# Hyperfine interaction induced dephasing of coupled spin qubits in semiconductor double quantum dots

###### Abstract

We investigate theoretically the hyperfine-induced dephasing of two-electron-spin states in a double quantum dot with a finite singlet-triplet splitting . In particular, we derive an effective pure dephasing Hamiltonian, which is valid when the hyperfine-induced mixing is suppressed due to the relatively large and the external magnetic field. Using both a quantum theory based on resummation of ring diagrams and semiclassical methods, we identify the dominant dephasing processes in regimes defined by values of the external magnetic field, the singlet-triplet splitting, and inhomogeneity in the total effective magnetic field. We address both free induction and Hahn echo decay of superposition of singlet and unpolarized triplet states (both cases are relevant for singlet-triplet qubits realized in double quantum dots). We also study hyperfine-induced exchange gate errors for two single-spin qubits. Results for III-V semiconductors as well as silicon-based quantum dots are presented.

###### pacs:

03.65.Yz, 76.30.–v, 71.70.Jp, 76.60.Lz## I Introduction

Spin qubits in quantum dots (QDs) or donors have been extensively investigated during the past decade.Hanson et al. (2007); Liu et al. (2010); Morton et al. (2011) Single spin qubitsLoss and DiVincenzo (1998) have been initialized, manipulated, and read out both electrically in gated dotsElzerman et al. (2004); Koppens et al. (2006); Nowack et al. (2007); Koppens et al. (2008); Pioro-Ladrière et al. (2008); Shaji et al. (2008); Morello et al. (2010); Pla et al. (2012) and optically in self-assembled dots.Press et al. (2008, 2010); Greilich et al. (2009) Two-qubit exchange gates have been demonstrated recently in gated QDs,Brunner et al. (2011); Nowack et al. (2011) so are two-electronKim et al. (2011) and two-holeGreilich et al. (2011) couplings in a pair of vertically stacked self-assembled QDs. Furthermore, a subspace formed by the singlet and the unpolarized triplet states of two electron spins has also been explored as a logical qubit,Levy (2002); Petta et al. (2005); Foletti et al. (2009); Maune et al. (2012) and coupling between two such qubitsvan Weperen et al. (2011) and their entanglementShulman et al. (2012) has been demonstrated recently.

Slow decoherence relative to the control speed is one of the key criteria for a scalable quantum information processor. In QDs made of III-V compounds or from natural Si, the dominant source of single-spin decoherence is the nuclear spin bath, coupled to the carrier spins by hyperfine (hf) interaction.Abragam (1983); Meier and Zakharchenya (1984); Merkulov et al. (2002); Hanson et al. (2007) The role of this hf coupling is clear: the energy splitting of an electron spin is affected by the fluctuating Overhauser field of the nuclear spins. The nuclear spin bath is quasistatic due to its dynamics being much slower than the dynamics of the electron spins. Therefore, the strongest effect of the hf interaction is an inhomogeneous broadening in the distribution of electron spin splitting due to the random but static orientation of the nuclear spins, causing a decay of the electron free induction signal on the time scale of .Merkulov et al. (2002); Hanson et al. (2007) The slow nuclear spin dynamics within the bath causes homogeneous broadening, or pure dephasing of the electron spin qubit, which is measurable in a spin echo experiment. This nuclear spin dynamics is either due to hf interaction only at lower applied fields,Deng and Hu (2006); Yao et al. (2006); Cywiński et al. (2009a, b); Bluhm et al. (2010a); Neder et al. (2011) or, at higher fields, due to dipolar interaction between nuclear spins.de Sousa and Das Sarma (2003); Witzel and Das Sarma (2006); Yao et al. (2006)

