Coupled superconducting flux qubits

(1)

(2)

(3)

Coupled superconducting flux qubits

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 5 oktober 2007 om 10.00 uur door

Jelle Hendrik PLANTENBERG

(4)

Prof. dr. ir. J. E. Mooij Toegevoegd promotor:

Dr. C. J. P. M. Harmans

Samenstelling van de promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. J. E. Mooij Technische Universiteit Delft, promotor Dr. C. J. P. M. Harmans Technische Universiteit Delft,

toegevoegd promotor

Prof. dr. A. V. Ustinov Universität Erlangen-Nürnberg, Duitsland Univ.-Prof. dr. A. Shnirman Universität Innsbruck, Oostenrijk

Prof. dr. G. Wendin Chalmers Tekniska H¨ogskola, Zweden Prof. dr. ing. D. H. A. Blank Universiteit Twente

Prof. dr. ir. H. S. J. van der Zant Technische Universiteit Delft

Prof. dr. ir. L. P. Kouwenhoven Technische Universiteit Delft, reservelid

Casimir PhD Series, Delft-Leiden, 2007-10 ISBN/EAN: 978-90-8593-036-5

Copyright c 2007 by Jelle H. Plantenberg.

(5)

Preface

After obtaining my MSc in Twente and while looking for a PhD position, the re-search in Hans Mooij’s group in Delft immediately sparked my interest. I clearly remember the feeling of excitement when the invitation to return to my city of birth arrived. Meeting the challenges in advancing the state of superconducting quantum bits turned out to be a difficult job. The initial focus of my research was to suppress the noise sensitivity of flux qubits, in order to extend the time available for computation. After obtaining some first spectroscopic responses from the gradiometer samples, several fabrication improvements were clearly called for. We developed different process improvements, enabling multi-layer samples, on-chip filtering and wafer-based processing to increase productivity. The latter was a great help, since looking back I measured at least ten differ-ent gradiometer microchips! Although the results improved and rendered clear spectroscopy maps and even coherent oscillations, the gradiometer designs did not increase the time available for computation. At this time I started the coupled-qubit project, in order to increase the range of possible results, but still allowing for single-gradiometer experiments. Expanding the setup was challeng-ing and a lot of fun at the same time. In the last year of my PhD everythchalleng-ing finally came together, when the last sample displayed Rabi oscillations for both gradiometer qubits. The months that followed were amazing, the realization of the two-qubit controlled-NOT quantum gate, the great discussions on the possibilities, and in the end even acquiring the induced quantum phase. I could not have envisioned for a grander finale!

(6)

Ruben Benard and Ammeret Rossouw, thank you for the discussions and good times; I hope you learned as much from it as I did. Alexander ter Haar and Pieter de Groot, many thanks for the joint development of experiments, insights and interpretations, and the good fun we had during the measurement runs. I acknowledge Frank Wilhelm, Lieven Vandersypen and Seth Lloyd for interesting and helpful discussions. Furthermore I would like to thank Raymond Schouten for the joy of electronics, the people from the Instrumentendienst/DEMO for their help in expanding the setup, Bram van der Enden, Mascha van Oossanen, Leo Lander, Willem den Braver, Wim Schot, Leo Dam and Remco Roeleveld for their technical support, and Ria van Heeren, Yuki French-Nakagawa and Ang`ele Fontijn for their help with administrative issues.

In my view the great results of the QT group can clearly be attributed to the open and friendly, yet competitive, atmosphere. For realizing this and for making my time in QT a very enjoyable one, I would like to specifically mention Jorden and Hubert of ”Kamer A”, the spin qubit team including Ivo, Frank, Katja, Laurens and Ronald, and from teams-for-other-subjects Silvano, Floris, Maarten, Gary and Pablo. On a more personal note I would like to thank Pieter, Karin, Katja, Ivo, Stijn and Freek (for enjoying the bbq’s), Sijmen and Daaf (for our sort-of-weekly dinners), the Processie (both free as in ’beer’ and free as in ’speech’) and the Heerendispuut Bracque (especially for our December weekends, concert visits, the RaBo and the bbq dinners), the Vriendenweekend (for the kazerne-sfeer), Huize onder den Linde (for Carnaval and the great Villa-weekend) and Mark, Steven, Witho and Arjan of de Campusmongolen (what a great new year’s eve), for the joyful times we spent when the dilution fridge was not running!

I would like to thank my paranimfen Bart Carelsen (BartC) and Martijn van der Bom (BOM) for accompanying me on the defense day, and my family, mom, dad, brother and sister for their support. Finally I would like to thank Agnes for her continued care and love throughout these demanding years!

(7)

(8)

(9)

Introduction

1.1 Quantum computing

Einstein is seen by many as the example genius physicist. Interestingly, one of his important contributions to physics is the result of an attempt to prove that some predictions resulting from quantum mechanics could only be wrong. When the ”gedanken” experiment of the Einstein-Podolsky-Rosen paradox [1] was actually carried out, demonstrating that quantum mechanical entanglement indeed produces larger correlations than is possible in any classical systems with only local interactions, many experiments testing the extents of quantum theory followed.

Quantum theory is very successful in explaining the behavior of microscopic particles, like photons, electrons and atoms. How the theory continues to hold in the world of larger objects is still an active field of investigation. Interesting experiments establishing the validity of different parts of quantum theory using larger-scale objects include the particle-wave duality of larger molecules [2] and macroscopic quantum tunneling in solid-state superconducting systems [3].

The ability to produce superconducting tunnel junctions with negligible dis-sipation allowed the study of quantum behavior in electronic circuits. These macroscopic objects can be engineered to mimic the quantum mechanical prop-erties of microscopic particles. In this thesis, such circuits are exploited to both look at the fundamental properties of such devices, as well as to investigate their suitability for use in quantum computers.

A normal computer handles information in elementary information carriers, bits, that can take only one of two values, 0 or 1. This means that the total amount of different values that an N -bit register can hold is 2N_{. In a}

(14)

α|0i + β|1i. As a consequence, an N -qubit register can in a sense contain all possible values 0 . . . 2N _{at the same time, by forming a 2}N _{dimensional Hilbert}

space. Unfortunately, harvesting this immense increased information density proves to be difficult. Quantum mechanics puts very fundamental constraints on the measurement of a quantum register, again allowing only 2N _possible

outcomes. Certain algorithms however can in a sense still be executed on all possible values simultaneously, before the measurement. No general recipe for creating a quantum algorithm that is faster than its classical counterpart has been developed yet, but several interesting algorithms were discovered. One of the most spectacular ones is Shor’s factorization algorithm, which relies on the quantum Fourier transform to factorize large numbers into their prime factors with an exponential speed-up compared to classical algorithms as a function of the number of bits. The successful realization of Shor’s algorithm using nuclear magnetic resonance techniques on specially designed molecules [5], proved that quantum mechanics can help to fundamentally speed up certain hard compu-tational tasks. Many proposals and experiments followed, trying to implement quantum computing elements in solid-state devices.

1.1.1 Solid-state quantum devices

In addition to nuclear magnetic resonance, ion-trap experiments have yielded promising results demonstrating quantum behavior [6, 7]. These systems how-ever, are also based on controlling single particles and may be limited in their ability to scale towards larger numbers of qubits. A different approach to obtain this desired scalability, is to try and implement quantum bits using standard fabrication techniques from the semiconductor industry. One of the major chal-lenges of this approach, is to devise successful ways of isolating a single degree of freedom for use as a quantum mechanical two-level system. This quantum bit needs to be decoupled from the many other degrees of freedom in the solid state, in order to retain its information. Any noise or coupling to the environment will lead to loss of quantum information, i.e. decoherence. This can occur through two processes, either by actual energy exchange with the environment, called relaxation, or by virtual energy exchange, yielding uncontrolled modulation of the qubit’s internal energy separation, referred to as dephasing. In addition to the required isolation, both local control to manipulate the qubit state, and an information extraction mechanism are needed to operate a qubit. These two requirements will inevitably connect the qubit to extra degrees of freedom, and thus to sources of decoherence.

