Quantum smoothing for classical mixtures

In quantum mechanics, wave functions and density matrices represent our knowledge about a quantum system and give probabilities for the outcomes of measurements. If the combined dynamics and measurements on a system lead to a density matrix $\rho(t)$ with only diagonal elements in a given basis $\{|n\rangle\}$, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states $|n\rangle$ with probabilities $\rho_{nn}(t)$. Fully equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states $|n\rangle$ improves our ability to retrodict what was the outcome of a projective state measurement at time $t$. Here, we show with experiments on a superconducting qubit that the smoothed probabilities do not, in the same way as the diagonal elements of $\rho$, permit a classical mixture interpretation of the state of the system at the past time $t$.Comment: 5 pages, 4 figure

Bath engineering of a fluorescing artificial atom with a photonic crystal

We demonstrate how the dissipative interaction between a superconducting qubit and a microwave photonic crystal can be used for quantum bath engineering. The photonic crystal is created with a step-impedance transmission line which suppresses and enhances the quantum spectral density of states, influencing decay transitions of a transmon circuit. The qubit interacts with the transmission line indirectly via dispersive coupling to a cavity. We characterize the photonic crystal density of states from both the unitary and dissipative dynamics of the qubit. When the qubit is driven, it dissipates into the frequency dependent density of states of the photonic crystal. Our result is the deterministic preparation of qubit superposition states as the steady-state of coherent driving and dissipation near by the photonic crystal band edge, which we characterize with quantum state tomography. Our results highlight how the multimode environment from the photonic crystal forms a resource for quantum control.Comment: 9 pages 7 figure

Correlations of the time dependent signal and the state of a continuously monitored quantum system

In quantum physics, measurements give random results and yield a corresponding random back action on the state of the system subject to measurement. If a quantum system is probed continuously over time, its state evolves along a stochastic quantum trajectory. To investigate the characteristic properties of such dynamics, we perform weak continuous measurements on a superconducting qubit that is driven to undergo Rabi oscillations. From the data we observe a number of striking temporal correlations within the time dependent signals and the quantum trajectories of the qubit, and we discuss their explanation in terms of quantum measurement and photodetection theory.Comment: 8 pages 5 figure

Mapping quantum state dynamics in spontaneous emission

The evolution of a quantum state undergoing radiative decay depends on how the emission is detected. We employ phase-sensitive amplification to perform homodyne detection of the spontaneous emission from a superconducting artificial atom. Using quantum state tomography, we characterize the correlation between the detected homodyne signal and the emitter's state, and map out the conditional back-action of homodyne measurement. By tracking the diffusive quantum trajectories of the state as it decays, we characterize selective stochastic excitation induced by the choice of measurement basis. Our results demonstrate dramatic differences from the quantum jump evolution that is associated with photodetection and highlight how continuous field detection can be harnessed to control quantum evolution.Comment: 8 pages, 8 figure

Homodyne monitoring of post-selected decay

We use homodyne detection to monitor the radiative decay of a superconducting qubit. According to the classical theory of conditional probabilities, the excited state population differs from an exponential decay law if it is conditioned upon a later projective qubit measurement. Quantum trajectory theory accounts for the expectation values of general observables, and we use experimental data to show how a homodyne detection signal is conditioned upon both the initial state and the finally projected state of a decaying qubit. We observe, in particular, how anomalous weak values occur in continuous weak measurement for certain pre- and post-selected states. Subject to homodyne detection, the density matrix evolves in a stochastic manner, but it is restricted to a specific surface in the Bloch sphere. We show that a similar restriction applies to the information associated with the post-selection, and thus bounds the predictions of the theory.Comment: 11 pages, 8 figure

Spectroscopy of the 1S0^1S_0-to-1D2^1D_2 clock transition in 176^{176}Lu+^+

High precision spectroscopy of the $^1S_0$-to-${^1}D_2$ clock transition of $^{176}$Lu is reported. Measurements are performed with Hertz level precision with the accuracy of the hyperfine-averaged frequency limited by the calibration of an active hydrogen maser to the SI definition of the second via a GPS link. The measurements also provide accurate determination of the $^1D_2$ hyperfine structure. Hyperfine structure constants associated with the magnetic octupole and electric hexadecapole moments of the nucleus are considered, which includes a derivation of correction terms from third-order perturbation theory.Comment: 8 pages, 3 figure

