Cavity Control over Heavy-Hole Spin Qubits in Inversion-Symmetric Crystals

Philipp M. Mutter∗ and Guido Burkard†Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

The pseudospin of heavy-holes (HHs) confined in a semiconductor quantum dot (QD) represents apromising candidate for a fast and robust qubit. While hole spin manipulation by a classical electricfield utilizing the Dresselhaus spin-orbit interaction (SOI) has been demonstrated, our work explorescavity-based qubit manipulation and coupling schemes for inversion-symmetric crystals forming aplanar HH QD. Choosing the exemplary material Germanium (Ge), we derive an effective cavity-mediated ground state spin coupling that harnesses the cubic Rashba SOI. In addition, we proposean optimal set of parameters which allows for Rabi frequencies in the MHz range, thus entering thestrong coupling regime of cavity quantum electrodynamics.

After decades of research, a full-scale universal quan-tum computer is still not available. Various platformsare investigated in pursuit of this ultimate goal includingultra-cold atoms [1], trapped ions [2, 3], superconduct-ing circuits [4, 5] and highly engineered semiconductorstructures [6, 7]. Recently, holes (missing electrons in thevalence band) confined in QDs defined in the group-IVmaterial Ge have attracted a fair amount of renewed in-terest due to state-of-the-art experiments [8–15], placingthem at the forefront of promising candidates for quan-tum information processing. There are two main pointsin favour of a qubit, the quantum analogue of a classi-cal bit, built from the pseudo-spin degree of freedom ofholes in Ge QDs. (i) They are reliable due to long co-herence times owing to the absence of a valley degree offreedom, low disorder and weak isotropic hyperfine in-teraction (HFI), the latter being a consequence of thep-symmetry of the orbital wave functions. While thereare other types of HFI in hole systems [16], these can besuppressed by working with isotopically pure samples.(ii) Qubit manipulation is potentially fast and may beperformed by all-electrical means. This is made possibleby the strong intrinsic SOI of holes in Ge, which can beused in an electric dipole spin resonance (EDSR) schemeto manipulate spins in the context of spintronics [17].EDSR by an alternating electric field in electron systemshas been found to enable fast single-qubit operations [18–20]. Theoretical investigations of HHs in semiconductorstructures include the detailed study of 1D-materials suchas nano- and hutwires, investigating decoherence and re-laxation times as well as EDSR [21–25].

Previous studies on HHs in planar QDs have identifiedthe Dresselhaus SOI as the primary driving force of HHspin manipulation by a classical electric field [26–28] asit couples the ground state to the first excited state withopposite spin. Holes in bulk inversion-symmetric crys-tals such as Ge, however, are not subject to this type ofSOI [29], putting at risk fast qubit manipulation. In thispaper we show that with the aid of the sizeable intrinsicRashba SOI of holes and an externally applied in-planemagnetic field the ground state HH spins may still berotated coherently by coupling the QD to a photonic mi-

crowave resonator in the framework of cavity quantumelectrodynamics. Moreover, we propose a set of systemparameters which allows for Rabi frequencies of severalMHz.

We consider a planar QD formed in a heterostruc-ture with an inversion-symmetric middle layer, such asGe/SiyGe1−y quantum wells [30–32]. As such we as-sume strong confinement along the growth direction (sayz) of the heterostructure such that only the lowest or-bital energy level is occupied along that direction. Thestrong confinement induces a splitting ∆ = 2~2γs/m0d

2z

between the HH and light hole (LH) bands, where γs isthe Luttinger parameter in spherical approximation, m0

is the bare electron mass and dz is the height of the QD,i.e., the thickness of the middle layer in a heterostruc-ture. Consequently, only the HH band is occupied at lowtemperatures, kBT � ∆, and we may adopt an effec-tive description of the states with Kramers index (‘spin’)Jz ≡ s = ±3/2. For parabolic and circular in-plane con-finement characterized by the energy scale ~ω0 and anout-of-plane magnetic field B the effective Hamiltoniandescribing a HH confined in a planar QD reads

H0 =π2x + π2

y

2m+

1

2mω2

0(x2 + y2) +1

2gzµBBσz, (1)

where π = p + eA, and A = B(−y, x, 0)/2 denotes thevector potential in symmetric gauge, m is the in-planeHH mass, gz > 0 is the out-of-plane HH g-factor, and σzthe Pauli matrix along the quantization axis. The eigen-states of this Hamiltonian are |n, l, s〉 ≡ |n, l〉|s〉, where|n, l〉 denote the Fock-Darwin states with principal (az-imuthal) quantum number n (l). The corresponding en-ergies are given by En,l,s = l~ωL+(n+1)~ω+sgzµBB/3,where ωL = eB/2m is the Larmor frequency and ω =√ω20 + ω2

L [33]. Note that the Larmor frequency can bequite sizeable for HH in planar QDs due to its dependenceon the inverse mass.

