Coupling of three-spin qubits to their electric environment

Maximilian Russ, Florian Ginzel, and Guido BurkardDepartment of Physics, University of Konstanz, D-78457 Konstanz, Germany

We investigate the behavior of qubits consisting of three electron spins in double and triple quan-tum dots subject to external electric fields. Our model includes two independent bias parameters, εand εM , which both couple to external electromagnetic fields and can be controlled by gate voltagesapplied to the quantum dot structures. By varying these parameters one can switch the qubit typeby shifting the energies in the single quantum dots thus changing the electron occupancy in eachdot. Starting from the asymmetric resonant (ARX) exchange qubit with a (2,0,1) and (1,0,2) chargeadmixture one can smoothly cross over to the resonant exchange (RX) qubit with a detuned (1,1,1)charge configuration, and to the exchange-only (EO) qubit with the same charge configuration butequal energy levels down to the hybrid qubits with (1,2,0) and (0,2,1) charge configurations. Here,(l,m, n) describes a configuration with l electrons in the left dot, m electrons in the center dot,and n electrons in the right dot. We first focus on random electromagnetic field fluctuations, i.e.,“charge noise”, at each quantum dot resulting in dephasing of the qubit and provide a completemap of the resulting dephasing time as a function of the bias parameters. We pay special attentionto the so-called sweet spots and double sweet spots of the system which are least susceptible tonoise. In the second part we investigate the coupling of the qubit system to the coherent quantizedelectromagnetic field in a superconducting strip-line cavity and also provide a complete map of thecoupling strength as a function of the bias parameters. We analyze the asymmetric qubit-cavitycoupling via ε and the symmetric coupling via εM .

I. INTRODUCTION

Qubits based on the spin of electrons trapped in quan-tum dots (QDs)1 are a leading candidate for enablingquantum information processing. They provide long co-herence times2–12, together with a scaleable architec-ture for a dense qubit implementation. Semiconductormaterials like gallium arsenide (GaAs)13 and silicon14

are the most common choices as the host material forQDs. One common feature of these implementations isthe need for control with electric fields at the nanoscalewhich unavoidably couples the qubit system to electricalnoise1. Dominating sources of decoherence are nuclearspins15–17, spin-orbit interaction18,19, and charge noisefrom either the environment or the confining gates20–25.The effect of the first and second source of decoherencecan be drastically reduced by using silicon as the hostmaterial due to its highly abundant nuclear spin freeisotope and a weak spin-orbit interaction21. Using ac-tive noise suppression methods such as quantum errorcorrection26 and composite pulse sequences27–29 leavescharge noise coupled to the spin as the remaining prob-lem to be taken care of. Thus, additional passive suppres-sion methods are needed such as optimal working points(sweet spots)30,31 which vary in effectiveness for differentqubit implementations.

Qubit implementations using single or multiple QDsto encode a single qubit show high-fidelity gate opera-tions, long decoherence times together with fast qubitcontrol allowing for many operations during the qubitlifetime6,32–35. An advantage of multi-spin qubit encod-ings consists in their improved protection against certaintypes of noise36 together with faster gate operations37–41.This ultimately leads to the three-spin- 12 qubits (seeFig. 1); the exchange-only (EO) qubit allowing for full

FIG. 1. Schematic illustration of a three-spin qubit coupled toa noisy electric environment. The environment can affect theelectron spins directly through the gate voltages Vi with i ∈{1, 2, 3} of each quantum dot (QD) or the exchange coupling(green cloud) between the electron spins through the gate-controlled tunnel hopping (tl and tr).

qubit control with only the exchange interaction37, theresonant exchange (RX) qubit with permanently act-ing exchange interaction and control through resonantdriving22,24,34, and the always-on exchange-only (AEON)qubit with symmetric gate control25. Robustness againstcharge noise can be achieved by operating the qubit onsweet spots22 where the qubit energy splitting is ex-tremal with respect to one noisy parameter or doublesweet spots25,42 where both noisy parameters are opti-mized. In this paper we provide a full analysis of chargenoise for three-spin- 12 qubits (Fig. 1) and present opti-

mal working points. We go beyond previous work24,25

by exploring the full (ε, εM ) parameter space. Moreover,

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we include two noisy tunneling parameters mapping theresulting dephasing time in this parameter space. Singlesweet spots (SSS) and double sweet spots (DSS) are pre-sented for both types of noisy parameters and combinedto provide the best working points.

Two-qubit gates are provided by the exchangeinteraction25,28,37,43–45, Coulomb interaction22,46,47 andcavity-mediated coupling42,48 while the range of the lat-ter is only limited by the extension of the cavity. Thislong-ranged coupling technique can be realized withinthe approach of cavity electrodynamics (cQED) by cou-pling the qubit capacitely to a superconducting strip-line cavity49,50 adapted for spin qubits42,48,51–59. Im-plementing a three-spin qubit in a triple quantum dot(TQD) coupled to a cavity is possible for two distinct se-tups; a longitudinal coupling or asymmetric setup and atransversal or symmetric coupling42. In this paper bothof these setups are discussed, going beyond previous workfor the asymmetric implementation42 and providing a mi-croscopic description for both implementations.

This paper is organized as follows. In section II, wedefine the three-spin qubit states, discuss the differentregimes in parameter space where each qubit implemen-tation is located and their conversion into each other.In section III, we analyze in detail the optimal workingpoints best suitable for operating the qubit in the pres-ence of charge noise coupled to the qubit through detun-ing and tunnel parameters. Subsequently in section IV,we present two setups for coupling three-spin qubits to asuperconducting strip-line cavity in order to find opera-tion points with a strong and controllable coupling. Weconclude in section V with a summery and outlook.