For two uncoupled spins, spin product states such as and are the two-spin eigenstates. If the two spins are initially prepared in a singlet state () ,Johnson et al. (2005); Koppens et al. (2005); Petta et al. (2005) the random Overhauser field would strongly mix the and states, and the time scale on which this mixing leads to decay of the measured signal is , the same as that for a single-spin qubit.Johnson et al. (2005); Koppens et al. (2005); Petta et al. (2005) Similar to the single spin case,Koppens et al. (2008) the application of a Hahn echo pulse sequence removes the influence of the quasistatic nuclear fluctuations, and reveals the much slower decoherence of two independent single spins.Petta et al. (2005); Bluhm et al. (2010a)

For two coupled electrons in a uniform effective magnetic field, the singlet and triplet states are the two-electron eigenstates. At a finite exchange splitting in a double quantum dot (DQD), it was predictedCoish and Loss (2005) and then shown experimentallyLaird et al. (2006) that the hf-induced singlet-triplet mixing is suppressed, and the probability of the initialized singlet to remain in this state decays as a power-law towards a saturation value which approaches unity as is increased. The decoherence effect of nuclear spin pair flips due to inter-nuclear dipolar interactions has also been analyzed in the limit of much smaller than the Zeeman splitting of the polarized triplet states.Yang and Liu (2008) Recently, decoherence in the - subspace has been investigated at finite in experiments on GaAsDial et al. (2013); Higginbotham et al. (2013) and InGaAsWeiss et al. (2012) DQDs. In the former work the dominant role played by the charge noise was uncovered, while in the latter paper the effects of charge noise were minimized, and a lower bound on time due to interaction with nuclei was obtained.

In this paper we systematically investigate the hf-induced dephasing of two-spin states at finite values of exchange splitting . We focus on the case of dephasing within the - subspace, in which full control over the qubit state is possible because of the creation of a stationary Overhauser field gradient,Foletti et al. (2009) or a gradient of magnetic field generated by a proximal nanomagnet.Pioro-Ladrière et al. (2008); Petersen et al. (2013) We investigate both the limit of uniform effective field and of the finite effective field gradient, and we find that with an increasing magnitude of the gradient there is a smooth transition from strongly suppressed decoherence to decoherence that is similar to the case of single-spin qubits. We also study the hf-induced decay in an Hahn echo experiment, which is possible with a controllable field gradient.Dial et al. (2013) Lastly, we analyze the hf-induced exchange gate error, when the two spins in a DQD are treated as two single-spin qubits.

Our theoretical approach is based on first obtaining an effective pure dephasing Hamiltonian via an appropriate canonical transformation of the full hf Hamiltonian.Shenvi et al. (2005); Yao et al. (2006); Coish et al. (2008); Yang and Liu (2008); Cywiński et al. (2009a, b) The effective Hamiltonian is diagonal in the basis of the relevant states, allowing dephasing calculations for superpositions of these states. When dealing with terms in that are of second order in the transverse Overhauser field, we use the ring diagram theoryCywiński et al. (2009a, b); Cywiński et al. (2010) (RDT). The terms that are of first or second order in longitudinal Overhauser field are treated classically. We also compare the RDT results with calculations based on semiclassical averaging over the quasistatic transverse Overhauser fields, underlining its connection to the short-time RDT calculations.Neder et al. (2011); Cywiński (2011)