(15)

us-Quantum computing

ing standard techniques. The first incorporates the microscopic particles onto microchips, by designing circuits that allows for trapping and isolation, as well as control, read-out and coupling. For instance semiconductor quantum dots created in two-dimensional electron gases can isolate a single electron. The spin of this electron behaves as a quantum mechanical two-level system and re-cently there have been results demonstrating control [8] and state detection [9] of such spin-qubits. There are also efforts to trap atoms using micro-fabricated structures [10]. Alternatively, one can adapts actual macroscopic electromag-netic circuits to create quantum energy level structures that are suitable as a building blocks for quantum computing.

1.1.2 Josephson quantum circuits

In this thesis we explore the possibilities of using superconducting Josephson cir-cuits for scalable solid-state quantum computing. These circir-cuits employ collec-tive electro-magnetic modes of macroscopic components. Since quantum effects are suppressed by dissipation, these components need to have a purely reac-tive impedance. Superconducting materials lose their d.c.-resistance and expel magnetic fields below their critical temperature TC, by pairing electrons into

Cooper pairs. These pairs can move frictionless through a crystal lattice under the exchange of phonons. The superconductor protects the designed quantum circuit from unpaired normal electrons through the large excitation gap of the coherent superconducting Cooper pair condensate.

I S

S

Ψ2(θ2)

Ψ1(θ1)

Figure 1.1: Josephson tunnel junction. The superconducting wave functions Ψiinside

the two superconductors S have a certain quantum phase. When the barrier I separat-ing the two electrodes is thin enough for the wavefunctions to overlap, Cooper pairs can pass the barrier by non-dissipative quantum tunneling, resulting in a current.

(16)

phase of the superconducting condensate. The properties of these junctions are described by the two Josephson equations [11], linking the electrical circuit parameters of voltage difference and current, to the macroscopic phase of the superconductor.

The a.c. Josephson equation describes the voltage difference (potential) over the junction as a function of the evolution of the phase difference across the barrier,

V = Φ0 2π

∂ϕ

∂t, (1.1)

where Φ0 is the superconducting flux quantum h/(2e) ≈ 2 · 10−15 Wb. The

d.c. Josephson equation relates the current passing through the junction to the phase difference,

IS = ICsin (θ2− θ1) = ICsin ϕ, (1.2)

where IC is the critical current of the junction, representing the maximum

dissipation-free current that can flow and thus the strength of the coupling between the two superconductors, and θi are the phases of the wavefunction

describing the superconducting condensate on the two sides of the junction. To describe realistic implementations, the ideal Josephson tunnel junction has to be shunted by an electrical capacitance, inherently formed by the isolat-ing barrier, and a resistor, describisolat-ing the losses due to dissipative behavior. This model, called the Resistively Capacitively Shunted Junction (RCSJ) model, ac-curately describes the dynamics of classical junctions. The model describes the same physics as those of a driven pendulum, with phase difference corresponding to the angle with respect to the vertical, current to the applied force, voltage to the angular speed, resistance to the viscosity of the medium and capacitance to the mass of the pendulum. Note that these shunts can be tailored to accommo-date many different applications. For rapid-single-flux-quantum (RSFQ) logic [12] and programmable a.c.-voltage standards [13] the shunt resistor is important and relatively small, while for d.c.-voltage standards [14] and quantum circuits [15] the shunt capacitance is important and the resistor is almost infinite.

(17)

Quantum computing

d c

b a

Figure 1.2: Different Josephson junction based solid-state qubits. a, Charge qubit (Chalmers University of Technology). b Charge-phase qubit (CEA Saclay). c Flux qubit (Delft University of Technology). d Phase qubit (NIST Boulder / University of California, Santa Barbara).

N , expressed in Cooper-pairs. These two effects compete and their relative stengths are quantified by the Josephson energy EJ = Φ0IC/(2π), and the

charging energy EC = e2/(2C). Different implementations of superconducting

quantum bits have emerged [16], that have different ratios EJ/EC and are thus

better described in terms of either phase (EJ EC) or charge (EC EJ).

The first successful realization of a superconducting qubit consisted of a Cooper-pair box [17], depicted in Fig. 1.2a, which has a ratio EJ/EC smaller

than one, consequently charge is defined best. The potential over the junction is biased using gates to the regime where either an excess Cooper-pair or the absence thereof are energetically comparable. The coupling between these states is controlled by EJ. Since this implementation is most sensitive to charge,

both control and readout are realized using methods that are effective in this regime, using gates and single-electron transistors (SETs) respectively. The main source of decoherence for these qubits is charge noise, which is both present the environment and introduced through the electrical connections.

Since charge-noise appears to be relatively strong in solid-state implementa-tions, a method of improving the Cooper-pair box’s immunity was devised by rendering the the Josephson energy EJ and the charging energy ECcomparable

(18)

elec-trical (charge) and magnetic (flux) noise is suppressed to first order. Recently, it was proposed to increase the EJ/EC ratio of the Cooper-pair box even further

by shunting it with a transmission line and using only microwave-based control and detection methods [19].

Allowing EJ to exceed EC by one or two orders of magnitude, creates

junc-tions for which phase is a better quantum number than charge to describe the junction properties. These junctions are used to engineer flux qubits [20, 21], depicted in Fig. 1.2c. Because of the ratio, these qubits are relatively immune to electrical noise, but sensitive to magnetic noise. Although a first order sup-pression of flux noise can be obtained at a special biasing point (the degeneracy point), this method limits, as with the split Cooper-pair box, the flexibility of the quantum bit. In this thesis, a new design is investigated that allows for a geometric suppression of flux noise that at the same time retains its biasing flexibility.

Finally EJ can be made to exceed ECby three to four orders of magnitude,

where a single junction can already form a qubit when a current-bias close to the junction’s critical current is applied [22]. These phase qubits, depicted in Fig. 1.2d, have no special biasing points to reduce noise sensitivity, but are relatively insensitive to charge noise and may have convenient properties to allow for scaling towards larger systems.

1.1.3 Josephson quantum device fabrication

Compared to the first flux qubit devices [23], the fabrication of such circuits has advanced significantly. The need for on-chip dissipative components, reactive filtering and isolated state-detection circuits has called for several process im-provements. However, the basic shadow-evaporation technique [24] used for the actual Josephson tunnel junction fabrication has remained mostly unaltered. Here, a short summary of the clean room fabrication technique is given, high-lighting the process advances. Details of the actual recipe parameters are found elsewhere [25, 26].

(19)

Quantum computing

b, development a, patterning

e, junction

PdAu, Au 2x Al/AlOx/Al

i, 19x19mm chip j, qubit microchip d, 2nd_{Al layer} c, 1st_{Al layer + O} 2 100 nm f g 500 nm h, 400wafer

(20)

removed together with the unexposed resist in the lift-off step, using a chemical solvent, leaving only the desired pattern, see Fig. 1.3e.

The devices used in this thesis consist of several thin film metal layers. The first layer consists of PdAu contact pads, on-chip resistors and electron-beam alignment markers. The pads are used to connect the microchip to the electrical leads of the measurement setup, the resistors are used in filtering structures and the alignment markers allow for exact overlay of the electron-beam patterns of the qubit and detector layers. The second layer consists of thick Au heat sinks and additional contact pads. These two normal metal films are processed on full 400wafers containing 16 · 5 qubit microchips. The first layer also contains markers that are now used to dice the wafer into 12 separate 19 × 19mm chips. The third layer is a shadow-evaporated Al/AlOx/Al tri-layer, out of which the qubit junctions are formed. Also, the bottom plates of the on-chip capacitors used for filtering and produced with this layer. An isolation is added by growing an AlOx dielectric layer using an oxygen plasma. Finally, the SQUID detector is fabricated by using a second Al/AlOx/Al tri-layer. This last layer also contains the on-chip wiring, the capacitor top plates and the on-chip inductors used for filtering.