The Hamiltonian in Eq. (2) is separable in orbitaland spin parts, i.e., it does not mix hole states of dif-ferent pseudo-spin. Mixing between these states en-ters via the spin-orbit interaction, which for HHs ininversion-symmetric crystals is described by the Hamil-

arX

iv:2

008.

0943

6v1

[co

nd-m

at.m

es-h

all]

21

Aug

202

0

2

ωc

ϵ

3

s = 3/2s = − 3/2(n, l)

λb

3gcΠ±

2gcΠ±

gcΠ±

a

(0,0)

s = 3/2s = − 3/2

λR

(3,-3)

(3,3)(3,1)(3,-1)

(2,-2)(2,0)(2,2)

(1,1)(1,-1)

b

λb

n = 0

n = 1

n = 2

λR

gcΠ

FIG. 2. Coupling of the low-energy states. a Allowed transitions between the 20 lowest energy levels. In each spin block(s = ±3/2), we have dipole transitions, represented by arrows drawn solid (⇧+ = 1 + !L/!) or dashed (⇧� = 1 � !L/!)and coloured according to the magnitude of the transition matrix element. The Rashba SOI (dashed blue lines) and in-planemagnetic field (solid blue lines) mix the two spin blocks with coupling strengths �R and �b, respectively. b A sketch of thecoupling mechanism. A spin in the orbital ground state (n = 0) can be flipped by the combined e�ects of the SOI, the in-planemagnetic field and the cavity field. For the sake of clarity we do not show the azimuthal sublevels.

✏(B, b) = µB

pg2

zB2 + g2xb2 + 2~

X

±

(!L ± !)4

!3!0

"✓�b

~!0

◆2!!0

!Z � 2!L ⌥ 2!+

✓�R

~!0

◆23 (!L ± !)

2

!Z � 3!L ± 3!

#, (5a)

gs(B, b) = 24�R

~!0

✓�b

~!0

◆2 p!0

�2!Z!L + !2

0

� ⇥2!2

0

�!2

0 + 2!2L

�+ !Z!L

�3!2

0 + 4!2L

�⇤

!5/2 (!Z + ! � !L) (!Z + 2! � 2!L) (�!Z + ! + !L) (�!Z + 2! + 2!L)gc, (5b)

where we defined the out-of-plane Zeeman energy ~!Z =gzµBB. A contour plot of the relative e�ective couplingstrength gc/gc as a function of in- and out-of-plane mag-netic field components is shown in Fig. 3a. We find ex-cellent agreement between the analytical approximationand numerical results obtained from exact diagonaliza-tion of the total Hamiltonian for the energies as a func-tion of both out-of-plane and in-plane magnetic fields(Figs. 3b and 3c). Note that, for the sake of clarity,we display the energy splitting ✏ only up to quadraticorder in the perturbation parameters in Eq. (5a). Asa side result, we mention that linearisation in the mag-netic field (valid for B . 1 T) yields an e�ective out-of-plane g-factor ge�

z , which is reduced not only by theRashba SOI but also by the in-plane magnetic field(Fig. 3c), an e�ect that further closes the gap betweenmeasured and theoretically predicted values [10]. Re-garding the e�ective spin coupling, we do not displaythe contributions stemming from the in-plane Zeemanenergy. We find the ratio gs(gx 6= 0)/gs(gx = 0) to de-viate by less than five percent from unity for all mag-netic field values of interest except near a resonance atB⇤ = ~!0/µB

pg2

z + 2gzm0/m. This divergence is non-

physical as it describes the point where the unperturbedeigenstates |0, 0, +3/2i and |1,�1,�3/2i align in energy.The in-plane Zeeman term HZ in combination with thedipole transitions induced by the cavity couples thesestates, and the perturbative approach is not valid in thisregion (shown as a box in Fig. 3a).