Development of a NbN Deposition Process for Superconducting Quantum Sensors

We have carried out a detailed programme to explore the superconducting characteristics of reactive DC-magnetron sputtered NbN. The basic principle is to ignite a plasma using argon, and then to introduce a small additional nitrogen flow to achieve the nitridation of a Nb target. Subsequent sputtering leads to the deposition of NbN onto the host substrate. The characteristics of a sputtered film depend on a number of parameters: argon pressure, nitrogen flow rate and time-evolution profile, substrate material, etc. Crucially, the hysteresis in the target voltage as a function of the nitrogen flow can be used to provide a highly effective monitor of nitrogen consumption during the reactive process. By studying these dependencies we have been able to achieve highly reproducible film characteristics on sapphire, silicon dioxide on silicon, and silicon nitride on silicon. Intrinsic film stress was minimised by optimising the argon pressure, giving NbN films having Tc = 14.65 K. In the paper, we report characteristics such as deposition rate, Residual Resistance Ratio (RRR), film resistivity, transition temperature, and stress, as a function of deposition conditions. In order to enhance our understanding of the microwave properties of the films, we fabricated a wide range of microstrip NbN resonators (half wavelength, quarter wavelength, ring resonators). In the paper, we provide an illustrative result from this work showing a 2.1097 GHz resonator having a Q of 15,962 at 3.3 K.Comment: 4 pages 13 figures, Submitted to Proc. 24TH International Symposium On Space Terahertz Technology, Groningen, 8-10 APRIL, 201

Laser spectroscopy of 176^{176}Lu+^+

We perform high resolution spectroscopy on $^{176}$Lu$^+$ including the $^1S_0\leftrightarrow{^3}D_1$ and $^1S_0\leftrightarrow{^3}D_2$ clock transitions. Hyperfine structures and optical frequencies relative to the $^1S_0$ ground state of four low lying excited states are given to a few tens of kHz resolution. This covers the most relevant transitions involved in clock operation with this isotope. Additionally, measurements of the $^3D_2$ hyperfine structure may provide access to higher order nuclear moments, specifically the magnetic octupole and electric hexadecapole moments.Comment: 13 pages, 5 figures, correct an incorrect g-factor used in the evaluation of Zeeman shifts on 1S0-to-3D2 transition

Dynamic polarizability measurements in 176^{176}Lu+^+

We measure the differential polarizability of the $^{176}$Lu$^+$ $^1S_0$ -to- ${^3}D_1$ clock transition at multiple wavelengths. This experimentally characterizes the differential dynamic polarizability for frequencies up to 372 THz and allows an experimental determination of the dynamic correction to the blackbody radiation shift for the clock transition. In addition, measurements at the near resonant wavelengths of 598 and 646 nm determine the two dominant contributions to the differential dynamic polarizability below 372 THz. These additional measurements are carried out by two independent methods to verify the validity of our methodology. We also carry out a theoretical calculation of the polarizabilities using the hybrid method that combines the configuration interaction (CI) and the coupled-cluster approaches, incorporating for the first time quadratic non-linear terms and partial triple excitations in the coupled-cluster calculations. The experimental measurements of the $|\langle ^3D_1|| r || ^3P_J\rangle|$ matrix elements provide high-precision benchmarks for this theoretical approach.Comment: 11 pages, 5 figures, and 13 page supplemental materia

Oscillating magnetic field effects in high precision metrology

We examine a range of effects arising from ac magnetic fields in high precision metrology. These results are directly relevant to high precision measurements, and accuracy assessments for state-of-the-art optical clocks. Strategies to characterize these effects are discussed and a simple technique to accurately determine trap-induced ac magnetic fields in a linear Paul trap is demonstrated using $^{171}\mathrm{Yb}^+$Comment: 10 pages, 6 figure