Coherent control via weak measurements in $^{31}$P single-atom electron and nuclear spin qubits
J. T. Muhonen J. P. Dehollain A. Laucht S. Simmons R. Kalra F. E. Hudson D. N. Jamieson J. C. McCallum K. M. Itoh A. S. Dzurak A. Morello




[Current address: ]Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
muhonen@amolf.nl
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


[Current address: ]QuTech & Kavli Institute of Nanoscience, TU Delft, 2628 CJ Delft, The Netherlands
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


[Current address: ]Department of Physics, Simon Fraser University, Burnaby BC V5A 1S6, Canada
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


[Current address: ]School of Mathematics & Physics, University of Queensland, Brisbane QLD 4072, Australia.
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


Centre for Quantum Computation and Communication Technology, School of
Physics, University of Melbourne, Melbourne VIC 3010, Australia


Centre for Quantum Computation and Communication Technology, School of
Physics, University of Melbourne, Melbourne VIC 3010, Australia


School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi,
223-8522, Japan


Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


a.morello@unsw.edu.au
Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney NSW 2052, Australia


The understanding of weak measurements and interaction-free measurements has greatly expanded the conceptual and experimental toolbox to explore the quantum world. Here we demonstrate single-shot variable-strength weak measurements of the electron and the nuclear spin states of a single ^31P donor in silicon. We first show how the partial collapse of the nuclear spin due to measurement can be used to coherently rotate the spin to a desired pure state. We explicitly demonstrate that phase coherence is preserved throughout multiple sequential single-shot weak measurements, and that the partial state collapse can be reversed. Second, we use the relation between measurement strength and perturbation of the nuclear state as a physical meter to extract the tunneling rates between the ^31P donor and a nearby electron reservoir from data, conditioned on observing no tunneling events. Our experiments open avenues to measurement-based state preparation, steering and feedback protocols for spin systems in the solid state, and highlight the fundamental connection between information gain and state modification in quantum mechanics.



Coherent control via weak measurements in ^31P single-atom electron and nuclear spin qubits
    A. Morello
    December 30, 2023
===========================================================================================



The quantum measurement postulate, as found in quantum mechanics textbooks, implicitly describes projective (von Neumann) measurements, where a measurement apparatus is coupled to a quantum system and, upon performing the measurement, returns a unique value a_k for some observable Â of the quantum system. If the system was initially in the state |ψ⟩, the act of measurement leaves it in the state |ϕ_k⟩, the eigenstate of the observable Â with eigenvalue a_k. The non-deterministic and non-unitary process through which the act of measurement transforms the initial state |ψ⟩ into the final state |ϕ_n⟩ is known as ”wavefunction collapse”, and has been the subject of a century of debate and controversy.

However, as was already appreciated by von Neumann <cit.>, the projective measurement is only a limiting case. One can also have a detector which is only partially correlated with some observable of the quantum system, and therefore returns only partial information on the system state. Accordingly, the wavefunction needs not be fully projected onto an eigenstate, but is only weakly disturbed by the measurement process. The implications and applications of such “weak measurements” and corresponding partial collapse of the quantum state have gained considerable attention, especially in the context of quantum information processing. Recent experiments on superconducting qubits have demonstrated partial wavefunction collapse <cit.>, measurement reversal <cit.>, stabilized Rabi oscillations using quantum feedback <cit.>, direct observation of quantum trajectories <cit.>, reduction of decoherence via “uncollapsing” <cit.>, and observation of the back-action steering from a variable strength measurement <cit.>.

In this Letter, we describe how to apply the principles of weak quantum measurements to the electron and nuclear spin states of an individual ^31P donor atom in silicon. In the context of quantum measurement, the ^31P atom provides access to many key features, in particular related to negative-result measurements <cit.> and quantum steering <cit.>. In particular, we show that weak measurements can be used to phase coherently control the state of the ^31P nuclear spin, and that it is possible to preserve phase coherence through the sequential measurement and control steps. This aspect of weak measurements has not been explicitly clarified in the recent literature, and opens avenues to measurement based state preparation, EPR steering and possible feedback protocols in these systems. As a further demonstration of the applicability of the weak measurement toolbox to the ^31P system, we show how the tunneling rate of the electron to a nearby electron reservoir can be extracted from a dataset conditioned on having no tunneling events, in a spirit similar to the Elitzur-Vaidman bomb <cit.>.

Figure <ref>(a) shows a scanning electron microscope image of our device, which is fabricated on an isotopically enriched ^28Si substrate <cit.>, and where the ^31P atom is introduced via single-ion implantation <cit.>. This system has gained considerable attention in the field of solid-state quantum information processing, since it contains two natural qubits (the electron spin, with S=1/2 and basis states |↑⟩, |↓⟩, and the ^31P nucleus, with spin I=1/2 and basis states |⇑⟩, |⇓⟩) that exhibit extremely long coherence times <cit.>, high quantum gate fidelities <cit.> and can be efficiently entangled with each other <cit.>. 

At its core, the quantum state of the ^31P system is accessible through the measurement of the z-projection of the electron spin, where z is the axis along which a strong external magnetic field B_0 (≈ 1.5 T in the present experiment) is applied. The donor is placed in close proximity (≈ 25 nm <cit.>) to a cold (T ≈ 100 mK) electron reservoir. Under suitable biasing conditions, the donor-bound electron can tunnel into the cold reservoir if and only if it is in the excited |↑⟩ state. The positively charged donor left behind after this tunneling event shifts the bias point of a nearby single-electron transistor (SET) and switches it to a high conductance state. Conversely, a |↓⟩ electron cannot escape the donor, leaving the SET in a near-zero conductance state. This spin-dependent tunneling process <cit.> thus gives rise to a single-shot measurement, with fidelity in excess of 90% <cit.>. This mechanism provides a near-ideal negative-result measurement for the |↓⟩ state, which is identified by the absence of a signal in the SET current. 

The ^31P nuclear spin couples to the electron through the Fermi contact hyperfine interaction A 𝐈·𝐒, with A ≈ 97 MHz in this specific device. As a consequence, the electron spin can have two possible resonance frequencies, ν_ e1,2 = γ_ eB_0 ∓ A/2 [Fig. <ref>(c)], where γ_ e≈ 28 GHz/T is the electron gyromagnetic ratio. Single-shot nuclear readout <cit.> is obtained by initializing the |↓⟩ state and applying a microwave π-pulse at e.g. ν_ e1, where subsequently measuring the electron |↑⟩ state indicates that the nuclear spin state was |⇓⟩. Since we work in the limit γ_ eB_0 ≫ A, the hyperfine interaction can be approximated with A I_z S_z, and therefore commutes with the S_z electron spin observable. This means that the readout of the z-projection of the nuclear spin is of quantum nondemolition type <cit.>, and can be repeated to achieve a readout fidelity approaching 99.9% <cit.>, well beyond that of a single-shot electron readout. The use of an electron π-pulse is just a limiting case, where one gains maximum information about the nuclear spin state. Here instead we explore the more general case where the electron rotation angle is θ≠π <cit.>, which causes the subsequent electron readout to provide only partial information on the nuclear state. This realizes a tunable weak measurement, with strength controlled by the electron rotation angle θ. We show below that, as a result of a weak nuclear measurement conditioned on measuring electron |↓⟩, the nuclear state can be coherently rotated to an arbitrary pure state. This could be extended to provide an interesting implementation of EPR steering <cit.> with spins in the solid state, by applying ESR pulses simultaneously on both ν_ e1 and ν_ e2 <cit.>. Our experiments were conducted by exciting only one ESR frequency at a time, and therefore we will refrain from using the term “steering” to describe the process.  



Let us assume that the nuclear spin is initially in the state |ψ_ n0⟩ = (|⇓⟩ + |⇑⟩)/√(2), while the electron spin is initialized in its ground state |↓⟩. We then apply a microwave pulse at frequency ν_ e2 to produce a rotation by an angle θ of the electron spin, conditioned on the nuclear spin being in the |⇑⟩ state. The full electron-nuclear state then becomes |Ψ_ en⟩ = [ |⇓↓⟩ + cos(θ/2) |⇑↓⟩ + sin(θ/2) |⇑↑⟩]/√(2). A readout of the electron spin state will then produce |↑⟩ with probability P_↑= sin^2(θ/2)/2 and leave the nuclear spin state |⇑⟩. More interestingly, with probability P_↓ = [1+cos^2(θ/2)]/2 the electron readout will produce |↓⟩ and leave the nuclear spin in a coherent superposition state |ψ_ n⟩ = [|⇑⟩+cos(θ/2)|⇓⟩]/[1+cos^2(θ/2)], which has therefore been rotated from the original state |ψ_ n0⟩ using only electron spin resonance (ESR) pulses and electron spin measurements. Importantly, as we show below, this rotation is fully coherent and can be used the prepare any nuclear spin superposition state. The rotation is probabilistic in the sense that it can fail (if the outcome of the electron readout is |↑⟩), but in case of a success (heralded by the |↓⟩ electron readout) the steered state is fully deterministic.

A more complete description of the process is obtained through a density matrix formalism <cit.>. The initial nuclear spin state is

    ρ_0 = |ψ_ n0⟩⟨ψ_ n0| = 1/2[ 1 1; 1 1 ].

After the θ rotation of the electron spin (initially |↓⟩) conditioned on the |⇑⟩ nuclear state, and a |↓⟩ electron readout, the nuclear spin is left in the state

    ρ(θ) = 1/1+cos^2(θ/2)[ cos^2(θ/2)  cos (θ/2);  cos (θ/2)          1 ],

which notably is a pure state for all values of θ. This readily generalizes to multiple electron rotation and measurement steps. For example, after two sequential applications of the sequence, the nuclear spin state is (conditional on reading |↓⟩ at both steps)

    ρ(θ_1,θ_2)    =   1/1+cos^2(θ_1/2)cos^2(θ_2/2)
       ×   [ cos^2(θ_1/2)cos^2(θ_2/2)   cos (θ_1/2)cos (θ_2/2);  cos (θ_1/2) cos (θ_2/2)                        1 ],

assuming phase coherence is preserved at the intermediate electron readout step (see below).

An interesting scenario appears if the second electron rotation is applied at ν_ e1 instead of ν_ e2, so that the rotation is conditioned on the nuclear |⇓⟩ state. Calling ϕ the rotation angle of the microwave pulse at ν_ e1, the final state becomes

    ρ(θ,ϕ)    =   1/cos^2(ϕ/2)+cos^2(θ/2)
       ×   [          cos^2(θ/2)  cos (θ/2)cos (ϕ/2); cos (θ/2) cos (ϕ/2)          cos^2(ϕ/2) ].

If we set ϕ = θ, the final state is ρ(θ,θ) = ρ_0. This is known as “measurement reversal” <cit.>: the second weak measurement of the nuclear spin erases the effect of the first one. 

Figure <ref> shows experimental data obtained with full quantum state tomography, i.e. measurement of all three nuclear spin components σ_z = (ρ_1,1-ρ_2,2), σ_x = (ρ_1,2+ρ_2,1), and σ_y = (ρ_1,2-ρ_2,1). 

The left column of Fig. <ref> is the result of a single nuclear rotation step, consisting of an ESR pulse at ν_ e2 inducing a rotation of angle θ around the x-axis to the electron spin state, followed by single-shot electron readout, and postselection on the |↓⟩ outcome. The dashed lines, in excellent agreement with the data, show the expected nuclear state, on the basis of the density matrix description presented above, without any free fitting parameters.

The middle column in Fig. <ref> illustrates the application of two sequential rotation steps, conducted for simplicity with the same ESR rotation angle θ on ν_ e2 at both steps. The fact that the data (especially the σ_x-component) follows the theoretical predictions indicates that the nuclear state remains coherent throughout the sequence, which contains two weak nuclear measurements. In other words, the partial collapse of the nuclear state after the first weak measurement is a phase coherent, predictable process, although the evolution is non-unitary. A minimum requirement for observing this effect is that the dephasing time of the nuclear spin qubit has to be longer than the electron readout time. The ^31P nuclear spin qubit in ^28Si already has an intrinsically long dephasing time (T_2^∗≈ 0.5 ms <cit.>), but here we further extend it by applying two NMR refocusing pulses during the 3 ms electron readout step (see Fig. 1(d)). We also frequency-modulate the NMR source to track the resonance frequency of the nuclear spin qubit during the electron readout phase, since the change in the donor electrostatic potential under readout conditions causes a Stark shift of the resonance frequency <cit.>. 



On the right column of Fig. <ref> we present the so-called measurement reversal <cit.>, which requires a rotation by θ on ν_ e2 and rotation by ϕ = θ on ν_ e1. As predicted, we recover the original state each time (again, conditional on obtaining |↓⟩ at each electron readout step). Note that when θ = π, the nuclear measurement becomes fully projective and the probability of a successful reversal becomes zero (all success probabilities are presented in supplementary Fig. 1). The data points around θ = π are thus only statistical fluctuations. 

We now explore the possibility of performing a weak electron spin measurement, and the effects that such a measurement has on the nuclear spin. The spin-dependent tunneling mechanism that provides a discrimination between the |↑⟩ and |↓⟩ states yields a fully projective measurement only in the limit Γ_↑,out t_ m→∞, where t_ m is the measurement time and Γ_↑,out is the tunnel-out rate for a |↑⟩ electron, defined such that the probability for a |↑⟩ electron to have tunnelled out of the donor after time t_ m is P_↑,out(t_ m) = 1 - exp(-Γ_↑,out t_ m). For a finite value of Γ_↑,out t_ m, the absence of a tunnel-out event constitutes only a weak |↓⟩ measurement. 

The effect on the nuclear spin of a weak electron measurement can be captured quantitatively in the density matrix formalism, by modifying Eq. <ref> to include the probability 1-P_↑,out(t_ m) that an |↑⟩ does not tunnel out within the measurement time <cit.>.


    ρ(θ,t_ m)    =   1/1+cos^2(θ/2)+(1-P_↑,out(t_ m))sin^2(θ/2)
       ×   [ cos^2(θ/2)+(1-P_↑,out(t_ m))sin^2(θ/2)                              cos (θ/2);                              cos (θ/2)                                      1 ].


Hence, the expectation value of σ_z as a function of measurement time, conditioned on measuring |↓⟩ (no tunneling) is 

    ⟨σ_z (t_ m) ⟩  = cos^2(θ/2)+exp(-Γ_↑,out t_ m)sin^2(θ/2)-1/cos^2(θ/2)+exp(-Γ_↑,out t_ m)sin^2(θ/2)+1,

which for θ=π reduces to a particularly simple form

    ⟨σ_z (t_ m) ⟩  = exp(-Γ_↑,out t_ m)-1/exp(-Γ_↑,out t_ m)+1.

Solving for Γ_↑,out as a function of ⟨σ_z (t_ m)⟩ we find

    1/Γ_↑,out = -t_ m/ln(1+⟨σ_z (t_ m)⟩/1-⟨σ_z (t_ m)⟩).


In Fig. <ref> we show the results of an experiment where we perform the above mentioned protocol, i.e., we prepare the nucleus in |ψ_ n0⟩ = (|⇓⟩ + |⇑⟩)/√(2), the electron in |↓⟩ and then apply an electron π-pulse at ν_ e1, thus leaving the electron-nuclear system in the Bell state <cit.> |Φ^+⟩ = (|↓⇓⟩ + |↑⇑⟩)/√(2). We then bring the electron towards the readout position for a time t_ m = 1.5 ms and, conditional on having no tunneling events, we subsequently measure the nuclear polarization ⟨σ_z ⟩. The experiment is repeated at different values of the gate voltage V_ DG, which controls the donor electrochemical potential μ_ D relative to the Fermi level of the electron reservoir <cit.>, and thereby tunes the donor-reservoir tunnel rate Γ_↑,out. For V_ DG≳ 0.2 V the |↑⟩ state is well below the Fermi level and neither the |↓⟩ nor the |↑⟩ states have a significant probability of tunneling out, i.e. the measurement strength vanishes: the absence of a tunneling event does not imply a |↓⟩ state. Accordingly, we find ⟨σ_z ⟩≈ 0 in that limit, i.e. the nuclear polarization has not been perturbed from the initial value. For V_ DG < 0.2 V, 1/Γ_↑,out becomes shorter and ⟨σ_z ⟩ veers towards negative values, which indicates that the electron |↓⟩ measurement is becoming stronger, thus turning the initial |Φ^+⟩ Bell state towards |↓⇓⟩. Using Eq. <ref> we can extract the numerical value of 1/Γ_↑,out, and compare it [Fig. <ref>(b)] to the tunnel time extracted directly from tunneling probabilities. The two methods agree almost perfectly, confirming the validity of our approach. The non-monotonic behavior of Γ_↑,out(V_ DG) is related to modulations in the density of states of the electron reservoir <cit.>.

Unlike the weak nuclear measurement describe earlier, this process using weak electron measurement does not preserve the purity of the nuclear spin state. Also, the use of a maximally entangled |Φ^+⟩ Bell state as the starting point of the sequence is inconsequential for this particular experiment – the same result would be obtained starting from an incoherent mixture of |↓⇓⟩ and |↑⇑⟩, though the perfect correlation between the two spins is obviously required. Nonetheless, the process provides a curious example of interaction-free measurement <cit.> in the solid state.



In conclusion, we have shown the application of several concepts and tools of weak single-shot measurements to a model solid-state spin system. In particular, we have demonstrated the ability to coherently control a nuclear spin using only ESR pulses and electron spin readout, and we have shown how to measure tunnel rates without any tunneling events. In the future, these techniques can be applied to a variety of interesting problems, such as the study of qubit dynamics under driving and weak measurement <cit.>, past quantum states of a monitored system <cit.> and the use of steering to improve qubit initialization. 

We thank K. Mølmer and R. Ruskov for insightful comments. This research was funded by the Australian Research Council through a Discovery Project (DP150101863) and the Centre of Excellence Quantum Computation and Communication Technology (CE11E0001027), the US Army Research Office (W911NF-13-1-0024) and the Commonwealth Bank of Australia. We acknowledge support from the Australian National Fabrication Facility, and from the laboratory of Prof Robert Elliman at the Australian National University for the ion implantation facilities. The work at Keio has been supported in part by KAKENHI (S) No. 26220602, Core-to-Core Program by JSPS, and Spintronics Research Network of Japan.  




35
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[von Neumann(1932)]Neumann1932
  author author J. von
  Neumann, @noop  title Mathematische Grundlagen
  der Quantenmechanik (publisher Springer, address Berlin, year 1932)NoStop
[Katz et al.(2006)Katz,
  Ansmann, Bialczak, Lucero,
  McDermott, Neeley, Steffen,
  Weig, Cleland, Martinis,  and Korotkov]Katz2006
  author author N. Katz, author M. Ansmann,
  author R. C. Bialczak, author E. Lucero, author
  R. McDermott, author
  M. Neeley, author M. Steffen, author E. M.  Weig, author A. N. Cleland, author J. M. Martinis,  and author A. N. Korotkov, http://www.sciencemag.org/content/312/5779/1498.abstract journal journal Science volume
  312, pages 1498 (year 2006)NoStop
[Katz et al.(2008)Katz,
  Neeley, Ansmann, Bialczak,
  Hofheinz, Lucero, Oâ€™Connell, Wang, Cleland,
  Martinis, and Korotkov]Katz2008
  author author N. Katz, author M. Neeley,
  author M. Ansmann, author R. C. Bialczak, author
  M. Hofheinz, author
  E. Lucero, author A. Oâ€™Connell, author H. Wang, author A. N.  Cleland, author J. M.  Martinis,  and author A. N.  Korotkov, http://link.aps.org/doi/10.1103/PhysRevLett.101.200401 journal journal Phys. Rev. Lett. volume 101, pages 200401 (year
  2008)NoStop
[Vijay et al.(2012)Vijay,
  Macklin, Slichter, Weber,
  Murch, Naik, Korotkov, and  Siddiqi]Vijay2012
  author author R. Vijay, author C. Macklin,
  author D. H. Slichter, author S. J. Weber, author
  K. W. Murch, author
  R. Naik, author A. N. Korotkov,  and author I. Siddiqi, http://dx.doi.org/10.1038/nature11505
  journal journal Nature volume 490, pages 77 (year 2012)NoStop
[Murch et al.(2013)Murch,
  Weber, Macklin, and Siddiqi]Murch2013
  author author K. W. Murch, author S. J. Weber,
  author C. Macklin,  and  author I. Siddiqi, http://dx.doi.org/10.1038/nature12539 journal journal Nature volume 502, pages
  211 (year 2013)NoStop
[Weber et al.(2014)Weber,
  Chantasri, Dressel, Jordan,
  Murch, and Siddiqi]Weber2014
  author author S. J. Weber, author A. Chantasri,
  author J. Dressel, author A. N. Jordan, author
  K. W. Murch,  and author
  I. Siddiqi, http://dx.doi.org/10.1038/nature13559 journal journal Nature volume 511, pages
  570 (year 2014)NoStop
[Zhong et al.(2014)Zhong,
  Wang, Martinis, Cleland,
  Korotkov, and Wang]Zhong2014
  author author Y. P. Zhong, author Z. L. Wang,
  author J. M. Martinis, author A. N. Cleland, author
  A. N. Korotkov,  and author
  H. Wang, http://dx.doi.org/10.1038/ncomms4135 journal journal Nat. Commun. volume 5,  (year 2014)NoStop
[Hatridge et al.(2013)Hatridge, Shankar, Mirrahimi, Schackert, Geerlings, Brecht, Sliwa, Abdo, Frunzio, Girvin, Schoelkopf, and Devoret]Hatridge2013
  author author M. Hatridge, author S. Shankar,
  author M. Mirrahimi, author F. Schackert, author
  K. Geerlings, author
  T. Brecht, author K. M. Sliwa, author B. Abdo, author L. Frunzio, author S. M. Girvin,
  author R. J. Schoelkopf,  and  author M. H. Devoret, http://science.sciencemag.org/content/339/6116/178.abstract journal journal Science volume
  339, pages 178 (year 2013)NoStop
[Groen et al.(2013)Groen,
  Ristè, Tornberg, Cramer,
  de Groot, Picot, Johansson,  and DiCarlo]Groen2013
  author author J. P. Groen, author D. Ristè,
  author L. Tornberg, author J. Cramer, author
  P. C. de Groot, author
  T. Picot, author G. Johansson,  and author L. DiCarlo, http://link.aps.org/doi/10.1103/PhysRevLett.111.090506 journal journal Phys. Rev. Lett. volume 111, pages 090506 (year
  2013)NoStop
[Dicke(1981)]Dicke1981
  author author R. H. Dicke, http://aapt.scitation.org/doi/10.1119/1.12592 journal journal Am. J. Phys volume
  49, pages 925 (year 1981)NoStop
[Wiseman et al.(2007)Wiseman, Jones, and Doherty]Wiseman2007
  author author H. M. Wiseman, author S. J. Jones,
   and author A. C. Doherty,  https://doi.org/10.1103/PhysRevLett.98.140402 journal
  journal Phys. Rev. Lett. volume 98,  pages 140402 (year 2007)NoStop
[Cavalcanti and Skrzypczyk(2017)]Cavalcanti2016
  author author D. Cavalcanti and author P. Skrzypczyk, http://stacks.iop.org/0034-4885/80/i=2/a=024001
  journal journal Reports on Progress in Physics  volume 80, pages 024001 (year 2017)NoStop
[Elitzur and Vaidman(1993)]Elitzur1993
  author author A. C. Elitzur and author L. Vaidman, doi:10.1007/BF00736012 journal journal Foundations of Physics volume 23,  pages 987 (year 1993)NoStop
[Itoh and Watanabe(2014)]Itoh2014
  author author K. M. Itoh and author H. Watanabe, https://doi.org/10.1557/mrc.2014.32 journal journal MRS Communications volume 4, pages 143 (year 2014)NoStop
[van Donkelaar et al.(2015)van Donkelaar, Yang, Alves, McCallum, Hougaard, Johnson, Hudson, Dzurak, Morello, Spemann, and Jamieson]Donkelaar2015
  author author J. van
  Donkelaar, author C. Yang,
  author A. D. C. Alves, author J. C. McCallum, author
  C. Hougaard, author
  B. C. Johnson, author
  F. E. Hudson, author
  A. S. Dzurak, author
  A. Morello, author D. Spemann,  and author D. N. Jamieson, http://stacks.iop.org/0953-8984/27/i=15/a=154204 journal
  journal Journal of Physics: Condensed Matter volume 27, pages 154204 (year
  2015)NoStop
[Tyryshkin et al.(2012)Tyryshkin, Tojo, Morton, Riemann, Abrosimov, Becker, Pohl, Schenkel, Thewalt, Itoh, and Lyon]Tyryshkin2012
  author author A. M. Tyryshkin, author S. Tojo,
  author J. J. L. Morton, author H. Riemann, author
  N. V. Abrosimov, author
  P. Becker, author H.-J. Pohl, author T. Schenkel, author M. L. W.  Thewalt, author K. M.  Itoh,  and author S. A.  Lyon, http://dx.doi.org/10.1038/nmat3182 journal journal Nat. Mater. volume
  11, pages 143 (year 2012)NoStop
[Saeedi et al.(2013)Saeedi,
  Simmons, Salvail, Dluhy,
  Riemann, Abrosimov, Becker,
  Pohl, Morton, and Thewalt]Saeedi2013
  author author K. Saeedi, author S. Simmons,
  author J. Z. Salvail, author P. Dluhy, author
  H. Riemann, author N. V. Abrosimov, author P. Becker, author H.-J.  Pohl, author J. J. L.  Morton,  and author M. L. W.  Thewalt, http://science.sciencemag.org/content/342/6160/830.abstract journal journal Science volume
  342, pages 830 (year 2013)NoStop
[Muhonen et al.(2014)Muhonen, Dehollain, Laucht, Hudson, Kalra, Sekiguchi, Itoh, Jamieson, McCallum, Dzurak, and Morello]Muhonen2014
  author author J. T. Muhonen, author J. P. Dehollain, author A. Laucht,
  author F. E. Hudson, author R. Kalra, author
  T. Sekiguchi, author
  K. M. Itoh, author
  D. N. Jamieson, author
  J. C. McCallum, author
  A. S. Dzurak,  and author
  A. Morello, http://dx.doi.org/10.1038/nnano.2014.211 journal journal Nat. Nanotech. volume 9, pages 986 (year 2014)NoStop
[Muhonen et al.(2015)Muhonen, Laucht, Simmons, Dehollain, Kalra, Hudson, Freer, Itoh, Jamieson, McCallum, Dzurak, and Morello]Muhonen2015
  author author J. T. Muhonen, author A. Laucht,
  author S. Simmons, author J. P. Dehollain, author
  R. Kalra, author F. E. Hudson, author S. Freer, author K. M.  Itoh, author D. N. Jamieson, author J. C. McCallum, author A. S. Dzurak,  and author A. Morello, http://stacks.iop.org/0953-8984/27/i=15/a=154205
  journal journal Journal of Physics: Condensed
  Matter volume 27, pages 154205
  (year 2015)NoStop
[Dehollain et al.(2016a)Dehollain, Muhonen, Blume-Kohout, Rudinger,
  Gamble, Nielsen, Laucht,
  Simmons, Kalra, Dzurak, and  Morello]Dehollain2016b
  author author J. P. Dehollain, author J. T. Muhonen, author R. Blume-Kohout, author K. M. Rudinger, author J. K. Gamble, author E. Nielsen,
  author A. Laucht, author S. Simmons, author
  R. Kalra, author A. S. Dzurak,  and author A. Morello, http://iopscience.iop.org/article/10.1088/1367-2630/18/10/103018/meta
  journal journal New Journal of Physics volume 18, pages 103018 (year
  2016a)NoStop
[Simmons et al.(2011)Simmons, Brown, Riemann, Abrosimov, Becker, Pohl, Thewalt, Itoh, and Morton]Simmons2011
  author author S. Simmons, author R. M. Brown,
  author H. Riemann, author N. V. Abrosimov, author
  P. Becker, author H.-J. Pohl, author M. L.  Thewalt, author K. M.  Itoh,  and author J. J.  Morton, http://www.nature.com/nature/journal/v470/n7332/full/nature09696.html
  journal journal Nature volume 470, pages 69 (year 2011)NoStop
[Dehollain et al.(2016b)Dehollain, Simmons, Muhonen, Kalra, Laucht, Hudson, Itoh, Jamieson, McCallum, Dzurak, and  Morello]Dehollain2016a
  author author J. P. Dehollain, author S. Simmons,
  author J. T. Muhonen, author R. Kalra, author
  A. Laucht, author F. Hudson, author K. M.  Itoh, author D. N. Jamieson, author J. C. McCallum, author A. S. Dzurak,  and author A. Morello, http://dx.doi.org/10.1038/nnano.2015.262 journal journal Nat. Nanotech. volume
  11, pages 242 (year
  2016b)NoStop
[Mohiyaddin et al.(2013)Mohiyaddin, Rahman, Kalra, Klimeck, Hollenberg, Pla, Dzurak, and Morello]Mohiyaddin2013
  author author F. A. Mohiyaddin, author R. Rahman,
  author R. Kalra, author G. Klimeck, author
  L. C. Hollenberg, author
  J. J. Pla, author A. S. Dzurak,  and author A. Morello, http://pubs.acs.org/doi/abs/10.1021/nl303863s journal
  journal Nano Letters volume 13,  pages 1903 (year 2013)NoStop
[Elzerman et al.(2004)Elzerman, Hanson, Willems van Beveren,
  Witkamp, Vandersypen, and Kouwenhoven]Elzerman2004
  author author J. M. Elzerman, author R. Hanson,
  author L. H. Willems van
  Beveren, author B. Witkamp,
  author L. M. K. Vandersypen,   and author L. P. Kouwenhoven,  http://dx.doi.org/10.1038/nature02693 journal
  journal Nature volume 430, pages 431 (year 2004)NoStop
[Morello et al.(2009)Morello, Escott, Huebl, Willems van Beveren, Hollenberg, Jamieson,
  Dzurak, and Clark]Morello2009
  author author A. Morello, author C. C. Escott,
  author H. Huebl, author L. H. Willems van Beveren,
  author L. C. L. Hollenberg,
  author D. N. Jamieson, author A. S. Dzurak,  and author R. G. Clark, http://link.aps.org/doi/10.1103/PhysRevB.80.081307 journal
  journal Phys. Rev. B volume 80,  pages 081307 (year 2009)NoStop
[Morello et al.(2010)Morello, Pla, Zwanenburg, Chan, Tan, Huebl, Mottonen,
  Nugroho, Yang, van Donkelaar,
  Alves, Jamieson, Escott,
  Hollenberg, Clark, and Dzurak]Morello2010
  author author A. Morello, author J. J. Pla,
  author F. A. Zwanenburg,
  author K. W. Chan, author K. Y. Tan, author
  H. Huebl, author M. Mottonen, author C. D.  Nugroho, author C. Yang, author J. A. van
  Donkelaar, author A. D. C. Alves, author D. N. Jamieson,
  author C. C. Escott, author L. C. L. Hollenberg, author R. G. Clark,  and author A. S. Dzurak, http://dx.doi.org/10.1038/nature09392 journal journal Nature volume 467, pages
  687 (year 2010)NoStop
[Pla et al.(2013)Pla,
  Tan, Dehollain, Lim,
  Morton, Zwanenburg, Jamieson,
  Dzurak, and Morello]Pla2013
  author author J. J. Pla, author K. Y. Tan,
  author J. P. Dehollain, author W. H. Lim, author
  J. J. L. Morton, author
  F. A. Zwanenburg, author
  D. N. Jamieson, author
  A. S. Dzurak,  and author
  A. Morello, http://dx.doi.org/10.1038/nature12011 journal journal Nature volume 496, pages
  334 (year 2013)NoStop
[Braginsky et al.(1980)Braginsky, Vorontsov, and Thorne]Braginsky1980
  author author V. B. Braginsky, author Y. I. Vorontsov,  and author K. S. Thorne, http://science.sciencemag.org/content/209/4456/547.abstract journal journal Science volume
  209, pages 547 (year 1980)NoStop
[Blok et al.(2014)Blok,
  Bonato, Markham, Twitchen,
  Dobrovitski, and Hanson]Blok2014
  author author M. S. Blok, author C. Bonato,
  author M. L. Markham, author D. J. Twitchen, author
  V. V. Dobrovitski,  and author
  R. Hanson, http://dx.doi.org/10.1038/nphys2881 journal journal Nat. Phys. volume 10, pages
  189 (year 2014)NoStop
[sup()]supp
  @noop  title See supplemental material at [url
  will be inserted by publisher] for additional figures and text on steering
  probabilities, pulse sequences and density matrix calculations.  NoStop
[Korotkov and Jordan(2006)]Korotkov2006
  author author A. N. Korotkov and author A. N. Jordan, http://link.aps.org/doi/10.1103/PhysRevLett.97.166805
  journal journal Phys. Rev. Lett. volume 97, pages 166805 (year
  2006)NoStop
[Laucht et al.(2015)Laucht,
  Muhonen, Mohiyaddin, Kalra,
  Dehollain, Freer, Hudson,
  Veldhorst, Rahman, Klimeck,
  Itoh, Jamieson, McCallum,
  Dzurak, and Morello]Laucht2015
  author author A. Laucht, author J. T. Muhonen,
  author F. A. Mohiyaddin,
  author R. Kalra, author J. P. Dehollain, author
  S. Freer, author F. E. Hudson, author M. Veldhorst, author R. Rahman, author G. Klimeck, author K. M.  Itoh, author D. N. Jamieson, author J. C. McCallum, author A. S. Dzurak,  and author A. Morello, http://advances.sciencemag.org/content/1/3/e1500022.abstract journal journal Science Advances volume 1,  (year 2015)NoStop
[Möttönen et al.(2010)Möttönen, Tan, Chan, Zwanenburg, Lim, Escott, Pirkkalainen, Morello, Yang, van Donkelaar, Alves, Jamieson,
  Hollenberg, and Dzurak]Mottonen2010
  author author M. Möttönen, author K. Y. Tan, author K. W. Chan,
  author F. A. Zwanenburg,
  author W. H. Lim, author C. C. Escott, author
  J.-M. Pirkkalainen, author
  A. Morello, author C. Yang, author J. A.  van Donkelaar, author A. D. C.  Alves, author D. N. Jamieson, author L. C. L. Hollenberg,  and author A. S. Dzurak, http://link.aps.org/doi/10.1103/PhysRevB.81.161304
  journal journal Phys. Rev. B volume 81, pages 161304 (year
  2010)NoStop
[Ruskov et al.(2007)Ruskov,
  Mizel, and Korotkov]Ruskov2007
  author author R. Ruskov, author A. Mizel,   and author A. N. Korotkov,  https://doi.org/10.1103/PhysRevB.75.220501 journal
  journal Phys. Rev. B volume 75,  pages 220501 (year 2007)NoStop
[Gammelmark et al.(2013)Gammelmark, Julsgaard, and Mølmer]Gammelmark2013
  author author S. Gammelmark, author B. Julsgaard,  and author K. Mølmer, http://link.aps.org/doi/10.1103/PhysRevLett.111.160401 journal journal Phys. Rev. Lett. volume 111, pages 160401 (year
  2013)NoStop







 

 

SUPPLEMENTARY MATERIAL: Coherent control via weak measurements in ^31P single-atom electron and nuclear spin qubits




§ SUCCESS PROBABILITIES


Performing a conditional weak measurement is necessarily a probabilistic process. As mentioned in the main text, the success probability for a single measurement [starting from the nuclear spin state in equation (1) of the main text] is P_1 = [1+cos(θ/2)]/2. It is however notable that, as this probability depends on the nuclear spin populations at the start of the measurement, the success probability of two sequential weak measurements is not simply this value squared. Rather, the success probability for n sequential weak measurements in our case is P_n = [1+cos(θ/2)^2n]/2 if all measurements are performed with electron spin rotation θ on the same electron spin resonance frequency.

For the measurement reversal (two weak nuclear measurements, each using a different ESR frequency) the success probability reads P_rev = cos^2(θ/2) which is notably zero for θ=π, as should be expected (one cannot reverse a projective measurement). These predictions together with data are plotted in Fig. 4.





§ DENSITY MATRIX CALCULATIONS


Below we refer to the nuclear spin state with the thick arrow (⇑ or ⇓) and the electron spin state with the narrow arrow (↑ or ↓). The Pauli operators are σ^i where i=e,n refers to either electron or nuclear spin, respectively. We start from the state Φ = 1/√(2)(|⇑⟩+|⇓⟩)⊗|↓⟩, i.e., in density matrix form (in the basis  |⇑↑⟩|⇑↓⟩|⇓↑⟩|⇓↓⟩)

    ρ_0 = 1/2[ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1;   ].

The conditional rotation matrix reads U(θ) = |⇓⟩⟨⇓|⊗ I + |⇑⟩⟨⇑|⊗ R(θ) where R(θ) is the rotation matrix

    R(θ) =
    [  cos(θ) -sin(θ);  sin(θ)  cos(θ);         ].

Hence, after the initialization step and the conditional electron spin rotation of an angle θ the system state is

    ρ_θ = U(θ)ρ_0 U^†(θ) = 1/2[       sin^2(θ/2) cos(θ/2)sin(θ/2)                0         sin(θ/2); cos(θ/2)sin(θ/2)       cos^2(θ/2)                0         cos(θ/2);                0                0                0                0;         sin(θ/2)         cos(θ/2)                0                1;                  ].

which is an entangled electron-nuclear state for all θ≠ 0,2π (according to the PPT criterion).

If we then just simply trace out the electron (no conditioning), we obtain the nuclear spin state as

    ρ_n^u = _2(ρ_θ) =  1/2[ sin^2(θ/2) + cos^2(θ/2)                cos(θ/2);                cos(θ/2)                       1;                         ]
    = 1/2[        1 cos(θ/2); cos(θ/2)        1;          ],

showing that the expectation value of σ_z^n remains constant independently of θ, but the off-diagonal elements decay as a function of the measurement strength. In the limiting case of θ=π, we are left with a classical mixture of up and down nuclear spin states.

More interestingly, tracing out the electron conditionally on measuring |↓⟩ we obtain

    ρ_n^c = Tr_2[ρ_θ(I⊗|0⟩⟨0|)]
    = 1/1+cos^2(θ/2)[ cos^2(θ/2)   cos(θ/2);   cos(θ/2)         1,;            ]

which is the state mentioned in the main text. The second measurement is then simply done by repeating the process starting from this state

    ρ_θ^(2) = U(θ)(ρ_n^c⊗|↓⟩) U^†(θ)

and tracing out similarly. For the measurement reversal, we need the rotation matrix for the other electron spin resonance frequency, which reads  U(θ) = |⇓⟩⟨⇓|⊗ R(θ) + |⇑⟩⟨⇑|⊗ I. Otherwise the procedure is the same. Expectation values for the nuclear spin components for one or two measurements with rotation θ are plotted in Fig. 5. The measurement reversal should just preserve all three components. These are also plotted with the data in main figure 2.



Finally, if we also add a finite electron tunnel-out probability to process described above, we obtain

    ρ_n^c    =   Tr_2 {ρ_θ[ I⊗( |0⟩⟨0|+exp(-Γ t)|1⟩⟨1|) ] }
       =   1/1+cos^2(θ/2)+exp(-Γ t)sin^2(θ/2)[ cos^2(θ/2)+exp(-Γ t)sin^2(θ/2)                       cos(θ/2);                       cos(θ/2)                              1;                                ].

Note that, unlike all the previous states, this one is not pure unless exp(-Γ t)sin^2(θ/2) = 0.



§ NOTES ON EPR STEERING


The use of the word “steering” in the context of quantum systems is somewhat ambigous in the existing literature. The experiments in this paper demonstrate coherent control of a qubit state by measuring another, correlated, qubit state. This is in many contexts called steering and this usage of the word indeed makes intuitive sense; one is steering the nuclear spin (qubit) by weakly measuring it via the electron (ancilla).

However, it is also common that the word steering - in the quantum context - exclusively refers to what is more exactly known as EPR steering. In the operational definition of Wiseman et al., EPR steering consists of a “game” where Alice must convince Bob that she has shared with him an entangled state. To do so, she wants to show Bob that she has the ability to control his quantum state by choosing which measurement to perform at her end. This, in turn, can be formalized in experimentally testable EPR steering inequalities.  

A demonstration of EPR steering could be conducted on the ^31P electron-nuclear system, where “Alice” is the electron spin and “Bob” is the nuclear spin, by following three steps:

(i) Initialize the electron-nuclear system in a maximally entangled Bell state, for example |Φ^+⟩ = (|↓⇓⟩ + |↑⇑⟩)/√(2), as described in the main text. 

(ii) Define different measurement axes for the electron spin. This requires an unconditional electron spin rotation, which could be obtained by simultaneously applying ESR pulses of rotation angle θ on both ν_ e1 and ν_ e2, before a projective electron spin measurement. This is the key difference between EPR steering and the experiments shown in the main text, where all electron spin rotations were conditional on the nuclear spin state. This is because simultaneous excitation of ν_ e1 and ν_ e2 was not feasible in our setup. 

(iii) Conditioned on measuring electron spin |↓⟩, perform nuclear state tomography. 

Supplementary figure 6 shows the expected nuclear spin components as a function of θ. At θ = 0 the electron spin measurement is along the z-axis and therefore the subsequent measurement of σ_z^n could be predicted by unity accuracy, whereas the measurement of σ_x^n is completely undetermined. At θ = π/2 the electron spin measurement is along the x-axis, and now the reverse is true. This simple simulation captures the essence of EPR steering. The state of Bob's particle tracks exactly the choice of measurement basis made by Alice.

We note that the violation of Bell's inequality has already been demonstrated with the electron-nuclear system studied here, and it is known that the requirements for EPR-steering are less strict than those for Bell inequalities. Therefore, using e.g. two separate microwave sources to excite ν_ e1 and ν_ e2 simultaneously, it should be possible to demonstrate EPR steering in the ^31P system.




