Decoherence of spin qubits due to a nearby charge fluctuator in gate-defined double dots
The effects of a nearby two-level charge fluctuator on a double-dot two-spin qubit are studied theoretically. Assuming no direct tunneling between the charge fluctuator and the qubit quantum dots, the Coulomb couplings between the qubit orbital states and the fluctuator are calculated within the Hund-Mulliken framework to quadrupole-quadrupole order in a multipole expansion. We identify and quantify the coupling term that entangles the qubit to the fluctuator and analyze qubit decoherence effects that result from the decay of the fluctuator to its reservoir. Our results show that the charge environment can severely impact the performance of spin qubits, and indicate working points at which this decoherence channel is minimized. Our analysis also suggests that an ancillary double-dot can provide a convenient point for single-qubit operations and idle position, adding flexibility in the quantum control of the two-spin qubit.
pacs:03.67.Lx, 73.21.La, 85.35.Gv, 85.75.-d
The wide-spread interest in quantum information processing in recent years has been a critical driving force in the research of electron spins localized in semiconductor quantum dots (QDs).HanRMP While these two-level systems are attractive candidates for implementation of scalable systems due to their compatibility with conventional microelectronic technology, their quantum control at the single- and few-qubit level remains a significant experimental challenge. Furthermore, as all solid state systems, they are inherently less isolated from their environment as compared with atomic systems.
An attractive platform to study quantum control and the related problem of decoherence is the system of gate-defined lateral QDs, in which several of the major breakthroughs in spin qubit technology have emerged in recent years. While isolating a single electron in a QD was achieved only in 2000,Ciorga rapid progress has been made since then. Long singlet-triplet relaxation times of the order of milliseconds were measured for a single dot,Hanson and a lower bound on the spin coherence time (dominated by pure dephasing) exceeding was established, using spin-echo techniques in a double dot system.Petta The single-spin relaxation and decoherence time scales have since been pushed to the order of 1 s Amasha and 0.1 msYacobyPC respectively.
The relative isolation of QD electron spins, which is indicated by these long coherence times, renders their manipulation and readout particularly challenging. This is accomplished by using Pauli spin blockade to convert spin to charge information so that fast measurement of spin states becomes possible.Ono ; Elzerman ; DiCarlo In addition, coherent exchange of two-electron spins in a double dot system,Petta and driven Rabi oscillations of single electron spins using oscillating magneticKoppens and electricNowack fields have been demonstrated as well.
Electron spin relaxation via spin-orbit interaction was shown to be an insignificant decoherence channel,Golovach and it has been generally accepted that the nuclear spins in the surrounding host material are the main source for the electron spin decoherence in III-V host materials such as GaAs and InAs.Merkulov This has led to intensive experimentalPetta ; Johnson ; Koppens1 ; Bayer ; Tarucha ; Steel and theoreticalCoish ; Klauser ; Yao ; Liu ; Witzel ; Witzel2 ; Deng ; Lukin studies of the nuclear environment, and various proposals for alleviating its adverse effects on the electron spin qubit, among which dynamical nuclear spin polarization was suggestedBurLos ; HuSSC ; Ramon and demonstrated.PettaPol ; Reilly ; Foletti
In contrast, the effects of the charge environment on QD spin qubits have only recently started to receive some theoretical attention.Coish ; Hu ; Culcer Charge noise in lateral gated devices can originate from various sources. Suggested mechanisms include gate leakage currents via localized states, charge traps near the quantum point contacts (QPCs), donor centers near the gate surface, Johnson noise from the gate electrodes, and switching events in the doping layer, typically located at an interface 100 nm below the surface.Petta ; Pioro Measurement of the background charge fluctuation in GaAs quantum dots has shown a linear temperature dependence characteristic of noise.Jung Random telegraph noise in GaAs lateral gated structures was measured and characterized by Pioro-Ladrière et al..Pioro This noise was attributed to electrons that tunnel from the gate and are trapped near the QPC, causing fluctuations in the conductance with typical frequency of 1 Hz. Applying a positive gate bias during the device cooldown significantly reduces the noise by reducing the density of ionized donors near the surface, thereby suppressing the electron tunneling.Pioro Furthermore, background charge fluctuations were suggested as a possible source for the bistable behavior observed in the coupled electron-nuclear spin system,Koppens1 and telegraph noise induced by the QPCs was also measured recently in double and triple coupled QDs.Taubert