After investigating the test-structures both optically and electrically, the 19 × 19mm chip is diced once more into five separate 6 × 6mm microchips that contain a single quantum bit circuit.

1.1.4 Entanglement and Bell inequalities

(21)

Quantum computing

fake coins, constant true coins, balanced

a b

Figure 1.4: The four different coins that can be differentiated a single pass. a, The two true coins, with one heads and one tails. b The two fake coins with either two heads or two tails.

affect the other.

One of the famous states that results in such correlations are the Bell states [28], of which |β00i = (|00i + |11i)/

√

2 is an example, where |00i corresponds to both qubits in state |0i and |11i leaves both qubits in state |1i. Measurement of the first qubit has equal probability of finding 0 or 1 as the outcome. Surpris-ingly, a consecutive measurement of the second qubit is not random, but fully correlated with the first measurement, always yielding the same result. In itself this may not seem so special, since the particles may have ’agreed’ beforehand on a certain outcome. Nevertheless, the Bell experiment is set-up in such a way that it allows one to make the distinction between this ’pre-arrangement’ (or local hidden-variable theory) case and a true quantum mechanical situation, by measuring the correlation along several projection axes.

1.1.5 A quantum algorithm: Deutsch-Jozsa

One of the first algorithms that demonstrated the possible fundamental compu-tational advantages of quantum computers, is the Deutsch-Jozsa algorithm [29]. In its simplest form it can be executed on a two-bit quantum computer and it can evaluate in a single measurement whether a function is bit-wise balanced or constant. For a classical computer in the worst case scenario, half the basis needs to be measured in order to make this same determination.

(22)

computer has to perform two measurements (look at each side) in order to determine what coin it has received. A quantum computer on the other hand, can determine in a single evaluation if a fake or true coin was presented at its input.

1.2 Thesis overview

(23)

Bibliography

[1] A. Einstein, B. Podolsky & N. Rosen, Phys. Rev. 47, 777 (1935). [2] M. Arndt et al., Nature 401, 680 (1999).

[3] A.J. Legget & A. Garg, Phys. Rev. Lett. 54, 857 (1985).

[4] M.A. Nielsen & I.L. Chuang, ”Quantum computation and quantum infor-mation”, Cambridge University Press, ISBN 0-521-63235-8 (2000). [5] L.M.K. Vandersypen et al., Nature 414, 883 (2001).

[6] J. Chiaverini et al., Science 308, 997 (2005). [7] W. H¨ansel et al., Nature 438, 643 (2005). [8] F.H.L. Koppens et al., Nature 442, 766 (2006). [9] R. Hanson et al., Phys. Rev. Lett. 94, 196802 (2005). [10] e.g. S. Seidelin et al., Phys. Rev. Lett. 96, 253003 (2006). [11] B.D. Josephson, Phys. Lett. 1, 251 (1962).

[12] K.K. Likharev & V.K. Semenov, IEEE Trans. Appl. Supercond. 1, 3 (1991). [13] C.A. Hamilton et al., IEEE Trans. Instrum. Meas. 44, 223 (1995).

[14] F.L. Lloyd et al., IEEE Electon. Dev. Lett. 8, 449 (1987). [15] e.g. M.H. Devoret et al., cond-mat/0411174 (2004).

(24)

[17] Y. Nakamura et al., Nature 398, 786 (1999). [18] D. Vion et al., Science 296, 886 (2002). [19] J. Koch et al., cond-mat/0703002 (2007). [20] J.E. Mooij et al., Science 285, 1036 (1999). [21] J.R. Friedman et al., Nature 406, 43 (2000).

[22] J.M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002). [23] C.H. van der Wal et al., Science 290, 773 (2000). [24] G.J. Dolan, Appl. Phys. Lett. 31, 337 (1977).

[25] L. Zonnenberg, MSc Thesis, Delft Univ. Technology (2004). [26] C.M. Huizinga, MSc Thesis, Delft Univ. Technology (2005). [27] A.R. Calderbank et al., Phys. Rev. Lett. 78, 405 (1997). [28] J.S. Bell, Physics 1, 195 (1965).

(25)

2

Persistent-current

qubits

2.1 The persistent-current flux qubit

In the introduction we presented a number of different superconducting qubit implementations. One of these, the persistent-current flux qubit (or flux qubit in brief) is the subject of this thesis. The flux qubit consists of a closed su-perconducting ring interrupted by three or more Josephson junctions [1, 2, 3]. Since the EJ/EC ratio is taken much larger than 1, the eigenstates are best

described using the phase, current or flux parameter. In this chapter we present theory describing the persistent-current qubit, followed by a detailed analysis of the decoherence sources that limit its operation.

2.1.1 The qubit circuit

The persistent-current flux qubit circuit is a superconducting ring interrupted by at least three Josephson junctions. Intuitively, the behavior of this circuit can already be understood by investigating Fig. 2.1a. When an external flux bias Φex = BexA is applied starting from zero magnetic field, the ring responds by

generating a magnetic flux opposing this external frustration whose magnitude depends on circuit parameters. When the external flux exceeds half a super-conducting flux quantum Φ0 = h/(2e), the ground state |0i of the ring inverts

its response and generates a paramagnetic flux, aligned with the external field. The excited state |1i has the opposite behavior. This difference in flux response between the ground and excited state, forms the basis for control, coupling and measurement of this superconducting qubit.

(26)

Bex E0 E1 |1i ∆ E 1 2Φ0 Φex |0i ϕ3 ϕ2 ϕ3 C3 Ibias Vbias,13 N23 N13 C2 Vbias,23 M C1 b a

Figure 2.1: Operating principle and electrical schematic of the flux qubit. a The flux qubit is biased using an external magnetic field Bex, that induces a flux Φex= Bex· A

in the superconducting loop. Around Φex= 1₂Φ0 the ring forms a quantum two-level

system. b ϕi represent the phase differences over the Josephson junctions and Nij

correspond to the excess numbers of Cooper pairs on the islands between capacitances i and j. Magnetic bias is provided using an external field coupling into the qubit ring through the mutual inductance M , and electrical bias may be provided using the gates Vbias,ij. L(~ϕ, ~˙ϕ) = T (~˙ϕ) − U (~ϕ) =X i e2 2EC,i Φ0 2π 2 ˙ ϕ2i − X i EJ,i(1 − cos ϕi) , (2.1)

where EJ,i is the Josephson energy, EC,i is the charging energy and ϕi are the

(27)

The persistent-current flux qubit −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

Figure 2.2: Energy landscape of the three junction flux qubit. The potential energy surface (dark represents low energy, light corresponds to high energy) at f = 0.48, f = 0.5 and f = 0.52 (left to right) of the three junction qubit reveals the two wells of lowest energy. The tunnel barrier between the two states is formed by the path of lowest action that connects the two wells. The height of this barrier can be changed by tuning the relative size of the smaller α-junction.