The e�ective spin coupling gs exhibits an exception-ally strong dependence on the QD height dz via the HH-LH splitting �, gs / 1/�3 / d6

z. An increase in theQD height by a factor of three may thus increase the ef-fective spin coupling strength by two to three orders ofmagnitude. However, in the present model this restrictsthe in-plane magnetic field to small values such that theperturbative approach is valid and orbital e�ects may beneglected. The regime of large in-plane magnetic fieldsand a relatively large QD height may show particularlystrong spin-photon couplings and presents a promisingavenue for future research. On the other hand, we finda inverse dependence of the coupling strength on the dotradius, stemming from the behaviour of the momentummatrix elements in HSO. For the same reason, we see astrong dependence on the e�ective hole mass, with in-creased coupling for larger masses. The in-plane heavy

FIG. 1. Coupling of a quantum dot to a resonator. Photons ofenergy ~ωc are confined in a cavity and interact with a planarquantum dot with renormalized ground state spin splitting ε.The dotted square shows the interaction region, the strengthof the effective spin-photon coupling being gs.

tonian [26, 28, 34, 35],

HSO = iα(σ+π

3− − σ−π3

+

)+ ξ(b)

(σ+π

2− + σ−π

2+

), (2)

where σ± = (σx ± iσy)/2 with in-plane Pauli matricesσx/y and π± = πx ± iπy. The first term in Eq. (2) isobtained in second-order perturbation theory from theLuttinger-Kohn Hamiltonian including the Rashba termαR (π × 〈Ez〉ez) · J, where J is a vector containing spin-3/2 matrices, αR is the Rashba coefficient and 〈Ez〉 is theaveraged electric field along the growth direction experi-enced by the hole in an asymmetric quantum well. Onefinds α = 3γsαR〈Ez〉/2m0∆, and we define the quan-tity λR = α (m~ω0)

3/2 as the characteristic energy scaleof the cubic Rashba SOI. The second term in Eq. (2) isonly present when an in-plane magnetic field is appliedto the system and stems from the combined effects of theSOI and HH-LH mixing. Due to in-plane degeneracy, wemay choose a coordinate system such that the additionalin-plane field is along x, resulting in a total magnetic fieldB = (b, 0, B) and ξ(b) = 3γsκµBb/m0∆, where κ is themagnetic Luttinger parameter in the context of the enve-lope function approximation. We define the energy scaleof the magnetic coupling strength as λb = ξ(b)m~ω0.

Note that applying an in-plane magnetic field also in-troduces an in-plane Zeeman term Hx

Z = gxµBbσx/2 tothe effective HH Hamiltonian via the cubic spin-3/2 op-erator terms. The in-plane g-factor gx is typically muchsmaller than its out-of-plane counterpart, gx � gz, andwe may treat Hx

Z as a perturbation. On the other hand,we completely disregard the orbital effects of the in-planemagnetic field due to strong out-of-plane confinement(of the order of 10 meV for a typical dot height of 10nm). The latter is valid as long as ~eb/2m � Uz =~2π2/2md2z, where dz is the height of the QD. We canexpress the condition in terms of the HH-LH splitting ∆,b� 104∆[eV] T.

As depicted in Fig. 1, we aim to couple the groundstate spins to a cavity mode of frequency ωc, describedby the Hamiltonian Hc = ~ωca†a, where a† and a are thephoton creation and annihilation operators, respectively.The interaction of the cavity photons with the QD holesin the Fock-Darwin-spin basis {|ν〉 ≡ |n, l, s〉} is of theform

HI = ~gc(a+ a†

)∑

ν,ν′

Πνν′ |ν〉〈ν′|, (3)

where gc = e√ω0/2ε0εrmV ωc [36, 37] for a cav-

ity of volume V and relative permittivity εr, andΠνν′ ≡ 1√

~ω0m〈n, l, s|πx|n′, l′, s′〉 are the dimensionless

momentum matrix elements for linearly polarized lightalong x, i.e., along the in-plane magnetic field.

