DOI: 10.5061/dryad.931zcrjtf
title: Schrödinger cat states of a nuclear spin qudit in silicon

description: High-dimensional quantum systems are a valuable resource for quantum
 information processing. They can be used to encode error-correctable
 logical qubits, for instance in continuous-variable states of oscillators
 such as microwave cavities or the motional modes of trapped ions. Powerful
 encodings include ‘Schrödinger cat’ states, and superpositions of widely
 displaced coherent states, which also embody the challenge of quantum
 effects at the large scale. Alternatively, recent proposals suggest
 encoding logical qubits in high-spin atomic nuclei, which can host
 hardware-efficient versions of continuous-variable codes on a
 finite-dimensional system. Here we demonstrate the creation and
 manipulation of Schrödinger cat states using the spin-7/2 nucleus of a
 single antimony (123Sb) atom, embedded and operated within a silicon
 nanoelectronic device. We use a coherent multi-frequency control scheme to
 produce spin rotations that preserve the SU(2) symmetry of the qudit, and
 constitute logical Pauli operations for logical qubits encoded in the
 Schrödinger cat states. The Wigner function of the cat states exhibits
 parity oscillations with a contrast up to 0.982(5), and state fidelities
 up to 0.913(2). These results demonstrate high-fidelity preparation of
 nonclassical resource states and logical control in a single atomic-scale
 object, opening up applications in quantum information processing and
 quantum error correction within a scalable, manufacturable semiconductor
 platform.
description: # Creation and manipulation of Schrodinger cat states of a nuclear spin
 qudit in silicon This is accompanying data for the journal article
 'Creation and manipulation of Schrodinger cat states of a nuclear
 spin qudit in silicon'
 ([https://arxiv.org/abs/2405.15494)](https://arxiv.org/abs/2405.15494\))
 **Outline of the dataset** The dataset consists of the following *
 paper_data - folder containing the data used to produce figures in the
 main text, divided into subfolders according to the date that the data was
 taken * Sb_Main - Jupyter notebooks to evaluate the data and make the
 figures for the main text. For each figure of the main text, a separate
 notebook is stored in the corresponding folder. E.g. 'Figure 3 - Cat
 state creation' contains the notebook to evaluate the data and create
 the plot for figure 3. * PlottingHelper.py - Functions that are imported
 in each notebook to facilitate fitting curves * 'Supplementary'
 - folder containing data used to produce figures of the supplementary
 material. This folder contains all supplementary data in
 'data_supplement', and evaluation notebooks for each figure of
 the supplement. * QCoDeS - Python package required to open data files.
 Note that this is not the most recent version of Qcodes. See below for
 information how to install QCoDeS * package-list.txt - A full list of
 packages and their versions that were used to evaluate the notebooks. If
 at some point you encounter an error message, please compare the version
 of the package that is causing a problem to the version in
 package-list.txt **Outline of the data folders** Data folders are saved as
 `\paper_data\main\{dataset}` and `\paper_data\supp\{dataset}` for the data
 used in the main text and the supplementary, respectively. The data
 folders/files are supposed to be opened using QCoDeS `load_data()`
 function. Most data sets have the "ESR_up_proportions" as the
 main measurement result. For each data point, it is a list of 8 numbers,
 where each entry in the list corresponds to the measured electron up
 proportion after applying an electron spin resonance (ESR) pulse at a
 frequency that corresponds to a nuclear spin state. From these 8 numbers,
 the evaluation script then determines the nuclear spin state as the entry
 with the highest ESR up proportion. Once the nuclear spin state is
 determined, the evaluation script then derive other parameters such as the
 parity, or I_z expectation value, from the nuclear spin state.
 **Replicating the data analysis** All measurements were performed and
 analysed using the QCoDes data framework. Each jupyter notebook contains
 the analysis to a specific figure in the manuscript. The notebooks can be
 accessed and executed for example by using jupyter notebook or jupyter
 lab. We recommend using anaconda to install both Python and jupyter
 Notebook, see
 [https://www.anaconda.com/download](https://www.anaconda.com/download) .