X

i

ϕi+ ϕL+ 2πf = 2πn, (2.2)

where ϕL represents the phase induced over the geometric and kinetic

self-inductances by the circulating current and is generally small enough compared to the junction phases to be disregarded for the circuits we investigate here, f = Φex/Φ0 represents the external magnetic frustration and n is the integer

number of fluxoids, or 2π phase quanta, present in the ring. We now consider a ring with three junctions, where the third junction i = 3 is a factor α in area compared to the other two, typically around α = 0.7. This results in a Josephson energy that is reduced by α, while the charging energy is increased by a factor 1/α. The decreased Josephson energy reflects the weaker coupling between the two superconductors on each side of that junction IC,3= αIC,{1,2},

(28)

0.45 0.5 0.55 −15 −10 −5 0 5 10 15 0.45 0.5 0.55

Figure 2.3: Eigenenergies calculated in the phase and charge basis. The two calcu-lations with EJ/EC = 50 and α = 0.75 yield very similar results, especially for the

lowest two levels. Around the magnetic frustration f = 0.5 a two-level system is formed that can be used as a quantum bit.

eliminate the phase of this third junction using the phase quantization Eq. 2.2, the Lagrangian is conveniently described by the half-sum ϕ+ = (ϕ1+ ϕ2)/2 and

half-difference ϕ−= (ϕ2− ϕ1)/2 of the two remaining phases,

L(ϕ+, ˙ϕ+, ϕ−, ˙ϕ−) = e2 8EC,i Φ0 2π 2 ( ˙ϕ++ ˙ϕ−)2+ ( ˙ϕ+− ˙ϕ−)2+ α( ˙ϕ−)2 − EJ(cos(12(ϕ++ ϕ−)) + cos(12(ϕ+− ϕ−))) + αEJcos(ϕ−− 2πf ). (2.3)

The potential energy landscape formed by the three cos-terms is depicted in Fig. 2.2, displaying two local minima around f = 1

2 corresponding to the two lowest energy states forming the two-level system.

As suggested by this picture, the basic properties of this circuit can be explored if only the phase difference ϕ− is considered, since the two wells of

lowest energy are located on the line ϕ+= 0 and the connecting path resulting

(29)

The persistent-current flux qubit

wells. The influence of this inter-well coupling becomes clear when considering the energy spectrum of the qubit system.

The Legendre transformation of this simplified Lagrangian transforms the time derivative of phase to static charge and yields the Hamiltonian of the system, H(n−, ϕ−) = 8EC(n−− ng)2 1 + 2α − 2EJcos( 1 2ϕ−) − αEJcos(ϕ−− 2πf ), (2.4) where n−= (n2− n1)/2 the half-difference of the island charges, and ng

repre-senting an induced offset in charge along this n− direction. Just as a magnetic

bias induces a phase frustration f , an electrical bias, for instance applied through an electrical gate, induces a charge frustration ng, although most of the time

a gate is not present since its influence on a flux qubit is much smaller. The eigenvalues and wavefunctions of the system can be represented in either the phase or charge basis by transforming n → −i∂

∂ϕ or ϕ → − i∂

∂n. The latter

trans-formation is relatively easy, since the cos functions only couple integer charge states. This can be understood by considering,

eiϕj_|n ji = |nJ+ 1i, e−iϕj_|n ji = |nJ− 1i, e(iϕj+k)_|n ji = eik|nJ+ 1i, (2.5)

and using cos θ = 1 2(e

iθ_{+ e}−iθ_{) [4]. The phase basis on the other hand is the}

most natural basis when considering the ratio EJ/EC> 1. Both methods yield

in principle the same results, as is depicted in Fig. 2.3. When calculating the energies in the phase basis, the phase-space [−π . . . π] is quantized and the grid density determines the accuracy. For the charge-space calculation, only integer states are considered and the accuracy increases by allowing for larger charge differences n−.

The lowest two energy levels can be used as the quantum-mechanical two-level system defining a qubit. For flux frustrations smaller than half a flux quantum f < 0.5, the ground state carries a superconducting persistent current Ip = −1/Φ0∂E/∂f that generates a diamagnetic phase that counteracts the

externally induced one, resulting in a total phase accumulationP

iϕi= 0 along

(30)

to the magnetic moment of the qubit m = IpA times the external magnetic

field B, yielding E = ±IpBA= ±IpΦ. This means ∂E/∂Φ = ±Ip, depending

on whether the ground or excited state is considered. Flux biases larger than 1

2Φ0 (f > 0.5) result in a ground state persistent current that paramagnetically aligns its phase with the external field and completes the phase along the ring to 2π. The excited states exhibit exactly opposite behavior, completing the phase to 2π for f < 1/2 and towards 0 for f > 1/2. In other words, away from f = 0.5, the eigenstates approach the classical states and can be equivalently expressed by stating that there is an extra phase of plus or minus π along the circumference of the loop, that the loop carries a left or right circulating per-sistent current, or that the loop either contains an excess fluxoid or not. At the flux degeneracy point f = 1/2, the two classical current states have equal energy, i.e. they are degenerate. However, quantum-mechanical coupling repels the two classical current states by forming symmetric and anti-symmetric su-perpositions, resulting in an anti-crossing in the energy spectrum with a level separation denoted by ∆.

The value of the persistent current can be estimated by considering Eq. 2.4 in the classical limit with EC = 0. The minima of the potential energy are

located at ϕ− = ±ϕ∗ with ϕ∗ = arccos 1/2α. These phases can be used to

calculate energies Ei = −2EJcos(±12ϕ

∗_{) − αE}

Jcos(±ϕ∗− 2πf ). Using the

trigonometric product identity cos(s − t) − cos(s + t) = 2 sin(s) sin(t) and the identity sin(arccos t) = √1 − t2_{, the energy difference between the two states}

becomes E = 2Φ0ICpα2− 1/4(f −1/2), where only the first term of the Taylor

expansion of sin(2πf ) is taken around f = 1/2. From this energy separation for the classical current states, the persistent current is extracted to be,

Ip=12 ∂E ∂Φ = IC

p

α2_{− 1/4.} _(2.6)

The three-junction flux qubit has several advantages; all the properties of the qubit are determined by the junctions, no self-inductance is needed. This re-sults in very small footprints on the microchip, which increases noise immunity since there is only a small loop to pick-up noise. Also, as is shown in Fig. 2.3, the anharmonicity, defined as the difference between E01and E12, can be very

large, especially when compared to other superconducting qubit implementa-tions. This prevents unwanted population of the higher states and allows for fast state manipulation.

(31)

The persistent-current flux qubit −1 −0.5 0 0.5 1 0.49 0.51 0.49 0.51

Figure 2.4: Ground and excited state wavefunctions in the phase basis. These wave-functions are calculated by quantizing the phase space ϕ−= [−π . . . π] into 128 steps.

When the wavefunctions of the ground and excited state are calculated in the phase basis as a function of external frustration, they show that the two states exhibiting a phase ϕ− of either +π or −π are indistinguishable at the

degener-acy point, depicted in Fig. 2.4. At this point the first derivative to f is also zero, suppressing the response to flux to first order. When the two eigenstates ap-proach the classical persistent-current states, their difference in phase emerges. In contrast, in the charge basis the eigenstates are sharpest in the degeneracy point and the wavefunction broadens when moving away from this point, see Fig. 2.5a. In this basis the effect of the charge bias is also most prominent as is shown in Fig. 2.5b, shifting the wavefunctions in the direction of the electrostatic frustration.

The energy level splitting at the degeneracy point ∆ depends slightly on the charge bias. It modulates periodically with ng and the amplitude is set by the

parameters of the junctions. At integer values of 2ngthe first derivative is zero,

suppressing the response to charge in first order.

(32)

−3 −2 −1 0 1 2 3 0.49 0.51 0.49 0.51 −3 −2 −1 0 1 2 3 −1 0 1 −1 0 1

Figure 2.5: Ground and excited state wavefunctions in the charge basis & effect of electrical biasing on the wavefunctions at the degeneracy point. a, These wavefunctions are calculated by allowing a charge difference n−of up to 32 Cooper pairs. The gray

scale represents the probability to find the system in a state |n−i. Black represents

zero probability and lighter shades increased probability. b, The gray scale sets the probability of finding the system in state |n−i, with lighter shades corresponding to

larger probabilities.

in first order and as a consequence will not exert uncontrolled influence on the qubit state.

