Understanding and Suppressing Dephasing Noise in Semiconductor Qubits

Understanding and suppressing dephasing noise in semiconductor qubits

Félix Beaudoin

Department of Physics McGill University, Montréal

July 2016

A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics ©2016 – Félix Beaudoin all rights reserved. Thesis advisor: Professor William A. Coish Félix Beaudoin

Understanding and suppressing dephasing noise in semiconductor qubits

Abstract

Magnetic-field gradients and microwave resonators are promising tools to realize a scalable quantum-computing architecture with spin qubits. Indeed, magnetic-field gradients allow fast selective manipulation of distinct qubits through electric-dipole spin resonance and coherent coupling of spin qubits to a microwave resonator. On the other hand, microwave resonators are useful for quantum state transfer and two-qubit gates between distant qubits, and qubit readout. In this thesis, we take a theoretical approach to understand and suppress pure-dephasing mechanisms relevant to spin qubits in the presence of the above-mentioned devices, recently introduced to improve scalability. We first focus on dephasing of a spin qubit in the presence of a magnetic-field gradient. We predict that hyperfine coupling of the qubit to an environment of nuclear spins precessing under the influence of a magnetic-field gradient leads to a new qubit dephasing mechanism. We show that in realistic conditions, this new mechanism can dominate over the usual dephasing processes occurring in the absence of a gradient. This result is relevant to spin qubits in GaAs or silicon quantum dots, or at single phosphorus donors in silicon. A magnetic-field gradient may also expose spin qubits to charge noise. We thus also study microscopic charge dephasing mechanisms coming from two-level fluctuators. These mechanisms α typically lead to qubit coherence decay of the form exp[−(t/T2) ]. Focusing on processes coupling charge fluctuators to electron or phonon baths, we find distinct dependencies of T2 and α on temperature depending on the nature of the fluctuator-bath interaction. These predictions may be useful for experimental identification of physical processes leading to charge dephasing of semiconductor qubits, and offer a new perspective to better understand the results of a recent experiment [Dial et al. Phys. Rev. Lett. 110:146804 (2013)]. Finally, we develop and assess a new protocol for quantum state transfer between a qubit and a resonator that has a high fidelity even in the presence of strong dephasing from low-frequency noise caused, e.g., by nuclear-spin or charge noise. In addition, upon a small modification of our state-transfer protocol, we obtain a method for fast quantum nondemolition readout of a qubit through the resonator output field. This new approach leads to a high-fidelity readout even when resonator damping is stronger than the qubit-resonator coupling. These two improved quantum operations (state transfer and readout) are particularly relevant for spin qubits coupled to microwave resonators, since spin-resonator coupling is typically weaker than qubit dephasing and resonator damping.

i Thesis advisor: Professor William A. Coish Félix Beaudoin

Understanding and suppressing dephasing noise in semiconductor qubits

résumé

Les gradients de champ magnétique et les résonateurs micro-ondes sont des outils prometteurs pour la réalisation d’une architecture extensible de calcul quantique fondée sur les qubits de spin. En effet, les gradients de champ magnétique permettent la manipulation rapide et sélective de qubits distincts grâce à la résonance dipolaire électrique de spin, ainsi que le couplage cohérent à un résonateur micro-ondes. Pour leur part, les résonateurs micro-ondes sont utiles pour réaliser des transferts d’états quantiques et des portes logiques entre qubits éloignés, ainsi que pour lire l’état quantique des qubits. Dans cette thèse, on entreprend une approche théorique pour comprendre et réduire le déphasage des qubits de spin en présence des outils mentionnés ci-dessus, favorables à l’extensibilité. On s’intéresse d’abord au déphasage d’un qubit de spin en présence d’un gradient de champ magnétique. On prédit que le couplage hyperfin du qubit à un environnement de spins nucléaires en précession sous l’effet du gradient mène à un nouveau mécanisme de déphasage. Dans des conditions réalistes, ce mécanisme peut dominer les processus de déphasage habituels qui surviennent sans le gradient. Ce résultat s’applique aux qubits de spin dans des boîtes quantiques en GaAs ou en silicium, ainsi qu’aux qubits de spin dans des donneurs uniques de phosphore dans le silicium. Un gradient de champ magnétique peut également exposer les qubits de spin au bruit de charge. On s’intéresse donc aux mécanismes microscopiques de déphasage de charge provenant de fluctuateurs à deux niveaux. Ces mécanismes mènent typiquement à un amortissement de la cohérence du α qubit de la forme exp[−(t/T2) ]. En se concentrant sur les processus couplant les fluctuateurs de charge à des bains d’électrons ou de phonons, on trouve des dépendances en température pour T2 et α qui se distinguent selon la nature de l’interaction fluctuateur-bain. Ces prédictions pourraient être utiles à l’identification expérimentale des processus physiques menant au déphasage de charge dans les qubits semiconducteurs, et offrent une nouvelle perspective pour mieux comprendre les résultats d’une expérience récente [Dial et al. Phys. Rev. Lett. 110:146804 (2013)]. Enfin, on développe et on caractérise un protocole pour réaliser un transfert d’états quantiquesde haute fidélité entre un résonateur et un qubit même en présence d’un fort déphasage provenant de bruit à basse fréquence causé, par exemple, par des spins nucléaires ou du bruit de charge. Grâce à une légère modification du protocole de transfert d’états proposé, on obtient de surcroît une méthode de lecture rapide et non destructive d’un qubit à travers le champ sortant du résonateur auquel il est couplé. Cette approche mène à une lecture haute fidélité même lorsque l’amortissement du résonateur est plus fort que son couplage au qubit. Ces deux opérations quantiques améliorées (le transfert d’états et la lecture) conviennent particulièrement aux qubits de spin couplés à un résonateur micro-ondes, puisque les couplages spin-résonateur sont typiquement faibles par rapport au déphasage du qubit et à l’amortissement du résonateur.

ii Statement of Originality

The author declares that the following elements of this thesis constitute original scholarship and an advancement of knowledge:

• The analytical expressions for the qubit coherence factor in the presence of hyperfine coupling to a nuclear-spin environment exposed to an inhomogeneous magnetic field. All the specific predictions resulting from these expressions for realistic devices (quantum dots in GaAs or silicon, phosphorus donors in silicon).

• The expression for the critical longitudinal magnetic field above which the motional-averaging regime is reached, again due to the presence of an inhomogeneous magnetic field. The expression for the critical transverse magnetic-field gradient beyond which dephasing ofa spin qubit in a single quantum dot becomes Markovian.

• The prediction that, for nuclear spins in an ideal narrowed state, a magnetic-field gradient can cause dephasing of a spin qubit in a GaAs or silicon quantum dot that is faster than dephasing due to the usual flip-flop and dipolar mechanisms.

• The prediction that, in a realistic setting, a magnetic-field gradient can reduce the Hahn-echo dephasing time by almost an order of magnitude for a spin qubit in a GaAs double quantum dot.

• The prediction of a Markovian regime for dephasing of a spin qubit in a silicon single quantum dot due to nuclear spins in an inhomogeneous magnetic field. The prediction of a breakdown of the Gaussian approximation for dephasing of a spin qubit at a single phosphorus donor impurity in silicon, again in the presence of a magnetic-field gradient.

• The result that the coherence factor of a qubit undergoing charge dephasing due to Gaussian noise from two-level fluctuators approximately takes the form of a compressed exponential α exp[−(t/T2) ], and the evaluation of the error that follows from this approximation. The universal relation between α for Hahn echo and α for free-induction decay in the fast-noise regime.

• The analytical temperature dependencies (in the slow-noise and fast-noise regimes) of T2 and α for charge dephasing of a qubit due to fluctuators whose switching dynamics is caused by any of the following mechanisms: direct tunneling and cotunneling with an electron reservoir, and direct, sum, and Raman phonon emission and absorption processes (through either piezoelectric or deformation coupling mechanisms).

iii • The Hamiltonian-engineering protocol (named SQUADD in the manuscript) that leads to high-fidelity quantum state transfer between a qubit and a resonator even under strong

dephasing due to inhomogeneous broadening. The exact and large-np analytical expressions

for the fidelity of the state transfer as a function of the number of decoupling π pulses np (neglecting cavity damping).

• The result that, under SQUADD, the dynamics of a state transfer between a resonator and × a collective mode of an ensemble of √N qubits is well approximated within a closed 4 4 subspace, up to corrections ∼ O(1/ N). The analytical expression that follows for the fidelity of the state transfer.

• The analytical expression for error due to deterministic pulse imperfections in a phase- alternated implementation of SQUADD.

• The idea of using Hamiltonian engineering to generate the longitudinal qubit-resonator interaction appeating in Chapter 4, which leads to a fast quantum nondemolition readout.

• The analytical expression for the signal-to-noise ratio of the above-mentioned readout in the presence of qubit switching. When the readout is obtained through Hamiltonian engineering, the relation between this signal-to-noise ratio and the cavity damping and pulse interval.

iv Contribution of co-authors

This manuscript is composed of three articles: two of them are already published [1, 2] while the third has been submitted for publication on June 24th, 2016 [3].

In the article presented in chapter 2:

Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient Félix Beaudoin and W. A. Coish Phys. Rev. B. 88, 085320 (2013). [1]

I performed all the calculations under the supervision of Prof. Coish. I also wrote the manuscript with the assistance of my supervisor.

In the article presented in chapter 3:

Microscopic models for charge-noise-induced dephasing of solid-state qubits Félix Beaudoin and W. A. Coish, Phys. Rev. B 91, 165432 (2015). [2]

I performed all the calculations under the supervision of Prof. Coish. I also wrote the manuscript with the assistance of my supervisor.

In the article presented in chapter 4:

Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED Félix Beaudoin, Alexandre Blais, and W. A. Coish, submitted for publication (June 24th, 2016) [3]

I performed all the calculations under the supervision of Prof. Coish. I also wrote the manuscript with the assistance of my supervisor. Prof. Blais provided guidance for the cavity-QED aspects of this work, especially concerning qubit readout. He also assisted us through critical reading of the manuscript.

v Acknowledgments

My first words of acknowledgments unquestionably go to my supervisor, Prof. William A. Coish, who helped me throughout the completion of this work. Bill has a phenomenal physical intuition that is always enlightening. He also took a lot of time for critical reading of my manuscripts which, I think, always led to significantly improved work. In addition, Bill also gave me a lot of freedom, leaving me time to finish interesting work that I had started during my master’s degree with Alexandre Blais and collaborators at BBN Raytheon Technologies and the University of Syracuse. I would also like to thank my Ph. D. advisory committee members, Jack Sankey and Tami Pereg- Barnea. The research presented here also benefited from immensely appreciated collaborators. I would first like to mention Alexandre Blais, whose criticisms and intuitions related with cavity QEDwere of significant help in realizing the research presented in Chapter 4. Alexandre also generously received Bill and I in fruitful and enjoyable visits at Université de Sherbrooke. The research presented in this thesis also hugely benefited from Michel Pioro-Ladrière, with whom we had decisive discussions that led us to investigate the coupling of single spins to microwave resonators. Michel’s students, in particular Dany Lachance-Quirion, also assisted me enormously regarding realistic parameters and possible experimental applications of some of the work presented here. I would also like to thank Xiaoya (Judy) Wang, Nicolas Didier, and Benjamin d’Anjou for helpful discussions about the various aspects of my research that interesected with their own work. My life as a Ph. D. student involved a lot of work, but always in good company. I would like to thank my former office colleagues during the first year of my Ph. D. in R-421. Specifically, I would like to thank Vadim Nemytov, Ahmed Khorshid, Mohammed Harb, and Hadi Razavipour for their good mood and pleasant conversation. J’aimerais également remercier mes collègues de bureau durant les dernières années de mon doctorat : Vincent Michaud-Rioux, Antoine Roy-Gobeil, Lauren Gingras, Benjamin d’Anjou, Zeno Schumacher et Marc-Antoine Lemonde. Votre présence était en soi une excellente raison de se rendre au bureau tous les jours. Par ailleurs, travailler ne se fait pas sans manger, et manger perd de son intérêt sans la présence d’amis avec qui plaisanter en dégustant du fromage, des ouiches lorraines ou de la bouffe chinoise. Je parle ici évidemment de Jean-René Souquet, Alexandre Baksic, Nicolas Didier, Patrick Hofer, Olivier Landon-Cardinal, Luke Gobia... Merci pour votre conversation qui permettait de se reposer de la physique, parfois pour mieux y revenir ! J’aimerais aussi mentionner ma famille : mes frères Didier et Charles, mes parents Mireille et

vi Guy, mes grands-parents Monique et Gilles, ainsi que ma chère grand-maman Margot. Je vous remercie chaleureusement pour tout ce que vous m’avez toujours donné. Je suis privilégié d’avoir grandi parmi vous. Enfin, le plus beau merci du monde à Brigitte.

vii Contents

1 Introduction 1 1.1 Pure dephasing and dynamical decoupling ...... 4 1.2 Magnus expansion ...... 11 1.3 Outline of this thesis ...... 21

2 Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient 27 2.1 Introduction ...... 28 2.2 Hamiltonian and exact solution ...... 31 2.3 Simplified coherence factor ...... 34 2.4 Coherence measurement protocols ...... 39 2.5 Physical realizations ...... 43 2.6 Conclusions ...... 51

3 Microscopic models for charge-noise-induced dephasing of solid-state qubits 53 3.1 Introduction ...... 54 3.2 Two-level fluctuators ...... 56 3.3 Functional form of the coherence factor ...... 60 3.4 Electron baths ...... 65 3.5 Phonon baths ...... 69 3.6 Coherence time and stretching parameter from microscopic models ...... 75 3.7 Conclusions ...... 83

4 Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED 84 4.1 Introduction ...... 85 4.2 Hamiltonian engineering ...... 86 4.3 Qubit-cavity state transfer ...... 88 4.4 Collective modes in qubit ensembles ...... 91 4.5 Pulse errors ...... 97

viii 4.6 Qubit readout ...... 98 4.7 Conclusions ...... 100

5 Conclusion 101

Appendix A Appendices of “Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient” 104 A.1 Hyperfine coupling constants in a double quantum dot ...... 104 A.2 Mapping to a singlet-triplet qubit ...... 106 A.3 Validity of the Magnus expansion ...... 107 A.4 Validity of the Gaussian approximation ...... 110 2 A.5 Σs for relevant geometries ...... 112

Appendix B Appendices of “Microscopic models for charge-noise-induced dephasing of solid-state qubits” 114 B.1 Corrections to the leading-order Magnus expansion and the Gaussian approximation 114 B.2 Electron-phonon coupling strength ...... 119 B.3 Fluctuator equilibration rate for the phonon sum process ...... 120

Appendix C Appendices of “Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED” 121 C.1 Evolution under SQUADD ...... 121 C.2 Finite bandwidth and counter-rotating terms ...... 123 C.3 Spectrum of the 4-dimensional Hamiltonian for state transfer to a collective mode in a qubit ensemble ...... 126 C.4 Readout signal-to-noise ratio ...... 127

References 134

ix 1 Introduction

Quantum superpositions are at the root of some of the most remarkable effects arising from quantum mechanics. In particular, quantum superpositions are essential for quantum parallelism and quantum interference which, together, lead to spectacular advantages of quantum computers over their classical counterparts [4]. These advantages are however offset by decoherence, the phe- nomenon through which quantum superpositions are turned into mundane statistical mixtures. A quantum system decoheres when its evolution is perturbed in an unpredictable and often irreversible manner by an external environment. The timescale over which this process occurs, called the coherence time, is determined by the microscopic nature of the interaction between the quantum system and its environment. Therefore, to suppress noise sources and minimize their negative effects, it is useful to possess an accurate microscopic description of the system-environment coupling.

Such a microscopic theory of system-environment coupling has led to several successes in the field of spin qubits in semiconductors [9]. Indeed, decoherence of spin qubits in GaAs [illustrated in Fig. 1.1(a)] is known to be predominantly due to hyperfine interaction with nuclear spins inthe host material [10–12]. Knowledge of the hyperfine-interaction and nuclear-spin Hamiltonians has led to microscopic calculations of the decoherence dynamics of spin qubits [13–17] whose predictions

1 (a) (b)

Figure 1.1 – Simplified sketches of semiconductor devices used to confine single electrons and manipulate their spin. (a) Double quantum dot in an AlGaAs/GaAs heterostructure [5]. A two-dimensional electron gas (2DEG, shown in blue) is formed near the AlGaAs/GaAs interface. Electrostatic gates (shown in yellow) are used to deplete the population of conduction electrons in the areas of the 2DEG below the dashed circles. By applying an appropriate voltage configuration, a single electron (blue hemisphere) can be allowed to hop in these otherwise-forbidden areas of the 2DEG below the dashed circle. The spin of this electron can subsequently be manipulated. Similar quantum dots have also been realized using silicon as the host material [6, 7]. (b) Single electron at a phosphorus donor in silicon [8]. The phosphorus donor (blue dot) provides a potential that can tightly confine a single electron (pale blue hemisphere). A nearby single-electron transistor (yellow) is used to let single electrons tunnel in or out of the bound state at the phosphorus donor. In (a) and (b), the electron appears on the surface for visibility reasons. However, in practice, the electron is confined (a) in the 2DEG below the dashed circles or (b) at a phosphorus donor implanted wellbelowthe silicon surface.

have often been successfully tested by experiment [18, 19]. Since the dynamics of the nuclear-spin

environment are typically much slower than those of the qubit, echo-based dynamical decoupling

sequences have been used to drastically prolong the coherence time of spin qubits [19]. To further

prolong coherence times, spin qubits have been realized in silicon [see Fig. 1.1(b)], which contains

fewer nuclear spins [20].* This small number of nuclear spins can be reduced even more through

isotopic purification. As shown in Table 1.1, reducing the fraction of spinful nuclei has increased

qubit coherence times by orders of magnitude relative to GaAs.

Coherence times much longer than single-qubit operation times have thus been achieved using

spin qubits. Although this is an important step toward fault-tolerant quantum computing, scala-

bility will also become a challenge in the long term. A useful tool in that respect is a magnetic-field

gradient generated with a micromagnet [25]. Indeed, such an inhomogeneous magnetic field allows distinct spin qubits to have different Zeeman splittings, opening the door to selective driving of distinct qubits through electric-dipole spin resonance. Magnetic-field gradients have also been theoretically proposed [26, 27] and recently applied [28] to achieve coherent coupling between a

*The only spinful isotope of silicon, 29Si, has a natural abundance of 4.7 %. In contrast, all the isotopes of gallium and arsenic carry spin.

2 ∗ echo CPMG Type of spin qubit T2 T2 T2 Reference e− spin in GaAs quantum dot 10 ns 30 µs 200 µs [19] e− spin in Si quantum dot (natural) 300 ns – – [6] e− spin in Si quantum dot (purified) 100 µs 1 ms 30 ms [7] e− spin at P donor in Si (natural) 50 ns 200 µs – [8] e− spin at P donor in Si (purified) 200 µs 1 ms 500 ms [21]

Table 1.1 – Coherence times typically achieved in varieties of spin qubits that are relevant to this work. Coherence times are measured for a qubit that is subject to common echo-based dynamical decoupling ∗ echo sequences. T2 : free-induction decay time, i.e., coherence time in the absence of refocusing pulses. T2 : CPMG coherence time including a single refocusing pulse. T2 : coherence time under the Carr-Purcell-Meiboom- Gill sequence [22, 23], which contains N refocusing pulses applied with a constant time interval. In Ref. [19], refocusing pulses are realized on a timescale ∼ 1 ns using the exchange interaction present in singlet-triplet qubits. In Refs. [7, 8, 21], refocusing pulses are realized using standard electric-dipole spin resonance, which can generate qubit rotations on a timescale ∼ 100 ns [8]. Using a magnetic-field gradient generated with a micromagnet, this manipulation time may be reduced to ∼ 1 ns [24]. single spin qubit and a microwave resonator. Coupling to a resonator is useful to shuttle quantum states between distant qubits and to mediate two-qubit interactions through virtual photons [29]. In addition, the output signal of a driven resonator can be monitored to achieve qubit readout [30, 31].

Despite its clear advantages for scalability, the introduction of an inhomogeneous magnetic field may also convey important drawbacks. Indeed, an inhomogeneous magnetic field may introduce unwanted nuclear-spin dynamics [32] which, in turn, may impact on qubit coherence. In addition, a magnetic-field gradient may generate important correlation between spin and charge degrees-of- freedom, exposing the spin qubit to charge noise [33–37]. This additional charge noise may reduce the fidelity of quantum operations involving the qubit and, possibly, a microwave resonator [28].

Crucially, charge noise has been shown experimentally to be a significant source of decoherence in both GaAs [38] and silicon [39]. It is thus important to understand the dynamics of decoherence caused by the above noise sources (magnetic-field inhomogeneities and charge noise), which are increasingly relevant in new devices with improved scalability. In addition, quantum operations involving a spin qubit and a resonator should be made robust to the low-frequency noise arising from nuclear-spin and electric-field fluctuations. These two goals – understanding and suppressing decoherence sources relevant to the novel devices described here – will thus constitute the subject of this thesis.

3 1.1 Pure dephasing and dynamical decoupling

The decoherence mechanisms relevant to this thesis will mostly involve pure dephasing. In this pure- dephasing scenario, the qubit-environment interaction suppresses the qubit coherences (off-diagonal density-matrix elements of the qubit), but preserves the qubit populations (diagonal density-matrix elements of the qubit). Pure dephasing occurs when the qubit-environment coupling Hamiltonian is longitudinal with respect to the eigenbasis of the qubit. Under quite standard assumptions which will be detailed here, simple expressions then relate the dynamics of qubit coherence to the evolution of the environment for a given qubit control sequence [40–43]. When the dynamics of the environment are well established from a microscopic Hamiltonian (as in the case of a nuclear-spin environment), this allows to make strong testable predictions about the qubit dephasing dynamics [1]. When the physical mechanisms driving the evolution of the environment are more nebulous

(as in the case of charge noise), distinct models may lead to different predictions for the qubit- dephasing dynamics, allowing to distinguish between them [2].

1.1.1 Dephasing from Gaussian noise

We consider a qubit coupled to an environment through the Hamiltonian (taking ℏ = 1)

1 H (t) = ξ(t)σ , (1.1) QE 2 z

where σz is the Pauli matrix whose eigenstates are those of the free qubit Hamiltonian, HQ = ωqσz/2, with ωq the qubit splitting. In the simplest case, ξ(t) is a random function of time, thus describing classical noise. However, ξ(t) may also be a non-commuting operator evolving in the interaction

iH t −iH t picture associated with the free Hamiltonian HE of the environment, ξ(t) = e E ξ(0)e E . In addition to the qubit-environment coupling, we introduce the control Hamiltonian Hc(t), which describes refocusing pulses applied on the qubit. The full Hamiltonian is then, in the interaction picture that elimates the free qubit and environment Hamiltonians,

H(t) = HQE(t) + Hc(t). (1.2)

4 1 ) t

( 0 s -1

t1 t2 t3 t

Figure 1.2 – Sign function s(t) for a hypothetical echo-based dynamical decoupling sequence in the limit of instantaneous π pulses. The π pulses are applied at times {t1, t2, t3, ...}.

It is useful to move to the toggling frame using the time-dependent unitary transformation

[ ∫ ] t UT(t) = T exp −i dt1Hc(t1) . (1.3) 0

The toggling frame allows to incorporate Hc(t) into the transformed Hamiltonian

† ˙ † HT(t) = UT(t)H(t)UT(t) + iUT(t)UT(t). (1.4)

Assuming that Hc(t) generates a train of instantaneous π rotations of the qubit state around an axis orthogonal to z on the Bloch sphere (thus producing σz → −σz) at times tj, j ∈ N, Eq. (1.4) becomes

1 H (t) = s(t)ξ(t)σ . (1.5) T 2 z

In Eq. (1.5), s(t) is the sign function that alternates between s(t) = 1 at times tj, as illustrated in Fig. 1.2.

As mentioned above, pure dephasing will lead to decay of the qubit coherences, motivating the following definition of a qubit coherence factor:

⟨σ+(t)⟩ C(t) = , σ+ ≡ σx + iσy, (1.6) ⟨σ+(0)⟩ where the expectation value is taken with respect to the initial state of the qubit-environment system. The expectation value may also include an ensemble average, for example when ξ(t) changes from one repetition of an experiment to the next. To evaluate σ+(t), we assume that

5 the time-evolution operator is well approximated by the zeroth-order term of the Magnus expansion corresponding to HT(t). The Magnus expansion and general conditions under which corrections to its zeroth-order term can be neglected will be the topic of Section 1.2. Assuming that these requirements are fulfilled, σ+(t) then takes the form expected for classical noise, for which

[HT(t1),HT(t2)] = 0 ∀ t1, t2,

[ ∫ ] [ ∫ ] i t i t σ+(t) = exp dt1s(t1)ξ(t1)σz σ+(0) exp − dt1s(t1)ξ(t1)σz (1.7) 2 0 2 0 L(t) = e σ+. (1.8)

Equation (1.8) involves the Liouvillian

∫ i t L(t)· = dt1s(t1)ξ(t1)[σz, ·] , (1.9) 2 0 where the centerdot (“·”) represents the relevant operator upon which L(t) is applied. Since

[σz, σ+] = 2σ+, the exponential in Eq. (1.8) can be Taylor-expanded and applied on σ+ term by term to give

iϕ(t) σ+(t) = e σ+, (1.10) where ϕ(t) is the phase accumulated by the qubit due to interaction with its environment

∫ t ϕ(t) = dt1s(t1)ξ(t1). (1.11) 0

Substituting Eq. (1.10) into Eq. (1.6) then gives, for a separable qubit-environment initial state,

C(t) = ⟨eiϕ(t)⟩. (1.12)

From Eq. (1.12), decay of the coherence factor arises from fluctuations in the accumulated phase

ϕ(t) due to the dynamics of the environment.

6 To make analytical progress, we can express C(t) in terms of the cumulant expansion

C(t) = exp [K(x, t)] x=1, (1.13) where we have introduced the cumulant-generating function

∞ ∑ xn K(x, t) = log⟨ex iϕ(t)⟩ ≡ K (t) . (1.14) n n! n=1

The n-th order cumulant is then given by

n n d d ⟨ x iϕ(t)⟩ Kn(t) = n K(x, t) = n log e . (1.15) dx x=0 dx x=0

There are many relevant physical situations in which all cumulants of order n > 2 will vanish in the above expansion. The noise ξ(t) is then said to be Gaussian. When, in addition, ⟨ξ(t)⟩ = 0, the

2 only non-vanishing cumulant is K2(t) = −⟨ϕ (t)⟩. Subtituting this into Eq. (1.14), and substituting the resulting expression into Eq. (1.13) then gives [40–43]

[ ∫ ∫ ] 1 t t C(t) = exp − dt1 dt2 s(t1)s(t2)⟨ξ(t1)ξ(t2)⟩ , (1.16) 2 0 0 where we have used the definition of ϕ(t), Eq. (1.11).

We now descibe common physical situations in which the above expression for Gaussian noise is appropriate. Gaussian noise arises naturally using the following spin-boson model [44], in which we consider a qubit coupled linearly to a bath of bosonic modes through the Hamiltonian [45, 46]

∑ ∑ [ ] ω † 1 † H(t) = Q σ + Ω b b + g b + g∗b σ + H (t), (1.17) 2 z k k k 2 k k k k z c k k

(†) where ωQ is the qubit splitting, and Ωk, bk , and gk are the frequency, annihilation (creation) operator, and coupling strength of bosonic mode k, respectively. In Eq. (1.17), the coupling between the qubit and the bath is purely longitudinal (commuting with the free Hamiltonian of the qubit).

The above model thus provides a simple testbed for dynamical decoupling sequences aiming to mitigate pure qubit dephasing [42, 46]. Indeed, moving to the frame that simultaneously eliminates

7 Hc(t) and the free Hamiltonians of the qubit and the bosonic modes leads to the qubit-environment coupling Hamiltonian of Eq. (1.5), with the noise operator given by

∑ [ ] † −iΩkt ∗ iΩkt ξ(t) = gkbke + gkbke . (1.18) k

−H /k T −H /k T When the bosonic bath is initially in thermal equilibrium, ρE(0) = e B B /tr[e B B ], with ∑ † ⟨ m ⟩ HB = k Ωkbkbk, Wick’s theorem can be used to evaluate moments of the form ϕ (t) . It can then be shown that cumulants of order n > 2 vanish exactly in the expansion of Eq. (1.13), corresponding to Gaussian noise. However, away from thermal equilibrium, higher-order cumulants in Eq. (1.13) can become finite, and non-Gaussian noise may be expected [47]. Deviations from

Gaussian behavior may also be observed if the coupling between the qubit and the bath is nonlinear [47–53]. Nonlinear coupling between the qubit and the bath occurs, e.g., for qubits at an optimal working point† [50–52].

Non-Gaussian corrections can also be significant when the bath is formed by an ensemble of subsystems with a finite amount of energy levels (instead of harmonic oscillators). This situation corresponds to a qubit coupled to, e.g., nuclear spins [1] or two-level charge fluctuators [2]. When the bath is formed by a single two-level system [55–57], the qubit coherence factor can display pronounced non-Gaussian behavior characterized by plateau-like features [58]. These non-Gaussian corrections are especially strong for slow noise, i.e. when the noise correlation time is larger than the inverse noise amplitude [58]. In the opposite limit of large number of independent subsystems

(spins or fluctuators) that have the same statistics, the central limit theorem implies that ξ(t) is approximately Gaussian. In addition, it has been shown that non-Gaussian corrections are suppressed in the limit of large number of dynamical decoupling pulses [41, 59]. To quantify departure from the Gaussian approximation, it is often possible to evaluate terms of order greater than 2 in the cumulant expansion of Eq. (1.14)[41, 47, 50]. Such deviations from the Gaussian approximation will be considered in Chapters 2 and 3 for small numbers of nuclear spins or charge fluctuators.

For the rest of this Chapter, we will assume that the noise ξ(t) is Gaussian. In this situation,

†The equivalent term “sweet spot” can also be found in the literature in lieu of “optimal working point” [54].

8 we will see that Eq. (1.16) provides a simple relation between the qubit coherence factor and the autocorrelation function of ξ(t) which can be exploited, e.g., to probe the dynamics of the environment [60, 61].

1.1.2 Stationary noise, noise spectrum, and filter functions

Equation (1.16) can be written in an especially informative form under the assumption of stationary noise

⟨ξ(t1)ξ(t2)⟩ = ⟨ξ(t1 − t2)ξ(0)⟩. (1.19)

A physical situation in which Eq. (1.19) is satisfied occurs when the interaction-picture evolution of the noise operator ξ(t) is well approximated by the solution of a Lindblad master equation

ρ˙E(t) = LEρE(t), where the time-independent Liouvillian LE is independent of the qubit. This may describe, e.g., a situation in which the environment of the qubit is itself coupled to a larger bath with which it exchanges energy through a Markovian process.‡ The autocorrelation function of Eq. (1.19) is then evaluated using the standard formula for multitime averaging [62]   [ ]  L (t −t ) trE ξ e E 1 2 ξρE(t2) , t1 ≥ t2, ⟨ ⟩ ξ(t1)ξ(t2) =  [ ] (1.20)  L (t −t ) trE ξ e E 2 1 ρE(t1) ξ , t1 < t2,

L t where ξ ≡ ξ(0), ρE(t) = e E ρE(0), and where we follow the convention that LE acts on the product of all the operators on its right. When the environment is initially in a steady state of LE,

LEρE(0) = 0, then ρE(t1) = ρE(t2) = ρE(0). Equation (1.20) then immediately yields a time-local autocorrelation function, satisfying Eq. (1.19). Thus, quite intuitively, stationary noise is obtained when the density matrix of the environment is a constant of motion. This discussion will become especially relevant in the treatment of charge noise arising from two-level fluctuators, presented in

Chapter 3.

‡For example, the qubit may be interacting with environmental charge fluctuators which are themselves coupled to an electron or phonon reservoir, as in Chapter 3. Within the general formalism presented here, unitary evolution of the environment is retrieved when taking LE· = −i [HE, ·].

9 When ⟨ξ(t1)ξ(t2)⟩ is local in time, it is natural to introduce the noise spectrum

∫ ∫ 1 ∞ ∞ S(ω) ≡ dt eiωt⟨ξ(t)ξ(0)⟩ ⇒ ⟨ξ(t)ξ(0)⟩ = dω e−iωtS(ω). (1.21) 2π −∞ −∞

Substituting Eqs. (1.19) and (1.21) into Eq. (1.16) then gives [40–43]

[ ∫ ∞ ] − F (ωt) C(t) = exp dω S(ω) 2 , (1.22) −∞ ω where

∫ 2 t 2 ω iωt F (ωt) = dt s(t)e (1.23) 2 0 is the filter function. While the noise spectrum depends only on the evolution of the environment, the filter function is entirely determined by the pulse sequence applied on the qubit through s(t).

Expressions for F (ωt) using common dynamical decoupling sequences are given, e.g., in Ref. [41].

Following Eq. (1.22), F (ωt) acts as a filter for qubit dephasing, suppressing noise-spectrum contributions at frequencies at which F (ωt) vanishes. This is illustrated in Fig. 1.3, which displays a hypothetical noise spectrum as the solid red line and the filter functions for free-induction decay

(no π pulse) and Hahn echo (a single π pulse) as the dotted purple line and the dashed blue line, respectively. While the filter function for free-induction decay is largest at ω = 0, employing Hahn echo allows to filter out the zero-frequency contribution of the noise. A vanishing filter functionat

ω = 0 is a common feature that is also shared by many other useful dynamical decoupling schemes such as the Carr-Purcell [22] and Uhrig [42] sequences. These sequences are employed to make a qubit immune to quasistatic pure-dephasing noise, for which S(ω) ∝ δ(ω).

Another important feature of filter functions is that their weight concentrates more andmore around ω = 0 as t increases. Noise spectra are often approximately flat, S(ω) ≃ S0, for |ω| ≪ 1/τc, where τc is the noise correlation time. In the Markovian limit, t ≫ τc, Eq. (1.22) then leads to

[ ∫ ∞ ] ≃ − F (ωt) ≃ − C(t) exp S0 dω 2 exp[ πS0t], (1.24) −∞ ω

10 aso n ftecniin ne hc t edn re ufcst cuaeydsrb the describe toaccurately suffices ex- order Magnus leading the its of which discussion dynamics. under general system-environment brief conditions a the give of we and section, pansion present the In operator. evolution Section in discussion The 1.2 obtained been has spectrum noise multidimensional a involving [ recently formula general stationary more yet non-Gaussian a For approximation. noise, Gaussian the to due is spectrum noise and function under spectrum, flat a with environment an to here. coupled made qubit assumptions a the of time decoupling coherence dynamical using the of increase impossibility to the demonstrates it sequence, pulse byEq.( the of independent given form the of t functions filter all by obeyed rule sum the used have we where line: purple Dotted frequency. high logarithmic. at decay (no power-law decay and frequency zero at – 1.3 Figure ≫ ial,w ealta h ipiiyo q ( Eq. of simplicity the that recall we Finally, τ ansexpansion Magnus c orsod oepnnildcyo h ui oeec atr qain( Equation factor. coherence qubit the of decay exponential to corresponds , 47 π us) ahdbu line: blue Dashed pulse). ]. itrfntosadniesetu.Sldrdln:hpteia os pcrmwt peak a with spectrum noise hypothetical line: red Solid spectrum. noise and functions Filter

2 S(ω), F (ωt)/ω 1.1 eiso edn-re ansepnino h system-environment the of expansion Magnus leading-order a on relies F ( ωt ∫ ) −∞ /ω ∞ 2 ω dω o aneh asingle (a echo Hahn for ab units) (arb. F 11 ω ( ωt 1.22 2 ) = eaigtechrnefco otefilter the to factor coherence the relating ) t, π 1.23 [ ) 41 π .Tu,teMroinlimit, Markovian the Thus, ]. us) h cl fthe of scale The pulse). F ( ωt ) /ω 2 o free-induction for 1.24 y being ) (1.25) xsis axis 1.2.1 First few terms of the Magnus expansion and convergence

Finding the time-evolution operator that describes the dynamics of an arbitrary quantum system generally amounts to solving a differential equation of the form

V˙ (t) = A(t)V (t). (1.26)

For a general closed quantum system, we take A(t) = −iH(t), with H(t) the system Hamiltonian, and Eq. (1.26) then becomes Schrödinger’s equation for the unitary time-evolution operator V (t).

