High Fidelity Single-Shot Singlet-Triplet Readout of Precision Placed Donors in Silicon
M. A. Broome T. F. Watson D. Keith S. K. Gorman M. G. House J. G. Keizer S. J. Hile W. Baker M. Y. Simmons





Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia




In this work we perform direct single-shot readout of the singlet-triplet states in exchange coupled electrons confined to precision placed donor atoms in silicon. Our method takes advantage of the large energy splitting given by the Pauli-spin blockaded (2,0) triplet states, from which we can achieve a single-shot readout fidelity of 98.4±0.2%. We measure the triplet-minus relaxation time to be of the order 3 s at 2.5 T and observe its predicted decrease as a function of magnetic field, reaching 0.5 s at 1 T.



High Fidelity Single-Shot Singlet-Triplet Readout of Precision Placed Donors in Silicon
    M. A. Broome, T. F. Watson, D. Keith, S. K. Gorman, M. G. House, J. G. Keizer, S. J. Hile, W. Baker, M. Y. Simmons
    December 30, 2023
======================================================================================================================



An increased ability to control and manipulate quantum systems is driving the field of quantum computation forward <cit.>. The spin of a single electron in the solid-state has long been utilised in this context <cit.>, providing a superbly clean quantum system with two orthogonal quantum states that can be measured with over 99% fidelity <cit.>. As a natural next step, the coupling of two electrons at separate sites has been studied in gate-defined quantum dots <cit.>, as well as in donor systems <cit.>. In addition to being the eigenstates for two coupled spins, the singlet-triplet (ST) states of two electrons can form a qubit subspace, and have previously been utilised for quantum information processing <cit.>. Unlike in gate-defined quantum dots, donor systems do not require electrodes to confine electrons. The resulting decrease in physical complexity makes donor nano-devices very appealing for scaling up to many electron sites <cit.>.



In the (1,1) charge configuration the ST states are eigenstates if the exchange coupling is greater than any difference in Zeeman energy between the two spins. The singlet and three triplet states are split only by the Zeeman energy in the cases of |T^+⟩=|↑↑⟩ and |T^-⟩=|↓↓⟩, and an exchange energy, J, for the singlet |S⟩=(|↑↓⟩-|↓↑⟩)/√(2) and |T^0⟩=(|↑↓⟩+|↓↑⟩)/√(2) states. However, in the (2,0) configuration all triplet states split from the singlet |S(2,0)⟩ by a larger exchange interaction, Δ_ST, measured in previous works to be >5meV for donors <cit.>. The triplet states are therefore blocked from tunnelling from the (1,1)→(2,0) charge configuration, known as Pauli spin-blockade.

Typically, direct ST readout is performed by charge discrimination between the (1,1) and (2,0) states below the ST energy splitting Δ_ST. However, this relies on the charge sensor having a large enough differential capacitive coupling to each dot to discriminate between the two charge states. This is not possible in some architectures due to symmetry constraints, in particular for donors it is advantageous for multiple donor sites to be coupled equally to a charge sensor for independent readout and/or loading. The tightly confined electron wavefunction at each donor site therefore necessitates that they are equidistant from the charge sensor. As a consequence (1,1)↔(2,0) charge transfer signals are often too small to detect directly in this architecture.

Until now single-shot readout of ST states in donors has been limited to strongly coupled systems where the ST states comprise both the ground and excited valley-orbit states <cit.>. Furthermore, this method <cit.> has limited fidelity as it relies on spin dependent tunnel rates that cannot be independently controlled. Here we utilise an alternative technique to perform single-shot readout of ST states across two coupled donor sites in a regime suitable for quantum computing applications. Importantly, there is no need for any capacitive difference between the charge sensor and the two donor sites, as our method does not utilise a direct (1,1)-(2,0) charge transfer signal. Instead, we utilise an energy selective readout technique relying on relaxation of the metastable triplet state in the (1,1) configuration when pulsed into the (2,0) region. The method has been previously demonstrated in a time-averaged fashion <cit.>, however we employ threshold discrimination analysis (c.f. single-spin readout <cit.>) for single-shot readout with fidelity greater than 98%—close to fault tolerant thresholds for surface-code quantum computation <cit.>.