The lowest energy state that the ground state is cou-pled to by the in-plane magnetic field term ∝ b in Eq. (2)is |2,±2,∓3/2〉 and this state can transition via the com-bined effects of the Rashba SOI, the in-plane magneticfield and the electric dipole coupling to the orbital groundstate with opposite spin, thereby creating an effectiveground state spin-photon coupling. For the spin to flipwe need an odd number of transitions due to the RashbaSOI and in-plane magnetic field. Since the ground statespins are not directly coupled, we expect the minimumnumber of spin-orbit induced transitions required to bethree. A graphical overview of the allowed transitions inthe low energy part of the system is given in Fig. 2a,and an exemplary sequence of transitions realizing aground-state spin-flip is shown in Fig. 2b. To supportour qualitative reasoning mathematically, we considerthe first four orbital levels, n ∈ {0, 1, 2, 3}. Includingspin and azimuthal sublevels, this amounts to a 20 × 20Hamiltonian matrix. In this state space we perform aSchrieffer-Wolff transformation on the total QD Hamil-tonian HQD = H0 + HSO + Hx

Z to decouple the n = 0subspace from higher energy states to quartic order in theperturbation parameters λR/~ω0 and λb/~ω0. The SOImixes the eigenstates |n, l,±3/2〉 of the Hamiltonian H0

yielding the perturbed eigenstates |n, l,±〉. We proceedto apply the same transformation to the part describ-ing matter in the interaction Hamiltonian HI . A de-tailed description of the procedure is given in the supple-mentary material, Ref. [38]. Projecting the transformedHamiltonian onto the orbital ground state, we obtainan effective Rabi-type Hamiltonian in the logical basis{| ↑ 〉 = |0, 0,+〉, | ↓ 〉 = |0, 0,−〉

},

H =ε

2σz + ~ωca†a+ ~gs

(a+ a†

)σy. (4)

The renormalized energy splitting ε and the effectiveground-state spin coupling gs are given by

3

s = 3/2s = − 3/2(n, l)

∝ λb

3gcΠ±

2gcΠ±

gcΠ±

a

(0,0)

s = 3/2s = − 3/2

∝ λR

(3,-3)

(3,3)(3,1)(3,-1)

(2,-2)(2,0)(2,2)

(1,1)(1,-1)

b

n = 0

n = 1

n = 2

∝ λR

gcΠ−

λb

∝

FIG. 2. Coupling of the low-energy states. a Allowed transitions between the 20 lowest energy levels (each level shown twicefor clarity). In each spin block (s = ±3/2), we have dipole transitions, represented by arrows drawn solid (Π+ = (ω+ ωL)/ω0)or dashed (Π− = (ω − ωL)/ω0) and coloured according to the magnitude of the transition matrix element. The Rashba SOI(dashed blue lines) and in-plane magnetic field (solid blue lines) mix the two spin blocks with coupling strengths λR and λb,respectively. b A sketch of the coupling mechanism. A spin in the orbital ground state (n = 0) can be flipped by the combinedeffects of the SOI, the in-plane magnetic field and the cavity field. For the sake of clarity we do not show the azimuthalsublevels.

ε(B, b) = µB√g2zB

2 + g2xb2 + 2~

∑

±

(ωL ± ω)4

ω3ω0

[(λb~ω0

)2ωω0

ωZ − 2ωL ∓ 2ω+

(λR~ω0

)23 (ωL ± ω)

2

ωZ − 3ωL ± 3ω

], (5a)

gs(B, b) = 24λR~ω0

(λb~ω0

)2 √ω0

(2ωZωL + ω2

0

) [2ω2

0

(ω20 + 2ω2

L

)+ ωZωL

(3ω2

0 + 4ω2L

)]

ω5/2 (ωZ + ω − ωL) (ωZ + 2ω − 2ωL) (−ωZ + ω + ωL) (−ωZ + 2ω + 2ωL)gc, (5b)

where we defined the out-of-plane Zeeman energy ~ωZ =gzµBB. A contour plot of the relative effective couplingstrength gs/gc as a function of in- and out-of-plane mag-netic field components is shown in Fig. 3a. We find ex-cellent agreement between the analytical approximationand numerical results obtained from exact diagonaliza-tion of the total Hamiltonian for the energies as a func-tion of both out-of-plane and in-plane magnetic fields(Figs. 3b and 3c). Note that, for the sake of clarity,we display the energy splitting ε only up to quadraticorder in the perturbation parameters in Eq. (5a). Asa side result, we mention that linearisation in the mag-netic field (valid for B . 1 T) yields an effective out-of-plane g-factor geffz , which is reduced not only by theRashba SOI but also by the in-plane magnetic field(Fig. 3c), an effect that further closes the gap betweenmeasured and theoretically predicted values [13]. Re-garding the effective spin coupling, we do not displaythe contributions stemming from the in-plane Zeemanenergy. We find the ratio gs(gx 6= 0)/gs(gx = 0) to de-viate by less than five percent from unity for all mag-netic field values of interest except near a resonance at

