Quantum Phase-Slip Element

• Quantum Phase-Slip Element is a superconducting circuit component defined by coherent 2π tunneling of the superconducting phase, forming a dual counterpart to Josephson junctions.
• It is realized in quasi-one-dimensional nanowires where localized phase suppression creates nonlinear charge dynamics, enabling effects such as Coulomb blockade and current quantization.
• Applications span quantum metrology, digital logic, and qubit architectures, with experiments demonstrating interference phenomena and coherent control in phase-slip devices.

A quantum phase-slip element, often called a quantum phase-slip junction or phase-slip junction, is a superconducting circuit element in which the fundamental nonlinear process is the quantum tunneling of the superconducting phase by $2\pi$. It is realized most commonly by a quasi-one-dimensional superconducting nanowire, or more abstractly by an effective circuit component with phase-slip energy $E_{\mathrm{QPS}}$, critical voltage $V_c$, and a charge- or electric-flux-periodic constitutive law dual to that of a Josephson junction. In the microscopic picture the modulus of the superconducting order parameter briefly vanishes at a localized spacetime point while the phase slips by $2\pi$; in the circuit picture the same event acts as a nonlinear charge-domain process and can generate Coulomb blockade, Bloch-type dynamics, and current quantization (Arutyunov et al., 2016, Arutyunov et al., 2016, Wang, 2021).

1. Physical basis and microscopic mechanism

In a quasi-one-dimensional superconductor, the condensate is described by a complex order parameter of the form

$\Delta = |\Delta| e^{i\phi},$

or equivalently $\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$. A phase slip is a topological singularity in which the amplitude $|\Delta|$ or $|\Psi|$ is momentarily suppressed to zero and the phase changes by $\pm 2\pi$. In a current-carrying nanowire this changes the phase winding and produces a voltage pulse; repeated slips yield finite resistance or, in the coherent regime, a well-defined quantum tunneling amplitude between distinct fluxoid states (Arutyunov et al., 2016, Arutyunov et al., 2016, Lehtinen et al., 2012).

The distinction between thermally activated phase slips and quantum phase slips is central. Near $T_c$, thermally activated phase slips dominate. At $E_{\mathrm{QPS}}$0, thermal activation is suppressed, but quantum tunneling of the phase remains possible. This quantum regime is favored in ultra-narrow wires, in materials with low critical temperature and high normal-state resistivity, and in geometries with transverse dimensions comparable to relevant coherence scales. Titanium and strongly disordered NbN and NbSi are repeatedly used for this reason, while high-$E_{\mathrm{QPS}}$1 YBa$E_{\mathrm{QPS}}$2Cu$E_{\mathrm{QPS}}$3O$E_{\mathrm{QPS}}$4 nanowires demonstrate that phase-slip dynamics can persist with crossover temperatures in the 12–13 K range (Arutyunov et al., 2016, Constantino et al., 2018, Lyatti et al., 2019).

The phase-slip event is commonly interpreted as the dynamic counterpart of Cooper-pair tunneling through a static Josephson weak link. In a Josephson junction the weak link is fixed in space and Cooper pairs tunnel across it; in a phase-slip element the wire is structurally homogeneous, but the order parameter locally collapses and the phase tunnels. This establishes the standard charge–phase duality: Josephson elements are naturally phase- and magnetic-flux-centered, whereas phase-slip elements are charge- and electric-flux-centered (Arutyunov et al., 2016, Arutyunov et al., 2016).

2. Duality with the Josephson junction

The canonical Josephson-junction Hamiltonian used in the QPS literature is

$E_{\mathrm{QPS}}$5

whereas a short superconducting nanowire in the QPS regime is written as

$E_{\mathrm{QPS}}$6

The formal substitutions are

$E_{\mathrm{QPS}}$7

or, in equivalent notation, $E_{\mathrm{QPS}}$8 depending on convention. The identity of these Hamiltonians is the standard statement of the fundamental quantum duality between Josephson tunneling and quantum phase slips (Arutyunov et al., 2016, Arutyunov et al., 2016).

