Coherent control of three-spin states in a triple quantum dot

Journal name:
Nature Physics
Volume:
8,
Pages:
54–58
Year published:
DOI:
doi:10.1038/nphys2149
Received
Accepted
Published online

Spin qubits involving individual spins in single quantum dots or coupled spins in double quantum dots have emerged as potential building blocks for quantum information processing applications1, 2, 3, 4. It has been suggested that triple quantum dots may provide additional tools and functionalities. These include encoding information either to obtain protection from decoherence or to permit all-electrical operation5, efficient spin busing across a quantum circuit6, and to enable quantum error correction using the three-spin Greenberger-Horn-Zeilinger quantum state. Towards these goals we demonstrate coherent manipulation of two interacting three-spin states. We employ the Landau–Zener–Stückelberg7, 8 approach for creating and manipulating coherent superpositions of quantum states9. We confirm that we are able to maintain coherence when decreasing the exchange coupling of one spin with another while simultaneously increasing its coupling with the third. Such control of pairwise exchange is a requirement of most spin qubit architectures10, but has not been previously demonstrated.

At a glance

Figures

  1. Device, three-spin states spectrum, and spin arch.
    Figure 1: Device, three-spin states spectrum, and spin arch.

    a, Electron micrograph of a device identical to the one measured. Gates 1 and 2 are connected to high-frequency lines for the application of fast voltage pulses (δV 1, δV 2) in addition to d.c. voltages (V 1, V 2). Gate C tunes the (1,1,1) region size by shifting the centre dot addition line. b, Calculated energies versus detuning ε for the three-spin states for a 22-mV-wide (1,1,1) region (that is, |ε+ε|=22mV), neglecting the hyperfine interaction. The Zeeman splitting, Ez, originates from an applied 60mT field. The detuning line is describing a −45° angle with respect to the V 1axis in the V 1V 2 plane. The states shown in grey are split by the tunnel couplings TRC and TLC(not drawn to scale) from the Δ′ states. The Δ′1/2 state is also drawn for a midsized (1,1,1) region (green dash–dotted line) and for a narrow (1,1,1) region (green dotted curve). c, Calculated energy diagram including the effect of hyperfine interaction resulting from the proximity of the four lowest energy three-spin states with Sz>0 (states with Sz<0 are excluded for simplicity). Dotted red circles indicate pairs of states coupled by the hyperfine interaction. The dotted red circle at ε=0 represents the hyperfine interaction between Δ′1/2 and Q1/2 (the meaning of the remaining dotted red circles is clear). The single Δ′1/2−Q3/2 anticrossing (SA) and double Δ′1/2−Q3/2 anticrossing (DA) pulses are drawn. d, Numerical derivative of the left QPC conductance with respect to V 2in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1) for a 9-mV-wide (1,1,1) region. The extent of the (1,1,1) region along the detuning line (approximately joining the centres of the two charge transfer lines) is measured by a projection onto the gate voltage axis that is on the same side as the QPC detector used in the measurement. It is the resulting gate voltage range that is set equal to |ε+ε|, and this is used for comparison between regimes of (1,1,1) regions with different widths. Black is low, orange is medium and yellow is high. The pulse shape is in the Supplementary Information. The detuning line makes a −51.3° angle with respect to the V 1axis in the V 1V 2 plane, permitting both sides of the spin arch to be observed. The dashed line is the theoretical fit (with detuning-dependent interdot couplings included).

  2. LZS oscillations from the two [Delta][prime]1/2-Q3/2 qubits for a wide (1,1,1) region.
    Figure 2: LZS oscillations from the two Δ′1/2−Q3/2 qubits for a wide (1,1,1) region.

    a,b, Numerical derivative of the conductance with respect to detuning showing LZS oscillations versus pulse duration τ. Black is low, red is medium and yellow is high. Panel a shows measurements with the right QPC for  |ε+ε|=27mV along V 1; the pulse goes across the (1,0,2) to (1,1,1) charge transfer line at B=60mT. Both V 2 and V 1 are swept to detune parallel to the pulse direction in the V 1V 2 plane. Panel b shows measurements with the left QPC for |ε+ε|=41.5mV along V 2; the pulse goes across the (2,0,1) to (1,1,1) charge transfer line at B=60mT. c,d, Probability of ending in the Δ′1/2 state as a function of τ with fits for T2*. For the right QPC (c) the pulse goes from (1,0,2) to (1,1,1) and |ε+ε|~50mV along V 1. For the left QPC (d) the pulse goes from (2,0,1) to (1,1,1) and |ε+ε|=27mV along V 2. The experimental data are shown as points, whereas the theoretical fits are shown as red lines. The values of T2* extracted from the single parameter fit to the LZS model are indicated.