2.1.2 The pseudo-spin-

1

2

model

The qubit behavior of the lowest energy levels of the persistent-current qubit around the flux degeneracy point can be described by a spin-1

2 particle in an external magnetic field. The two eigenstates of such a particle either align or oppose the externally applied magnetic bias [17]. The flux qubit levels are now restricted to this two-level manifold, as is depicted in Fig. 2.6. The classical magnetic frustration energy is mapped to the externally applied flux using = 2Ip(Φex− 12Φ0), and ∆ is equal to the energy level separation ν at the crossing of the classical persistent-current states at = 0.

The use of the spin-1

2 model generalizes the description among the different quantum bit implementations and promotes comparison in the constant search for improved coherence. Also, it allows direct application of most of the quantum information processing techniques and algorithms that have been developed.

The dynamics of the pseudo-spin-1

(33)

The persistent-current flux qubit 0.46 0.48 0.5 0.52 0.54 −6 −4 −2 0 2 4 6

Figure 2.6: Qubit energy levels restricted to the two-level manifold of a single spin in a magnetic field. The energy level separation at f = 0 is equal to ∆ and the slopes away from this point are proportional to the persistent currents of the states.

Hqb = −12H~0(~λ) · ~ˆσ, (2.7)

= −1

2(ˆσz+ ∆ˆσx) , (2.8)

= −1

2ν ˆτk, (2.9)

where ~λ = (, ∆) represents the control parameters, ~ˆσ are the Pauli spin matrices denoting the laboratory frame flux basis with flux along the z-axis, and ~ˆτ are the Pauli matrices spanning the qubit eigenbasis with the eigenstates along the z-axis. The qubit energy splitting is equal to ν =|H~ˆ0(λ) |=

√

2_{+ ∆}2 _{and the}

basis transformation is realized by,

ˆ σz ˆ σx = cos ϑ − sin ϑ sin ϑ cos ϑ · ˆ τk ˆ τ⊥ , (2.10)

(34)

ϕ θ |0i z |1i y x

Figure 2.7: Geometric representation of a qubit state as a point on the Bloch sphere. The state is described by the azimuth ϕ and elevation θ and the eigenstates correspond to the north and south poles.

states, pointing along the flux axis. At the degeneracy point = 0 the two classical states are equal in energy, and the angle ϑ = ±1

(35)

The persistent-current flux qubit

|Φi = α|0i + β|1i = eiγ cosθ 2|0i + e iϕ_sinθ 2|1i (2.11) where θ, ϕ and γ are real numbers and |α|2_{+ |β|}2 _{= 1, since the sum of all}

probabilities should yield one. The global phase factor γ has no measurable effect on the qubit state and can be set to 0. Rotating a state 2π in θ gives the same probabilities since,

|Φ0_{i = cos(π + θ/2)|Φi + e}iϕ_{sin(π + θ/2)|Φi}

= − cos(θ/2)|Φi − eiϕsin(θ/2) = −|Φi. (2.12)

This phase factor γ = π has no influence on single qubit probabilities and is not observable in any single qubit experiment. Using a second qubit, a quan-tum interference protocol using the four states of a coupled system can reveal this extra single qubit phase. In chapter 6 this quantum phase will be probed successfully using a two-qubit system with individual state detectors.

The Bloch sphere representation is completed by aligning the external mag-netic field with the poles, coinciding with the z-axis. The qubit state vector precesses around this field at the Larmor angular frequency ωL = E/~. To

induce transitions, that is to change θ, an oscillating magnetic field is applied in the x-y plane matching the Larmor frequency. Viewed in a frame rotating at this same frequency, the rotating frame approximation allows the microwaves to be expressed as a static field to the qubit vector in the transverse plane, forcing a second precession around this new field, effectively changing θ and thus the relative weights of the two eigenstates [16]. To understand this approximation, the resonant microwave field can be defined as δΦµw= Aysin ωLt+Axcos ωLt =

A sin (ωLt + ϕµw), which couples to the qubit through δ = 2IpδΦµw. This field

has an oscillating component in the x−y plane of the qubit eigenbasis due to the finite energy gap ∆, in amplitude adhering to the basis transformation Eq. 2.10,

H_µw∗ = µw 2√2_{+ ∆}2 − −∆ −∆ . (2.13)

The oscillating ˆσz field can be represented by two counter-rotating harmonic

(36)

direction. Neglecting the fast counter rotating terms under the rotating wave approximation, this radiation induces Rabi rotations under the resonance con-dition ωµw = ωL with a Rabi frequency ωR linearly dependent on the applied

amplitude A [17]. The microwave phase ϕµw determines the rotation axis

lo-cation in the x-y plane. This allows the transition to be represented in the eigenbasis as an effective rotation,

eiωRτ2 (cos ϕµwσˆx+sin ϕµwσˆy)₌

cosωRτ

2 (sin ϕµw+ i cos ϕµw) sin ωRτ

2

(− sin ϕµw+ i cos ϕµw) sinωR₂τ cosωR₂τ

(2.14) where τ is the microwave pulse length and where the effects of the angle between the two bases and the microwave amplitude are combined in the induced Rabi frequency ωR. This method of excitation works optimally at the degeneracy

point = 0 and reduces in efficiency when moving away from this point. Note that far from the degeneracy point f 6= 0.5, one could use an oscillating electrical field Helec= Bxcos(ωµwt)ˆσx to excite the qubit [14, 15].

Using this model, the eigenenergies and eigenvectors can by obtained by diagonalizing the Hamiltonian of Eq. 2.8. For the eigenenergies this yields E0,1 =

∓1 2

√

2_{+ ∆}2_{= ∓}1

2ν and for the normalized eigenvectors of the system,

|Ψ0i = 1 √ 2ν2_{+ 2ν} −∆ ν + , |Ψ1i = 1 √ 2ν2_{− 2ν} ν − ∆ , (2.15)

which reduce to the eigenvectors in the flux basis (1 0)T _{and (0 1)}T _{at ϑ = 0}

or || ∆, and to the anti-symmetric and symmetric superpositions (−1 1)T

and (1 1)T _{at ϑ = ±}1

2π or || ∆. Another way of describing the basis transformation is,

U−1· ˆσz· U = ˆτk,

U−1· ˆσx· U = ˆτ⊥, (2.16)

where U = (Ψ0Ψ1) contains the two eigenvectors. This transformation is more

(37)

Flux qubits in an electromagnetic circuit

Csq

Isq

Vsq

Figure 2.8: SQUID magnetometer used for qubit state detection. Applying a d.c.-pulse Isqswitches the SQUID to the voltage state for only one of the two qubit states, which

is detected using the switching voltage Vsq. Alternatively, an a.c.-pulse can be applied

to determine the LJCsq resonance frequency, which depends on the qubit state since

the SQUID Josephson inductance LJ is modulated by the flux threading the SQUID

ring.

Eq. 2.4. This can prove useful in analyzing the behavior of the higher levels or the dynamics further away from the degeneracy point f = 0.5 than where the pseudo-spin-1

2 approximation is valid.

2.2 Flux qubits in an electromagnetic circuit

In order to use persistent-current qubits for quantum computing applications, they need to be initialized, manipulated, coupled and detected. Here several possibilities for state detection and qubit-qubit coupling are presented, with a focus on the techniques used for the experiments in this thesis.

2.2.1 Qubit state detection

(38)

applying an a.c.-current to detect the resonance frequency [21]. The latter has the advantages that this detection is dispersive and has no on-chip dissipation as well as that it can be non-destructive and quantum-mechanically projective [22]. For the purposes of this thesis however, the d.c.-current method is used.

