Building a bigger Hilbert space for superconducting devices, one Bloch state at a time

Abstract

Superconducting circuits for quantum information processing are often described theoretically in terms of a discrete charge, or equivalently, a compact phase/flux, at each node in the circuit. Here we revisit the consequences of lifting this assumption for transmon and Cooper-pair box circuits, which are constituted from a Josephson junction and a capacitor, treating both the superconducting phase and charge as noncompact variables. The periodic Josephson potential gives rise to a Bloch band structure, characterized by the Bloch quasicharge. We analyze the possibility of creating superpositions of different quasicharge states by transiently shunting inductive elements across the circuit and suggest a choice of eigenstates in the lowest Bloch band of the spectrum that may support an inherently robust qubit encoding.

Charge quantization and detector resolution

“…Note in particular, when E S = 0, the array acts as a perfect superinductor, which seems to break charge quantization (along the lines of Refs. [12,16]). However, thanks to the ϕ-dependence, it is possible to render H low 2π-periodic even in this regime.…”

“…There remained a conundrum: charge quantization seemed broken even in the limit of large inductance, when the transport is dominated by the JJ [12]. Recently, a resolution was proposed by advocating continuous charge and noncompact phase irrespective of the presence or absence of a linear inductance [16]. At the same time, there emerged an opposite school of thought in several different contexts, where charge quantization is preserved.…”

“…To summarize, the existing literature appears ambiguous. We here aim to build a bridge between the theories of quantized [17,23] versus continuous charge [12,16,37] by developing and illustrating the following two statements (see also Fig. 1):…”

Observation of Josephson Harmonics in Tunnel Junctions

“…Note that the series inductance does not break the 2π periodicity of the Hamiltonian, despite the occurrence of ϕ 2 . This is in contrast to a shunt inductance, for which the flux ϕ and the charge n would need to be treated as non-compact operators with continuous spectrum R, see [85][86][87][88]. For a series inductance, though, due to the two symmetry properties of ϕ ∆ (ϕ), the two ϕ-dependent terms of Eq.…”

Compact description of quantum phase slip junctions

“…Charge and phase being canonically conjugate, the quantization unit of N fixes the periodicity (compactness) of φ. While phase compactness and possible consequences of it are themes that have been studied long time ago [12][13][14] , they have seen a revival in recent years in various contexts 15 , such as to understand charge noise sensitivity of quantum circuits [16][17][18][19][20] , quantum dissipative phase transitions [21][22][23][24][25][26][27][28] , the validity of the spin-boson paradigm 29 , geometric aspects of current measurements 30 or flux-driving 9,10 , as well as in topological phase transitions defined in the transport degrees of freedom [31][32][33][34][35][36][37][38][39][40][41][42] . In particular, while symmetries and constraints are equivalent for a closed quantum system 6 , the community seems to be still divided on that subject for open quantum systems [26][27][28] .…”