At branch level, this duality is expressed by replacing the Josephson current–flux relation with a voltage–charge relation. In the electromagnetic-flux-distribution model, an ideal Josephson junction is written as

$E_{\mathrm{QPS}}$9

while an ideal QPS junction is written as

$V_c$0

For a practical QPS junction, dual to the RCSJ model, the constitutive law becomes

$V_c$1

with $V_c$2 and $V_c$3 the series resistance and inductance, respectively (Wang, 2021).

The same duality underlies the definition of the critical voltage. In the coherent-QPS literature one writes

$V_c$4

dual to the Josephson critical current $V_c$5. Under microwave irradiation the Josephson relation $V_c$6 is replaced by the dual current quantization rule

$V_c$7

which is the phase-slip analogue of Shapiro steps (Shaikhaidarov et al., 2022).

A recurrent qualification is that the duality is exact at the level of the low-energy structure, but not always at the level of naive unconstrained circuit quantization. Compact formulations argue that treating a QPS junction as an ordinary nonlinear capacitor in an unconstrained Hilbert space leads to overcounting of states and inconsistencies in the treatment of environmental coupling and time-dependent flux bias (Koliofoti et al., 2022).

3. Circuit-theoretic descriptions

One line of development formulates Josephson and QPS circuits within a unified electromagnetic-flux-distribution model. In that framework, charge $V_c$8 is reinterpreted as electric flux stored in nodes, while magnetic flux $V_c$9 remains associated with loops. A non-capacitive branch acts as an electric-flux pump,

$2\pi$0

and a non-inductive branch acts as a magnetic-flux pump,

$2\pi$1

The node and loop conservation laws are then written as

$2\pi$2

Josephson circuits become magnetic-flux-distribution systems,

$2\pi$3

whereas QPS circuits become electric-flux-distribution systems,

$2\pi$4

Within this formulation the dual substitutions are explicit: current $2\pi$5 voltage, magnetic flux $2\pi$6 electric flux $2\pi$7, inductance $2\pi$8 capacitance, and critical current $2\pi$9 critical voltage (Wang, 2021).

A second line of work emphasizes compactification of the QPS Hilbert space. The standard Mooij–Nazarov Hamiltonian uses a continuous phase variable $\Delta = |\Delta| e^{i\phi},$0 and an integer phase-slip number $\Delta = |\Delta| e^{i\phi},$1,

$\Delta = |\Delta| e^{i\phi},$2

or equivalently

$\Delta = |\Delta| e^{i\phi},$3

The compact description restricts $\Delta = |\Delta| e^{i\phi},$4 to $\Delta = |\Delta| e^{i\phi},$5 and replaces the naive doubled description by a constrained wavefunction with boundary conditions linking $\Delta = |\Delta| e^{i\phi},$6 and the discrete slip index. The resulting compact Hamiltonian is

$\Delta = |\Delta| e^{i\phi},$7

This reformulation is used to unify Aharonov–Bohm and Aharonov–Casher effects, to specify admissible inductive couplings to an environment, and to clarify the computational Hilbert space of qubit architectures containing QPS elements (Koliofoti et al., 2022).

For dynamical simulation, the phenomenological branch equation with sine nonlinearity, series $\Delta = |\Delta| e^{i\phi},$8, series $\Delta = |\Delta| e^{i\phi},$9, and additive noise remains widely used. In that approximation both Josephson and QPS circuits reduce to coupled second-order differential equations whose nonlinearity is periodic in magnetic flux for Josephson elements and periodic in electric flux $\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$0 for QPS elements (Wang, 2021).

4. Device architectures and experimental realizations

Theoretical circuit archetypes were introduced by replacing tunnel junctions in Cooper-pair-box and Cooper-pair-transistor circuits with phase-slip wires. The resulting QPS-box and QPS-transistor were proposed specifically to demonstrate Coulomb blockade caused by coherent phase slips, while a tunable phase-slip flux qubit was proposed by replacing a single phase-slip junction with a charge-SQUID composed of two phase-slip junctions and a gated island (Hriscu et al., 2010, Zhao et al., 2013).

