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arXiv:1506.02773 · Physical Review A

Origin and implications of an A2 -like contribution in the quantization of circuit-QED systems

Moein Malekakhlagh, Hakan E. Türeci

Published Jan 25, 2016• 38 citations
arXivHTMLPDF

Abstract

By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the ${A}^{2}$ contribution. Although in atomic cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit-QED system consisting of a single superconducting transmon qubit coupled to a multimode superconducting microwave resonator. A complete derivation of the quantum Hamiltonian of an open circuit-QED system consisting of a transmon qubit coupled to a leaky transmission line cavity is presented. We introduce a complete set of modes that properly conserves the current in the entire structure and present a sum rule for the dipole transition matrix elements of a multilevel transmon qubit coupled to a multimode cavity. Finally, an effective multimode Rabi model is derived with coefficients that are given in terms of circuit parameters.

DOI
10.1103/physreva.93.012120

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By placing an atom into a cavity, the electromagnetic mode structure of the cavity is modified. In cavity QED, one manifestation of this phenomenon is the appearance of a gauge-dependent diamagnetic term, known as the A2{A}^{2}A2 contribution. Although in atomic cavity QED, the resulting modification in the eigenmodes is negligible, in recent superconducting circuit realizations, such corrections can be observable and may have qualitative implications. We revisit the canonical quantization procedure of a circuit-QED system consisting of a single superconducting transmon qubit coupled to a multimode superconducting microwave resonator. A complete derivation of the quantum Hamiltonian of an open circuit-QED system consisting of a transmon qubit coupled to a leaky transmission line cavity is presented. We introduce a complete set of modes that properly conserves the current in the entire structure and present a sum rule for the dipole transition matrix elements of a multilevel transmon qubit coupled to a multimode cavity. Finally, an effective multimode Rabi model is derived with coefficients that are given in terms of circuit parameters.
We present a Heisenberg-Langevin formalism to study the effective dynamics of a superconducting qubit coupled to an open multimode resonator, without resorting to the rotating wave, two level, Born or Markov approximations. Our effective equations are derived by eliminating resonator degrees of freedom while encoding their effect in the Green's function of the electromagnetic background. We account for the openness of the resonator exactly by employing a spectral representation for the Green's function in terms of a set of non-Hermitian modes. A well-behaved time domain perturbation theory is derived to systematically account for the nonlinearity of weakly nonlinear qubits like transmon. We apply this method to the problem of spontaneous emission, capturing accurately the non-Markovian features of the qubit dynamics, valid for any qubit-resonator coupling strength. Any discrete-level quantum system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When inside a cavity, these quantities can be strongly modified with respect to vacuum. Generally, this modification can be captured by including only the closest resonant cavity mode. In circuit-QED architecture, with substantial coupling strengths, it is however found that such rates are strongly influenced by far off-resonant modes. A multimode calculation over the infinite set of cavity modes leads to divergences unless an artificial cutoff is imposed. Previous studies have not pointed out what the source of this divergence is. We show that unless the effect of A2A^2A2 is accounted for up to all orders exactly, any multimode calculations of circuit-QED quantities is bound to diverge. Subsequently, we present the calculation of finite radiative corrections to qubit properties that is free of an artificially introduced high frequency cut-off.
Multimode cavity-QED systems can be leveraged to explore a wide range of physical phenomena; however, a complex multimode environment makes systematic characterization of light-matter interactions challenging. Here we present a general measurement protocol, applicable to both atomic and synthetic cavity-QED systems, that enables the determination of coupling to individual photonic modes. The method leverages measurements of the AC-Stark and Kerr effects, along with known detuning dependencies, to eliminate the need for single-photon resolution, independent photon-number calibration, or insertion-loss calibration. We demonstrate the method using a superconducting transmon qubit coupled to a one-dimensional microwave resonator lattice. We validate the consistency of the extracted light-matter couplings ggg determined at multiple qubit detunings, and from the self-Kerr and cross-Kerr shifts for three photon modes, which provide separate measurements of ggg for each of the three modes.