  3. Coherent three-spin state manipulation with a narrow (1,1,1) region
    Figure 3: Coherent three-spin state manipulation with a narrow (1,1,1) region

    a, Stability diagram in the presence of a pulse (drawn as a white line for a given (V 1, V 2)), showing coherent LZS oscillations in the (2,0,1) region with features parallel to both charge transfer lines. The colour map (black is low, red is medium and yellow is high) corresponds to the numerical derivative of the left QPC conductance with respect to V 2in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1). The (1,1,1) region is tuned to a width of ~5mV with gate C. B=25mT. The stability diagram also shows LZS oscillations involving (2,0,2) and (1,1,2). b, Calculated dPΔ′1/2/dV 2 map zooming mainly into the (2,0,1) region of the stability diagram from a. The dashed line shows where the addition line is expected, although it is not part of the calculation. B=40mT. c,d, Traces of dPΔ′1/2/dV 2 versus τ. For c the data points are extracted from Fig. 4b (40mT, white line) at V 2=−1.0,751V. For d the data points are extracted from Fig. 4b (40mT, blue line) at V 2=−1.074V. The fits (red lines) use B=60mT. The values of T2* extracted from the fits are indicated.

  4. Magnetic field dependence of coherent three-spin state manipulation with a narrow (1,1,1) region.
    Figure 4: Magnetic field dependence of coherent three-spin state manipulation with a narrow (1,1,1) region.

    a, Energy spectra for the three-spin states for different magnetic fields. The colour code for the states is the same as in Fig. 1b. From left to right we have: B=5mT and |ε+ε|=3.9mV; B=25mT and |ε+ε|=5.1mV; B=40mT and |ε+ε|=5.6mV; and B=60mT and |ε+ε|=4.6mV. b, Coherent oscillations shown in the τV 2 plane as the numerical derivative of the left QPC conductance with respect to V 2 (black is low, red is medium and yellow is high) in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1). The (1,1,1) region is tuned to a width of ~5mV with gate C. V 1 is swept proportionally to V 2 to detune parallel to the pulse direction. The magnetic field and (1,1,1) region sizes from left to right are as in a. The white dot in the B=40mT map indicates a coherent oscillation resulting from a DA pulse reaching past the far Δ′1/2−Q3/2 anticrossing. c, Calculated dPΔ′1/2/dV 2 maps (black is low, red is medium and yellow is high) in the τε plane for the same experimental settings as for b. The magnetic field and (1,1,1) region sizes from left to right are as in a. To keep the fringes clearer, dephasing is not included. The very rapid oscillations in the upper right corner of the figures are an artefact due to the large exchange energy past the far Δ′1/2−Q3/2 anticrossing. At B=5mT, the anticrossings have merged, so there is only one boundary in the diagram (white dashed line). At B=60mT, two Δ′1/2−Q3/2 anticrossings are recovered (see the two white dashed lines). At B=25mT, for both theory and experiment, dotted white curves are drawn as a guide to the eye for the peak of an oscillation in between the two Δ′1/2−Q3/2 anticrossings. We note that a small DNP effect20 is present which depends on the size of the (1,1,1) region, details of pulse shape and pulse orientation. In b,c (25 and 40mT), it is found that a DNP ~20mT is required to properly describe the period of oscillations. This is why the stability diagram in Fig. 3b is calculated at 40mT rather than 25mT, and why the fits in Fig. 3c,d are calculated at 60mT instead of 40mT.

