APhO 2019 Problem 1

Physics · electromagnetism

Problem

Theory Q1-1 English(Official) RF reflectometry for spin readout for silicon quantum computing Introduction Developing the idea of quantum computing into a practical technology is one of the largest outstanding challenges in science and technology. A promising path is to manipulate individual electrons in silicon transistors by time-dependent electromagnetic fields. In this question, we investigate the use of radio frequency (RF) reflectometry and single-electron transistors to readout the state of quantum bits in silicon-based quantum computer prototypes. Part A and Part B discuss radio wave transmission through cables and transmission lines, part C is devoted to conditions for wave reflection, part D introduces the single-electron transistor, and parts E and F introduce and ask you to optimise the method of reflectometry. Part A: Lumped element model of a co-axial transmission line (2.0 points) When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode). Diagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket. Consider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius aa, covered by an outer co-axial copper shield with inner radius bb. A dielectric of dimensionless relative permittivity εr\varepsilon_r and dimensionless relative permeability μr\mu_r separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding. A.1 At what speed do electromagnetic waves propagate in the co-axial cable? 0.2 pt A.2 If there is a charge Δq\Delta q on a length Δx\Delta x of the inner core of the co-axial cable, and the outer shield is grounded, find the electric field in the region between the inner core and the shield. 0.2 pt Theory Q1-2 English(Official) A.3 Find the capacitance per unit length, CxC_x, of the co-axial cable. You may wish to consider a length Δx\Delta x of the cable. 0.3 pt A.4 Find the inductance per unit length, LxL_x, of the cable. 0.3 pt A lumped element model of the cable is constructed by considering the inductance and capacitance of short sections of the cable. The inductance is assumed to be a property of the inner core, and the capacitance links the core with the shielding. A diagram of the lumped element model is shown below. Circuit diagram of lumped element model of coaxial cable. A.5 i. Show that the impedance Z0Z_0 of a semi-infinite length of cable is Z0=Lx/CxZ_0 = \sqrt{L_x/C_x}. 1.0 pt ii. Find b/ab/a if the cable has impedance Z0=50ΩZ_0 = 50\Omega and is made using a dielectric material with εr=4.0\varepsilon_r = 4.0 and μr=1.0\mu_r = 1.0. Part B: Hypothetical transmission line with return along a grounded plane (1.0 points) An alternative hypothetical transmission line is shown in the diagram below. The input signal is sent through a very thin conductor of radius aa, which is a distance d≫ad \gg a from a highly conductive grounded plane. The material surrounding the conductor has dimensionless relative permittivity εr\varepsilon_r and dimensionless relative permeability μr\mu_r. The return current flows along the grounded plane. Diagram of a hypothetical transmission line showing C - the conductor of radius aa, at a distance d≫ad \gg a from P - the grounded conducting plane. The conductor is embedded in a material with dimensionless relative permeability εr\varepsilon_r and dimensionless relative permittivity μr\mu_r. B.1 Find an expression for the characteristic impedance of this hypothetical transmission line. 1.0 pt Part C: Basics of RF reflectometry (1.2 points) An electromagnetic wave can propagate in a transmission line in two opposite directions. For each direction of propagation, the characteristic impedance Z0Z_0 can be used to relate the voltage V0V_0 and current I0I_0 amplitudes as in the Ohm's law, Z0=V0/I0Z_0 = V_0/I_0. Consider an interface between two transmission lines, with characteristic impedances Z0Z_0 and Z1Z_1. A schematic diagram of the circuit is shown below. Circuit diagram of a transmission line of impedance Z0Z_0 connected to a transmission line of impedance Z1Z_1. The physical size of the interface is much smaller than the wavelength. When a signal ViV_i sent into the transmission line with impedance Z0Z_0 reaches the interface it is partially transmitted into the second transmission line, resulting in a signal VtV_t in that line which propagates forward. Some of the signal may also be reflected, resulting in a backward propagating signal in the initial transmission line VrV_r. C.1 Find the reflectance of the interface Γ=Vr/Vi\Gamma = V_r/V_i. 1.0 pt C.2 State the condition(s) for the signal ViV_i to have gained a π\pi phase change on reflection. 0.2 pt Part D: The single electron transistor (3.3 points) A single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitively to the quantum dot, while the two other electrodes — the source and the drain — are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure. Circuit diagram representation of an SET. QD is the quantum dot, S is the source, D is the drain and G is the gate. Theory Q1-4 English(Official) The capacitance of the gate is CgC_g and the capacitance of the tunnel junctions is Ct≪CgC_t \ll C_g. Consider CgC_g to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at VgV_g. D.1 Consider a state of the SET in which the quantum dot contains nn electrons. 