The device shown in Fig. <ref> was fabricated using scanning tunnelling microscope (STM) hydrogen lithography. The patterned donor sites, L, and R consist of 2 and 1 phosphorus atoms respectively, determined by examining the size of the lithographic patches <cit.> and their charging energies (see Supplementary Material). Gates, {G_L,G_M,G_R} control electron numbers at L and R, whereas G_SET is predominantly coupled to the single-electron transistor (SET) charge sensor. The SET is composed of approximately 1000 phosphorus atoms and is 19±1 nm from L and R, allowing for electron loading and unloading, see Fig. <ref>b. The SET is operated with a 2.5 mV source-drain bias and has a charging energy of ∼5 meV. Further details of the fabrication methods have been published previously <cit.>. All data herein was taken inside a dilution refrigerator at 100 mK (electron temperature ∼200 mK).

Figure <ref>c shows the SET current as a function of V_GL and V_GR near the (1,1)-(2,0) charge transition. No change in current is observed across the interdot transition (red line), and therefore ST readout cannot be performed here. Instead, we utilise two alternative tunnelling routes to the S(2,0) ground state near the (2,0)-(2,1) charge transition shown schematically in Fig. <ref>a. Importantly, by monitoring the SET current in real time we can distinguish the two different tunnelling routes. At the readout position (white circle marker in Fig. <ref>c) the SET current is high when electrons are in the (2,0) and low in the (2,1) charge state. When initialising here in a S(1,1) state, an electron on R can tunnel directly to L forming S(2,0), because the SET is not sensitive to inter-donor transitions no charge transfer signal is observed (see left of Fig. <ref>a). However, when initialising in any T(1,1) state at the readout position, tunnelling to T(2,0) is prohibited due to Pauli spin blockade <cit.>. Now the S(2,0) ground state is reached via an electron first tunnelling onto L to form (2,1) (singlet state on left donor site, see arrow 1 in Fig. <ref>a) followed by an electron tunnelling from R to the SET, forming the S(2,0), shown by arrow 2 in Fig. <ref>a. This process results in a `dip' in SET current <cit.> which is used as the readout signal, see Fig. <ref>b.



This charge transfer signal—given by movement of an electron to and from the SET—is as significant as for single-spin readout <cit.>. Furthermore, the large ST energy splitting Δ_ST for donors—much larger than in gate-defined quantum dots—is reproducible as it is not influenced by surrounding electrostatic gates <cit.>. The timescales for readout are dependent on the electron tunnel-on time from the SET to L, τ_(1,1)→(2,1) (arrow 1 in Fig. <ref>a), and tunnel-off time from R to the SET τ_(2,1)→(2,0) (arrow 2 in Fig. <ref>a). These were determined by analysing 100,000 readout traces to be τ_(1,1)→(2,1)=1.15±0.03 ms and τ_(2,1)→(2,0)=5.3±0.2 ms. 

Following from previous works on single-shot spin readout <cit.>, the assignment of singlet or triplet state to each readout trace comprises two separate parts, (i) electrical readout and (ii) state-to-charge conversion (STC), which we discuss in detail below.

(i) Electrical readout.—Here we determine whether a given SET current trace can be assigned as having a dip during the readout phase (time spent at the readout position), or not. In the experiment the SET current passes through a room temperature current amplifier, hence the resulting voltage is relevant (see Fig. <ref>b). During the readout phase a trace is assigned as having a voltage dip if its minimum value, V_m≤V_t. A Monte Carlo simulation of 10,000 readout traces with added white Gaussian noise equivalent to the experimental signal-to-noise ratio is shown in Fig. <ref>c <cit.>. This histogram shows the simulated minimum voltages V_m from which we deduce the fidelity of assigning either a dip (triplet) or no dip (singlet), F_T or F_S respectively, to each trace using the equations,

    F_T    = 1-∫_V_t^∞N_T(V_m)dV_m 
    
    F_S    = 1-∫_-∞^V_tN_S(V_m)dV_m,

where V_m is the minimum voltage and N_i is the fraction of each state i. The results are shown in Fig. <ref>d along with the calculated electrical readout visibility 𝒱_ER=F_T+F_S-1. From Eqs. 1 and 2 we specify the optimum voltage threshold, V_t, where 𝒱_ER is maximised. In total, 500 independent simulations were run (each 10,000 simulated traces) allowing the assignment of errors shown in Table <ref>.