In addition, Eq. (1.26) can describe an open quantum system subject to Markovian relaxation and dephasing. In this situation, we take A(t) = L(t), with L(t) the Liouvillian of the system, and

Eq. (1.26) becomes the Linblad master equation for the propagator V (t) [63].

In both relevant situations (closed quantum system and Lindblad master equation), the solution of Eq. (1.26) may be expressed using the ansatz

V (t) = exp[Ω(t)]. (1.27)

Substituting Eq. (1.27) into Eq. (1.26), the problem becomes finding the solution of a differential equation for Ω(t). Magnus showed [64] that this problem may be solved by expanding Ω(t) in a series

∑∞ Ω(t) = Ωk(t), (1.28) k=0 whose first few terms are [65]

∫ t Ω0(t) = dt1 A(t1), (1.29) 0∫ ∫ 1 t t1 Ω1(t) = dt1 dt2 [A(t1),A(t2)] , (1.30) 2 0 0 ∫ ∫ ∫ { } 1 t t1 t2 Ω2(t) = dt1 dt2 dt3 [A(t1), [A(t2),A(t3)]] + [[A(t1),A(t2)] ,A(t3)] . (1.31) 6 0 0 0

12 Equation (1.28) constitutes the Magnus expansion. The inequalities

∫ ∫ t t

dt1A(t1) ≤ dt1∥A(t1)∥, ∥[A, B]∥ ≤ 2∥A∥∥B∥, (1.32) 0 0 valid for the 2-norm ∥ · ∥ of arbitrary operators A and B, lead to the following bound for each term of the Magnus expansion:

(∫ ) t k+1 ∥Ωk(t)∥ ≤ λk dt1∥A(t1)∥ . (1.33) 0

In Eq. (1.33), λk is a purely numerical prefactor associated with term k of the Magnus expansion. From Eq. (1.33), convergence of the Magnus expansion can be expected heuristically when ∫ t ∥ ∥ ≲ 0 dt1 A(t1) 1. More rigorously, it has been shown [66] that when V (0) = 1 and A(t) is a bounded operator,§ a sufficient (but not necessary) criterion for convergence of the Magnus expansion is

∫ t dt1∥A(t1)∥ < π. (1.34) 0 ∫ t ∥ ∥ ≪ This convergence will typically be very rapid when 0 dt1 A(t1) 1, i.e., for sufficiently short times relative to 1/ max ∥A(t)∥.

The convergence criterion expressed by Eq. (1.34) has the benefit of generality; however, it is also too stringent to be useful in many practical situations. For example, the full dynamics of qubit dephasing may involve coherence factors C(t) ≪ 1, which imply ∥Ω(t)∥ ≫ 1, typically leading to violation of the convergence criterion of Eq. (1.34). Therefore, it is often preferable to check for convergence using the specific form of A(t) and of the resulting expansion terms Ωk(t) for a given problem. This approach, which will be taken throughout this thesis, often gives a less stringent criterion than the general Eq. (1.34). In this thesis, the Magnus expansion will be employed to solve problems in which A(t) oscillates quickly on the timescale set by the amplitude of 1/∥A(t)∥. Due to the successive integrals appearing in Ωk(t), these oscillations will lead to correction terms that grow with time. Truncation of the Magnus expansion to leading order will then

§This approach also holds when A(t) = L(t) is a superoperator, since superoperators acting on N- dimensional operators can be written as N 2-dimensional operators in Liouville space [67].

13 be possible without significant introduction of error for times such that these growing corrections remain negligible [1, 2, 68].

1.2.2 Average liouvillian theory

When A(t) is periodic, A(t + T ) = A(t), the Magnus expansion can be written in a particularly useful form. For times that are multiples of the period T , t = mT with m ∈ N, substituting

Eq. (1.28) into Eq. (1.27) then gives ( ) ∑∞ V (t) = exp t Ωk , (t = mT ) (1.35) k=0 ∑ where the first few terms of the expansion k Ωk are ∫ 1 T Ω = dt A(t ), (1.36) 0 T 1 1 ∫0 ∫ 1 T t1 Ω1 = dt1 dt2 [A(t1),A(t2)] , (1.37) 2T 0 0 ∫ ∫ ∫ { } 1 T t1 t2 Ω2 = dt1 dt2 dt3 [A(t1), [A(t2),A(t3)]] + [[A(t1),A(t2)] ,A(t3)] . (1.38) 6T 0 0 0

Thus, for A(t) periodic, V (t) stroboscopically takes the form of the propagator for a time-independent ∑ Hamiltonian or Liouvillian given by k Ωk. In addition, the first term Ω0 of the above expansion is simply the time average of A(t), hence the name “average Liouvillian theory” [69]. For a closed quantum system with A(t) = −iH(t), the above approach reduces to average Hamiltonian theory [70, 71].

A major advantage of average Liouvillian theory lies in a less stringent convergence criterion relative to the usual Magnus expansion. Indeed, following Eqs. (1.36) to (1.38), terms of order k ∫ ∼ T ∥ ∥ k+1 typically grow like ( 0 dt1 A(t1) ) , i.e., their norm grows with the period T , which may be much shorter than t. A sufficient criterion for convergence is then

∫ T dt1∥A(t1)∥ < π, (1.39) 0 ∫ T ∥ ∥ ≪ and convergence will typically be very rapid when 0 dt1 A(t1) 1. When Eq. (1.39) is satisfied,

14 average Liouvillian theory leads to a convergent expansion even at very long times, t ≫ T .

Average Liouvillian theory is especially useful to the theoretical treatment of dynamical decoupling. Indeed, many common schemes like the Carr-Purcell sequence rely on the periodic application of π pulses, leading to a periodic Hamiltonian in the toggling frame. By carefully designing an appropriate sequence of qubit π pulses, unwanted interactions may be averaged out order by order using Eqs. (1.36) to (1.38)[72]. Equation (1.39) then implies that convergence of the expansion of Eq. (1.35) will be guaranteed for sufficiently short pulse interval relative to the inverse normof the Hamiltonians of the interactions to eliminate. This Hamiltonian-engineering approach will be taken in Chapter 4 to realize high-fidelity quantum operations involving a qubit and a resonator.

Fulfillment of Eq. (1.39) implies that convergence of average Liouvillian theory is guaranteed at all times; however, it does not necessarily imply that the first few terms of the expansion of

Eq. (1.35) suffice for accurate description of system dynamics. For example, evenif ∥Ω1∥ ≪ ∥Ω0∥,

∥Ω1∥t may still be of order unity if ∥Ω1∥t ≫ 1. This may lead to significant corrections to the zeroth-order expression for V (t), since Ω1t appears in the argument of an exponential. Therefore, when ∥Ω0∥t ≫ 1, special care must always be taken to make sure that correction terms do not lead to large error when truncating the expansion of average Liouvillian theory [3].

1.2.3 Example: motional averaging in the dynamics of a driven qubit under

periodic energy-splitting modulations

In this section, we illustrate the typical range of validity of average Hamiltonian theory by comparing its predictions with an exact approach using a simple concrete example. We consider a qubit with bare energy splitting ωq under the influence of a sinusoidal drive with strength ε and angular frequency ω. In addition, the qubit energy splitting is modulated over time, leading to the

Hamiltonian

ω + χ(t) H(t) = ε cos ωt σ + q σ , (1.40) x 2 z

15 +χ ) t

( 0 χ −χ

τ/2 3τ/2 5τ/2 7τ/2 9τ/2 t

Figure 1.4 – Periodic modulation of the qubit energy splitting over time. The qubit splitting toggles between ωq + χ and ωq − χ at times τ/2, 3τ/2, 5τ/2, ... where χ(t) represents the splitting modulation. We consider a splitting modulation with period

T = 2τ, such that

χ(t + 2mτ) = χ(t), m ∈ Z. (1.41)

Within a single period, this modulation takes the form     χ if 0 ≤ t < τ/2,  χ(t) = −χ if τ/2 ≤ t < 3τ/2, (1.42)     χ if 3τ/2 ≤ t < 2τ.

The function χ(t) specified by Eqs. (1.41) and (1.42) is illustrated in Fig. 1.4. The time average ∫ 2mτ of the qubit splitting over m modulation periods is 0 dt[ωq + χ(t)] = ωq. We consider a drive that is resonant with this time-averaged qubit splitting, thus taking ω = ωq. Moving to the frame rotating with the free Hamiltonian of the qubit, H0 = ωqσz/2, and neglecting terms that lead to rapid oscillations at frequency 2ωq using a rotating-wave approximation (thus assuming ε ≪ 2ωq) then yields the interaction-picture Hamiltonian

ε χ(t) H (t) ≃ σ + σ . (1.43) I 2 x 2 z

Since HI(t) has the periodicity of χ(t), T = 2τ, it is natural to use average Liouvillian theory to evaluate the time-evolution operator at times t = mT = 2mτ. Because the quantum system considered here is closed, time evolution is given by a unitary operator U(t). We thus write the

16 Magnus expansion as ( ) ∑∞ (k) (k) U(t) = exp −it H , H ≡ iΩk, (t = mT ) (1.44) k=0 where the first few terms for Ωk are obtained by substituting A(t) → −iHI(t) in Eqs. (1.36) to

(1.38). Using Eq. (1.43) for HI(t) then leads to

(0) ε (1) H = σx, H = 0, (1.45) 2 ( ) χ2τ εχτ H(2) = −ετ σ + σ . (1.46) 48 x 16 z

The leading term, H(0), is the average Hamiltonian [70, 71] for the system considered here. In

Eq. (1.45), H(1) vanishes because the interaction-picture Hamiltonian given by Eq. (1.43) has the following property: HI(t) = HI(T −t), corresponding to a symmetric cycle [73]. For such symmetric cycles, all odd orders vanish in the Magnus expansion, leading to H(1) = 0 [74].

Sustituting Eqs. (1.45) and (1.46) into Eq. (1.44) leads to the the following unitary evolution operator

( ) θ U ≡ U(2mτ) = exp −i m v · σ , (1.47) m 2

where σ = σxˆx + σy ˆy + σzˆz. Within the Bloch sphere representation, Equation (1.47) describes a rotation of the qubit state vector by an angle

√( ) ( ) χ2τ 2 2 εχτ 2 2 θ ≃ 2mετ 1 − + (1.48) m 24 8

around the axis defined by the unit vector v = vxˆx + vy ˆy + vzˆz, with

1 − χ2τ 2/24 vx ≃ √ , vy = 0, (1.49) (1 − χ2τ 2/24)2 + (εχτ 2/8)2 εχτ 2/8 vz ≃ −√ . (1.50) (1 − χ2τ 2/24)2 + (εχτ 2/8)2

17 Using Eqs. (1.47) to (1.50), it is then straightforward to evaluate the qubit expectation value

⟨σz(t)⟩ for t = 2mτ. Assuming that the initial state of the qubit is given by the density matrix

ρ(0) = |↑⟩⟨↑| = (1 + σz)/2, we have

1 ⟨σ (t = 2mτ)⟩ = tr[σ U ρ(0)U † ] = tr[σ U σ U † ]. (1.51) z z m m 2 z m z m

Substituting Eq. (1.47) into Eq. (1.51) and using the identity

( ) ( ) ( ) θ θ θ exp −i m v · σ = cos m 1 − i sin m v · σ (1.52) 2 2 2 then gives

( ) ( ) θ ( ) θ ⟨σ (t)⟩(2) ≃ cos2 m + v2 − v2 sin2 m , (t = 2mτ) (1.53) z 2 z x 2

with θm, vx, and vz given by Eqs (1.48) to (1.50). The superscript “(2)” is used in Eq. (1.53) to indicate that the expression was derived using the truncated expansion H(0) + H(1) + H(2) within

(0) average Hamiltonian theory. Including only H in the expansion rather leads to vx = 1, vz = 0, and

( ) ( ) θ θ ⟨σ (t)⟩(0) = cos2 m − sin2 m = cos (mετ) , (1.54) z 2 2

Thus, within leading-order average Hamiltonian theory, the qubit simply undergoes rotations around the x axis at the Rabi frequency ε; the effect of qubit-splitting modulations averages out in the limit τ → 0. According to Eqs. (1.48) to (1.50), the leading correction term H(2) modifies the Rabi frequency and adds a small z component to the rotation axis for finite τ.

As stated by Eq. (1.39), the expansion of Eq. (1.44) will converge for sufficiently short pulse intervals relative to 1/ max ∥HI(t)∥. The 2-norm [65] of HI(t) is given by √ [ ( )] √ † 1 ∥H (t)∥ = max eig H (t)H (t) = ε2 + χ2, (1.55) I I I 2 where eig(x) is the energy spectrum of x. Substituting Eq. (1.55) into Eq. (1.39) then leads to the

18 convergence criterion

√ ε2 + χ2 τ < π. (1.56)

We thus expect rapid convergence when max(ε, χ)τ ≪ π. This can be verified explicitly by comparing the results of average Hamiltonian theory, Eqs. (1.53) and (1.54), with an exact treatment.

Within time-intervals between instantaneous switching events (occuring at times τ/2, 3τ/2, ∫ T − t 5τ/2, ..., Fig. 1.4), HI(t) is time-independent. Thus, U(t) = exp[ i 0 dtHI(t)] exactly takes the form of a product of qubit rotation operators associated with each interval (within the regime of validity of the rotating-wave approximation). Substituting this product of rotations into Eq. (1.51) then leads to an exact solution for ⟨σz(t)⟩ that is compared with the results of average Hamiltonian

¶ theory in Fig. 1.5. In Fig. 1.5(a), we plot the exact dynamics of ⟨σz(t)⟩ for χτ = 5 (solid blue line), χτ = 2 (dashed purple line), and χτ = 1 (dotted red line), for χ/ε = 10. As expected from the discussion below Eq. (1.56), the exact solution approaches the prediction from leading-order average Hamiltonian theory (solid black line) as max(ε, χ)τ = χτ decreases relative to π. The leading correction term in average Hamiltonian theory, H(2), can be used to evaluate deviations from leading-order predictions for t = 2mτ, as shown in Fig. 1.5(a) by the black dots for χτ = 2.

Fig. 1.5(a) shows that the effect of qubit splitting becomes negligible when τ ≪ 1/χ. This can be understood heuristically in terms of motional averaging, an effect closely related to the motional narrowing of spectral lines frequently observed in nuclear magnetic resonance [32]. When the qubit splitting fluctuates on a timescale that is much more rapid than the amplitude of the fluctuations, only the time-averaged splitting becomes relevant. The qubit can then be considered to be resonant with the drive, leading to the Rabi oscillations shown in Fig. 1.5(a). In the example presented here, we have considered periodic splitting fluctuations; however, periodicity is not necessary for motional averaging to arise. As will be seen in Chapters 2 and 3, decreasing the correlation time of an environment can suppress qubit dephasing through motional averaging. Motional averaging was recently studied experimentally using a superconducting qubit whose splitting undergoes fast

¶For this periodic two-dimensional problem, the product of qubit rotations can be evaluated analytically, but the resulting expressions are unnecessary to the present discussion. Details of this fully-analytical approach are presented in Appendix C for a similar problem.

19 (a) 1 0.8 0.6 0.4 χτ = 5

i χτ = 2 ) 0.2 t ( 0 χτ = 1 z

σ (0)

h -0.2 hσz(t)i (2) -0.4 hσz(t)i , χτ = 2 -0.6 -0.8 -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 εt/2π (b) 0.5 0.4 (2)

i 0.3 )

t 0.2 ( z 0.1 σ 0 -0.1 i−h )

t -0.2 (

z -0.3 σ h -0.4 -0.5 0 20 40 60 80 100 120 140 εt/2π

Figure 1.5 – (a) Time evolution of the qubit expectation value ⟨σz(t)⟩ under the Hamiltonian given by Eq. (1.43), taking χ/ε = 10. Exact solution for χτ = 5 (solid blue line), χτ = 2 (dashed purple line), (0) and χτ = 1 (dotted red line). Solid black line: leading-order average Hamiltonian theory, ⟨σz(t)⟩ [from (0) Eq. (1.54), ⟨σz(t)⟩ is independent of χ and τ]. Black dots: solution including leading corrections in average Hamiltonian theory due to H(2), Eq. (1.53), for χτ = 2. (b) Difference between the exact expectation (2) value, ⟨σz(t)⟩, and the expectation value obtained within second-order average Hamiltonian theory, ⟨σz(t)⟩ [Eq. (1.53)], for χτ = 1.

20 random telegraph noise in a setting similar to the example presented here [75].

As explained in the final paragraph of Section 1.2.2, for sufficiently long times, large deviations can arise between the exact solution and the results of a truncated expansion within average

Hamiltonian theory, even when the convergence criterion of Eq. (1.39) is satisfied. This is illustrated in Fig. 1.5(b), in which we plot the difference between the qubit expectation value ⟨σz(t)⟩ using

(2) the exact solution and using the second-order treatment, ⟨σz(t)⟩ [Eq. (1.53)], for χτ = 1. In Fig. 1.5(b), the truncated expansion of average Hamiltonian theory only gives accurate predictions for finite time, limited to a few Rabi oscillations of the qubit.

1.3 Outline of this thesis

In this thesis, we use the formalism presented above to provide a theoretical description of pure dephasing processes relevant to spin qubits in semiconductors. Here, we provide a brief description of the problems studied in each chapter of this work, and give some background useful to understand them.

1.3.1 Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient

As mentioned above, magnetic-field gradients are useful for electric-dipole spin resonance, single- site addressability, and coupling to a microwave resonator. It is thus relevant to consider the impact of an inhomogeneous magnetic field on qubit coherence.

Figure 1.6 illustrates how a magnetic field gradient may reduce qubit coherence in a simple but instructive one-dimensional geometry. In Fig. 1.6(a), we consider a qubit consisting of a single electron or hole spin coupled to a nuclear-spin environment through the hyperfine interaction [12].

Neglecting nonsecular terms of the hyperfine interaction by means of a rotating wave approximation

(which will be justified in Chapter 2) then leads to the qubit-environment coupling Hamiltonian

H(t) = δhz(t)Sz, δhz(t) = hz(t) − ⟨hz(t)⟩, (1.57)

In Eq. (1.57), Sz = σz/2 is the projection along z of the qubit spin operator (assuming S = 1/2) and

21 (b) ) ω ( S

−γB 0 γB ω

Figure 1.6 – Dephasing noise generated by a nuclear-spin environment in an inhomogeneous magnetic field produced, e.g., with a micromagnet. (a) Sketch of the physical system. Dark blue curve: probability distribution |ψ(z)|2 associated with the envelope part of the wavefunction of the electron or hole carrying the spin used as a qubit. Full red circles: nuclear spins precessing in an inhomogeneous magnetic field B(z) = Bx(z)ˆx + Bˆz (blue arrows) that has a strong uniform z component and a slanting contribution in the x direction. Precession of the nuclear spins generates fluctuations in the Overhauser field δhz(t) experienced by the spin qubit due to its secular hyperfine coupling with the nuclear spins. (b) Sketch ofthe noise spectrum S(ω) [introduced in Eq. (1.21)] associated with the noise operator ξ(t) ≡ δhz(t). Red arrows represent Dirac deltas. The structures appearing for |ω| > γB, where γ is the nuclear gyromagnetic ratio, are due to the x component of the magnetic field.

∑ z z z h = k AkIk is the Overhauser field, in which Ak and Ik are the hyperfine coupling strength and z projection of the nuclear spin operator for spin k, respectively. The hyperfine coupling strength Ak

2 is proportionnal to |ψ(rk)| , where ψ(rk) is the envelope part of the electron or hole wave function at the position rk of nuclear spin k.

The Hamiltonian in Eq. (1.57) has the same form as HQE(t) in Eq. (1.1), which led to pure dephasing of the qubit. Thus, the same steps can be taken as in Section 1.1. Here, dephasing will be caused by the noise operator ξ(t) ≡ δhz(t), whose time evolution is generated by the inhomogeneous mag- ∫ ∞ iωt z z netic field illustrated in Fig. 1.6(a). The noise spectrum S(ω) = −∞ dωe ⟨δh (t)δh (0)⟩/2π that follows from these dynamics (under the Gaussian approximation and to zeroth order in the Magnus expansion) is illustrated schematically in Fig. 1.6(b) for nuclear spins in an infinite-temperature thermal state. In Fig. 1.6(b), the central arrow represents a Dirac delta function located at ω = 0 due to quasistatic noise. Here, quasistatic noise arises from uncertainty in δhz associated with the thermal state of the nuclear spins. In addition to this zero-frequency component, the noise spectrum has contributions at finite frequency due to the oscillations in δhz(t) arising from the transverse field gradient Bx(z)ˆx. As shown in Fig. 1.6(b), these oscillations lead to additional delta functions for |ω| > γB, where γ is the gyromagnetic ratio of the nuclear spins. More specifically, these delta

22 √ 2 2 functions are located at Larmor frequencies ωk = γ B + Bx(zk) of distinct nuclear spins, which experience different magnetic fields due to the gradient. These finite-frequency contributions to the noise spectrum will cause qubit dephasing that may be difficult to suppress simply by using a

Hahn echo. In addition, the peculiar structure of this spectrum, which vanishes for 0 < |ω| < γB, will give rise to interesting long-time behavior, as the weight of the filter function for free-induction decay or Hahn echo becomes concentrated around ω = 0.

In Chapter 2, we investigate these dephasing effects arising from a magnetic field gradient within realistic models describing the devices shown in Fig. 1.1. We show that for all these devices, a magnetic-field gradient of ∼ 1 T/µm (achieved experimentally using micromagnets [25]) provides a qubit dephasing mechanism that can be dominant over other known dephasing processes in these systems, in the absence of a gradient. We also show that this problem can be avoided using a moderate external magnetic field to reach a motional averaging regime in which the nuclear spins precess very quickly, suppressing their impact on qubit coherence. In addition to the thermal nuclear-spin state mentioned above, we also consider nuclear spins in a narrowed state with reduced fluctuations along z [76]. Narrowing of the nuclear-spin state has recently been achieved with a spin qubit in a GaAs quantum dot [77] and has led to a significant increase in the qubit free-induction decay time, opening the door to experimental observation of the effects predicted in Chapter 2.

1.3.2 Microscopic models for charge-noise-induced dephasing of solid-state

qubits

In newly-developed silicon qubits, nuclear spins can be eliminated prior to device fabrication through isotopic purification [7, 21], thus suppressing the dephasing mechanism discussed in Chap- ter 2. However, a magnetic-field gradient used, e.g., for electric-dipole spin resonance or coupling to a microwave resonator, may still expose the qubit to charge noise by generating an artificial spin-orbit coupling [33, 34]. Charge noise is often due to bistable charge fluctuators present in the material that hosts the qubit [78–81]. While the interactions that drive nuclear-spin dynamics are well established, the physical mechanisms at the origin of fluctuator switching are more nebulous. Thus, to better understand the processes leading to charge dephasing, we have developed microscopic models of fluctuator switching that can be tested using a simple set of measurements.

23 In a recent experiment [38], Dial et al. characterized charge noise afflicting a singlet-triplet qubit in GaAs. Singlet-triplet qubits consist of a double quantum dot containing two electrons (one in each dot) whose spins are coupled through the exchange interaction. The strength of this exchange coupling depends on the electric potential difference between the two quantum dots, making the qubit sensitive to electric-field fluctuations. In Ref. [38], the authors measured the qubit coherence factor C(t) under Hahn echo in a regime in which dephasing due to charge noise is dominant. The authors found that C(t) is well described by a compressed exponential,

α C(t) = exp[−(t/T2) ], (1.58)

where T2 is the coherence time and α the stretching parameter. When α = 1, Eq. (1.58) describes exponential decay of the qubit coherence, corresponding to the Markovian limit introduced above

Eq. (1.24). Conversely, α ≠ 1 corresponds to non-Markovian dephasing. In Ref. [38], the authors measured T2 and α as a function of temperature. As shown in Fig. 1.7(a), the authors found that T2 decreases, roughly obeying a power law, as temperature increases. In addition, as shown in Fig. 1.7(b), α crosses over from a non-Markovian regime (α > 1) at low temperature to an approximately Markovian regime (α ≃ 1) for 100 mK < T < 150 mK, corresponding to nearly exponential decay of C(t).

In Chapter 3, we explain how such measurements of the temperature dependence of T2 and α may shed light on the physical mechanisms at the origin of charge-noise-induced dephasing. We consider microscopic mechanisms driving the switching dynamics of a charge-fluctuator environment, focusing on electron tunneling and phonon absorption and emission. We show that different mechanisms lead to distinct dependencies of T2 and α as a function of temperature. These dependencies may help to experimentally identify the physical processes at the origin of charge noise in semiconductor qubits. In particular, we explain how the experiment of Dial et al. may be better understood by combining the fluctuator-induced dephasing mechanisms introduced in Chapter 3 with Markovian dephasing of the singlet-triplet qubit due to a phonon bath [82–84].

24 (a) 101 (b) 1.8 1.6 100 1.4 s) µ ( α 1.2

2 −1 T 10 1.0 0.8 10−2 0.6 40 100 200 300 50 100 150 200 250 300

T (mK) T (mK)

Figure 1.7 – Temperature dependence of the coherence time T2 and stretching parameter α, which charac- α terize decay of the coherence factor C(t) = exp[−(t/T2) ] of a singlet-triplet qubit in GaAs exposed to charge noise. (a) Coherence time T2 for Hahn echo. (b) Stretching parameter α, also for Hahn echo. Experimental error bars become very large for T > 150 mK (not shown). Thus, the strong variations in α shown above for T > 150 mK may be attributed to experimental uncertainty. The data shown in the plots is taken from Ref. [38].

1.3.3 Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED

In cavity QED, atoms are strongly coupled to microwave photons trapped in a superconducting cavity to study fundamental features of light-matter interaction [85]. In an analogous setup named circuit QED, solid-state qubits are coupled to photons trapped in a microwave resonator, leading to a very successful architecture for quantum computation [86]. Indeed, a microwave resonator has been employed in many useful quantum operations such as state transfer between a qubit and a quantum memory [87], two-qubit gates between distant qubits [29], and qubit readout through the resonator output field [30]. Strong coupling to microwave resonators is now routinely realized with superconducting qubits [88]; however, coherent coupling between a single spin and a resonator has only been accomplished very recently, and the coupling achieved is of the same order of magnitude as cavity damping and qubit dephasing [28]. An important obstacle in realizing high-fidelity quantum operations between a spin qubit and a resonator is the presence of dephasing noise that is stronger than the qubit-cavity coupling g. Indeed, intrinsic coupling between a single spin and the magnetic field in a resonator isg/ tiny( 2π < 1 kHz [27]). Several approaches exist to increase the coupling strengths to g/2π ∼ 1 MHz, but they all involve some correlation of the spin with a charge degree-

25 of-freedom either through spin-orbit coupling [89, 90], exchange coupling [91], or a magnetic-field gradient [26–28], thus exposing the spin qubit to charge noise mechanisms such as those discussed in Chapter 3. High-fidelity quantum operations involving the resonator would be especially difficult in materials in which an important fraction of nuclei carry spin. For example, in GaAs (in which ∗ ∼ all nuclei carry spin), the free-induction decay time, T2 10 ns, is much shorter than the expected timescale of coherent qubit-resonator oscillations, π/g ∼ 1 µs [27].

A feature that spin and charge noise spectra often have in common is a strong quasistatic

(zero-frequency) contribution [19, 38]. As seen in Section 1.1.2, qubit dephasing arising from such low-frequency noise can be avoided using most dynamical decoupling strategies, such as the Carr-

Purcell sequence. However, the formalism described in Section 1.1.2 describes a qubit that is coupled only to its dephasing environment (not to a resonator). In the presence of an additional qubit-resonator coupling, as will be seen in Chapter 4, refocusing π pulses rapidly excite the cavity, leading to significant error in, e.g., a qubit-resonator quantum state transfer.

In Chapter 4, we develop an approach based on Hamiltonian engineering that avoids unwanted excitation of the cavity and admits high-fidelity spin-resonator state transfer even in the presence of the strong dephasing that is typical of spin qubits. This approach may be useful for all kinds of spin qubits described in the beginning of this Chapter, but may also be applied to quantum state transfer between a resonator and a collective mode in a spin ensemble, in which strong dephasing due to inhomogeneous broadening has been observed [87]. Finally, a simple modification of our decoupling scheme leads to a fast quantum nondemolition readout of the qubit through the resonator output field [92]. We will find that this readout can have a high signal-to-noise ratio and a high single-shot fidelity even for cavity damping that is weaker than the qubit-cavity coupling. This is particularly useful for semiconductor spin qubits, for which the predicted couplings, g/2π ∼ 1 MHz [26, 27], are typically smaller than the cavity damping rates achieved experimentally, κ/2π = 2-10 MHz [31, 93].

26 2 Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient

This chapter is the integral text from:

Magnetic-field gradients are important for single-site addressability and electric- dipole spin resonance of spin qubits in semiconductor devices. We show that these advantages are offset by a potential reduction in coherence time due to the non-uniformity of the magnetic field experienced by a nuclear-spin bath interacting with the spin qubit. We theoretically study spins confined to quantum dots or at single donor impurities, considering both free-induction and spin-echo decay. For quantum dots in GaAs, we find that, in a realistic setting, a magnetic-field gradient can reduce the Hahn-echo coherence time byalmost an order of magnitude. This problem can, however, be resolved by applying a moderate external magnetic field to enter a motional averaging regime. For quantum dots in silicon, we predict a cross-over from non-Markovian to Marko- vian behavior that is unique to these devices. Finally, for very small systems such as single phosphorus donors in silicon, we predict a breakdown of the common Gaussian approximation due to finite-size effects.

27 2.1 Introduction

One of the most intriguing features of spins in semiconductor nanostructures is our ability to systematically determine decoherence mechanisms for these systems from microscopic models.

Although complex spin dynamics in the presence of the relevant interactions can be difficult to evaluate, these interactions can be known to a high degree of precision and irrelevant terms systematically dropped from the Hamiltonian. Indeed, for electron spin qubits confined to quantum dots in GaAs or natural Si, the dominant contribution to dephasing is known to be the Fermi contact hyperfine interaction between the electron spin and nuclear spins in the surrounding host material [6, 10, 19, 94].

The contact hyperfine interaction divides naturally into two contributions: a secular (diagonal) part that commutes with the electron Zeeman term, for which dynamics are reversible by Hahn echo [19], and a non-secular (off-diagonal) part, the so-called flip-flop interactions, whichmay lead to a more complex dynamics. In the presence of a large electron Zeeman splitting compared to the total hyperfine interaction strength, perturbative expansions can be used to calculate the contribution of flip-flop interactions to decoherence [14, 95]. Alternative techniques such as cluster expansions or high-order partial resummations have also been used to estimate dynamics due to flip-flops in the low-field limit16 [ –18, 96, 97].

In addition to hyperfine interactions, dipole-dipole coupling between nuclear spins can induce in- ternal dynamics in the nuclear-spin bath, which can act back on the electron spin through hyperfine interactions, resulting in decoherence. This process of electron-spin decoherence due to fluctuations in the nuclear-spin bath (spectral diffusion [98]) is independent of the applied magnetic field for moderately large fields and is not fully reversible by spin-echo [18, 99], thereby typically limiting achievable coherence times.

Electron spins confined in silicon using either quantum dots[6] or at single phosphorus donors [8,

100] may have significant advantages over spins in GaAs quantum dots, since natural silicon contains only 4.7% nuclear-spin-carrying 29Si, a figure that can be further reduced by isotopic purification.