1.5 pt i. Find the electrical potential φn\varphi_n on the QD. ii. Find the amount of energy ΔEn\Delta E_n that is necessary to bring an additional electron from the source or the drain onto the QD. If ΔEn<0\Delta E_n < 0 then electrons will spontaneously tunnel into the quantum dot until such a number N>n\mathcal{N} > n is reached that ΔEN≥0\Delta E_{\mathcal{N}} \geq 0. The equilibrium number of electrons N\mathcal{N} and the corresponding addition energy ΔEN\Delta E_{\mathcal{N}} can be controlled by choosing the appropriate voltage VgV_g. D.2 Find an expression for the maximal possible value Ec=max⁡ΔEN(Vg)E_c = \max \Delta E_{\mathcal{N}}(V_g) of the equilibrium addition energy that can be achieved by tuning the gate voltage of the SET. 0.5 pt If ΔEN=0\Delta E_{\mathcal{N}} = 0 then tunnelling of electrons does not require extra energy and SET is in a highly conductive ON state. If ΔEN>0\Delta E_{\mathcal{N}} > 0, then the conductance of the SET is reduced (high-resistance OFF). For the number of electrons on the quantum dot to remain well-defined, certain conditions need to be satisfied. Firstly, if electrons in the source or drain have thermal energies sufficient to move spontaneously onto the quantum dot, the contrast between the ON and OFF states will disappear. D.3 Find a condition on the temperature of the electrons so that electrons cannot move onto the quantum dot by thermal excitation. 0.5 pt Secondly, tunnelling of electrons onto or off the dot limits the lifetime of their energy states. This tunnelling can be modelled using an effective resistance of the tunnel junction with the characteristic tunnelling time equal to the characteristic time for charging or discharging the quantum dot through the junction. D.4 i. Estimate the tunnelling time for a quantum dot in terms of capacitance CtC_t and effective resistance RtR_t of the tunnel junction. 0.8 pt ii. Find a condition on the effective resistance RtR_t so that the electrons in the quantum dot retain sufficiently well-defined energy for the ON and OFF states to remain distinct. Part E: RF reflectometry to readout SET state (1.0 points) The state of the SET is sensitive to electrical potentials created by nearby elements of the quantum circuit (such as quantum bits), and distinguishing between ON and OFF states provides a way to readout the information produced by the quantum computer. The SET in the ON state can be modelled by a resistance RON=100kΩR_{ON} = 100 k\Omega while in the OFF state we can assume the SET to be a complete insulator (neglecting any capacitative connection between the source and the drain via the SET). While it is possible to determine the state of the SET by measuring the response to an input signal through the source, it is faster to do so using RF reflectometry to measure both the amplitude and phase of the reflected signal, i.e. determined the reflectance Γ\Gamma. The change in reflectance due to switching of an SET between ON and OFF states is ΔΓ=∣ΓON−ΓOFF∣,(1)\Delta \Gamma = |\Gamma_{ON} - \Gamma_{OFF}|, \quad (1) where ΓON\Gamma_{ON} and ΓOFF\Gamma_{OFF} are the reflectances in two different states. Circuit diagram of transmission cable of impedance Z0Z_0 connected to an SET. E.1 Find the change in reflectance ΔΓ\Delta \Gamma between the conductive and insulating states for a typical SET connected to a co-axial cable with impedance of 50Ω50\Omega. 0.2 pt In order to increase the change in reflectance, and hence the sensitivity of the RF reflectometry, the circuit is modified by inclusion of an inductor. The intrinsic capacitance due to the device geometry C0≈0.4pFC_0 \approx 0.4 pF is also taken into account. The RF reflectometry is conducted using a signal of angular frequency ωrf\omega_{rf}. Modified SET circuit. E.2 Estimate the value of the inductance L0L_0 that can result in the change in reflection on the order of one. Calculate your estimate for L0L_0 numerically for ωrf/(2π)=100MHz\omega_{rf}/(2\pi) = 100 MHz and compute the corresponding ΔΓ\Delta \Gamma. 0.8 pt Part F: Charge sensing with a single lead quantum dot (1.5 points) For a scalable quantum computing architecture, the number of wires reaching each individual quantum bit need to be minimized. A promising alternative to an SET for charge sensing in silicon quantum computing is a Single Lead Quantum Dot (SLQD). In many ways it is similar to an SET, but does not have the source and drain leads. The gate is the only electrode, through which the electron energy states of the quantum dot are controlled and also through which RF reflectometry is conducted. Like an SET, a SLQD has an OFF in which the SLQD behaves as a total insulator. In contrast to an SET, the ON state of the SLQD is capacitive, with capacitance CqC_q. In order to maximize the difference in reflectance ΔΓ\Delta \Gamma of the SLQD, the following circuit is constructed. The parasitic capacitance C0≈0.4pFC_0 \approx 0.4 pF is fixed by circuit geometry, but the value of L0L_0 and the operating frequency can be changed to optimize the performance. The characteristic impedance of the transmission line is Z0=50ΩZ_0 = 50\Omega. Circuit diagram of the SLQD readout circuit connected to the transmission line. F.1 Suggest ωrf\omega_{rf} and ZC=L0/C0Z_C = \sqrt{L_0/C_0} that allow ΔΓ∼1\Delta \Gamma \sim 1 for given C0C_0 and CqC_q. 1.0 pt Optimal values of L0L_0 are relatively large and not always technically feasible. Hence, other types of circuit elements may be needed to improve sensitivity of the reflectometry readout circuit. F.2 Assume that L0L_0 (and hence ZCZ_C) is fixed. Draw a circuit diagram showing where to place an additional element in the SLQD readout circuit and specify the parameter(s) of this element such that ΔΓ∼1\Delta \Gamma \sim 1 can still be achieved without requiring a large inductance. 0.5 pt

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