For moderate values of singlet-triplet splitting, the shortest singlet-triplet dephasing times due to hf interaction derived in this paper are on the order of a microsecond (millisecond) for typical GaAs (Si) QDs. It is important to point out that there are many other decoherence channels beyond hf coupling with nuclear spins for electrons in a DQD. For example, at finite , the orbital degree of freedom is not completely frozen, so that its fluctuations can lead to spin decoherence. In particular, charge noise could be an important, even dominant, source of dephasing, as suggested by theoryCoish and Loss (2005); Hu and Das Sarma (2006); Culcer et al. (2009); Gamble et al. (2012) and seen in experiment.Shulman et al. (2012); Dial et al. (2013); Higginbotham et al. (2013) In fact, the singlet-triplet dephasing time ns due to charge noise in GaAs observed recentlyDial et al. (2013) is an order of magnitude smaller than the shortest hf-induced dephasing times predicted for this material in this paper. Our results presented in this paper are most relevant in an experimental situation where charge noise is not dominant. However, despite the possibly critical role of charge noise in DQDs, we would like to stress that this noise can, in principle, be removed to a large degree (e.g., by different designs of samples or the gate circuitry), while the presence of nuclear spins is unavoidable in III-V materials, since all isotopes of the III-V elements have finite nuclear spins (unlike Si, where isotopic purification could, in principle, suppress the nuclear-induced spin decoherence). Furthermore, it has recently been shown that and states of an optically controlled self-assembled QD molecule can be manipulated in a regime in which their splitting is insensitive (in the first order) to charge fluctuations.Weiss et al. (2012) The existence of such “sweet spots” was also predictedStopa and Marcus (2008); Li et al. (2010) in gated QDs. Besides charge noise, the different charge distributions for the singlet and triplet states also allow electron-phonon interaction to cause dephasing in the - subspace,Roszak and Machnikowski (2009); Hu (2011); Gamble et al. (2012) although this dephasing channel should be weak in a double dot if is not too large.Hu (2011)

In addition to pure dephasing, the - qubit can undergo longitudinal relaxation through phonon emission, which is allowed by spin-orbit couplingStano and Fabian (2006); Raith et al. (2012); Borhani and Hu (2012) and/or hyperfine mixing between the states.Raith et al. (2012) Such dissipative relaxation processes lead to both transitions between and , and leakages out of the qubit subspace (to states, for example). The characteristic time scales are most often much longerBorhani and Hu (2012) than the time scales considered in this paper, although recent calculations suggest that in some parameter regimes the - transitions in GaAs DQDs can occur at the microsecond time scale.Raith et al. (2012) One can, however, exploit the large anisotropy of the relaxation rates with respect to the direction of applied magnetic field, and suppress these processes by an appropriate choice of the in-plane -field direction.Raith et al. (2012)

We mention that although the single qubit decoherence is often theoretically studied in the literature in various contexts, there are few concrete analyses of multiqubit decoherence simply because the multiqubit decoherence problem is technically difficult due to the many possible decoherence channels for the entangled system when even a few qubits are coupled together. Our current work demonstrates that the simplest spin system in which entanglement can occur, namely, a system of just two exchange-coupled electron spins weakly interacting with an environment of nuclear spins, is theoretically challenging, even though the full Hamiltonian for the problem is completely known. Real experimental situations are obviously far more complex because it is unlikely that all the environmental influences would be completely known. For example, as we mentioned above, for coupled spin qubits in semiconductor quantum dots, there would be, in addition to the nuclear-induced Overhauser noise, other decoherence mechanisms such as charge noise, stray fluctuating magnetic fields arising from random impurity spins in the semiconductor and from microwaves and other random fluctuations in the background. However, the eventual construction of a fault-tolerant practical quantum computer necessarily requires understanding (and if possible, mitigating) all multiqubit decoherence mechanisms since the quantum error correction threshold is small, which implies that only the smallest amount of decoherence can be efficiently eliminated by the error correction protocols. Our current work should be construed as a first step toward the goal of a comprehensive understanding of multiqubit decoherence in one of the most practical and widely studied quantum computer architecture proposals, namely, the spin quantum computer in semiconductor quantum dots. It is somewhat sobering that even this first step of understanding Overhauser noise induced two-qubit decoherence in semiconductor quantum dots is already a very challenging problem.