Generally, single-spin qubits in solids rely on the exchange interaction to perform fast two-qubit operations. Furthermore, a number of recent works have utilized two-spin singlet and unpolarized triplet states in biased configuration to encode a logical qubit, which offer better control as compared with single spin states.Taylor However, such exchange-coupled spin qubits are vulnerable to dephasing induced by charge noise, since exchange coupling is electrostatic in nature, and singlet and triplet states generally have different charge distributions.Hu In the same spirit, the effects of charge noise on the coherence of spin qubits in Silicon double dots were studied very recently.Culcer In addition, the effects of a single, randomly positioned, charge impurity on a three-spin encoded qubit in a triple QD were studied by calculating the impurity-induced changes in the qubit orbital levels.Puerto Finally, electron-phonon interaction can also lead to dephasing in an exchange coupled double quantum dot because two-spin singlet and triplet states have different charge distributions.HuArx ; Roszak
There are many types of charge impurities and defects that can generate electrical fluctuations that affect spin qubits in solid states. In this paper we carry out a microscopic calculation focused on the Coulomb coupling between a biased two-spin qubit and a nearby trapped charge fluctuator represented by a two-centered two-level-system (TLS), utilizing a multipole expansion up to and including the quadrupole-quadrupole order. One scenario for such a two-center defect may be for an electron to be trapped around two donor nuclei that have potential wells somewhat lower than other donor nuclei nearby, so that this electron would oscillate between the sites until the charge motion is relaxed by the background charge fluctuations or phonon emissions. Using a master equation formalism we use the calculated qubit-TLS couplings to study the dynamics of the open system that is formed by the spontaneous emission of the TLS coupled to a reservoir. Thus we obtain quantitative estimates of the decoherence and dephasing effects on the spin qubit during various gate operations, and when idle. This analysis enables us to determine optimal working points at which the qubit’s sensitivity to charge fluctuations is reduced.
It is important to note that this work is only an initial step in the quantitative analysis of the effects of charge fluctuations on spin qubits. The focus here is on a quantitative evaluation of the qubit-TLS entangling term. We are particularly interested in clarifying how the TLS-qubit coupling could lead to qubit decoherence due to the background charge fluctuations, with the TLS acting as an intermediary between the spin qubit and the charge environment. The TLS coupling to the environment is dealt at a rudimentary level, serving only to demonstrate the applicability of the presented theory in estimating charge-induced spin decoherence. Building on the results given in this paper, the next step should benefit from the extensive work that has been carried out in recent years on charge-environment-induced decoherence in superconducting qubits.Astafiev ; Martinis ; Shnirman ; Paladino ; Bergli ; Paladino1 ; Faoro
The paper is organized as follows. In section II, we derive the Coulomb coupling between the qubit and the TLS, using the Hund-Mulliken approach to calculate the qubit orbital states and a multipole expansion for the Coulomb interaction. In section III we use these results to study the decoherence effects due to charge fluctuations mediated by the qubit-TLS coupling. After deriving the master equations for the system density matrix in section III.1, we present and analyze in sections III.2-III.3 the resulting dynamics during various single-qubit operations for singlet-triplet qubits. In section III.4 we discuss a convenient working point at which the effective exchange energy is zero and quantify the dephasing time. A summary of our results and a brief discussion on possible extensions of this work are given in section IV. In Appendix A we calculate the system concurrence, showing the conditions for qubit-TLS entanglement. In appendix B we provide details of the qubit’s orbital Hamiltonian. Appendix C lists the full expressions for the qubit-TLS coupling terms, and finally appendix D presents an analytical solution to the master equation for the case of no TLS tunneling.