In cavity quantum electrodynamics (QED), the key physical systems include Fabry-Pérot cavity resonators and multilevel cold atoms. Since ultra-strong electromagnetic interaction between photons and quantized cavity modes occurs in cavity QED, many novel quantum optical effects such as optical Rabi oscillation, quantum collapse/revival of atomic-level population and atom-field quantum entanglement can be realized in such quantum microcavities. As is known in quantum optics, these schemes can be generalized to circuit QED, which is a circuit analog of cavity QED. Fundamental devices in circuit QED are superconducting coplanar waveguide microwave resonators and mesoscopic Cooper-pair boxes. Here, the coplanar waveguide resonator with photons stored can be identified as a resonant cavity and the Cooper-pair boxes with large electric dipoles play the role of multilevel cold atoms. Thus, all the quantum optical effects in conventional cavity QED can also be exhibited in the present superconducting circuit QED system. We consider the theoretical working mechanism of superconducting charge qubits based on tunable Josephson junction coupling. It can be found that an analog of optical Rabi oscillation in Cooper-pair boxes can be achieved by tuning an applied external magnetic field. Time evolution of energy levels in a Cooper-pair box, which is manipulated by a time-dependent magnetic flux through its Josephson-junction ring, is studied within the framework of time-dependent quantum mechanics.
Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number nnn scales as n\sqrt{n}n​ for nnn up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.
Waveguide circuit quantum electrodynamics (waveguide circuit QED) studies light-matter interaction with superconducting circuits in one dimension. In circuit QED, natural atoms are replaced by superconducting qubits consisting of a non-linear Josephson junction, resulting in an anharmonic energy spectrum just like real atoms. With superconducting qubits, it is possible to study quantum optical phenomena and reach new regimes hard to achieve with real atoms due to weak coupling to the electromagnetic field. The reduction to one dimension in waveguide QED increases the electromagnetic field's directionality, which results in reduced losses. In this thesis, we first introduce circuit quantisation, giving the basis for the next part, where we investigate a transmon, a charge-insensitive artificial atom, coupled to a semi-infinite transmission line. An atom coupled to a semi-infinite waveguide is referred to as an atom in front of a mirror and is the subject of all appended papers. We proceed by summarising Paper I and III's main results: in Paper I, we investigate the spontaneous emission of a transmon coupled to a semi-infinite transmission line, where we take time-delay effects into account. We find that the system dynamics strongly depend on the coupling strength to the transmission line and the atom's position with respect to the electromagnetic field, leading to the Purcell effect or the convergence to a dark state with finite excitation probability. In the high-impedance regime, which we investigate in Paper III, the properties of the transmon coupled to the high-impedance transmission line change drastically. It becomes highly reflective and creates its own cavity with the mirror, resulting in the emergence of cavity modes and vacuum Rabi oscillations in the spontaneous emission dynamics. In the next chapter of the thesis, we demonstrate how to quantise an electromagnetic field and derive a light-matter interaction Hamiltonian within dipole approximation. We then give an introduction to open quantum systems and derive the quantum-optical master equation in Lindblad form. Furthermore, we introduce the dressed state picture, where the interaction of light and matter is so strong that the individual energy levels of light and matter are no longer separable. Both the quantum optical master equation and the dressed state picture are relevant in Paper II and IV. In Paper II, an experimental collaboration, we perform several experiments to characterise and discriminate different decay rates of a superconducting qubit coupled to the end of a transmission line. One experiment measured the atomic fluorescence spectral density, which shows an asymmetry for off-resonant driving, resulting from pure dephasing: an effect that we explain in more detail in this thesis and Paper II. In Paper IV, we theoretically investigate amplification mechanisms realised by different set-ups of an atom coupled to a semi-infinite waveguide. In the considered systems, the amplification of a probe field happens either due to population inversion between the pure states or dressed states or multi-photon processes. We find that compared to an open waveguide, we can achieve a higher gain in the amplification with a semi-infinite waveguide.
The quantum Rabi model (QRM) is a cornerstone in the study of light-matter interactions within cavity and circuit quantum electrodynamics (QED). It effectively captures the dynamics of a two-level system coupled to a single-mode resonator, serving as a foundation for understanding quantum optical phenomena in a great variety of systems. However, this model may produce inaccurate results for large coupling strengths, even in systems with high anharmonicity. Moreover, issues of gauge invariance further undermine its reliability. In this work, we introduce a renormalized QRM that incorporates the effective influence of higher atomic energy levels, providing a significantly more accurate representation of the system while still maintaining a two-level description. To demonstrate the versatility of this approach, we present two different examples: an atom in a double-well potential and a superconducting artificial atom (fluxonium qubit). This procedure opens new possibilities for precisely engineering and understanding cavity and circuit QED systems, which are highly sought-after, especially for quantum information processing.