 All evaluation was done using python 3.9.18, matplotlib 3.5.0, numpy
 1.24.3 . **Install QCoDeS** To install the QCoDeS version supplied in this
 package: 1. Open command prompt as admin 2. Navigate to the QCoDeS folder
 3. run `python setup.py develop` 4. (optional) If additional packages are
 required, use pip install to install missing packages manually. Then try
 step 3 again **Troubleshooting** * File not found: Although we use
 `os.normpath()` to make data paths available for all operating systems, we
 have previously encountered issues where the data path could not be found.
 In this case, the full data path can be inserted manually. * If other
 issues arise, please contact the authors. ## Data set explanations: ###
 Figure 1 This code is used solely for numerical simulations and plotting
 of the Wigner spheres in figure 1 of the manuscript. It does not contain
 any experimental data. ### Figure 2 **Figure2.ipynb** This file is used to
 create figure 2a and figure 2b. Here, we sweep
 "Pulse_Duration_0_0_0" and measure
 "ESR_up_proportions_0_0_0_0_0_0". The state for each
 "sample" is measured and and used to derive the I_z expectation
 value ("exp_values_Iz") and the state probabilities
 ("state_probabilities"). Standard errors are also computed and
 stored in variables ending on "_std".
 **Fig2_wigner_tomography.ipynb** This file imports data from the density
 matrix reconstruction routine to then plot the Wigner spheres. **Used data
 sets** * `paper_data/main/#003_Global_Rotation_ionized_10-30-30` - ESR up
 proportions as a function of pulse duration *
 `paper_data/main/density_matrix_reconstruction/Fig2/scs_04pi.npy` -
 Density matrix of state at 0 pulse duration *
 `paper_data/main/density_matrix_reconstruction/Fig2/scs_24pi.npy` -
 Density matrix of state at pi/2 pulse duration *
 `paper_data/main/density_matrix_reconstruction/Fig2/scs_44pi.npy` -
 Density matrix of state at pi pulse duration ### Figure 3
 **Figure3.ipynb** This file creates figure 3b,c,f,g of the manuscript.
 Each data set is analysed similar to figure 2. All subfigures are plotted
 together at the end of the analysis script.
 **Fig3_wigner_tomography.ipynb** This file imports data from the density
 matrix reconstruction routine to then plot the Wigner spheres. **Used data
 sets** * `paper_data/main/#003_ladder_climbing_09-36-37` - ESR up
 proportions for the initial state of the Givens rotation method *
 `paper_data/main/#025_ladder_climbing_13-58-19` - ESR up proportions as a
 function of target state of the Givens rotation method *
 `paper_data/main/#014_Scarani_Protocol_17-05-01` - ESR up proportions as a
 function of analysis phase phi. Used to derive parity for the cat state
 created with Givens rotations *
 `paper_data/main/#005_cat_state_oat_population_10-04-09` - ESR up
 proportions for the initial state *
 `paper_data/main/#009_cat_state_oat_population_10-42-10` - ESR up
 proportions for the state after pi/4 covariant rotation *
 `paper_data/main/#008_cat_state_oat_population_10-18-54` - ESR up
 proportions for the state after pi/2 covariant rotation *
 `paper_data/main/#007_cat_state_oat_population_09-22-47` - ESR up
 proportions for the state after pi/2 covariant rotation and the one-axis
 twisting phase update *
 `paper_data/main/#010_cat_state_oat_population_11-00-14` - ESR up
 proportions for the state after pi/4 covariant rotation after the one-axis
 twisting phase update *
 `paper_data/main/#002_cat_state_oat_population_10-36-11` - ESR up
 proportions for the state after pi/2 covariant rotation after the one-axis
 twisting phase update *
 `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_00.npy` - Density matrix of the initial state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_01.npy` - Density matrix of the second state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_02.npy` - Density matrix of the third state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_03.npy` - Density matrix of the fourth state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_04.npy` - Density matrix of the fifth state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_05.npy` - Density matrix of the sixth state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_06.npy` - Density matrix of the seventh state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/Givens_rotation/lc_07.npy` - Density matrix of the final state of the Givens rotation method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/scs_04pi.npy` - Density matrix of the initial state of the one-axis twisting