Ii qubit-TLS Coulomb coupling
We consider the Coulomb interaction between a qubit formed from the singlet and unpolarized triplet spin states of two electrons in a double dot and a nearby two level system (TLS), assuming no qubit-TLS tunnel coupling. To properly describe a biased double quantum dot, we use the Hund-Mulliken model to calculate the qubit orbital states. With Coulomb interaction being spin-independent, the interaction Hamiltonian can generally be written in the form:delta
where we have
Here are the Coulomb matrix elements, where the left superscript denotes the qubit state (Singlet/Triplet) and the right one denotes the TLS state (Left/Right). While the coupling should not directly affect the qubit spin state, the coupling effectively renormalizes the qubit exchange energy. The coupling acts to entangle the qubit and the TLS and therefore leads to qubit spin decoherence when the TLS is coupled to a larger reservoir representing the background charge fluctuations. Appendix A formalizes this last statement, showing that the concurrenceHilWoo of qubit-TLS system under the time evolution of is nonzero only for a nonzero coupling.
The TLS’s we consider are sufficiently removed from the spin qubit so that there is no exchange coupling between the spins and the single electron in the TLS. With no electrons tunneling between the qubit and the TLS, the two charge distributions are separated in space, and the Coulomb interaction between them can be described systematically using a multipole expansion approach In the following we use this model to evaluate the Coulomb coupling terms .
ii.1 Hund-Mulliken approach for the qubit orbital Hamiltonian
The sensitivity of an exchange-coupled spin qubit to a remote charge fluctuator comes from the different charge distributions the singlet and triplet states have.Coish Thus we first construct the two-electron orbital states by extending the Hund-Mulliken approachBurLos to a biased dot configuration.
We start by approximating the orbitals for the two quantum dots by those of two harmonic wells centered at
where . Here is the harmonic oscillator frequency, and the magnetic compression factor, , is given by with the Larmor frequency . The phase factor in Eq. (3) involving the magnetic length results from a gauge transformation, and is the orbital shift due to the electric field. The direction wavefunction is taken as the ground state (with associated energy ) of a finite potential well of width
with , , .
To simplify the Hund-Mulliken calculation, the single particle single-dot states are orthonormalized, , where , is the wavefunction overlap (, with the Bohr radius associated with the harmonic QD confinement potential), and . The orthonormalized orbitals are then used to construct 4 two-particle states: the two doubly occupied singlets, , , the separated singlet state, , and the separated triplet state, . (we neglect the doubly occupied triplet states as their energy is typically much higher for the gate-defined structure we have in mind.Johnson ; Hanson ; Koppens ) In the basis of these two-particle states the orbital Hamiltonian is given as
where the diagonal elements include the Coulomb interactions, and and are the single- and double-hopping matrix elements, respectively. Calculational details of the orbital Hamiltonian in a biased configuration are given in appendix B. Figure 1 shows the energy diagram near the to charge transition, which results from diagonalization of Eq. (5), where we also included the polarized triplet states splitted by the Zeeman interaction, , with and the Bohr magneton. For this figure and throughout this work we have considered mT (), dot confinement ( nm), and half interdot distance (in units) , corresponding to the experimental parameters in typical gate-defined double dot systems.Petta ; Koppens The bias shift in the figure is normalized to the Bohr radius:
and it is proportional to the interdot bias gate potential.
Since the orbital Hamiltonian, Eq. (5), does not connect the triplet state with any of the singlet states, the combined two-particle orbital-spin triplet state can be written as:
The diagonalization of the singlet block of yields a hybridized singlet state that is predominantly the separated singlet, at negative or zero bias.bias In the basis of the three singlet states the lowest-energy orbital-spin singlet state is
where , and are the components of the lowest lying singlet eigenstate of the orbital Hamiltonian Eq. (5). The exchange energy is defined as the difference between the triplet and this singlet state (see inset of Figure 1).