Representative experimental realizations span transport devices, interference devices, nanowire spectroscopic elements, and qubits:

Architecture	Observed signature	Source
Junctionless Cooper pair transistor in Ti wires	Coulomb blockade with gate modulation; blockade disappears above $\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$1 and under strong magnetic field	(Arutyunov et al., 2016)
QPS interference transistor	Gate-periodic Coulomb blockade voltage, dual to dc SQUID	(Hongisto et al., 2011)
NbN CQPS nanowires	Zero conductance below a critical voltage; critical voltages up to 5 mV	(Constantino et al., 2018)
YBCO phase-slip nanowires	Energy-level quantization; crossover temperature 12–13 K; excited-state lifetime exceeds 20 ms at 5.4 K	(Lyatti et al., 2019)
TiN phase-slip qubit	Qubit frequency $\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$2 GHz, lifetimes $\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$3, operation above 300 mK	(Purmessur et al., 10 Feb 2025)

In the junctionless Cooper pair transistor, two thin titanium wires in the QPS regime replace the two tunnel junctions of a conventional superconducting SET. The device exhibits Coulomb blockade whose gap is modulated by the gate potential, and the blockade disappears above the critical temperature and can be suppressed by strong magnetic field. Those observations were used to argue against an explanation based on unintended static tunnel barriers (Arutyunov et al., 2016).

The QPS interference transistor introduced in NbSi is a direct dual of the dc SQUID. It uses two narrow nanowire segments in series, a wider gated central segment, and a high-ohmic Cr environment. Its gate-modulated blockade voltage obeys

$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$4

which reduces in the symmetric case to

$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$5

This is the charge-domain counterpart of the dc-SQUID critical-current modulation law (Hongisto et al., 2011).

The YBCO nanowire experiments extend the notion of the phase-slip element into the high-$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$6 regime. Ultra-thin YBa$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$7Cu$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$8O$\Psi(\mathbf r)=|\Psi|e^{i\phi(\mathbf r)}$9 nanowires exhibit switching-current distributions consistent with quantized levels in a tilted washboard potential, a crossover temperature to the quantum regime of 12–13 K, and an excited-state lifetime exceeding 20 ms at 5.4 K. A plausible implication is that phase-slip elements need not be restricted to low-$|\Delta|$0 materials if the plasma-frequency and gap scales are favorable (Lyatti et al., 2019).

The 2025 TiN phase-slip qubit demonstrates a fully operational qubit based on a phase-slip junction. Its Hamiltonian is written in the fluxon basis,

$|\Delta|$1

and operation is reported at zero flux, with coherent control, readout, $|\Delta|$2, and temperatures exceeding 300 mK (Purmessur et al., 10 Feb 2025).

5. Coherent phenomena, interference, and applications

One of the most direct coherent-QPS signatures is the dual of the AC Josephson effect. In a superconducting nanowire placed in an inductive environment and irradiated with microwaves, current plateaus appear at

$|\Delta|$3

Direct sharp current steps were reported in NbN nanowires, clear up to 26 GHz with current values 8.3 nA, establishing the dual Shapiro effect and strengthening the case for QPS-based quantum current standards (Shaikhaidarov et al., 2022).

Quantum interference of phase slips is another defining phenomenon. In long Josephson-junction arrays acting as “slippery” wires, coherent phase slips at different locations interfere through the Aharonov–Casher effect, and slow fluctuations of offset charges broaden the corresponding energy-level shifts into linewidths of order 100 kHz. This provided spectroscopy-based evidence for coherent phase slips in a many-junction environment (Manucharyan et al., 2010).

The phase-slip process also supports higher-order tunneling events. Thin-wire superconducting loops have shown quantum paired phase slips, in which the phase changes by $|\Delta|$4 and the winding number changes by $|\Delta|$5. In a reported regime, paired phase slips were exponentially more probable than single ones, which is significant because paired slips preserve winding-number parity and have been discussed in connection with parity-protected qubits (Belkin et al., 2014).