Although the long coherence times achievable for semiconductor spin qubits are certainly an ad-

28 2

Wavefunction Single dot Double dot Sec. V, we apply our model to the physical systems shown 2 in Fig. 1 and make testable predictions for experiments 1 r ri ψi(r) exp | − | ∝ − 2 r0 that we expect to be realizable with present technology. , We conclude in Sec. VI with a discussion of the most im- {| ↑ | ↓} portant results presented here and possible implications Logical spin qubit , or {| ↑ | ↓} for future experiments on spin dynamics in these systems. |↑↓±|↓↑ √2 Wavefunction Donor

1 r ri II. HAMILTONIAN AND EXACT SOLUTION ψi(r) exp | − | ∝ − 2 r0 Logical spin qubit , We consider a single electron spin interacting with a {| ↑ | ↓} nuclear-spin bath through hyperfine coupling. In general, Figure 2.1 – Various physical systems considered in this chapter. We study spin dynamicsthe coupling for both may arise from the Fermi contact term34 single- and double-dotFIG. geometries. 1: (Color In online) the double-dot Various case, physical we consider systems either considered (i) a single electron(most relevantshared for electrons in III-V semiconductors), by the two dots orin (ii)√ this an electron paper. in We each study dot,√ withspin spin dynamics states expressedfor both in single- the singlet-tripletpurely anisotropic basis, interactions (relevant for, e.g., hole {| ⟩ | ↑↓⟩ − | ↓↑⟩ | ⟩ | ↑↓⟩ | ↓↑⟩ } S = ( and)/ 2 double-dot, T0 = ( geometries.+ )/ 2 In. Thethedouble-dot associated electron case, we envelope con- wavefunctionsspins in III-V can semiconductors28), or both (relevant for, be either hydrogenicsider (relevant either for (i)a donor a single impurity) electron or Gaussian shared (relevant by the two for a dots quantum or dot) in dimension 35 d = 1, 2, or 3. e.g., spins in silicon ). We will, however, consider only (ii) an electron in each dot, with spin states expressed in the cases where spin dynamics is dominated by the sec- the singlet-triplet basis, S =( )/√2, T0 = ( + )/√2 . The{| associated | ↑↓ electron − | ↓↑ envelope| wave- ular hyperfine coupling (the part that commutes with functions| ↑↓ | ↓↑ can be either} hydrogenic (relevant for a donor impu- the electron Zeeman term). We allow for a position- vantage, control and scalability are also required for potential applications in quantum information x rity) or Gaussian (relevant for a quantum dot) in dimension dependent external magnetic field Bext(r)=B (r) ˆx + B ˆz , B being a constant, as illustrated in Fig. 2(a). We processing. A usefuld =1 tool, 2, inor this 3. respect is a magnetic-field gradient. Indeed, electron-spin reso- further take Bext(r) and the electron envelope wavefunc- 3 2 nance of single electron spins in lateral dots in GaAs has been demonstrated using inhomogeneoustion ψ(r) to satisfy d rBext(r) ψ(r) = B ˆz [this is x | | of the spin qubit. The second technique, a Hahn echo true, e.g., when B (r) is an odd function of z,while magnetic fields generated by cobalt micromagnets [25, 101]. Recent proposals suggestψ to(r) useis these an even function of z, as in Fig. 2(a)]. Adding (HE), involves a refocusing pulse applied at time t after | | gradients to achieveinitial strong preparation. coupling of single This electron procedure spins reversesto a coplanar-waveguide the evolu- resonatora secular [27 hyperfine], coupling gives (setting = 1), tion of the spin qubit under a static secular hyperfine in particular in Si where coherence times are longer. Interactions with a resonator can be used for coupling, allowing for a retrieval of coherence at time 2t. ˆ ˆz ˆz x ˆx ˆz H =(b + h )S + γk bkIk + bIk , (1) a qubit readout [31In, 90 this] or case, to mediate the presence interactions of an between inhomogeneous distant spin transverse qubits [89], an important magnetic field then leads to a finite nuclear-spin-bath cor- k step toward scalability.relation time, preventing the full recovery of spin-qubit ˆz ˆz where h = k AkIk is the Overhauser field, with sums coherence. over nuclear spins k, b = g∗µBB is the electron Zeeman In this chapter, we calculate the effect of such an inhomogeneous magnetic field on the coherence x x Our analysis of a transverse magnetic-field gradient on splitting, and γkbk = γkg∗µBB (rk). The nuclear gyro- of (electron or heavy-hole)the decoherence spin qubits. dynamics These results of spin can qubits be applied applies to a tonumber sev- of experimentallymagnetic ratios are thus given by γkg∗µB,withµB the eral different systems: spins in quantum dots formed in, Bohr magneton and g∗ the effective g-factor. Finally, relevant scenarios (see Fig. 2.1). We account for the secular hyperfine interaction along with 2 e.g., either III-V semiconductors or Si, and to single P Ak ψ(rk) is the hyperfine coupling strength at nu- donor impurities in Si (Si:P) (see Fig. 1). In each case, ∝ | | the electron- and nuclear-Zeeman terms due to an inhomogeneous magnetic-field cleargradient. site k This=0, 1, 2,.... These couplings are calculated we find that a magnetic-field gradient can decrease the as functions of k for several relevant wavefunctions in inhomogeneity cancoherence lead to dynamics time, relative in the nuclear-spin to known systemcoherence and times a consequent in the decayAppendix of electron- A. absence of a magnetic-field gradient. In addition to elec- Eigenstates of Hˆ , Eq. (1), are simultaneous eigenstates spin coherence, similar to the case of spectral diffusion from nuclear dipolar interactions. We tron spins, our dephasing model is directly applicable to of Sˆz, which we denote , .Sincethereisnoin- 28–32 {| ↑ | ↓} further show that ahole magnetic-field spins in III-V gradient semiconductors, can result inand the mayfailure give in-of two teractiontypical techniques between used nuclear spins, the eigenstates of Hˆ can sight into dephasing from the nuclear quadrupolar inter- further be written as a product of nuclear-spin states of 33 to mitigate decoherenceaction due due to to the inhomogeneous nuclear environment: strain in nuclear InAs nanowires. spin state narrowingdefinite and Hahn angular momentum along the direction of an ef- The remainder of this article is divided as follows. In σ echo. State narrowing involves the preparation of the nuclear-spin bath in a narrowfective distribution field hk , [see Fig. 2(b)], for each nuclear spin Sec. II, we introduce the Hamiltonian accounting for Zee- at site k with electron spin σ = , . These states can man terms coupling to the spin qubit and nuclear-spin be obtained from an electron-spin-state-dependent↑ ↓ rota- y / 29 / ˆ ↑ ↓ bath for an inhomogeneous magnetic field, as well as the ˆ iIk φk z tion Rk↑ ↓ =e− applied to Ik -eigenstates, Ik,mk , secular hyperfine coupling. We then derive an exact for- where | mula for the spin coherence factor. In Sec. III, we in- x troduce a perturbative method to obtain simple expres- / γkbk φ↑ ↓ = arctan , (2) sions for the coherence factor with associated parametric k bγ 1 A k 2 k dependences. In Sec. IV, we obtain analytical expres- ± sions for the electron spin coherence factor in the context as illustrated in Fig. 2(b). This gives the exact of free-induction decay for an initial narrowed nuclear- electron-spin coherence factor C(t), defined by Sˆ+(t) = spin state and for Hahn echo with an initial infinite- Sˆ+(0) C(t), for two protocols: (i) free-induction decay temperature thermal state of the nuclear-spin system. In and (ii) Hahn echo. of eigenstates of the nuclear field operator. Ideally, this corresponds to an exact eigenstate ofthe component of the nuclear field along an applied magnetic field [13, 76, 102–104]. The presence of a transverse magnetic-field gradient destroys this dephasing-free nuclear-spin state, leading to free-induction decay (FID) of the spin qubit. The second technique, a Hahn echo (HE), involves a refocusing pulse applied at time t after initial preparation. This procedure reverses the evolution of the spin qubit under a static secular hyperfine coupling, allowing for a retrieval of coherence at time 2t. In this case, the presence of an inhomogeneous transverse magnetic field then leads toa finite nuclear-spin-bath correlation time, preventing the full recovery of spin-qubit coherence.

Our analysis of a transverse magnetic-field gradient on the decoherence dynamics of spin qubits applies to several different systems: spins in quantum dots formed in, e.g., either III-V semiconductors or Si, and to single P donor impurities in Si (Si:P) (see Fig. 2.1). In each case, we find that a magnetic-field gradient can decrease the coherence time, relative to known coherence times inthe absence of a magnetic-field gradient. In addition to electron spins, our dephasing model is directly applicable to hole spins in III-V semiconductors [68, 105–108], and may give insight into dephasing from the nuclear quadrupolar interaction due to inhomogeneous strain in InAs nanowires [109].

The remainder of this chapter is divided as follows. In Sec. 2.2, we introduce the Hamiltonian accounting for Zeeman terms coupling to the spin qubit and nuclear-spin bath for an inhomogeneous magnetic field, as well as the secular hyperfine coupling. We then derive an exact formula forthe spin coherence factor. In Sec. 2.3, we introduce a perturbative method to obtain simple expressions for the coherence factor with associated parametric dependences. In Sec. 2.4, we obtain analytical expressions for the electron spin coherence factor in the context of free-induction decay for an initial narrowed nuclear-spin state and for Hahn echo with an initial infinite-temperature thermal state of the nuclear-spin system. In Sec. 2.5, we apply our model to the physical systems shown in

Fig. 2.1 and make testable predictions for experiments that we expect to be realizable with present technology. We conclude in Sec. 2.6 with a discussion of the most important results presented here and possible implications for future experiments on spin dynamics in these systems.

30 2.2 Hamiltonian and exact solution

We consider a single electron spin interacting with a nuclear-spin bath through hyperfine coupling.

In general, the coupling may arise from the Fermi contact term [110] (most relevant for electrons in III-V semiconductors), purely anisotropic interactions (relevant for, e.g., hole spins in III-V semiconductors [105]), or both (relevant for, e.g., spins in silicon [111, 112]). We will, however, consider only the cases where spin dynamics is dominated by the secular hyperfine coupling (the part that commutes with the electron Zeeman term). We allow for a position-dependent external

x magnetic field Bext(r) = B (r) ˆx+B ˆz, B being a constant, as illustrated in Fig. 2.2(a). We further ∫ 3 2 take Bext(r) and the electron envelope wavefunction ψ(r) to satisfy d rBext(r)|ψ(r)| = B ˆz [this is true, e.g., when Bx(r) is an odd function of z, while |ψ(r)| is an even function of z, as in

Fig. 2.2(a)]. As illustrated in Fig. 2.2(a), this means that the external magnetic field experienced by the electron is aligned with the quantum dot principal axis of symmetry, such that the g-tensor is diagonal and only its z component is relevant. Adding a secular hyperfine coupling gives* (setting

ℏ = 1),

∑ ( ) ˆ ˆz ˆz x ˆx ˆz H = (b + h )S + γk bkIk + bIk , (2.1) k ∑ ˆz ˆz ∗ where h = k AkIk is the Overhauser field, with sums over nuclear spins k, b = g µBB is the x ∗ x I ∗ I electron Zeeman splitting, γkbk = γkg µBB (rk), and γk = γk/g µB, where γk is the nuclear ∗ gyromagnetic ratio. The nuclear gyromagnetic ratios are thus given by γkg µB, with µB the Bohr

∗ 2 magneton and g the effective g-factor. Finally, Ak ∝ |ψ(rk)| is the hyperfine coupling strength at nuclear site k = 0, 1, 2,... These couplings are calculated as functions of k for several relevant wavefunctions in Appendix A.1.

Eigenstates of Hˆ , Eq. (2.1), are simultaneous eigenstates of Sˆz, which we denote {| ↑⟩, | ↓⟩}. ∑ * x ˆx z ˆz ˆ Anisotropic hyperfine interactions of the form k(AkIk + AkIk )Sz can easily be accounted for within x z our theoretical framework if each nuclear spin is rotated by an angle θk = arctan(Ak/Ak) around the ˆy − ˆy y-axis by applying the operators Rk(θ) = exp( iθkIk ). We retrieve√ a Hamiltonian that has the exact → x 2 z 2 same form as Eq. (1), but with modified coupling constants Ak (Ak) + (Ak) , gyromagnetic ratios → x x z ′ x → x z − x z x x γk γk(bkAk + bAk)/bAk, and transverse fields bk b(bkAk bAk)/(bAk + bkAk). However,∑ this approach ˆy ˆz will not reproduce exactly the same Hamiltonian in the presence of terms proportional to k AkIk S , but should lead to similar qualitative behavior, especially if the anisotropic hyperfine coupling terms are weak.

31 (a) x (b) ↑ hk z γkb Ak Bzˆ |ψ(r)| ↓ hk ↑ z φ L k r0 rk R ↓ φk l x B (r)xˆ x x γkbk

Figure 2.2 – (a) Example of a device studied in this chapter. A constant longitudinal magnetic field Bzˆ and an inhomogeneous transverse magnetic field Bx(r)xˆ are applied to an electron spin interacting with a nuclear-spin bath. The light blue circles illustrate the range r0 of the electron wavefunction ψ(r) in the specific case of a double quantum dot with| left( L⟩) and right (|R⟩) single-particle orbital states separated ↑/↓ by a distance l. (b) Total effective field hk experienced by nuclear-spin k, including Knight shifts coming from the secular hyperfine interaction (of strength Ak) with the electron spin, which can be in state | ↑⟩ or | ↓⟩. This can be viewed as an electron-spin-dependent rotation of the nuclear spin precession axis by ↑/↓ x an angle ϕk , which also depends on the Zeeman field γk(bk, 0, b). Thus, because of the inhomogeneous magnetic field, the precession axis is slightly different for each nuclear spin. This leads to a finite correlation time for collective states of the nuclear-spin bath, preventing the full recovery of electron spin coherence.

Since there is no interaction between nuclear spins, the eigenstates of Hˆ can further be written as a

product of nuclear-spin states of definite angular momentum along the direction of an effective field σ ↑ ↓ hk , [see Fig. 2.2(b)], for each nuclear spin at site k with electron spin σ = , . These states can be ↑/↓ − ˆy ↑/↓ ˆ iIk ϕk z obtained from an electron-spin-state-dependent rotation Rk = e applied to Ik -eigenstates,

|Ik, mk⟩, where ( ) x ↑ ↓ γ b ϕ / = arctan k k , (2.2) k 1 bγk 2 Ak as illustrated in Fig. 2.2(b). This gives the exact electron-spin coherence factor C(t), defined by

⟨Sˆ+(t)⟩ = ⟨Sˆ+(0)⟩C(t), for two protocols: (i) free-induction decay and (ii) Hahn echo.

For both Hahn echo and free-induction decay, we take an initial nuclear-spin state of the form ∑ ∏ M | ⟩⟨ | | ⟩ | j ⟩ ρˆI(0) = j=1 pj j j , with j = k ψk , i.e., a statistical mixture containing M tensor-product states. In this chapter, we will focus on two possible initial states: narrowed states and infinite-

temperature thermal states. Narrowed nuclear-spin states are defined as a mixture of eigenstates

32 {| ⟩} ˆz z j of h having the same eigenvalue hn for all j, i.e. that satisfy the equation

ˆz| ⟩ z | ⟩ h j = hn j . (2.3)

j j | ⟩ | ⟩ ′ For numerical evaluation, we generate such a state by taking ψk = Ik, mk and pj = δjj (i.e., j ˆz M = 1), with the eigenvalues mk of Ik uncorrelated for different k and chosen randomly from a uniform distribution. To evaluate thermal averages over infinite-temperature states, we take | j ⟩ pj = 1/M with ψk a random spin coherent state with quantization axis sampled uniformly on the unit sphere independently for each k, j. The value of M is set by increasing this value until the coherence factor has converged to within approximately 1%; we find a value M ∼ 100 is typically sufficient for convergence.

In the case of free-induction decay, the time-evolution operator is simply Uˆ(t) = e−iHtˆ and the ∑ coherence factor is given by C(t) = j pjCj(t), with

∏ ⟨ j | ˆ↑ ˆ↑ † ˆ↓ ˆ↓ ˆ↓ †| j ⟩ Cj(t) = ψk Rk[Ek(t)] QkEk(t)(Rk) ψk , (2.4) k

↑/↓ − ˆy ↑/↓− ↓/↑ ˆ iIk (ϕk ϕk ) and where we have introduced Qk = e and

∑Ik ↑/↓ − ↑/↓ ˆ imkhk t| ⟩⟨ | Ek (t) = e Ik, mk Ik, mk , (2.5) mk=−Ik √( ) ↑/↓ x 2 2 hk = γkbk + (bγk Ak/2) . (2.6)

In the case of Hahn echo, a refocusing pulse (a π-pulse, with unitary given by the Pauli matrix

σˆx) is applied at time t, and again at time 2t to return the spin to its initial state. Thus, the time-evolution operator is given by Uˆ(2t) = e−iσˆxHˆ σˆxte−iHtˆ and the electron-spin coherence factor is ∏ ⟨ j | ˆ↑ ˆ↑ † ˆ↓ ˆ↓ † ˆ↑ ˆ↑ ˆ↓ ˆ↓ ˆ↓ †| j ⟩ Cj(2t) = ψk Rk(Ek) Qk(Ek) QkEkQkEk(Rk) ψk . (2.7) k

ˆ↑/↓ Note that we have dropped the explicit t-dependence on Ek in Eq. (2.7). Knowing the distribution of hyperfine couplings, magnetic-field distribution, and gyromagnetic

33 ratios for the nuclear spins, Eqs. (2.4) and (2.7) can be used to compute the exact electron-spin coherence factor numerically. The calculation involves a product of O(N) square matrices of dimension 2I + 1, N being the number of nuclear spins within r0, the effective (donor or quantum-dot) Bohr radius. Therefore, for large systems with ≳ 106 nuclear spins, the computation time can become rather long. More importantly, Eqs. (2.4) and (2.7) do not give any physical insight into the coherence time of the electron spin with respect to relevant physical parameters. For these two reasons, in the rest of this chapter, we seek approximate expressions for the coherence factor which have a simpler form.

2.3 Simplified coherence factor

In this section, we approximate C(t) using a Magnus expansion. The Magnus expansion is a perturbation theory in the amplitude of Knight-shift fluctuations ∼ Ak relative to the typical rate √ ∼ 2 x 2 of nuclear-spin fluctuations ωk = γk b + (bk) . We will be able to truncate the expansion for the time-evolution operator at leading order in Ak/ωk ≪ 1, giving expressions for both free- induction decay and Hahn echo. In addition, we will invoke a Gaussian approximation, valid for a large uncorrelated nuclear-spin bath, to obtain simple approximate expressions for the coherence factor. We ultimately show that, at leading order in the Magnus expansion, the coherence factor for a nuclear-spin bath in a narrowed state is only affected by fluctuations of transverse components of the nuclear field, hˆx, hˆy, whereas all components (hˆx, hˆy, hˆz) are equally important for an infinite- temperature thermal state. This approach generalizes that applied in the recent work of Ref. [68] on hole-spin dynamics.

34 2.3.1 Time-evolution operators

We first move the Hamiltonian of Eq.(2.1), Hˆ = Hˆ0+Vˆ , to the interaction picture with perturbation

z z Vˆ = hˆ Sˆ and taking Hˆ0 to be the Zeeman terms, yielding

∑ ˆ − ˆ iH0t iH0t z ˆ Vˆ (t) = e Vˆ e = Sˆ Akvk(t) · Ik, (2.8) k [ ] − x z − x z 2 x 2 vk(t) = nknk(cos ωkt 1), nk sin ωkt, (nk) + (nk) cos ωkt , (2.9)

x where nk = γk(bk, 0, b)/ωk. In the Magnus expansion [71, 113], the time-evolution operator is recast in the form Uˆ(t) = ˆ ∑∞ −iHM(t) ˆ ˆ ˆ (n) ˆ (n) e , where HM(t) is given by a series expansion, HM(t) = n=0 H (t). The term H (t) contains n + 1 time integrals over the rapidly oscillating perturbation Vˆ (t). We therefore expect rapid convergence of the Magnus expansion in the limit of large ωk, which sets the oscillation frequency for Vˆ (t). Explicit formulas for the lowest orders of the Magnus expansion are given in the literature [71, 113]. Criteria for the convergence of the Magnus expansion are discussed for specific physical systems considered here in Appendix A.3.

At leading order in the Magnus expansion, Uˆ(t) ≃ e−iHˆ (0)(t). As will be shown below, this leading- order analysis is sufficient to describe the dynamics of the coherence factor in several different spin systems. In the case of free-induction decay,

∫ t ∑ ˆ (0) ˆ ˆz FID · ˆ H (t) = V (t1)dt1 = S hk (t) Ik, (2.10) 0 [ ( k ) ] − FID x z − sin ωkt x 1 cos ωkt z 2 x 2 sin ωkt hk (t) = Ak nknk t , nk , (nk) t + (nk) . (2.11) ωk ωk ωk

For Hahn echo, π-pulses are applied at times t and 2t, leading to Sˆz → σˆxSˆzσˆx = −Sˆz. Therefore,

[∫ ∫ ] ∑ t 2t ∑ ˆ (0) ˆz ˆ · − ˆz HE · ˆ H (2t) = S AkIk vk(t1)dt1 vk(t1)dt1 = S hk (t) Ik, (2.12) k 0 t k A bx [ ] HE k k z z − y − x z hk (t) = 2 nkfk (t), fk (t), nkfk (t) , (2.13) ωk y − − z − where fk (t) = 2 cos ωkt cos 2ωkt 1, fk (t) = sin 2ωkt 2 sin ωkt. (2.14)

35 In both cases (free-induction decay and Hahn echo), we have found an approximate evolution ∑ − ˆz ˆ ˆ ·ˆ operator of the form exp[ iS X(t)], where X(t) = k hk(t) Ik has the simple form of independent x ∀ FID time-varying effective fields hk(t) on each nuclear spin. As expected, if bk = 0 k, hk (t) fluctuates ∗ only along zˆ, resulting in pure dephasing on a time scale T2 for a random initial nuclear-spin state. This decay will not occur, however, if the nuclear bath is initially in a narrowed state. For the ideally narrowed initial state defined by Eq. (2.3), dephasing will arise entirely from the transverse

x,FID y,FID x components, hk (t), hk (t), in the presence of finite bk. Finally, for Hahn echo, the evolution x operator only deviates from the identity for non-zero values of bk.

2.3.2 Gaussian approximation and finite-size effects

The steps taken above have allowed us to obtain a simpler approximate evolution operator in the interaction picture. Nevertheless, applying this simplified evolution operator to find the coherence factor C(t) typically leads to complicated expressions, especially for the case of large nuclear spin,

I > 1/2. Therefore, we will take advantage of the large number of nuclear spins usually present in semiconductor devices to introduce the Gaussian approximation.

We will use the same symbol, Hˆ (0)(τ), for the leading-order term in the Magnus expansion for both Hahn echo and free-induction decay. For the case of free-induction decay, we take τ = t and for Hahn echo, τ = 2t. The transverse spin is then given by

⟨Sˆ+(τ)⟩ = eiϕ(τ)⟨eiL0(τ)Sˆ+⟩, (2.15) where ϕ(τ) = bt for free-induction decay and ϕ(τ) = 0 for Hahn-echo. We have also introduced

(0) (0) L0(t)Oˆ = [Hˆ (τ), Oˆ], i.e., L0(t) is the Liouvillian superoperator associated with Hˆ (τ). In both ∑ ˆ (0) ˆ (0) ˆz ˆ ˆ · ˆ Eqs. (2.10) and (2.12), H (τ) has the form H (τ) = S X(τ), where X(τ) = k hk(τ) Ik acts only on nuclear-spin degrees of freedom. This important property allows us to calculate each power in the Taylor series expansion of eiL0(τ)Sˆ+, giving eiL0(τ)Sˆ+ = exp[iXˆ(τ)]Sˆ+. For an initial state of the form ρˆS(0) ⊗ ρˆI(0), with ρˆS(0) and ρˆI(0) respectively the initial spin-qubit and nuclear-spin

+ + states, we find ⟨Sˆ (τ)⟩ = ⟨Sˆ (0)⟩Cχ(τ). This defines the coherence factor for a nuclear-spin bath

iϕ(τ) initially in state ρI (0) = χ: Cχ(τ) = e ⟨exp[iXˆ(τ)]⟩χ. Since dephasing arises from fluctuations

36 ∑ ˆ ˆ − ⟨ ˆ ⟩ · ˆ in the nuclear field, we define δχX(τ) = X(τ) X(τ) χ = k hk(τ) δχIk, and the coherence factor becomes ⟨ ⟩ iϕ(τ) i⟨Xˆ(τ)⟩χ iδχXˆ(τ) Cχ(τ) = e e e . (2.16) χ

⟨ ˆ ⟩ ≃ − 1 ⟨ ˆ 2⟩ The Gaussian approximation then corresponds to taking exp[iδχX(τ)] χ exp[ 2 [δχX(τ)] χ]. This approximation is justified when the following two conditions are met:

1. There are no correlations between nuclear spins, i.e. ⟨δ Il1 δ Il2 ⟩ = ⟨δ Il1 ⟩ ⟨δ Il2 ⟩ ∀ k ≠ χ k1 χ k2 χ χ k1 χ χ k2 χ 1

k2, with l1, l2 ∈ {x, y, z}.

2. N ≫ 1, where N is the number of nuclear spins within the range of the electron wavefunction.

The first condition is always met for the infinite-temperature states introduced in Section 2.2. As will be seen later in this section, it can also be satisfied for the ideally narrowed state defined by Eq. (2.3). However, the number of nuclear spins N needed to adequately satisfy the second criterion can be very large. Indeed, as discussed in Appendix A.4, subleading corrections to the

Gaussian approximation are typically only suppressed by ∼ O (1/N α), with α < 1 (α = 1/4 for free-induction decay and α = 1/8 for Hahn echo). Thus, non-Gaussian corrections can have an important effect in small systems such as single phosphorus donors in silicon, where N ∼ 102. As will be shown in Section 2.5.2, in situations where the Gaussian approximation predicts Gaussian

(∼ e−t2 ) decay, the exact solution will rather exhibit exponential (∼ e−t) behavior.

For the rest of this section, we assume that N is sufficiently large to avoid these finite-size effects. We can then proceed to the calculation of explicit coherence-factor formulas foreach considered nuclear-spin state. We first consider an infinite-temperature thermal state, which we ∑ | ⟩⟨ | model as described in Section 2.2, i.e. by a statistical mixture ρˆI(0) = j pj j j of M states ∏ | ⟩ | j ⟩ j = k ψk , with each nuclear spin randomly oriented in space. In this state, nuclear spins are uncorrelated, justifying use of the Gaussian approximation. Calculating the moments involved in Eq. (2.16) and taking M → ∞, we find that the only nonvanishing contributions come from ⟨ ˆx 2⟩ ⟨ ˆy 2⟩ ⟨ ˆz 2⟩ 1 (Ik ) th = (Ik ) th = (Ik ) th = 3 Ik(Ik + 1), and the coherence factor reduces to [ ] 1 ∑ C (τ) ≃ eiϕ(τ) exp − I (I + 1)|h (τ)|2 . (2.17) th 6 k k k k

37 ˆx ˆy ˆz Thus, for an infinite-temperature thermal state, all components of the nuclear field(hk, hk, hk) α appear with the same prefactor, with each individual value of hk (τ) set by the field distribution and coherence measurement scheme (free-induction decay or Hahn echo).

By definition, an ideally narrowed state ρˆn satisfies Eq. (2.3). Most generally, this state can be ∑ ∑ | ⟩⟨ | | ⟩⟨ | written as ρˆn = j pj j j + i≠ j ρij i j , where the sums are performed over all the degenerate | ⟩ ˆz z iϕ(τ)⟨ ˆ ⟩ eigenstates j of h with energy hn. We can then use the fact that Cn(τ) = e exp[iX(τ)] n = ∑ ∑ iϕ(τ) iϕ(τ) ⟨ ˆ ⟩ ⟨ | ˆ | ⟩ e j pjCj(τ)+e i≠ j ρijCij(τ), where Cj(τ) = exp[iX(τ)] j and Cij(τ) = j exp[iX(τ)] i . Assuming no special phase relationship between the eigenstates |j⟩, we can drop the second term in Cn(τ), which amounts to dropping the coherences in ρˆn [95]. Then, for each tensor- ∏ | ⟩ | j ⟩ product state j = k Ik, mk , nuclear-spin fluctuations on different sites are uncorrelated (i.e.,

⟨ ˆz ˆz ⟩ ∝ ′ δjIk δjIk′ j δk,k ), so we apply the Gaussian approximation to calculate each Cj(τ). We further | ⟩ ⟨ˆz⟩ ⟨ ˆx 2⟩ ⟨ ˆy 2⟩ find that all the moments involved vanish in each state j , except for Ik j, (δjIk ) j, (δjIk ) j, ⟨ ˆx ˆy⟩ ⟨ ˆy ˆx⟩ ⟨ ˆx ˆy⟩ ⟨ ˆy ˆx⟩ δjIk δjIk j, and δjIk δjIk j. Furthermore, we find that δjIk δjIk j and δjIk δjIk j cancel each ⟨ˆz⟩ j ⟨ x 2⟩ ⟨ y 2⟩ 1 − j 2 other out, and that Ik j = mk and (δjIk ) j = (δjIk ) j = 2 [Ik(Ik + 1) (mk) ]. The result is further simplified for N ≫ 1. Indeed, the number of available eigenstates |j⟩ of hˆz scales exponen-

j tially with N, such that states that have isotropic distributions of mk’s are overwhelmingly more j j 2 probable than states for which mk depends on the site k. We can thus replace every (mk) by ∑ † 2 2 its expectation value E(m ) = m p(m)m , with p(m) a probability distribution over accessible ⟨ x 2⟩ ⟨ y 2⟩ 1 values of m. Taking p(m) to be uniform, we find (δjIk ) j = (δjIk ) j = 3 Ik(Ik + 1). Inserting the calculated moments in Eq. (2.16) and considering a single realization j of a narrowed state, we obtain [ ] ∑ 1 ⊥ C (τ) ≃ eiϕn(τ) exp − I (I + 1)|h (τ)|2 , (2.18) n 6 k k k k

⊥ n ∑ − hz z where hk (τ) is the projection of hk(τ) in the x y plane and ϕn(τ) = ϕ(τ) + A k hk(τ), with ∑ n ⟨ |ˆz| ⟩ j z hz = j h j = k Akmk. Strikingly, hk(τ) appears only in the phase ϕn(τ), and therefore does not contribute to the decay of the coherence factor. This is a direct consequence of the vanishing ∑ † j 2 → 2 Deviations from the predictions of the approximation (mk) m p(m)m were numerically checked to be negligible even for systems as small as single phosphorus donors in silicon, where N ≃ 250. This is not surprising, since this approximation works when there is a large number of nuclear-spin configurations that lead to the same average polarization in a small region of space. That number of configurations, and thus the quality of the approximation, grows exponentially with N, in contrast with the Gaussian approximation, which improves as a power law in N.

38 variance in hˆz for a narrowed state.

2.4 Coherence measurement protocols

In the previous section, we found compact expressions for C(τ) for a nuclear-spin bath initially in an infinite-temperature thermal or narrowed state. We now use these results to obtain characteristic features of the coherence dynamics under a magnetic-field gradient. In other words, we replace → FID → HE hk(τ) hk (τ) for free-induction decay and hk(τ) hk (τ) for Hahn echo. In each case, we study both short-time and long-time behavior. We first consider free-induction decay for an initially narrowed nuclear-spin state and show that at long time (t ≳ max[1/γ∆bx, 1/γb], with ∆bx

x the typical range of bk experienced within the envelope wavefunction), the coherence factor decays ∼ − ∇ 2 as a Gaussian, C(t) exp[ (t/T2 ) ]. We then address the case of Hahn-echo decay with infinite- temperature thermal states. Although for short times, we find the Hahn-echo decay envelope tobe ∼ − ∇ 4 of the form exp[ (t/T2e) ], for long times coherence dynamics is very rich, displaying revivals, incomplete decay due to motional averaging, or exponential decay (reflecting a Markovian limit), depending on the particular physical setting and associated parameters.

2.4.1 Free-induction decay with narrowed states

Substituting Eq. (2.11) for hk(τ) into Eq. (2.18), which gives the coherence factor for a bath initially in a narrowed state, we obtain [ ] ∑ I (I + 1) C (t) ≃ eiϕn(t) exp − k k A2(nx)2g (t) , (2.19) n 6 k k k ( k ) ( ) 2 − 2 z 2 − sin ωkt 1 cos ωkt gk(t) = (nk) t + . (2.20) ωk ωk

39 The term that grows as ∼ t2 in the argument of the exponential in C(t) dominates for t ≳ max(1/γ∆bx, 1/γb), resulting in an approximate Gaussian decay,

[ ( ) ] ∇ 2 ≃ iϕn(t) − Cn(t) e exp t/T2 , (2.21) √ 1 1 1 ∑ = ν I (I + 1)γ2Σ2, (2.22) ∇ b 6 s s s s s T2 s ∑ 2 x 2 Σs = (bks Aks ) , (2.23) ks ∑ where s is taken over all nuclear species s in the material, such that νs, Is, and γs are, respectively, 2 the relative abundance, nuclear spin, and gyromagnetic ratio of species s. We have also defined Σs as a sum over nuclei ks belonging to the same species s, which depends on the geometry of the magnetic field through bx and on the electron wavefunction through A . Explicit formulas for Σ2 are given ks ks s in Appendix A.5 for various geometries. Thus, in this typical narrowed-state free-induction decay √ ∇ ∼ x scenario, we find Gaussian decay with inverse decay time 1/T2 (A/ N)(γ∆b /b). As mentioned in Section 2.3.2 and as will be shown explicitly in Section 2.5.2, this behavior results from the

Gaussian approximation, which breaks down when the number N of nuclear spins interacting with the electron spin is not sufficiently large.

2.4.2 Hahn echo

We now consider the coherence factor when the nuclear-spin bath is in an infinite-temperature → HE thermal state. The relevant coherence factor is given by Eq. (2.17). Substituting hk(t) hk (t) from Eq. (2.13) into Eq. (2.17), we find

[ ( )] 8 ∑ (A γ bx)2 ω t C (2t) ≃ exp − I (I + 1) k k k sin4 k . (2.24) th 3 k k ω4 2 k k

This sum can be simplified further in various physically meaningful cases which are discussed in this section.

The most straightforward way to reduce Eq. (2.24) is through a short-time approximation. Ex-

40 4 panding sin (ωkt/2) to leading order in ωkt/2 yields

[ ( ) ] ≃ − ∇ 4 Csh(2t) exp t/T2,e , (2.25) ( ) 1/4 1 1 ∑ = ν I (I + 1)γ2Σ2 . (2.26) ∇ 6 s s s s s T2,e s

∇ Thus, under the short-time approximation, decay occurs within a dephasing time T2,e, defined ∇ such that Csh(2T2,e) = 1/e. As will be shown in Section 2.5.1, this dephasing mechanism can dominate other processes (electron-nuclear flip-flops and nuclear dipolar interactions), which have given rise to decay measured in GaAs singlet-triplet qubits [14, 18, 19]. In addition, we further note that a gradient in the z component of the magnetic field would have no effect on Hahn- echo dephasing. Indeed, such a gradient can easily be incorporated in our model with the simple → z ∇ replacement γkb γkbk. Since T2,e is independent of b, z gradients do not contribute; in order for a longitudinal gradient to contribute to dephasing, nonsecular terms in the hyperfine interaction would need to be included [114].

As in recent calculations for hole-spin echo dynamics [68], for large b, a motional-averaging regime ≫ x ≃ can be reached in Hahn echo. Indeed, when b ∆b , we have ωks γsb and then Eq. (2.24) predicts recurrences in the coherence factor with a period ∼ 1/γsb. For sufficiently large ωk ∝ b, Eq. (2.24) shows that C(t) ∼ 1 for all time, in which case the motional-averaging regime has been reached. ≳ This occurs for b bc, where ( ) 1/4 ∑ Σ2 b = ν I (I + 1) s . (2.27) c s s s γ2 s s Thus, while a transverse magnetic-field gradient can enhance dephasing, this enhancement canbe controlled or eliminated with b large enough to reach the motional-averaging regime. In GaAs, for a single electron shared by two dots of radius r0 = 25 nm separated by l = 200 nm with

6 N = 4.4 × 10 nuclei within r0 (parameters taken from Ref. [19]) with an added transverse gradient x| ∗ ∼ ∂zB z=0 = 1 T/µm (a typical value from Refs. [25, 101, 115]), we find Bc = bc/g µB 300 mT. ∼ 2 For a single dot with the same properties, we find Bc 200 mT. To evaluate Σs, we have used Eqs. (A.25) and (A.24), respectively, for double- and single-dot geometries. ≪ x ≃ x Finally, when b ∆b we replace ωk γkbk in Eq. (2.24). The time-dependence of the resulting

41 sum over k then strongly depends on geometry. Therefore, we have derived explicit formulas for that sum limiting ourselves to single electrons in single and double dots in 2D, with Gaussian orbital

x wavefunctions. In both cases, we take bk from Eq. (A.23) and Ak, respectively, from Eq. (A.1) or (A.4) for single and double dots. Assuming an isotropic distribution of nuclear spins, we calculate an average of the sum over k appearing in Eq. (2.24) with respect to the angular degree of freedom in the position of each nucleus. For t ≳ 1/γ∆bx, for single dots the decay becomes exponential, with − ∇ Cth(2t) = exp( t/T2M). For a Gaussian electron wavefunction, the Ak’s are given by Eq. (A.1) and we obtain for b ≪ ∆bx, t ≳ 1/γ∆bx,

C (2t) = exp(−t/T ∇ ), (2.28) th √ 2M 1 π ∑ I (I + 1) A2 = ν s s s , (2.29) ∇ 24 s γ δbx N T2M s s where As is the total hyperfine coupling strength for nuclear spins of species s. We have also x ≡ x| defined δb r0 ∂zb z=0, the variation of bx over the single-dot Bohr radius r0. This is to be distinguished from ∆bx, the variation of bx over the length-scale of the whole device, which becomes x x| ∆b = l ∂zb z=0 in the case of a double dot with separation l. Thus, Eq. (2.29) shows that the decay process becomes Markovian, leading to a pure exponential decay of Cth(t), when the correlation ∼ x ∇ time of the nuclear-spin bath 1/γ∆b is short compared to the decay time T2M, i.e., when ∇ x x T2M > 1/γ∆b . This allows us to determine a critical field gradient, δbM, beyond which the leading contribution to dephasing is exponential. Comparing Eqs. (2.25) and (2.29) for a homonuclear

(single-isotope) spin bath, we find

( ) 1 π 1/3 √ A δbx = √ I(I + 1) √ . (2.30) M 2 3 2 γ N

Thus, we find Markovian decay when the broadening of nuclear spin precession frequencies dueto √ ∼ x ∼ the gradient γδbM exceeds the nuclear-field fluctuations A/ N. This behavior is not obtained in our model for double dots. Indeed, in that geometry, all nuclei that interact significantly with x| the electron spin are subject to a finite magnetic field, with average value (l/2) ∂zB z=0, with l the inter-dot spacing, the sign depending on which dot a nuclear-spin occupies. Averaging over

42 the angular degree-of-freedom and evaluating the sum in Eq. (2.24), we find that the double-dot geometry prevents the occurrence of a Markovian regime at long times, and that we rather have limt→∞ Cth(2t) = C0, C0 being a constant. In other words, in a double quantum dot, a very strong transverse field gradient alone can lead to motional averaging, in contrast to the single-dot case where a longitudinal field is needed. This qualitative difference in behavior between singleand double dots is illustrated in Fig. 2.4(c).