Main

Following the spin qubit proposal by Loss and DiVincenzo10 and the electrostatic isolation of single spins in quantum dots (QDs)11 and double quantum dots (DQDs)12, coherent manipulation was demonstrated in two-level systems based on single-spin up and down states2 as well as two-spin singlet and triplet states1. Here we demonstrate coherent manipulation of a two-level system based on three-spin states. We employ the triple quantum dot (TQD) device layout shown in Fig. 1a, consisting of multiple metallic gates on a GaAs/AlGaAs heterostructure. The gates are used to electrostatically define three QDs in series within a two-dimensional electron gas 110 nm below the surface. The QDs are surrounded by two quantum point contact charge detectors (QPCs)13. The QPC conductance identifies the number of electrons in each QD and its derivative with respect to a relevant gate voltage maps out the device configuration stability diagram. We tune the device to the qubit operating electronic configuration, (NL,NC,NR)=(1,1,1), between two spin-to-charge conversion regimes (1,0,2) and (2,0,1), where L, C and R refer to the left, centre and right QDs respectively. The detuning, ε, controls the energy difference between configurations (1,0,2), (1,1,1) and (2,0,1). The exchange coupling, J, depends on ε and the tunnel couplings.

Figure 1: Device, three-spin states spectrum, and spin arch.
Device, three-spin states spectrum, and spin arch.

a, Electron micrograph of a device identical to the one measured. Gates 1 and 2 are connected to high-frequency lines for the application of fast voltage pulses (δV 1, δV 2) in addition to d.c. voltages (V 1, V 2). Gate C tunes the (1,1,1) region size by shifting the centre dot addition line. b, Calculated energies versus detuning ε for the three-spin states for a 22-mV-wide (1,1,1) region (that is, |ε+ε|=22mV), neglecting the hyperfine interaction. The Zeeman splitting, Ez, originates from an applied 60mT field. The detuning line is describing a −45° angle with respect to the V 1axis in the V 1V 2 plane. The states shown in grey are split by the tunnel couplings TRC and TLC(not drawn to scale) from the Δ′ states. The Δ′1/2 state is also drawn for a midsized (1,1,1) region (green dash–dotted line) and for a narrow (1,1,1) region (green dotted curve). c, Calculated energy diagram including the effect of hyperfine interaction resulting from the proximity of the four lowest energy three-spin states with Sz>0 (states with Sz<0 are excluded for simplicity). Dotted red circles indicate pairs of states coupled by the hyperfine interaction. The dotted red circle at ε=0 represents the hyperfine interaction between Δ′1/2 and Q1/2 (the meaning of the remaining dotted red circles is clear). The single Δ′1/2−Q3/2 anticrossing (SA) and double Δ′1/2−Q3/2 anticrossing (DA) pulses are drawn. d, Numerical derivative of the left QPC conductance with respect to V 2in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1) for a 9-mV-wide (1,1,1) region. The extent of the (1,1,1) region along the detuning line (approximately joining the centres of the two charge transfer lines) is measured by a projection onto the gate voltage axis that is on the same side as the QPC detector used in the measurement. It is the resulting gate voltage range that is set equal to |ε+ε|, and this is used for comparison between regimes of (1,1,1) regions with different widths. Black is low, orange is medium and yellow is high. The pulse shape is in the Supplementary Information. The detuning line makes a −51.3° angle with respect to the V 1axis in the V 1V 2 plane, permitting both sides of the spin arch to be observed. The dashed line is the theoretical fit (with detuning-dependent interdot couplings included).

In this paper we concentrate on two scenarios. In the first scenario, at each point in the stability diagram the exchange coupling to the centre spin from one or both of the edge spins is minimal (that is, one edge spin resembles a passive spectator). This configuration is used as a control to confirm that our device maps onto two-spin results in this limit9. In the second scenario a true three-interacting-spin regime is achieved. (Results from a third intermediate regime are shown in the Supplementary Information.)

The energy level spectrum of a TQD (ref. 14) consists of quadruplets Q with total spin S=3/2 separated by the Zeeman energy in a magnetic field and doublets Δ′ and Δ with S=1/2. The two states of our qubit consist of one of the quadruplets, Q3/2, and one of the doublets, Δ′1/2, where

with , and where JLC ( JRC) is the exchange coupling between the left (right) and centre spins. (Other three-spin states are described in more detail in the Supplementary Information.)

