# Bismuth in silicon qubits: the role of EPR cancellation resonances

## Abstract

We investigate theoretically and experimentally the electron paramagnetic resonance (EPR) spectra of bismuth doped silicon (Si:Bi) at intermediate magnetic fields, T. We identify a previously unexplored EPR regime of “cancellation-resonances”- where the non-isotropic part of , the Ising part of the hyperfine coupling, is resonant with the external field-induced splitting. We show this regime has interesting and experimentally accessible consequences for spectroscopy and quantum information applications. These include reduction of decoherence, fast manipulation of the coupled nuclear-electron qubit system and line narrowing in the multi-qubit case. We test our theoretical analysis by comparing with experimental X-band (9.7 GHz) EPR spectra obtained in the intermediate field regime.

###### pacs:

03.67.Lx,03.67.-a,76.30-v,76.90.+d,Following Kane’s suggestion (1) for using phosphorus doped silicon as a source of qubits for quantum computing, there has been intense interest in such systems (2). The phosphorus system (P) is appealing in its simplicity: it represents a simple electron-spin qubit coupled to a nuclear-spin qubit via an isotropic hyperfine interaction of moderate strength ( MHz). However, recent developments (3); (5); (4) point to Si:Bi (bismuth doped silicon) as a very promising new alternative. Two recent studies measured spin-dephasing times of over 1 ms at K which is longer than comparable (non-isotopically purified) materials, including Si:P (3); (4). Another group implemented a scheme for rapid (on a timescale of s) and efficient (of order ) hyperpolarization of Si:Bi into a single spin-state (5).

Bismuth has an atypically large hyperfine constant GHz and nuclear spin . This makes its EPR spectra somewhat more complex than for phosphorus and there is strong mixing of the eigenstates for external field T. Mixing of Si:P states was studied experimentally in (6), by means of electrically detected magnetic resonance (EDMR), but at much lower fields T. Residual mixing in Si:Bi for T, where the eigenstates are pure uncoupled eigenstates of both and , was also proposed as important for the hyperpolarization mechanism of illuminated Si:Bi (5). In (4) it was found that even a reduction in the effective paramagnetic ratio (where is the transition frequency) lead to a detectable reduction in decoherence rates.

Below we present an analysis of EPR spectra for Si:Bi and test the results with experimental spectra. We identify a series of regimes for which , explaining them in a unified manner as a series of EPR “cancellation resonances”; some are associated with avoided level-crossings while others, such as a maximum shown in ENDOR (7) spectra at T in (4) is of a quite different origin. These cancellation resonances represent, to the best of our knowledge, an unexplored regime in EPR spectroscopy, arising in systems with exceptionally high and . They are somewhat reminiscent of the so-called “exact cancellation” regime, widely used in ESEEM spectroscopy (7); (8), but differ in essential ways: they affect both electronic and nuclear frequencies rather than only nuclear frequencies; they concern only the non-isotropic component of the interaction (and are thus not “exact”; indeed the T point is not even a full cancellation). They have important implications for the use of Si:Bi as a coupled electron-nuclear qubit pair: we show all potential spin operations may be carried out with fast EPR pulses (on nanosecond timescales) where in contrast, most operations for Si:P require slower NMR pulses (on microsecond timescales). A striking spectral signature is reduced sensitivity to certain types of ensemble averaging, giving an analog to the ultra-narrow lines well-known in “exact cancellation”, as well as the reduction of decoherence. Further details are found in (12).

We model the Si:Bi spin system approximately by a Hamiltonian including an isotropic hyperfine coupling term:

(1) |

where represents the frequency of the external field and represents the ratio of the nuclear to electronic Zeeman frequencies. For , there are 20 eigenstates which can be superpositions of high-field eigenstates ; but since , the basis is at most mixed into a doublet with constant . One can thus write the Hamiltonian for each sub-doublet as a dimensional matrix (where ):

(2) |

and where is the rescaled field, represent Pauli matrices in the two-state basis and is the identity operator. It becomes clear that whenever , the quantum states become eigenstates of . Thus at , the eigenstates, , assume Bell-like form: ( ). In contrast, “exact cancellation” results in a simple superposition of the nuclear states, which also allows other types of manipulations (9). Since , only states with can yield resonances where the field-splitting term is eliminated. In this case, they occur at , corresponding to applied field Tesla; below we show that all points which are minima occur midway between these resonances. But the maximum at the T resonance, and seen in experiments (4) is shown to be of a different type.