2.5 Physical realizations

In the previous sections, we have found the main features of the coherence dynamics under a magnetic-field gradient. The purpose of the present section is to predict whether these features could be measured in common materials used in current-day experiments. In particular, we find conditions under which this decoherence mechanism dominates over the leading dephasing sources known in these materials in the absence of a magnetic-field gradient. We focus on quantum dots in both GaAs and Si, and also investigate the case of single P donors in Si. Unless otherwise specified, for numerical evaluation, we have used hyperfine coupling constants and gyromagnetic ratios for the considered materials from the literature [12, 116].

2.5.1 Gallium arsenide quantum dots

∇ We first check how the inverse decay time, 1/T2 , for an initially narrowed nuclear-spin bath under ff a magnetic-field gradient compares with the inverse dephasing times due to flip-flops(1/T2 ) and dd ff nuclear-spin dipole-dipole couplings (1/T2 ). 1/T2 has been calculated from a Schrieffer-Wolff expansion [14] for b > A, which for GaAs roughly corresponds to B > 4 T. The derivations of ff ∝ 2 ∇ ∝ Ref. [14] result in 1/T2 1/b , while Eq. (2.22) predicts 1/T2 1/b. Thus, for sufficiently large b, dephasing due to the gradient will always dominate over flip-flop decay. This is illustrated in

Fig. 2.3(a), where the dephasing rates are compared assuming typical parameters [25, 101, 115]

∇ for a GaAs quantum dot and a magnetic-field gradient of 1 T/µm. In this particular case, 1/T2 dominates over the entire magnetic-field range considered.

In addition to flip-flops, nuclear dipolar interactions could also potentially lead to decay ona

43 B (T) (a) (b) Magnus 4 5 6 7 8 91011 Short time 0.4 1 ) 1 −

s 0.3 ) 6 t 10 (2 0.2 ∇ 0.5 × th

( 1/T2 C

2 0.1 ﬀ dd 1/T2 1/T2 /T 1 0 0 100 150 200 250 300 0 200 400 ∗ b = g µBB (µeV) 2t (ns) (c) 1 (d) x 0.8 B (r) ) t 0.6 z (2

th 0.4 C 0.2 0 012345678910 2t (µs)

Figure 2.3 – Dephasing due to a magnetic-field gradient in GaAs quantum dots. (a) Comparison ofthe ﬀ hyperfine-induced dephasing times due to the flip-flop (T2 , black solid line) and nuclear dipolar interactions dd ∇ (T2 , gray region), with the dephasing time, T2 , arising from the B-field gradient. These times have been evaluated for FID with a bath of N = 106 nuclear spins that are prepared in a narrowed state. We consider x| a single electron spin in a single quantum dot with radius r0 = 40 nm and Bx(z) = z ∂zB z=0, with x| ﬀ ∂zB z=0 = 1 T/µm, typical for experiments [25, 101]. Solid black line: inverse dephasing time 1/T2 due dd to flip-flop hyperfine interaction14 [ ]. Shaded grey area: predicted inverse dephasing times 1/T2 due to nuclear dipole-dipole interactions, depending on dot size and shape [18]. Dashed blue line: inverse dephasing ∇ time, 1/T2 , due to a magnetic-field gradient, from Eq.2.22 ( ). (b) Coherence factor for infinite-temperature thermal-state HE decay in a singlet-triplet qubit with dot radii r0 = 25 nm and spacing l = 200 nm. We × 6 x| also use N = 4.4 10 , B = 45 mT (parameters given in the experiment of Ref. [19]), and ∂zB z=0 = 0.25 T/µm. The transverse magnetic field has the shape illustrated in Fig. 2.2. We compare the full leading-order Magnus expansion of Eq. (2.24) (solid black line) to the short-time approximation of Eq. (2.25) (dashed red line). (c) Full leading-order Magnus expansion results of Eq. (2.24) for various B values considered in the experiment of Ref. [19]: 45 mT (dotted red line), 95 mT (thick black line), 195 mT (dashed blue line), and 495 mT (thin green line). The dashed vertical line indicates the experimental dephasing time T2,e, such that C(2T2,e) = 1/e, measured for B = 45 mT in Ref. [19]. Experimental dephasing times are longer for stronger magnetic fields. (d) Sawtooth-shaped transverse field distribution, for which leakage outofthe S-T0 subspace vanishes.

44 dd shorter time scale than the mechanisms considered here. Inverse dephasing times 1/T2 due to dipole-dipole interactions have been predicted in Fig. 14 of Ref. [18] to be between 104 s−1 and

105 s−1, with smaller values for small dot radii. For example, with a quantum-dot Bohr radius of dd × 4 −1 × 4 −1 r0 = 40 nm, as in Fig. 2.3(a), 1/T2 lies between 2 10 s and 5 10 s [see Fig. 2.3(a)] de- ∝ 2 pending on the dot thickness and the crystal orientation. On the other hand, assuming N πr0y0, ∇ ∇ (where y0 is the dot thickness), Eqs. (2.22) and (2.26) predict that 1/T2 and 1/T2e are both 2 x 2 x 2 independent of r0. Indeed, for single dots, Σ ∼ (Aδb ) /N ∼ (Ar0∂zb ) /N ∼ 1/y0, with r0 con- ﬀ ∝ ∝ 2 tributions canceling out. Therefore, since 1/T2 1/N 1/r0, the gradient dephasing mechanism dominates both flip-flop and dipole-dipole decay for sufficiently large dots with a fixed gradient,a

∇ ﬀ dd situation that corresponds to the parameters of Fig. 2.3(a), where 1/T2 exceeds 1/T2 , 1/T2 by dd nearly an order of magnitude. Additionally, 1/T2 is expected to be independent of b [99]. For large dots, gradient-induced dephasing should then dominate for a wide range of b. In summary, ﬀ ∝ 2 we have the following power-law hierarchy for the various mechanisms studied here: 1/T2 1/b , ∇ ∝ ∇ ∝ 1/T2 1/b, and 1/T2 1. For large dots it may then be possible to experimentally identify regimes where each mechanism dominates from a measurement of the inverse dephasing time as a function of b.

We now consider the case of GaAs singlet-triplet qubits, in which Hahn-echo dephasing dynamics without externally applied magnetic-field gradients has been extensively studied [19, 117]. As explained in Appendix A.2, the Hamiltonian for a system of two electron spins in the {|S⟩, |T0⟩} = √ √ {(| ↑↓⟩ − | ↓↑⟩/ 2), (| ↑↓⟩ + | ↓↑⟩/ 2)} basis can be approximately mapped to the single-spin Hamil- tonian of Eq. (2.1) when b > ∆bx. The results of Section 2.4.2 therefore apply equally well to the coherence of a singlet-triplet qubit [where, for a singlet-triplet qubit, C(t) measures coherence between the states | ↑↓⟩ and | ↓↑⟩]. However, a finite value of ∆bx ≠ 0 will typically lead {| ⟩ | ⟩} x x − x to spin flips, resulting in leakage out of the S , T0 subspace. Here, ∆b = bR bL, where ∫ x ≡ ∗ x | |2 bα dr g µBB (r) ψα(r) and ψL(R)(r) is a single-particle envelope function localized on the left (right) dot. The magnetic-field configuration illustrated in Fig. 2.2(a) leads, for example, to

∆bx ≠ 0 and consequently to finite leakage due to spin flips. One way to circumvent this leakage

(as discussed below) would be to arrange a “sawtooth” magnetic-field configuration, as shown in

Fig. 2.3(d).

45 We now make predictions for Hahn-echo dephasing under a magnetic-field gradient, using parameters from an experiment by Bluhm et al. [19], in which the Hahn-echo dephasing time for a singlet-triplet spin qubit in GaAs is measured as a function of a magnetic field B, with no applied x x| gradient, bk = 0. We consider the effect of a weak transverse gradient ∂zB z=0 = 0.25 T/µm, x x ∗ 2 corresponding to ∆B = ∆b /g µB = 50 mT. From Eq. (2.27), with Σs taken from Eq. (A.26), we estimate the critical field for motional averaging tobe Bc ∼ 225 mT. Thus, it is possible to set

x ∆B < B < Bc, such that leakage outside the S-T0 subspace remains small, but loss of electron- spin coherence due to the gradient is observed. In that regime, as illustrated in Fig. 2.3(b), the dynamics of the coherence factor is well-described by Eq. (2.25), which assumes a short-time expan- ∇ ≃ sion. Equation (2.25) predicts decay in a time T2,e 116 ns, while the Hahn-echo measurements without the gradient yield a dephasing time T2,e ∼ 600 ns for B = 45 mT. Thus, our model predicts that introducing a transverse magnetic-field gradient of only 0.25 T/µm in that experiment would severely decrease coherence times. However, as illustrated by Fig. 2.3(c), taking B ≫ Bc would drive this system to a motional-averaging regime, avoiding rapid decay from the magnetic-field gradient.

Finally, we stress that the results of the previous paragraph correspond to a best-case scenario for the geometry of Fig. 2.2(a) using a singlet-triplet qubit, where all leakage out of the computational subspace has been neglected. With the magnetic-field distribution Bx(r) of Fig. 2.2(a), there would be no leakage due to spin flips for a bonding/antibonding molecular state ψ(r) with ∫ 3 2 x equal weight on right and left dots, chosen such that d r|ψ(r)| B (r) = 0. For an electron in

∇ one of the states ψ(r), T2,e and Bc would be comparable to the values obtained above assuming localized states in the left/right dot, ψL,R(r). Alternatively, localized orbital states ψL,R(r) combined with the sawtooth field distribution Bx(r) of Fig. 2.3(d) would also avoid leakage provided ∫ 3 2 x d r|ψL,R(r)| B (r) = 0.

2.5.2 Silicon

We now apply our model to electron-spin qubits in Si, considering quantum dots and single P donors.

Mainly because the density of nuclear spins is much lower than in GaAs, the hyperfine field in Siis much smaller. Indeed, in natural Si, which contains 4.7% 29Si nuclei, we typically have [6, 11, 116]

46 A ∼ 100 neV. Without a magnetic-field gradient, for the isotopic concentration of natural silicon,

Witzel et al. (see Fig. 23 of Ref. [118]) have calculated a Hahn-echo dephasing time T2,e between

100 and 300 µs in both quantum dots and single donors (including hyperfine interaction and dipole- dipole couplings)Ê. These values are also supported by experiments performed on phosphorus-doped silicon at low concentration [119, 120]. Neglecting dipole-dipole and flip-flop interactions, our model predicts dephasing times due to a moderate magnetic-field gradient that are shorter than this

100-µs timescale. We will therefore neglect contributions to dephasing from flip-flop and dipolar interactions in the remainder of this section, focusing on the dominant magnetic-field-gradient mechanism.

Silicon quantum dots

∇ We first discuss results for narrowed-state free-induction decay. Fig. 2.4(a) shows 1/T2 as predicted from Eq. (2.22) for B ranging from 30 mT to 10 T. For B < 100 mT and for the parameters described

∇ in the caption of Fig. 2.4, T2 is smaller than 10 µs, i.e. at least an order of magnitude shorter than ∇ 2T2,e taken from Ref. [118], illustrated by the shaded gray area. However, for B > 1 T, T2 can be pushed beyond 100 µs, providing a way to avoid most of this additional dephasing. Fig. 2.4(b) shows the coherence dynamics of a spin in a single quantum dot. Interestingly, for b ∼ δbx (with x ≡ x| ∼ −t2 δb r0 ∂zb z=0), the result deviates from pure Gaussian ( e ) behavior. The coherence factor C(t) displays additional envelope modulations at the nuclear-spin precession frequency, ∼ γb. This effect is illustrated in the lower plot ofFig. 2.4(b). The agreement between the Magnus expansion and the exact solution is very good and even reproduces these small oscillations. Precise conditions for the validity of the Magnus expansion are derived in Appendix A.3. We find qualitatively similar results to those presented in Fig. 2.4(b) for a single spin in a double dot.

Because of the smallness of A in silicon, the critical gradients required for both motional averaging and exponential decay in HE decay that were obtained in Section 2.4.2 are much smaller than in

GaAs. Indeed, for a single dot, exponential decay from the onset of a Markovian regime would x ∗ x ∗ ≃ require δb /g µB > δbM/g µB 2 T in GaAs, while in silicon such a regime is obtained for x ∗ x ∗ ≃ δb /g µB > δbM/g µB 20 mT. Since flip-flop interactions are negligible in silicon dots even for B ∼ 1 mT, a transition from the non-Markovian to the Markovian regime could, in principle,

47 3 (a) 10 (b) 0 20 40 1 | ) )

1 2 t ( − 10 0.5 s C | 3 101 10 1 | × ) Ex. ( 0 t 10 S-T0 qubit ( 0.5 Mag. ∇ 2 C

Single dot | 0 /T

1 0.1 1 10 0 5 10 15 B (T) t (µs) (c) Singledot Doubledot 1 )]

t 0.8 Exact

(2 0.6 Magnus

C 0.4

e [ 0.2 R 0 0 5 10 15 0 10 20 30 2t (µs) 2t (µs)

Figure 2.4 – Dephasing due to a magnetic-field gradient in Si quantum dots. (a) Inverse decay times x| for narrowed-state FID from Eq. (2.22). We take a magnetic-field gradient ∂zB z=0 = 1T/µm and a quantum-dot Bohr radius r0 = 15 nm, with inter-dot spacing l = 80 nm in the double-dot case. We also take a total hyperfine coupling A = 210 neV and N = 104 from Refs. [6, 116]. Shaded gray area: range of total HE inverse dephasing times 2T2,e in Si without a magnetic field gradient [118]. (b) Coherence dynamics for narrowed-state FID in a single dot. Top: B = 100 mT, bottom: B = 30 mT. Dashed red line: Magnus expansion results, from Eq. (2.19), solid black line: exact solution, from Eq. (2.4). (c) Coherence dynamics for infinite-temperature thermal-state HE, with B → 0. Solid lines: Magnus expansion prediction, from Eq. (2.24). Dots: exact solution, from Eq. (2.7). Blue: ∆Bx = 20 mT, red: ∆Bx = 80 mT, black: x x x ∗ x ∆B = 400 mT, with ∆B = ∆b /g µB and ∆b defined in Section 2.4.1.

48 x ∗ be observed by tuning δb /g µB from ∼ 20 mT to a value > 100 mT, as illustrated in the left plot of Fig. 2.4(c), though sustaining such a large gradient on a length-scale of ∼ 15 nm may pose a challenge with current technology. Additionally, the right-hand-side plot of Fig. 2.4(c) shows that, consistent with Section 2.4.2, the coherence properties of a double dot in the long-time limit are radically different from those of a single dot. Indeed, for large ∆Bx, rather than reaching a

Markovian regime, the gradient can induce motional averaging. Both the single- and double-dot calculations shown here are performed with B = 0 to emphasize that the gradient itself is the source of that behavior, though calculations for a small but finite field of B ∼ 1 mT (required to justify the secular-hyperfine coupling assumption) give similar results.

Phosphorus donors in silicon

We now turn to single phosphorus donor spin qubits in silicon [8, 100, 121] (Si:P). In this system, the logical qubit is the spin of a single electron bound to the phosphorus atom. The P nucleus has a finite spin I = 1/2. When b is much larger than the strength of the coupling between the electron spin and the donor nuclear spin, ASI, the system is well-described by the Hamiltonian of ∗ Eq. (2.1). The relevant crossover magnetic field is ASI/(g µB) ∼ 1 mT, which can be achieved with a moderate applied magnetic field [121].

First, we calculate the evolution of coherence in narrowed-state FID. As shown in Fig. 2.5(a), ∇ ≃ Eq. (2.19) (solid black line) predicts Gaussian decay with time constant T2 65 µs, much smaller ‡ than 2T2,e ∼ 600 µs measured in the absence of a magnetic-field gradient [119, 120]. However, the corresponding exact solution, calculated from Eq. (2.7) and illustrated by the red line deviates from a Gaussian in the long-time limit. Indeed, after a short-lived Gaussian-dominated behavior, the exact solution exhibits an exponential tail. Yet, according to Eq. (A.14), the Magnus expansion

x x ∗ should converge rapidly for up to δB = δb /g µB ∼ 600 mT, which is overwhelmingly larger than δBx = 3 mT, as chosen here. We thus turn to the other approximation introduced in the derivation of Eq. (2.19): the Gaussian approximation.

To rigorously check that non-Gaussian contributions are responsible for this divergence in be-

‡ ∇ In the absence of measured narrowed-state decay in Si, we compare the FID decay time T2 to the 1/e-decay time 2T2,e for Hahn echo accounting for the full 2τ duration between intialization and echo.

49 (a)1 (b) 1 0.8 0.8 )] t |

) 0.6 0.6 t (2 ( C C

| 0.4 0.4 e[

0.2 R 0.2 0 0 01234 0 10 20 30 40 50 ∇ t/T2 2t (µs)

Figure 2.5 – Coherence dynamics of an electron spin qubit in a single P donor in Si. We take N = 250, | A = 210 neV, and ∂zBx z=0 = 1 T/µm. The system corresponds to a single dot with r0 = 3 nm, d = 3 and q = 1. (a) Narrowed-state FID with B = 200 mT. Thick black curve: prediction from the Magnus expansion under the Gaussian approximation, Eq. (2.19). Thin red curve: exact solution from Eq. (2.4). Blue dots: Magnus expansion without Gaussian approximation, from Eq. (2.32). Gray dashed lines, from top to bottom: exact coherence decay, respectively, for N = 125, 500, and 2000. (b) Hahn-echo decay with initial thermal states for N = 250, with varying B. From bottom to top: B = 0, 10 mT, 20 mT and 100 mT. Points: exact solution of Eq. (2.7). Solid lines of corresponding color: prediction from the Magnus expansion, Eq. (2.24).

1 29 havior, we notice that for I = 2 , which is the case for a bath of Si nuclei, the exponential in ˆ Eq. (2.16), eiδχX(τ), can be expanded using the identity

v · σˆ eiv·σˆ = cos v + i sin v, (2.31) v where v is an arbitrary vector and σˆ is the vector of Pauli matrices (ˆσx, σˆy, σˆz). Doing so, the coherence factor can be expressed as a product, ( ) ∑ ∏∞ | | z j | | ibt+i hz mj hk hkmk hk Cj(t) = e k k k cos + 4i sin . (2.32) 4 |hk| 4 k=1

We thus find an expression for the coherence factor under the leading-order Magnus expansion that includes non-Gaussian corrections if we replace hk(t) by its value given in Eq. (2.11). The result for single donors is given by the blue dots in Fig. 2.5(a). Strikingly, the leading-order Magnus expansion overlaps with the exact solution, clearly showing the breakdown of the Gaussian approximation for small nuclear-spin baths.

Though finding a criterion for the validity of the Gaussian approximation directly fromEq.(2.32)

50 is not a straightforward task, an order-of-magnitude estimate is sufficient here. As explained in

Appendix A.4, non-Gaussian corrections are suppressed only as a weak power of N (∼ 1/N 1/4).

Thus, N must be fairly large for the Gaussian approximation to work, a few hundreds typically not being sufficient. Indeed, for N = 250, we have N 1/4 ≃ 3.97 ∼1. As illustrated by the dashed gray lines in Fig. 2.5, as N increases, the exponential tail due to the non-Gaussian corrections fades away, with dephasing becoming almost entirely Gaussian for N ∼ 2000. To generate this plot, as

N is increased, we have adjusted A to keep the ratio N 5/6γb/A constant, such that the criterion for convergence of the Magnus expansion expressed by Eq. (A.14) remains satisfied.

Finally, Fig. 2.5(b) gives the evolution of the coherence factor under Hahn echo for a nuclear-spin bath initially in an infinite-temperature thermal state. From Eq.(2.27), we find that the critical magnetic field for the onset of motional averaging is Bc ∼ 19 mT. This is confirmed by the exact solution of Eq (2.7), as shown by the points in Fig. 2.5(b). In past experiments [8], an external field B ∼ 1 T has been used. For magnetic-field gradients of ∼ 1 T/µm, these experiments would be deep in the motional averaging regime, making dephasing due to the magnetic-field gradient completely reversible by Hahn echo. On a separate note, for Hahn echo, the predictions from the combination of Magnus expansion and Gaussian approximation closely fit with the exact solution, as illustrated by the solid lines. This does not contradict the results of Appendix A.4, since the criteria found there are sufficient but not necessary conditions for non-Gaussian contributions to be negligible.

Finally, from Eq. (2.24), we obtain a criterion for Markovian Hahn-echo decay due to a magnetic- field gradient that is very similar to Eq.(2.30), except that Ak values are taken for d = 3, q = 1.

x We find that Markovian behavior in single donors requires δB ∼ 100 mT. With r0 = 3 nm, this x| would require a gradient ∂zB z=0 > 30 T/µm, which could be technically very difficult to achieve. Larger quantum dots such as those discussed in Section 2.5.2 would thus be more appropriate to observe that behavior.

2.6 Conclusions

We have calculated the free-induction decay and Hahn-echo dynamics of an electron spin interacting with a nuclear-spin bath in the presence of an inhomogeneous magnetic field. Accounting for only

51 secular hyperfine coupling, we have given exact solutions for spin coherence in terms ofaproduct of O(N) matrices. Additionally, we have found closed-form simple analytical expressions within a leading-order Magnus expansion and Gaussian approximation.

In the case of free-induction decay, the coherence factor typically decays as ∼ e−t2 . In the case of Hahn echo, we have found ∼ e−t4 dephasing if the longitudinal magnetic field is below a known threshold, B ≲ Bc, above which this system enters a motional-averaging regime characterized by an incomplete decay of spin coherence. For electron spins in single quantum dots, a large transverse magnetic-field gradient decreases the nuclear-spin correlation time, leading to Markovian

(exponential) rather than ∼ e−t4 echo dephasing. In contrast, for spins in double quantum dots we rather obtain motional averaging and an associated incomplete coherence decay in the limit of a large uniform magnetic-field gradient.

We have further investigated the relevance of the above results to three physical systems: quantum dots in GaAs or Si and single P donors in Si. In each case, the inverse dephasing time due to the

∇ magnetic-field gradient mechanism 1/T2 can be much larger than those due to electron-nuclear ﬀ dd ∇ ≫ ﬀ dd flip-flop (1/T2 ) and nuclear dipole-dipole interactions (1/T2 ), i.e. 1/T2 1/T2 , 1/T2 . This ∇ dominance of 1/T2 can be confirmed in GaAs singlet-triplet qubits [see Fig. 2.3(c)]. Additionally, we have found that single quantum dots in Si would be best suited to observe a cross-over between non-Markovian and Markovian dephasing. Finally, we have shown that the Gaussian approximation fails to correctly predict decoherence in small systems such as single P donors in Si. This result highlights the difference between spin baths and bosonic baths, for which Wick’s theorem allows for exact suppression of all non-Gaussian contributions.

The model developed here applies directly to heavy-hole spin qubits, for which flip-flop terms can be suppressed through confinement and the secular-hyperfine limit can be reached evenfor very weak external magnetic fields [68]. Our results may also give insight into dephasing in spin- orbit qubits in InAs nanowires [109], where strain leads to inhomogeneous nuclear quadrupolar interactions that may decorrelate the nuclear-spin system as in the case of a magnetic-field gradient.

52 3 Microscopic models for charge-noise-induced dephasing of solid-state qubits

This chapter is the integral text from:

Several experiments have shown qubit coherence decay of the form α exp[−(t/T2) ] due to environmental charge-noise fluctuations. We present a

microscopic description for temperature dependences of the parameters T2 and α. Our description is appropriate to qubits in semiconductors interacting with spurious two-level charge fluctuators coupled to a thermal bath. We find dis-

tinct power-law dependences of T2 and α on temperature depending on the nature of the interaction of the fluctuators with the associated bath. We consider fluctuator dynamics induced by first- and second-order tunneling witha continuum of delocalized electron states. We also study one- and two-phonon processes for fluctuators in either GaAs or Si. These results can beusedto identify dominant charge-dephasing mechanisms and suppress them.

53 3.1 Introduction

One of the most challenging obstacles to the realization of solid-state quantum computing devices is decoherence caused by charge noise [7, 21, 35, 36, 38, 122]. Charge fluctuations in solid- state devices can arise from several sources, such as Johnson noise from electrical wiring [21, 123,

124], evanescent-wave Johnson noise from metallic gates [125, 126], or 1/f noise [127]. The most widely accepted explanation for 1/f noise is the presence in the host sample of bistable localized charge states [128]. Such two-level fluctuators involve tunneling between two spatial configurations with nearly equal potential energy and are routinely observed in amorphous materials [129–132].

These fluctuators have been observed as spurious resonances in the spectrum of superconducting phase qubits [133], and have been the subject of an extensive literature in the Josephson qubit community [134–138]. Similar two-level fluctuators consisting of a charge hopping between localized states have been observed in the environment of various other solid-state devices, including lateral gated heterostructures [78–81] and self-assembled quantum dots [139, 140]. Two-level fluctuators have thus been considered an important source of qubit dephasing in several theoretical studies [57,

141–145].

Despite the ubiquitousness of two-level charge fluctuators in the solid state, their physical nature can be expected to change from one system to the next. In addition, the microscopic mechanisms causing transitions within pairs of states can hardly be assumed to be universal. For example, the fluctuators can interact with a phonon bath[134, 146]. Alternatively, charge traps near metallic gates or itinerant bands can undergo tunneling [147–150]. To minimize the consequent deleterious effects on qubit coherence, it is important to be able to discriminate between different fluctuator baths (e.g., phonons or electrons) from a simple set of measurements.

Any experiment that is designed to measure qubit coherence will typically reveal information about the local environment and may shed light on charge dynamics. Qubit coherence is described by the coherence factor, which empirically often takes the form [38, 151]

α C(t) = exp[−(t/T2) ]. (3.1)

54 Here, the coherence time, T2, and stretching parameter, α, parametrize the decay of qubit coherence. When α = 1, Eq. (3.1) describes exponential decay, arising from Markovian evolution of the qubit.

For α ≠ 1, Eq. (3.1) describes a non-Markovian stretched-exponential (α < 1) or compressed- exponential (α > 1) decay.

The analysis of coherence measurements giving the above empirical form is often based on phenomenological techniques. In the presence of classical Gaussian dephasing noise, C(t) can be written as a simple function of the associated noise spectrum. An analytical form for the noise spectrum is then chosen to best fit the measured coherence factor, C(t) [38, 151]. For example, choosing a

1/f-like spectrum S(ν) ∝ 1/νβ, with β = α − 1 > 0, exactly yields a coherence factor described by

Eq. (3.1).

In this chapter, rather than assuming a 1/f-like spectrum, we begin from a generic microscopic model of fluctuator dynamics. This model results in a coherence factor that closely approximates the compressed-exponential form given in Eq. (3.1). From this model, we find closed-form expressions for the coherence time and stretching parameter, T2 and α. These results allow us to predict a crossover from the non-Markovian to the Markovian regime as temperature T is varied. In addition, we find that different microscopic mechanisms giving rise to fluctuator dynamics typically leadto distinct power-law dependences for T2(T ). In combination with complementary theoretical studies of T2(T ) in the Markovian regime (see, e.g., Refs. [82–84]), this will help to better understand and suppress microscopic sources of dephasing.

This chapter is divided as follows. In Sec. 3.2, we present the general features of the fluctuator model used throughout the chapter. This fluctuator model is used in Sec. 3.3 to show that the qubit coherence factor is well approximated by the compressed exponential form, Eq. (3.1). In Secs. 3.4 and 3.5, we find analytical expressions for the fluctuator equilibration time and the corresponding noise amplitude for fluctuators coupled to electron or phonon baths, respectively. These expressions are then used in Sec. 3.6 to find the temperature dependence of the qubit coherence time T2 and the stretching parameter α for the microscopic mechanisms considered in Secs. 3.4 and 3.5. We conclude by illustrating an application of this theory to recent experiments.

55 3.2 Two-level fluctuators

We consider an ensemble of two-level fluctuators coupled to a qubit. Each fluctuator is itself coupled to an independent thermal bath, allowing equilibration [see Fig. 3.1(a)]. The qubit is subject to a train of fast π-pulses. In the toggling frame [73], which accounts for dynamics induced by qubit rotations, the Hamiltonian is then

∑ [ ] ∑ ˆ ˆ ˆ n ˆ n ˆ n ˆ n H = HQ(t) + HF + HFB + HB + HQF(t), (3.2) | n {z } |n {z }

≡Hˆ0(t) ≡Vˆ where

1 1 Hˆ (t) = ℏω s(t)ˆσz, Hˆ n = ℏω τˆz, (3.3) Q 2 Q F 2 n n 1 Hˆ n (t) = ℏΩ s(t)ˆσzτˆz. (3.4) QF 2 n n

z z Here, we have introduced the Pauli matrices σˆ and τˆn for the qubit and for the n-th fluctuator, respectively. The qubit and fluctuator energy splittings are ℏωQ and ℏωn, respectively, and the qubit-fluctuator couplings are ℏΩn. The sign function, s(t), alternates between s(t) = 1 at times tm = t1, t2, t3, . . . , ts−1, accounting for a sequence of fast π-pulses, ending at t = ts [see Fig. 3.1(d)]. Here we will focus on free-induction decay (no π-pulse) and Hahn echo (a single π-pulse) [152], but this notation also allows for a direct extension to other pulse sequences, including, e.g., Carr- ∼ z z Purcell [22] or Uhrig dynamical decoupling [46]. Retaining only the Ising-like terms σˆ τˆn in the qubit-fluctuator Hamiltonian is justified within a secular approximation, in which the qubitand typical fluctuator splittings are assumed to be large compared to the relevant couplings, ℏωQ, ℏωn ≫ ℏ ˆ n ˆ n Ωn. The fluctuator-bath interaction HFB and bath Hamiltonian HB are left unspecified for now. Microscopic forms for these Hamiltonians are considered in Secs. 3.4 and 3.5, where we analyze fluctuator equilibration dynamics for specific physical systems. ˆ ′ ≡ ˆ − ⟨ ˆ ⟩ ˆ ′ ≡ ˆ ⟨ ˆ ⟩ To set up a perturbative expansion, we define V V V F and H0(t) H0(t) + V F , where

Hˆ0 and Vˆ are defined in Eq. (3.2). The expectation values ⟨·⟩F are taken with respect to the initial state of the fluctuators. We then move to the interaction picture, taking Vˆ ′ as a perturbation (i.e.,

56 1 -1

Figure 3.1 – (a) A qubit (Q) is coupled to an ensemble of independent fluctuators. Each fluctuator (Fn) is itself coupled to an independent bath (Bn). (b) A two-level fluctuator. Because of the interaction with its bath, the two-level fluctuator is excited at the rate γ↑ and relaxes at the rate γ↓. (c) Two-level fluctuators can be, e.g., two localized states (represented by the green wave functions) between which a charge can tunnel. (d) The qubit evolves under the influence of sharp control π-pulses.

∫ ˆ ˆ † ˆ − t ′ ˆ ′ ′ ℏ for a general operator O, OI(t) = U0 (t)OU0(t),U0(t) = exp[ i 0 dt H0(t )/ ]). We thus have

1 Vˆ ′(t) = ℏξˆ(t)s(t)ˆσz, (3.5) I 2 and we have introduced the noise operator

∑ [ ] ˆ z − ⟨ z⟩ ξ(t) = Ωn τˆn,I(t) τˆn F . (3.6) n

Our goal is to evaluate the coherence factor parametrized by a pulse sequence s,

s C (ts) = ⟨Sˆ+(ts)⟩ / ⟨Sˆ+(0)⟩ , (3.7)

x y where ⟨Sˆ+(ts)⟩ = [⟨σˆ (ts)⟩ + i⟨σˆ (ts)⟩] /2 is the off-diagonal element of the qubit density matrix in the σˆz eigenbasis. Under quite general conditions, Eq. (3.7) can be accurately evaluated using a

Magnus expansion [64, 65, 71]. The leading-order term in the Magnus expansion describes dynamics ˆ ′ under the action of the time average of VI (t). This leading-order term will always dominate at

57 sufficiently short time or for sufficiently rapid fluctuations in the noise operator (seeAppendix

B.1). Assuming a large number of independent fluctuators, ξˆ(t) becomes a source of Gaussian noise due to the central-limit theorem. Conditions for Gaussian noise to dominate over the leading non-Gaussian corrections to the qubit coherence factor are discussed in Appendix B.1. We will also assume that the noise is stationary, i.e., that the fluctuators are in a steady state. If, in addition, the initial state of the fluctuators and the qubit is separable, the coherence factor is givenby

∫ ∫ 1 ts ts s − dt1 dt2s(t1)s(t2)g(t1−t2) C (ts) = e 2 0 0 , (3.8)

g(t) = ⟨ξˆ(t)ξˆ(0)⟩, (3.9) where s → ∗ for free-induction decay and s → e for Hahn echo. In Appendix B.1, we consider subleading corrections to the leading-order Magnus expansion and Gaussian approximation. These corrections set limits on the range of validity of Eq. (3.8).

In the frequency domain, Eq. (3.8) becomes [40–43]

[ ∫ ∞ ] s − S(ν) s C (ts) = exp dν 2 F (νts) , (3.10) −∞ ν

s where the noise spectrum S(ν) and filter function F (νts) are given by

∫ 1 ∞ S(ν) = dt eiνtg(t), (3.11) 2π −∞ ∫ 2 ts 2 s ν iνt F (νts) = dts(t)e . (3.12) 2 0

A natural way to describe a compressed-exponential decay [Eq. (3.1)] is to postulate a 1/f-like noise spectrum [7, 38, 41, 151], A S(ν) = , (3.13) |ν|β with a general exponent β. Such a spectrum can also be justified from noise-spectroscopy measurements [61, 153]. Inserting the 1/f-like spectrum, Eq. (3.13), into Eq. (3.10) leads directly to a

s compressed-exponential decay [Eq. (3.1)] with stretching parameter α and coherence time T2 given

58 by

α = 1 + β, (3.14) ( ∫ ) ∞ s −1/α s F (x) T2 = 2A dx α+1 . (3.15) 0 x

s T2 exists when the integral in Eq. (3.15) converges, i.e., when α < 2 for free-induction decay (since F ∗(x) ∝ x2 for x → 0) and when α < 4 for Hahn echo (since F e(x) ∝ x4 when x → 0). One consequence of Eq. (3.13) is that the stretching parameter α depends only on the noise spectrum through the exponent β [Eq. (3.14)], not on the pulse sequence s. This procedure provides a

s satisfying and useful relationship between the stretching parameter α, coherence time T2 , and pulse sequence s. However, ultimately, Eq. (3.13) amounts to a (non-unique) reparametrization of the observed compressed-exponential decay and does not necessarily provide additional insight into the relevant physical processes or further predictive power. An alternative approach, which we take here, is to directly evaluate fluctuator dynamics from plausible microscopic interactions.

s Equation (3.8) shows that for a given pulse sequence, C (ts) is entirely determined by the autocorrelation function g(t) of the fluctuator-induced noise. To evaluate this autocorrelation function, we consider the regime where the fluctuator dynamics are described by a Markovian master equation. The evolution of a fluctuator is Markovian when the fluctuator equilibrates with itslocal

n n bath on a time scale τn that is long compared to the bath correlation time τcB. Typically, τcB is set ≫ n ∀ by the inverse bandwidth of bath excitations. When τn τcB n, the evolution of the fluctuators is described by a Lindblad-form master equation. Assuming, as illustrated in Fig. 3.1(a), that each fluctuator is coupled to an independent bath, the reduced density matrix, ρˆn, for fluctuator n evolves according to

ρˆ˙n(t) = Lnρˆn(t), (3.16)

L · − i ˆ n · nD + · nD − · n = ℏ[HF, ] + γ↑ [ˆτn ] +γ↓ [ˆτn ] , (3.17)

D ˆ ˆ ˆ ˆ ˆ † − 1 ˆ † ˆ ˆ ˆ ˆ † ˆ n n where [X]O = XOX 2 (X XO+OX X) and where γ↑ and γ↓ are the excitation and relaxation rates for fluctuator n [see Fig. 3.1(b)]. In the above equation, and throughout this chapter, the

59 centerdot (“·”) represents an arbitrary operator upon which the relevant superoperator is applied.