Figure 1b illustrates the three-spin energy spectrum as a function of detuning (zero detuning is defined as the centre of the (1,1,1) regime as shown). Experimentally we can tune the (1,1,1) region size by using gate C primarily15. The eigenvalues of the four lowest states relevant for our experiments are:

The hyperfine interaction16 couples the state Δ′1/2 to the state Q3/2(Q1/2) at their anticrossing (asymptotic approach), see Fig. 1c. (Q1/2 and Δ1/2 are also hyperfine coupled.) Figure 1c also illustrates the two types of experiment we describe in this paper. With the single anticrossing (SA) pulse, based on the methodology in ref. 9, the system starts in the Δ′1/2 state in the (2,0,1) (or (1,0,2)) regime and then a pulse is applied to reach the (1,1,1) regime. The pulse rise time (see Supplementary Information) ensures that Landau–Zener (LZ) tunnelling creates a coherent superposition of Q3/2 and Δ′1/2 on passage through the anticrossing. After a state evolution time, τ, the pulse steps down, completing the spin interferometer on the return passage through the anticrossing. The probability of the Δ′1/2 state occupation, PΔ′1/2, is directly obtained by this projection back into the (2,0,1) (or (1,0,2)) regime, where the required spin-to-charge information conversion is achieved by the Pauli blockade17 of the Q3/2 state. An experiment with a double anticrossing (DA) pulse is also illustrated in Fig. 1c. The sequence is similar, with the important distinction that a larger pulse enables LZ tunnelling processes through both anticrossings before again projecting back in the (2,0,1) regime having passed through both anticrossings twice. Important calibration information is obtained if the pulse time is longer than the coherence time (that is, τ>T2*) where the mixing at the Δ′1/2−Q3/2 anticrossing is detected independently of coherence effects. Figure 1d plots this against magnetic field for a 9-mV-wide (1,1,1) regime midway between the narrow and wide (1,1,1) regimes. The two anticrossings form a ‘spin arch’ which is used to extract the coupling parameters for the model.

The distinction between our two regimes is now clear. In the case of a wide (1,1,1) region, close to zero detuning, both JLC and JRC~0, so EΔ′1/2EΔ1/2EQ1/2. Away from zero detuning only two of the spins are coupled: right–centre (left–centre) at negative (positive) detuning. Experiments using DA pulses in this regime involve coupling to not only Q3/2 but also to Q1/2. Thus this regime is not suitable for a two-level system involving three interacting spins. As a control experiment, however, in Fig. 2 we plot the coherent Landau–Zener–Stückelberg (LZS) oscillations obtained in this regime for both positive and negative detuning with a SA pulse. These compare to the first LZS experimental results with DQDs from ref. 9, later described theoretically in refs  18, 19. The degree of LZ tunnelling, that is, the relative size of A and B in the coherent A|Δ′1/2right fence+Beiϕ(t)|Q3/2right fencestate, depends on the speed, v, through the anticrossing: PLZ=e−(2πΔ2/planckv) , where 2Δ is the energy splitting at the anticrossing. The visibility of the oscillations is a balance between this speed and T2*. For an infinite T2*, a rise time ~0.2μs would produce a 50/50 superposition (see also ref. 9). Experimentally it is found that a 6.6ns pulse rise time (or 3.3ns Gaussian time constant) leads to oscillations with the highest visibility. The value of T2*, obtained from a single parameter fit to the data, ranges from 5 to 18ns, consistent with previous DQD experiments where T2* was limited by fluctuations in the nuclear field environment1.

Figure 2: LZS oscillations from the two Δ′1/2−Q3/2 qubits for a wide (1,1,1) region.
LZS oscillations from the two [Delta][prime]1/2-Q3/2 qubits for a wide (1,1,1) region.

a,b, Numerical derivative of the conductance with respect to detuning showing LZS oscillations versus pulse duration τ. Black is low, red is medium and yellow is high. Panel a shows measurements with the right QPC for  |ε+ε|=27mV along V 1; the pulse goes across the (1,0,2) to (1,1,1) charge transfer line at B=60mT. Both V 2 and V 1 are swept to detune parallel to the pulse direction in the V 1V 2 plane. Panel b shows measurements with the left QPC for |ε+ε|=41.5mV along V 2; the pulse goes across the (2,0,1) to (1,1,1) charge transfer line at B=60mT. c,d, Probability of ending in the Δ′1/2 state as a function of τ with fits for T2*. For the right QPC (c) the pulse goes from (1,0,2) to (1,1,1) and |ε+ε|~50mV along V 1. For the left QPC (d) the pulse goes from (2,0,1) to (1,1,1) and |ε+ε|=27mV along V 2. The experimental data are shown as points, whereas the theoretical fits are shown as red lines. The values of T2* extracted from the single parameter fit to the LZS model are indicated.