The present paper is organized as follows. In Sec. II, we describe the Hamiltonian of the two electrons in a DQD applicable in the regime of our interest. At finite , the Hamiltonian of two electrons is given in the basis in Sec. II.1. The state of the nuclear bath is described in Sec. II.2. In Sec. II.3, we derive an effective Hamiltonian in the - basis. The dephasing of an - qubit due to the various terms in the effective Hamiltonian is calculated for DQDs made of various materials in Sec. III. We identify the dominant dephasing mechanisms for various types of DQDs (GaAs, Si, or InGaAs) in Sec. III.5, and discuss the existence of optimal value of at which the coherence time is predicted to be maximal. In Sec. IV, we clarify the effect of a finite interdot field gradient, and analyze the decay of the Hahn echo signal. The SWAP gate error due to the hf-induced dephasing processes is investigated in Sec. V, with a focus on the effect of inhomogeneous broadening. A description of experimental protocols to measure - decoherence is given in Appendix A. Additional technical details are provided in Appendices B–E.

## Ii Two electron spins in a double quantum dot

In this section we define the starting point of our calculations. Specifically, we first identify the four relevant lowest-energy two-electron states in a DQD, then project the total Hamiltonian onto the basis that is a product of these four electronic states and the nuclear spin Hilbert space. We also discuss the semiclassical description of the nuclear spin reservoir. Starting from the general Hamiltonian, depending on the particular physical problems, we derive the effective Hamiltonian that couples the two levels that we are interested in to the nuclear spin reservoir.

### ii.1 Low-Energy Two-Electron Hamiltonian in a DQD

The system we consider in the current study consists of two electrons located in two (weakly) tunnel-coupled QDs, labeled left (L) and right (R), deep in the (1,1) charge configuration.Hu and Sarma (2000); Petta et al. (2005); Coish and Loss (2005); Taylor et al. (2007); Li et al. (2012) The total Hamiltonian for the coupled electron-nuclear spin system can be written as

(1) |

where the three terms represent the electronic, hyperfine, and nuclear part of the Hamiltonian, respectively. Below we discuss each of these terms in a two-electron DQD.

Deep in the (1,1) regime of the charge stability diagram, the four lowest-energy two-electron states, including one singlet and three triplet states , are well approximated by the Heitler-London states. Specifically, and , and . The orbital parts are symmetric and antisymmetric combinations of and states, which are the single-electron ground state orbital of the potentials for the L and R dots. States in the doubly occupied and configurations have significantly higher energies, and are not included explicitly in our consideration. The only role they play is to lower the energy of the singlet with respect to by the exchange splitting ( for typical values of magnetic fields used in experiments), when the DQD has a finite tunnel coupling.Petta et al. (2005); Koppens et al. (2005)

In the presence of a magnetic field, which has both a uniform and a gradient component, the electronic Hamiltonian in the basis is

(2) |

where and (the sign convention is such that the positive field lowers the energy of in GaAs, where the effective g-factor is negative).

The two electrons couple to the environmental nuclear spins through the contact hf interaction, which takes the general form

(3) |

where are the spin operators of the two electrons at positions , are the spin operators of nuclei at site , and is the hf constant corresponding to the species (e.g., Ga, Ga, and As in GaAs, or Si in Si) of the nucleus at site . The values of hf constants for relevant nuclei are given in Table 1. is the volume of the primitive unit cell, and using a single-electron wave function is the envelope function, the hf-interaction energy (i.e., the Knight shift) of the -th nucleus interacting with one electron is . , where

Projecting Hamiltonian (3) onto the basis (see Appendix B for details), we obtainCoish and Loss (2005); Taylor et al. (2007); Särkkä and Harju (2008)

(4) |

where

(5) | |||||

(6) |

Here, denotes the hf coupling between an electron in the L(R) orbital and a nuclear spin at . Here we have neglected the overlap between the L and R orbital wavefunctions. Keeping the finite overlaps amounts to small quantitative corrections to the matrix elements of , and neglecting them does not cause any qualitative change (for the general form of and its derivation, see Appendix B). The same holds for the corrections to two-electron hf interaction terms brought by tunneling-induced admixture of singlet to singlet state [we assume interdot bias, or detuning, to be such that charge state is the more strongly coupled doubly-charged state]. The main influence of these corrections is to diminish the hf interaction between the singlet and the nuclei as the amplitude of state, which is hf-coupled to other states, decreases. As long as this decrease is small, i.e., we are far enough from anticrossing of and singlet states, the modifications brought by this effect are not qualitative.