II. QUBIT

We consider the spins of three electrons in a linearlyarranged triple quantum dot (TQD) (Fig. 1) where eachQD has a single available orbital, whereas, higher or-bitals are energetically unfavorable due to a strong con-finement. Additionally, we restrict ourselves to the spindegree of freedom (DOF) only, hence, we either considera material with no valley DOF or a strong valley split-ting surpassing the energy of the exchange splittings andthen treat the valley as an orbital DOF. Further we as-sume that the TQD is connected to an electric environ-ment (schematically illustrated in Fig. 1) via the gatevoltages Vi of QD i with i ∈ {1, 2, 3} and via the gate-controlled tunnel barriers, consisting of either randomelectric fluctuations or a coherent quantized electric field.As a model of the TQD we use the three-site extended

FIG. 2. Energy landscape of the ground-state energy gap ωas a function of the detuning parameters ε and εM in unitsof the on-site Coulomb repulsion U . Maneuvering throughthe (ε, εM ) plane one can access various parameter regimesthat allow the use of different qubit implementations in dif-ferent charge configurations (l,m, n), where l electrons are inthe left, m electrons in the center, and n electrons in the rightQD. We further highlight the double sweet spots (DSS) (blackdots), the location of the exchange-only (EO) qubit, the reso-nant exchange (RX) qubit (dashed triangle), the asymmetricresonant exchange (ARX) qubit, and the left and right hybrid(Hl,r) qubit.

Hubbard model

HHub =

3∑i=1

[U

2ni(ni − 1) + Vini

]

+∑〈i,j〉

UCninj +∑σ=↑,↓

(tijc†i,σcj,σ + h.c.

) ,(1)

where c†i,σ (ci,σ) creates (annihilates) an electron in QDi with spin σ. We define the number operator ni =∑σ c†i,σci,σ and the gate-controlled pairwise hopping ma-

trix elements tij with i, j ∈ {1, 2, 3}. Here, we con-sider symmetric, spin-conserving nearest-neighbor hop-ping, t13 = t31 = 0, t12 = t21 ≡ tl/

√2, and t23 = t32 ≡

tr/√

2. We also include the Coulomb repulsion of two

electrons in the same QD U and in neighboring QDs UCwhich leads to a static energy shift in the dots.

Restricting ourselves to the subspace of the three-spinHilbert space with total spin S = Sz = 1/2, we identifysix relevant states, two states with a (1,1,1) charge con-figuration, and one each with a (2,0,1), (1,0,2), (1,2,0),

3

(0,2,1) charge configuration,

|0〉 ≡ 1√2

(c†1,↑c

†2,↑c†3,↓ − c

†1,↓c†2,↑c†3,↑

)|vac〉

= |s〉13 |↑〉2 , (2)

|1〉 ≡ 1√6

(2c†1,↑c

†2,↓c†3,↑−c

†1,↑c†2,↑c†3,↓−c

†1,↓c†2,↑c†3,↑

)|vac〉 ,

=

√2

3|t+〉13 |↓〉2 −

1√3|t0〉13 |↑〉2 , (3)

|2〉 ≡ c†1,↑c†1,↓c†3,↑ |vac〉 = |s〉11 |↑〉3 , (4)

|3〉 ≡ c†1,↑c†3,↑c†3,↓ |vac〉 = |↑〉1 |s〉33 , (5)

|4〉 ≡ c†1,↑c†2,↑c†2,↓ |vac〉 = |↑〉1 |s〉22 , (6)

|5〉 ≡ c†2,↑c†2,↓c†3,↑ |vac〉 = |s〉22 |↑〉3 , (7)

where |vac〉 denotes the vacuum state, |s〉ij denotes the

singlet state, |t0〉ij denotes the Sz = 0 triplet state, and

|t+〉ij denotes the Sz = +1 triplet state. Here, (l,m, n)describes a configuration with l electrons in the left dot,m electrons in the center dot, and n electrons in theright dot. An additional leakage state with S = 3/2 andSz = 1/2 is not coupled since charge noise conserves thetotal spin S.

We introduce a new set of detuning parameters Vtot =(V1 + V2 + V3)/3, ε ≡ (V1 − V3)/2 and εM ≡ V2 − (V1 +V3)/2 + UC . The parameter Vtot merely shifts the totalenergy, hence, contributes to neither the dynamics of thequbit nor the decoherence. The asymmetric detuningε is the energy difference between the left QD and theright QD, and the symmetric detuning εM is the energydifference between the center QD and the mean of theouter QDs. Defining the charging energy U ≡ U − UCwe find for the Hamiltonian Eq. (1) in the S = Sz = 1/2basis {|0〉 , |1〉 , |2〉 , |3〉 , |4〉 , |5〉} the matrix representation

H =

0 0 1

2 tl12 tr

12 tr

12 tl

0 0√

32 tl −

√3

2 tr −√

32 tr

√3

2 tl12 tl

√3

2 tl ε−εM+U 0 0 0

12 tr −

√3

2 tr 0 −ε−εM+U 0 0

12 tr −

√3

2 tr 0 0 ε+εM+U 0

12 tl

√3

2 tl 0 0 0 −ε+εM+U

.

(8)

We assume tl,r to be real since any complex phase onlycontributes a global phase to the eigenstates.

Depending on the position in the (ε, εM )-plane dif-ferent qubit implementations are realized (Fig. 2). Di-rectly in the center of the (1,1,1) charge occupancy re-gion, the exchange-only (EO) qubit37 and the always-on exchange-only (AEON) qubit25 are located. Stillin the (1,1,1) charge occupancy, but in the area withεM � ε which allows transitions into the (2,0,1) and(1,0,2) charge states, we find the resonant exchange (RX)qubit (white dashed triangle). The asymmetric reso-nant exchange (ARX) qubit is located deeper into theregime with εM � ε and a strong mixture of (2,0,1) and(1,0,2) charge configurations24. Due to mirror-symmetry

four DSS can be found (black dots) in the corner of thediamond-shaped (1,1,1)-charge configuration area. Atthe bottom left (right) in the (ε, εM )-plane in Fig. 2only two neighboring QDs are occupied by three elec-trons giving rise to the double quantum dot (DQD) hy-brid qubit39–41 formed in QD 1 and QD 2 (QD 2 andQD 3).