B∗ = ~ω0/µB√g2z + 2gzm0/m. This divergence is non-

physical as it describes the point where the unperturbedeigenstates |0, 0,+3/2〉 and |1,−1,−3/2〉 align in energy.The in-plane Zeeman term Hx

Z in combination with thedipole transitions induced by the cavity couples thesestates, and the perturbative approach is not valid in thisregion (shown as a box in Fig. 3a).

The effective spin coupling gs exhibits an exception-ally strong dependence on the QD height dz via the HH-LH splitting ∆, gs ∝ 1/∆3 ∝ d6z. An increase in theQD height by a factor of three may thus increase the ef-fective spin coupling strength by two to three orders ofmagnitude. However, in the present model this restrictsthe in-plane magnetic field to small values such that theperturbative approach is valid and orbital effects may beneglected. The regime of large in-plane magnetic fieldsand a relatively large QD height may show particularlystrong spin-photon couplings and presents a promisingavenue for future research. On the other hand, we finda inverse dependence of the coupling strength on the dotradius, stemming from the behaviour of the momentummatrix elements in HSO. For the same reason, we see a

4

0.0 0.5 1.0 1.5 2.02.0

2.5

3.0

3.5

4.0

a2

3

4

0.05 0.1 0.2 0.25

0 1 2

−0.8

−0.4

0

b |0,0,−⟩

|0,0,+⟩|1, − 1,−⟩ |2, − 2,−⟩

0 1 2 3

0.150.05 0.10 0.15 0.20 0.25

0.2 b = 4 T

ϵ/μBB

c

9876

0 1 2 3 40 1 2 3 45

6

7

8

9

5

−0.2

−0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.0008-0.0006-0.0004-0.00020.0000

0.0002

[T] B

[meV]

E

[T] B

[T] b

[T]

b

We consider the singlet-triplet Hamiltonian,

H = �hJ

2�z +

�gµBB

2�x =

1

2

0BBB@

�hJ �gµBB

�gµBB hJ,

1CCCA , (A1)

where J is the exchange energy in Hz. The time evolution may be calculated exactly, yieldingthe singlet probability PS,

PS(t, B) = 1 � sin2(!t) sin2(2⇠),

! =

p(hJ)2 + (�gµBB)2

2~, ⇠ = arctan

�gµBB

hJ +p

(hJ)2 + (�gµBB)2

!.

(A2)

The formula contains twice the frequency we want. However, we can make use of thetrigonometric identity sin2(x/2) = (1 � cos x)/2 to rewrite the singlet probability,

PS(t, B) =1 + cos2 2⇠

2+

1

2sin2 2⇠ cos!t, (A3)

which now is a genuinely sinusoidal oscillation (and not sin2) with frequency ! = 2! aroundthe value (1 + cos2 2⇠)/2. I believe that when you perform a FFT on the data, this isthe frequency you get. Thus, we obtain S-T0-oscillations with frequency f = !/2⇡ =p

(hJ)2 + (�gµBB)2/h in accordance with your results. I also believe that when you takeinto account the exchange energy, the fit to the data will be even better.

Note that we only reach PS = 0 for J = 0 due to the detuning between singlet andtriplet states introduced by a non-vanishing exchange energy (the singlet is energeticallymore favourable than the triplet for J > 0). At J = 0 the oscillation takes on a particularlysimple form,

PS(t, B) =1 + cos!t

2. (A4)

To summarize: All factors of 1/2 in the Hamiltonian and the Pauli operators we use arecorrect. The confusion simply stemmed from the fact that a sin (or cos) oscillation onlyhas half the period of a sin2 oscillation. However, as both describe the same quantity, thismust be compensated by twice the frequency in a sinusoidal oscillation compared to a sin2

oscillation.

L/R(x, y) = e⌥i ya

2l2B 00(x ± a, y). (A5)

✏/µBB ' ge�z gs/gc

7

We consider the singlet-triplet Hamiltonian,

H = �hJ

2�z +

�gµBB

2�x =

1

2

0BBB@

�hJ �gµBB

�gµBB hJ,

1CCCA , (A1)

where J is the exchange energy in Hz. The time evolution may be calculated exactly, yieldingthe singlet probability PS,

PS(t, B) = 1 � sin2(!t) sin2(2⇠),

! =

p(hJ)2 + (�gµBB)2

2~, ⇠ = arctan

�gµBB

hJ +p

(hJ)2 + (�gµBB)2

!.