In Figs 3 and 4 we show results for experiments with DA pulses in a narrow (1,1,1) regime, where JLC and JRC are finite throughout and two well-defined qubit states exist between the two anticrossings (that is, simulations based on experimentally extracted parameters confirm that Δ′1/2 has moved far enough below the Q1/2 state such that no experimental features are related to interactions with the Q1/2 state). The energy level diagrams for this regime are shown in Fig. 4a. The stability diagram, measured in the presence of a fixed amplitude DA pulse at 25mT, is shown in Fig. 3a. The results reveal LZS resonances parallel to both charge transfer lines, consistent with theoretical simulations (Fig. 3b) and confirming that coherence is maintained as the Δ′1/2 state is transformed from one dominated by coupling between left and centre spins to one dominated by right and centre spins, effectively demonstrating coherent pairwise exchange control.

Figure 3: Coherent three-spin state manipulation with a narrow (1,1,1) region
Coherent three-spin state manipulation with a narrow (1,1,1) region

a, Stability diagram in the presence of a pulse (drawn as a white line for a given (V 1, V 2)), showing coherent LZS oscillations in the (2,0,1) region with features parallel to both charge transfer lines. The colour map (black is low, red is medium and yellow is high) corresponds to the numerical derivative of the left QPC conductance with respect to V 2in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1). The (1,1,1) region is tuned to a width of ~5mV with gate C. B=25mT. The stability diagram also shows LZS oscillations involving (2,0,2) and (1,1,2). b, Calculated dPΔ′1/2/dV 2 map zooming mainly into the (2,0,1) region of the stability diagram from a. The dashed line shows where the addition line is expected, although it is not part of the calculation. B=40mT. c,d, Traces of dPΔ′1/2/dV 2 versus τ. For c the data points are extracted from Fig. 4b (40mT, white line) at V 2=−1.0,751V. For d the data points are extracted from Fig. 4b (40mT, blue line) at V 2=−1.074V. The fits (red lines) use B=60mT. The values of T2* extracted from the fits are indicated.

Figure 4: Magnetic field dependence of coherent three-spin state manipulation with a narrow (1,1,1) region.
Magnetic field dependence of coherent three-spin state manipulation with a narrow (1,1,1) region.

a, Energy spectra for the three-spin states for different magnetic fields. The colour code for the states is the same as in Fig. 1b. From left to right we have: B=5mT and |ε+ε|=3.9mV; B=25mT and |ε+ε|=5.1mV; B=40mT and |ε+ε|=5.6mV; and B=60mT and |ε+ε|=4.6mV. b, Coherent oscillations shown in the τV 2 plane as the numerical derivative of the left QPC conductance with respect to V 2 (black is low, red is medium and yellow is high) in the presence of a pulse across the charge transfer line between (2,0,1) and (1,1,1). The (1,1,1) region is tuned to a width of ~5mV with gate C. V 1 is swept proportionally to V 2 to detune parallel to the pulse direction. The magnetic field and (1,1,1) region sizes from left to right are as in a. The white dot in the B=40mT map indicates a coherent oscillation resulting from a DA pulse reaching past the far Δ′1/2−Q3/2 anticrossing. c, Calculated dPΔ′1/2/dV 2 maps (black is low, red is medium and yellow is high) in the τε plane for the same experimental settings as for b. The magnetic field and (1,1,1) region sizes from left to right are as in a. To keep the fringes clearer, dephasing is not included. The very rapid oscillations in the upper right corner of the figures are an artefact due to the large exchange energy past the far Δ′1/2−Q3/2 anticrossing. At B=5mT, the anticrossings have merged, so there is only one boundary in the diagram (white dashed line). At B=60mT, two Δ′1/2−Q3/2 anticrossings are recovered (see the two white dashed lines). At B=25mT, for both theory and experiment, dotted white curves are drawn as a guide to the eye for the peak of an oscillation in between the two Δ′1/2−Q3/2 anticrossings. We note that a small DNP effect20 is present which depends on the size of the (1,1,1) region, details of pulse shape and pulse orientation. In b,c (25 and 40mT), it is found that a DNP ~20mT is required to properly describe the period of oscillations. This is why the stability diagram in Fig. 3b is calculated at 40mT rather than 25mT, and why the fits in Fig. 3c,d are calculated at 60mT instead of 40mT.