method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/scs_14pi.npy` - Density matrix of the second state of the one-axis twisting method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/scs_24pi.npy` - Density matrix of the third state of the one-axis twisting method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_08rotation.npy` - Density matrix of the fourth state of the one-axis twisting method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_28rotation.npy` - Density matrix of the fifth state of the one-axis twisting method * `paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_48rotation_80s.npy` - Density matrix of the final state of the one-axis twisting method11:46 20.05.2024 ### Figure 4 **Figure4.ipynb** This file creates figure 4a,b,c,e of the manuscript. Each data set is analysed similar to figure 2. All subfigures are plotted together at the end of the analysis script. **Fig4_wigner_tomography.ipynb** This file imports data from the density matrix reconstruction routine to then plot the Wigner spheres for figure 4d. **Used data sets** * `paper_data/main/#003_Global_Rotation_ionized_generalized_ramsey_13-55-16` - ESR up proportions as a function of wait time in a Ramsey experiment * `paper_data/main/#019_Global_Rotation_ionized_generalized_echo_11-44-45` - ESR up proportions as a function of wait time in a Hahn Echo experiment * `paper_data/main/#012_ionized-cat-coherence-generalized-ramsey_17-49-09` - ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 7/2 * `paper_data/main/#009_ionized-cat-coherence-generalized-ramsey_13-46-19` - ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 5/2 * `paper_data/main/#008_ionized-cat-coherence-generalized-ramsey_11-25-47` - ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 3/2 * `paper_data/main/#015_ionized-cat-coherence-generalized-ramsey_18-59-39` - ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 1/2 * `paper_data/main/#006_ionized-cat-coherence_16-52-05` - ESR up proportions as a function of wait time in a global Ramsey experiment for subspace 5/2 * `paper_data/main/#005_ionized-cat-coherence_15-21-26` - ESR up proportions as a function of wait time in a global Ramsey experiment for subspace 3/2 ### Supplementary figures **Figure S1** This code (`wigner_representation.ipynb`) is used solely for numerical simulations and plotting of the Wigner spheres in figure S1 of the manuscript. It does not contain any experimental data. **Figure S2-S6** This code (`cat_phase.ipynb`) is used solely for numerical simulations and plotting of figure S2-S6 of the manuscript. It does not contain any experimental data. **Figure S7** `CR_power_dependence.ipynb` is used to create figure S4a,b. It analyses Rabi oscillations (I_z expectation value vs. pulse duration) for different powers of the radiofrequency sent to the on-chip antenna. `CRSimulation.ipynb` is used to create figure S4c. It is purely a numerical simulation of the covariant rotation operation and does not contain any experimental data. Used data sets: * `paper_data/supp/#004_global_rotation_11-51-20` - ESR up proportions as a function of pulse duration in a Rabi oscillation for power 11.43mV & 22.86mV output voltage of the radiofrequency source. * `paper_data/supp/#006_global_rotation_13-14-17` - ESR up proportions as a function of pulse duration in a Rabi oscillation for power 34.29mV & 45.71mV output voltage of the radiofrequency source. * `paper_data/supp/#013_global_rotation_17-57-55` - ESR up proportions as a function of pulse duration in a Rabi oscillation for power 45.71mV & 57.14mV output voltage of the radiofrequency source. **Figure S8** `SubspaceRotations.ipynb` analyses Rabi oscillations within different subspaces. data_1 contains Rabi oscillations for 7/2 and 5/2 subspace, data_2 contains Rabi oscillations for 3/2 and 1/2 subspace. Used data sets: * `paper_data/supp/#025_subspace_rotation_22-06-43` - ESR up proportions as a function of pulse duration in a Rabi oscillation for subspace 7/2 and 5/2 * `paper_data/supp/#005_subspace_rotation_07-52-34` - ESR up proportions as a function of pulse duration in a Rabi oscillation for subspace 3/2 and 1/2 **Figure S9** `figure_different_size_cats.ipynb` analyses cat states in different subspaces. Here, we plot the reconstructed density matrix, the wigner representation and the measured parity oscillations at the end of the script. Used data sets: * `paper_data/supp/#014_Scarani_Protocol_17-05-01` - State probabilities as a function of analysis phase phi in a parity oscillation for subspace 7/2 * `paper_data/supp/density_matrices/rho72.npy` - Density matrix