The diagonal terms for the two polarized triplet states in are the longitudinal Overhauser field along the external field direction . On the time scale of interest to the present study, it is quasistatic.Merkulov et al. (2002); Taylor et al. (2007) We therefore express the field operator in terms of its ensemble average and fluctuations:

(7) | |||||

(8) |

where denotes an average over the nuclear bath.

In Hamiltonian , the Overhauser field difference between the two dots, , couples and states. It can be an important control for universal manipulation of an - qubit.Foletti et al. (2009) Again we split this term into the average and the fluctuationsYang and Liu (2008):

(9) |

Here the mean field can be built up through dynamical nuclear spin polarization (DNP),Foletti et al. (2009) while the fluctuation can be reduced with respect to its “natural” high-temperature value during the DNP process.Bluhm et al. (2010b)

In experiments where the Overhauser field in the DQD is prepared by DNP through multiple sweeps across the - anticrossing,Petta et al. (2008); Foletti et al. (2009); Bluhm et al. (2010b) both finite and are established. Typically both and are of the order mT (or eV) for GaAs. In Ref. Foletti et al., 2009, reaches above mT, with reaching approximately mT. Note that an equivalent role can be played by an external magnetic field gradient (leading to finite ), which could be established using a nanomagnet located close to the DQD,Tokura et al. (2006); Pioro-Ladrière et al. (2008); Petersen et al. (2013) with reported values of mT field difference between the two dots.

The total Hamiltonian for the two electron spins and the hyperfine interaction now takes the form

(10) | |||||

Here we have combined the quasistatic mean-field Overhauser terms with the external magnetic field, with replaced by

(11) |

and replaced by

(12) |

Nuclear species | (eV) | (neV) |
---|---|---|

Ga | ||

Ga | ||

As | ||

In | ||

In | ||

Si |

The last term in is the nuclear Zeeman energy:

(13) |

where is the Zeeman splitting of the nucleus of species at site . Typically, these splittings are smaller than the electronic one by three orders of magnitude (see Table 1). Note that including the finite values of in the case of multiple isotopes (as is the case in III-V based QDs) is crucial for description of Hahn echo decay of a single electron spinCywiński et al. (2009a, b); Bluhm et al. (2010a) (or an - qubitNeder et al. (2011) at , which is equivalent to two independent single spins), while the nuclear Zeeman energies generally have much smaller influence on dephasing during the free evolution of the qubit. Below, we will show that these statements also hold for the - decoherence at finite .

### ii.2 The nuclear bath and its semiclassical description

As we have discussed at the end of the previous section, one of the key features of a coupled electron-nuclear-spin system is the smallness of the intrinsic nuclear energy scales (both the Zeeman energies and the dipolar interactions among the nuclei). Consequently, at experimentally realistic temperatures, the equilibrium nuclear density operator is proportional to unity, . When nuclear spins are dynamically polarized,Petta et al. (2008); Vink et al. (2009); Latta et al. (2009); Xu et al. (2009); Foletti et al. (2009); Bluhm et al. (2010b); Frolov et al. (2012); Petersen et al. (2013) the direction of the polarization in each dot is along the applied field () direction. The components of the nuclear spins transverse to this direction are randomized on a time scale of s due to intra-nuclear dipolar interactions,Merkulov et al. (2002); Khaetskii et al. (2002) so that for experiments in which the total data acquisition time is much longer than this time scale, the appropriate nuclear density matrix is diagonal in the basis of eigenstates of . A semiclassical description of the nuclear reservoir is thus valid for at least some situations.Neder et al. (2011) Here we discuss some of the most important characteristics of the nuclear reservoir.

