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Photon Induced Decoherence of a Transmon Superconducting Charge Qubit an MPHYS Project Report

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Photon Induced Decoherence of a Transmon Superconducting Charge Qubit an MPHYS Project Report Photon Induced Decoherence of a Transmon Superconducting Charge Qubit An MPHYS project report Supervisor: Dr. Peter Leek Candidate number: 154180 Project number: CMP1502 Word count: 4793 Contents 1 Introduction 1 2 Theoretical background 2 2.1 Cavity field quantization . 2 2.2 Coherent cavity excitations . 2 2.3 Interaction with a qubit . 3 2.4 The Jaynes-Cummings Hamiltonian . 3 2.5 Interpreting the JC Hamiltonian . 3 2.6 The effect of cavity photons on T2 times . 3 2.7 The transmon qubit . 4 3 Experimental setup 4 3.1 Refrigeration . 4 3.2 Readout . 5 3.3 Device parameters . 5 4 Measurements 5 4.1 A note on errors . 5 4.2 Measuring !q, !c ............................................... 6 2g2 4.3 Measuring the cavity shift ∆ ........................................ 6 4.4 Measuring the qubit shift . 7 4.5 Limits on the effective cavity temperature . 8 4.6 Measuring the effective qubit temperature . 8 4.7 Measuring the T2 time of the qubit . 9 4.8 The effects of cavity photons on qubit decoherence times . 10 5 Conclusion 10 A Appendix 11 A.1 The coherent cavity states . 11 A.2 Transmon mechanics . 12 A.3 Signal Processing . 13 A.3.1 Signal generation . 13 A.3.2 Signal measurement . 13 A.4 Lorentzian broadening . 14 Abstract As well as having long enough coherence times, it is desirable for a candidate architecture for quantum com- The properties of a transmon qubit inside a 3D supercon- putation to be scalable in the number of qubits, without ducting microwave cavity are investigated in the disper- sacrificing control fidelity or the ability to prepare them sive limit. It is found that the behaviour of the system in an ensemble ground state. One promising field for maps well to a two-state quantum bit (qubit) modelled by improved qubit designs is the study of superconducting the Jaynes-Cummings Hamiltonian. The effective tem- circuits. Because these are solid state systems they lend perature of the qubit is measured as 138:5±20mK, corre- themselves to scalability in much the same way as mod- sponding to an average occupancy of state j1i of 10±4%. ern electronics. These superconducting systems use the An upper limit of 144:5mK is placed on the effective charge states of a small superconducting island as the −10 cavity temperature for low power (10 W) experiments, qubit states, in what is known as a charge qubit. Charge corresponding to a single photon occupancy of less than qubits were first suggested in 1997 [3] and significant work 3.1%. The effects of decoherence are investigated both at has since been done on improving their T2 coherence times base temperature and when photons are artificially put [2]. into the cavity. The T2 decoherence times of the qubit are found to be on the order of µs when the cavity con- In this study we consider a particular type of charge tained an average of ∼ 0:1 photons. The T2 times are qubit known as a transmon [11] coupled to a 3D super- found to decrease with the inverse photon number in a conducting microwave cavity. Such systems have been dramatic fashion, consistent with theoretical predictions. demonstrated to be capable of T2 times of ∼100µs [4], It is found that for the device investigated an average approaching the regime where the T2 time is sufficiently number of photons of 0.36 is sufficient to decrease the long compared to the gate operation time (103 −104 times T2 time to the sub-microsecond regime and that in low the gate operation time, corresponding to T2 ∼ 20−200µs power experiments we have a potential T2 time ∼ 8µs. for typical systems [4] [13]). 1 Introduction An important source of decoherence is the presence of photons in the cavity, either as a result of thermal excita- The search for ever better implementations of two-state tion or from the measurement signals themselves. As well quantum systems (or quantum bits) has been a quickly as this, thermal excitations of the qubit itself contribute growing area of research in many fields of physics. The significantly to decoherence. We investigate the effects of ability to accurately control a collection of such quantum photons in the cavity on the coherence of the qubit system systems is essential to building a quantum computer, a and put limits on the effective temperature of the system device which carries out logical operations on qubit states components in an effort to better understand how signif- (we denote the single qubit states j0i and j1i). Any sys- icant each of these effects is. Low effective temperatures tem possessing a conserved observable