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arXiv:1605.06268 · Physical review. B./Physical review. B

Some implications of superconducting quantum interference to the application of master equations in engineering quantum technologies

S. N. A. Duffus, Kieran Niels Bjergstrom, V. M. Dwyer, John H. Samson, Timothy P. Spiller, Alexandre M. Zagoskin, William J. Munro, Kae Nemoto, M. J. Everitt

Published Aug 24, 2016• 5 citations
arXivHTMLPDF

Abstract

In this paper we consider the modeling and simulation of open quantum systems from a device engineering perspective. We derive master equations at different levels of approximation for a superconducting quantum interference device (SQUID) ring coupled to an ohmic bath. We demonstrate that the master equations we consider produce decoherences that are qualitatively and quantitatively dependent on both the level of approximation and the ring's external flux bias. We discuss the issues raised when seeking to obtain Lindbladian dissipation and show, in this case, that the external flux (which may be considered to be a control variable in some applications) is not confined to the Hamiltonian, as often assumed in quantum control, but also appears in the Lindblad terms.

DOI
10.1103/physrevb.94.064518

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In this paper we explore the quantum behaviour of a SQUID ring which has a very strong Josephson coupling energy. We show that that the eigenfunctions of the Hamiltonian for the ring can be used to create Schrodinger cat states. We also show that the ring potential may be utilised to squeeze coherent states. With the SQUID ring as a strong contender as a device for manipulating quantum information, such properties may be of great utility in the future. However, as with all candidate systems for quantum technologies, decoherence is a fundamental problem. In this paper we apply an open systems approach to model the effect of coupling a quantum mechanical SQUID ring to a thermal bath. We use this model to demonstrate the manner in which decoherence affects the quantum states of the ring.

•Jul 24, 2003•0 citations
In this paper we consider the modeling and simulation of open quantum systems from a device engineering perspective. We derive master equations at different levels of approximation for a superconducting quantum interference device (SQUID) ring coupled to an ohmic bath. We demonstrate that the master equations we consider produce decoherences that are qualitatively and quantitatively dependent on both the level of approximation and the ring's external flux bias. We discuss the issues raised when seeking to obtain Lindbladian dissipation and show, in this case, that the external flux (which may be considered to be a control variable in some applications) is not confined to the Hamiltonian, as often assumed in quantum control, but also appears in the Lindblad terms.
High fidelity models, which are able to both support accurate device characterization and correctly account for environmental effects, are crucial to the engineering of scalable quantum technologies. As it ensures positivity of the density matrix, one preferred model of open systems describes the dynamics with a master equation in Lindblad form. In practice, Linblad operators are rarely derived from first principles, and often a particular form of annihilator is assumed. This results in dynamical models that miss those additional terms which must generally be added for the master equation to assume the Lindblad form, together with the other concomitant terms that must be assimilated into an effective Hamiltonian to produce the correct free evolution. In first principles derivations, such additional terms are often canceled (or countered), frequently in a somewhat ad hoc manner, leading to a number of competing models. Whilst the implications of this paper are quite general, to illustrate the point we focus here on an example anharmonic system; specifically that of a superconducting quantum interference device (SQUID) coupled to an Ohmic bath. The resulting master equation implies that the environment has a significant impact on the system's energy; we discuss the prospect of keeping or canceling this impact and note that, for the SQUID, monitoring the magnetic susceptibility under control of the capacitive coupling strength and the externally applied flux results in experimentally measurable differences between a number of these models. In particular, one should be able to determine whether a squeezing term of the form $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}+\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{P}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ should be present in the effective Hamiltonian or not. If model generation is not performed correctly, device characterization will be prone to systemic errors.
In this paper we explore the quantum behavior of a superconducting quantum-interference device (SQUID) ring which has a significant Josephson coupling energy. We show that the eigenfunctions of the Hamiltonian for the ring can be used to create macroscopic quantum superposition states of the ring. We also show that the ring potential may be utilized to squeeze coherent states. With the SQUID ring as a strong contender as a device for manipulating quantum information, such properties may be of great utility in the future. However, as with all candidate systems for quantum technologies, decoherence is a fundamental problem. In this paper we apply an open systems approach to model the effect of coupling a quantum-mechanical SQUID ring to a thermal bath. We use this model to demonstrate the manner in which decoherence affects the quantum states of the ring.
We develop a ``chain-boson model'' master equation, within the Born-Markov approximation, for a few superconducting quantum interference devices (SQUIDs) coupled into a chain and exchanging their angular momenta with a low-temperature phonon bath. Our master equation has four generators; we concentrate on the damping and diffusion and use them to study the relaxation and decoherence of a Heisenberg SQUID chain whose spectrum exhibits critical-point energy-level crossings, entangled states, and pairs of resonant transitions. We note that at an energy-level crossing, the relevant bath wavelengths are so long that even well-spaced large SQUIDs can partially exhibit collective coupling to the bath, dramatically reducing certain relaxation and decoherence rates. Also, transitions into entangled states can occur even in the case of an independent coupling of each SQUID to the bath. Finally, the pairs of resonant transitions can cause decaying oscillations to emerge in a lower-energy subspace.
We present a multilevel quantum theory of decoherence for a general circuit realization of a superconducting qubit. Using electrical network graph theory, we derive a Hamiltonian for the circuit. The dissipative circuit elements (external impedances, shunt resistors) are described using the Caldeira-Leggett model. The master equation for the superconducting phases in the Born-Markov approximation is derived and brought into the Bloch-Redfield form in order to describe multilevel dissipative quantum dynamics of the circuit. The model takes into account leakage effects, i.e., transitions from the allowed qubit states to higher excited states of the system. As a special case, we truncate the Hilbert space and derive a two-level (Bloch) theory with characteristic relaxation (T1){(T}_{1})(T1​) and decoherence (T2){(T}_{2})(T2​) times. We apply our theory to the class of superconducting flux qubits; however, the formalism can be applied for both superconducting flux and charge qubits.
We consider the Lindblad-Kossakowski quantum master equation describing the dynamics of an open quantum systems in the form originally proposed by Davies and Spohn. This equation contains dissipative corrections, accounting for the interaction with the environment, whose expression strongly depends on the adopted Markov approximation. In the case where a control is present, the rigorous derivation of the Markov approximation in the standard case, the weak coupling limit, shows that the control appears not only in the coherent part of the equation but also in the dissipative correction. This complicates the analysis of the dynamics but also offers the opportunity of indirectly affecting the interaction with the environment through the control. In this paper we study this scenario for a finite dimensional quantum system interacting with a (Bosonic) bath of harmonic oscillators. We prove several control theoretic properties of this system and discuss how the control can be used to effectively shape the influence of the environment and obtain desired features of the dynamics.
In this paper we explore the quantum behaviour of a SQUID ring which has a very strong Josephson coupling energy. We show that that the eigenfunctions of the Hamiltonian for the ring can be used to create Schrodinger cat states. We also show that the ring potential may be utilised to squeeze coherent states. With the SQUID ring as a strong contender as a device for manipulating quantum information, such properties may be of great utility in the future. However, as with all candidate systems for quantum technologies, decoherence is a fundamental problem. In this paper we apply an open systems approach to model the effect of coupling a quantum mechanical SQUID ring to a thermal bath. We use this model to demonstrate the manner in which decoherence affects the quantum states of the ring.