The Overhauser field is defined as . The maximal value of the Overhauser field (as felt by a single electron in a given orbital) in a fully polarized nuclear bath is

(14) |

where denotes the nuclear sites, is the -th nuclear spin, denotes the nuclear species, and is the average number of nuclei of this species in the unit cell (i.e., in both III-V compounds and in Si, we have ), and are the hf couplings of nuclei of species given in Table 1. With the envelope functions normalized as , where is the volume of the Wigner-Seitz unit cell, we have then , as stated before. We also define the number of unit cells , in which the probability of finding an electron described by wavefunction has appreciable magnitude

(15) |

This definition implies that

(16) |

where the sum over is over all the Wigner-Seitz unit cells (we assume that the envelope function is practically constant within each cell).

On a time scale on which the nuclear spins can be considered static, we can replace the quantum averages over the nuclear bath by classical averages over the values of the static Overhauser field described by a Gaussian probability distribution,Merkulov et al. (2002); Khaetskii et al. (2002)

(17) |

where

(18) |

where is the transverse component of , and is the average value of the longitudinal Overhauser field (in the direction). The width of the distribution of is given by

(19) |

Under realistic experimental conditions, the nuclear spin bath is in the thermal state at the high-temperature limit, . This state is isotropic, so that the variance of the projection of any given spin is . A *narrowed* state of the bath, in which the variance is reduced from this “high temperature” value, is also often considered, and it can be created in experiments.Barthel et al. (2009); Bluhm et al. (2010b); Vink et al. (2009); Latta et al. (2009); Xu et al. (2009); Frolov et al. (2012) We account for the possibility of narrowing of the distribution of fields by introducing a narrowing factor , defined as the ratio between actual and its high temperature value given by the above expression. We thus have for

(20) |

where we have defined the “typical” energy of hf interaction . For GaAs is of the same order of magnitude as the maximal Overhauser field . For silicon with a fraction of the spinful Si nuclei, .

For typical achievable values of nuclear polarization, the width of the distribution of the transverse Overhauser field is given by an analogous formula, albeit without the factor. In other words,

(21) |

In III-V compounds, the number of nuclei interacting appreciably with an electron is (the factor of appears because there are two nuclei per unit cell), and the typical value for a GaAs QD with spins is a few mT (eV). For silicon, the result depends also on the concentration of the spin- Si nuclei, given by (with for natural silicon). With , we obtain

(22) |

where the value of is given in Table 1 (note that we are using here a different definition for compared to Ref. Assali et al., 2011, where an -dependent quantity was used).

In the following we will be mostly interested in the distribution of the *difference* of the Overhauser fields between the two dots,

(23) |

which could have a finite average and/or a narrowed distribution. Using the values of for the two dots (L and R), we introduce

(24) |

where

(25) |

and we have taken the natural (non-narrowed) values for , i.e., we have used Eq. (19) with . The factor accounts now for possibly reduced standard deviation of the difference of the Overhauser field in the two QDs. It should be noted that such a narrowing can be achieved by enforcing a correlation between the values of and , that is by modifying the joint probability distribution for the two fields, without affecting the distribution of each one of them considered separately. The state narrowing obtained in experiments on singlet-triplet qubits in DQDs is of this nature.Bluhm et al. (2010b) The standard deviation of the transverse components of the Overhauser field difference, , is, analogous to the single-electron case from Eq. (21), defined as .

### ii.3 Effective Hamiltonian in the - subspace

One focus of the present paper is the decoherence of - qubits. Here we derive the effective Hamiltonian in the basis of and states in the presence of a large external magnetic field. It is directly applicable to experiments on - qubits whenever exchange splitting is large enough.

