None of the work in this thesis would have been possible without the support and help of the members of the Piktori group, to whom I owe many thanks. The recovery time of the superconducting circuit after the optical pulse places a limit on the repetition rate of the transducer.
INTRODUCTION
SUPERCONDUCTING METAMATERIALS FOR CIRCUIT QUANTUM ELECTRODYNAMICS IN THECIRCUIT QUANTUM ELECTRODYNAMICS IN THE
NON-MARKOVIAN REGIME
BACKGROUND: SUPERCONDUCTING QUBITS
- Superconducting Qubit Basics
- Qubit Frequency Tuning
- Qubit State Preparation
- Qubit Readout
- Qubit Characterization
To perform qubit readout, we simply interrogate the readout resonator and its frequency allows us to infer the state of the qubit. After a variable time delay 𝜏 we apply another 𝜋2 pulse and read out the state of the qubit.
COLLAPSE AND REVIVAL OF AN ARTIFICIAL ATOM COUPLED TO A STRUCTURED PHOTONIC RESERVOIR
- Introduction
- Slow-Light Metamaterial Waveguide
- Non-Markovian Radiative Dynamics
- Time-Delayed Feedback
- Conclusion
SEM image of a Q1qubit showing the long, thin readout capacitor (false green color), the XY control line, the Z flux line, and the coupling capacitor to the readout resonator (false dark blue color). Results of measurements of the time domain dynamics of the qubit population as a function of 𝜔0.
UTILIZATION OF METAMATERIAL WAVEGUIDE FOR 2D CLUSTER STATE GENERATION
Furthermore, exploiting the fast flux control of the qubit's transition frequency provides several additional benefits for generating multidimensional cluster states. This directly improves the reliability of the entanglement between photonic qubits in the time bin that takes place via the time-delayed feedback mechanism.
APPENDIX: DETAILS OF DEVICE DESIGN, FABRICATION, MEASUREMENT SETUP, AND MODELING
Fabrication and Measurement Setup Device FabricationDevice Fabrication
TWPA Pump
Capacitively Coupled Resonator Array Waveguide Fundamentals Band Structure AnalysisBand Structure Analysis
First, we substitute in ˆ𝐻|𝜓i = 𝐸|𝜓i the following ansatz for the quantum states of the compound qubit waveguide system, i.e. the bare resonator frequency is again chosen to be 4.8GHz, and the calculated delays are for the center frequencies of the passband . For the remainder of the analysis, we focus on the qubit-photon bound state of the system.
The wave function of the bound state with energy𝐸 can be obtained by first substituting Eq. The qubit and photon components of the bound state can be calculated from the normalization condition for|𝜓𝐸i,.
Physical Implementation of Finite Resonator Array Geometrical Design of Unit CellGeometrical Design of Unit Cell
CAD diagram showing the end of the final resonator array including the boundary matching circuit (which in this case includes the first two resonators) and the first unit cell.b. Transmission spectrum of the entire resonator array consisting of 22 unit cells and 2 boundary-matching resonators at either end of the array (for a total of 26 resonators). The capacitance between the Xmon capacitor and the rest of the unit cell was designed to be ~2 fF, resulting in the unit cell coupling of 𝑔 qubits.
In principle, however, more resonators could have been used to tune the finite structure to the gates, thus reducing the ripples in the gates. 0;𝐶𝑔was obtained from the 𝐵-parameter of the 𝐴 𝐵𝐶 𝐷 matrix (which contains information about the series impedance of the unit cell circuit).
Disorder Analysis
Given that the effect of tapering the circuit parameters at the limit is to optimally couple the normal modes of the structure to the source and load impedances, the ripples in the passband are simply overlapping low-𝑄 resonances of the normal modes. Therefore, we can extract the normal mode frequencies from the maximum of the ripples in the passband, which will be shifted with respect to the normal mode frequencies of a structure without disorder. Here, the disorder is in the bare frequencies of the (unit cell) resonators that make up the metamaterial waveguide and is the coupling between nearest-neighbor resonators in the resonator array.
For each perturbation level, we performed simulations of 500 different perturbation realizations and for each different perturbation realization we calculated the standard deviation in the free spectral range of the wave, ΔFSR. Note that the minimum value of ΔFSR at 𝜎 =0 is determined by the intrinsic frequency dispersion of the normal mode of the undisturbed array of resonators.
Modeling of Qubit Q 1 Coupled to the Metamaterial Waveguide
The eigenstate with energy outside the passband corresponds to the bound state of the system |𝑏i d. Furthermore, in the model we coupled the qubit to the first, third and fourth resonators of the array (as opposed to only the third resonator), with couplings 𝑔1 = 2.2 MHz, 𝑔. 𝐽/(𝐸𝑏−𝜔 . 0) where 𝑏𝐸 is the energy of the bound state; this theoretical photonic wave function is plotted in the top panel.