III. CHARGE DEPHASING

Recent progress of spin qubits using purified siliconas the host material show exceptionally long T1 and T2times emphasizing the importance of charge noise. Theuse of isotopically purified Si eliminates nuclear spins asthe leading source for decoherence and leaves charge noiseas the main cause of decoherence21. Charge noise or elec-trical noise originates from random charge fluctuations ofthe material or from the control and confinement voltagesgiving rise to fluctuating energies. Formally, we describethis by substituting q → q + δq in which the parame-ter q ∈ {ε, εM , tl, tr} is affected by random fluctuationsδq ∈ {δε, δεM , δtl, δtr}. There are two effects of deco-herence for charge noise, longitudinal and transversal de-phasing, where the first causes the energy gap betweenthe qubit states |e〉 and |g〉 to fluctuate while the secondone gives rise to transitions between the qubit states.These can further be divided into decoherence due to de-tuning parameters22–25 (ε, εM ) and decoherence due tocharge noise coupled to the qubit by tunneling (tl, tr).

The remainder of this section is organized as follows.First we present a generalized framework extending pre-vious models describing charge noise coupled to the qubitto different control parameters. In the next step, wetake only individual control parameters into account. Westart with longitudinal and transversal noise originatingfrom the detuning parameters ε and εM and identifysweet spots22–25, i.e., working points in which the qubitslifetime is highly increased due to vanishing coupling infirst order of the qubit to the noise. Subsequently, wefocus on the effects of noise coupled to the qubit via fluc-tuations of the tunnel amplitude and show that thereexist no sweet spots for both noisy tunneling parameterssimultaneously. In the last part of this section, we takeall separately discussed effects of the noisy parametersinto account in order to present the best working points.

A. General framework

In general, we can write the qubit Hamiltonian in itseigenbasis24, {|g〉 , |e〉}, as

Hnoise =~2

[(ω + δωz)σz + δωxσx + δωyσy] , (9)

with the unperturbed energy gap ~ω between the qubitstates. Here, |e〉 and |g〉 are the two lowest eigenstates of

4

H in Eq. (8). The longitudinal effect of the charge noiseup to second order is

δωz =∑q

(ωqδq +

ωq,q2δq2)

+1

2

∑p 6=q

ωp,qδpδq, (10)

where we used the definitions ωq ≡ ∂ω∂q and ωp,q ≡ ∂2ω

∂p∂q

with p, q ∈ {ε, εM , tl, tr}. For the transversal effects weconsider

δωx =∑q

δωx,qδq, (11)

with δωx,q ≡ 〈g|H1,q |e〉 and δωy = 0 due to real valuedtunneling. Here, H1,q is the part of the Hamiltonianfrom Eq. (8) associated with the perturbation in q, thus,H1,q = ∂

∂qH δq. Considering only longitudinal noise one

can calculate the pure dephasing in a Ramsey free decayapproach54

f(t) ≡⟨

eiφ(t)⟩

= e−〈φ(t)2〉 = e−t

2/T 2ϕ (12)

with φ(t) ≡∫ tt0dt′δωz(t

′) in which we used δq = δq(t).

For the first equality we used Gaussian distributed noisewith zero mean, while for the second equality we useda spectral density exponent60 γ = 1. This allows us tocalculate for a given spectral density of the noise Sq(ω) =Aq|ω|−γ the associated dephasing time24

Tϕ =~[∑

q

ω2q

2Ap log r +

ω2q,q

4A2q log2 r

+1

2

∑p 6=q

ω2p,q

2ApAq log2 r +

1

8ωp,pωq,qApAq

]− 12

.

(13)

Here, we assumed independent and uncorrelated noisefor each noisy parameter p, q ∈ {ε, εM , tl, tr} and usedthe ratio r ≡ ωUV/ωIR of lower frequency cutoff ωIR andupper frequency cutoff ωUV needed to ensure convergenceof the intergral.

With this in mind, we formally define the expression“sweet spot” initially introduced as best points for opera-tions due to a vanishing coupling of the qubit to the noisein first order. Assuming first order noise effects to be thedominant ones these points yield the longest life-time ofthe qubit according to Eq. (13), and therefore, the idealoperation points for the qubit. Taking only first order ef-fects into account we obtain for the best working pointsthe condition ∑

q

ωqδq = 0, (14)

which is in general only possible for each ωq = 0 withq ∈ {ε, εM , tl, tr}. We now define a single “sweet spot”(SSS) if this condition is fulfilled for one noisy parame-ter. Analogously, we define a “double sweet spot” (DSS)

if ωq = ωp = 0 with q 6= p ∈ {ε, εM , tl, tr}. Consideringa total of four noisy parameters we can also introduce“triple” and “quadruple sweet spots” in which Eq. (14)is for three noisy parameters or completely fulfilled. Un-fortunately, we find that there exist no such quadruplesweet spots in a three-spin- 12 qubit in maximally threeQDs.

The second term in the Hamiltonian (see Eq. (9)) δωxleads to random rotations of the qubit around the x-axiswith timescales on the order of ms for realistic chargenoise (assuming

√〈δq2〉 = 1µeV at 1 Hz)60. Somewhat

more devastating for the qubit is the fact that transver-sal charge noise changes the orientation of the eigenstatesand therefore the energy gap giving rise to an additionalterm for the dephasing in second order of the fluctuations.This becomes clear when expanding the eigenenergy dif-ference from Eq. (9),

ω =√

(ω0 + δωz)2 + δω2x + δω2

y

' ω0 + δωz +δω2

x

2ω0+δω2

y

2ω0+O(δω3). (15)

Inside the (1,1,1) charge configuration regime and awayfrom the charge transition lines, the states |0〉 and |1〉defined in Eq. (3) and Eq. (4) are nearly the qubitstates while the coupling of the other states can betaken into account using a low-energy Schrieffer-Wolffapproximation61. We obtain for the resonance frequencybetween the qubit eigenstates25

~ω =√J2 + 3j2, (16)

with the mean J ≡ (Jl + Jr)/2 and the half of the dif-ference j ≡ (Jl − Jr)/2 of the left and right exchangeinteraction and respectively between the left (right) QDand the center QD,

Jl =2t2lU/[U2 − (ε− εM )2

], (17)

Jr =2t2rU/[U2 − (ε+ εM )2

]. (18)

Utilizing this, we find a closed analytical expression forthe longitudinal fluctuation

ωq = (J∂qJ + 3j∂qj)/ω, (19)

and the transversal effect

δωx,q = (J∂qj − j∂qJ)/ω, (20)

in the (1,1,1) charge regime. For εM � 0 (RX regime)these results converge with the expressions consideringonly the RX qubit24, since there the influence of thestates |4〉 and |5〉 becomes negligible.

B. Detuning noise

Longitudinal dephasing Tϕ due to low frequency chargenoise originating from the detuning parameters ε and εM

5

FIG. 3. Dephasing time Tϕ given by Eq. (13) due to longitudinal noise as a function of the detuning parameters ε and εM . Inthe top row ((a) and (b)) we plot Tϕ resulting from charge noise in the two detuning parameters ε and εM , in the center row((c) and (d)) we plot Tϕ resulting from charge noise in the two tunneling parameter tl and tr, and in the bottom row ((e) and(f)) we plot Tϕ resulting from charge noise from all four parameters combined, where we choose the parameter settings identicalin each column. The left column shows results for weak tunneling and strong noise while in the right column, results for strongtunneling and weak noise are plotted. Parameters are set as follows; tl = 22µeV, tr = 15µeV, Aq = 1µeV2 where q = ε, εMin (a) and (e), and Aq = (10−1 µeV)2, where q = tl, tr in (c) and (e), for the left column and tl = 220µeV, tr = 150µeV,Aq = (10−2 µeV)2 where q = ε, εM in (b) and (f), and Atl = Atr = (10−3 µeV)2, where q = tl, tr in (d) and (f), for the rightcolumn. To include a large frequency bandwidth we globally set the ratio of the lower and higher frequency cut-off r = 5× 106.The black dots indicate the DSS.

6

is usually seen as the dominant source for decoherence.For reasons of simplicity we set in this subsection δtl =δtr = 0 and only consider charge noise originating fromthe detuning parameters δε 6= 0 and δεM 6= 0. The effectof this can be drastically reduced by working on SSS22 orDSS23–25. They fulfill the condition ωε = ωεM = 0 suchthat the longitudinal coupling given in Eq. (10) vanishesand only second order effects remain.

In Figs. 3 (a)-(b), we plot the resulting dephasing timeTϕ given in Eq. (13) as a function of the detuning pa-rameters considering only longitudinal noise originatingfrom ε and εM for different parameter settings. Wefind in total five DSS for the three-spin qubit markedas black dots. Two DSS are already known, one in-side the (1,1,1) charge configuration regime25 and oneat the transition between the (2,0,1) and (1,0,2) chargestates24, while the other three DSS appear at the remain-ing three charge transitions, located on the left between(2,0,1) and (1,2,0), on the right right between (1,0,2) and(0,2,1), and bottom between (1,2,0) and (0,2,1) in thefigures due to symmetry considerations. For symmetrictunneling (tl = tr = t) the five DSS are approximatelylocated at (ε, εM ) = (0, U), (0,−U), (−U, 0), (U, 0), and(0, 0), while for asymmetric tunneling (tl 6= tr) all DSSexcept the center one are slightly shifted due to a shift ofthe charge transitions. Comparing these DSS with eachother, the four DSS located at the charge transitions areunfavorable for a small tunneling and strong noise due tostrong higher order effects limiting the coherence of thequbit. Considering stronger tunneling between the QDsand weaker noise, the higher order effects are greatly re-duced due to softening of the charge transitions. If forsome reasons working on the DSS is unpractical, e.g. cou-pling the qubit to a cavity, one should favor in the casetl 6= tr the working points given by ε = εM (diagonal or-ange line seen in the in Fig. 3 (a) and see Appendix C).

Comparing the resulting dephasing times consideringnoise coupled to the qubit through only one of the de-tuning parameters ε or εM (plotted in Fig. 6 (a) andFig. 6 (e)), we find that the results are mirror symmetricto each other with the symmetry axis given by ε = εM .Single sweet spots are found on a straight vertical (hor-izontal) line passing through the center; serpentine ver-tical (horizontal) line for tl 6= tr (a comparison of sym-metric and asymmetric tunnel coupling can be found inAppendix C).

Considering transversal noise we cannot easily find ananalytical expression for the Tϕ using the free decaymodel from Eq. (12). Thus, we have calculated δωx =|δωx,ε| + |δωx,εM | for δε = δεM 6= 0 and δtl = δtr = 0which is a good measure for the coupling of noise to thequbit. In Fig. 7 (a) and (b) we plot the resulting δωxfor different parameter settings. Note, that in these fig-ures, δωx rather than the dephasing time is plotted, thus,small values lead to a longer lifetime of the qubit. Sincetransversal noise leads to transitions between the qubitstates, this is also a first indication for the strength ofthe coupling between a qubit and a microwave cavity.

Comparison of the results obtained for transversal chargenoise (Fig. 7 in Appendix B) and qubit-cavity couplingstrength (Fig. 5) show a high level of agreement as ex-pected.

Inside the (1,1,1) charge configuration regime and con-sidering only noisy detuning parameters ε and εM thesweet spot condition from Eq. (14) simplifies to

J∂εJ + 3j∂εj = J∂εMJ + 3j∂εM j = 0 (21)

with ∂ε,εMJ = ±J2l (ε−εM )/2t2lU+J2

r (ε+εM )/2t2rU and∂ε,εM j = ±J2

l (ε− εM )/2t2lU − J2r (ε− εM )/2t2rU . There

exist only a single complete solution (DSS) in this regimefor ε = εM = 025. In contrast to the other 4 DSS theposition of the center DSS is unaffected by the strength ofthe tunneling couplings tl and tr, thus, more convenientfor symmetric gate operations using the tunnel couplingsas qubit control parameters.

C. Tunnel noise

Symmetric qubit control by tuning the tunneling cou-pling between the QDs for qubit control has been pro-posed since the very beginning62 and recent experimentsin Si/SiGe12 and GaAs63 indicate that symmetric op-erations lead to longer coherence times. Working atthe symmetric operation points reduces the coupling tothe charge noise originating from the detuning parame-ters, here, ε and εM , hence, operating on a sweet spotrelative to these parameters. However, this opens an-other channel for noise coupling to the qubit systems viafluctuations in the tunnel amplitude, since the tunnel-ing is now gate-controlled, time-dependent, and strong,narrow-band filtering cannot be applied as effective asfor the static case. Thus, the tunneling of the electronsis susceptible to charge noise.

