Creation and manipulation of Schrödinger cat states of a nuclear spin qudit in silicon
Data files
May 27, 2024 version files 137.96 MB

README.md

SchrodingerCatStatesData.zip
Abstract
Highdimensional quantum systems are a valuable resource for quantum information processing. They can be used to encode errorcorrectable logical qubits, for instance in continuousvariable 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 highspin atomic nuclei, which can host hardwareefficient versions of continuousvariable codes on a finitedimensional system. Here we demonstrate the creation and manipulation of Schrödinger cat states using the spin7/2 nucleus of a single antimony (^{123}Sb) atom, embedded and operated within a silicon nanoelectronic device. We use a coherent multifrequency 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 highfidelity preparation of nonclassical resource states and logical control in a single atomicscale object, opening up applications in quantum information processing and quantum error correction within a scalable, manufacturable semiconductor platform.
README: 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)
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
 packagelist.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 packagelist.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 .
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:
 Open command prompt as admin
 Navigate to the QCoDeS folder
 run
python setup.py develop
 (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_103030
 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_093637
 ESR up proportions for the initial state of the Givens rotation method 
paper_data/main/#025_ladder_climbing_135819
 ESR up proportions as a function of target state of the Givens rotation method 
paper_data/main/#014_Scarani_Protocol_170501
 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_100409
 ESR up proportions for the initial state 
paper_data/main/#009_cat_state_oat_population_104210
 ESR up proportions for the state after pi/4 covariant rotation 
paper_data/main/#008_cat_state_oat_population_101854
 ESR up proportions for the state after pi/2 covariant rotation 
paper_data/main/#007_cat_state_oat_population_092247
 ESR up proportions for the state after pi/2 covariant rotation and the oneaxis twisting phase update 
paper_data/main/#010_cat_state_oat_population_110014
 ESR up proportions for the state after pi/4 covariant rotation after the oneaxis twisting phase update 
paper_data/main/#002_cat_state_oat_population_103611
 ESR up proportions for the state after pi/2 covariant rotation after the oneaxis 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 oneaxis twisting method 
paper_data/main/density_matrix_reconstruction/Fig3/SNAP/scs_14pi.npy
 Density matrix of the second state of the oneaxis twisting method 
paper_data/main/density_matrix_reconstruction/Fig3/SNAP/scs_24pi.npy
 Density matrix of the third state of the oneaxis twisting method 
paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_08rotation.npy
 Density matrix of the fourth state of the oneaxis twisting method 
paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_28rotation.npy
 Density matrix of the fifth state of the oneaxis twisting method 
paper_data/main/density_matrix_reconstruction/Fig3/SNAP/phase_48rotation_80s.npy
 Density matrix of the final state of the oneaxis 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_135516
 ESR up proportions as a function of wait time in a Ramsey experiment 
paper_data/main/#019_Global_Rotation_ionized_generalized_echo_114445
 ESR up proportions as a function of wait time in a Hahn Echo experiment 
paper_data/main/#012_ionizedcatcoherencegeneralizedramsey_174909
 ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 7/2 
paper_data/main/#009_ionizedcatcoherencegeneralizedramsey_134619
 ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 5/2 
paper_data/main/#008_ionizedcatcoherencegeneralizedramsey_112547
 ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 3/2 
paper_data/main/#015_ionizedcatcoherencegeneralizedramsey_185939
 ESR up proportions as a function of wait time in a generalised Ramsey experiment for subspace 1/2 
paper_data/main/#006_ionizedcatcoherence_165205
 ESR up proportions as a function of wait time in a global Ramsey experiment for subspace 5/2 
paper_data/main/#005_ionizedcatcoherence_152126
 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 S2S6
This code (cat_phase.ipynb
) is used solely for numerical simulations and plotting of figure S2S6 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 onchip 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_115120
 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_131417
 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_175755
 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_220643
 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_075234
 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_170501
 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_160123
 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_155706
 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_112536
 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_ionizednmrramsey_202623
 Flip probabillities as a function of wait time a twolevel 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_testsequentialaesrreadout_130500
 ESR up proportion for different ESR resonances, measured multiple times (repetitions)
Figure S14
FigMaxwellDemon.ipynb
analyses a Rabi oscillation within a twolevel 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_neutralnmrrabi_100028
 ESR up proportions as a function of NMR pulse duration, with Maxwell's Demon set to on 
paper_data/supp/#009_neutralnmrrabi_104211
 ESR up proportions as a function of NMR pulse duration, with Maxwell's Demon set to off 
paper_data/supp/#004_checknuclearspinSETsaved_095653
 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_ionizedcatT1_115615
 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_171837
 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_ionizednmrspectrum_175752
 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_153446
 ESR up proportion as a function of analysis phase phi in a parity oscillation measurement for device A 
paper_data/supp/#024_ionizednmr_162445
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#002_ionizednmrspectrum_094315
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#006_ionizednmrspectrum_100646
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#008_ionizednmrspectrum_102238
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#010_ionizednmrspectrum_105459
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#012_ionizednmrspectrum_112446
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#014_ionizednmrspectrum_114158
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#017_ionizednmrspectrum_115229
 Flip probabilities of the nucleus as a function of NMR frequency for device B 
paper_data/supp/#024_subspace_rotation_131728
 ESR up proportion as a function of covariant rotation pulse duration for device B 
paper_data/supp/#007_cat_expectation_140054
 ESR up proportion as a function of analysis phase phi in a parity oscillation measurement for device B 
paper_data/supp/#010_ionizednmr_165539
 Flip probabilities of the nucleus as a function of NMR frequency for device C