quantity, with at are also of critical importance to the task of preparing least two sufficiently spaced eigenvalues, is a candidate the qubit in state j0i, which it is essential to be able to for implementing a qubit. As a result, realizations of do in order to carry out logical operations. qubits are possible in many areas of physics (for example successful implementations of qubits have used ion traps, Section 2 covers some of the theory behind the cou- NMR and photon polarization states [1]). pling of a qubit to a microwave cavity and explains how this enables measurement of the state of the system, some A good candidate for an architecture for quantum of the theory behind the transmon qubit design is also ex- computing has to fulfil many requirements. In order for plained. In section 3 the experimental setup that enables operations to be carried out successfully on a set of qubits, measurements is explained. Then in section 4 the results they must remain in a well-defined quantum state for the and analysis of experiments that characterize the proper- duration. The coupling of such systems to their envi- ties of the qubit-cavity system are presented including an ronment causes them to lose this state information in a investigation into the aforementioned decoherence effects. process known as quantum decoherence. An important measure of the decoherence of a qubit is the T2 coherence time. The T2 time is the characteristic timescale on which the relative phase information for the state a j0i + b j1i is lost (i.e the timescale beyond which the phase difference between the complex numbers a and b is not well known). 1 2 Theoretical background quantities to operators (q ! q^,_q ! p^) and introducing the annihilation and creation operators: In order to model the interaction of a qubit with a mi- − 1 crowave cavity, we derive an appropriate Hamiltonian a^ = (2~!c) 2 (!cq^ + ip^) (6) (adapting a derivation given in [5]). The first stage is y − 1 a^ = (2 ! ) 2 (! q^ − ip^) (7) to derive the Hamiltonian describing quantized electro- ~ c c magnetic excitations of the cavity (without a qubit). We obtain an expression for the free cavity Hamiltonian: 1 2.1 Cavity field quantization H^ = ! (^aya^ + ): (8) c ~ c 2 We begin with the case of a radiation field confined to Where the operatora ^ya^ ≡ n^ is the number operator for a one-dimensional cavity with perfectly conducting walls photons in the cavity, each having energy ! . The elec- at z = 0 and z = L. ~ c tric and magnetic field operators then become: 1 2 ~!c y E^x(z; t) = (^a +a ^ ) sin(kz) (9) 0V 1 µ !3 2 B^ (z; t) = 0 0~ c i(^ay − a^) sin(kz) (10) y k V Figure 1: A cavity 2.2 Coherent cavity excitations We solve the (sourceless) Maxwell's equations within the cavity subject to the boundary condition that the A natural way to think about exitations of a cavity is in electric field must vanish at the cavity walls: terms of the photon number states jni defined byn ^ jni = n jni. The states jni are clearly energy eigenstates with ~ ~ 1 E(z = 0;L) = 0 (1) energies En = ~!c(n + 2 ). However from equation 9 we see that Assuming an electric field polarized along the x axis, we hnj E^x(z; t) jni = 0: (11) obtain the single-mode solution These states do not obey the correspondence principle, as 1 in the classical limit we require hE^ i to be given by equa- 2!2 2 x E (z; t) = c q(t) sin(kz): (2) tion 2. This classical limit corresponds to the operators x V 0 a;^ a^y being replaced by some continuous variables α; α∗ (i.e. the reverse of what we did in section 2.1). From Where wc is the frequency of the mode, k = wc=c is the corresponding wavenumber and V is the effective volume equations 9 and 10 we see that this replacement is nec- of the cavity. q(t) is some arbitrary time dependent func- essary to produce a classical field. One way to make this tion with the dimensions of length which will play the replacement is to seek states jαi such thata ^ jαi = α jαi, role of a canonical position. The magnetic field is then i.e. eigenstates of the annihilation operator. We call these polarized along y: states `coherent' states. It can be shown that (see ap- pendix A.1): 1 2 2 µ00 2!c 1 n By(z; t) = q_(t) cos(kz) (3) − 1 jαj2 X α k V 0 jαi = e 2 p jni (12) n=0 n! Note the appearance ofq _, which will play the role of a 1 ~!c 2 canonical momentum. The field energy is given by hαj E^x jαi = 2Re[α] sin(kz) (13) 0V Z 2 1 3 2 1 2 hαj n^ jαi ≡ n¯ = jαj (14) H = d ~r 0Ex(~r; t) + By (~r; t) : (4) 2 µ0 Equation 13 shows that the states jαi indeed correspond It is straightforward to show that to classical excitations of the cavity, equation 14 is the expression for the expected number of photons in the cav- 1 H = (_q2 + w2q2): (5) ity for a particular coherent state.
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