When a measurement current is passed through the SQUID circuit, the junc-tions switch to the non-zero voltage state when the critical current is exceeded. This critical current is a function of the magnetic flux enclosed by the ring, given by, IC,sq(fsq) = q I2 C,1+ I 2 C,2+ 2IC,1IC,2cos(2πfsq) = IC,1=IC,2 2IC|cos (πfsq)| , (2.17)

with IC,i the junction critical currents and fsq = Φring/Φ0 the magnetic

frus-tration in the SQUID ring. The switching of the SQUID is a stochastic process, governed by both thermal excitation and quantum tunneling. The statistics of these switching processes are related to the plasma frequency [23, 24],

2πfpl(i) =

p

2π/Φ0(IC/C) · 1 − i2

1/4

, (2.18)

where i = I/IC,sq is the normalized bias current through the SQUID, and the

barrier height in the SQUID washboard potential,

∆U (i) = 2Φ0IC 2π p 1 − i2_{− i arccos i} ≈ (4 √ 2 3 Φ0IC 2π )(1 − i) 3/2_. _(2.19)

Switching from the superconducting to the voltage state occurs when the phase particle, representing the state of the SQUID, escapes from a local mini-mum of the potential energy to the running state. The two escape methods of the SQUID are given by the rates,

Γth(T, i) = 2πfpl(i)e −∆U (i)_{kB T}

, Γq(i) = 2πfpl(i)e

(39)

Flux qubits in an electromagnetic circuit 1.55 1.6 1.65 1.7 1.75 0 0.2 0.4 0.6 0.8 1 1.55 1.6 1.65 1.7 1.75 0 5 10 15 20 25 30

Figure 2.9: Simulation of SQUID switching statistics. a, Switching probability Psw

in dependence of the current through the SQUID I, for the case of no qubit present (gray line), the qubit in the ground |0i and and the qubit in the excited state |1i. b, The derivative the of the escape chance yields the switching histogram. The critical current IC = 1.5 µA per junction and the shunt capacitance Csq = 6 pF, yielding a

plasma frequency fpl,0 = 3.4 GHz and a crossover temperature T∗= 165 mK. The

measurement pulse duration is τm= 5 ns.

where T∗ _{= hf}

pl/kB is the transition temperature, separating behavior

domi-nated by quantum tunneling for T < T∗ and thermal escape for T > T∗ [19]. The plasma frequency of the SQUID sets the crossover temperature and can be tuned by the value of the shunt capacitor. Increasing this value increases the effective mass, slows the dynamics and lowers the plasma frequency. This frequency is chosen such that the qubit energy splittings ν are at higher fre-quency, preventing energy exchange, while the SQUID escape is still dominated by quantum tunneling.

Here the detector is operated by short d.c.-current pulses, which for an ideal rectangular pulse yields a switching probability,

Psw(i, τm) = 1 − e−Γ(i)·τm, (2.21)

with τm the sample pulse width and i the amplitude normalized to the

(40)

a b c

Mgeo Mkin Mjos

d e

Msq(ib)

ib

fcp

Mqb(fcp)

Figure 2.10: Different coupling schemes for persistent-current qubits. Possible direct coupling mechanisms include a inductive, b kinetic and c Josephson coupling, whereas tunable schemes can for instance use d SQUID or e coupling-qubit flux transformers.

The rate depends on the qubit state, since the qubit eigenstate flux away from the degeneracy point induces a frustration fqb= M Ip/Φ0.

This determines the strength of the coupling that the SQUID should have to the qubit. The mutual inductance M between the qubit and the SQUID is in-creased until the escape histograms for ±Ipare clearly separated in dependence

of the sample pulse height and width. However, this also increases the SQUID’s negative impact on qubit coherence, as explained in section 2.3.2. Note that generally the visibility of the qubit signal is less than the predicted 100 %, pos-sibly related to relaxation of the qubit during the state detection process. This formed one of the reasons to pursue the dispersive state detection method.

2.2.2 Coupling of flux qubits

Since flux qubits are by design more magnetic than electric, the most natural way of coupling them is through the flux parameter. This allows two flux qubits to feel each others state, coupling them through a ˆσz-ˆσz interaction,

either through a direct mutual inductance or a (tunable) flux transformer. The energy stored in the coupling can be defined using the strength of both qubits and their mutual inductance J = M Ip,1Ip,2, where M may be controllable and

(41)

Qubit decoherence

be used [8]. By adjusting the bias current through a SQUID, it can transform the effective coupling from anti-ferromagnetic to ferromagnetic and even to zero [9], see Fig. 2.10d. Coupling through a third qubit that remains in the ground state, as depicted in Fig. 2.10e, allows for similar control [10, 11]. Theoretically there have been many more proposals, for instance enabling parametric coupling in the flux degeneracy point [12] or coupling through harmonic oscillator like structures [13].

An attractive feature of flux qubits is that for the case of direct coupling, M can be made relatively large by using kinetic or Josephson coupling and consequently J can be as large as the single qubit energy gap ∆. In chapter 5, results obtained with such a large coupling will be presented. By decreas-ing J , the eigenstates of the coupled quantum system become more like sdecreas-ingle qubit states. This allows for experiments that more closely resemble elemen-tary quantum computing operations. Results in the regime where J is an order of magnitude smaller than ∆ are presented in chapter 6, where a two-qubit controlled-NOT gate is demonstrated and characterized. Finally note that cou-pling can also be achieved through other control parameters like charge [14, 15] or EJ.

2.3 Qubit decoherence

So far no quantum algorithms have been executed on solid-state quantum com-puters. Recently, the first experiments revealing two-qubit interaction were performed [7, 25], followed by experiments on entanglement [6] and the demon-stration of the controlled-NOT gate [26]. One of the main limiters on the rate of progress is the fact that it proves to be hard to either control the many degrees of freedom present in the solid-state or isolate the qubits from them. In this section we first discuss the various noise sources that are known, followed by how these affect the coherence of the qubit.

2.3.1 Origins of noise

(42)

a b

Josephson fluctuations

vortices

nuclear spins traps

charge phonons noise magnetic circuit noise photons trapped c

Figure 2.11: Decoherence in superconducting qubits. a, Fluctuations in the biasing conditions through virtual energy exchange modulates the Larmor frequency, which results in uncertainty of the qubit phase ϕ. This effect is referred to as pure dephasing. b, Exchange of energy between the qubit and another object results in depolarization, the combined effect of relaxation (the qubit emits a photon) and excitation (the qubit receives a photon), disturbing the qubit parameter θ. c, There are many potential sources of dephasing and depolarization. Here, a distinction between circuit, macro-scopic and micromacro-scopic noise is made.

that has a white power spectrum, and amplifier and control signal generator noise that usually have both a 1/f shape and a white background. The theory to predict the effects of circuit noise is well-developed [27, 28]. In this chapter we will draw upon this theory and apply it to flux qubit bias, excitation and read-out schemes.

(43)

spin-Qubit decoherence

locking mechanism [32]. In general, this noise is harder to quantify, since the inherent randomness and the many unknowns necessitate more estimates and assumptions. This type of noise will be discussed in-depth in section 2.3.3.

In Fig. 2.11c, different noise sources are depicted. These have different ef-fects on the qubit, and may result in distortion of the qubit azimuth angle (or quantum phase) ϕ through dephasing, see Fig. 2.11a, or of the qubit elevation (corresponding to energy) θ through relaxation, as depicted in Fig. 2.11b. In this section these effects and the relative influences of the noise sources are analyzed. Qubit dephasing and relaxation

The ideal dynamics of the quantum bit, as described in this chapter in the previous sections, may be disturbed in different ways, limiting the number of coherent quantum operations that can be performed. There are many possi-ble sources of error, like preparation errors in the qubit initial state, errors in the excitation signals and measurement inaccuracies. Furthermore, undesired transfer of population of the qubit states out of the two-state Hilbert space and entanglement of the qubit with environmental degrees of freedom destroys the quantum information contained in the qubit. These effects combined are referred to as decoherence. Here we link the transverse (ϕ) and longitudinal (θ) degrees of freedom of the qubit to the degrees of freedom in the environment, described by a noise power spectral density [28, 33]. Using the Bloch-Redfield approximation [29], the decoherence dynamics resulting from these noise sources can be expressed in changes in quantum phase ϕ or elevation θ. These effects are stochastic in nature, and are described by average rates Γ1for longitudinal

energy relaxation and Γ2for transverse dephasing. The former is related to the

power spectral density of the fluctuations Sδν⊥(f ) by Fermi’s golden rule,

Γ1=T1−1= Γrel+ Γex ≈ kBT <hν

Γrel= πSδν⊥(ν) , (2.22)

where the relaxation dominates over excitation, the latter being exponentially suppressed at low temperature. This rate equation is valid for any spectral shape since only the density at the qubits energy resonance is important. For transverse dephasing, the noise has to be short-correlated and weak [30], in order to be able to express the dephasing rate as,