Using Eq. (3.17), it is then straightforward to evaluate g(t) with the usual multitime averaging formula. [62] Under the stationary-noise assumption, the autocorrelation function of the resulting noise becomes that of a mixture of independent Ornstein-Uhlenbeck processes [154, 155],

∑ −| − | − 2 t1 t2 /τn g(t1 t2) = ∆ξne . (3.18) n

Here, ∆ξn is the amplitude of the noise induced by fluctuator n and τn is the associated equilibration

n time. These parameters are related directly to the excitation (relaxation) rates γ↑(↓) and couplings

Ωn through

4γnγn 2 2 ↑ ↓ ∆ξn = Ωn n n 2 , (3.19) [γ↑ + γ↓ ]

n n 1/τn = γ↑ + γ↓ . (3.20)

∝ D z · We note that Eqs. (3.18) to (3.20) would be unchanged if a pure dephasing term [ˆτn] were added to Eq. (3.17).

As is well known, a mixture of Ornstein-Uhlenbeck processes, Eq. (3.18), can approximate 1/f noise with an appropriately chosen distribution of amplitudes and equilibration times [127, 128, 156–

158]. It is not, however, generally necessary to approximate a 1/f-like noise spectrum [Eq. (3.13)] to

s find a coherence factor C (ts) that approximates a compressed-exponential decay. As we illustrate numerically below, even a Lorentzian noise spectrum associated with a single equilibration time

τ = τn results in an approximate compressed-exponential decay over a wide parameter range.

3.3 Functional form of the coherence factor

Substituting the noise autocorrelation function [Eq. (3.18)] into the coherence factor [Eq. (3.8)] with the function s(t) for either free-induction decay (s → ∗) or Hahn echo (s → e) gives the

60 closed-form expressions [43, 98],

s s C (ts) = exp [−f (ts)] , (3.21) t ∑ f s(t ) = s − ∆ξ2τ 2 hs(t /τ ), (3.22) s T n n s n 2M n ∑ 2 1/T2M = ∆ξnτn, (3.23) n where

h∗(x) = 1 − e−x, (3.24)

he(x) = e−x − 4e−x/2 + 3. (3.25)

s s We define T2 to be the 1/e decay time of C (t) through

s s f (T2 ) = 1. (3.26)

s The form of C (ts) as given in Eq. (3.21) does not generally describe a pure compressed-exponential α decay, ∼ exp [− (t/T2) ]. However, we will show that Eq. (3.21) can approximate a compressed exponential over a wide parameter range. We therefore define a time-dependent stretching param-

s s s s α (ts) s eter α (ts) such that, instantaneously, f (ts) = (ts/T2 ) and introduce a typical value α of the stretching parameter at the 1/e decay time,

s s d log f (ts) s ≡ s s α (ts) = s , α α (T2 ). (3.27) d log(ts/T2 )

e e The functions f (ts) and α (ts) are shown in Figs. 3.2(a) and 3.2(b) assuming τn ≡ τ ∀ n. The s ≃ − s αs coherence factor can then be replaced by C (ts) exp[ (ts/T2 ) ] with small corrections when s s α (ts) varies slowly for ts in the vicinity of T2 . The coherence factor behaves very differently in either the “slow-noise” or “fast-noise” regime.

These two regimes are determined by the ratio of the correlation time τc [the decay time of the

s noise autocorrelation function g(t)] to the coherence time, T2 . We define the correlation time τc

61 through [67] ∫ ∞ ∑ dtg(t)t ∆ξ2τ 2 ≡ ∫0 ∑n n n τc ∞ = 2 , (3.28) 0 dtg(t) n ∆ξnτn where the second equality follows directly from Eq. (3.18).

s The slow-noise regime is given by T2 < τc. In this regime, g(t) is slowly-varying in Eq. (3.8) over ∼ s the time scale of interest ( T2 ). Expanding g(t) around t = 0 and keeping the leading nontrivial correction in Eq. (3.8) then gives the compressed-exponential form in Eq. (3.1), with α = αs and

s T2 = T2 for decoupling sequence s, consistent with known results for Gaussian spectral diffusion due to classical noise [98],

( ) ∗ ∗ ∑ 1 1 2 2 ∗ ≪ α = 2, 1/T2 = 2 n ∆ξn , (T2 τc), (3.29) ( ∑ ) 1 e e 1 2 3 e ≪ α = 3, 1/T2 = 12 n ∆ξn/τn , (T2 τc). (3.30)

s ≳ s s In the opposite (fast-noise) regime, T2 τc, we evaluate T2 and α from Eqs. (3.26) and (3.27). s ≳ Neglecting exponentially small corrections in T2 /τc 1, we find the coherence times ∑ (1 + ∆ξ2τ 2) T ∗ = ∑ n n n , (T ∗ ≳ τ ), (3.31) 2 ∆ξ2τ 2 c n∑ n n (1 + 3 ∆ξ2τ 2) e ∑ n n n e ≳ T2 = 2 , (T2 τc), (3.32) n ∆ξnτn and stretching parameters

αs = 1 + βs, (3.33) ∑ ∗ 2 2 ∗ ≳ β = ∆ξnτn, (T2 τc), (3.34) n∑ e 2 2 e ≳ β = 3 ∆ξnτn, (T2 τc). (3.35) n

In contrast with the result from an assumed 1/f-like spectrum in Sec. 3.2, here the stretching parameter αs is sensitive to the pulse sequence s. In fact, the parameters βs for echo and free- induction decay are related by a universal factor of three in the fast-noise regime, βe ≃ 3β∗.

Equations (3.29) to (3.35) provide a complete analytical description of both the coherence time

s s T2 and form of decay (through α ) in either the slow-noise or fast-noise regime. This description

62 2 can be related to a microscopic model of fluctuator dynamics through the noise amplitudes ∆ξn and s s equilibration times τn. In particular, T2 and α will inherit temperature dependences associated n with the fluctuator excitation (relaxation) rates γ↑(↓) through Eqs. (3.19) and (3.20). In the rest of this chapter, we will evaluate these temperature dependences for physically relevant microscopic mechanisms and connect fluctuator dynamics to qubit coherence through Eqs.(3.29) to (3.35).

s Since the qubit coherence time T2 and noise correlation time τc typically have distinct temperature dependences, tuning the bath temperature will typically induce a transition between the slow-noise s s ≳ (T2 < τc) and fast-noise (T2 τc) regimes. To describe the transition from the slow-noise to the fast-noise regime, it is useful to define a dimensionless parameter that controls a Markov approximation:

τ ∑ η ≡ c = ∆ξ2τ 2. (3.36) T n n 2M n

When η ≪ 1 (the fast-noise limit), a Markov approximation gives exponential decay (αs = 1), with ∗ ≃ e ≃ → ∞ T2 T2 T2M. In the opposite (slow-noise) limit, η , we recover the results of Eqs. (3.29) and (3.30).

While the coherence factor exhibits a simple form in either the slow-noise or fast-noise limit, it is less clear how to simply describe the decay in the intermediate regime η ∼ 1. It is, how-

s ever, straightforward to numerically verify the assumed compressed-exponential form {C (ts) = − s αs } exp [ (t/T2 ) ] . To simplify the analysis, we assume a single equilibration time for all fluctuators,

τn ≡ τ ∀ n, corresponding to a pure Lorentzian noise spectrum S(ν). In this case, Eq. (3.22) reduces to [ ] t f s(t ) = η s − hs(t /τ) . (3.37) s τ s

e − e αe In Figs. 3.2(c-e), we compare C (ts) with exp[ (ts/T2 ) ] for a fixed correlation time τ and a range s of η. For a given value of η, the maximum error made in replacing C (ts) by the compressed- exponential form is

≡ {| s − − s αs |} εmax max C (ts) exp[ (ts/T2 ) ] . (3.38) ts∈[0,∞[

63 ts/τ ts/τ − − 10 1 100 101 102 103 10 1 100 101 102 103 ∝ 3 (a) e ts) t/τ (b) ( ) /η 0

α s )

10 t s (

t 2 e

( −3 e 10 3 α f ∝(t/τ) 1 e e T2 Exact C (ts) e αe 0.08 exp[−(ts/T2 ) ] (f) 0.06 1 e )

s (c) (d) (e) c 0.04 max t ε ( 0.5 d e 0.02 C 0 0 − 1 2 1 2 1 2 10 2 1 102 e e e ts/T2 ts/T2 ts/T2 η

e Figure 3.2 – Approximate compressed-exponential form of C (ts) when τn ≡ τ ∀ n. (a) The slope of e e e ≃ − e αe f (ts) in log-log scale gives α (ts) [see Eq. (3.27)]. (b) We approximate C (ts) exp[ (ts/T2 ) ] by taking e ≡ e e e e ≃ e ≪ α α (T2 ). When α (T2 ) 3 (in the slow-noise limit, where T2 /τ 1 is in the blue area), decay is faster e e e ≃ e ≫ than exponential and T2 is given by Eq. (3.30). When α (T2 ) 1 (in the fast-noise limit, where T2 /τ 1 e ≃ is in the red area), the decay is purely exponential and T2 T2M is given by Eq. (3.23). (c-e) Comparison e of the exact (solid black line) and compressed-exponential (dashed red line) forms of C (ts). (c) η = 10, e ≃ − e αe e ≡ e s (d) η = 0.1, (e) η = 0.01. (f) Maximum error made by taking C (ts) exp[ (ts/T2 ) ] with α α (T2 ), Eq. (3.38). Dots correspond to (c-e).

64 Figure 3.3 – Tunneling processes between localized electron states and a continuum of delocalized states. (a) Direct tunneling. This first-order process is only allowed if |ϵn − µ| ≲ kBT , where ϵn is the energy of the localized state for fluctuator n and µ is the chemical potential of the electron reservoir. (b) Cotunneling between pairs of localized states forming a fluctuator n. This second-order process occurs if |ϵαn−ϵβn| ≲ kBT .

In Fig. 3.2(f), we plot εmax as a function of η. Dots in Fig. 3.2(f) indicate the three values of η corresponding to Figs. 3.2(c-e). The error, εmax, is maximized for η ≃ 0.1, the value taken for Fig. 3.2(d). Even in this worst case, the difference between the exact and compressed-exponential

e forms of C (ts) is small (εmax ≃ 0.06). Thus, while the microscopic analysis presented here leads, in general, to a complex functional form [Eq. (3.21)], this functional form will likely be indistin- guishable from a compressed exponential in many experiments.

3.4 Electron baths

In this section, we consider charge fluctuators described by Anderson impurities. These impurities can equilibrate through tunnel coupling to a continuum of delocalized electronic states in a reservoir

(the bath). The electron reservoirs are held in thermal equilibrium with occupation described by a Fermi-Dirac distribution nF(ϵ) = 1/{exp[(ϵ − µ)/kBT ] + 1} at a common temperature T and chemical potential µ. As illustrated in Fig. 3.3, we consider both first-order (direct tunneling) and second-order (cotunneling) processes. Qubit decoherence due to fluctuators tunnel-coupled to an electron reservoir has been considered previously in, e.g., Refs. [147–150, 159].

65 3.4.1 Direct tunneling

In the first-order process (direct tunneling), we assume that each impurity n is coupled to an independent bath through a Fano-Anderson model. [160] We then have

∑ ∑ ˆ n ˆ† ˆ ˆ n † HF = ϵndnσdnσ, HB = ϵkcˆknσcˆknσ, (3.39) σ kσ ∑ ( ) ˆ n ∗ † ˆ ˆ† HFB = tkncˆknσdnσ + tkndnσcˆknσ . (3.40) kσ

ˆ(†) (†) For each fluctuator n, we have introduced dnσ and cˆknσ, the annihilation (creation) operators for the localized and delocalized states, respectively. The corresponding eigenenergies are ϵn and ϵk.

The spin index is σ ∈ {↑, ↓} and tkn is the amplitude for tunneling between the impurity and the continuum. Assuming strong Coulomb blockade for each impurity (due, e.g., to a large on-site charging energy), we restrict to the space of singly-occupied (|α⟩ ≡ |σ⟩n) and empty (|β⟩ ≡ |0⟩n) states. Thus, each impurity n is a two-level fluctuator with splitting ℏωn = ϵn − µ. Each impurity can couple to the qubit through the Coulomb interaction [57, 161]. Under these assumptions,

Eqs. (3.39) and (3.40) correspond to the physical model of Eq. (3.2).

n n In direct tunneling, we find the excitation (γ↑ ) and relaxation (γ↓ ) rates of a given fluctuator using Fermi’s golden rule,

∑ n 2π | ⟨ | ˆ n | ⟩ |2 ℏ − γα→β = ℏ ρ(i) n βf HFB α i n δ( ωn + Ef Ei), (3.41) if where α and β are collective indices (including, e.g., both spin and orbital degrees of freedom) labeling the initial and final states of the fluctuator, i and f label the initial and final states of the bath with energies Ei and Ef , respectively, and ρ(i) is the probability for the bath to be initially in state i. In thermal equilibrium, this probability distribution is given by the Fermi-Dirac ∑ ∫ → distribution. We take the continuum limit k dϵDel(ϵ) when summing over the initial and final bath states i and f. Using Eqs. (3.19) and (3.20), we calculate the noise amplitudes ∆ξn and

n n the fluctuator equilibration rates 1/τn from γ↑ and γ↓ . Summing the rates of the transitions from

66 the reservoir to the degenerate eigenstates |↑⟩ and |↓⟩ then gives

2 ℏωn/kBT 2 8Ωne ∆ξn = ℏ , (3.42) (2 + e ωn/kBT )2 ℏ 1 2π 2 + e ωn/kBT = D (ϵ )|t (ϵ )|2 , (3.43) el n n n ℏω /k T τn ℏ 1 + e n B

where tn(ϵn) is the tunneling amplitude tkn in the continuum limit. Equation (3.42) implies that the fluctuators are frozen out and have exponentially small contribution to qubit dephasing when

ℏ|ωn| = |ϵn − µ| > kBT , as expected from Fig. 3.3. In the opposite (high-temperature) limit, ℏ| | 2 ≃ 8 2 kBT > ωn , we have ∆ξn 9 Ωn, giving a maximal contribution to qubit dephasing. In this high-temperature limit, Eq. (3.43) also gives an equilibration rate that is approximately constant with temperature.

3.4.2 Cotunneling

In the second-order tunneling process (cotunneling), we consider the case where two localized states with energies ϵnα and ϵnβ are coupled to the same electron reservoir n. We now have

∑ ∑ ( ) ˆ n ˆ† ˆ ˆ n ∗ † ˆ HF = ϵnldlnσdlnσ, V = tlkncˆknσdlnσ + H.c. , (3.44) lσ lkσ

∈ { } ˆ n − ∀ ∈ where l α, β . In this case, HB is again given by Eq. (3.39). When µ ϵln > kBT l {α, β}, direct tunneling is forbidden. However, the cotunneling process illustrated in Fig. 3.3 can still occur if ϵβn − ϵαn < kBT . Each fluctuator n is then described by a pair of localized states coupled to the same bath with fluctuator energy splitting ℏωn = ϵβn − ϵαn. The fluctuator- bath Hamiltonian corresponding to the (second-order) cotunneling process is obtained using the

Schrieffer-Wolff expansion. To leading order in Vˆn, the effective Hamiltonian for this process can be written as

[( ) ] ˆ n 1 1 ˆ n ˆ n HFB = n V , V , (3.45) 2 L0

67 1/τn Direct tunneling ∝ 1 Cotunneling ∝ T

Table 3.1 – Temperature dependence of the equilibration rates 1/τn for electronic baths when ℏωn < kBT in the case of first-order (direct) tunneling and second-order cotunneling. In both cases, 1/τn is independent of the fluctuator splitting ωn.

n· ˆ n ˆ n · where L0 = [HF + HB, ]. Using this fluctuator-bath Hamiltonian, we evaluate excitation and relaxation rates using Fermi’s golden rule, Eq. (3.41). As written, Eq. (3.45) contains formal divergences (zero denominators) corresponding to resonant cotunneling processes. These contributions can be systematically regularized [162], leading to exponentially small corrections in the limit

µ − ϵln > kBT ∀ l, which we assume here. Neglecting resonant cotunneling in this limit, from the inelastic cotunneling rates [163], we then find

( ) 2 1 1 ℏΓn ℏωn ≃ ωn coth , (3.46) τn π µ − ϵn 2kBT

Γn = 2πDel(µ)|tαn(µ)||tβn(µ)|/ℏ, (3.47)

2 2 2 ℏ ∆ξn = Ωn sech ( ωn/2kBT ) . (3.48)

Here, we have introduced ϵn ≡ (ϵαn + ϵβn)/2. Eq. (3.46) is valid up to corrections of order ∼ ℏ − 2 ωn/(µ ϵn). The difference between ∆ξn given in Eq. (3.48) and that given in Eq. (3.42) for direct tunneling arises from spin degeneracy [164, 165]. As in the case discussed below Eq. (3.42), 2 ℏ Eq. (3.48) implies that ∆ξn decays exponentially for ωn > kBT . However, from Eq. (3.46), for kBT > ℏωn the equilibration rate 1/τn now increases linearly with T .

Table 3.1 summarizes the distinct temperature dependences obtained for 1/τn due to the two processes discussed in this section. These will be useful in Sec. 3.6, when we evaluate the temperature

s s dependences of T2 and α .

68 3.5 Phonon baths

2 In this section, we evaluate the amplitude ∆ξn and equilibration time τn for fluctuators coupled 2 to independent phonon baths. For all processes considered in this section, ∆ξn is given simply by Eq. (3.48), valid in the absence of spin degeneracy. This expression can be derived from Eq. (3.19)

n n simply by assuming detailed balance between γ↑ and γ↓ . To evaluate τn, we will consider one- phonon direct, and two-phonon sum and Raman processes, as indicated schematically in Figs. 3.4(a- c).

Each fluctuator consists of two impurity states |α⟩ and |β⟩. These could be, e.g., two localized states in a double well, as illustrated in Fig. 3.1(c), or the ground and excited states of a single donor impurity. The energy splitting between states |α⟩ and |β⟩ for fluctuator n is ℏωn ≡ ϵβn −ϵαn. Thus, the fluctuator and bath Hamiltonians for fluctuator n are

∑ ∑ ˆ n ˆ† ˆ ˆ n ℏ † HF = ϵnldlnσdlnσ, HB = ωqλaˆqnλaˆqnλ, (3.49) lσ qλ

(†) where aˆqnλ annihilates (creates) a phonon with wave vector q in branch λ of the phonon bath n. We work within the regime of validity of the envelope-function approximation for the impurity. We also assume acoustic phonons with a linearized dispersion. We will focus on two materials: GaAs and silicon. For either material, ignoring anharmonic corrections, the fluctuator-bath interaction is then given by ∑ ∑ † † ˆ n n ˆ ˆ ′ HFB = AqλχSχ,ll′ (q)dnlσdnl σ(ˆaqnλ +a ˆ−qnλ), (3.50) σqλχ l≠ l′ where l, l′ ∈ {α, β}. In Eq. (3.50), we have introduced the electron-phonon coupling strength

d − p Aqλχ = Aqλχ iAqλ, (3.51) where d and p label the deformation and piezoelectric contributions, respectively. The form of these contributions is given in Appendix B.2 in terms of material parameters. In Eq. (3.50), we

69 have also introduced the form factor

∫ n | |2 n∗ n iq·r Sχ,ll′ (q) = dr αχφχ(r) Fχl (r)Fχl′ (r)e , (3.52)

where φχ(r) is the Bloch amplitude with wave vector kχ corresponding to the degenerate conduction-

n band minimum (valley) χ, and Fχl(r) is the corresponding envelope function for impurity state l ∑ n of fluctuator n. αχ is the coefficient for valley χ appearing in the wave function χ αχFχl(r)φχ(r) of impurity state l.

The coupling between pairs of impurity states is suppressed if they are separated by more than the impurity size, ℓimp, describing the extent of the envelope Fχl(r) [see Eq. (3.55), below]. Here, we assume ℓimp satisfies

ℓimp < ℏvλ/kBT ∀ λ, (3.53)

where vλ is the phase velocity of branch λ. Under the above condition, the typical phonon wavelength 2π/qth ∼ hvλ/kBT is much longer than the spacing between coupled impurity states. The

n form factor Sχ,ll′ (q) defined in Eq. (3.50) can then be approximated in the small-q (long-wavelength) limit,

n 2 χn S ′ (q) ≃ i|αχ| q · ℘ ′ , (3.54) χ,ll ∫ ll χn | |2 n∗ n ℘ll′ = dr r φχ(r) Fχl (r)Fχl′ (r), (3.55)

χn ′ where ℘ll′ is the transition dipole matrix element between states l and l . To obtain Eq. (3.54), we have used the first non-vanishing term of a Taylor expansion around q = 0. This amounts to neglecting phonon-bottleneck effects166 [ ], which suppress the contribution from short-wavelength

(high-energy) phonons having a typical wavelength on the order of the impurity spacing. For vλ = 3070 m/s (the smallest phase velocity among all the relevant branches in GaAs and silicon) and T = 100 mK, Eq. (3.53) implies that these bottleneck corrections can be neglected when

ℓimp < 2 µm. At higher temperature or in the presence of a non-thermal source of phonons, it may be necessary to account for the full q-dependence in Eq. (3.52). This can be done, in principle, although the

70 resulting temperature dependences will generally be more complicated, not described by the robust power laws we find here in the low-temperature limit.

3.5.1 Direct (one-phonon) processes

Figure 3.4 illustrates the fluctuator-phonon processes considered in this section. In the leading- order process, the fluctuator absorbs or emits a phonon with frequency ωqλ = ωn [see Fig. 3.4(a)]. The equilibration rate corresponding to this process is obtained from the coupling Hamiltonian,

Eq. (3.50), using Fermi’s golden rule, Eq. (3.41). In GaAs, the conduction band has a unique minimum (a single valley), such that αχ = δχ,1 in Eq. (3.54). In contrast, the conduction-band √ minimum of bulk silicon is six-fold degenerate. For silicon, we take αχ = 1/ 6 ∀ χ, consistent with the ground state for donor impurities [167, 168]. Other choices of αχ would not change the final temperature dependence of the equilibration rate. We also assume the transition dipole χn n ∀ n matrix element to be valley-independent, ℘ll′ = ℘ll′ χ. Valley-independence of ℘ll′ amounts to n neglecting anisotropy of the envelope functions Fχl(r) and thus of the effective mass [167]. With the above assumptions, we find the equilibration rate for the direct process [ ] ( ) ( ) ( ) ( ) ( ) 1 1 ω 4 4 4 ee 2 ω 2 9π ω ℏω = Ξ2 n + 1 + ζ2 14 n n |℘n |2 coth n , (3.56) D ℏ 3 αβ τn 3 vLA 35 3 ε vLA matωD kBT where ζ = vLA/vTA, with vLA and vTA the phase velocities of the longitudinal and transverse acoustic branches, respectively. Equation (3.56) assumes the piezoelectric tensor for a zincblende- structure material, such as GaAs. For this structure, the only non-vanishing tensor element is e14

(in Voigt notation). Silicon is not piezoelectric, resulting in e14 = 0. We have also introduced the Debye frequency ωD, the elementary charge e, the mass per lattice atom mat, and the static dielectric constant ε. In GaAs, Ξ = a(Γ1c) ≃ −8.6 eV, where a(Γ1c) is the volume deformation

1 potential for the conduction-band minimum. In silicon, Ξ = Ξd + 3 Ξu, where Ξd and Ξu are deformation potentials at zone boundaries [168, 169].

71 Figure 3.4 – Coupling of a fluctuator consisting of two localized electron states interacting with a phonon bath. We consider transitions up to second order in the electron-phonon interaction. (a) Direct phonon absorption. (b) Excitation due to the two-phonon sum process. (c) Raman excitation. (a-c) We also include the corresponding relaxation processes (not shown). (d) Equilibration rate for a fluctuator coupled to phonons through the deformation and piezoelectric mechanisms, calculated with Eqs. (3.56) to (3.58). Solid red line: GaAs lattice. Dashed blue line: silicon lattice. In GaAs [168, 170, 171], Ξ = a(Γ1c) = −8.6 eV, 2 e14 = −0.16 C/m , vLA = 5210 m/s, vTA = 3070 m/s, ℏωD/kB = 360 K, and ε = 12.9 ε0. In silicon, 1 ℏ Ξ = Ξd + 3 Ξu, Ξd = 5 eV, Ξu = 8.77 eV, e14 = 0, vLA = 9040 m/s, vTA = 5400 m/s, and ωD/kB = 640 K. | n | n ℏ ℏ For both GaAs and silicon, we take ℘αβ = 1 nm, ℘0 = 10 nm, ωn = 10 neV, and ωγ = 1 meV.

72 3.5.2 Two-phonon processes

We now consider the second-order processes stemming from the coupling Hamiltonian, Eq. (3.50).

We first consider the two-phonon sum process. In this case, two phonons with frequencies satisfying

ωqλ + ωq′λ′ = ωn are simultaneously absorbed or emitted [see Fig. 3.4(b)]. We also include the Raman process, in which a phonon in mode qλ is absorbed and another is emitted in mode q′λ′, with the constraint ωqλ − ωq′λ′ = ωn [see Fig. 3.4(c)]. Both of these second-order processes require the presence of an auxiliary third level |γ⟩n, with energy splittings relative to states |α⟩n and | ⟩ ℏ n ℏ n β n denoted by ωαγ and ωβγ. We obtain the effective Hamiltonians for these second-order ˆ n ˆ n processes using the leading-order Schrieffer-Wolff expansion, Eq.3.45 ( ), taking V = HFB from n Eq. (3.50). In general, resonant denominators arise in these second-order processes for ωqλ = ωαγ, n ℏ n ωβγ. We neglect contributions from these resonances, which are exponentially suppressed for ωαγ, ℏ n ≫ ωβγ kBT [172, 173]. We then evaluate the corresponding fluctuator equilibration rates using

Fermi’s golden rule, Eq. (3.41). For ℏωn < kBT , we find the temperature and fluctuator-splitting dependences of the sum and Raman processes shown in Table 3.2. Explicitly, the equilibration rate for the Raman process is [ ( ) ( ) ( ) 1 (2π)7(℘n)4 15π4 Ξ4 k T 11 18π2 ee 2 1 + 4 ζ2 k T 9 ≃ 0 B + Ξ 14 3 B τ R (ℏωn)2m2 ω6 11 v8 ℏ 175 ε v6 ℏ n γ at D LA LA ] ( ) ( ) 27 ee 4 (1 + 4 ζ2)2 k T 7 + 14 3 B , (3.57) 4 ℏ 8575 ε vLA

n n n −1 n 4 | n |2| n |2 | n · n∗ |2 where we have introduced ωγ = (1/ωαγ + 1/ωβγ) and (℘0 ) = ℘αγ ℘γβ + ℘αγ ℘γβ . The equilibration rate for the sum process is given by Eq. (B.36) in Appendix B.3. Comparing

Eq. (B.36) with Eq. (3.57), we immediately see that the prefactors are identical up to a factor of order one. Thus, using the ωn and T dependences summarized in Table 3.2, the condition for the Raman process to dominate over the sum process can be shown to be ℏωn < kBT . In other words, the Raman process always dominates over the sum process for fluctuators that participate significantly to qubit dephasing. Thus, we neglect the sum process in the rest of this chapter, regardless of the material. In contrast, the condition for the Raman process to dominate over the direct process does depend on the relevant material parameters.

73 Deformation (∼ Ξ) Piezoelectric (∼ e14) D ∝ 4 × ∝ 2 × Direct, 1/τn ωn T ωn T Σ ∝ 9 × 2 ∝ 5 × 2 Sum, 1/τn ωn T ωn T R ∝ 11 ∝ 7 Raman , 1/τn T T

Table 3.2 – Power-law dependences of each contribution to the fluctuator equilibration rates 1/τn on ωn and kBT for the electron-phonon interaction when ℏωn < kBT .

In Fig. 3.4(d), we plot the total equilibration rate,

1 1 1 = D + R , (3.58) τn τn τn as a function of temperature. The solid red (dashed blue) line shows the equilibration rate for a fluctuator in GaAs (silicon). For either material, Fig. 3.4(d) illustrates a typical crossover from a ≃ D ∝ low-temperature rate dominated by the direct (one-phonon) process 1/τn 1/τn T to a high- ≃ R ∝ 7 temperature rate dominated by the two-phonon Raman process (1/τn 1/τn T for piezoelectric ≃ R ∝ 11 coupling and 1/τn 1/τn T for deformation-potential coupling; see Table 3.2). From Eq. (3.56), the piezoelectric contribution dominates in the direct (one-phonon) process

D when ωn < ωcrit where

√ ( ) 12 4 e|e | v ωD = 1 + ζ2 14 LA . (3.59) crit 35 3 ε Ξ

For the Raman process, the piezoelectric mechanism dominates [see Eq. (3.57)] when T < Tcrit, where

( ) 1/4 ℏ D 1 11 ωcrit Tcrit = . (3.60) 2π 35 kB

ℏ D From Eq. (3.60), kBTcrit < ωcrit. Thus, for fluctuators that contribute significantly to qubit dephasing (having ℏωn < kBT ), if the piezoelectric contribution dominates in the Raman process

(T < Tcrit), then it also dominates for direct absorption and emission: ℏωn < kBT < kBTcrit < ℏ D ωcrit. Using the GaAs parameters given in Fig. 3.4, the piezoelectric contribution then dominates in both the direct (one-phonon) and two-phonon Raman processes if T < 1.0 K. Thus, in GaAs,

74 the crossover from piezoelectric to deformation-potential mechanisms occurs at

Tcrit = 1.0 K [GaAs]. (3.61)

This feature is indeed visible in Fig. 3.4(d). Quite significantly, Tcrit depends only on material parameters and is therefore completely independent of the details of the fluctuators themselves.

In summary, all qualitative differences between the results for GaAs and silicon inFig. 3.4(d) arise for T < Tcrit(GaAs), where the piezoelectric contribution dominates in GaAs.

3.6 Coherence time and stretching parameter from microscopic models

2 In this section, we use the expressions for ∆ξn and 1/τn found from microscopic models in Secs. 3.4 s s and 3.5 to find the temperature dependences of T2 and α . We first proceed numerically, which allows us to access the full temperature range. We then find explicit analytical expressions in either ≫ s ≲ s the slow-noise (τc T2 ) or fast-noise (τc T2 ) regime. We finally discuss implications for the interpretation of experiments.

3.6.1 Numerical evaluation

For numerical evaluation, we take the fluctuator frequency ωn to vary inhomogeneously between fluctuators, but take all other parameters (tunnel couplings, form factors, fluctuator-qubit couplings) to be approximately independent of n. Taking the continuum limit of Eqs. (3.22) and ∑ ∫ → → → (3.23) for a large number of fluctuators [ n dω, ∆ξn ∆ξ(ω), τn τ(ω)] then gives

∫ ∞ s ts 2 2 s f (ts) = − dωD(ω)∆ξ (ω)τ (ω)h [ts/τ(ω)], (3.62) T2M 0 ∫ ∞ 2 1/T2M = dωD(ω)∆ξ (ω)τ(ω), (3.63) 0

s where D(ω) is the fluctuator density of states. The qubit coherence time T2 is then given directly from the numerical solution of Eq. (3.26) and the stretching parameter αs is given by Eq. (3.27).

The resulting temperature dependences strongly depend on the density of states D(ω). Here, we assume a near-constant density of states D(ω) ≃ D(0) for ω ≲ kBT/ℏ, where the integrand carries

75 −ℏ appreciable weight [the integral in Eq. (3.62) is cut off by ∆ξ2(ω) ∼ e ω/kB T at large frequency].*

In systems where a non-constant density of states is expected or measured, this could easily be incorporated in Eq. (3.62), above.

e e We use the numerical method described above to evaluate T2 and α as a function of temperature T accounting for the two-phonon Raman [Figs. 3.5(a),(b)] and direct-tunneling [Figs. 3.5(c),(d)] processes. A measurement of the distinct temperature dependences shown in Fig. 3.5 could be used to distinguish different microscopic mechanisms. In Figs. 3.5(a),(b), the red solid lines show the temperature dependences expected in GaAs, where piezoelectric coupling to phonons dominates for

T < Tcrit ≃ 1.0 K, but the deformation mechanism dominates for T > Tcrit. The blue dashed lines in Figs. 3.5(a),(b) show the expected behavior for silicon, where only the deformation mechanism

e is relevant. The transition between distinct power-law dependences in T2 shown in Figs. 3.5(a),(c) e ∼ occurs in the crossover regime, when τc/T2 1. Unlike Tcrit, discussed above, the temperature scale determining this crossover is generally non-universal, depending on the specific details of the

e fluctuators and their coupling to the qubit. The distinct upturnin T2 at large T in Fig. 3.5(a) is due to motional averaging; the Raman mechanism leads to a strong reduction in the noise cor-

7 11 relation time at large T (τc ∝ 1/T or τc ∝ 1/T ), which cannot be compensated by the slow growth in the noise amplitude (∝ T ) for a constant density of states. The result is a fast averaging

e of the noise and a resulting increase in coherence time T2 . It should be possible to observe such an upturn experimentally when other high-temperature qubit-dephasing mechanisms can be suppressed. These mechanisms may arise, e.g., from direct coupling of the qubit to phonons, resulting in exponentially-activated pure dephasing from single-phonon absorption and emission [82], or from strongly temperature-dependent pure-dephasing rates due to multi-phonon processes [83, 84].

For all processes investigated here, there is a crossover, as a function of temperature, from the e ≪ e ≃ e ≫ fast-noise (Markovian) limit, τc/T2 1, in which α 1, to the slow-noise limit, τc/T2 1, where αe ≃ 3 (see Sec. 3.3). Strikingly, for the Raman process, the crossover is from the slow-noise to the fast-noise limit with increasing temperature [Fig. 3.5(b)]. In contrast, the tunneling process

*The assumption of a constant density of states in Eq. (3.62) at low temperature is consistent, e.g., with capacitance-probe spectroscopy experiments, where the inhomogeneous broadening of shallow donor levels in the dopant layer of a GaAs/AlGaAs heterostructure has been measured to be ∼ 1 meV (1 meV/kB ∼ 10 K)[174, 175]. It is also a standard assumption for two-level systems in glasses [176].

