Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving

Shruti Puri; Samuel Boutin; Alexandre Blais

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Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving

Shruti Puri, Samuel Boutin, Alexandre Blais

arXiv:1605.09408v3May 30, 2016quant-ph

TL;DR

The paper addresses cat-state preparation and manipulation in Kerr-nonlinear resonators, where photon loss and Kerr-induced distortion challenge quantum-information applications. It uses a two-photon drive to engineer the resonator eigenspace, prepare cats, cancel distortion, and implement logical gates. The resulting preparation is robust to single-photon loss, and the proposed protocols support fast, high-fidelity initialization and manipulation.

Problem
Photon loss and Kerr-induced deformation complicate cat-state preparation and manipulation, while existing approaches can require dissipation or expose the field to additional decoherence channels.
Method
The paper engineers a Kerr-nonlinear resonator using a two-photon drive, with drive control for cat preparation, distortion cancellation, and logical operations.
Results
The engineered eigenspace is robust against single-photon loss, supports fast high-fidelity cat-state initialization and manipulation, and enables a universal set of logical gates.
Takeaways & Limitations
A Josephson-parametric-amplifier-based setup offers a minimal hardware approach for preparing and manipulating microwave cat states for scalable quantum computation.
Takeaways & Limitations
The analysis neglects dephasing noise because its bit-flip rate decreases exponentially with the coherent-state amplitude.

Abstract

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Cat states of the microwave field stored in high-Q resonators show great promise for robust encoding and manipulation of quantum information. Here we propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. We show that this preparation is robust against single-photon loss. We moreover find that it is possible to remove undesirable phase evolution induced by a Kerr nonlinearity using a two-photon drive of appropriate amplitude and phase. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of the two-photon driven Kerr-nonlinear resonator.

INTRODUCTION

The paper proposes two-photon driving of a Kerr-nonlinear resonator to prepare and stabilize cat states, including under single-photon loss. It also develops faster preparation, Kerr-distortion cancellation, and universal logical gates in the engineered subspace.

Motivation and approach: The proposed preparation adiabatically converts |0⟩ or |1⟩ into the corresponding even or odd cat state without requiring dissipation.The approach instead relies on adiabatically turning on the two-photon drive.
Kerr-distortion correction: A two-photon drive with appropriate amplitude and phase cancels Kerr-induced cat-state distortion and associated dephasing during qcMAP.This addresses deterministic phase evolution and nondeterministic phase errors arising from the qubit-induced Kerr nonlinearity and photon loss.
Universal quantum logic gates: The engineered subspace supports a universal gate set for coherent-state logical encoding, with high-fidelity operations achievable using realistic parameters.The encoding maps |+α⟩ and |−α⟩ to logical states, relying on their quasi-orthogonality for large α.
Motivation and approach: Two-photon driving makes out-of-phase coherent states and their even-odd cat superpositions degenerate eigenstates of the Kerr-nonlinear resonator.For the Hamiltonian considered, the coherent-state amplitude satisfies α=(E_p/K)^1/2, and the even-odd cat states are also eigenstates.
Robustness to loss: Single-photon loss causes parity-changing jumps and decoherence but does not cause leakage from the degenerate cat-state subspace.In the regime κ/(8|Kα_0|^2)≪1, the coherent states remain degenerate eigenstates of the effective Hamiltonian.
Robustness to loss: Numerical steady-state fidelities reach 99.91% at κ/(8|Kα_0|^2)∼1/16 and decrease to 96.55% at κ/(8|Kα_0|^2)∼1/4.The larger loss-to-nonlinearity ratio visibly deforms the coherent states and reduces fidelity relative to the ideal steady state.

Adiabatic initialization of cat states:

The protocol adiabatically maps vacuum and single-photon states into even- and odd-parity cat states using a time-dependent two-photon drive. Transitionless driving accelerates preparation while retaining high fidelity under single-photon loss.

Adiabatically increasing the two-photon drive maps |0⟩ to an even cat state and |1⟩ to an odd cat state while preserving parity.
99.9% fidelity is obtained without photon loss, decreasing to 98.3% for K/κ = 250 at t = 6.5/K.
Transitionless driving uses an auxiliary counter-adiabatic Hamiltonian to follow instantaneous eigenstates during nonadiabatic parameter changes.
99.9% fidelity for κ = 0 and 99.5% for K/κ = 250 are achieved at t = 1.3τ with the auxiliary drive.
The orthogonal two-photon drive makes initialization approximately five times faster, achieving 99.9% fidelity for κ = 0 and 99.5% for κ = K/250.