(ii) State-to-charge conversion.—Next we determine the optimum readout time, Δ t, following the work on single-shot spin readout in <cit.> and <cit.>. The rate equation model described therein accounts for errors caused by, relaxation of excited states; triplet states failing to cause a tunnelling event before Δ t; and a singlet state causing a tunnelling event within Δ t. As inputs to the model, the tunnelling out time of the triplet state from (1,1)→(2,1), τ_T,out is assigned the same value as τ_(1,1)→(2,1), as shown in Fig. <ref>e. The tunnel time, τ_S,out, is also found experimentally by counting the number of tunnelling events occurring after a time much greater than τ_T,out (here we used 8 ms) and attributing them to the exponential decay of the singlet state. Only 0.033% of the 100,000 readout traces showed tunnelling after this time, giving an estimate of τ_S,out=16600±8300s. Using these characteristic tunnelling times, we implement a rate equation model to determine the optimum readout time <cit.>, Δ t, based on the probability of successfully assigning a voltage dip a triplet or singlet state, α and β respectively (see Supplementary Material). 

The resulting assignment probabilities α and β are shown as a function of the readout time in Fig. <ref>f. Similar to electrical readout, the visibility of state-to-charge conversion is calculated as 𝒱_STC=α+β-1, and the optimum readout time is chosen where 𝒱_STC is maximised and was found to be Δ t=18.4±0.7 ms.





Table <ref> gives a summary of the fidelity calculations, where the final measurement fidelity is given by, F_M=(α F_T+β F_S)/2=98.4±0.2%. Owing to the large energy separation between the S(2,0) and T(2,0) states, thermal broadening of the Fermi-distribution in the SET (T_e=200mK) has a negligible effect on the readout fidelity. As such, the 𝒱_STC is reported very close to unity. 

For this device the ST readout fidelity was limited by low electrical visibility 𝒱_ER, itself restricted by a relatively low SET peak current of 30 pA. Here we have a signal-to-noise ratio of 6.4, but with an increase of SET signal equivalent to SNR=8, we estimate achieving fidelities >99%. Single-shot charge detection of this quality has previously been demonstrated in donor systems using DC biased SETs <cit.> and with rf-reflectometry, which significantly improves SNR further still <cit.>. Nonetheless, high fidelity ST readout can be maintained over a large range of magnetic fields—as demonstrated in Fig. <ref>d—because Δ_ST is independent of B_z.

Singlet-triplet dynamics.—As a demonstration of this readout technique, here we map out the |S(2,0)⟩↔|T^-⟩ anti-crossing as a function of B_z. Figure <ref>a shows the two electron eigenspectrum with the addition of the |S(2,0)⟩ state as a function of detuning, ϵ (see arrow in Fig. <ref>c). To observe ST mixing we initialise deterministically in |S(2,0)⟩ by performing ST readout. Next we apply a 50 ms pulse along the detuning axis toward the |S(2,0)⟩↔|T^-⟩ anti-crossing, allowing sufficient time for mixing (see Fig. <ref>b). Two of the triplet states {|T^+⟩, |T^-⟩} are split by the Zeeman energy from the |S(1,1)⟩ and |T^0⟩ states, such that the position mixing between |S(2,0)⟩↔{|T^-⟩,|T^+⟩} changes with B_z. The two-level pulse scheme is shown schematically by the blue arrows in Fig. <ref>c along with the ST mixing positions shown by the red dashed lines (not to scale). Finally, we pulse back to the ST readout position (circle marker) for 25 ms where we measure the triplet state probability, the results are shown in Fig. <ref>d. 

In addition to a clear |S(2,0)⟩↔|T^-⟩ mixing point for ϵ>0, indicated by the high triplet probability in Fig. <ref>d (yellow), a faint feature related to |S(2,0)⟩↔|T^+⟩ mixing can also be seen at detuning values ϵ<0. Mixing between |S(2,0)⟩↔|T^+⟩ is suppressed due to fast charge relaxation from |S(2,0)⟩→|S(1,1)⟩ in this region. The position of |S(2,0)⟩↔|T^-⟩ mixing  in ϵ remains linear as a function of B_z, indicating a small value of tunnel coupling (t_c in Fig. <ref>a), and hence no `spin-funnel' shape is seen as reported in similar experiments <cit.>.

Finally, using the |S(2,0)⟩↔|T^-⟩ mixing point to randomly load the |T^-⟩ state, we measure its T_1 lifetime using the three-level pulsing protocol shown by the green arrows in Fig. <ref>c. Relaxation of the |T^-⟩ state occurs whilst inside the charge region enclosed by the dashed lines in Fig. <ref>c, i.e. where only inter donor-site tunnelling is allowed. This position, indicated by the star marker in Fig. <ref>c, lies at ϵ=10mV, ensuring that |S(2,0)⟩ remains the ground state for B_z≤2.5 T. The results for T_1 are shown in Fig. <ref>e and f. The observed decrease in 1/T_1 as a function of increasing B_z follows as a result of the decreasing energy gap between the excited |T^-⟩ state and |S(2,0)⟩ ground state <cit.>. Previous theoretical studies of triplet state relaxation in donors coupled along [001] predict a dependence on exchange energy as approximately, 1/T_1∼J^3 <cit.>, and should be the focus of future experimental work.