In analogy to detuning noise, considering longitudinalnoise through the tunnel parameters tl and tr can alsodrastically be reduced by working on sweet spots. Set-ting the noisy tunnel parameters δtl 6= 0 and δtr 6= 0 andignoring noise coupled to the qubit through the detun-ing parameters δε = δεM = 0 we again find preferableworking points and single sweet spots associated witheither tl or tr. The resulting dephasing is plotted inFig. 3 (c)-(d) for the same parameter settings used pre-viously. We find the best working points deep inside the(1,1,1) charge configuration regime, however, unlike inthe case of detuning noise, there is no trace of DSS inthe entire observed regime. The best working point wefind is located at the center DSS (ε = εM = 0) whichis marginally better than the surrounding area while theother DSS at the charge transitions appear very unfavor-able at first sight. A zoom-in, however, reveals a steepvalley with a long dephasing time which is broadened bylarger tunneling couplings. Therefore, the lifetime of thequbit at the DSS located at the charge transitions arelimited by higher order effects. Strong tunnel coupling(see Fig. 3 (d)) drastically increases the lifetime at these

7

points due to softening of the charge transitions challeng-ing the center DSS.

For the case δtr = 0 (see Fig. 6 (e)-(f) in Appendix A)we find single sweet spots nearby the charge transi-tion associated with tr, thus, (1, 1, 1) ↔ (1, 0, 2) and(1, 1, 1)↔ (1, 2, 0) since at these lines in parameter spacehopping from the left QD to the center QD is energet-ically highly unfavorable. The opposite case δtl = 0 isshown in Fig. 6 (g)-(h) in Appendix A.

Inside the (1,1,1) charge configuration regime takingonly noisy tunneling into account the sweet spot condi-tion can be simplified to

Jl(2Jl − Jr) = Jr(2Jr − Jl) = 0. (22)

This condition is only fulfilled in the trivial case Jl =Jr = 0, thus, there exist no DSS for tunneling noise. Sin-gle sweet spot corresponding to the tunneling parametertl (tr) require Jr,(l) = 2Jl,(r) which simplifies for ε = 0

to tr,(l) = ±√

2tl,(r). However, the best working pointsare given for an overall symmetric configuration includ-ing both tunneling and detuning. Since the DSS are alllocated at high symmetry points, the optimal workingpoints are given by tl = tr. Preferring points of opera-tion nearby the states |3〉 and |4〉 (|2〉 and |5〉) the optimalratio is tl/tr > 1 (tl/tr < 1). However, the benefit is notvery large compared to operating on a DSS.

D. Combination

Combining all effects, we plot in Fig. 3 (e)-(f) the de-phasing time Tϕ taking into account all four noisy pa-rameters ε, εM , tl, tr. Note that we put less weight to thetunneling parameters due to their smaller strength com-pared to the detuning parameters. As a result, we findthat the previous areas with long coherence times con-sidering only detuning noise of the sweet spots becomeless pronounced and softened due to the absence of DSSfor tunneling. The center DSS still remains as the opti-mal point of operation in terms of pure coherence time,however, only slightly better than the surrounding areain the parameter space.

IV. CAVITY QUANTUM ELECTRODYNAMICS(C-QED)

While the coupling to the uncontrolled fluctuations ofthe electric field at a three-spin qubit leads to dephasing,a controlled coupling to a quantized electromagnetic fieldin a microwave cavity can be used to couple qubits overlong distances. We consider three-spin qubits realized ina linear TQD embedded in a superconducting transmis-sion line resonator with a single photon mode near theresonance frequency of the qubit. Analogous to Sec. III,we calculate the qubit-cavity coupling for the full (ε,εM )-plane including all charge configuration numerically, and

FIG. 4. (a) Schematic illustration of a qubit implemented ina TQD coupled to the cavity and the architecture for a (b)asymmetric and (d) symmetric qubit-cavity coupling. Thecenter conductor of the superconducting transmission line res-onator is on the potential Vcav while the outer conductorsare connected to the ground to screen off surrounding fields.The corresponding potential (green) and electric field (blue) isshown for the asymmetric (c) and symmetric (e) arrangementas a function of the position x.

subsequently, we approximate the center of the (1,1,1)charge configuration analytically in order to analyze theresults. To generalize our previous analysis42 to the fullrange of charge states studied in the previous sections,we extend the existing formalism to include all six rele-vant states given by Eqs. (3)-(7). We model the dipolarinteraction64 between the qubit and the cavity with

HQC = −eE · x = −eE · x (a+ a†) (23)

and define the qubit-cavity coupling strength as

g ≡ −e 〈0|E · x |1〉 , (24)

where a† (a) creates (annihilates) a photon with fre-quency ωph of the cavity mode. Note, that in this paperthe formalism using E · x is more convenient than theequivalent formalism64 A · p used in previous works42,53.Here, E is the quantized electric field, E = E(a + a†),and A is the quantized electromagnetic vector potential.

In Fig. 4 (a) the basic implementation is schemati-cally shown together with the two architectures discussedin this work which we label asymmetric and symmetriccoupling corresponding to the affected detuning param-eter. In this setting, the qubit is placed in the anti-node

8

of the electromagnetic field of the cavity to achieve thestrongest coupling. The vacuum coupling strength of theinteraction is g0, here, defined as the coupling strengthof the qubit if the dipole moment 〈0| x |1〉 ∼= al+ar, thus,the full length of the TQD, where al,(r) is the distancebetween the left (right) QD and the center QD (sketchedin Fig. 4 (b) and (d)). We find

g0 = −eE(al + ar) (25)

with E = |E| =√~ωph/2ε0εv, where e is the elec-

tron charge, ε0 (ε) is the (relative) dielectric constantof the vacuum (material) and v is the volume of thecavity64. Using realistic parameter settings for a siliconTQD (v = 3 cm × 5µm × 5µm, ε ≈ 12, ωph = 4.7 GHz,and al + ar = 60 nm) we obtain g0 = 2π × 1.96 MHz.Note, that this is the pure vacuum coupling strength andno field enhancement was included, which can furtherenhance the strength drastically65,66. To make connec-tion with experiments, it is sometimes more convenientto express the vacuum coupling strength in terms of ca-pacitance and impedance, thus, E = νωph

√Z0/π~/2w

with ν = Ccon/(Ccon +CTQD). Here, ν is the relative ca-pacitance of the TQD, CTQD, and the capacitance of theconnection to the resonator, CCon, Z0 is the characteris-tic impedance and w the distance in which the voltagedrop occurs48. Recent experiments show high impedanceresonators giving rise to a vacuum coupling strength g0in the order of 2π × 100 MHz66.

We first construct the real-space wave-functions of thestates |0〉 , |1〉 , |2〉 , |3〉 , |4〉, and |5〉, needed for the transi-tion dipole matrix element. For this we use the formalismof orthonormalized Wannier orbitals42, which transformsoverlapping single-electron wave-functions |φi〉 into an or-thonormal basis of maximally localized67 wave-functions|Φi〉 with i ∈ {1, 2, 3}. Here, the overlaps betweenthe pure single-electron wave-functions are denoted asSl ≡ 〈φ1|φ2〉, Sr ≡ 〈φ2|φ3〉, and 0 = 〈φ1|φ3〉 due tothe linear arrangement42. As a result we obtain for theposition operator in the basis {|0〉 , |1〉 , |2〉 , |3〉 , |4〉 , |5〉}

x =

0 0 1√2x12

1√2x32

1√2x23

1√2x21

0 0√

32x12 −

√32x32 −

√32x23

√32x21

1√2x21

√32x21 x11−x22 −x31 0 0

1√2x23 −

√32x23 −x13 x33−x22 0 0

1√2x32 −

√32x32 0 0 x22−x33 x31

1√2x12

√32x12 0 0 x13 x22−x11

,

(26)

where xij ≡ 〈Φi| x |Φj〉 denotes the transition dipole ma-trix element between the Wannier orbitals. Under theassumption of equal confinement potentials in each QDthese transition dipole matrix elements can always bechosen real42.

A. Asymmetric architecture

Placing the TQD inside the cavity such that the elec-tric field aligns with the long axis of the qubit42 (seeFig. 4 (b)), leads to a standard dipole coupling interac-tion between the qubit and the cavity. States with anasymmetric charge configuration interact with the elec-tromagnetic field of the cavity via their coupling to theopposite charge state, e.g., |2〉 ↔ |3〉 and |4〉 ↔ |5〉, whilecreating or annihilating a photon in the process. Hence,the qubit-cavity interaction gA in Eq. (23) can be simpli-fied to

HA = −eE x(a+ a†), (27)

and the qubit-cavity coupling strength for the asymmet-ric architecture becomes

gA ≡ −eE 〈g| x |e〉 , (28)

where |g〉 is the ground and |e〉 the first excited state. Fora (1,1,1) charge configuration these states coincide with|0〉 and |1〉. In Fig. 5 (left column) the qubit-cavity cou-pling is calculated numerically and plotted for various pa-rameter settings. The weakest qubit-cavity coupling canbe found inside the (1,1,1) charge configuration regimewhich is expected due to the symmetric electron distri-bution. The strongest coupling is located near the fourouter DSS, since at these points a charge transfer onlyrequires a small variation of the detuning parameters toproduce a large dipole moment. The asymmetric imple-mentation favors a charge transition associated with atransfer of one electron from the left QD to the rightQD42, thus, |2〉 ↔ |3〉 and |3〉 ↔ |4〉 resulting in a strongcoupling along a vertical line above and underneath the(1,1,1) charge regime in Fig. 5.

Deep inside the (1,1,1) charge configuration regime theground states are |0〉 and |1〉 which are |0〉 and |1〉 hy-bridized by a small admixture of the other charge states(|2〉, |3〉, |4〉, |5〉), hence, x = eS xe−S ≈ x + [S, x] +12 [S, [S, x]] + · · · . Here, S is the same Schrieffer-Wolfftransformation matrix used to derive the qubit splittingin the low-energy subspace given in Eq. (16). As a result,we obtain a closed analytical expression for the qubit-cavity coupling strength in Eq. (28),

gA/g0 =−√

3

2

[Jltl

Re(x12)

2(al + ar)− Jrtr

Re(x23)

2(al + ar)

]−√

3

4

[(ε− εM )

U

J2l

t2l

x11al + ar

+(ε+ εM )

U

J2r

t2r

x33al + ar

].

(29)

Here, the first (second) term in the second line resemblesthe matrix element of a DQD in the left (right) QD andthe center QD53 which compensate each other at the EODSS located at ε = 0 and εM = 0. Due to the signchange in the first term the overall matrix element isnonzero at the EO DSS as may be expected from general

9

considerations. For a completely symmetric setup, ε = 0,al = ar ≡ a, Sl = Sr ≡ S0, and tl = tr ≡ t which leads toJl = Jr ≡ J0 = 2t2U/(U2 − ε2M ), Re(x12) = −Re(x23) ≈−3aS0, and x11 = −x33 ≈ −a, Eq. (29) simplifies to

gA,0/g0 = −√

3

2

εM J0U2 − ε2M

+

√3

2

3 J0 S0

2t. (30)

The first term is identical to the expression for a sim-ple charge model gA,0/g0 = −

√(∂εJ)2 + 3(∂εj)2/2 for

this choice of parameters48 and approaches zero at theDSS while the second term remains finite. The gen-eral expression, however, is given in Eq. (29). Intro-ducing ξ ≡ S0/t, we find zero qubit-cavity coupling at

ξ =√

2/3 εM/(U2−ε2M ), e.g., for the exchange-only DSS

εM = 0 the condition is ξ = 0, thus, S0 � t.

B. Symmetric architecture

Alternatively, one can place the TQD in the cavity suchthat the center QD is connected to the transmission linewhile the outer two QDs are connected with the groundplane42 (see Fig. 4 (d)). In this scheme, the electric fieldis not alined continuously with the x-axis or other axis,but rather, it changes sign and strength in the center.Fig. 4 (e) shows the expected electric field as a function ofposition which without screening effects can be describedas a jump function. To model the electric field, we use

E(x) =1

π

{tan−1

[x

T (al + ar)

]+π(al − ar)2(al + ar)

}, (31)

where T is a screening parameter that softens the step(see Fig. 4 (e)). Note that E is an operator here becauseit is a function of the position operator, hence, we obtainfor the qubit-cavity interaction

HS = −e E(x) x (a+ a†). (32)

This Hamiltonian can be understood as a generalizationof the single-mode dipole interaction64 Hdip = −eE · x,in which the electric field E(x) can be dependent of theposition x associated with the architecture of the qubitinside the cavity. For the qubit-cavity strength for thesymmetric architecture we find

gS ≡ −e 〈g| E(x) x |e〉 , (33)

with |g〉 again being the ground state and |e〉 the first ex-cited state. Unfortunately, gS cannot be expressed in aclosed analytical form in the general case. In Fig. 5 (rightcolumn), the results are numerically calculated and plot-ted for the same parameter settings as for the asymmet-ric architecture (left column). We find the weakest valuesfor the qubit cavity coupling again deep inside the (1,1,1)charge configuration regime, which is expected due to thelarge energy required to enable a charge transition. In thevicinity of the expected charge transition areas which in-cludes the DSS we find the strongest coupling strength.

For ε = 0 and tl = tr (for tl 6= tr slightly shifted) thesymmetric coupling gS vanishes since for this architecturea charge transition between (1,0,2) and (2,0,1) or (1,2,0)and (0,2,1) is unfavorable with both outer QDs being atthe same potential. In contrast to the asymmetric ar-chitecture, the symmetric implementation should favora charge transition associated with an electron trans-fer only between the left (right) and center QD, thus,|2〉 ↔ |4〉 (|3〉 ↔ |5〉). Thus, we expect a strong re-sponse seen in a horizontal line from left to right in the(ε, εM ) parameter plane through the center (see Fig. 5).We believe the reason for the absence of this line in thenumerical results (Fig. 5, right column) is the need oftwo simultaneous charge transfers, hence, a two photonprocess which is beyond the scope of this model.

Inside the (1,1,1) charge configuration regime andassuming a large screening T > 1, thus, justifyingan expansion of the position dependent electric field,E(x)/E ≈ x/πT (al + ar) + (al − ar)/2(al + ar) we findan analytical expression for the qubit-cavity coupling de-fined in Eq. (33)

gS/g0 =al − ar

2(al + ar)2〈g| x |e〉+

1

πT (al + ar)2〈g| x2 |e〉

(34)

The first term 〈g| x |e〉 is the asymmetric coupling givenin Eq. (29) and for the second term we obtain analogously

〈g| x2 |e〉 =

√3

4

(J2l

t2l

x2112− J2

r

t2r

x2332

)+

√3

4

[Re(x12) +

(ε− εM )

U

Jltl

x11√2

]2−√

3

4

[Re(x23)− (ε+ εM )

U

Jrtr

x33√2

]2.

(35)

For a completely symmetric setup, ε = 0, al = ar ≡ a,Sl = Sr ≡ S0, and tl = tr ≡ t which leads toJl = Jr ≡ J0, Re(x12) = −Re(x23) ≈ −3aS0 andx11 = −x33 ≈ −a. Thus, in this fully symmetric case,both Eq. (35) and Eq. (34) yield gA = 0. This resultis consistent with previous results using a simple phe-nomenological approach42.

V. CONCLUSION

In this work we have analyzed different types of three-spin- 12 qubits in an electric environment, either coupledto charge noise or to coherent electric fields in a super-conducting strip-line cavity. The first coupling needs tobe minimized or eliminated in order to achieve long-livedqubits. On the other hand we want to maximize andcontrol the coupling between the qubit and the electricfield of the cavity in order to acquire the strong couplingregime needed for a fast long-distance two-qubit gate42.

In the case of a fluctuating electromagnetic environ-ment we have provided an extended description consid-ering external electric fluctuations coupled to the qubit

10

FIG. 5. The qubit-cavity coupling strength g in units of the vacuum coupling strength g0 as a function of the detuningparameters ε and εM for the asymmetric coupling (left column) and symmetric coupling (right column). The parameters arechosen as follows; top row ((a) and (b)) tl = tr = 20µeV, center row ((c) and (d)) tl = 22µeV and tr = 15µeV, and bottomrow ((e) and (f)) tl = tr = 200µeV. The interdot distances al and ar and the overlaps Sl and Sr are set by the strength of thetunneling parameters42,68. The black dots denote the DSS.

11

through four distinct noisy parameters, two detuning pa-rameters, and two tunneling parameters. We presentedand discussed the best suitable working points which takeall these couplings into account and minimize the impactlimiting the coherence time at the detuning sweet spot.However, no quadruple sweet spot was found suppress-ing first order noise effects of all four noisy parameterssimultaneously. We found that charge noise coupled tothe tunneling parameters is the limiting factor due tothe possibility of working on one of the five double sweetspots (DSS) for noisy detuning parameters. Four of thefive DSS are located each at the crossover regions be-tween connecting asymmetric charge configurations withthe fifth sitting in the center of the (1,1,1) charge config-uration regime. We have presented a full map of the de-phasing time in the (ε, εM ) parameter plane taking eitherthe effect of all four noisy parameters, pairs of two noisyparameters or each noisy parameter individually into ac-count. The optimal strategy depends on the strengthof the noise and the strength of each tunneling param-eter, however, it appears that a symmetric implementa-tion (tl = tr) typically provides the best result exactlyat the DSS while a slightly asymmetric implementation(tl 6= tr) elongates the favorable area surrounding theDSS.

In the second part of the paper, we have presented afull description of the coupling between the qubit anda high-finesse transmission line cavity taking both ba-sic alignments of connecting the physical qubit and thecavity into account, an asymmetric one being the intu-itive where the first and the last QDs are on oppositepotentials with a constant electric field. For the sym-metric coupling both outer QDs are on the same poten-tial while the center QD is connected with the transmis-sion line of the cavity. For both alignments we have pro-vided a detailed map of the coupling strength in parame-ter space and derived analytical results inside the (1,1,1)charge regime fully agreeing and extending previous re-sults where they exist. Additional features only appear-ing in the extended model were discussed. Best work-ing points for the asymmetric alignment were locatednearby the (2, 0, 1) ↔ (1, 0, 2) and (1, 2, 0) ↔ (0, 2, 1)charge transitions (the exact position depending on theparameter setting) featuring the top and bottom DSSas favorable choices. For the symmetric alignment thesepoints turn out to be less favorable within the scope ofour model and working points nearby the (1,1,1) chargetransitions should be favored in order to obtain decentcoupling strength combined with long coherence of thequbit. However, for the symmetric architecture we ex-pect additional influence of two-photon processes whichare beyond the scope of this paper.