As shown in Eq. (10), and states are coupled to the polarized triplet states by the transverse Overhauser field . In a finite external magnetic field, such that , dephasing between and states can be faithfully described by an effective Hamiltonian in the subspace of these two states, treating the coupling to the states perturbatively.Winkler (2003) With the zeroth-order Hamiltonian given by the first matrix in Eq. (10), the condition for the perturbative treatment is

(26) |

The effective Hamiltonian in the subspace is then:

(27) | |||||

which contains second-order effective interactions among the nuclei. In particular, comes from the virtual flip flops between and :

(28) | |||||

with . represents flip flops between and via virtual transitions through . It consists of a Hermitian and an anti-Hermitian part:

(29) |

where

(30) | |||||

(31) | |||||

with

Fig. 1 gives a cartoon that depicts the virtual flip-flop transitions contributing to and .

Hamiltonian in Eq. (27) contains terms that are linear in the nuclear spin operators but are second-order in the hyperfine coupling strength (within our approximation of neglecting the orbital overlap when calculating the hf interactions, ):

(32) |

They influence the - coherence in a way identical to what the longitudinal Overhauser field does to a single spin. Averaging over a thermal distribution of the Overhauser field felt by a single spin, , leads to a strong inhomogeneous broadeningMerkulov et al. (2002) and a dephasing time of , where is the spread of the values of the longitudinal Overhauser field (see Sec. II.2 for precise definition). This time is of the order of 10 ns in GaAs QDs for a single electron spin, while dephasing due to these -linear terms is strongly suppressed in the case of an - qubit, because the interaction strength is significantly reduced, i.e., the qubit-nuclei couplings in Eq. (32) are for the values of considered here.

Note that , , and are all fluctuations due to nuclear spins, and average to zero in a thermal nuclear spin reservoir. In the absence of these fluctuations, the universal control of the - qubit can be achieved via tuning of and .Foletti et al. (2009) In the context of decoherence when is finite (more precisely, when ), we have two interesting regimes to consider. The first is when , i.e., the DQD is in a uniform total field. In this limit, the DQD has left-right symmetry, so that the singlet and triplet states are system eigenstates. Since and are states, with no magnetic moment in either quantum dots, they are not directly affected by the Overhauser field . The - qubit made up from these two states should thus have a significantly longer inhomogeneous broadening time () than a single spin. We will study this regime in detail in Sec. III. The other regime is when , i.e., a magnetic field gradient is present (whether due to nuclear spin polarization or applied externally). Now the left-right symmetry of the DQD is broken, and the true eigenstates of the system are superpositions of and states. The electron spin densities in the two dots do not vanish anymore, so that can affect the two-spin coherence directly, and the system acquires single-spin qubit characteristics. This regime will be studied in Sec. IV.

An extreme case is when , which is a regime already investigated in existing experimental Petta et al. (2005); Koppens et al. (2005); Johnson et al. (2005); Bluhm et al. (2010a); Maune et al. (2012) and theoretical Witzel and Das Sarma (2008); Neder et al. (2011) studies. Here, the exchange coupling is effectively turned off. The dynamics of the two independent electron spins are determined by the Overhauser fields in the respective dots. The interdot nuclear spin flip flops are completely suppressed, so that all interactions involving such flip flops, and , vanish in this limit. In addition, now so that they do not affect the two-spin dynamics. The remaining hf-mediated interaction is due to intradot flip flops:

(33) |

When , the two-spin eigenstates are spin product states. Within the subspace, it is more convenient to consider the Hamiltonian in the basis of states. The resulting Hamiltonian is of pure dephasing form in this product basis:

(34) |

In a free evolution experiment with an initial singlet state, the ensemble coherence will now decay in due to the term.Petta et al. (2005); Bluhm et al. (2010b) On the other hand, in a Hahn echo experiment,Petta et al. (2005); Bluhm et al. (2010a) the influence of is removed, and the signal decay is due to from Eq. (33). Since this interaction is a sum of two commuting terms from two uncoupled dots, the appropriately defined - decoherence function is a product of the two single-dot decoherence functions.Bluhm et al. (2010a); Neder et al. (2011) This observation establishes a one-to-one correspondence between single-spin Hahn echo decayKoppens et al. (2008) due to hf-mediated interactions considered theoretically in Refs. Yao et al., 2006; Cywiński et al., 2009a, b; Cywiński et al., 2010, and the singlet-triplet Hahn echo decay.Petta et al. (2005); Bluhm et al. (2010a)