High fidelity single-shot readout of individual and multiple qubit states is a prerequisite for the observation of post-classical multi-qubit phenomena, in particular two-qubit entanglement <cit.>. In the original  Kane proposal for scalable donor based quantum computing architectures, the single-shot measurement of ST states is suggested to facilitate the readout of nuclear spins <cit.> and is advantageous over previously used readout techniques <cit.> as it does not require the high frequency manipulation of the electron spin. Furthermore, ST readout can be used to measure single electron spin qubits <cit.> at lower magnetic fields and higher temperatures easing the constraints on microwave electronics and cryogenic cooling <cit.>. Finally, encoding qubits using ST states <cit.> allows for an all electrical approach for control, in particular, multiple qubits can be coupled by utilising  the inherent electric dipole coupling given by the (1,1)-(2,0) charge configurations <cit.>. The results obtained herein, in addition to the reduced complexity of electron confinement in donors, makes a compelling case for further research on the scaling of multiple forms of donor based quantum computing architectures. 



This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project no. CE110001027) and the US National Security Agency and US Army Research Office (contract no. W911NF-08-1-0527). M.Y.S. acknowledges an ARC Laureate Fellowship.




1

Nielsen:2011
M. A. Nielsen and I. L. Chuang,
Quantum computation and quantum information,
Cambridge university press, 2010.

Ladd:2010vn
T. D. Ladd et al.,
Nature 464, 45 (2010).

RevModPhys.86.153
I. M. Georgescu, S. Ashhab, and F. Nori,
Rev. Mod. Phys. 86, 153 (2014).

RevModPhys.87.307
B. M. Terhal,
Rev. Mod. Phys. 87, 307 (2015).

petta2004
J. R. Petta, A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C. Gossard,
Phys. Rev. Lett. 93, 186802 (2004).

johnson2005triplet
A. Johnson et al.,
Nature 435, 925 (2005).

hanson2007
R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K.
  Vandersypen,
Rev. Mod. Phys. 79, 1217 (2007).

pla2012
J. J. Pla et al.,
Nature 489, 541 (2012).

morello2010
A. Morello et al.,
Nature 467, 687 (2010).

buch2013
H. Buch, S. Mahapatra, R. Rahman, A. Morello, and M. Y. Simmons,
Nature Commun. 4, 2017 (2013).

watson2014
T. F. Watson et al.,
Nano Lett. 14, 1830 (2014).

watson2015
T. F. Watson, B. Weber, M. G. House, H. Büch, and M. Y. Simmons,
Phys. Rev. Lett. 115, 166806 (2015).

koppens2007
F. H. L. Koppens et al.,
J. Appl. Phys. 101, 081706 (2007).

maune2012
B. M. Maune et al.,
Nature 481, 344 (2012).

watson2015b
T. F. Watson, B. Weber, H. Büch, M. Fuechsle, and M. Y. Simmons,
Applied Physics Letters 107 (2015).

House:2015rz
M. G. House et al.,
Nat Commun 6 (2015).

dehollain2014
J. P. Dehollain et al.,
Phys. Rev. Lett. 112, 236801 (2014).

petta2005
J. R. Petta et al.,
Science 309, 2180 (2005).

taylor2007
J. M. Taylor et al.,
Phys. Rev. B 76, 035315 (2007).

Prance2012
J. R. Prance et al.,
Phys. Rev. Lett. 108, 046808 (2012).

laird2010
E. A. Laird et al.,
Phys. Rev. B 82, 075403 (2010).

Kim:2015jt
D. Kim et al.,
Npj Quantum Information 1, 15004 EP  (2015).

weber2014
B. Weber et al.,
Nature Nanotech. 9, 430 (2014).

Sachrajda2012
S. A. Studenikin et al.,
Applied Physics Letters 101 (2012).

PhysRevB.92.125434
J. D. Mason et al.,
Phys. Rev. B 92, 125434 (2015).

Harvey2015
P. Harvey-Collard et al.,
arXiv preprint arXiv:1512.01606  (2015).

elzerman2004
J. M. Elzerman et al.,
Nature 430, 431 (2004).