The two-electron states can be expressed in terms of the single particle orthonormal states , , which are more convenient when calculating the qubit-TLS couplings. The two-particle orbital-spin states are given by the slater determinants:
where the number index on the left-hand-side denotes the first or second electron. The triplet and hybridized singlet states can be built as
ii.2 Multipole expansion for the qubit-TLS interaction
We model the TLS as a single electron moving in a double well, each of which has a wave function similar to those of the qubit orbitals, Eq. (3)
where is the Bohr radius of the (identical) TLS centers, is half the distance between them, and is the TLS ground state wavefunction similar to that of the qubit, Eq. (4), with potential , and width .spherical
The most general Coulomb interaction operator between the qubit and the TLS is given by
are the electron charge density operators for the qubit () and TLS (), where denote the qubit orbital state (symmetric or antisymmetric combination), and denote the TLS state. We use the dielectric constant for GaAs, , and consider only static dielectric constant for screening, since we assume the space near the double dot is completely depleted (i.e. no nearby 2DEG). In addition, we take the TLS inter-site distance to be sufficiently large so as to have a relatively small tunnel coupling, limiting our study to slow TLS’s. We can therefore neglect contributions to the qubit-TLS coupling coming from off-diagonal TLS charge densities.TLStun To reduce clutter we thus write .
The Coulomb matrix elements of interest are , and , where are the triplet and singlet states given in Eqs. (8)-(9). Assuming the creation operators for the electrons in the QDs commute with those in the TLS (i.e., no tunneling between the qubit and the TLS) we find
We calculate the Coulomb interaction terms by evaluating the electrostatic energy associated with placing the TLS charge distribution in the potential , that is due to the qubit charge distribution, expanding the latter in spherical harmonicsJackson
where is the dielectric constant, and are the charge, dipole, and quadrupole electric moments, respectively, associated with the qubit charge distribution. Combining this with the Taylor expansion for the potential
and using to denote the charge, dipole, and quadrupole electric moments, respectively, of the TLS charge distribution, we obtain up to and including quadrupole-quadrupole order:
In Eq. (16) is the vector connecting the qubit and TLS centers, and the two dots lie along the -axis. The centers of the TLS are aligned along the axis , so the angular dependence of the qubit-TLS interaction is specified by the four angles . The system geometry is depicted in Figure 2.
The electrical monopole (charge) for both the qubit and TLS is just .monopole The qubit dipole moments are, by construction, in the direction and are found to be
where the upper (lower) sign corresponds to (). For the mixed qubit orbital Coulomb matrix elements we only need to consider the sum (see Eq. (II.2) for the off-diagonal matrix element). In the case of the qubit dipole moments this sum is
Notice that for unbiased dots () the diagonal qubit dipole moments, vanish, since in this case the double dot is symmetric () thus the charge distribution has mirror symmetry around the plane and the dipole moment is identically zero. The introduction of bias allows one of the double occupied states to mix more strongly into the ground singlet state so that a finite dipole moment emerges.
Using the TLS wavefunctions Eq. (10) we find the TLS dipole moment as
where the plus (minus) sign corresponds to (). The quadrupole moments are found to have only diagonal elements. For the qubit charge distribution they are
In Eqs. (20) the upper (lower) sign refers to (), and
Similarly, the quadrupole moments of the TLS charge distribution are
where is the square extension of the z-direction TLS wavefunction, similar to Eq. (22). Note that the TLS quadrupole moment matrix elements are given in the rotated frame and have the same values for .
Using the above results we obtain the qubit-TLS coupling terms, Eqs. (2) to quadrupole-quadrupole order where the nonvanishing contributions are
The first (second) subscript in each term denotes contribution from the particular multipole moment: monopole (q), dipole (d), and quadrupole (Q) of the qubit (TLS) charge distribution. The explicit expressions for the various coupling terms are rather lengthy and are deferred to appendix C. Note that the dipole-charge, , dipole-dipole, , , and dipole-quadrupole, contributions are nonzero only for a biased dot configuration when the qubit dipole moment is nonzero.