(44)

I Dλ V HV(f ) SV,1(f, T ) SV,2(f, T ) SI,1(f, T ) V V SI,2(f, T ) GV,ng GI, HI(f ) SV,3(f, T ) _S_I,3_{(f, T )}

Figure 2.12: Coupling of circuit noise sources. The influence of a noise source at a certain temperature S(f, T ) on the qubit is modified by transfer functions H and the qubit sensitivity Dλ. The combined influence of the noise sources at different

temperatures and of different power spectral density can be analyzed by adding up their impacts.

with the pure dephasing rate

Γϕ= lim

f →0πSδνk(f ) . (2.24)

Additionally, the spectral shape Sδνk(f ) of the noise has to be smooth from f ≈ 0 up to frequencies of order Γϕ, of which Johnson-Nyquist (ohmic) noise is a

well-known example. This relatively simple rate equation is the very instructive on what properties of the noise spectral density have an influence on qubit dynamics. However, for dephasing the noise has to be well-behaved, limiting its application. In general, there are different approaches available to translate various spectral shapes into rates [5]. 1/f noise for instance results in difficulties when considering the low frequency limit f ≈ 0. Nevertheless, the spectral density at relevant low frequencies (for instance the repetition frequency of the experiment) can already yield an insight into its impact on qubit dephasing, and different methods are available tailored to different spectral shapes in order to render quantitative results.

2.3.2 Engineering decoherence due to circuit noise

(45)

Qubit decoherence

readout circuit needs to interact with the qubit in order to perform a mea-surement, but on the other hand the qubit needs to be fully decoupled during the manipulation phase of an experiment. A bias circuit presents even another problem, since it needs to couple a noise-free d.c.-signal to the qubit in order to exert control, but it should prevent the qubit from transferring energy back to it.

In Fig. 2.12 the coupling of circuitry to the flux qubit is depicted schemati-cally. A flux qubit can be controlled through magnetic and electrostatic energy bias, and ng respectively, but the former is by design more effective for flux

qubits. Another control parameter could for instance modulate the Josephson energy of one of the junctions [1], although this requires more complex designs or the use of different circuit elements [8]. Electromagnetic noise sources in the circuits are described using their equivalent power spectral density voltage fluc-tuations SV,i(f, T ). In order to translate this noise to actual qubit decoherence,

the noise is converted to energy spectral density in the two angles that describe the qubit state,

Sδν⊥(f ) = λn X λ=λ0 D2_λ,⊥· G2V,λ· nR X i=0 H_V,i2 (f ) · SV,i(f, T ), Sδνk(f ) = λn X λ=λ0 D2_λ,k· G2 V,λ· nR X i=0 H_V,i2 (f ) · SV,i(f, T ), (2.25)

for fluctuation in θ and ϕ respectively, where HV,i(f ) are the transfer functions

that can be tailored to have certain properties using passive components like capacitors, inductors, waveguides and microwave filters, and GV,λ converts the

electromagnetic circuit parameters to qubit bias parameters. Finally, the qubit control is converted to actual change in the qubit energy state by Dλ, which

is generally a non-linear dependence, as illustrated for example by Fig. 2.1a, resulting in a sensitivity to energy fluctuations depending on the static bias point λ0.

(46)

Qubit sensitivity to control parameter noise, Dλ

For a quantitative analysis, we make use of a general framework [5]. It al-lows for a systematic study of the influence of noise in any external parameter λ on the properties of the qubit. These parameters can represent any fluctua-tion, whether they originate from circuit-induced or uncontrolled environmental noise.

First, we assume a variation δλ to act as a perturbation on the qubit Hamil-tonian Eq. 2.7 around a point λ0= (0, ng,0). Note that other control parameters

could easily be incorporated. To first order this perturbation can be written as,

H = −1 2 ~ H0(λ0) + ∂ ~H0 ∂λ δλ ! · ~ˆσ, (2.26)

Using the basis transformation Eq. 2.15, this equation can be rotated into the qubit eigenbasis,

H = −1

2h (ν ˆτk+ δνkτˆk+ δν⊥τˆ⊥) , (2.27)

where δνkτˆk couples the control parameter variations to the longitudinal axis and δν⊥τˆ⊥ couples the variations to the transverse axis of the qubit. In the qubit eigenbasis there is never a static component in the ˆτ⊥ direction, whereas ν ˆτkcorresponds to the static parallel component giving rise to the qubit Larmor precession. Defining the derivatives,

Dλ,⊥= 1 h ∂ ~H0 ∂λ ! ⊥ = 1 h hΨ0| ∂H ∂λ|Ψ1i , (2.28) and, Dλ,k= 1 h ∂ ~H0 ∂λ ! k = 1 h hΨ0| ∂H ∂λ|Ψ0i − hΨ1| ∂H ∂λ|Ψ1i , (2.29)

(47)

Qubit decoherence

δνk=Dλ,kδλ

δν⊥=Dλ,⊥δλ (2.30)

where only the first order terms are given. The derivatives describe the sensi-tivity of the qubit parameters ϕ and θ to the control parameters λ, converting the control fluctuations to energy changes in the Hamiltonian. By changing the static bias conditions of the qubits, the sensitivity to control fluctuations is modulated. For example the first derivative to vanishes at Φex = 12Φ0, as depicted in Fig. 2.1a, increasing the intrinsic qubit protection to changes in this parameter. Note that the analysis of the influence of second order coupling is analogous, by taking the second derivatives of the Hamiltonian towards the control parameters, Dλ,⊥,2 = 1 h ∂2H~0 ∂λ2 ! ⊥ , (2.31) Dλ,k,2 = 1 h ∂2_H_~ 0 ∂λ2 ! k , (2.32)

and adding these terms to Eq. 2.30 as Dλ,⊥,2δλ2 and Dλ,k,2δλ2.

Exerting control using electromagnetic circuits, G{V,I},λ

Several approaches are available to convert the electromagnetic circuit parame-ters voltage V and current I to qubit control parameparame-ters. Once effective external flux Φexis known, the effective energy bias is set for the three-junction qubit

in the spin-1

2 basis by = 2Ip(Φex− 1

2Φ0), see also Eq. 2.6. Different qubit lay-outs may result in slightly modified expressions, for instance for the gradiometer qubit that is presented in the next chapter. The current I in a control circuit can be related to the effective external flux Φexthrough the mutual inductance

M . This mutual inductance between the two circuits can be obtained for ar-bitrary layouts using numerical tools, like the field solver FastHenry [34]. The mutual inductance converts the current to flux Φex= M I, resulting in a transfer

function,

(48)

The transfer function for capacitive coupling of a circuit voltage V to induced charge bias ng, to allow for modulation of the qubit energy splitting at the flux

degeneracy ∆, yields,

GV,ng = Ccross

e , (2.34)

with e the electron charge. For more complex circuits, for example enabling modulation of EJ by replacing one of the junctions with a small SQUID or a

Josephson field-effect device, similar approaches can be followed to determine the transfer functions G.