(A2)

The formula contains twice the frequency we want. However, we can make use of thetrigonometric identity sin2(x/2) = (1 � cos x)/2 to rewrite the singlet probability,

PS(t, B) =1 + cos2 2⇠

2+

1

2sin2 2⇠ cos!t, (A3)

which now is a genuinely sinusoidal oscillation (and not sin2) with frequency ! = 2! aroundthe value (1 + cos2 2⇠)/2. I believe that when you perform a FFT on the data, this isthe frequency you get. Thus, we obtain S-T0-oscillations with frequency f = !/2⇡ =p

(hJ)2 + (�gµBB)2/h in accordance with your results. I also believe that when you takeinto account the exchange energy, the fit to the data will be even better.

Note that we only reach PS = 0 for J = 0 due to the detuning between singlet andtriplet states introduced by a non-vanishing exchange energy (the singlet is energeticallymore favourable than the triplet for J > 0). At J = 0 the oscillation takes on a particularlysimple form,

PS(t, B) =1 + cos!t

2. (A4)

To summarize: All factors of 1/2 in the Hamiltonian and the Pauli operators we use arecorrect. The confusion simply stemmed from the fact that a sin (or cos) oscillation onlyhas half the period of a sin2 oscillation. However, as both describe the same quantity, thismust be compensated by twice the frequency in a sinusoidal oscillation compared to a sin2

oscillation.

L/R(x, y) = e⌥i ya

2l2B 00(x ± a, y). (A5)

✏/µBB ' ge�z gs/gc

7

We consider the singlet-triplet Hamiltonian,

H = �hJ

2�z +

�gµBB

2�x =

1

2

0BBB@

�hJ �gµBB

�gµBB hJ,

1CCCA , (A1)

where J is the exchange energy in Hz. The time evolution may be calculated exactly, yieldingthe singlet probability PS,

PS(t, B) = 1 � sin2(!t) sin2(2⇠),

! =

p(hJ)2 + (�gµBB)2

2~, ⇠ = arctan

�gµBB

hJ +p

(hJ)2 + (�gµBB)2

!.

(A2)

The formula contains twice the frequency we want. However, we can make use of thetrigonometric identity sin2(x/2) = (1 � cos x)/2 to rewrite the singlet probability,

PS(t, B) =1 + cos2 2⇠

2+

1

2sin2 2⇠ cos!t, (A3)

which now is a genuinely sinusoidal oscillation (and not sin2) with frequency ! = 2! aroundthe value (1 + cos2 2⇠)/2. I believe that when you perform a FFT on the data, this isthe frequency you get. Thus, we obtain S-T0-oscillations with frequency f = !/2⇡ =p

(hJ)2 + (�gµBB)2/h in accordance with your results. I also believe that when you takeinto account the exchange energy, the fit to the data will be even better.

Note that we only reach PS = 0 for J = 0 due to the detuning between singlet andtriplet states introduced by a non-vanishing exchange energy (the singlet is energeticallymore favourable than the triplet for J > 0). At J = 0 the oscillation takes on a particularlysimple form,

PS(t, B) =1 + cos!t

2. (A4)

To summarize: All factors of 1/2 in the Hamiltonian and the Pauli operators we use arecorrect. The confusion simply stemmed from the fact that a sin (or cos) oscillation onlyhas half the period of a sin2 oscillation. However, as both describe the same quantity, thismust be compensated by twice the frequency in a sinusoidal oscillation compared to a sin2

oscillation.