ACKNOWLEDGMENTS

We acknowledge funding from ARO through Grant No.W911NF-15-1-0149.

Appendix A: Longitudinal noise coupled onlythrough a single noisy parameter

In Fig. 6 the dephasing times are plotted consider-ing the case where charge noise is coupled to the qubitthrough only a single noisy parameter for two differentparameter settings. Comparing the plots, we find thatthe resulting dephasing times for detuning noise in ε andεM are mirror symmetric to each other with the symme-try axis given by ε = εM , while there is no such symmetryaxis for tunneling noise in the general case tl 6= tr. TheSSS for detuning noise are located on a serpentine ver-tical or horizontal line with the crossing points given bythe DSS. For tunneling noise (tl or tr) we find that theSSS are located on a narrow curve connecting the topDSS and the right DSS as well as the right DSS and thebottom DSS. A zoom in, however, reveals that the DSSare not directly located on the line, more precisely, theSSS for tunneling noise in tl (tr) are slightly shifted tothe right (left) in parameter space. In contrast to detun-ing noise, there exist no crossing points for the SSS intunneling noise.

Appendix B: Transversal noise

In Fig. 7 we plot the transversal effect of chargenoise for the same parameter settings as for longitudi-nal noise. Since the dephasing time cannot be calcu-lated easily for transversal noise, we plot instead δωx =∑q 6=p |δωx,q|+ |δωx,p| with q, p ∈ {ε, εM , tl, tr} and δωx,q

given in Sec. III A which is a good measure for the sus-ceptibility of the noise.

In Fig. 7 the combined effects of only noisy detuning(top row), only noisy tunnel coupling (center row), andthe combined effects of two noisy detuning and tunnel-ing parameters (bottom row) are shown. Comparisonleads to similar results as for longitudinal noise. Thewell protected spot in the center of the (1,1,1) chargeconfiguration considering only detuning noise fades awayin the bottom row due to the influence of tunneling noiseand becomes as well protected against transversal chargenoise as its surrounding are. Since the outer DSS arelocated at charge transitions, they are very susceptibleto transversal noise and, therefore, will dephase muchfaster than other points. A zoom-in, however, revealsthat the outer DSS are not located at a maximum, al-though, much more susceptible than the center DSS totransversal charge noise.

Appendix C: Dephasing for symmetric tunnelcoupling

In Fig. 8 we plot the resulting dephasing time Tϕ forsymmetric tunnel couplings tl = tr = t (left column) andasymmetric tunnel coupling tl 6= tr (right column) taking

12

FIG. 6. Dephasing time Tϕ due to longitudinal noise for each noisy parameter ε, εM , tl, and tr individually in the (ε, εM )-plane.Each row shows the dephasing time due to a single noisy parameter (from top to bottom: ε, εM , tl, tr), while we choose theparameter settings identical in each column. The left column contains the results for weak tunneling and strong noise, while theright column comprises the results for strong tunneling and weak noise. Parameters are set as follows; tl = 22µeV, tr = 15µeV,Aq = 1µeV2 where q = ε in (a) and q = εM in (c), and Aq = (10−1 µeV)2, where q = tl in (e) and q = tr in (g), for the leftcolumn and tl = 220µeV, tr = 150µeV, Aq = (10−2 µeV)2 where q = ε in (b) and q = εM in (d), and Aq = (10−3 µeV)2, whereq = tl in (f) and q = tr in (h), for the right column. The black dots represent DSS.

13

FIG. 7. Impact of transversal noise as a function of the detuning parameters ε and εM . In this figure δωx rather than thedephasing time is plotted, thus, small values lead to longer coherence times of the qubit. In the top row ((a) and (b)) weconsider charge noise only from the two detuning parameters ε and εM , in the center row ((c) and (d)) we consider chargenoise only from the two tunneling parameters tl and tr, and in the bottom row ((e) and (f)) we consider charge noise from allfour parameters simultaneously, while we choose the parameter settings identical in each column. The left column contains theresults for weak tunneling and strong noise, while the right column comprises the results for strong tunneling and weak noise.The black dots indicate DSS. Parameters are set as follows; tl = 22µeV, tr = 15µeV, δq = 1µeV where q = ε, εM in (a) and(e), and δq = 10−1 µeV, where q = tl, tr in (c) and (e), for the left column and tl = 220µeV, tr = 150µeV, δq = 10−2 µeVwhere q = ε, εM in (b) and (f), and δq = 10−3 µeV, where q = tl, tr in (d) and (f), for the right column.

14

FIG. 8. Comparison of the dephasing time Tϕ as a function of the detuning parameters ε and εM for symmetric and asymmetrictunnel coupling. In the top row ((a) and (b)) we consider charge noise only from ε, in the center row ((c) and (d)) we considercharge noise only from εM , and in the bottom row ((e) and (f)) we consider charge noise both detuning parameters, ε andεM , simultaneously, while we choose the parameter settings identical in each column. The left column comprises the resultsfor symmetric tunneling, while the right column repeats the results for asymmetric tunneling given in Fig. 3 (a) and Fig. 6 (a)and (c). The black dots denote DSS. Parameters are set as follows; tl = tr = t = 20µeV for the left column and tl = 22µeV,tr = 15µeV for the right column. Further we set Aq = 1µeV2 where q = ε in (a) and (b), q = εM in (c) and (d), and q = ε, εMin (e) and (f).

15

into account only the noise from the detuning parame-ters. Comparing these two situations, we find that in thecase of only a single noisy parameter ε (εM ) and symmet-ric tunneling the SSS can be found on a straight vertical(horizontal) line through the center DSS in contrast tothe serpentine vertical (horizontal) lines for asymmetrictunneling. Taking both noisy parameters into account,

this leads to a single crossing point of the two lines at thecenter DSS in the symmetric case (tl = tr = t) while wefind an elongated area in the asymmetric case (tl 6= tr)for the center DSS. Therefore, the asymmetric case al-lows for a greater flexibility in choosing the point of op-eration while still being protected against longitudinalcharge noise.

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