of a cat state for subspace 7/2 * `paper_data/supp/#010_Scarani_Protocol_16-01-23` - State probabilities as a function of analysis phase phi in a parity oscillation for subspace 5/2 * `paper_data/supp/density_matrices/rho52.npy` - Density matrix of a cat state for subspace 5/2 * `paper_data/supp/#014_Scarani_Protocol_15-57-06` - State probabilities as a function of analysis phase phi in a parity oscillation for subspace 3/2 * `paper_data/supp/density_matrices/rho32.npy` - Density matrix of a cat state for subspace 3/2 * `paper_data/supp/#005_Scarani_Protocol_11-25-36` - State probabilities as a function of analysis phase phi in a parity oscillation for subspace 1/2 * `paper_data/supp/density_matrices/rho12.npy` - Density matrix of a cat state for subspace 1/2 **Figure S12** `Figure S12.ipynb` analyses T2 coherence times in different qubit subspaces. Note that here we analyse flip probabilities instead of state probabilities, i.e. instead of measuring every state of the nucleus, we only look at flipping events between adjacent levels. Used data sets: * `paper_data/supp/#005_ionized-nmr-ramsey_20-26-23` - Flip probabillities as a function of wait time a two-level Ramsey experiment for all combinations of adjacent states **Figure S13** `nuclear state.ipynb` plots "ESR_up_proportions" after flipping the state of the electron conditional on the state of the nucleus. This experiment represents the usual readout procedure. Used data sets: * `paper_data/supp/#005_test-sequential-aesr-readout_13-05-00` - ESR up proportion for different ESR resonances, measured multiple times (repetitions) **Figure S14** `FigMaxwellDemon.ipynb` analyses a Rabi oscillation within a two-level subspace of the nucleus for the case where the Maxwell's Demon state initialisation is used and for the case where it's not used. Used data sets: * `paper_data/supp/#007_neutral-nmr-rabi_10-00-28` - ESR up proportions as a function of NMR pulse duration, with Maxwell's Demon set to on * `paper_data/supp/#009_neutral-nmr-rabi_10-42-11` - ESR up proportions as a function of NMR pulse duration, with Maxwell's Demon set to off * `paper_data/supp/#004_check-nuclear-spin-SET-saved_09-56-53` - Current trace of single electron transistor as a function of time **Figure S15** `cat_T1.ipynb` analyses ESR up proportions/state probability of a cat state after a wait time. Used data sets: * `paper_data/supp/#009_ionized-cat-T1_11-56-15` - ESR up proportions of a cat state as a function of wait time after creating a cat state **Figure S16** `Figure16.ipynb` analyses the ionisation shock of the nucleus depending on state. Used data sets: * `paper_data/supp/#016_ionisation_shock_17-18-37` - Number of readout successes of sequential readout events before reinitialision **Figure S17** `other_devices.ipynb` analyses the NMR spectrum, Rabi oscillations, and cat state parity for three different devices. The heading "Scarlett" and "Venus" in the readout script refer to different dilution refrigerators. Used data sets: * `paper_data/supp/#023_ionized-nmr-spectrum_17-57-52` - Flip probabilities of the nucleus as a function of NMR frequency for device A * `paper_data/supp/#003_Global_Rotation_ionized_10` - ESR up proportion as a function of covariant rotation pulse duration for device A * `paper_data/supp/#009_cat_state_OAT_15-34-46` - ESR up proportion as a function of analysis phase phi in a parity oscillation measurement for device A * `paper_data/supp/#024_ionized-nmr_16-24-45` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#002_ionized-nmr-spectrum_09-43-15` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#006_ionized-nmr-spectrum_10-06-46` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#008_ionized-nmr-spectrum_10-22-38` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#010_ionized-nmr-spectrum_10-54-59` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#012_ionized-nmr-spectrum_11-24-46` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#014_ionized-nmr-spectrum_11-41-58` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#017_ionized-nmr-spectrum_11-52-29` - Flip probabilities of the nucleus as a function of NMR frequency for device B * `paper_data/supp/#024_subspace_rotation_13-17-28` - ESR up proportion as a function of covariant rotation pulse duration for device B * `paper_data/supp/#007_cat_expectation_14-00-54` - ESR up proportion as a function of analysis phase phi in a parity oscillation measurement for device B * `paper_data/supp/#010_ionized-nmr_16-55-39` - Flip probabilities of the nucleus as a function of NMR frequency for device C