Electromagnetic noise sources, S_{V,I}(f, T )

Many electronic noise sources can be modeled using the symmetric Johnson-Nyquist voltage fluctuations for a resistor R, given by,

SV(f, T ) = 2R · hf · coth

hf 2kBT

, (2.35)

which is related to the well-known voltage power spectral density at high tem-perature (or low frequency) SV(T ) = 4kBT R by the Laurent series of the

hy-perbolic cotangent coth(x) = 1/x +1 3x + o(x

3_{). As is depicted in Fig. 2.12, the}

thermal noise of a resistor can be equivalently expressed using this series voltage source, or a parallel current source,

SI(f, T ) = 2 R · hf · coth hf 2kBT . (2.36)

Depending on the bias circuit design, either approach may be more con-venient and they may be freely interchanged as long as the voltage source is directly in series with the resistor or the current source is directly parallel to it. Engineering the transfer from source to control, H_{V,I}(f )

In order to optimize the relaxation and dephasing induced by control circuits, the transfer functions H{V,I}(f ) can be customized. The spectra of different

(49)

Qubit decoherence HRLC I (f ) H_Isq(f ) HRL I (f ) HCL I (f )

Figure 2.13: Different circuit couplings. Depending on the function of an electromag-netic circuit, it can be coupled in different ways. Depicted are d.c.-couplings HRL

I (f )

and HIRLC(f ), a.c.-coupling H CL

I (f ) and a tunable measurement coupling H sq I (f ).

using this approach. Fig. 2.13 depicts typical transfer circuits, for instance an excitation line that incorporates d.c.-blocks and a static bias control employing a resonant filter stage. The simplest coupling is purely inductive and is most conveniently described using a parallel current noise source. This parallel R-L circuit results in current fluctuations through the inductance equal to,

HIRL(f ) = ZRkL ZL = R R + i2πf L== 1 1 + i2πf τRL , (2.37)

where τRL= L/R is the characteristic time of the R-L low-pass filter, and 1/τRL

the -3dB frequency. Using the fluctuations of Eq. 2.36 and converting to control parameter changes using Eq. 2.33 yields,

Sδ(f ) = G2I,(H RL I (f )) 2_S I(f, T ) = 16RM2_I2 pkBT (2πf L)2_{− i2πf LR − R}2, = f 1/τRL 16M2_I2 pkBT R , (2.38)

with coth(x) = 1/x, valid for low frequency (dephasing) or high temperature. The fluctuations decrease for smaller M (isolating the qubit further), smaller Ip

(50)

to prevent relaxation. A microwave excitation line can also be decoupled for low frequencies using for instance the C-L coupling of Fig. 2.13,

H_VCL(f ) = 1 ZL+ ZC

= i2πf C

1 − (2πf )2_LC, (2.39)

which indeed goes towards zero for f → 0, decoupling the source for low fre-quencies. For a parallel R-L-C circuit, as is used for local static bias control in chapters 3 and 4, the transfer function gives,

H_IRLC(f ) = ZRkLkC ZL = R (2πf )2_{RLC − i2πf L + R} = 1 1 + i2πf_2ζ +_ff 0 2, (2.40)

with resonance frequency 2πf0 = p1/LC and damping factor ζ = R/(2L).

Transfer functions of more complicated structures, like microwave filter struc-tures and transmission lines, can be determined either up to some extent ana-lytically, or by using readily available electronic or radio-frequency engineering software. Finally, for qubit state detection using the SQUID magnetometer, see Fig. 2.8, the transfer function HI(f, Ib, fsq) depends on the current Ib running

through the SQUID and its magnetic frustration fsq. The transfer function can

be tuned to zero in first order by symmetrizing the SQUID junctions using an optimal bias current I_b∗ [35], isolating the noise sources. The current through the SQUID Ib and the circulating current J within the SQUID are related to

the internal and external phases 2πfsq and ϕs, by,

Ib(fsq, ϕs) = 2ICcos 2πfexsin ϕs+ ∆ICsin 2πfsqcos ϕs,

J (fsq, ϕs) = 2ICsin 2πfexcos ϕs+ ∆ICcos 2πfsqsin ϕs. (2.41)

The first equation can be solved for ϕs(Ib, fsq), which is substituted into the

second equation to yield J (Ib, fsq). Maximizing this result yields the optimal

(51)

Qubit decoherence

decoupling the external and internal degrees of freedom of the SQUID as seen from the qubit when the mutual inductance of both squid branches is equal. This current vanishes when the SQUID has identical critical currents ∆IC= 0, but

an optimal current is defined for all asymmetries ∆IC, leaving only a coupling

that is of second order in the current fluctuations HI(f, Ib∗+ δIb, fsq) ∝ δIb2.

For SQUIDs that couple asymmetrically, with different mutual inductances M1

and M2 for the branches containing the junctions I1 and I2, this approach is

similar, although Eq. 2.42 has a less elegant form.

2.3.3 Decoherence due to macroscopic and microscopic

noise

The influence of the set of global and microscopic noise sources can be described using mostly the same equations as for the circuit noise coupling either electro-statically or magnetically. Their couplings to the qubit circuit H(f ) cannot be engineered though, so it is important to either understand their origin in order to eliminate them, or to know their spectral content to allow for engineering of the qubit sensitivity in a way that minimizes their influence.

Charge fluctuators

Charge noise has always been one of the major contributors to environmental noise in the solid-state [36]. It produces a noise spectral density with a 1/f shape, and is generally associated with a large amount of electrical two-level fluctuators (TLFs). A TLF, e.g. an electron, can switch between two states, generating random telegraph noise at a characteristic frequency. Such random telegraph signals originating from a single TLF are frequently observed in solid-state systems [37, 38]. The resulting noise spectrum of a single TLF has a Lorentzian shape. The overall 1/f character is recovered by combining a large amount of these Lorentzian spectra of different characteristic frequency.

Direct measurements of charge noise using SETs are available and yield noise power spectral densities in the order of Sq ≈ 10−6..−7e2/Hz at frequencies below

a few kHz [39]. For charge qubit implementations, this noise generally exceeds the noise level of the circuit noise resulting from electronics and resistors.

(52)

a pure dephasing rate for 1/f noise of Γϕ= πDng,kpAq ≈ 10

−6 _{Hz, where the}

derivative towards charge was taken to be 300 MHz, an average value between maximum and minimum (zero) sensitivity. This free-induction dephasing rate leads to coherence times in the µs range. This value is significant compared to the best results obtained for flux qubits [35], and exploratory work towards electrical gate biasing for flux qubits was undertaken. For example, all samples discussed in this thesis were also equipped with the possibility for electrical biasing, but since other decoherence mechanisms were clearly dominant, this control parameter has so far not been characterized experimentally using flux qubits.

Magnetic fluctuators

Magnetic fluctuations may be of local origin, related to two-level systems in the vicinity of the quantum circuit (as for charge noise), or of a global nature. Historically, there is a lot of evidence for flux noise and comparing these results provides some insight in the question whether this is a global or local form of noise. In one of the first experiments, critical current noise (which we address later) and flux noise where identified separately, yielding SΦ= 10−10..−11Φ20/Hz

at 1 Hz for devices differing six orders of magnitude in area. More recently, lower values have been found [43, 44], reaching down into the 10−12..−13_{range. These}

results dispute the notion of a global magnetic field noise, but the question of how local the noise sources are, remains valid.

One source contributing to magnetic noise consists of trapped superconduct-ing fluxoids that can hop between different positions [42]. These fluxoids trap in the superconducting wires connected to the device, or in on-chip passives like capacitor plates. These fluxoids would indeed contribute flux noise to most lay-outs used in the experiments mentioned in the previous paragraph. Very small (or gradiometric) structures should possess an increased immunity to noise gen-erated by this source, if the separation between the qubit and the fluctuators becomes large compared to the device dimensions.

A recent proposal links magnetic fluctuators directly to charge fluctuators [32]. The magnetic moments of electrons in defect states, which they occupy for a wide distribution of times, generate 1/f flux noise. This proposal would link the effective noise experienced by a structure more to length then to area, since defects close to the wires of the device contribute the most.

Coupled superconducting flux qubits