Realization with superconducting circuits:

Superconducting Josephson parametric amplifiers can implement the two-photon-driven Kerr resonator, whose engineered subspace supports cat-state stabilization and universal logical gates. The scheme stabilizes against Kerr-induced evolution and achieves high simulated gate fidelities under photon loss.

Realization with superconducting circuits:: A flux-pumped SQUID terminating a λ/4 resonator supplies both Kerr nonlinearity and a two-photon drive in a Josephson parametric amplifier.
Realization with superconducting circuits:: With K/2π = 750 KHz, a cat state with α0 = 2 can be encoded in 63.6 ns using transitionless driving.
Stabilization of cat states against Kerr induced: The two-photon drive stabilizes cat states against Kerr-induced rotation, dephasing, amplitude damping, and loss of fidelity.
Stabilization of cat states against Kerr induced: A two-photon drive corrects distortions produced by the qubit-induced Kerr nonlinearity in the qcMAP protocol.
Universal quantum logic gates:: Logical Z rotations use a single-photon drive to lift the degeneracy between coherent-state logical states.
Universal quantum logic gates:: The simulated Z gate reaches 99.9% fidelity without loss and 99.5% for K/κ = 250, while the X gate reaches 99.7% and 98.6%, respectively.
Universal quantum logic gates:: Strong confinement makes X rotations difficult, requiring temporary drive removal or a detuning satisfying δx ≪ 2Ep.
Universal quantum logic gates:: A bilinear coupling between two driven resonators realizes an entangling σ̄z1σ̄z2 interaction with 99.99% fidelity without loss and 94% for K/κ = 250.

DISCUSSION

Two-photon driving engineers a Kerr-nonlinear resonator’s eigenspace into coherent-state pairs that support cat-state preparation and manipulation. The approach also generalizes to n-component cats through n-photon driving.

DISCUSSION: Two-photon driving engineers two out-of-phase coherent states as a Kerr-nonlinear resonator eigenspace robust against single-photon loss.The work presents this state engineering as a practical route for correcting Kerr-induced effects in applications such as the qcMAP gate.
DISCUSSION: The engineered resonator supports fast, high-fidelity cat-state initialization and manipulation using a minimal Josephson-parametric-amplifier setup.The discussion frames this as a hardware-efficient platform for quantum computation and quantum algorithms based on multi-component cats.
DISCUSSION: The Hamiltonian H = −K a†^n a^n + E_p(a†^n + a^n) has n coherent states as degenerate eigenstates, enabling initialization of n-component cat states.A Josephson parametric amplifier can supply the required nonlinearity and n-photon drive through flux modulation at n times the resonator frequency.
DISCUSSION: The effective-Hamiltonian analysis uses displacement by α_0 and chooses α_0 to cancel linear terms, leaving a residual parametric-drive contribution.The coherent-state eigenvalue condition follows from selecting α_0 consistently with the drive amplitude and Kerr nonlinearity.
DISCUSSION: For κ ≪ 8K|α_0|^2, single-photon loss preserves |±α_0⟩ as degenerate effective-Hamiltonian eigenstates despite inducing squeezing around them.The associated steady state is the equal mixture of the two coherent states and is unique under the stated quantum-jump dynamics.

Cat state decoherence under single-photon loss:

Single-photon loss causes transitions between even- and odd-parity cats, reducing Wigner-fringe contrast during cat-state evolution. For the stated initialization protocol, the estimated phase error corresponds to 98.4% fidelity.

Cat state decoherence under single-photon loss:: Single-photon loss causes decoherence of cat-state superpositions by inducing transitions between even- and odd-parity cat states.These transitions reduce the contrast of the Wigner-function fringes.
Cat state decoherence under single-photon loss:: γ = 2κ|α_0|^2 gives the phase-decay rate for the cat-state coherence under single-photon loss.The rate depends on the loss rate κ and the squared coherent-state amplitude.
Cat state decoherence under single-photon loss:: 98.4% fidelity results from the estimated initialization phase error of 0.016 for the specified time-dependent two-photon drive.This estimate agrees closely with the numerically estimated fidelity of 98.3%.

Additional Hamiltonian for faster than adiabatic

A time-dependent auxiliary two-photon drive is used to reconcile short- and long-time behavior during cat-state preparation. The supplied passages describe the operator-matching rationale but do not state a quantitative speed or fidelity outcome.