1 coupling, this overlap was not high enough in the simulations relative to the coupling of a qubit to a metamaterial waveguide (extracted from separate measurements in the passband). In addition, in the model, we connected the qubit to the first, third and fourth resonators of the array with capacitive assemblies𝐶.
Modeling of Qubit Coupled to Dispersion-less Waveguide in Front of MirrorMirror
Note that to capture the background transmission levels as well as the qubit interaction with the background transmission, we included a small direct coupling capacitance of 0.75 fF between the first and last resonators of the array. In simulations without this background transmission, near-band-cubic mode splitting and confined-state signatures outside the pass band were significantly weaker. However, for intermediate Γ1D such as Γ1D/2𝜋=0.6 MHz, 1.8 MHz, the shapes of the population dynamics curves are sensitive to𝜙.
In addition, we also plotted similar comparisons between this ideal model of the observed time-delayed feedback phenomenon, and the data shown in Figs. Quantification of the non-Markovianity of the discussed model under various parameters is presented in Ref.
QUANTUM TRANSDUCTION
The last two chapters of Part 2 will address some of the nanofabrication challenges in realizing our transducer device. As will be seen in Chapter 6, our transducer device is designed on a thin-film lithium niobate on silicon-on-insulator (LN on SOI) platform. An etching process for etching lithium niobate on silicon-on-insulator and the challenges involved are discussed in Chapter 9.
The integration of niobium-based superconducting circuits with a converter device requires the fabrication of niobium qubits on silicon-on-insulator substrates. The fabrication process of niobium-based superconducting qubits on silicon-on-insulator substrates is developed in Chapter 8.
BACKGROUND: CAVITY OPTOMECHANICS AND 1D OPTOMECHANICAL CRYSTALS
Cavity Optomechanics Hamiltonian
The optical frequency can be written as 𝜔𝑐(𝑥) =2𝜋∗𝑐/(2(𝐿+𝑥)) where c is the speed of light and L+x is the effective length of the Fabry-Perot cavity. Identify the position operator of the mechanical mode as ˆ𝑥=𝑥. 2𝑚𝜔𝑚) is the zero-point motion of the mechanical oscillator, we arrive at the following Hamiltonian. Although we derived the Hamiltonian in Equation 5.3 starting for a Fabry-Perot cavity with a mechanically compliant mirror, it is applicable to a wide variety of optomechanical systems with cavities, including the specific nanomechanical 1D optomechanical crystals that we will explore in this thesis. to use.
We can rewrite the Hamiltonian in eq 5.3 in a rotating frame at the laser frequency𝜔𝐿 as. The optical field consists of a large coherent part (𝛼) with small quantum fluctuations (𝛿𝑎ˆ), so we linearize the Hamiltonian by making the substitution ˆ𝑎=𝛼+𝛿𝑎ˆ.
Equations of Motion
For the purpose of quantum transduction, we are particularly interested in the optomechanical damping rate𝛾𝑜𝑚 at the mechanical frequency (𝜔 =𝜔𝑚). Intuitively, we convert phonons from the mechanical state to photons in the optical cavity. Intuitively, we can think of 𝛾𝑜𝑚 as the 'good damping rate' where phonons in the mechanical state are converted to photons in the optical state, which we.
In the center of the nanobeam we have the 'defect region' where we break the translational symmetry of the mirror region by adiabatically tuning the dimensions. A photoelastic contribution (𝑔 . 0,PE) where the strain induced by the mechanical shear causes a change in the refractive index.
DESIGN OF A WAVELENGTH-SCALE
PIEZO-OPTOMECHANICAL QUANTUM TRANSDUCER
- Introduction
- Piezo Cavity Design
- Optomechanics Design
- Full Device Design
- Efficiency and Added Noise
- Conclusion
The dimensions of the piezo box (outlined in Fig. 6.2b) are designed to support a periodic mechanical mode whose periodicity is similar to that of the IDT fingers. IDT also minimizes the energy participation of the electric field in the qubit in the lossy piezo region, given by the relation 𝜁𝑞 = 𝐶. Right shows in-plane (breathing) and left shows out-of-plane (lam wave) components of the optimized mechanical state.
For𝑁 IDT fingers, the length𝑙𝑝of the piezo region is given by𝑙𝑝 =𝑁 𝑝/2, where𝑝is the IDT period. In the design presented above, we minimize the dimensions of the piezo cavity so that most of the energy in the mechanical state lives in the OMC region.
