coherent transient phenomena in optics. Here we study experimentally and by numeric simulations how a square shape pulse gradually transforms into a smooth sech shape pulse of well-defined pulse area, depending on the parameters such as the pulse amplitude, duration, propagation distance etc. The SIT experiments for circularly polarized light are performed in the R1−3∕2) line of a 30 ppm ruby (α-Al2O3:Cr3+) at 1.7 K in a magnetic field of BIIc = 4.5 T, which corresponds to effective absorption coefficient, αL =14.5. In such a magnetic field and temperature range, a 30 ppm ruby is in the so-called superhyperfine limit resulting in a very long decoherence (phase memory) time, TM = 50 μs. We show, in good quantitative agreement with the simulations, how SIT soliton is formatted and how this results in extremely slow pulse peak propagation velocity of ∼300 m∕s, which is to date, the slowest pulse propagation ever observed in a SIT experiment. We also show that for accurate quantitative description of the observed SIT and soliton pulse shapes, the simulations need to account for variation of the incident pulse amplitude across the beam spatial profile. Potential implications of the SIT effect on classical- and quantum information storage will be discussed.
In this paper we derive the environmental spectral density for flux, phase and charge qubits when each of them is coupled to an environment with a resonance. From the spectral density we obtain the characteristic spontaneous emission (relaxation) lifetimes T1T_1T1​ for each of these qubits, and show that there is a substantial enhancement of T1T_1T1​ beyond the resonant frequency of the environment.