PhysRevA.86.032324
A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland,
Phys. Rev. A 86, 032324 (2012).

fuhrer2009
A. Fuhrer, M. Fuechsle, T. C. G. Reusch, B. Weber, and M. Y. Simmons,
Nano Lett. 9, 707 (2009).

Ono1313
K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha,
Science 297, 1313 (2002).

johnson2005
A. C. Johnson, J. R. Petta, C. M. Marcus, M. P. Hanson, and A. C. Gossard,
Nature 435, 925 (2005).

koppens2006
F. H. L. Koppens et al.,
Nature 442, 766 (2006).

Veldhorst:2015qv
M. Veldhorst et al.,
Nature 526, 410 (2015).

house2016
M. G. House et al.,
Phys. Rev. Applied 6, 044016 (2016).

PhysRevB.82.241302
M. Borhani and X. Hu,
Phys. Rev. B 82, 241302 (2010).

kane1998silicon
B. E. Kane,
nature 393, 133 (1998).

pla2013
J. J. Pla et al.,
Nature 496, 334 (2013).

vandersypen2016
L. Vandersypen et al.,
arXiv preprint arXiv:1612.05936  (2016).

kim2014quantum
D. Kim et al.,
Nature 511, 70 (2014).

wu2015singlet
X. Wu,
Singlet-triplet electron spin qubit in Si/SiGe double quantum
  dot,
PhD thesis, The University of Wisconsin-Madison, 2015.

Shulman202
M. D. Shulman et al.,
Science 336, 202 (2012).

Nichol:2017tg
J. M. Nichol et al.,
npj Quantum Information 3, 3 (2017).















§ SUPPLEMETARY INFORMATION





§ CHARGING ENERGY CALCULATION


In this section we provide a full derivation for the calculation of donor-island charging energies. From this analysis, along with examining the extent of the STM images, we can determine with high probability that the number of the donors at sites L and R are 2 and 1 respectively. 

The charging energy, of a quantum dot (QD) a, ϵ_c^a, is calculated by establishing the charging energy of another QD in its vicinity, b which is capacitively coupled to a <cit.>. The charging energies of a and b are related through their mutual charging energy, ϵ_m, given by,

    ϵ_m = α_g^i δ V_g^i,

where α_g^i is the lever arm, or conversion factor, of voltage applied to gate g to the energy of QD i and δ V_g^i is the voltage shift of the potential of QD i due to a charging event on QD j. Importantly, the mutual charging energy by definition must be equal for both QDs a and b, that is to say we can write,

    α_g^a δ V_g^a = α_g^b δ V_g^b.

In a similar vein, the charging energy of a QD itself is given by the voltage difference of gate g between two charge transitions, Δ V_g^i, multiplied by its corresponding lever arm,

    ϵ_c^i = α_g^i Δ V_g^i.

Now it is not necessary to know the level arms α_g^i, as we can eliminate it by combining equations Eq. <ref> and Eq. <ref>, 

    ϵ_c^a/Δ V_g^aδ V_g^a = ϵ_c^b/Δ V_g^bδ V_g^b.


In the cases where multiple charging events occur within the range Δ V_g^i, we may have the condition that ϵ_c^a ≠ϵ_c^b and it is necessary to count the number of charging events. In this case, the measured Δ V_g^i is actually the sum of the true voltage change which we denote ΔV̂_g^i and the number of charging events, n^j of the other QD, that is,

    Δ V_g^i = ΔV̂_g^i + n^j δ V_g^i,

Substituting Eq. <ref> into Eq. <ref> we have the ratio of charging energies for the two QDs a and b,

    ϵ_c^a/ϵ_c^b  = δ V_g^b(Δ V_g^a - n^b δ V_g^a)/δ V_g^a(Δ V_g^b - n^a δ V_g^b).


The above equation relates the charging energies of QDs a and b. In our device, we use an SET island, which is a quantum dot with a small charging energy, such that we have the case Δ V_g^D≫Δ V_g^S (where we have denoted the donor-island as `D' and the SET as `S'). When the condition Δ V_g^D≫Δ V_g^S holds there will be multiple charging events of the SET i.e. n^S≠0, and exactly zero for the donor (n^D=0) within the voltage ranges Δ V_g^D and Δ V_g^S, respectively. As a result, we can simplify Eq. <ref> to,

    ϵ_c^D = ϵ_c^Sδ V_g^S/Δ V_g^S(Δ V_g^D/δ V_g^D - n^S).