ii.3 Qubit-TLS coupling terms
In order to present graphically the qubit-TLS Coulomb interaction terms, Eqs. (24)-(26), we consider a generic system geometry (i.e. no a-priori knowledge of the relative orientations of the two subsystems). Since giving statistics of various parameters does not shed clear light on the interaction, we give a representative value of the qubit-TLS interaction by averaging over all possible values of qubit-TLS orientation parameters . To obtain the correct total values of , this angular averaging should be performed after the addition of the individual contributions from the multipole expansion (i.e., the terms given in Eqs. (C-1)-(C-9)). Figure 3 shows the angle-averaged values of the coupling terms vs qubit-TLS distance at the bias point corresponding to the singlet anticrossing (see Figure 1). In addition to the double-dot parameters given in Fig. 1, we use here and throughout the paper, unless specified otherwise, QD thickness nm, and vertical confinement potential meV. For the TLS, we take nm, meV, TLS center Bohr radius nm, and half TLS centers distance nm. The latter are chosen to characterize -doped dopants in the insulator with a typical small radius and a fairly large inter-center distance.
The figure demonstrates the convergence of the multipole expansion for each of the three couplings, as increases. For the above set of parameters we expect higher order contributions in the multipole expansion to be insignificant for qubit-TLS distances exceeding 100 nm (for ), 40 nm (for ), and 30 nm (for ). As explained below, it is the coupling that is responsible for the spin qubit decoherence effects, thus we expect our results to be accurate down to nm.
Figure 4 shows the Coulomb couplings and the qubit exchange energy as functions of the QD bias shift for qubit-TLS distances of nm. While the bias dependence of the coupling is minimal, both and strongly depend on the qubit bias, suggesting that the qubit is substantially more susceptible to decoherence due to charge fluctuations at and above the anticrossing point where the component increases significantly in the ground singlet state and the qubit charge distribution acquires a strong dipole component. It is seen that for our parameter choice, becomes comparable to the exchange energy at nm, below which we anticipate sizable spin decoherence effects due to charge coupling. Inspection of the leading terms in the and couplings (Eqs. (C-4)-(C-9)) shows that only the latter scales with the TLS centers distance , thus the ratio decreases with the charge fluctuator size, leading to increased qubit-TLS distances at which the qubit is susceptible to decoherence. We note that the qubit-TLS distance at which the Coulomb terms become appreciable roughly scales linearly with the size of the dots, which is consistent with the basic characteristics of a multipole expansion.
The angular averaging procedure that was used to produce figures 3 and 4 was tested by randomly taking values for and using these random 4-vectors to calculate the interaction terms. This calculational mode is useful in later evaluation of decoherence effects such as gate errors. We then calculate the error (or any other decoherence effect) for many randomly selected qubit-TLS geometries and average at the end of the calculation. The results shown in Figs. 3 and 4 were reproduced to within an error by averaging over 10,000 random runs.
Iii Effects of charge fluctuations on a double-dot spin qubit
In this section we examine the effects of the qubit-TLS coupling on the performance of a double-dot two-spin qubit. To do so we employ a master equation formalism to study the dynamics of the coupled qubit-TLS system due to the spontaneous emission of the TLS. Specifically we study the effects of the charge coupling on dephasing of the spin qubit and on the fidelity of specific single-qubit operations.
iii.1 Master equation for the qubit-TLS system
We consider the master equation describing the qubit-TLS system, with the TLS coupled to a reservoir that results in its spontaneous emission
Here is the density matrix of the qubit-TLS system, are the Lindblad operators, and the qubit-TLS Hamiltonian is
where is given in Eq. (1), and () superscript denotes an operator on the qubit (TLS) subsystem. In Eq. (28) , with the magnetic field inhomogeneity between the dots arising from either application of an inhomogeneous , different factors in the two dots, or inhomogeneous nuclear polarizations. where is the TLS level splitting and is the tunnel coupling between the two centers. The latter is a function of the TLS Bohr radius and center separation , found using Eq. (10) to be
We assume coupling of the TLS to a cold bath in the vacuum state through spontaneous emission, described by a single Lindblad operator , where is the spontaneous emission rate. Measurements of the relaxation time of charge qubits in lateral GaAs double dots yielded ns,PettaPRL corresponding to .
Transforming Eq. (27) to the interaction picture, with , we find
Here we have defined , and , where , and the upper (lower) sign corresponds to ().
Inserting into Eq. (29) we find three separable sets of differential equations for the matrix elements of . The first set is
The equations for take the same form as those above, with , . The third set of equations reads: