Proposal for a cavity-induced measurement of the exchange coupling in quantum dots

In spin qubit arrays the exchange coupling can be harnessed to implement two-qubit gates and to realize intermediate-range qubit connectivity along a spin bus. In this work, we propose a scheme to characterize the exchange coupling between electrons in adjacent quantum dots. We investigate theoretically the transmission of a microwave resonator coupled to a triple quantum dot (TQD) occupied by two electrons. We assume that the right quantum dot (QD) is always occupied by one electron while the second electron can tunnel between the left and center QD. If the two electrons are in adjacent dots they interact via the exchange coupling. By means of analytical calculations we show that the transmission profile of the resonator directly reveals the value of the exchange coupling strength between two electrons. From perturbation theory up to second order we conclude that the exchange can still be identified in the presence of magnetic gradients. A valley splitting comparable to the inter-dot tunnel coupling will lead to further modifications of the cavity transmission dips that also depend on the valley phases.

This raises the question whether microwave cavity transmission can be used to probe the exchange coupling between neighboring quantum dots. At the surface, it appears that the answer is negative because, despite its many advantages, a coupled microwave resonator is not well-suited to measure the exchange coupling between two electrons in a DQD. This is because the exchange interaction emerges deep in the (1,1) charge sector where the electron number is fixed and charge transitions between the two QDs are extremely unlikely. The electric dipole of a DQD in this regime and hence the coupling to the cavity field are extremely low.
In this paper, we show that in fact the exchange coupling can be probed by a cavity in a triple quantum dot (TQD). We study theoretically the transmission of a microwave resonator coupled to a TQD occupied by two electrons close to the (1, 0, 1) ↔ (0, 1, 1) charge transition ( Fig. 1). Here, (n l , n r , n c ) denote the numbers of electrons in the left (l), center (c) and right (r) QD. In the (0, 1, 1) configuration the short-ranged exchange interaction couples the two electrons and splits the spin singlet and triplet states in energy. The resonator transmission T exhibits a response to the avoided level crossings of the charge transition. We show that it is possible to extract the exchange coupling J from the cavity response during a sweep of the left dot potential. We further derive the effect of an inhomogeneous magnetic field on T and show that the measurement scheme also works in the presence of a lifted valley degeneracy.
This article is organized as follows. In Sec. II we introduce the model for the coupled TQD and resonator. In Sec. III we describe the cavity-induced measurement of the exchange interaction. In Secs. IV and V, we discuss the effect of magnetic gradients and a lifted valley degeneracy on the transmission profile. In Sec. VI we summarize our results.
Schematic depiction of the system. (a) Energy levels of the TQD. The on-site potentials are assumed to be set such that the right dot (r) is permanently occupied by one electron while a second electron can tunnel between the left (l) and center (c) dot. In the (0, 1, 1) charge configuration the exchange energy J between singlet and triplet states emerges due to the wavefunction overlap. The single-electron physics of the TQD is characterized by Zeeman splittings B l , Bc, Br, tunneling t lc and energy detunings ε, δ. (b) Sketch of a microwave resonator with embedded TQD. The electric field couples to the dipole moment associated with the charge transition (1, 0, 1) ↔ (0, 1, 1).

II. CAVITY-COUPLED TQD MODEL
To model the TQD we introduce the Hamiltonian H TQD which incorporates the electrostatic potential H el , the inter-dot tunneling H t and the Zeeman effect H j z in the left (j = l), center (j = c) and right (j = r) QD. In all three dots only the lowest orbital is considered. Explicitly, where n j = σ c † jσ c jσ denotes the total occupation number operator in QD j and c ( †) jσ annihilates (creates) an electron with spin σ in QD j. The potentials E j can be tuned electrically. The Coulomb repulsion between electrons in adjacent dots, U 1 , and in the same dot, U 2j , are determined by the inter-dot distance and the QD radius.
Tunneling between QDs is included with Here, t lc(cr) is the tunneling matrix element between the left and center (center and right) QD and h.c. denotes the Hermitian conjugate. Note that H t includes only spin conserving tunneling. Spin-orbit interaction (SOI) can lead to spin-flip tunneling [71,72], this is commented on in Sec. IV B.
In QD j, the spin Hamiltonian is of the form H j z = B j · S j , where S j is the spin operator at site j. The local magnetic fields B j are given in energy units [71] and comprise a homogeneous external field B extẑ along the zaxis and potentially an inhomogeneous contribution from a static Overhauser field or a micromagnet [3,71]. To quantify the inhomogeneity we define the longitudinal (b zj ) and transverse (b xj ) magnetic field differences for j = l, c, α = x, z, For the remainder of this work we assume that E j , j = l, c, r are adjusted such that two electrons are confined to the TQD. Furthermore, we define ε = E c − E r and δ = E l − E c − U 1 and assume that the TQD is operated in the regime |δ| |ε| + U 1 U 2c , U 2r . Using the notation (n l , n c , n r ) for the charge configuration, the operating regime allows two stable charge configurations, (1, 0, 1) and (0, 1, 1), i.e. the right QD is always occupied by one electron while the second electron can be either in the left or center QD. States with doubly occupied QDs, (0, 2, 0) and (0, 0, 2), are split off by a large spectral gap of the order U 2c(r) − U 1 and have very low occupation probability. Consequently, the Hamiltonian can be reduced to the low-energy subspace of states with single occupation. This can be accomplished by a Schrieffer-Wolff transformation [73,74] and results in [6] where τ z = |1, 0, 1 1, 0, 1| − |0, 1, 1 0, 1, 1| is the Pauli z operator and τ x = |1, 0, 1 0, 1, 1| + h.c. is the Pauli x operator for the two available charge configurations. In the low energy subspace U 1 is but an offset of the left dot potential δ and the two-electron dynamics is captured in the exchange energy The resonator is modeled as a single mode harmonic oscillator, H res = ω 0 a † a with annihilation (creation) operator a ( †) , choosing = 1. The electric field E couples to the dipole er of the DQD via H dip = eE ·r [75,76] where e is the electron charge. In the present case this can also be written as H dip = g(a + a † )d /2 with g = 2eE 0 r 0 and d = n c − n r . The matrix elements include the projection E 0 of the electric field E to the DQD-axis and the distance r 0 between the left and right QD. The interaction must also be transformed into the low energy subspace, A comprehensive sketch of the system is shown in Fig. 1. We consider a setup where, additionally, a coherent driving field with frequency ω p and amplitude |a in | enters the cavity through port 1 (see Fig. 1b). At port 2 the transmitted field a out is measured. Port i = 1, 2 has the leakage rate κ i , the total cavity leakage rate is κ = κ 1 + κ 2 . The total Hamiltonian is then Input-output theory [77] is used to compute the stationary state of the output field a out = √ κ 2 a and the normalized transmission T = |a out /a in | 2 of the resonator, following the lines of Refs. [49,78]. The Hamiltonian is transformed into the eigenbasis of H TQD , defined by , and further into a rotating frame to remove the time dependence from H p . We apply a rotating wave approximation (RWA). We choose a rotating frame that allows to observe transitions between states adjacent in energy. Solving the quantum Langevin equations for a and the TQD ladder operators yields [49] where γ is the dephasing rate of the DQD states and d nm are the matrix elements of d = U TQD d U † TQD . From Eq. (9) it follows that the cavity transmission shows a dip if the TQD is tuned to an avoided crossing (AC) whose splitting matches ω p .
The TQD is assumed to have the finite temperature T dot , with the thermal population

III. PROPOSED TRANSMISSION-BASED MEASUREMENT OF EXCHANGE
It is well known that some Hamiltonian parameters that govern the single-electron dynamics in a quantum dot system can be extracted from the cavity transmission T [49][50][51][52][53][54][55]. It is desirable to have a similarly simple way to characterize the exchange J between two electrons in adjacent QDs in the (0, 1, 1) charge configuration. As discussed in App. A the transmission in the (0, 1, 1) regime with t lc = 0 carries information about the exchange J that can be classically measured. However, there the cavity response has a visibility of 10 −5 under realistic conditions since the dipole moment of the electron charge is very small in this regime, due to the small contribution of the (0, 2, 0) and (0, 0, 2) charge states.
To evade the problem of the small dipole moment we propose to sweep the electrostatic potential δ of the left dot through the (1, 0, 1) ↔ (0, 1, 1) charge transition. The dipole moment of this transition allows for a sufficiently visible cavity response which depends on the twoelectron spin state. This allows to extract the energy splitting of J between the (0, 1, 1) singlet and unpolarized triplet states, |S cr and |T cr 0 . We first discuss the case without magnetic gradients, b z,l = b z,c = 0, b x,l = b x,c = 0. Here, it is straightforward to derive T explicitly from Eqs. (8) and (9), as shown in App. B. The cavity response has two contributions, one due to the tunneling between the singlet states at the two sites, the other due to the tunneling between the triplet states. As a function of δ and t lc the responses exhibit the characteristic arc shape of an AC (Fig. 2 (a)). During a sweep of the left dot potential δ two pairs of resonances are thus observed at Due to the finite exchange J the transition between the singlets is shifted, directly revealing the value of J [Eq. (10)]. This is illustrated by the gold (J = 0) and orange (J = 0) curves in Fig. 2(b). Note that the two arcs intersect at (δ, t lc ) = −J/2, 1 4 4ω 2 p − J 2 . If 2t lc ≈ ω p J is chosen the transmission dips are hard to resolve individually. Thus, we recommend to choose 2t lc < ω p for the measurement. Without the gradients no spin-flip processes are present and thus B ext enters only via the populations. We recommend to chose k B T dot B ext for a significant population in the singlet state.
The result of this section is based on the Hamiltonian Eq. (4) which relies on the assumption that the electrons cannot interact unless they occupy adjacent QDs. Realistically, it is possible that a superexchange J s couples electrons occupying the left and right dots. Repeating the previous derivation with a small J s we find that in this case three arcs emerge in the transmission profile: The transition between the singlets is visible at δ 1s ≈ −(J − Js 4 ) ± ω 2 p − 4t 2 lc , the T 0 states at δ 2s ≈ Js 4 ± ω 2 p − 4t 2 lc and the response associated with the T ± remains at δ 2 ≈ ± ω 2 p − 4t 2 lc .

IV. DISCUSSION OF MAGNETIC GRADIENTS
In quantum information applications it may be required to include a micromagnet into the QD device device to perform fast gate operations [4-6, 34, 79] or to realize spin-photon coupling [31][32][33]. A longitudinal gradient b zl , b zc mixes the singlet and unpolarized triplet states T 0 while a transverse gradient b xl , b xc allows spinflip transitions to the spin-polarized triplets T ± .

A. Longitudinal magnetic gradient
To discuss the role of the longitudinal gradient we assume b xl = b xc = 0 and treat b zl /J, b zc /J as a perturbation of H TQD , Eq. (4). We apply non-degenerate perturbation theory to derive the corrections up to second order. Two prime effects of b zl , b zc are found.
Due to the refined energy splitting between the S and T 0 states the response from the tunneling of the singlets is shifted to the tunneling of the T 0 states is now observed at The responses due to the tunneling of the T ± states are not affected and still appear as specified by Eq. (11). Furthermore, additional transmission dips can be observed, stemming from S-T 0 transitions. These appear near Both effects are visible in the example of Fig. 2 With the gradient, the exchange can be reconstructed by identifying the transmission dips in the measurement and solving their respective equation for J. However, the example Fig. 2(b) also highlights that a longitudinal magnetic gradient can potentially be detrimental. If the responses from singlet and triplet states cannot be distinguished clearly.

B. Transverse magnetic gradient
Similarly to Sec. IV A, here, we assume that b zl = b zc = 0 and treat b xl , b xc with second order nondegenerate perturbation theory. The resulting expressions are expanded around the charge transition, assum- Due to the admixture of T ± states the responses due to the tunneling of the singlets is shifted as a function of B ext to while the tunneling of the T ± states is now observed in separate transmission dips at The response due to the tunneling of the T 0 triplet is not affected in this case and remains as given by Eq. (11). Note that the corrections in Eqs. (17) and (18) become singular for B ext = ±2t lc ± J/2. There, the nondegenerate perturbation theory breaks down. Furthermore, b xl , b xc allow for a number of additional spin-flip transitions. The associated dipole moments are ∝ (b xl /2B ext ) 2 , however. Thus, we propose to choose a large magnetic field B ext 2t lc for the measurement. This eliminates undesired responses and makes sure the analytical results from perturbation theory can be applied.
The effects of the transverse magnetic gradient alone and in conjunction with the longitudinal gradient are shown in Fig. 2(b) (dark red and black curves). In this example Eqs. (17) and (18) are only a coarse approximation since B ext is close to 2t lc + J.
To include these processes we use a modified Hamiltonian H TQD → H SOI TQD (Eq. 4) with a complex spin-flip tunneling terms f ij , .
By treating f ij as perturbation and including terms up to second order we find that the effect of the SOI is of similar form as the transverse magnetic gradient. The explicit expressions are presented in App. C. Note that in the presence of the spin-flip terms J is not given by Eq. (5). The SOI can furthermore contribute to the position-dependent part of the spin splitting [82]. This effect can be directly incorporated into b zl , b zc .

V. VALLEY DEGREE-OF-FREEDOM
For spin qubits realized in silicon the valley pseudospin [58][59][60][61] can be described with a pseudospin operator V j with ladder operators V j± = V jx ± iV jy . In each singly occupied dot j the valley Hamiltonian is given by H j v = ∆ j e iϕj V j+ + h.c. [69,83]. The valley splitting ∆ j and phase ϕ j can differ between the dots [62][63][64].
In the valley eigenbasis of all dots the valley phase differences δϕ lc = (ϕ l − ϕ c )/2 and δϕ cr = (ϕ c − ϕ r )/2 can be viewed as the angles between the valley pseudospins in adjacent dots and parametrize the ratio of of valley conserving (t ij cos δϕ ij ) and valley-flip tunneling (t ij sin δϕ ij ) between these dots [49,51].
The low-energy Hamiltonian is analogous to Eq. (4).
The exchange contribution in H v (011) is known in the literature for various combinations of interaction terms [84]. In the case of |B c(r) | b zc and |∆ c(r) | |∆ c − ∆ r | and B ext , b zc , b xc = 0, we find The low-energy Hamiltonian has 32 relevant basis states, forming six supersinglets and ten supertriplets in each charge configuration [85,86]. First, we consider the effect of the lifted valley degeneracy for the limit of a large valley splitting which is comparable to the Zeeman splitting, ∆ j ≈ B ext J and 4t lc < ∆ l + ∆ c − |∆ l − ∆ c |. In this limit it is possible to treat J and the magnetic gradients as perturbations and approximate the eigenenergies of H v TQD near the ACs. Knowledge about the valley phase differences and thus the occurrence of valley-flip tunneling is of vital importance for the interpretation of the results. The splitting of the ACs at the charge transition is determined by δϕ lc . On the other hand, δϕ cr determines which (0, 1, 1) states can couple to the (0, 2, 0), (0, 0, 2) subspace and are thus shifted in energy by the exchange interaction.
We find that the valley-conserving tunneling between the left and center dot gives rise to up to twelve pairs of transmission dips with a dipole moment ∝ t lc cos(δϕ lc ). The tunneling between states without spin polarization is observed in the cavity response near while the spin-polarized states are observed near Analogously, the valley-flip tunneling between l and c gives rise to responses with dipole moment ∝ t lc sin(δϕ lc ) near and also near  Fig. 2(b). For clarity the curves are offset by 1 each.
These results are illustrated in Fig. 3.
In the opposite limit of a small valley splitting, ∆ j ≈ J B ext we treat ∆ j and J as perturbations. As a simplification, only the lowest spin state is considered, which is justified if B ext k B T dot . Due to the valley-conserving tunneling between the left and center QD up to four arc-shaped transmission dips emerge in the δ-t lc plane, when the condition is satisfied. The valley-flip tunneling similarly gives rise to cavity responses if Consequently, the exchange interaction can be identified from both, a sweep of δ and t lc .

VI. CONCLUSIONS
In this article we have proposed a scheme to measure the exchange coupling J between two electrons in neighboring QDs from the transmission of a dipole-coupled microwave resonator. The exchange interaction between adjacent QDs emerges in a regime where charge transitions are extremely unlikely, resulting in a very low dipole moment. Our proposal circumvents this hindrance by introducing an empty third QD. The required dipole moment is obtained by sweeping through a charge transition where one electron can tunnel into the additional (left) QD. The relative position of the observed transmission dips reveals the value of J.
Exact analytical expressions for the transmission T and the position of the transmission dips during a sweep of the left dot potential δ were derived. Furthermore, we applied perturbation theory up to second order to discuss corrections due to magnetic field gradients and weak spin-orbit interaction. A transverse magnetic field gradient b xj has only small effects if the external magnetic field B ext is sufficiently high. A longitudinal gradient b zj , however, can obstruct the measurement, since it alters the singlet-triplet splitting.
The proposed measurement scheme also works in the case of a lifted valley degeneracy, e.g. in silicon QDs. Approximate expressions for the position of the transmission dips are presented in the limits of large and small valley splitting. In both cases the valley phase differences δϕ ij have a crucial role. The phase differences parametrize the valley-conserving and valley-flip tunneling and thus determine which transitions couple to the cavity field.
Our results can be applied to simplify and speed up the characterization of the short-range interaction between spin qubits, of multi-spin qubit devices and the interaction in longer spin chains. With the addition of the estimation of J to their range of applications microwave resonators become even more significant a component of spin qubit devices. and it is However, the leading contribution to the dipole moment associated with this transition is of the order of b xc t 2 cr /(min(U 2r , U 2c ) − U 1 − |ε|) 3 , resulting in an extremely low visibility of the corresponding cavity response. An analogous result can be obtained with lifted valley degeneracy ∆ j J, but the dipole moment and visibility are of the same order of magnitude.
Appendix B: Explicit expression for T Without magnetic gradients the transmission according to equation Eq. (8) can be directly computed, The two contributions to the cavity response, are stemming from the singlet (S) and triplet (T) states. We have defined a J = (4t lc ) 2 + (2J + 2δ) 2 and a = (4t lc ) 2 + (2δ) 2 . The associated dipole moments are