Figure <ref> shows the measurement of all required parameters {ϵ_c^S,δ V_g^S, δ V_g^D, Δ V_g^S, Δ V_g^D}, where we have chosen to measure along the right gate, G_R, i.e. g→R. Using Eq. <ref> for the 1→2 electron transitions for both donor-islands, we find charging energies of 65±8 and 43±5 meV and for L (2P) and R (1P), respectively. These values are consistent with theoretical <cit.> and previously measured <cit.> charging energies for 2P and 1P donor-islands respectively.



§ DERIVATION OF STATE-TO-CHARGE CONVERSION

Assuming we have detected a dip below the voltage threshold (see electrical readout section of main text), we now ask what is the probability that this event resulted from the presence of a T(1,1) state and not from the state S(1,1), causing a tunnelling event (1,1)→(2,1) (both scenarios would give rise to an equivalent voltage dip). This section was previously derived by Buch <cit.> for the purpose of single spin-readout, however, we repeat it here for completeness.

The calculated electrical readout fidelities F_T and F_S give indicate the fraction of T and S states at the end of the readout time, but for quantum computation applications it is important to ascertain the loss in fidelity due to the change in these states during this time. This can be done by estimating the fraction of each state, N^0_T and N^0_S at the beginning of the readout phase. 

We define the initial time at the start of the readout phase as the elapsed time t=0, such that the system dynamics can be described by coupled rate equations in the basis {S,T},

    d N̅(t)/d t = [         -1/τ_S,out              1/T_1;                  0 -1/τ_T,out - 1/T_1 ]N̅(t),

where τ_S,out and τ_T,out are the characteristic tunnelling out times of the singlet and triplet respectively. The relaxation time of the triplet |T^-⟩ state, T_1 in the above equation, was measured at B_z=2.5T to be T_1=3.0±0.3 s (see Fig. 3e of main text). The above equation has the two solutions,

    N̅_T(t) = N̅^0_T e^-T_1 + τ_T,out/τ_T,out T_1 t,


    N̅_S(t) = N̅^0_S T̃^2 + N̅^0_T τ_S,outτ_T,out/T̃^2 e^t/τ_S,out - N̅^0_T τ_S,outτ_T,out/T̃^2 e^-T_1 + τ_T,out/τ_T,out T_1 t,

where N̅_T(t) and N̅_S(t) are the remaining fraction of triplet and singlet states at time t and T̃^2=T_1τ_S,out-T_1τ_T,out+τ_S,outτ_T,out, and T_1 is the characteristic relaxation time of the excited triplet state. Importantly, here we have used the relaxation of the triplet-minus state as described in the main text.



Next we define the fraction of states that successfully generate a voltage dip by time t as, 

    N^'(t) = 1 - N̅_T(t) - N̅_S(t) = α(t) N̅^0_T + (1 - β(t)) N̅^0_S

equivalently, the faction of states that do not generate a voltage dip at time t is given by 1-N^'(t), or,

    1-N^'(t) = N̅_T(t) + N̅_S(t) = (1 - β(t)) N̅^0_T + α(t) N̅^0_S.

Here, α represents the probability of the triplet state tunnelling to the SET from the double-donor system (with correction for its T_1 relaxation), while β represents the probability of the singlet state not tunnelling to the SET i.e. at t=0 the probability the singlet state has not tunnelled to the SET is unity. 

From the above set of relationships we can calculate the probability that the detected dip at time t can be assigned to a T or S state. We denote these as α(t) and β(t), respectively,

    α(t)=1/T̃^2[T̃^2-τ_S,outτ_T,out/e^t/τ_S,out-T_1τ_S,out/e^t/τ_T,oute^t/T_1+T_1τ_T,out/e^t/τ_T,oute^t/T_1],


    β(t) = e^-t/τ_S,out.

Now, to optimise the state-to-charge conversion fidelity we find the maximum visibility given by,

    𝒱_STC=α(t)+β(t)-1.




2

hile2015
S. J. Hile et al.,
Appl.Phys. Lett. 107, 093504 (2015)

saraiva2015
A. L. Saraiva et al.,
J. Phys.: Condens. Matter 27, 154208 (2015)
	
weber2014
B. Weber et al.,
Nature Nanotech. 9, 430 (2014).

fuechsle2012
M. Fuechsle et al.,
Nature Nanotech. 7, 242, (2012)

Buch2013
H. Buch, et al.,
Nature Commun. 4, 2017 (2013).