76 0 (a) 10 ∝ T -4 6 3 (b) − ∝T 10 2 2.5 e

(s) −4 10 2 α

e ∝T 2 10

T −6 1.5 10 ∝ T -8/3 1 0.01 0.1 1 0.1 1 10−4 (c) ∝ T -1 3 (d) 10−5 2.5 1 3 e (s) - / 2 α e −6 ∝ T 2

T 10 1.5 10−7 1 0.001 0.01 0.1 1 0.01 0.1 1 T (K) T (K)

e e Figure 3.5 – Hahn-echo coherence time T2 and corresponding stretching parameter α (a) Coherence time from the Raman phonon process in GaAs (solid red line) and silicon (dashed blue line). (b) Corresponding stretching parameter. (c) Coherence time for the direct tunneling process. (d) Corresponding stretching parameter. (a-b) Material parameters take the same values as given in the caption of Fig. 3.4 for GaAs and n n ℏ n 2 × 3 −1 silicon. For the Raman process, we choose ℘αβ = 0, ℘0 = 100 nm, ωγ = 200 µeV, and D(0)Ωn = 1 10 s ∀ e ∼ e ≃ ∼ n. These numbers are chosen to give T2 1 µs and α 1 for T 100 mK, consistent with Ref. [38]. The decay is given by a compressed exponential with αe > 1 at low temperature, but becomes exponential e 2 × 2 −1 (α = 1) at higher temperature. (c-d) For the direct tunneling process, we choose D(0)Ωn = 1 10 s and 2π | |2 6 −1 ∀ e e ℏ Del(ϵαn) tn(ϵαn) = 10 s n. The behaviors of T2 (T ) and α (T ) are radically different for the Raman mechanism and tunneling mechanism, which makes them easily distinguishable. As explained in the main text, various other qubit dephasing channels can become relevant at higher temperatures, possibly obscuring the crossovers seen here in any given experiment.

leads to a crossover from fast- to slow-noise with increasing temperature [Fig. 3.5(d)]. In the case

of the Raman process, the fast-noise limit is naturally reached at large temperature because of the

7 11 rapid decrease of the noise correlation time (τc ∼ 1/T or τc ∼ 1/T ) in combination with an

s increase in T2 due to motional averaging (see the discussion above). For the tunneling process, the correlation time saturates at high temperature τc ∼ τn ∝ const. (see Table 3.1), while the

amplitude of the noise increases as progressively more fluctuators satisfying ℏωn ≲ kBT contribute, s s ≫ leading to a decrease in T2 and a corresponding transition to the slow-noise limit τc/T2 1 at high temperature.

77 ≪ s ≫ s ≲ s Fast noise (Markovian, τc T2 ) Slow noise (τc T2 ) Crossover (τc T2 ) ∗ e ∗ e s ∝ s Process 1/T2 = 1/T2 = 1/T2M 1/T2 1/T2 β η α Direct tunneling ∝ T ∝ T 1/2 ∝ T 1/3 ∝ T ↑ Cotunneling ∝ 1 ∝ T 1/2 ∝ T 2/3 ∝ T −1 ↓ Direct (deformation) ∝ T −17/2 ∝ T 1/2 ∝ T 2 ∝ T −39/2 ↓ Direct (piezoelectric) ∝ T −4 ∝ T 1/2 ∝ T 4/3 ∝ T −11 ↓ Raman (deformation) ∝ T −10 ∝ T 1/2 ∝ T 4 ∝ T −21 ↓ Raman (piezoelectric) ∝ T −6 ∝ T 1/2 ∝ T 8/3 ∝ T −13 ↓

s ≃ − s αs Table 3.3 – Temperature dependence of the coherence factor C (ts) exp[ (ts/T2 ) ] for a qubit coupled to fluctuators interacting with either an electron bath (first two rows) or a phonon bath (last fourrows). We give the coherence-time temperature dependences for both free-induction decay (s → ∗) and Hahn echo → ≪ s ≫ s (s e) in the limits of fast noise (Markovian, τc T2 ) and slow noise (τc T2 ). In the crossover regime, s s − ≲ s we also give the temperature dependence of β = α 1 for τc T2 . The last column indicates whether αs increases (↑) or decreases (↓) as a function of temperature T . All these results are obtained for a near- constant fluctuator density of states D(ω) ≃ D(0) for ω ≲ kBT/ℏ. Different densities of states could easily be accounted for using Eqs. (3.65) to (3.67). Predictions for the two-phonon sum process are absent since, for ℏωn < kBT , these processes are always negligible relative to the Raman processes (see Sec. 3.5).

3.6.2 Slow- and fast-noise regimes

As described above, given sufficient microscopic information, it is possible to make quantitative

s predictions for the temperature dependence of the qubit coherence time T2 and stretching parameter αs. To do this, we would need a good description of the relevant transition dipole matrix elements

n ℘αβ or tunnel couplings tαn(ϵ) as well as the fluctuator density of states and microscopic material- specific parameters. When the specific impurities associated with charge noise can be identified, it may be possible to estimate or measure these quantities. In many experiments, however, it may be difficult to establish the specific source of charge noise and the associated parameters. Inthiscase,

s we can still make strong analytical predictions about the scaling of T2 with temperature in either ≲ s ≫ s the fast-noise (τc T2 ) or slow-noise (τc T2 ) regime. n We allow the qubit-fluctuator couplings Ωn, dipole matrix elements ℘αβ, etc. to vary generally with n. However, to make analytical progress, we assume that these parameters are approximately

2 independent of ωn for ωn ≲ kBT/ℏ where ∆ξ (ωn) is appreciable. To determine the simple scaling

2 −ℏωn/k T behavior, we replace the exponential dependence ∆ξ (ωn) ∼ e B with a hard cutoff at ℏ ≪ s ωn = kBT . Taking the continuum limit of Eq. (3.23) for the fast-noise limit (τc T2 ) then gives ∫ kBT/ℏ 1 1 1 ∝ ∗ = e = dω D(ω)τ(ω, T ). (3.64) T2 T2 T2M 0

78 With the same assumptions, we perform the continuum limit in Eqs. (3.29) and (3.30) for the ≫ s slow-noise limit (τc T2 ), giving [ ] ∫ ℏ 1/2 1 kBT/ ∗ ∝ dω D(ω) , (3.65) T2 0 [ ] ∫ 1/3 kBT/ℏ 1 ∝ D(ω) e dω . (3.66) T2 0 τ(ω, T )

From Eqs. (3.34) to (3.36) for βs and η, we also have, in the fast-noise regime

∫ kBT/ℏ s ∝ ∝ 2 ≲ s β η dω D(ω)τ (ω, T ). (τc T2 ) (3.67) 0

∗ In the slow-noise limit, the inhomogeneously broadened decay time T2 is independent of the fluctuator equilibration time τn. This decay time is therefore independent of the specific microscopic mechanism giving rise to fluctuator dynamics and can be used to measure the frequency dependence

a of the fluctuator density of states. Indeed, taking D(ω) = D0ω , Eq. (3.65) gives

∗ a+1 ∝ 2 ≫ s 1/T2 T , [τc T2 ] (3.68)

− ∗ where we have assumed a > 1. Thus, the scaling with temperature of 1/T2 in the slow-noise regime can be used to determine a under the assumption that fluctuator parameters other than ω n ≲ ℏ (i.e. Ωn, ℘αβ, etc.) are approximately frequency-independent for ω kBT/ . In Tables 3.1 and 3.2, we give the ω and T dependences of 1/τ for all fluctuator-bath processes considered in this chapter. Substituting these dependences into Eqs. (3.64) to (3.66) and assuming a

s constant fluctuator density of statesa ( = 0) gives the power-law scalings for T2 shown in Table 3.3. These scalings are consistent with those obtained numerically in Fig. 3.5. Similar tables could easily be built for different values of a, i.e., for non-constant fluctuator densities of states.

In Fig. 3.6, we plot βe = αe − 1 as a function of temperature for the Raman process [Fig. 3.6(a)] and direct tunneling [Fig. 3.6(b)]. We evaluate Eqs. (3.62) and (3.63) numerically with the same assumptions and parameters as described in the caption of Fig. 3.5. These numerical results are represented in Fig. 3.6 by circles and triangles. The analytical predictions of Table 3.3 are also

79 (a) (b) 100 100 ∝ T −21 −2 −2 e 10 10 β ∝ T −13 −4 ∝ T −4 10 10

− − − 0.1 1 10 610 410 2 100 T (K) T (K)

Figure 3.6 – Parameter βe = αe − 1. (a) Raman process. Black circles (black triangles): βe for GaAs (silicon), from the numerical method of Eqs. (3.26), (3.27), and Eqs. (3.62), (3.63). Solid red line (dashed ≲ s blue line): analytical temperature dependence for τc T2 from Table 3.3 for GaAs (silicon). (b) Direct tunneling. Red circles: numerical method. Solid black line: analytical prediction. Assumptions and microscopic parameters are the same as in Fig. 3.5. plotted as straight lines. As expected from the discussion above Eq. (3.31), these analytical results only substantially deviate from exact numerical calculations when βe ≃ 3η ∼ 1, corresponding to ∼ e → ∞ τc T2 . Indeed, when η (the slow-noise limit), Eq. (3.67) predicts an unbounded growth of βs, while, from Eqs. (3.29) and (3.30), βs saturates to 1(2) for free-induction decay (Hahn echo).

s However, for τc > T2 , Eq. (3.67) and the corresponding power laws in Table 3.3 still give the trends in αs [increasing (↑) or decreasing (↓)] shown in Table 3.3.

In Table 3.3, all processes we have considered can be distinguished from a combined measure-

s s ment of the temperature dependence of T2 and β . From this table, it should be possible to rule s s out specific fluctuator noise mechanisms based on a measurement of T2 and α as a function of temperature.

3.6.3 Relevance to experiment

To assess the usefulness of the approach described here, we now consider an application to a recent experiment. In Ref. [38], Dial et al. have observed coherence decay as a function of temperature for a qubit defined by singlet and triplet spin states in a two-electron double quantum dot inGaAs.

These measurements revealed an approximate linear dependence of the inhomogeneously broadened ∗ ∝ − ∗ ∼ ∼ decay time, T2 A BT , with α = 2 for temperatures between 50 mK and 250 mK. This ∗ a+1 ∝ 2 − behavior may be compatible with any dependence T2 1/T given by Eq. (3.68) with a > 1.

80 ∗ Thus, a more precise measurement of T2 as a function of temperature may establish the specific form of the fluctuator density of states in this experiment.

Under the assumption of a constant fluctuator density of states [D(ω) ∼ ωa with a = 0], we attempt to apply the results of Table 3.3 to describe the experimental results of Ref. [38]. In e e e ∝ −γ ∼ Ref. [38], the authors measured T2 (T ) and α (T ) and found that: (i) T2 (T ) T with γ 2 for the whole temperature range of the experiment, (ii) βe decreases monotonically as T increases from

∼ 50 mK to ∼ 150 mK, and (iii) βe ≲ 0.7 for the whole temperature range, corresponding to the fast- ≲ e e ≃ e ∼ noise regime, in which τc T2 . In this regime, Eqs. (3.32) and (3.35) yield T2 (1+β )T2M T2M e up to a correction O(β T2M). The first column of Table 3.3 should then accurately reflect the trend e e → in T2 (T ) in the fast-noise regime, consistent with β 0. For all phonon mechanisms, we find that e ≲ e T2 increases with temperature for τc T2 , while the data from Ref. [38] exhibit the opposite trend. e The only mechanism in Table 3.3 for which T2 correctly decreases when T increases in the fast-noise regime is direct tunneling. However, for this process, βe increases monotonically with temperature, in contradiction with the experimental data of Ref. [38]. Therefore, under the assumption of a constant density of states, none of the physical processes displayed in Table 3.3 can, alone, explain all the observations listed above.

One of the assumptions behind Table 3.3 may be violated in the context of Ref. [38]. Here, we review the assumptions and limitations leading to this table. To begin with, it may be that the true fluctuator density of states was not constant in the experiment ofRef.[38]. A precise measurement

∗ of T2 in the slow-noise regime can be used to establish the true frequency dependence of the fluctuator density of states through Eq. (3.68). In addition, for phonon mechanisms, we have assumed a long-wavelength limit to establish the low-frequency behavior of the fluctuator equilibration rates.

From Eq. (3.53), this assumption may be violated for fluctuators with large extended orbital states, or at high temperatures, leading to phonon-bottleneck effects [166]. Finally, we have assumed that the dominant dephasing mechanism results from coupling to charge fluctuators. It is, of course, possible that other decay channels become relevant. For example, in the presence of an independent

81 extrinsic Markovian dephasing process, the coherence factor takes the form

[ ( ) ] αs s − ts − ts C (ts) = exp ′ s . (3.69) T2 T2

s s In the above equation, T2 and α are the decay time and stretching parameter for the fluctuator ′ processes presented here, while T2 is the decay time due to an additional Markovian dephasing process acting directly on the qubit. At high temperature, many extrinsic dephasing mechanisms

(not related to charge fluctuators) may become relevant (these may be due, e.g., to coupling to phonons [83, 84, 177]). The first term in Eq. (3.69) may then dominate over the second. To ensure that the fluctuator mechanisms presented in this chapter are the dominant source of dephasing, it may be necessary to understand and suppress alternative sources of dephasing (by, e.g., working at sufficiently low temperature). Alternatively, when these alternate sources of dephasing arewell

s understood, a combined formula such as Eq. (3.69) could be used to extract the values of T2 and αs associated with fluctuator dynamics, even in the presence of extrinsic dephasing mechanisms.

To further illustrate how Eq. (3.69) can be used to identify interactions at the origin of fluctuator

s e dynamics, we apply it to the analysis of the data from Ref. [38]. We take T2 = T2 to be the Hahn- echo decay time for one of the fluctuator processes of Table 3.3 in the slow-noise limit (in which

e ′ e α = 3). When T2 < T2 , the contribution to qubit decay of the extrinsic Markovian process dominates over the contribution of the fluctuators. We then find the qubit decay time T2 including both fluctuator and extrinsic processes. We do so by setting the argument of the exponential in Eq. (3.69) equal to one and solving for ts ≡ T2 using an expansion in increasing powers of

′ e T2/T2 . Substituting the resulting expression for T2 in the definition of the stretching parameter α, Eq. (3.27), we find the form of β including both processes (fluctuator and extrinsic) to leading order ′ e ≃ ′ ∝ −δ in T2/T2 . We take T2 T2 T for the extrinsic dephasing mechanism, with δ the exponent e ∝ −γ obtained from the experiment of Ref. [38], and T2 T , with γ the appropriate exponent for the ′ e relevant fluctuator mechanism from Table 3.3. We then find, to leading order in T2/T2 ,

β ∝ T 3(γ−δ). (3.70)

The decreasing trend for β(T ) observed in Ref. [38] from ∼ 50 mK to ∼ 150 mK is thus reproduced

82 for γ < δ. For δ = 2, as written in Ref. [38], the decreasing trend for β(T ) is consistent with all the fluctuator mechanisms from Table 3.3 in the slow-noise limit except for the Raman processes (from either piezoelectric or deformation mechanisms). However, for 100 mK< T < 200 mK, Kornich et al. have predicted Markovian decay of singlet-triplet coherence at a rate ∝ T 3 due to two-phonon processes including spin-orbit coupling (see Fig. 3 of Ref. [84]). This behavior is compatible with the experimental data of Ref. [38] in the relevant temperature range (100 mK< T < 200 mK).

Taking δ = 3 in Eq. (3.70) implies that the observed decreasing trend for β(T ) with T < 100 mK becomes compatible with all the fluctuator mechanisms in Table 3.3 except the Raman process due to deformation coupling to phonons. With the help of Eq. (3.70) and knowing δ from a precise measurement of T2(T ) in the fast-noise regime, γ could be estimated through a precise measurement of β as a function of T , allowing for further identification of fluctuator processes.

3.7 Conclusions

We have described qubit dephasing due to two-level fluctuators undergoing equilibration dynamics with either electron or phonon reservoirs. Even for a Lorentzian noise spectrum, which arises naturally for two-level fluctuators, the qubit coherence factor is well approximated byacom- − s αs pressed exponential exp[ (ts/T2 ) ]. In contrast with the situation for 1/f noise [41, 151], here the stretching parameter αs depends on the chosen pulse sequence s and obeys a universal relation, e − ∗ − ≃ s ≳ (α 1)/(α 1) 3, in the fast-noise regime, in which T2 τc. We have determined the explicit temperature dependences for the stretching parameter αs and

s coherence time T2 from several microscopic mechanisms giving rise to fluctuator equilibration dynamics. These mechanisms include direct tunneling and cotunneling between localized electronic states and an electron reservoir. We have also considered coupling of two-level charge fluctuators to a phonon bath. In the latter case, we have allowed for direct phonon absorption and emission, as well as the two-phonon sum and Raman processes. We have found that different fluctuator-bath

s s processes lead to distinct temperature dependences for T2 and α . A measurement of the predicted temperature dependences should thus allow to experimentally distinguish between physical processes at the origin of fluctuator noise, providing an additional tool to suppress charge noise.

83 4 Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED

This chapter is the integral text from:

Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED Félix Beaudoin, Alexandre Blais, and W. A. Coish, submitted for publication (June 24th, 2016) [3]

Quantum state transfer into a memory, state shuttling over long distances via a quantum bus, and high-fidelity readout are important tasks for quantum technology. Realizing these tasks is challenging in the presence of realistic couplings to an environment. Here, we introduce and assess protocols that can be used in cavity QED to perform high-fidelity quantum state transfer and fast quantum nondemolition qubit readout through Hamiltonian engineering. We show that high-fidelity state transfer between a cavity and a single qubit canbe performed, even in the limit of strong dephasing due to inhomogeneous broadening. We generalize this result to state transfer between a cavity and a logical qubit encoded in a collective mode of a large ensemble of N physical qubits. Under a decoupling sequence, we show that inhomogeneity in the ensemble couples two collective bright states to only two other collective modes, leaving the remaining N − 3 single-excitation states dark. Moreover, we show that large signal-to-noise and high single-shot fidelity can be achieved in a cavity-based qubit readout, even in the weak-coupling limit. These ideas may be important for novel systems coupling single spins to a microwave cavity.

84 4.1 Introduction

Spin qubits encoded in collective modes of ensembles [87, 178, 179] and single spins in quantum dots [28, 90, 180] can be coupled to microwave cavities for cavity quantum electrodynamics (QED) experiments [85]. Spin qubits show promise for use as long-lived quantum memories, but often suffer from weak qubit-cavity coupling relative to the inhomogeneously broadened linewidth [181].

Inhomogeneous broadening typically originates from nuclear-spin or electrical (charge) noise [7, 21,

182, 183]. While nuclear-spin noise [19] can often be controlled through isotopic purification, strong coupling of a single spin to the electric field of a cavity mode typically requires a strong correlation of spin and charge degrees of freedom [27, 89, 91, 184, 185]. This correlation makes the spin qubit susceptible to low-frequency charge noise [38, 186]. An alternative strategy is to enhance the weak magnetic coupling of a spin qubit (which may be otherwise insensitive to charge noise) by coupling to the collective mode of a large spin ensemble [187, 188]. However, spatial inhomogeneities in such ensembles can result in an inhomogeneous linewidth that is comparable to the qubit-cavity coupling [87, 189].

It is well known that the effects of inhomogeneous broadening can be eliminated through a suitable dynamical decoupling sequence. To determine the quality of a cavity-QED scheme, the coupling is therefore often compared with the inverse qubit coherence time under a train of decoupling π-pulses [27, 89, 91], rather than the inhomogeneous linewidth. However, for a qubit coupled to a cavity, a sequence of π-pulses typically generates unwanted cavity excitations on the same timescale as coherent qubit-cavity oscillations, severely reducing the fidelity of, e.g., quantum state transfer between a qubit and a cavity.

In this chapter, we show that these limitations can be overcome by engineering appropriate time- averaged Hamiltonians [72, 190–192] through a combination of qubit dynamical decoupling and control of the qubit-cavity coupling. In particular, we introduce and quantitatively characterize protocols for a high-fidelity quantum state transfer between a qubit and cavity, and for afast quantum nondemolition qubit readout. Our readout protocol yields a large signal-to-noise ratio even in the weak-coupling regime, in which the qubit-cavity coupling is small compared to the cavity

85 damping rate. Moreover, we show that control of the qubit-cavity coupling makes high-fidelity quantum state transfer possible even in the strong-dephasing limit, in which the inhomogeneous linewidth dominates the qubit-cavity coupling. This result applies even to logical qubits encoded in the collective mode of an ensemble of physical qubits (relevant to, e.g., spin or atomic ensembles that are routinely used for quantum memories [87, 193, 194]). Inhomogeneous broadening across an uncontrolled ensemble of N physical qubits would typically lead to coupling of the logical qubit to ∼ N collective modes [189, 195, 196]. However, remarkably, for our pulse sequence we find that the leading corrections in average Hamiltonian theory couple only four distinct collective modes in the large-N limit. This may allow for very high-fidelity storage-and-retrieval or even coherent manipulation of quantum information in the ensemble through revivals.

This chapter is organized as follows. In Section 4.2, we introduce the Hamiltonian engineering protocol studied throughout this work. In Section 4.3, we evaluate the fidelity of a quantum state transfer between a cavity and a single physical qubit under the Hamiltonian-engineering protocol presented here, and show that errors can be strongly suppressed, even in the strong-dephasing limit

(in which the inhomogeneous broadening is larger than the qubit-cavity coupling). In Section 4.4, we generalize this result to state transfer between a cavity and a collective mode of a large ensemble of physical qubits. In Section 4.5, we present a scheme to avoid accumulation of error arising under deterministic over (under)-rotations of the qubit during imperfect π pulses. Finally, in Section 4.6, we show how a simple modification of the dynamical decoupling sequence introduced here leadstoa readout protocol yielding high signal-to-noise and single-shot fidelity in the weak-coupling regime.

4.2 Hamiltonian engineering

We first consider a single qubit coupled to a cavity. With the cavity and the qubit onresonance and working in a rotating frame within the rotating-wave approximation,* the system is described by a Jaynes-Cummings Hamiltonian:

† HJC(t) = ξσz/2 + g(t)(a σ− + aσ+), (4.1)

*See Appendix C.2 for a discussion of the effect of counter-rotating terms and finite-bandwidth control.

86 π π π π ) t ( g

τ/2 3τ/2 5τ/2 7τ/2 t Figure 4.1 – SQUADD (SQUare wave And Dynamical Decoupling): the Carr-Purcell sequence is applied to a qubit coupled to a cavity while turning off the coupling g(t) [Eq. (4.4)] after each odd-numbered π-pulse to prevent unwanted cavity excitations. where we have allowed for a tunable qubit-cavity coupling g(t) (setting ℏ = 1). In addition, the qubit is controlled via Hc(t), giving the total Hamiltonian

H(t) = HJC(t) + Hc(t). (4.2)

2 In HJC(t), we take ξ to be a Gaussian random variable with zero mean and variance (∆ξ) that describes inhomogeneous broadening in the qubit-cavity detuning. Most decoupling schemes rely entirely on qubit control. However, electrical control of g(t) is now possible in several architectures [26, 28, 91, 197, 198]. By modulating Hc(t) and g(t) sufficiently quickly, we can eliminate unwanted terms and generate useful time-averaged Hamiltonians.

To average away unwanted terms, we move to the toggling frame [73], which incorporates Hc(t) into the transformed system Hamiltonian,

† HT(t) = Uc (t)HJC(t)Uc(t), (4.3)

∫ T − t ′ ′ where Uc(t) = exp[ i 0 dt Hc(t )]. To reduce dephasing due to the random detuning ξ, a natural 1 choice for Uc(t) is the Carr-Purcell sequence: a train of sharp π-pulses applied at times (m + 2 )τ, with m ∈ N (Fig. 4.1). In this case,    1 † 2 ξσz + g(t)[a σ− + aσ+], n(t) even, HT(t) = (4.4)   − 1 † 2 ξσz + g(t)[a σ+ + aσ−], n(t) odd,

87 with n(t) the number of π-pulses applied before time t. For n(t) even, the qubit-cavity interaction

† is described by a co-rotating term, preserving the total number of excitations, Nex ≡ a a + σ+σ−. However, for n(t) odd, the interaction is rather given by a counter-rotating term, which does not conserve Nex. For fixed g(t) = g, the counter-rotating term leads to simultaneous excitation of the qubit and cavity on a time scale comparable to the state-transfer time. This flow of excitations can be blocked simply by taking g(t) = 0 for n(t) odd. With this choice, Nex is a constant of motion, allowing for coherent state transfer between the qubit and the Hilbert space spanned by the vacuum and first excited state of the cavity. As we show below, taking g(t) = g ∀ t would rather lead to a fast qubit readout via the cavity.

4.3 Qubit-cavity state transfer

In the rest of this chapter, we will use the acronym SQUADD (SQUare wave And Dynamical

Decoupling) to describe the simultaneous square-wave modulation of g(t) and sequence of π-pulses shown in Fig. 4.1. If this sequence were not applied, inhomogeneous broadening ∆ξ of the order of g ≡ maxt[g(t)] would result in a state-transfer error (infidelity) of order 1. However, under SQUADD with a sufficiently short period 2τ, the system dynamics are accurately described by the average Hamiltonian

∫ 2τ (0) 1 † H = dt HT(t) = g(a σ− + aσ+), (4.5) 2τ 0 ∫ ≡ 2τ (0) where we have introduced the average coupling, g 0 dt g(t)/2τ. In Eq. (4.5), H is the leading ∫ T − t ′ ′ term in average Hamiltonian theory, which recasts the evolution operator U(t)= exp[ i 0 dtHT(t )] in terms of a Magnus expansion [65, 70]: ( ) ∑∞ U(t) = exp −it H(k) . (4.6) k=0

Complete state transfer is achieved at a final transfer time tf , where

π gt = gn τ = , (4.7) f p 2

88 and np is the total number of π-pulses. Importantly, this condition for state transfer is independent of the precise shape of g(t), as long as g(t) = 0 for n(t) odd. The Magnus expansion converges rapidly when

∫ 2τ dt∥HT(t)∥2 ≲ max(g, ∆ξ)2τ ≪ 1. (4.8) 0

Equivalently, for a fixed transfer time tf = npτ = π/2 g, np ≫ π max(1, ∆ξ/g). Equation (4.5) gives an exact description of the time evolution in the limit τ → 0. However, in practice, τ will always be limited by the bandwidth of g(t) and the duration of π pulses. Thus, to characterize the performance of SQUADD, we evaluate the average fidelity

∫ ⟨ | †M | ⟩⟨ | | ⟩ F = dψ ψ U0 ( ψ ψ )U0 ψ . (4.9)

The integral in Eq. (4.9) represents an average with respect to the Haar measure dψ (a uniform average over the Bloch sphere) for the ensemble of states of the form |ψ⟩ ≡ |ψ⟩q|0⟩c, where |ψ⟩q is an arbitrary pure qubit state and |0⟩c is the cavity vacuum. We have also introduced the unitary operator U0 describing an ideal state transfer: U0|ψ⟩q|0⟩c = |g⟩q|ψ⟩c, with |g⟩q the qubit ground state. In addition, M is the completely positive trace-preserving map that describes the actual state transfer for finite τ, accounting for an average over the random detuning ξ and a finite cavity damping rate κ. We first consider the case κ = 0, then generalize to finite κ, below.

When g(t) = 0 for n(t) odd, HT(t) preserves the number of excitations Nex. For fixed detuning ξ, the map M can then be expressed in terms of a unitary, introducing a trivial phase on the state

† with Nex = 0 and a ξ-dependent SU(2) rotation in the two-dimensional space of Nex = 1. We take g(t) = g identically for n(t) even and expand to leading (fourth) order in τ = π/gnp. For max(∆ξ, g)τ ≪ 1 [equivalently, np ≫ π max(1, ∆ξ/g)], the error is then well-approximated by [ ] ( ) ( ) ( ) ( ) 1 π 2 ∆ξ 4 1 ∆ξ 2 π 4 1 − F ≃ + . (4.10) 6 4 g 3 g 2np

− ∝ 4 The error (1 F 1/np) is thus strongly suppressed with an increasing number of π-pulses, as

†See Appendix C.1 for the exact solution for state transfer using SQUADD.

89 shown in Fig. 4.2. Given a finite off/on ratio goﬀ /g [where g(t) = goﬀ for n(t) odd], we find a

2 correction to the error of order ∼ (goﬀ /g) . This term would ultimately limit the saturation fidelity at large np whenever goﬀ ≠ 0.

Taking goﬀ /g = 0, a small error can be reached when np ≫ 1, even for strong dephasing, √ ∗ ∗ gT2 = 2g/∆ξ < 1, where T2 is the qubit free-induction decay time (dephasing time) due to inhomogeneous broadening ∆ξ. This result is apparent in Fig. 4.2, which gives the average error

1 − F as a function of the total pulse number np = tf /τ = π/gτ obtained from the exact solution described above (solid purple line), along with the large-np expansion of Eq. (4.10) (dashed black ∗ line). Here, we have chosen gT2 = 1/10. Even for this choice of parameters, placing the system in the strong-dephasing regime, errors smaller than 1% are reached with np ∼ 40 pulses, at the onset of the validity criterion for Eq. (4.10): np > π∆ξ/g ∼ 40. Consequently, the usual weak-dephasing ∗ ≪ ≪ ∗ criterion (1/T2 g) has been traded for a fast-control requirement (τ T2 ). Fast π-pulses in this limit have already been demonstrated with isolated spin qubits (not coupled to cavities) [19, 24, 38], and could in principle be made even faster for single spins by taking advantage of exchange coupling and the magnetic field gradient generated by a micromagnet [199, 200]. Since g(t) can be controlled electrically when these systems are coupled to cavities [28], fast pulsing of g(t) may be possible in the very near future (we give an analysis of finite-bandwidth control for g(t) and the influence of counter-rotating terms in Appendix C.2).

When np → ∞, inhomogeneous broadening becomes irrelevant and the fidelity will ultimately be limited by cavity damping at rate κ (we neglect the homogeneous qubit decay under the decoupling sequence when κT2 > 1). We find that the error saturates at

( ) π κ κ2 1 − F = + O (4.11) 6 g g2

when np → ∞. To illustrate this, we numerically solve the Lindblad master equation generated by a Liouvillian L accounting for both Hamiltonian evolution under Eq. (4.4) and cavity damping. As shown in Fig. 4.2, cavity damping does indeed lead to a saturation of the error as a function of np at 1 − F ∼ κ/g (blue dots: κ/g = 1, red triangles: κ/g = 1/100).

As a concrete example, a coupling g/2π ≃ 1 MHz has been predicted for spin qubits in GaAs

90 1

0.1 F

− ∗ = 1 10

1 gT2 / 0.01 κ/g = 1 κ/g = 1/100 κ/g → 0 0.001 10 100 = tf = π np τ gτ − ∗ Figure√ 4.2 – Suppression of state-transfer error 1 F with increasing number of pulses np for gT2 = 2g/∆ξ = 1/10. Dashed black line: Eq. (4.10). Solid purple line: exact solution, without cavity damping. Blue dots: exact numerical master-equation simulation including cavity damping, with κ/g = 1. Red triangles: κ/g = 1/100.

∗ ≃ double quantum dots [27], leading to gT2 0.05 due to hyperfine coupling to nuclear spins [19]. Even in this case, SQUADD could enable coherent coupling between a single spin and a cavity.

In addition, SQUADD could improve state transfer between a single spin confined in a carbon nanotube and a coplanar-waveguide resonator. In a recent experiment on this system, g/2π = ∗ ≃ 1.3 MHz, κ/2π = 0.6 MHz, and T2 60 ns have been reported [28]. With these parameters, a large state-transfer error 1 − F ≃ 0.42 results from Eq. (4.9) without π-pulses. Using SQUADD, n = 10 (τ = π ≃ 40 ns) suffices to reduce the error from pure dephasing to 0.004. The total p gnp error is then 1 − F ≃ 0.18, limited by the large κ/g ratio in this experiment.

4.4 Collective modes in qubit ensembles

In this section, we consider the application of SQUADD to quantum state transfer between a cavity and a collective mode of an ensemble of N physical qubits. We account for leading corrections in √ the Magnus expansion and show that, up to corrections ∼ O(1/ N), this system evolves in a closed

4-dimensional subspace. Using this approach, we retrieve an expression similar to Eq. (4.10) for

91 the state-transfer fidelity.

For single spins coupled to microwave cavities, the κ/g ratio can be large [28, 201], limiting the fidelity achievable through SQUADD. However, the effective coupling can be significantly enhanced by encoding a logical qubit into a large number of physical qubits. Indeed, an ensemble of N | ⟩ qubits coupled to a common cavity mode hosts an excitation out of the ground state g q = | ⟩ ⊗ | ⟩ · · · | ⟩ g 1 g 2 g N that is annihilated by the collective lowering operator v u ∑N u∑N 2 gi − t g b = √ σ , g ≡ i (4.12) i av N i=1 Ngav i=1 where gi is the coupling for qubit i. For N ≫ 1, the logical qubit encoded in the subspace of √ | ⟩ | ⟩ † | ⟩ ≡ g q and e q = b g q couples to the resonator with an ensemble coupling gens Ngav [187]. However, an inhomogeneity in qubit-cavity detunings across the ensemble may lead to leakage from the collective mode b to many dark states [189, 195, 196]. When ∆ξ ≳ gens, leakage due to dephasing will typically result in an error of order one.

Errors due to inhomogeneous broadening in an ensemble can be suppressed through SQUADD.

The toggling-frame Hamiltonian for a qubit ensemble is   ∑ ∑  1 z † − + 2 i ξiσi + i gi(a σi + aσi ), n(t) even, HT(t) = (4.13)  ∑  − 1 z 2 i ξiσi , n(t) odd.

We thus consider an ensemble of qubits with couplings gi(t) and detunings ξi from the cavity. As in the single-qubit case, we assume that gi(t) = gi ∀ i for n(t) even and gi(t) = 0 ∀ i for n(t) odd. The time-dependent Hamiltonian in Eq. (4.13) describes rapid periodic modulation for small pulse interval τ. For max(gens, ∆ξ)τ ≪ 1 [equivalently, np ≫ max(π, ∆ξ/gens)], the Magnus expansion converges rapidly, allowing us to truncate the expansion at leading and first subleading order.