To gain further insight, Fig. 4b and c show experimental and theoretical plots of the pulse duration dependence of LZS oscillations at different magnetic fields. Two boundaries, marked with horizontal white dashed lines, can be observed at fields above 25mT. The region between the boundaries corresponds to the regime between the two anticrossings, while the resonances correspond to LZS oscillations. It can be seen (for example, curved dotted lines) that the resonances double back on themselves. This is a direct observation of tracking the resonance across the maximum in the Δ′1/2 versus detuning curve (see Figs 4a and 1b). We speculate that operating at this spot may provide more protection from charge noise, as the energy levels become locally flat versus detuning.

Although the frequency of coherent oscillations grows with field, owing to the increased spacing between the two qubit levels, it seems as if the experiment and theory differ by 20mT for experimental data at 40mT and by 15mT for data at 25mT. We attribute this to a dynamic nuclear polarization effect (DNP; ref. 20). To make this quantitative we extract horizontal slices in Fig. 4b at 40mT (blue and white lines) and fit them to obtain T2*. The data are consistent with a 20mT DNP effect. It is found experimentally that the values of T2* for the three-spin qubit experiments in Fig. 3c,d (8–15ns) are within error identical to the values from the two-spin qubit experiments. This is consistent with T2* being dominated by local uncorrelated nuclear field fluctuations as both sets of qubit states differ by the same total spin21. Finally, we note that we also observe a resonance beyond the second anticrossing, marked with a white spot in Fig. 4b. This is a non-trivial feature corresponding to a resonance condition of two interacting spin interferometers, one between the two anticrossings and a second, beyond the second anticrossing.

In conclusion, we have demonstrated coherent control of a qubit based on three-interacting-spin states. We have confirmed that there is no detectable change in the coherence time in the three-spin experiments compared with the two-spin experiments. We have realized the pairwise control of exchange for a three-spin system by pulsing the detuning energy of a triple quantum dot. The same technique should carry over when more quantum dots are added in series to increase the number of qubits. Pairwise control of exchange, as demonstrated here, will then be useful for building complex quantum algorithms based on electron spin qubits in quantum dots.

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Acknowledgements

We thank D. G. Austing, W. Coish and E. Laird for discussions and O. Kodra for programming. A.S.S. and M.P-L. acknowledge funding from NSERC. G.G., A.K, M.P-L., and A.S.S. acknowledge funding from CIFAR. G.G. acknowledges funding from the NRC-CNRS collaboration.

Author information

  1. These authors contributed equally to this work

    • L. Gaudreau &
    • G. Granger

Affiliations

  1. Institute for Microstructural Sciences, National Research Council Canada, Ottawa, Ontario K1A 0R6, Canada

    • L. Gaudreau,
    • G. Granger,
    • A. Kam,
    • G. C. Aers,
    • S. A. Studenikin,
    • P. Zawadzki,
    • Z. R. Wasilewski &
    • A. S. Sachrajda
  2. Département de physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada

    • L. Gaudreau &
    • M. Pioro-Ladrière

Contributions

Z.R.W. developed and grew the 2DEG heterostructure free of telegraphic noise; A.K. fabricated the triple quantum dot device capable of reaching the few-electron regime; P.Z., L.G. and S.A.S. designed and built the high-frequency lines up to 50GHz at millikelvin temperatures; P.Z., L.G. and G.G. ran the cryogenic equipment; L.G., G.G., S.A.S. and M.P-L. developed the pulsing techniques; G.G., L.G. and S.A.S. performed the measurements; G.G., L.G. and G.C.A. analysed the data; G.C.A. performed theoretical simulations; G.G. and A.S.S. wrote the manuscript and Supplementary Information with input from all authors; G.G., L.G. and G.C.A made the figures; G.G., L.G., G.C.A., S.A.S., M.P-L. and A.S.S. participated in discussions concerning the experimental and theoretical results; and A.S.S. supervised the project.

Competing financial interests

The authors declare no competing financial interests.

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