NIOBIUM BASED TRANSMON QUBITS ON SILICON SUBSTRATES FOR QUANTUM TRANSDUCTION
- Introduction
- Qubit Design
- Device Fabrication
- Experimental Setup
- Qubit Characterization
- Qubit Response to Optical Illumination
- Conclusion
Note that there is no Z-line on the chip for tuning the flux to the frequency of the qubit. This allows us to characterize the optical response of the readout resonator without the influence of the qubit. It is clear that there is no measurable change in the spectrum of the readout resonator after the laser pulse.
We find a similar linear scaling of the qubit decoherence rate as a function of laser power (P), 𝛾∗. We further investigate the dependence of the population in the state of dynamic equilibrium 𝑃𝑒,𝑆 𝑆 as a function of the repetition rate and duration of the laser pulse, fig.
TOWARDS FABRICATING NIOBIUM BASED QUBITS ON SILICON-ON-INSULATOR SUBSTRATES
- Fabrication Challenges
- Development of a Sputtering Process for Niobium Thin Films
- Development of an Etching Process for Niobium Thin Films
- Protection of Nb Surface from VHF Attack
- Quality of Sputtered and Etched Niobium Thin Films
- Etch of VHF Release Holes
- Proposed Fabrication Process for Niobium Transmon Qubits on SOI Having developed a sputter and etch process for Nb thin films and having testedHaving developed a sputter and etch process for Nb thin films and having tested
Sputtering allows considerable control over the deposition parameters, which can be adjusted to change the tension of the deposited film. 8.2, we plot the voltage of the deposited 150 nm Nb film as a function of sputtering process pressure. This step can be done directly after the alumina etch without stripping the ZEP 520a resist.
The resist will only serve as an additional mask on top of the alumina mask for this step. In this layer we will deposit a 15 nm thick ALD alumina layer for protection of the Nb surface against VHF.
![Figure 8.1: Fabrication process for Al transmon qubits on SOI. Figure reproduced from [78].](https://thumb-ap.123doks.com/thumbv2/123dok/10412333.0/116.918.163.757.108.227/figure-fabrication-process-transmon-qubits-soi-figure-reproduced.webp)
APPENDIX: DEVELOPMENT OF AN ETCH PROCESS FOR LITHIUM NIOBATE ON SILICON-ON-INSULATOR
Dry Etch Process
Although the overall dependence of etching on these three parameters is complex, we see two trends: 1. For fixed ICP power, the LN etch rate can be increased by increasing RF power. After some investigation of the parameter space we arrive at the etching parameters from Table 9.1.
The mask we use for the dry etching is a thin film chromium (Cr) mask formed using e-beam lithography, e-beam evaporation and metal lift-off. Using this dry etching process, we fabricate a wavelength-scaled lithium niobate piezo-acoustic cavity as shown in Fig.
LN40° sidewall
Wet Etch Process
One of the problems with a Cr/Au mask is the formation of defects or holes that allow HF to penetrate through the mask layer. These defects are thought to originate after substrate cooling following electron beam evaporation of the masking layer [217]. Since the gold surface tends to be hydrophilic, HF can be absorbed into these defects and penetrate through the mask into the underlying LN.
In principle, dividing the evaporation into a larger number of stages will increase the robustness of the mask, but we find a two-stage evaporation sufficient. To test our Cr/Au mask and the anisotropy of the etch, we try to etch simple LN 'boxes'.
Proposed Hybrid Etch Process
This trench is etched most of the way using the wet etch described in Section 9.2, but we do not etch the trench through to the Si layer. By etching the trench most of the way with a wet etch, we can get smooth steep sidewalls as seen in Fig. Smooth side walls of a LN piezo 'box' etched with a wet etch most of the way followed by a dry etch to clean the last.
The damage is limited to the trench and will thus minimize the effect on the optical quality factors of the OMC patterned in silicon. In this step we need to ensure that the writing area not only covers the top of the LN 'box' but also extends into the trench.
BIBLIOGRAPHY
Low-loss waveguides in lithium niobate thin films with diamond blade optical cutting”. Fabrication of a multifunctional photonic integrated chip on lithium niobate-in-insulator using laser-assisted femtosecond chemical polishing. Chemical-mechanical coating lithography: A route to large-scale low-loss photonic integration in lithium niobate on insulators.
True switchable electro-optical delay lines of meter-scale lengths fabricated in lithium niobate on insulator using photolithography-assisted chemi-mechanical etching”. Ultra-high aspect ratio photonic crystal structures in lithium niobates fabricated by focused ion beam milling”.