† Because we have assumed g(t) = 0 for n(t) odd, the total number of excitations Nex = a a + ∑ + − (k) H i σi σi is a constant of motion. We thus project each H into the subspace 01 associated with Nex = 0 or 1. Explicitly, H01 is spanned by the states |g⟩q ⊗ |0⟩c, |g⟩q|1⟩c, and |g⟩1 ⊗

|g⟩2 · · · |e⟩j · · · |g⟩N ⊗ |0⟩c, where 0 and 1 label cavity Fock states, and where |g⟩j and |e⟩j label the

92 ground state and excited state of qubit j, respectively. We then have

g H(0) = ens (a†b + ab†), H(1) = 0, (4.14) 2 (2) † † † † † H = Ω1(b c + c b) + Ω2(a d + d a) + χa a. (4.15)

In Eqs. (4.14) and (4.15), we have introduced two new collective qubit lowering operators √ ∑ ∑ 2 2 √1 giξi − gi ξi c = σi , (gξ)av = , (4.16) N (gξ)av N i √ i 1 ∑ g ξ2 ∑ g2ξ4 d = √ i i σ−, (gξ2) = i i . (4.17) (gξ2) i av N N i av i

Equation (4.15) describes the coupling of modes a and b with the new modes c and d with strengths

√ Nτ 2 Nτ 2 Ω = − g (gξ) , Ω = − (gξ2) . (4.18) 1 48 av av 2 48 av

In addition, Eq. (4.15) contains a resonator shift by frequency

τ 2 ∑ χ = g2ξ . (4.19) 24 i i i

By construction, after projecting into H01, the collective qubit operators obey the commutation relations

[b, b†] = [c, c†] = [d, d†] = 1 + O (1/N) . (4.20)

Therefore, in the large-N limit, the Hamiltonian H(0) +H(2) can be expanded in the basis of single-

† excitation states |m⟩ ≡ m (|g⟩q ⊗ |0⟩c), where m ∈ {a, b, c, d}. However, this basis is typically non-orthogonal. To see this, we assume that the coupling strengths gi are uncorrelated with the detunings ξi, implying that, e.g., (gξ)av → gavξav for N → ∞. We also assume that the distribution

93 of qubit-resonator detunings is Gaussian with mean E[ξi] = 0. Projecting into H01, this gives

( √ ) ( √ ) [b, c†] = O 1/ N , [c, d†] = O 1/ N , (4.21) √ ( √ ) [b, d†] = 1/ 3 + O 1/ N , (4.22) all other relevant commutators between different modes being 0. Though [b, c†] and [c, d†] are suppressed in the large-N limit, [b, d†] always remains of order 1. This implies that

√ ( √ ) s ≡ ⟨b|d⟩ = ⟨0|[b, d†]|0⟩ = 1/ 3 + O 1/ N . (4.23)

To avoid the complications associated with the non-orthogonal basis {|a⟩, |b⟩, |c⟩, |d⟩}, we introduce a new set of single-excitation states {|a˜⟩, |˜b⟩, |c˜⟩, |d˜⟩}, where

|a˜⟩ = |a⟩, |c˜⟩ = |c⟩, (4.24) 1 1 1 1 |˜b⟩ = −√ |b⟩ + √ |d⟩, |d˜⟩ = √ |b⟩ + √ |d⟩. (4.25) 2(1 − s) 2(1 − s) 2(1 + s) 2(1 + s)

The states given in Eq. (4.24) and (4.25) form an orthonormal basis if we neglect overlaps ∼ ( √ ) O 1/ N . Writing a matrix representation of H(0) + H(2) in this basis, we find

 

 0 ω− 0 ω   +     ′  [ ]  ω− 0 ω 0  [ ( √ )]  +  H(0) + H(2) =   1 + O 1/ N . (4.26)    ′ ′   0 ω+ 0 ω−    ′ ω+ 0 ω− 0

In Eq. (4.26), we have introduced couplings between the orthonormal modes a˜, ˜b, c˜, and d˜, given by

√ [ ] √ ∓ 1 s gens 1 2 2 ′ 1 1 s 2 2 ω = ∓ g (ξ ) τ , ω = g ξ τ . (4.27) 2 2 48 ens av 48 2 ens av

( √ ) Assuming N ≫ 1, we neglect corrections ∼ O 1/ N . The Hamiltonian in Eq. (4.26) can then

94 ′ be represented graphically, as shown in Fig. 4.3(b). For ∆ξτ = 0, Eq. (4.27) yields ω = 0. The

Hamiltonian then has the structure of a Λ system, with a basis state |a˜⟩ coupled to the two basis states |˜b⟩ and |d˜⟩. These couplings are represented by the thick red arrows in Fig. 4.3(b). In this limit, the Hamiltonian in Eq. (4.26) has two bright eigenstates with energy gens/2. This is clearly seen in Fig. 4.3(a), which shows the eigenenergies of the Hamiltonian in Eq. (4.26) as a function of

∆ξτ, where ∆ξ is the standard deviation of the Gaussian distribution of qubit-resonator detunings.

For ∆ξτ = 0, all other N − 1 eigenstates are zero-energy dark states which do not couple to the resonator mode. In contrast, when ∆ξτ > 0, the Hamiltonian has the structure of a tight-binding problem on a ring; the basis states |˜b⟩ and |d˜⟩ become weakly coupled through an additional mode |c˜⟩. These additional hopping terms are represented by the thin blue arrows in Fig. 4.3(b).

Because of the simple tridiagonal form of the Hamiltonian in Eq. (4.26), we are able to find analytic expressions for the eigenenergies and eigenstates of H(0) + H(2) for ∆ξτ > 0. These expressions are given in Appendix C.3. As shown in Fig. 4.3(a) by the solid red lines, introducing couplings to the mode |c˜⟩ by turning on ∆ξ shifts the energies of the two initial bright states, and lifts the degeneracy between two of the initial dark states by coupling them to the resonator mode. These two effects will generate errors in the state transfer of the resonator quantum state into theensemble of qubits.

We characterize errors in SQUADD due to inhomogeneous broadening with the average fidelity

F , as defined in Eq. (4.9). We consider the initial state |ψ⟩ ≡ |g⟩q ⊗ |ψ⟩c, where |ψ⟩c is an arbitrary superposition of the cavity states |0⟩c and |1⟩c. We choose the evolution operator for an ideal state † transfer to be U0 = −ib a, where the −i phase factor appears because the state transfer described here is equivalent to an SU(2) rotation. We take the linear map M representing imperfect state transfer to correspond to the evolution operator under the effective time-independent Hamiltonian in Eq. (4.26). We perform a Taylor expansion of the resulting fidelity to leading (fourth) order in

τ, assuming max(gens, ∆ξ)τ ≪ 1. Using the condition for complete state transfer, τ = π/gensnp, this assumption becomes np ≫ max(π, ∆ξ/gens), resulting in

[ ( ) ( ) ] ( ) 8 + π2 ∆ξ 4 1 ∆ξ 2 π 4 1 − F ≃ + . (4.28) 18 2gens 18 2gens 2np

95 (a) gens/2

(b) ˜ |di ′ ω+ ω−

0 |a˜i |c˜i Energy ′ ω− ω+ |˜bi

−gens/2

0 0.2 0.4 0.6 0.8 1 ∆ξτ

Figure 4.3 – Spectrum of the time-independent effective Hamiltonian H(0) + H(2) for a qubit ensemble coupled to a resonator under SQUADD. (a) Eigenenergies as a function of ∆ξτ, where ∆ξ is the standard deviation of the Gaussian distribution of qubit-resonator detunings ξi, and τ is the dynamical-decoupling pulse interval. Solid red line: eigenenergies obtained√ by analytically diagonalizing the effective 4 × 4 Hamil- tonian in Eq. (4.26), dropping corrections ∼ O(1/ N) (see Appendix C.3). Black dots: exact numerical√ diagonalization of H(0) + H(2) given by Eqs. (4.14) and Eq. (4.15), which include corrections ∼ O(1/ N). ˜ ˜ We take gens = ∆ξ, and N = 1000. (b) Couplings between the basis states {|a˜⟩, |b⟩, |c˜⟩, |d⟩}, as given by Eq. (4.26).

96 ( √ ) We recall that we have dropped corrections of O 1/ N arising from overlaps between basis states. Ignoring numerical prefactors of order 1, Eq. (4.28) exactly corresponds to Eq. (4.10) for a single qubit, after the replacement g → gens. In Eq. (4.28), the numerical prefactors (obtained for N ≫ 1) differ from those obtained in Eq.(4.10) for N = 1 because the mode structure is not the same. Indeed, taking N = 1 in Eqs. (4.12), (4.16), and (4.17) leads to b = c = d = σ−. The overlap between excitations of any pair of modes ∈ {b, c, d} is then ⟨0|σ−σ+|0⟩ = 1, in contradiction with ( √ ) the overlaps obtained from Eqs. (4.21) to (4.23) when neglecting terms ∼ O 1/ N for N ≫ 1.

The above discussion shows that SQUADD is robust to inhomogeneous broadening, even when coupling a cavity to a collective mode. By modulating the detuning rather than the coupling, it may be possible to use a variation of SQUADD on ensembles of nitrogen vacancy (NV)-center spin qubits in diamond coupled to superconducting coplanar waveguides, for which ∆ξ ∼ gens has been reported [189].

This treatment of collective modes in qubit ensembles also demonstrates a clear advantage of our analytical approach over brute-force numerical methods for optimal control. Indeed, the time required for numerical exponentiation of the full system Hamiltonian grows exponentially with ensemble size, making the problem numerically challenging for N ≫ 1. In contrast, the analytical approach reveals a closed 4-dimensional subspace in the same large-N limit.

4.5 Pulse errors

In general, over-rotation or under-rotation of the qubit due to imperfect control can lead to an accumulation of errors as the number of pulses np is increased. A simple way to avoid accumulation of these pulse errors is to use a phase-alternated sequence [73], in which the qubit rotation direction alternates from one π-pulse to the next. Consequently, the (fixed, deterministic) error ε on the rotation angle of successive pulses cancels for n(t) even, but introduces a small over-rotation for n(t) odd. We evaluate the resulting correction δF to the state-transfer fidelity of Eq. (4.10) by ∫ T − t ′ ′ ≪ expanding U(t) = exp[ i 0 dt HT(t )] to leading order in ε and in τ. When max(g, ∆ξ)τ 1,

97 δF is then well-approximated by this leading correction:

( )( ) 1 4 1 π ∆ξ 2 δF ≃ − − √ sin √ ε . (4.29) 2 3 2 2 g

Thus, neglecting order-unity prefactors, pulse errors can be made negligible compared to the error [ ] ≪ 2 2 given in Eq. (4.10) when ε max (∆ξ/g) , 1 /np.

4.6 Qubit readout

† Recently, a longitudinal qubit-cavity interaction [∝ g(a + a)σz] has been considered theoretically and shown to produce a quantum nondemolition readout that is faster than the usual dispersive readout [92]. Here, we show how this type of interaction can be engineered through a simple modification of SQUADD. We investigate limits to the readout signal-to-noise and single-shot fidelity using this Hamiltonian-engineering approach, and find that these two measures canbe large even in the weak-coupling regime (g < κ).

If the Carr-Purcell sequence shown in Fig. 4.1 is applied with a fixed coupling, g(t) = g ∀ t, the counter-rotating term in Eq. (4.4) contributes. Although this is harmful to state transfer, this term can also generate otherwise useful quantum operations. Indeed, the evolution operator from leading-order average Hamiltonian theory is then

† −ig(a +a)σxtf /2 UR(tf ) = e = D(−iσxgtf /2), (4.30)

where D(α) is the displacement operator producing the coherent state |α⟩c ≡ D(α)|0⟩c [202]. The interaction appearing in Eq. (4.30) is longitudinal with respect to σx eigenstates, |⟩q. Applying

UR(tf ) on |⟩q then gives

UR(tf )|⟩q|0⟩c = |⟩q| α⟩c, (4.31)

with α ≡ −igtf /2. Thus, in combination with a qubit rotation conditioned on the cavity state [203],

UR(tf ) can be used to map a qubit state to a superposition of cavity coherent states; a Schrödinger’s cat state [204, 205]. Alternatively, the states | α⟩c can be resolved by homodyne detection of the

98 signal leaking from the cavity, enabling quantum nondemolition readout of the qubit in the basis

{|⟩q}. When a qubit is successively prepared and measured m ≫ 1 times to estimate an expectation value, the measurement statistics describing the mean of many independent repeated measurements become Gaussian. The performance of the readout proposed here is then well characterized by the signal-to-noise ratio (SNR). Indeed, the SNR compares the first two moments of the measurement operator

∫ √ tf † − M = i κ dt[aout(t) aout(t)], (4.32) 0

the integrated homodyne-detection signal for a measurement time tf , with aout(t) the output field leaking from the cavity. We then take

|⟨M⟩ − ⟨M⟩−| SNR ≡ + , (4.33) 2 2 1/2 [∆M+ + ∆M−]

2 2 2 where ∆M ≡ ⟨M ⟩ − ⟨M⟩ and where ⟨O⟩ = tr[O(tf )ρ], with ρ = |⟩⟨|q ⊗ |0⟩⟨0|c [92]. To evaluate the maximum achievable SNR, it is important to account for the first two nonvanishing orders in the Magnus expansion for the time-periodic Liouvillian: L ≃ L(0) + L(2). While L(0) generates the required conditional coherent-state displacement, L(2) results in qubit switching at a

2 2 rate Γ ≃ g τ κ/24 in the basis {|⟩q}. This qubit switching acts as a source of telegraph noise in the Langevin equation for the cavity field a(t) [62]. For g ≪ κ and κτ ≪ 1, this telegraph noise leads to a maximal value for the SNR:†

( ) √ 6g2 1/4 2 3 SNR ≃ ≃ √ . (4.34) κΓ κτ

Thus, for a short pulse interval, κτ < 1, this approach enables large SNR even in the weak-coupling regime (g < κ), reducing additional noise (beyond projection noise) in the evaluation of qubit expectation values from a soft average [206, 207].

In contrast to the case of many repeated measurements (described above), for a single-shot readout, the measurement statistics are non-Gaussian. Indeed, while the conditional probability

99 distribution describing the integrated signal ⟨M⟩ would simply describe a displaced Gaussian in the absence of switching, random switching events (e.g. qubit decay due to the mechanism described above) lead to significant bimodality [208]. A good measure of quality is then the single-shot fidelity.

In the same regime as above (g ≪ κ and κτ ≪ 1), we apply the methods of Refs. [208, 209] and find a large single-shot fidelity

[ ] (κτ)2 96 F ≃ 1 − log , (4.35) 1 192 (κτ)2 limited by the qubit switching mechanism described above.† For κτ = 0.1, this yields a single-shot fidelity of 99.95 %. This type of readout may be useful in several novel experimental settings where it is challenging to achieve strong coupling. For example, a spin qubit in a carbon nanotube has recently been successfully coupled to a microwave resonator, but the coupling achieved is marginal, g/κ ∼ 1 [28]. Alternative setups for semiconductor spin qubits in quantum dots or at single donor impurities coupled to microwave cavities have predicted couplings g/2π ≲ 1 MHz [27, 185, 201], typically smaller than the damping rate κ/2π = 2-10 MHz [31, 93].

4.7 Conclusions

Moving forward, the ideas presented here could lead to applications well beyond state transfer and readout. For example, going to second order in average Hamiltonian theory yields terms

∝ g2τ[a2 + (a†)2], which could be used to generate cavity squeezing. Such squeezing may be useful, e.g., to further improve qubit readout [92]. In addition, by monitoring the coherence of a state that is periodically swapped between a qubit and a bosonic mode, it may be possible to characterize noise processes affecting a harmonic system (e.g., a cavity or a magnon mode[210]). This may generalize noise spectroscopy methods [61] to linear systems for which direct application of π-pulses is impossible.

†See Appendix C.4 for a derivation of signal-to-noise and single-shot fidelity for the readout scheme proposed here.

100 5 Conclusion

In this thesis, we have made significant advances in understanding and suppressing dephasing sources that are increasingly relevant to novel semiconductor spin-qubit architectures involving magnetic-field gradients and microwave resonators. We have predicted the existence ofanew nuclear-spin-induced dephasing mechanism that arises in the presence of a magnetic field gradient.

We have shown that this mechanism can be dominant over other known dephasing processes in spin qubits in GaAs or silicon quantum dots, and at phosphorus donors in silicon. We have also developed microscopic models describing distinct physical processes leading to charge noise in semiconductor qubits, and explained how to distinguish them through measurements of the qubit coherence time and stretching parameter as a function of temperature. Finally, we have developed and assessed a protocol to achieve high-fidelity quantum state transfer between a qubit and a resonator even in the presence of strong dephasing due to low-frequency noise arising, e.g., from nuclear-spin or charge noise. A simple modification of this protocol has also led us to a readout schemethat yields high signal-to-noise and single-shot fidelity even for cavity damping that is stronger thanthe qubit-cavity coupling.

Clear pathways exist to build on the theoretical tools developed here and achieve even better

101 suppression of dephasing in semiconductor spin qubits. For example, in Chapters 2 and 3, all our predictions were made considering either free-induction decay or Hahn echo. It may be useful to consider more involved dynamical decoupling protocols, such as the Carr-Purcell sequence [22],

Uhrig dynamical decoupling [42] or concatenated dynamical decoupling [211]. In particular, knowing the noise spectrum of nuclear spins in a magnetic-field gradient, it may be possible to design a dynamical decoupling sequence that reduces qubit dephasing outside the motional-averaging regime discussed in Chapter 2. In addition, the protocol for quantum state transfer presented in Chapter 4 was designed to mitigate the effect of inhomogeneous broadening to first order in a Magnus expansion. It should in principle be possible to improve this protocol by designing a pulse sequence that also eliminates error at higher orders. In addition, the theoretical approach presented in Chapter 4 may be generalized to finite-frequency noise to consider state-transfer error arising from the finite- frequency contributions of the spectrum of, e.g., nuclear spins in an inhomogeneous magnetic field or charge noise from two-level charge fluctuators. On a separate note, it would be interesting to explore further the dynamics occurring in the effective 4 × 4 subspace arising in the description of state transfer to a collective mode of a qubit ensemble. This may open new avenues for suppression of error due to inhomogeneous broadening.

Beyond suppression of error due to dephasing, the theoretical approaches taken here may be used to study new and interesting physics. For example, it may be relevant to investigate the effect of an inhomogeneous magnetic field on the lifetime of nuclear-spin polarization [212–215].

In addition, it is well known that using dynamical decoupling techniques, a qubit can be used to probe its environment and measure its spectrum [47, 60, 61]. In the context of Chapter 2, this procedure may be used to experimentally measure the noise spectrum arising from nuclear spins in an inhomogeneous magnetic field. In addition to the thermal and narrowed states presented in

Chapter 2, more exotic nuclear-spin states such as spin-squeezed states [216] may be considered.

It may also be interesting to study theoretically the effect of dipole coupling between nuclear spins [217] or even charge fluctuators [218] on the relevant noise spectra. Finally, by monitoring the coherence factor of a quantum state swapped between a qubit and a bosonic mode under the dynamical decoupling sequence proposed in Chapter 4, it should become possible to experimentally probe the noise spectrum of interesting nearly-linear systems such as spin ensembles [179, 219] or

102 magnon modes [210]. With these systems, measurement of the noise spectrum would be challenging using standard spectroscopy methods [47, 60, 61], which rely on applying a train of π pulses on a qubit and measuring the resulting coherence factor, two tasks that would very difficult with nearly-linear systems.

103 A Appendices of “Enhanced hyperfine-induced spin dephasing in a magnetic-field gradient”

A.1 Hyperfine coupling constants in a double quantum dot

In this Appendix, we obtain expressions for the hyperfine coupling strengths Ak for three possible geometries: (i) one electron in a single dot; (ii) one electron in the (bonding or antibonding) delocalized molecular state of a symmetric double quantum dot; (iii) two electrons in a double quantum dot (each in the localized |L⟩ and |R⟩ states), in the S-T0 basis. Fig. 2.2 helps to visualize the problem in two dimensions.

Case (i) has already been treated in the literature [14], yielding for an electron wavefunction of

q the form ψ(r) = ψ(0) exp[−(r/r0) /2], in dimension d,

2 A( ) −(k/N)q/d Ak = Av0|ψ(rk)| = e , (A.1) d d N q Γ q with v0 the volume per nuclear spin and N the number of nuclei within the dot radius r0. We deal with the two cases involving a double dot in a very similar way, using left (σ = L) and

104 σ − σ q σ − σ σ right (σ = R) basis states ψσ(rk) = ψ(r ) exp[ (rk /r0) /2], where rk = rk r and the vectors r σ 2 2 2 σ locate the center of each dot. Using spherical coordinates, (rk ) = rk + l /4 + rkl Gd,k, with    k  (−1) d = 1,  GL/R = (A.2) d,k  cos θk d = 2,    sin θk cos φk d = 3.

For d = 2, θk is the polar angle that locates the nuclear spin k, while for d = 3, θk and φk are, respectively, the polar and azimuthal angles. As in Ref. [14], we define N as the number of nuclear spins within the Bohr radius r0 and label the spins in increasing order of rk, such that

1/d rk/r0 = (k/N) . Thus, for an electron in dot σ, the hyperfine couplings are given by

 [ ]   ( ) ( ) q/2 k 2/d k 1/d Aσ = Aσ exp − + η2 + 2η Gσ , (A.3) k 0  N N d,k 

≡ σ with η l/2r0, the dimensionless separation of the double dot, and A0 is obtained from the ∑ σ normalization condition k Ak = A. In the case of a single electron shared by two dots of the same size and at the same electrical

1 potential, the electron wavefunction is ψ(r) = √ [ψ (r) ψ (r)], treating both symmetric (+) 2 L R − σ and antisymmetric ( ) superpositions at the same time. In order to calculate A0 and thus obtain ∑ ≫ Ak , we assume an isotropic distribution of nuclear spins and N 1 and evaluate A = k Ak by ∑ converting k to an integral to obtain for d = q = 2 ( √ ) k cosh 2η N cos θk 1 2 A −k/N A = Av0|ψ(r )| ≃ e . (A.4) k k N eη2 1

In the above, we have defined Ak as the hyperfine coupling strength for nuclear spin k when the + electron wavefunction is ψ(r). As expected, Ak tends toward the single-dot expression [Eq. (A.1)] for η ∝ l → 0.

Hyperfine couplings for case (iii) are obtained in a similar way, however it is first requiredto map the Hamiltonian of a singlet-triplet qubit to Eq. (2.1). This is done in Appendix A.2.

105 A.2 Mapping to a singlet-triplet qubit

For moderate electron Zeeman spittings b > ∆bx, the model of Eq. (2.1) can be applied to describe two electron spins in the S-T basis [220], defined by |S/T ⟩ = √1 (| ↑↓⟩ ∓ | ↓↑⟩). Indeed, defining 0 0 2

τˆx = |S⟩⟨T0| + |T0⟩⟨S| and τˆz = |T0⟩⟨T0| − |S⟩⟨S| and assuming a negligible exchange coupling, the effective Hamiltonian describing the two electron spins and their nuclear-spin bath in a magnetic- field gradient is ∑ ∑ ˆ ˆ z ˆ x ˆz x ˆz x ˆx HST0 = HZ + HZ + δh τˆ + b γkIk + γkbkIk , (A.5) k k

ˆ α where HZ is the Zeeman coupling of the electrons for a magnetic field along direction α. Defining

|T+⟩ = | ↑↑⟩ and |T−⟩ = | ↓↓⟩, we find

ˆ z | ⟩⟨ | − | ⟩⟨ | HZ = b ( T+ T+ T− T− ) , (A.6) | ⟩ | ⟩ x | ⟩ − | ⟩ x x T+ + T− ∆b T+ T− Hˆ = b √ ⟨T0| − √ ⟨S| + H.c., Z 2 2 2

x ≡ x x x ≡ x − x x where b (bR + bL)/2, ∆b bR bL and bi is the electron Zeeman splitting along x in the localized orbital state of dot i. For a magnetic field Bx(z) that is an odd function of z, as shown in Fig. 2.2, bx = 0. The term ∝ ∆bx above typically does not vanish, and thus may lead to leakage

x out of the S-T0 subspace. This leakage can, however, be suppressed if b ≫ ∆b . In this limit, we neglect the term ∝ ∆bx and take τˆx → 2Sˆz in the Hamiltonian of Eq. (A.5), which approximately → L − R maps it to Eq. (2.1), provided we replace Ak δAk = Ak Ak . The coherence factor then corresponds to the off-diagonal element of the density matrix inthe {| ↑↓⟩, | ↓↑⟩} basis. Using the results of Appendix A.1, we find (d = q = 2) ( √ ) A − k +η2 k δA = 2 e ( N ) sinh 2η cos θ . (A.7) k N N k

We estimate corrections to the above mapping for finite ∆bx from the leakage probability, 1 − ⟨ ˆ ⟩ ˆ ≡ | ⟩⟨ | | ⟩⟨ | ⟨ ˆ ⟩ PST0 (t) , where PST0 S S + T0 T0 . We have evaluated PST0 (t) within time-dependent

x perturbation theory to leading order in ∆b /b and Ak/ωk. This calculation predicts the onset of √ a stationary leakage probability ∼ (∆bx/b)2 after a time ∼ N/A. In principle, this probability

106 could grow over time if other processes that can decorrelate the electron and nuclear spins were included, such as nuclear dipolar interactions. In that case, we expect the leakage probability to

x 2 grow on a time scale ∼ (b/∆b ) τc, with τc the correlation time of the nuclear-spin bath. In bulk

GaAs, τc ∼ 100 µs can be estimated from the NMR linewidth [221]. We therefore expect the approximate mapping to give an accurate representation of dynamics in the singlet-triplet basis for times t ≲ 100 µs whenever ∆bx/b ≲ 1.

A.3 Validity of the Magnus expansion

Using the Gaussian approximation, we showed in Section 2.3.2 that the coherence factor is of the ∑ ∼ | |2 form C(τ) exp[ k hk(τ) ], as expressed in Eq. (2.17) and Eq. (2.18). The field hk(t) is then obtained from the Hamiltonian of Eq. (2.1) with a zeroth-order Magnus expansion. To subleading order in that expansion, we would have, for α ∈ {x, y, z}

[ ] [ ] 2 2 α 2 ∼ α(0) α(0) α(2) α(1) (hk ) hk + 2hk hk + hk . (A.8)

In the rest of this Appendix, we drop the α index to ease notation. In the above expression, (n) ∼ x n+1 (0) ∼ x 2 hk (Akbk/ωk) , such that hk (Akbk/ωk) is the leading term, considered in the main (0) (2) (1) 2 x 4 body of the paper, while hk hk and [hk ] , of order (Akbk/ωk) , are subleading. Note that we x have dropped terms containing odd powers of Akbk because they cancel out when summed over k due to the symmetries of |ψ(r)|2 and Bx(r). We will now establish criteria for the leading term to dominate the subleading ones, and hence for the Magnus expansion to be valid.

The n-th order term in the Magnus expansion involves n + 1 integrals over time of vk(t), which is given by Eq. (2.9). Roughly speaking, each integral of a sine generates a factor cos ωkt/ωk in (n) hk , while each integral of a cosine generates a factor t sin ωkt. Thus, we find ( ) γA bx γA bxt γA bx 2 t h(0) ∼ k k + k k , h(1) ∼ k k , (A.9) k ω2 ω k ω ω k( ) k ( ) k k A3 γbx 3 A3 γbx 3 (2) ∼ k k k k 2 hk 2 t + t , (A.10) ωk ωk ωk ωk

107 such that

[ ] ( ) ( ) ( ) 2 A4 γbx 4 A4 γbx 4 A4 γbx 4 (1) (0) (2) ∼ k k k k 2 k k 3 hk + hk hk 3 t + 2 t + t . (A.11) ωk ωk ωk ωk ωk ωk

Using an order-of-magnitude estimate of the sum over k, this gives three critical times (t1, t2, t3) for which each of the above terms can be of order 1. In the limit where b ≫ ∆bx, we find

γ3N 3b7 γN 3/2b3 γ1/3Nb5/3 t ∼ , t ∼ , t ∼ , (A.12) 1 (A∆bx)4 2 (A∆bx)2 3 (A∆bx)4/3 while in the opposite limit (b ≪ ∆bx), we get

(γN∆bx)3 γN 3/2∆bx N(γ∆bx)1/3 t ∼ , t ∼ , t ∼ . (A.13) 1 A4 2 A2 3 A4/3

If tcrit = min{ti} is shorter than the predicted dephasing time from the leading-order Magnus expansion, we conclude that the approximation is not valid. This approach is especially convenient to fix a validity condition for narrowed-state free-induction decay, since there is no motional averaging ≫ x ∇ in that regime. For b ∆b , comparing T2 from Eq. (2.22) to the above three timescales sets three criteria on ∆bx, of which the most stringent is

∆bx N 5/6γb λ ≡ < . (A.14) b A

∼ 5/6 M Considering a worst-case scenario with λ 1, this criterion becomes b > A/γN = bmin. Such M M ∗ minimum values Bmin = bmin/g µB in the three main classes of devices studied here are displayed in Table A.1. These numbers allow us to conclude that keeping only the leading order in the Magnus expansion is a good approximation in all the free-induction cases considered here.

For Hahn-echo decay, it is less convenient to fix a boundary on parameters, because motional averaging can keep coherence finite for long times. Thus, we directly study the timescales beyond which the Magnus expansion can fail. These timescales are calculated in dots and donor impurities for the various parameter regimes studied in this paper and are presented in Table A.2. In situations

∇ where decay occurs in a finite time T2e , that decay time is always shorter than tcrit, except in single

108 M Material A (MHz) γ (MHz/T) N Bmin GaAs 1.3 × 105 60 4.4 × 106 50 mT Si 320 53 104 0.3 mT Si:P 320 53 250 60 mT

M Table A.1 – Minimum longitudinal fields Bmin below which the Magnus expansion can fail to describe narrowed-state FID.

Small longitudinal field limit (b ≪ ∆bx) GaAs ∆Bx = 25 mT ∆Bx = 100 mT ∆Bx = 200 mT

tcrit 800 ns 1.2 µs 1.5 µs Si ∆Bx = 20 mT ∆Bx = 80 mT ∆Bx = 400 mT

tcrit 4.5 µs 7.5 µs 13 µs Large longitudinal field limit (b ≫ ∆bx) GaAs B = 200 mT B = 295 mT B = 695 mT

tcrit 1.5 µs 3 µs 12 µs Si:P B = 10 mT B = 20 mT B = 100 mT

tcrit 0.5 µs 1.5 µs 20 µs

Table A.2 – Timescales tcrit below which the Magnus expansion is valid, according to Eqs. (A.12) and (A.13). Unless specified otherwise in the table, parameters for lateral dots are those taken fromFigs. 2.3 and 2.4, considering double dots.

109 donors. In situations where motional averaging is predicted, no discrepancy between the Magnus expansion and the exact numerical solution is seen. We emphasize that the analysis done here only gives us critical times tcrit below which the leading-order term in the Magnus expansion is sure to dominate, but does not determine where the subleading-order term becomes dominant. In other words, we believe these estimates give a definite range of applicability, but the approximations introduced may well be valid outside of that range.

A.4 Validity of the Gaussian approximation

In this Appendix, we find a simple criterion which, if respected, justifies the Gaussian approximation. In Section 2.3.2, we found that the coherence factor is related to ⟨eiδXˆ(τ)⟩, which can be expanded in its Taylor series

∑ n ∑ ∑ ˆ i ⟨eiδX ⟩ = hl1 ...hln ⟨δIˆl1 ...δIˆln ⟩. (A.15) n! k1 kn k1 kn n k1...kn l1...ln

Since we have taken a bath for which nuclei are uncorrelated, we have ⟨δIˆl1 δIˆl2 ⟩ = ⟨δIˆl1 ⟩⟨δIˆl2 ⟩ = k1 k2 k1 k2

0 ∀ k1 ≠ k2, by definition of δIˆk. To simplify the notation, in this Appendix, we define δXˆ ≡ δχXˆ ˆ ˆ and δIk ≡ δχIk and drop the χ index in expectation values. Thus, we have

∑ 2n ∑ ∏n ′ ′ iδXˆ(τ) i lq lq lq lq ⟨e ⟩ ≃ an h h ⟨δIˆ δIˆ ⟩, (A.16) (2n)! kq kq kq kq n kll′ q=1

n with an = (2n)!/2 n! the number of ways to group 2n terms into n pairs. In the above, we have dropped all moments of order greater than two, which lead to non-Gaussian contributions. The above sums can then be reorganized to yield [ ] ∑ ˆ 1 ′ ′ ⟨eiδX(τ)⟩ ≃ exp − hl (τ)hl (τ)⟨δIˆl δIˆl ⟩ , (A.17) 2 k k k k kll′ which is the Gaussian approximation used in Section 2.3.2 and throughout the paper.

To justify the omission of the moments of order larger than two, we invoke the large number N of ∑ nuclei. Indeed, there are roughly N times less elements in a sum of the form over moments k1...kn

110 of order 3 than in a similar sum over moments of order 2. Furthermore, in both thermal and narrowed states, we find that moments of odd order vanish. Thus, in order to find the conditions under which the Gaussian approximation is justified, we estimate the importance of the sum over fourth-order moments. Considering only products of these fourth-order moments, we find that the non-Gaussian contributions are of the order of

∑ 4n ∑ ∏n i b l′ l′′ l′′′ l′ l′′ l′′′ n lq q q q q1 q q q C4(τ) ∼ h h h h ⟨δIˆ δIˆ δIˆ δIˆ ⟩, (A.18) (4n)! kq kq kq kq q1 kq kq kq n q=1

n with bn = (4n)!/24 n! the number of ways to group 4n terms into strings of four. For a nuclear-spin bath initially in a narrowed state, this leads us to { } 1 ∑ [ ] C (t) ∼ exp (hx(t))4 + (hy(t))4 . (A.19) 4 24 k k k

≫ x For b ∆b , this becomes [ ( ) ] ∑ A bxt 4 C (t) ∼ exp k k . (A.20) 4 b k

4 x 3/4 Thus, non-Gaussian contributions grow like exp[(t/τ4) ], with 1/τ4 ∼ A∆b /N b, while Gaussian − ∇ 2 ∇ ∼ x 1/2 contributions led to exp[ (t/T2 ) ] decay. Comparing 1/τ4 and 1/T2 A∆b /N b, we find that ∇ ∼ 1/4 τ4/T2 N , and thus that non-Gaussian contributions only decay as a weak power law as N is increased.

In the case of Hahn echo, we rather find { } ∑ ∼ α 4 C4(2t) exp [hk (t)] . (A.21) kα

Taking b ≪ ∆bx, if the short-time expansion is valid, we can expand the sines and cosines in the α ∼ 8 ∇ ∼ 1/8 fields hk (t) to leading order in t. We obtain C4(2t) exp[(t/τ8) ], and we find τ8/T2e N . Another relevant limit is when b ≫ ∆bx, useful to describe motional averaging. We can then replace ωk → b and the argument of the exponential becomes an oscillating function with amplitude ∼ x 4 8 (Akbk) /b . Requiring this term to be smaller than the equivalent amplitude for the Gaussian

111 term, discussed in Section 2.4.2, we find that the following criterion must bemet

√ ∆bx Nγb λ = < . (A.22) b A

This criterion is easily satisfied in all materials studied here, even in single P donor impurities in

Si, in accordance with the fact that the Gaussian approximation does not fail to predict motional averaging in that material [see Fig. 2.5(b)].

2 A.5 Σs for relevant geometries

As shown in Section 2.4, both the free-induction and Hahn-echo dephasing times can be linked to ∑ the sum Σ2 = (A bx )2. In this section, we calculate Σ2 for various geometries. To simplify s ks ks ks s the notation, we will drop the species index s.

In all cases, we assume a transverse magnetic field of the form Bx(r) = βz, with β a constant, such that √ x x| rk cos θk x k x bk = zk ∂zb z=0 = δb = δb cos θk, (A.23) r0 N

where θk is the polar angle locating nucleus k and nuclei are labeled with increasing rk, as in x ≡ x| Appendix A.1. We have also defined δb r0 ∂zb z=0. Using Eqs. (A.1) and (A.23), we calculate Σ2 for a single dot assuming an isotropic distribution of nuclear spins, with N large enough to convert the sum into an integral. We obtain ( ) 2+d x 2 Γ 2 1 (Aδb ) (q ) Σ = 2+d . (A.24) d d q 2 d N q 2 Γ q

In the specific case of d = q = 2, this yields Σ2 = (Aδbx)2/8N.

For double dots with d = q = 2, we rather use Eqs. (A.4) and (A.7) for Ak. In the case of a single electron with a symmetric (+) or antisymmetric (−) delocalized wavefunction ψ(r) as in

112 Appendix A.1, we get

( ) Aδbx 2 3 4eη2/2(1 + η2) + e2η2 (1 + 4η2) Σ2 = . (A.25) eη2 1 16N

Different wavefunctions lead to different values of Σ2 because we have taken into account the overlap between the orbitals. For a singlet-triplet qubit, we rather have

1 + 4η2 − e−2η2 Σ2 = (Aδbx)2 . (A.26) 4N

113 B Appendices of “Microscopic models for charge-noise-induced dephasing of solid-state qubits”

B.1 Corrections to the leading-order Magnus expansion and the Gaussian

approximation

The discussion in Sec. 3.2 applies when the fluctuator dynamics are well described by the leading term in the Magnus expansion under a Gaussian approximation. In this Appendix, we derive the leading-order corrections to the formulas of Sec. 3.2 and find a simple condition for which these corrections can safely be neglected.

