"""KerrCavity: Kerr-nonlinear resonator model.
Hamiltonian:
.. math::
H = \\omega \\, \\hat{n} - K \\, \\hat{n}(\\hat{n} - I)
where :math:`\\omega` is the cavity frequency (GHz, ordinary) and :math:`K`
is the Kerr nonlinearity (GHz, positive). Eigenvalues are:
.. math::
E_n = \\omega n - K n(n-1)
The Kerr term shifts higher Fock levels down by :math:`K` per photon pair,
which gives the anharmonic energy ladder. Together with a two-photon parametric
drive, this ladder stabilises cat states.
Approximation
-------------
This is an effective single-mode model after adiabatic elimination of the SNAIL
or STS-SQUID that supplies the nonlinearity. The Kerr coefficient :math:`K`
captures the leading-order nonlinearity, and higher-order corrections are
ignored. The Hilbert space is truncated at ``levels`` Fock states. To prevent
truncation artefacts, choose ``levels >= 4 * (eps2 / K) + 10``.
References
----------
.. [1] Grimm et al., *Stabilization and operation of a Kerr-cat qubit*,
Nature 584, 205 (2020). arXiv:1907.12131.
.. [2] Hajr et al., *High-Coherence Kerr-Cat Qubit in 2D Architecture*,
PRX Quantum 5, 020347 (2024). arXiv:2404.16697.
"""
from __future__ import annotations
from typing import Any, ClassVar
from quchip.declarative.expr import PhysicsExpr
from quchip.declarative.ops import LocalOps
from quchip.declarative.parameters import UNBOUND, Scalar, parameter
from quchip.devices.fock import FockDevice
[docs]
class KerrCavity(FockDevice):
"""Kerr-nonlinear resonator supporting cat-qubit stabilisation.
Hamiltonian:
.. math::
H = \\omega \\, \\hat{n} - K \\, \\hat{n}(\\hat{n} - I)
The nonlinearity :math:`K` shifts the photon-number eigenenergies and makes
the cavity anharmonic. With a two-photon parametric drive at
:math:`2\\omega`, the steady state becomes a cat state with amplitude
:math:`\\alpha = \\sqrt{\\varepsilon_2 / K}`.
Parameters
----------
freq : float
Positive cavity frequency :math:`\\omega` in GHz. Can be a JAX tracer
for sweeps / gradients.
kerr : float
Non-negative Kerr nonlinearity :math:`K` in GHz. A positive value
shifts even-photon levels downward. Usually 1–100 MHz in
superconducting circuits.
levels : int
Fock-space truncation dimension. To prevent truncation artefacts,
choose at least ``4 * (eps2 / K) + 10``. Default 30.
label : str | None
Human-readable label. ``None`` gives an automatic label
``kerr_cavity_0``, ``kerr_cavity_1``, …
T1 : float or None, default None
Energy-relaxation time in ns. ``None`` disables T1 relaxation.
T2 : float or None, default None
Total 0-1 coherence time in ns. If both are set, ``T2 <= 2*T1``.
thermal_occupation : float or None, default None
Dimensionless mean bath occupation. ``None`` disables absorption.
Notes
-----
This Hamiltonian is diagonal in the Fock basis and does not itself define a
computational subspace. With a two-photon parametric drive, you can
engineer the steady state into a cat-code manifold spanned by the even and
odd cat states :math:`|C^+_\\alpha\\rangle` and
:math:`|C^-_\\alpha\\rangle`. Bit-flip errors in that manifold are
exponentially suppressed, :math:`\\sim e^{-2|\\alpha|^2}`, in the
stabilized regime. This class's inherited Pauli surface
(:attr:`computational` is ``False``) addresses the bare Fock ``|0>``,
``|1>`` subspace. See :meth:`physics_notes` for the caveat.
References
----------
.. [1] Grimm et al., Nature 584, 205 (2020). arXiv:1907.12131.
.. [2] Hajr et al., PRX Quantum 5, 020347 (2024). arXiv:2404.16697.
Examples
--------
>>> from quchip.devices.kerr_cavity import KerrCavity
>>> cav = KerrCavity(freq=5.0, kerr=1.0, levels=10, label="cav")
>>> cav.freq, cav.kerr, cav.levels
(5.0, 1.0, 10)
"""
_type_prefix: ClassVar[str] = "kerr_cavity"
_default_levels: ClassVar[int] = 30
tunable_param_names = ("freq", "kerr")
approximation = (
"Kerr-nonlinear cavity effective single-mode model; "
"SNAIL/STS-SQUID adiabatically eliminated."
)
# The inherited Pauli surface (sigma_x/y/z) addresses the bare Fock
# |0>, |1> subspace, not the cat-code manifold |C+_alpha>, |C-_alpha>.
# This class does not implement cat-basis Paulis.
computational = False
freq: Scalar = parameter(default=UNBOUND, positive=True, unit="GHz", symbol=r"\omega")
# Non-negative: a positive Kerr shifts even-photon levels downward.
kerr: Scalar = parameter(default=UNBOUND, nonnegative=True, unit="GHz", symbol="K")
[docs]
def local_hamiltonian(self, op: LocalOps, p: Any) -> PhysicsExpr:
"""Return :math:`H = \\omega \\hat{n} - K \\hat{n}(\\hat{n} - I)`.
The Kerr term :math:`\\hat{n}(\\hat{n}-I) = \\hat{n}^2 - \\hat{n}`
gives eigenvalue contributions :math:`-K n(n-1)` for the
:math:`n`-photon Fock state.
Returns
-------
PhysicsExpr
Declarative expression for the Hermitian operator
``H = omega*n - K*n*(n-1)`` (GHz), diagonal in the Fock basis.
Parameters
----------
op : LocalOps
Fock operator namespace.
p : ParameterNamespace
Symbolic ``freq`` and ``kerr`` values.
"""
n = op.n
return p.freq * n - p.kerr * (n @ (n - op.I))
[docs]
def physics_notes(self) -> list[str]:
"""Return declared Kerr-cavity approximation notes."""
notes = super().physics_notes()
notes.append("Kerr Hamiltonian: H = ω·n̂ − K·n̂(n̂−I)")
notes.append(
"computational=False: the inherited Pauli surface (sigma_x/y/z) addresses the "
"bare Fock |0>, |1> subspace, not the cat-code manifold |C+_alpha>, |C-_alpha>; "
"this class does not implement cat-basis Paulis."
)
return notes