Additional Hamiltonian for faster than adiabatic: At short times, the transitionless-driving operator is approximated by a difference of two-photon creation and annihilation terms.This reflects the near-vacuum regime where the coherent amplitude α_0(t) is small.
Additional Hamiltonian for faster than adiabatic: At long times, a single-photon jump connects even- and odd-parity cats, motivating an auxiliary drive scaled by the evolving coherent amplitude.The restricted coherent-state basis yields h ∼ (a†2 − a^2)/(2α_0(t)) when α_0(t) ≫ 1.
Additional Hamiltonian for faster than adiabatic: The chosen auxiliary two-photon drive interpolates between the short- and long-time limits to obtain the stated transitionless-driving Hamiltonian.The supplied passages identify the interpolation form but do not provide its complete readable expression.

I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR

Two-photon driving stabilizes coherent states and adiabatically maps vacuum and single-photon states to cat states. The protocol remains effective under single-photon loss, while faster initialization requires attention to the minimum energy gap and non-adiabatic errors.

I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: The unique steady state can take arbitrarily long to reach as |α_0| increases because the coherent-state overlap approaches zero.For large α_0, a superposition evolves into an incoherent mixture only after t ≫ 1/κ.
I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: With κ ≪ 8K|α_0|^2, an initially coherent state remains in |±α_0⟩ under two-photon driving despite single-photon loss.The supplementary Wigner-function simulation contrasts decay without the drive with state preservation when the drive satisfies the eigenstate condition.
I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: Resonator dephasing and two-photon loss are omitted from the main paper because they are typically negligible compared with single-photon loss.Their effects are discussed separately in the supplementary analysis.
I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: For κ_φ ≪ 4K|α_0|^2, dephasing still leaves |±α_0⟩ as degenerate eigenstates, while dephasing jumps flip between them at rate κ_φ|α_0|^2e^−2|α_0|^2.Two-photon loss does not itself cause phase flips or spin flips of the coherent or cat states, and its additional dephasing is negligible when κ_2ph ≪ K.
I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: For K < 0, the initial Fock states transform into cat-state eigenstates as the two-photon-drive amplitude increases, though they need not remain ground states.The adiabatic condition is Δ_minτ ≫ 1; introducing detuning increases the minimum gap from 2K to 4.3K in the stated protocol.
I. STABILIZATION OF COHERENT STATES IN A TWO-PHOTON DRIVEN KNR: 99.3% fidelity is obtained for adiabatic cat-state initialization with κ = K/250, compared with 99.9% for κ = 0.Reducing the evolution time to τ = 2K lowers the fidelities to 84.9% and 85.6%, respectively, because of non-adiabatic errors.

IV. PULSE OPTIMIZATION WITH GRAPE

GRAPE pulse optimization designs time-dependent two-photon drives for fast cat-state initialization under realistic amplitude and timing constraints.

Pulse design: GRAPE designs orthogonal two-photon-drive components Ep,x(t) and Ep,y(t) for non-adiabatic cat-state initialization.The time-dependent Hamiltonian uses Ep,x(t) for the real component and Ep,y(t) for the orthogonal component.
Pulse constraints: The optimized protocol restricts Ep,y to zero at the beginning and end, sets Ep,x(T) to 4K, and keeps both amplitudes below 6K.These endpoint conditions stabilize the target even cat state while limiting the drive to realistic amplitudes.
Performance: 99.95% fidelity is achieved for the target cat state at T = 0.3/K with modulation time steps of at least 1 ns.The pulse is optimized for the cat state C+ and respects a realistic drive-modulation timescale.
JPA implementation: The JPA implementation uses flux modulation at twice the resonator frequency, with a slowly varying envelope for cat-state initialization.The envelope is chosen to change the two-photon drive amplitude from zero to its maximum.
Performance: The optimized initialization completes in the short evolution time T = 0.3/K.The supplementary figure presents the corresponding optimized pulse shapes.

Fourth-order expansion to map to Cassinian oscillator Hamiltonian

The fourth-order expansion of the JPA cosine potential and rotating-wave approximation produce a Kerr oscillator with a tunable two-photon drive, while higher-order effects constrain large-cat preparation.

Hamiltonian mapping: Expanding the JPA cosine potential to fourth order and applying the rotating-wave approximation yields a time-dependent two-photon-driven Kerr Hamiltonian.The resulting Hamiltonian contains the resonator term, Kerr nonlinearity, and oscillating two-photon-drive terms.
Drive control: The two-photon drive strength is controlled by the flux-modulation amplitude, which cannot be increased arbitrarily without substantially shifting the resonator frequency.Even-parity cat initialization starts from vacuum and slowly increases the modulation amplitude.
Circuit constraints: The drive-to-Kerr ratio scales as Ep/K ∝ (EJ/EC)1/4, so larger cats require larger EJ/EC.Increasing this ratio also makes higher-order terms more important and can reduce cat-state fidelity.
Full-Hamiltonian simulation: For experimentally realistic parameters, a cat with |α|2 = 4.8 is obtained in 26.67 µs with 99.4% fidelity under the full Hamiltonian.The simulation uses EC/2π = 1.5 MHz, EJ/2π = 600 GHz, resonator frequency 3.75 GHz, K/2π = 750 KHz, and Ep/2π = 3.7 MHz.
Single-photon drive: A single-photon drive slightly displaces the coherent components and lifts their degeneracy by δz = 4Re[Ezα0].The two stable eigenstates are |α0 + ϵ⟩ and |−α0 + ϵ⟩ for small Ez.