Using the Magnus expansion, the interaction-picture time-evolution operator corresponding to the perturbation given by Eq. (3.5) is (taking t ≡ ts to simplify the notation) [ ] ∑∞ ˆ − ˆ (m) U(t) = exp i HM (t) . (B.1) m=1

114 ˆ (m) − Here, the term HM (t) results from m integrals of m 1 nested commutators involving the interaction-picture perturbation V ′(t). When the perturbation is sufficiently weak, contributions I ( ) ˆ (m) O ˆ ′ m with large m will be suppressed since HM (t) = VI (t) . Explicit expressions for the first few orders of the Magnus expansion can be found in the literature [65, 71, 113]. To evaluate the coherence dynamics of the qubit, we calculate

⟨σˆ+(t)⟩ = eiϕ(t)tr[Uˆ †(t)ˆσ+Uˆ(t)ˆρ(0)], (B.2) with ϕ(t) given by

∫ t ′ ϕ(t) = ωQ dt1s(t1), (B.3) 0 ∑ ′ ⟨ z⟩ ωQ = ωQ + Ωn τˆn . (B.4) n

Even-order terms in the Magnus expansion are proportional to (ˆσz)2m = 1 while odd-order terms are proportional to (ˆσz)2m+1 =σ ˆz. Thus, we have

† L Uˆ (t)ˆσ+Uˆ(t) = ei M(t)σˆ+, (B.5) [ ] 1 L (t)· = Hˆ odd(t)ˆσz + Hˆ even(t), · , (B.6) M 2 ξ ξ

∑ ∑ ˆ odd ∞ ˆ (2m+1) ˆ even ∞ ˆ (2m) where we have introduced Hξ (t) = m=0 Hξ (t) and Hξ (t) = m=0 Hξ (t). Each ˆ (m) ˆ ≡ ℏˆ Hξ (t) is the m-th order term in the Magnus expansion associated with Vξ(t) s(t) ξ(t). Ex- iL (t) + panding e M in a Taylor series around LM(t) = 0 and applying every resulting term on σˆ , we find a recursion relation that leads to

† L Uˆ (t)ˆσ+Uˆ(t) = ei ξ(t)σˆ+, (B.7) [{ } [ ]] 1 L (t)· = Hˆ odd(t), · + Hˆ even(t), · . (B.8) ξ 2 ξ ξ

Crucially, Lξ(t) does not act on the space of qubit operators. We define the coherence factor C˜(t) through

⟨σˆ+(t)⟩ = ⟨σˆ+(0)⟩C˜(t), (B.9)

115 where C˜(t) contains phase information, and is related to the coherence factor given in the main text through C(t) = |C˜(t)|. For an initially separable state ρˆ(0) =ρ ˆQ(0) ⊗ ρˆFB(0), Eq. (B.2) combined with Eq. (B.7) then gives

L C˜(t) = eiϕ(t)⟨ei ξ(t)⟩, (B.10) [( ) ] iLξ(t) iLξ(t) ⟨e ⟩ ≡ trFB e 1 ρˆFB(0) . (B.11)

Since the qubit experiences noise due to many uncorrelated fluctuators, we expect a cumulant

L expansion to converge rapidly. To perform the cumulant expansion, we rewrite ⟨ei ξ(t)⟩ in terms of

δLξ(t) = Lξ(t) − ⟨Lξ(t)⟩:

L ⟨L ⟩ L ⟨L ⟩ ⟨ei ξ(t)⟩ = ei ξ(t) ⟨eiδ ξ(t)⟩ = ei ξ(t) eχ, (B.12) ∑∞ ∑ ∏p (−1)p+1 ⟨[iδL (t)]mi ⟩ χ = ξ , (B.13) p mi! p=1 m1,...,mp i=1 where the sums over mi range from 1 to ∞. Eq. (B.13) defines an expansion in increasing powers ˆ of Lξ(t), while Lξ(t) is itself obtained from the Magnus expansion associated with s(t)ℏξ(t). Terms of common powers of ξˆ(t) can then be collected. Up to and including O(ξˆ(t)4), we find

i[ϕ(t)+ϕ3(t)] C˜(t) = e exp [−f2(t) − f4(t)] , (B.14) where ϕ(t) is given by Eq. (B.3),

⟨ ⟩ ⟨[ ]⟩ ⟨ ⟩ ˆ (3) i ˆ (2) ˆ (1) − 1 ˆ (1) 3 ϕ3(t) = Hξ (t) + 4 Hξ (t), Hξ (t) 6 [Hξ (t)] , (B.15) and

⟨ ⟩ 1 ˆ (1) 2 f2(t) = 2 [Hξ (t)] , (B.16) [ ⟨ ⟩ ⟨ ⟩ ] ⟨{ }⟩ 2 − 1 1 ˆ (1) 4 − ˆ (1) 2 1 ˆ (3) ˆ (1) f4(t) = 8 3 [Hξ (t)] [Hξ (t)] + 2 Hξ (t), Hξ (t) ⟨{ [ ]}⟩ ⟨[ ]⟩ i ˆ (1) ˆ (2) ˆ (1) i ˆ (2) ˆ (1) 2 + 24 Hξ (t), Hξ (t), Hξ (t) + 12 Hξ (t), [Hξ (t)] . (B.17)

116 The leading-order term in the expansion of Eq. (B.13) is given by Eq. (B.16) and corresponds to the first-order Magnus expansion under the Gaussian approximation. Eq.(B.17) gives the first subleading term in |C˜(t)|. The corrections given by Eq. (B.17) come both from the non-Gaussian nature of ξˆ(t) and from the fact that ξˆ(t) does not generally commute with itself at different times. ⟨ˆ ˆ ˆ ˆ ⟩ All the terms in Eq. (B.17) involve correlators of the form ξn1 (t1)ξn2 (t2)ξn3 (t3)ξn4 (t4) . Following from the definition given in Eq. (3.6) when the initial state of the fluctuators is factorizable [ρFB(0) = ∏ n ˆ n ρFB(0)], the operators ξn(t) for the noise due to single fluctuators have the following properties

⟨ξˆn(t)⟩ = 0, (B.18)

⟨ˆ ˆ ⟩ ⟨ˆ ˆ ⟩ ξn1 (t1)ξn2 (t2) = δn1,n2 ξn1 (t1)ξn1 (t2) , (B.19)

⟨[ξˆn(t1), ξˆn(t2)]⟩ = 0. (B.20)

The last property in Eq. (B.20) comes from Eq. (3.18). In addition, when the evolution of each fluctuator is given by a Markovian master equation of the form ofEq.(C.33), we find from the standard formula for multitime averaging [62]

⟨ξˆn(t1)ξˆn(t2)ξˆn(t3)ξˆn(t4)⟩ [ ] 4 ′4 −|t2−t3|/τn −|t1−t2|/τn −|t3−t4|/τn = ∆ξn + ∆ξn e e e (B.21) where we have introduced

γnγn(γn − γn)2 ′4 4 ↑ ↓ ↑ ↓ ∆ξn = 16Ωn n n 4 . (B.22) (γ↑ + γ↓ )

n n Substituting the detailed-balance relation γ↑ /γ↓ = exp(−ℏωn/kBT ) into Eq. (B.22), which we substitute again in Eq. (B.21), we find an approximate upper bound for the fourth-order correlation function in Eq. (B.21):

−| − | −| − | ⟨ˆ ˆ ˆ ˆ ⟩ ≲ 4 t1 t2 /τn t3 t4 /τn ξn(t1)ξn(t2)ξn(t3)ξn(t4) ∆ξne e , (B.23)

for ℏωn < kBT , neglecting factors of order 1. In Eq. (B.23), ∆ξn is given by Eq. (3.48). Substituting

117 Eqs. (B.18) to (B.20) and Eq. (B.23) into Eq. (B.17), we find expressions for the upper bound on s | ˜ | f4 (t), the first subleading correction to C(t) for dynamical-decoupling sequence s. Taking the fast- ≪ s ∼ s noise limit (τc T2 ), and taking a typical value t T2 , we drop exponentially small corrections ∼ s s in t/τn T2 /τc. The inequality for f4 (t) then becomes, for both free-induction decay and Hahn echo,

∑ s ≲ 4 3 ∀ ≪ s f4 (t) ∆ξnτnt s. (τc T2 ) (B.24) n

≫ s s In the opposite, slow-noise limit (τc T2 ), we expand the upper bound on f4 (t) in a Taylor series around t/τn = 0. Keeping only the leading term, we find

∑ ∗ ≲ 4 4 ≫ s f4 (t) ∆ξnt , (τc T2 ) (B.25) n ∑ ∆ξ4t5 |f e(t)| ≲ n . (B.26) 4 τ n n

Typically, approximately N fluctuators will contribute to qubit dephasing, with N defined by

(∑ ) ∗ ∆ξ2 2 (1/T )4 ≡ ∑n n ∑ 2,sl. N 4 = 4 4 , (B.27) n ∆ξn n ∆ξn

∗ where T2,sl. is the free-induction decay time in the slow-noise limit, Eq. (3.29). Assuming that τn varies slowly with n for ℏωn ≲ kBT , we replace τn → τc in Eqs. (B.24) to Eq. (B.26), τc being given by Eq. (3.28). Also using the definition of N given by Eq. (B.27), we find

( ) 3 s ≲ 1 τc t ∀ ≪ s f4 (t) ∗ ∗ s, (τc T2 ) (B.28) N T2,sl. T2,sl. ( ) 4 ∗ ≲ 1 t ≫ s f4 (t) ∗ , (τc T2 ) (B.29) N T2,sl. 5 e ≲ 1 t f4 (t) ∗ 4 . (B.30) N (T2,sl.) τc

As explained in Sec. 3.3, the leading term f2(t) in the combined Magnus and cumulant expansion

≃ s αs is well approximated by f2(t) (t/T2 ) . This leading term then dominates over the subleading

118 ∼ s contribution given by Eqs. (B.28) to (B.30) when (taking t T2 and neglecting factors of order 1)

≫ ∀ ≪ s N 1 s, (τc T2 ) (B.31) ≫ ≫ s N 1, (free-induction decay, τc T2 ) (B.32) ≫ e ≫ s N τc/T2 . (Hahn echo, τc T2 ) (B.33)

∗ To obtain Eq. (B.31), we have used Eqs. (3.23) and (3.29) to express T2,sl. in terms of τc and T2M, replacing again τn → τc. Similarly, to obtain Eq. (B.33), we have used Eqs. (3.29) and (3.30) to

∗ e express T2,sl. in terms of τc and T2 in the slow-noise limit.

Eq. (B.33) shows that the minimum number of fluctuators required for the leading term f2(t) e → ∞ to dominate over the subleading term can become arbitrarily large in the limit τc/T2 , corresponding to fluctuators with a vanishing equilibration rate. This result is consistent with the results of Ref. [58], in which the authors showed that the Hahn-echo coherence factor for a qubit coupled to a two-level fluctuator with a switching rate 1/τn ≪ ∆ξn shows a strong non-Gaussian behavior. Non-Gaussian corrections to the qubit coherence factor have also been considered in Ref. [41] for various dynamical decoupling sequences.

When the criteria given by Eqs. (B.31) to (B.33) are satisfied, the leading contribution to |C˜(t)|

(corresponding to the theory explained in Sec. 3.2) dominates over the subleading term.

B.2 Electron-phonon coupling strength

Introducing a deformation potential tensor Ξχ for each conduction band minimum, the deformation contribution to Aqλχ is [160, 168] √ 1 ℏ ∑ Ad = Ξχ (q ξj + q ξi ), (B.34) qλχ 2 2ρυω ij i qλ j qλ qλ ij

where ρ is the mass density of the sample and υ its volume. We have also introduced ξqλ, the vector indicating the propagation direction of the phonon mode qλ with angular frequency ωqλ. The effect of shear strains on the single conduction-band minimum of GaAs is negligible relative to the effect of volume dilations [168]. The deformation-potential tensor for GaAs thus reduces

119 to Ξij = δija(Γ1c) ≃ −8.6 eV. In silicon, there are six conduction-band minima at k-points along the six directions equivalent to [100], at roughly 85 % of the distance to the the Brillouin-zone boundary [168]. We label these minima as x, y, and z. Using these labels, the silicon deformation-potential tensor takes the form [168, 169]

∑ χ Ξij = δijΞd + (δχ,sxδixδjx + δχ,syδiyδjy + δχ,szδizδjz)Ξu, (Si) s= where Ξd ≃ 5 eV and Ξu ≃ 8.77 eV [168]. Crystalline silicon is not piezoelectric since the diamond lattice has inversion symmetry. In contrast, GaAs has a zincblende structure, for which the piezoelectric contribution is [222] √ z x y p ℏ e14 qxqyξqλ + qyqzξqλ + qzqxξqλ Aqλ = 2e 2 , (B.35) 2ρυωqλ ε q

where e is the elementary charge, e14 is the 14 element of the piezoelectric tensor in Voigt notation, and ε is the static dielectric constant of the material.

B.3 Fluctuator equilibration rate for the phonon sum process

The fluctuator equilibration rate for the phonon sum process is

[ ( ) ( ) 4 9 2 7 1 ≃ Ξ ωn 9 ee14 4 2 ωn Σ 8 + Ξ 1 + ζ 6 τn 560 v 1225 ε 3 v LA ( ) ( LA ) ] 27 ee 4 4 ω5 2π(℘n)4(k T )2 + 14 1 + ζ2 n 0 B . (B.36) 4 ℏ4 n 2 2 6 6125 ε 3 vLA (ωγ ) matωD

The symbols in the above equation are defined below Eqs.3.56 ( ) and (3.57).

120 C Appendices of “Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED”

C.1 Evolution under SQUADD

In this section, we derive the exact evolution operator that corresponds to the SQUADD state- transfer sequence, described in the main text. We assume that the qubit-resonator coupling strength g(t) is g for even n(t) and 0 for odd n(t), where n(t) is the number of qubit π pulses applied before time t. The time-evolution operator then breaks into segments associated with the intervals of duration τ between π pulses. In the subspace for which the total number of excitations Nex = † a a + σ+σ− has eigenvalue 1, the evolution operator for a single period of the decoupling sequence takes the form

U1 = Rn(Ωτ)Rz(−ξτ)Rn(Ωτ), (C.1)

121 with

√ Ω = g2 + ξ2/4, (C.2) g ξ n = x + z. (C.3) Ω 2Ω

In the above equations, ξ is the random qubit-resonator detuning introduced above Eq. (4.4). In

−iθn·τ /2 addition, in Eq. (C.1), we have introduced the operator Rn(θ) ≡ e , which applies an SU(2) rotation by angle θ around the axis set by the unit vector n in the space spanned by the vector of pseudospins τ = (τx, τy, τz). These pseudospins are defined by

τx = |g1⟩⟨e0| + |e0⟩⟨g1|, (C.4)

τy = i(|g1⟩⟨e0| − |e0⟩⟨g1|), (C.5)

τz = |e0⟩⟨e0| − |g1⟩⟨g1|. (C.6)

In Eqs. (C.4) to (C.6), g (e) labels the ground (excited) state of the qubit, while 0 or 1 is the number of photons in the cavity. The product of the three rotation matrices in Eq. (C.1) is itself a rotation matrix U1 = Rv(ϑ), where

√ ϑ cos = 1 − A2 − B2, (C.7) 2 A B v = √ x + √ z, (C.8) A2 + B2 A2 + B2 with

( ) 2g Ωτ ξτ ξ Ωτ ξτ Ωτ A = cos cos + sin sin sin , (C.9) Ω 2 2 2Ω 2 2 2 ( ) ξ Ωτ ξτ Ωτ ξτ Ωτ ξ2 − 4g2 Ωτ ξτ B = sin cos − cos sin cos + sin2 sin . (C.10) Ω 2 2 2 2 2 4Ω2 2 2

122 The evolution at the end of the full sequence containing np pulses (and thus np/2 periods) is given by

np/2 U(tf ) = U1 = Rv(npϑ/2), (C.11)

for even np. Eq. (C.11) gives us a closed-form analytical expression for the evolution operator under † the SQUADD sequence. Taking M(|ψ⟩⟨ψ|) = U(tf )|ψ⟩⟨ψ|U (tf ) in Eq. (3) of the main text, with

U(tf ) given by Eq. (C.11), we obtain the average state-transfer fidelity

[ ] 1 n ϑ n ϑ F = E 1 + v2 sin2 p + v sin p , (C.12) 3 x 4 x 4 where E[·] is an ensemble average over the detuning ξ.

A simple analytical expression can be obtained for F when gτ ≪ 1 and ξτ ≪ 1. In this situation, we substitute Eqs. (C.9) and (C.10) into Eqs. (C.7) and (C.8), and expand the resulting expressions for v and ϑ/2 to fourth order in τ. In addition, we use the condition for complete state transfer gtf = (g/2)npτ = π/2, where g is the time average of the coupling strength on a period of the sequence and tf is the transfer time. Assuming a Gaussian distribution of the detuning ξ with standard deviation ∆ξ, we obtain Eq. (4.10), valid for gτ ≪ 1 and ∆ξτ ≪ 1.

C.2 Finite bandwidth and counter-rotating terms

We evaluate the state-transfer fidelity using numerical simulations that take into account both the finite bandwidth of the coupling modulation g(t) and the counter-rotating terms in the Rabi

Hamiltonian. In these simulations, we find the evolution of the system under the toggling-frame

Hamiltonian    even HT (t), n(t) even, HT(t) =  (C.13)  odd HT (t), n(t) odd.

123 where

even 1 † † 2iωqt HT (t) = 2 ξσz + g(t)[a σ− + a σ+e + H.c.], † † odd − 1 2iωqt HT (t) = 2 ξσz + g(t)[a σ+ + a σ−e + H.c.], (C.14)

with ωq the qubit frequency (the qubit and the resonator are resonant). In contrast with Eq. (4.4), Eqs. (C.14) take into account the counter-rotating terms appearing in the Rabi Hamiltonian. These terms give rise to leakage outside the subspace containing zero or one excitation when g(ω) ≡ ∫ ∞ −∞ dt exp[iωt]g(t) has significant weight at ω = 2ωq. To take into account the finite bandwidth of the coupling modulations, weuse

−1 g(t) = F [gid(ω)f(ω)] , (C.15) where

F − 2 2 gid(ω) = [gid(t)], f(ω) = exp[ ω /2σf ]. (C.16)

(−1) In Eqs. (C.15) and (C.16), F is the (inverse) Fourier transform, gid(t) is the ideal square wave described in the main text, and f(ω) is a Gaussian filter with standard deviation σf which eliminates high-frequency components of gid(t). Evaluation of the transforms in Eq. (C.15) leads to

n∑p/2 g(t) = gsq(t − 2jτ), (C.17) j=0 where gsq(t) describes a single filtered square pulse centered around t = 0,

{ [ ( )] [ ( )]} ′ ′ g σf τ σf τ gsq(t) = erf √ t + − erf √ t − . (C.18) 2 2 2 2 2

In Eq. (C.18), τ ′ is the width of the square coupling pulse, which may differ from the pulse interval

124 τ. The time required for the coupling to rise from 10 % to 90 % of its final value is, using Eq. (C.18),

[ ( ) ( )] √ −1 9 −1 1 2 1.51919 tr = erf − erf ≃ . (C.19) 10 10 σf σf

Taking σf < ωq allows to filter out the effect of the counter-rotating terms. Crucially, thereis always the possibility for a separation between ωq, σf , and 1/τ:

1 ω > σ > . (C.20) q f τ

When Eq. (C.20) is respected, the square wave is well approximated while still suppressing the effect of the counter-rotating terms.

To verify this, we numerically evaluate the fidelity of a state transfer using Eq. (3) of themain text, considering evolution under the toggling-frame Hamiltonian of Eq. (C.13). When the pulse rise time tr is finite, g(t) becomes non-zero even for n(t) odd. To suppress the resulting unwanted ′ excitations of the qubit and cavity, we then take τ = τ −2tr, as illustrated in Fig. 4.1. Consequently, the time-averaged coupling g also decreases; the pulse interval that results in a complete state transfer is then obtained by numerically solving gnpτ = π/2 for a given value of g and np. Taking ∗ np = 100, gT2 = 1/10, and κ = 0, the resulting state-transfer error is shown in Fig. C.1 as a function of σf /g (blue dots). As σf /g increases, the error decreases, approaching the value given by Eq. (4.10) for perfectly square modulation of g(t) (dashed black line). Additional error due to finite pulse rise time becomes negligible (for this choice of parameters) for σf ≳ 500g. Even for small bandwidth,

σf < 500g, Fig. C.1 shows that large overall suppression of error due to inhomogeneous broadening is ∗ possible (without SQUADD, error would be of order 1 for gT2 = 1/10). In the simulations, we have taken ωq = 2000g. This choice allows us to filter out the effect of the counter-rotating terms inthe whole parameter range of the simulation, for which σf < ωq is always satisfied. To give a concrete example, taking g/2π = 1 MHz and σf = 100g, the parameters used in the simulation presented ∗ ≃ in Fig. C.1 correspond to σf /2π = 100 MHz, ωq/2π = 2 GHz, and T2 16 ns. Numerically solving gnpτ = π/2 then gives g/2π ≃ 0.42 MHz and τ ≃ 6 ns for np = 100. Even for this narrow

−3 bandwidth (which leads to tr ≃ 2.4 ns), our simulation yields a relatively small error, 1−F ≃ 10 .

125 Gaussian ﬁlter − Eq. (10) 10 3 F − 1

− 10 4 100 1000 σf /g

Figure C.1 – Error 1 − F due to finite pulse rise time in the square-wave modulation of g(t). Blue dots: finite-bandwidth modulation obtained using a Gaussian filter with standard deviation σf . Dashed black line: ∗ error for perfectly square modulation of g(t), given by Eq. (4.10). Parameters are: np = 100, gT2 = 1/10, ωq = 2000g and κ = 0.

C.3 Spectrum of the 4-dimensional Hamiltonian for state transfer to a

collective mode in a qubit ensemble

In this Appendix, we give analytical expressions for the energy spectrum of the Hamiltonian of

Eq. (4.26), which describes quantum state transfer between a cavity and a collective mode of √ an inhomogeneously broadened ensemble of physical qubits. Neglecting corrections ∼ O(1/ N), diagonalization of this Hamiltonian reveals two energy doublets: (i) a doublet of bright states

(which have finite overlap with the cavity state |a˜⟩ = |a⟩ = a†|0⟩), and (ii) a doublet of states which are dark (no overlap with |a˜⟩) for ∆ξτ = 0, but become bright for ∆ξτ > 0. The eigenenergies corresponding to these two doublets are √ √ ω2 + Σ2 ω2 − Σ2 E(i) = tot ,E(ii) = tot , (C.21) 2 2 respectively, where

2 2 2 ′ 2 ′ 2 ωtot = ω+ + ω− + (ω+) + (ω−) , (C.22) √ 2 ′ 2 − ′ 2 − ′ 2 ′ 2 Σ = [(ω+ + ω+) + (ω− ω−) ][(ω+ ω+) + (ω− + ω−) ]. (C.23)

126 ′ Expressions for ω and ω are given in Eq. (4.27).

C.4 Readout signal-to-noise ratio

We evaluate the optimal signal-to-noise ratio for the readout scheme introduced in the main text using standard input-output theory [62]. To define the signal-to-noise ratio, it is useful to introduce the operator [92]

∫ √ tf † − M = i κ dt[aout(t) aout(t)], (C.24) 0

corresponding to the integrated homodyne-detection signal for measurement time tf . In Eq. (C.24), we have introduced the output field aout(t) leaking from a cavity with damping rate κ. The operator M is in turn used to define the measurement signal X and noise Ξ,

X = |⟨M⟩+ − ⟨M⟩−|, (C.25)

2 2 1/2 Ξ = (∆M+ + ∆M−) , (C.26)

2 2 2 ∆M = ⟨M ⟩ − ⟨M⟩, (C.27)

where ⟨O⟩ ≡ tr[O(tf )ρ(0)] and ρ(0) ≡ |⟩⟨|q ⊗ |0⟩⟨0|c, with |⟩q the eigenstates of σx with eigenvalues 1, respectively. The signal-to-noise ratio is then simply

SNR = X/Ξ. (C.28)

To evaluate the SNR for a given measurement scheme, it is useful to relate M to the input field ain(t) and to the field a(t) inside the cavity. This relation is given by the input-output formula [62]

√ aout(t) = ain(t) + κ a(t). (C.29)

127 Assuming that the input is vacuum, substitution of Eq. (C.29) into Eq. (C.24) gives

∫ t f † ⟨M⟩ = iκ dt[⟨a (t)⟩ − ⟨a(t)⟩], (C.30) 0 ∫ ∫ tf tf −t1 [ ] 2 2 † ⟨M ⟩ = κtf + 2κ dt1 dt2 ⟨a (t1 + t2)a(t1)⟩ − ⟨a(t1 + t2)a(t1)⟩ + H.c. . (C.31) 0 0

2 Eqs. (C.30) and (C.31) relate ⟨M⟩ and ⟨M ⟩ – and thus the SNR – to simple expectation values and autocorrelation functions of the cavity field a(t). Employing standard formulas [62], these expectation values and autocorrelation functions are easily calculated knowing the time-evolution operator V (t, t0) of the qubit-cavity system. We find V (t, t0) by solving the (time-inhomogeneous) master equation

V˙ (t, t0) = L(t)V (t, t0). (C.32)

In Eq. (C.32), we have introduced the Lindbladian L(t) describing cavity damping at rate κ and unitary evolution under the qubit-cavity toggling-frame Hamiltonian, HT(t),

L(t)· = −i[HT(t), ·] + κD[a]·, (C.33) ( ) D · · † − 1 † · · † [a] = a a 2 a a + a a , (C.34) where the centerdot (“·”) represents an arbitrary operator upon which the relevant superoperator is applied. In Eq. (C.33), HT(t) is given by Eq. (4.4), taking ξ = 0 and g(t) = g ∀ t.

To evaluate V (t, t0) analytically, we assume that t − t0 = 2npτ. We then use the Magnus expansion [ ] ∑∞ (k) V (t, t0) = exp L (t − t0) . (C.35) k=0

As in average Hamiltonian theory, the terms L(k) are time-independent because L(t) is periodic,

L(t + 2τ) = L(t) ∀ t. Explicit expressions for L(k) are given in the literature [65].

To gain insight into the problem, we evaluate the SNR to leading order in the Magnus expansion,

Eq. (C.35). In this first approach, we neglect any qubit decay that may arise from higher-order

128 terms in the expansion. We then find

(0) V (t, t0) = exp[L (t − t0)], (C.36)

L(0)· = −i[H(0), ·] + κD[a]·, (C.37) g H(0) = (a + a†)σ . (C.38) 2 x

According to Eqs. (C.36) and (C.37), the qubit forever remains in its initial state |⟩⟨|q. For

κtf ≫ 1, the cavity correspondingly settles in the coherent state |α⟩⟨α|c = | ∓ ig/κ⟩⟨∓ig/κ|c. Since the cavity field leaks from the output port at rate κ/2, this steady state leads to X ∝ κtf ×g/κ ∝ gtf .

2 In addition, noise in the output field then entirely comes from shot noise: ∆M = κtf , giving [92]

√ √ X = 4gtf , Ξ = 2κtf ⇒ SNR ∝ tf . (C.39)

Therefore, in this ideal scenario, signal always accumulates faster than noise, making it possible to achieve arbitrarily large SNR simply by increasing tf . In practice, qubit relaxation leads to saturation of the signal and to enhancement of the noise, thus limiting the achievable SNR. Qubit relaxation can be intrinsic, coming from coupling of the qubit to a decay channel independent of the cavity. Higher-order corrections to the leading-order

Magnus expansion taken here also lead to qubit decay via the cavity. This can be seen by means of a short-time expansion of ⟨σx(t)⟩. Indeed, the term of order O (() t) in this short-time expansion gives decay at a rate analogous to that of Purcell decay:

2 2 d (2)† g τ Γ ≡ ⟨σx(t)⟩ ≃ tr{[L σx]ρ(0)} = κ, (C.40) dt t→0 24 where ρ(0) = |⟩⟨|q ⊗ |0⟩⟨0|c. The term of order O (() t) in the above expansion of ⟨σx(t)⟩ dominates over the correction term of order O (() t2) when κt < 256/[3(κτ)2].

To take qubit relaxation into account in the calculation of the SNR, we employ the Langevin equation for the cavity field a(t), considering the average Hamiltonian H(0) in Eq. (C.38). This

129 gives

κ g √ a˙(t) + a(t) = −i σ (t) − κa (t). (C.41) 2 2 x in

Eq. (C.41) has the form of the equation of motion of a Brownian particle with mass m, momentum p, and friction coefficient γ: p˙ + (γ/m)p = η(t) [62]. In Eq. (C.41), the fluctuating force η(t) comes from a combination of shot noise from the input field ain(t) and telegraph noise from the qubit through the Heisenberg-picture operator σx(t). Assuming that qubit-cavity coupling is turned on at time t = 0 and that the cavity interacts with its environment since t → −∞, the solution of

Eq. (C.41) is

∫ ∫ t √ t g ′ −κ(t−t′)/2 ′ ′ −κ(t−t′)/2 ′ a(t) = −i dt e σx(t ) − κ dt e ain(t ). (C.42) 2 0 −∞

For a qubit undergoing simultaneous excitation and relaxation at equal rates Γ/2 in the eigenbasis of σx, we have

⟨σx(t)⟩ = exp(−Γt), (C.43)

′ ′ ⟨σx(t)σx(t )⟩ = exp(−Γ|t − t |). (C.44)

Substituting Eqs. (C.42) to Eq. (C.44) into Eqs. (C.30) and (C.31), we evaluate the signal and noise using Eq. (C.25) and Eq. (C.26). We find ( ) − − 2gκ 1 − e Γtf 1 − e κtf /2 X = − , (C.45) κ/2 − Γ Γ κ/2

√ 4g2κ2 X2 Ξ = 2κt + f(t ) − , (C.46) f (κ2/4 − Γ2)Γ2 f 2

130 where we have introduced

( ) [ ] ( ) Γ 2Γ 2Γ ( ) f(t) = Γt − 1 + 1 − e−(Γ+κ/2)t − 1 − e−κt/2 1 − e−Γt Γ + κ/2 κ κ ( ) [ ( )] [ ( )] 2Γ Γ 4Γ2 Γ + 1 − e−κt/2 e−Γt + 1 − e−κt/2 − Γt − 3 − 4e−κt/2 + e−κt . (C.47) κ κ κ2 κ

To simplify the above expressions, we expand X and f(t) to leading order in Γtf around Γtf = 0. We also assume that the cavity has reached its steady state; we thus have Γtf ≪ 1 ≪ κtf . Therefore, in Eqs. (C.45) and (C.47), we drop exponentially small corrections in κtf ≫ 1. In Eq. (C.45), we also drop terms of order O(Γ/κ) or higher, which do not change the dependence of X and Ξ on tf . However, in Eq. (C.47), we keep the terms of order O(Γ/κ), since they grow faster than linearly

2 2 with tf , but drop corrections of order O(Γ /κ ) or higher. We then find √ 16 X ≃ 4gt , Ξ ≃ 2κt + g2Γt3 . (C.48) f f 3 f

2 Equation (C.48) shows that Ξ contains two terms: one from photon shot noise, ∝ κtf , and an ∝ 2 3 additional contribution from qubit switching, g Γtf . Therefore, including qubit switching, the noise grows faster than the signal (∝ tf ) as a function of tf . This is visible in Fig. C.2(a), in which we plot X(tf ) and Ξ(tf ) resulting from an exact numerical solution of the master equation given by Eq. (C.32). In Fig. C.2(a), X(tf ) and Ξ(tf ) are represented by the solid black line and the √ dotted red line, respectively. Using the dashed blue line, we also plot Ξ(tf ) = 2κtf , expected for pure photon shot noise, Eq. (C.39). Clearly, excess noise due to qubit decay determines the measurement time topt, that maximizes the SNR [shown by the double arrow in Fig. C.2(a)]. We evaluate topt. analytically by maximizing SNR = X/Ξ, with X and Ξ given by Eq. (C.48). We find √ √ 1 3 κΓ Γt ≃ , (C.49) opt. 2 2 g ( ) 6g2 1/4 SNR ≃ . (t = t ) (C.50) κΓ f opt.

Equation (C.50) provides a simple relationship between the maximal SNR and the cooperativity

C ≡ g2/κΓ, SNR ≃ (6C)1/4.

131 100 (a) (b) Signal (X) 1000 Noise (Ξ)

Best SNR SNR 100 Exact numerics Shot noise 2√3/√κτ 1 1000 10000 0.01 0.1 1 10

κtf κτ

Figure C.2 – Signal-to-noise ratio (SNR) for the proposed readout with g/κ = 1/10. (a) Dynamics of signal and noise accumulation for measurement time tf . Solid black line: measurement signal X, Eq. (C.25). Red dotted line: measurement noise Ξ, Eq. (C.26). Dashed blue line: Ξ for shot noise only, Eq. (C.39). The double arrow indicates the measurement time that optimizes the ratio SNR = X/Ξ. S and Ξ are evaluated for κτ = 0.2. (b) Maximal SNR as a function of κτ. In (a) and (b), X and Ξ are evaluated using a numerical solution of the master equation, Eq. (C.32). For g/κ = 1/10, a cavity Hilbert space of dimension 3 is sufficient for accurate numerical evaluation of X and Ξ.

Equation (C.50) gives the maximal SNR when the qubit undergoes switching in the eigenbasis

(2) of σx. As seen above, this can be due to the subleading term L in the Magnus expansion, Eq. (C.35), which leads the qubit to decay at the rate given in Eq. (C.40). When this mechanism is the dominant source of decay in the eigenbasis of σx, we substitute Eq. (C.40) into Eq. (C.50) to find the corresponding optimal SNR,

√ 2 3 SNR ≃ √ , (t = t ) (C.51) κτ f opt.

2 valid for Γtf ≪ 1 ≪ κtf ≪ 256/[3(κτ) ]. The last inequality arises from the short-time expansion performed in Eq. (C.40). Equation (C.51) implies that SNR > 1 is achievable even in the weak- coupling regime, g < κ. This result is shown in Fig. C.2(b), in which we plot the maximal SNR obtained from an exact numerical solution of Eq. (C.32) as the black dots for g/κ = 1/10. This numerical result is in good agreement with the optimal SNR given in Eq. (C.51), displayed as the dashed blue line.

We now discuss conditions under which the dynamics of the proposed readout are accurately described by the Magnus expansion [Eq. (C.35)], up to and including order L(2). The Magnus ∫ 2τ ∥L ∥ ≪ expansion converges rapidly when 0 dt (t) 2 1 [65]. For g < κ, the steady-state cavity

132 † 2 population is small: ⟨a a(t)⟩ = (g/κ) < 1 for κtf ≫ 1. In this situation, we can represent the

† (†) operators a and a by truncated matrices of small dimension, making ∥a ∥2 ∼ 1. This implies that

∥L(t)∥2 ∼ κ, and we conclude that truncating the Magnus expansion is justified for κτ < 1 under the assumption that g < κ. This statement is supported by Fig. C.2(b), which shows excellent agreement between the exact numerical solution (represented by the black dots) and Eq. (C.51)

(represented by the dashed blue line) for κτ < 1.

The SNR given in Eq. (C.51) is useful to describe the measurement of qubit expectation values through a soft average [206, 207]. However, to characterize single-shot readout, the full probability distribution of the measurement outcomes is needed. Indeed, qubit switching leads to a non-

Gaussian probability distribution which is not fully characterized by its first two moments, given in

Eq. (C.48). To evaluate the full probability distribution, we use a classical readout model that takes into account qubit switching at symmetric rates Γ/2, where Γ is given by Eq. (C.40)[208, 209].

This probability distribution leads to a single-shot fidelity that converges asymptotically to

[ ( )] (κτ)2 96 F = 1 − log + O log−1/2 κτ (C.52) 1 192 (κτ)2

as κτ → 0. Taking κτ = 0.1 then leads to F1 = 99.95 %, showing that the error due to the subleading term in the Magnus expansion is rapidly suppressed in the limit of short pulse intervals.

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