In high-impedance superconducting waveguides, a localized phase-slip impurity can mediate strong inelastic photon scattering. A single microwave photon incident on such a boundary element can split into many lower-energy photons with near-unit probability, and the measured decay rates were quantitatively explained in a quantum-impurity/Luttinger-liquid framework without adjustable parameters in the relevant regime (Kuzmin et al., 2020).

Beyond metrology and spectroscopy, QPS elements have been proposed as building blocks for charge-based digital logic and neuromorphic circuits. In one logic family, an overdamped QPSJ is driven above $|\Delta|$6 to produce a quantized current pulse with

$|\Delta|$7

and OR/XOR gates were presented in SPICE simulations. In neuromorphic proposals, the same $|\Delta|$8 pulse serves as the analogue of a spiking event, and QPSJs combined with magnetic Josephson junctions were used to simulate neuron and synaptic circuits (Goteti et al., 2018, Cheng et al., 2018).

6. Limitations, controversies, and open questions

Several unresolved issues recur across the literature. The first is the environmental requirement. Strong Coulomb blockade and coherent charge-domain behavior are associated with high-impedance environments, whereas low-Ohmic probes favor dissipative transport and can suppress clear QPS-junction behavior. In ultra-thin Ti nanowires contacted by low-Ohmic leads, the zero-resistance state can disappear without forming a clean Coulomb gap, and the standard rare-event QPS formulas cease to fit the data once $|\Delta|$9 is no longer much smaller than $|\Psi|$0 (Arutyunov et al., 2016, Lehtinen et al., 2012).

A second issue is theoretical consistency. The widespread statement that a QPS element is simply a nonlinear capacitor dual to a Josephson junction is useful operationally, but compact formulations argue that it becomes misleading when used literally in unconstrained node-flux quantization. The compact viewpoint insists on a reduced Hilbert space, wavefunction boundary conditions, and restricted inductive couplings to the environment; this remains an active conceptual point because it affects time-dependent flux driving, Aharonov–Casher physics, and the actual dimension of the computational space in QPS-based qubits (Koliofoti et al., 2022).

A third issue is microscopic versus phenomenological modeling. Circuit-level descriptions based on

$|\Psi|$1

are tractable and useful, but they omit material-dependent nanowire parameters such as coherence length, wire length, and disorder profile. This suggests that phenomenological models are best interpreted as effective descriptions whose predictive power depends on how reproducibly $|\Psi|$2, $|\Psi|$3, and the environmental impedance can be engineered (Wang, 2021).

Experimentally, parameter control remains difficult. NbN nanowires fabricated with similar nominal dimensions can show bulk-like superconductivity, phase-slip centers, incoherent QPS, or coherent current blockade, and the prominence of QPS effects is attributed to the differing importance of quantum fluctuations and to wire-to-wire variability. The 2025 phase-slip qubit explicitly identifies control of junction parameters as a longstanding obstacle to incorporating phase-slip junctions into superconducting qubits (Constantino et al., 2018, Purmessur et al., 10 Feb 2025).

Two additional complications are repeatedly reported. First, gate-modulated QPS devices can show single-electron components superposed on the expected $|\Psi|$4-periodic response, a phenomenon described as quasiparticle poisoning and plausibly linked to nonequilibrium quasiparticles generated during QPS events. Second, ultra-thin Ti nanowires can display negative magnetoresistance whose origin remains unclear; proposed explanations include rogue magnetic moments, field-dependent charge-imbalance regions, and electrode effects, but no consensus mechanism is established in the cited work (Arutyunov et al., 2016, Lehtinen et al., 2012).

Taken together, these limitations do not negate the status of the quantum phase-slip element as a fundamental superconducting circuit component. They delimit the regimes in which the element is best viewed as a coherent dual of the Josephson junction, a dissipative fluctuational nanowire, or a compact topological degree of freedom whose correct treatment depends as much on its electromagnetic environment as on the nanowire itself.