L/R(x, y) = e⌥i ya

2l2B 00(x ± a, y). (A5)

✏/µBB ' ge�z gs/gc B = 1 T

7

FIG. 3. Effective low-energy system. a The effective spin-photon coupling constant gs as a function of the in-plane (b)and out-of-plane (B) magnetic fields. The dotted rectangleindicates that the perturbative approach is not valid in thisregion (see text). b, c Comparison between the analyti-cal approximation (red solid line) and the numerical resultobtained by exact diagonalization of the total Hamiltonian,taking into account the 20 lowest energy states (blue dots).b Lowest perturbed energy states |n, l,±〉 as a function of theout-of-plane magnetic field B at b = 4 T. We find good agree-ment between approximation and numerics up to B ' 2 T,where the perturbative approach becomes inaccurate due toincreased mixing with higher orbital states. The zero-fieldsplitting is caused by the in-plane Zeeman energy and theRashba SOI. c Renormalized ground state spin energy separa-tion at B = 1 T as a function of the in-plane magnetic field b.The g-factor renormalization at b = 0 T is due to the RashbaSOI. The parameter values used for all plots are γs = 5.11,κ = 3.41, gz = 10, gx = 0.2, ∆ = 10 meV, m = 0.2m0,~ω0 = 1meV and ~αR〈Ez〉 = 10−11 eVm.

strong dependence on the effective hole mass, with in-creased coupling for larger masses. The in-plane HHmass in Ge is measured to be m = 0.09m0 [11] andis extrapolated to be even lower at low hole densities.However, it has been reported that it is possible to in-crease the HH mass in inversion-symmetric materials byapplying tensile strain to the system [39, 40]. Finally,gs depends linearly on the strength of the cubic RashbaSOI. The Rashba coefficient in planar Ge is reported tobe ~αR〈Ez〉 = 10−13 − 10−11 eVm [34, 41, 42]. In Genanowires values reach ~αR〈Ez〉 = 10−10 eVm by ap-

plying external gates [17]. Such schemes have also beenproposed for planar QDs in group III-V materials suchas GaAs [43], suggesting the possibility of increasing theRashba SOI artificially in planar group-IV QDs. Ideally,quantum engineering should therefore aim for relativelylarge ratios of QD height to QD radius (while maintainingthe confinement along z strongest), sizeable HH massesand a large Rashba coefficient. One can see from Fig. 3athat an optimized set of parameters as described aboveallows for spin-photon couplings exceeding gs ≈ gc/4.Another way in which one might attempt to artificiallyincrease the spin-photon coupling would be the applica-tion of a static electric field in the QD plane. However,for harmonic confinement this merely leads to a shift ofthe QD position, rather than additional transitions thatcould modify the cavity coupling.

Finally, we may estimate feasible spin rotationtimes. Superconducting resonators have typical couplingstrengths of G/2π = gcωc/2πω0 ' 1 − 10 MHz for aQD of lateral size

√~/mω0 ' 10 − 100 nm, while cav-

ity loss rates are around κc ' 1 MHz (yielding a qual-ity factor Q = fc/κc = 103 for typical resonator fre-quencies of fc = 5 − 10 GHz) [44]. One major ad-vantage of spin qubits especially in hole systems com-pared to charge qubits is their high robustness, reachingcoherence times τ = Γ−1 of several microseconds andeven milliseconds [45–48]. In optimal operation mode,defined by the optimization of parameters described inthe previous paragraph, we may reach spin-photon cou-pling strengths gs/2π ≈ gc/8π, thus entering the strong-coupling regime, gs > κc,Γ, allowing for coherent spinrotations with vacuum Rabi frequencies fR = gs/π inthe MHz range. As an application, two-qubit gates maybe implemented by harnessing the spin-photon couplingto obtain a controlled interaction between distant spins.For instance, the iSWAP gate may be performed in thedispersive regime in time τ = (4k + 1)π|δ|/2g2s with spinqubit-cavity detuning δ = ωc−ε/~ and k = 0, 1, 2, . . . [49].Consequently, HH systems in Ge allow for fast two-qubitlogic with typical gate operation times τ ∼ |δ| [kHz] ns.

In conclusion, we show that the pseudo-spin of HHs inplanar QDs in bulk inversion-symmetric materials can becoupled to a cavity. The qubit manipulation requires atilted magnetic field with respect to the QD plane andutilizes the intrinsic cubic Rashba SOI, which is sizeablein Ge. We find that within our model in-plane Zeemanenergies and in-plane static electric fields do not consider-ably enlarge the spin-photon coupling strength. Finally,we propose an optimal planar QD design which allows forcoherent Rabi oscillations in the MHz range in the strongcoupling regime and thereby fast long-distance two-qubitlogic. Our results consolidate HH spin qubits in Ge asprime candidates for a platform in quantum informationprocessing.

5

∗ philipp.mutter@uni-konstanz.de† guido.burkard@uni-konstanz.de

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