VII. EFFECT OF DETUNING

Detuning the two-photon drive modifies the coherent-state amplitude while preserving the encoded eigenstates when the detuning remains sufficiently small.

Hamiltonian: The detuned Hamiltonian includes a linear detuning contribution δa†a in addition to the Kerr and two-photon-drive terms.The displaced-frame form is obtained after removing the constant energy shift and cancelling the linear displacement terms.
Eigenstates: For |δ| ≪ 2Ep, the system eigenstates remain the coherent states |±α0⟩.The detuning changes the coherent amplitude through α0 = (2Ep + δ)/2K.
Encoded dynamics: Because coherent states are non-orthogonal, detuning produces nonzero off-diagonal matrix elements of a†a between |α0⟩ and |−α0⟩.The matrix element is ⟨α0|a†a|−α0⟩ = −|α0|2e−2|α0|2.

VIII. EVOLUTION DURING THE GATE OPERATIONS

The engineered cat-state gates exhibit predicted oscillatory dynamics, while gate fidelity decreases as single-photon-drive or detuning errors grow beyond the regime supporting coherent-state eigenstates.

Single-qubit gates: Single-qubit Z-rotation dynamics show a period of π/4Ezα0 under H0 + Hz with single-photon loss.The corresponding probability is measured for evolution from an even cat state toward |α0⟩.
Single-qubit gates: Single-qubit X-rotation dynamics show a period of πe2|α0|2/4δx|α0|2 under H0 + Hx with single-photon loss.The system is initialized in |α0⟩ and monitored for occupation of |−α0⟩.
Gate errors: Gate performance decreases as Ez and δ increase, because the coherent-state mapping requires Ez ≪ 4K|α0|3 and δ ≪ 2Ep.Small oscillations indicate that the Hamiltonian eigenstates are no longer coherent states.
Two-qubit gate: The entangling gate reaches the specified entangled state with periodicity π/8|Ezzα2|.The evolution is tracked from a product cat state under the two-qubit interaction.
Figure S5: Figure S5 compares probability traces for single-qubit Z and X rotations and the two-qubit entangling operation under stated drive, detuning, and loss parameters.The panels use different initial states and parameters for each gate family.

IX. FIDELITY OF A CAT STATE IN KNR WITH AND WITHOUT TWO-PHOTON DRIVE

The section compares cat-state fidelity under single-photon loss in linear and Kerr-nonlinear resonators, showing that a suitable two-photon drive suppresses Kerr-induced rotation and dephasing. It also describes the qcMAP protocol and derives the drive parameters using an expanded Jaynes–Cummings model.

Fidelity comparison: Cat-state fidelity decreases faster in a Kerr-nonlinear resonator than in a linear resonator under single-photon loss.The lossy-KNR fidelity is defined relative to a lossless KNR to isolate non-deterministic errors.
Fidelity comparison: A two-photon drive with appropriately chosen amplitude stabilizes the cat state against Kerr-induced rotation and dephasing.The stabilization is confirmed by the time dependence of fidelity for the driven KNR.
qcMAP gate protocol: The qcMAP sequence entangles the qubit and resonator, applies displacement and photon-number-conditioned qubit operations, and returns the system to a disentangled cat-state superposition.The ideal sequence evolves to (|2α0⟩+|0⟩)⊗|g⟩ before a final displacement by −α0.
Drive engineering: The required drive is derived by expanding the Jaynes–Cummings interaction to fourth order, assuming g2/∆−g4/∆3 ∼ g2/∆ and an infinite-T1 qubit.At (g2/∆)Tgate=π/2, comparison with the main-text Hamiltonian gives Ep=−iKα2, with K=g4/∆3 and ⟨σz⟩=±1.
Drive engineering: The coherent states |±α0⟩ and corresponding qubit-resonator states are Hamiltonian eigenstates when the drive amplitude satisfies the stated condition.Numerical simulations use a slightly smaller |Ep| because of higher-order Jaynes–Cummings contributions.