It is standard practice to represent two-state quantum systems using vectors on the Bloch sphere (8). We define a parameter where represents the vector sum magnitude of spin and components. Denoting as the inclination to the -axis, and ; then Eq.2 can also be written:

(3) |

Straightforward diagonalisation gives the pair of eigenstates, for each , at arbitrary magnetic fields :

(4) |

where:

(5) |

and the corresponding eigenenergies:

(6) |

In Fig.1 the simple (but exact) expression Eq.6 reproduces the spin spectra investigated in e.g. (3) and (5). Eqs.5 are valid for all states except the unmixed states ( and ). For , there is no coupling: these two states are unmixed for all magnetic fields, thus and and Eq.6 simplifies drastically to give .

For the doublets, the are the dominant coefficients at high-field. Then, the
limit corresponds
to angle , so and and the
states become uncoupled. The cancellation resonances correspond
to so , while .
The EPR emission transitions
are dipole allowed at all
fields. Their
relative intensities

.
Since ,
variations in line intensities arise from mixing of the states. Thus,

(7) |

If mixing is significant transitions (of intensity ) and transitions (of intensity ), EPR-forbidden at high field, become strong, with relative intensities:

(8) |

Forbidden lines disappear at high fields; as , one can see from Eq.5 that since at high fields. Eqs.6, 7 and 8 are exact so are in complete agreement with numerical diagonalisation of the full Hamiltonian.

In Fig.2a we test the equations against experimental spectra at Tesla and microwave frequency 9.67849 GHz. The long spin relaxation times at low temperatures means that the EPR spectra are easily saturated, complicating the analysis of the line intensities. We therefore used a temperature of 42 K so as to measure unsaturated resonances. The shorter relaxation times at these elevated temperatures may be due to the presence of significant numbers of conduction electrons that are no longer bound to Si:Bi donors. We measure a very broad microwave absorption centred on zero magnetic field (subtracted from Fig.2a) which we attribute to these conduction electrons. The comparison with experiment shows excellent agreement with the positions of the resonances, which are far from equally spaced in the low-field regime. Experimental lines are found to be Gaussians of width mT; this was attributed in (3) to the effects of Si in this sample; samples with enriched Si are expected to give much narrower linewidths.

For Fig.2b, we generated the spectra, convolved with mT Gaussians to obtain the EPR spectra of all lines (both allowed and forbidden) at all frequencies below 10GHz. We indicate the main dipole allowed lines as well as indicating the approximate position of the main resonances. The spectra show a striking landscape of transitions which show maxima or minima where and double-valued EPR resonant fields (i.e. transitions with EPR resonances at two different magnetic fields). No Boltzmann factor has been included in the simulation. The nuclear field splittings are unresolved and extremely small; they do not affect line intensities significantly. To simplify our discussion, we neglect the tiny nuclear shifts , but include them whenever spectroscopically significant.

The well-studied 4-state , Si:P system can be mapped onto a two-qubit basis. With a 20-eigenstate state-space, the Si:Bi spectrum is more complex, but we can identify a natural subset of 4 states, which represents an effective 2-coupled-qubit analogue:

As hyperpolarization initialises the spins in state and this state has both the electron and nuclear spins fully anti-aligned with the magnetic field, it can be identified with the state. The other states are related to it by either one or two qubit flips, just as in the Si:P basis.

To have a universal set of gates for quantum information it is known to be sufficient to be able to perform arbitrary single qubit manipulations and a control-NOT (CNOT) gate (11). In the two qubit system described here, arbitrary electronic qubit-only manipulations can be performed with radiation pulses exciting transitions between states and , while single nuclear-qubit rotations correspond to and . The CNOT gate (for example using the nuclear spin as a control qubit) is even simpler as it requires only a pulse connecting (12).