## Iii Calculations of - dephasing in the absence of interdot effective field gradient

In this section, we study two-spin decoherence within the - subspace in the absence of interdot magnetic field gradient, i.e., (practically, this is true as long as ). With (note that the magnitudes of and are negligible compared to , as shown in Appendix C), we can perform a second canonical transformation to diagonalize Hamiltonian (27), treating the off-diagonal terms as a perturbation. We now obtain the final form of the effective Hamiltonian in a uniform effective magnetic field,

(35) | |||||

where

(36) | |||||

(37) |

To evaluate , we use a type approximation where we neglect the commutators of nuclear operators. The commutator in the above equation thus vanishes, so that

(38) |

where

With , and are two-spin eigenstates. The dynamics of the - coherence is quantified by the following decoherence function

(39) |

Here is the initial nuclear density operator, while and are the operators appearing on the diagonal in the effective Hamiltonian from Eq. (35). In Appendix A we discuss how the above quantity can be measured in electrically controlled - qubits. Some of the information contained in function may also be indirectly inferred from the optical spectra of two-electron states in coupled self-assembled InGaAs quantum dots.Weiss et al. (2012)

Our main goal is to quantify each of the dephasing processes as we vary and . There are three kinds of hf-related terms appearing in : (1) and , the -linear terms, (2) , the square of the longitudinal Overhauser field, and (3) and , which are second order in transverse Overhauser field. Below we discuss their individual contributions to - dephasing dynamics. Such a treatment makes sense when the time scales on which various terms operate are very different. If the importance of these interactions is comparable, treating various interactions as independent would introduce a quantitative error when the commutator of the two competing terms is non-negligible. In the following, we will present results for the envelope , in which we have removed the fast oscillating part of from Eq. (39):

(40) |

### iii.1 Dephasing due to and

While and leads to dephasing that is completely analogous to the hf-induced inhomogeneous broadening of a single spin free evolution, the magnitude of the dephasing here is strongly suppressed because of the reduced coupling between the nuclei and the qubit. From Eq. (39), keeping only and in the Hamiltonian, we obtain

(41) |

Neglecting the wavefunction overlap, , with

(42) |

For a large number of nuclei (), as discussed in Sec. II.2, tracing over the nuclear spin density matrix in Eq. (41) can be approximated by averaging over a classical Gaussian distribution of energy splittings with variance

(43) | |||||

where the variance for the -th spin, , is approximated by its value of at vanishing nuclear polarization (this is a good approximation at small , since corrections are of the order ). The value of depends on the distribution of the nuclear couplings, i.e., the shape of electron wavefunction, but it can be roughly estimated as , as shown in Appendix D. The resulting coherence decay is then given by

(44) |

with

(45) |

where and are the numbers of unit cells in the L and R dots (as defined in Eq. (15)). The typical values of this time for GaAs and Si are shown in Fig. 2.

The very long shown in Fig. 2 is a clear illustration of the strongly suppressed electron-nuclear spin interaction, which is a consequence of the highly symmetric nature of the singlet and unpolarized triplet states. In the following we generally neglect the contributions from and . Other consequences of the reduced effective Knight shifts experienced by the nuclei will be discussed at the end of Sec. III.3.

### iii.2 Dephasing due to the second-order longitudinal Overhauser field

-induced dephasing can be calculated classically because of the quasistatic nature of the longitudinal Overhauser field. We can rewrite in terms of the classical Overhauser field , where ,

(46) |

As in Sec. II.2, we treat as a Gaussian random variable, and obtain the relevant decoherence function by evaluating the Gaussian integral:

(47) |

where

(48) |

with a distribution width , where is defined in Eq. (24). We then obtain (see also Ref. Yang and Liu, 2008)

(49) |

where we have defined is defined by , giving us . The characteristic decay time scale