Schrödinger cat states of a nuclear spin qudit in silicon
Data files
May 27, 2024 version files 137.96 MB
Abstract
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.
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
- 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 .
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_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