The electronic flips are EPR allowed at all fields for both Bi and P donors so can be performed in a time on the order of 10 ns (3). The nuclear transitions, however, require a slower, (of order microseconds) NMR pulse for Si:P. For Si:Bi, on the other hand, at the resonance the nuclear and electronic transition strengths become exactly equal as may be verified by setting and in Eqs.(7) and (8). Time-dependent calculations (12) show that the duration of a pulse is also equalized.

The resonances yield textbook level anti-crossings
as well as “Bell-like” eigenstates.
Were it to become possible to vary the external field
sufficiently fast to produce sudden, rather than adiabatic evolution of the eigenstates
it would be possible to transfer the former to the high-field regime.
Unfortunately, ramping magnetic fields (up or down) sufficiently fast to violate adiabaticity,
though not impossible, would require some of the fastest magnetic field pulses ever
produced (eg T/s obtained by (10)).
However, we show that adiabatic magnetic field sweeps already achievable by
ordinary laboratory magnetic pulses ( T/ms) suffice to already achieve new
possibilities.

The frequency minima at 5-8 GHz: fields at which are expected to lead to a reductions of decoherence, since sensitivity to magnetic fluctuations is minimised; a measurable reduction has been seen (4) by varying the ratio by . In Fig.(2) we see several transitions have a minimum frequency. These minima (in effect of ) occur for:

(9) |

Thus ; the consequence is that the minima lie exactly midway in angular coordinates between cancellation resonances. For example, for the line () the minimum is at so T. Here, the doublet has passed its resonance point at T (for which ) by an angle and the resonance is at an equal angular distance before its resonance at T: thus while . Both doublets are quite close to the Bell-like form.

Line narrowing: an interesting and unexpected consequence has applications to studies with larger numbers of spins. A pair of Bi atoms, interacting via a spin-exchange term of the form will result in splitting of the EPR spectral lines (with an energy splitting of order ). However this is suppressed near the cancellation resonances. Analogously to “exact cancellation”, this makes the system less sensitive to ensemble averaging. For exact cancellation, this means the averaging over different orientations in powder spectra; here it means magnetic perturbations including spin-spin interactions. Fig.3 (left panel) plots the signal for GHz for a single pair of Si:Bi atoms and clearly shows the line splitting away from the resonances. The right panels show the effects of averaging many spectra each corresponding to different (with an average GHz and width GHz). While typical spectra show a broad feature of width , at the cancellation resonance the line width remains strikingly narrow (close to the single atom line width).

A frequency maximum at and T is marked with a in Fig.2b. We can show that a point which is a maximum (in effect of ) implies:

(10) |

This condition does not correspond to the elimination of the field splitting terms; instead it implies , thus equalizing the Bloch angle for the associated energy levels. In this sense it is somewhat different to the other cancellation resonances; nevertheless, it still provides a point and thus some potential for reducing broadening and decoherence. In (12), it is shown that the resonance offers new possibilities for copying and storing qubit states. At , the most drastic cancellation resonance occurs, since both and terms in Eq.2 are eliminated, leaving only the isotropic term. Although there is no or line narrowing here, there is a possibility of driving, by a second order process, simultaneous qubit rotations e.g. see (12).

Conclusions: In the intermediate-field regime ( T) the exceptionally large values of and for Si:Bi generate a series of cancellation resonances. They are associated not only with level crossing structures but also with more subtle and not previously studied effects: both line broadening and decoherence effects may be reduced; also, if the electronic and nuclear spins of Si:Bi are used as a 2-coupled qubit system, the cancellation resonances allow a universal set of quantum gates to be performed with fast EPR microwave pulses, eliminating the need for slower radio frequency addressing of the nuclear qubit. One scheme would envisage the following stages: (1) hyperpolarization of the sample into state (in which the 2-qubit system is initialized as ) at T. (2) A magnetic field pulse ( T/ms, of duration lower than decoherence times) would reduce to T. (3) As the pulse ramps up, a series of EPR pulses would execute a series of gates and operations on the system. (4) As the magnetic pulse decays, the system is restored to the high- limit, leaving it in the desired superposition of and basis states. Thus, given the capability to rapidly (ms) switch from the high to intermediate field regime, Si:Bi confers significant additional possibilities for quantum information processing relative to Si:P.

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