Proposal for a Loophole-Free Bell Test with Electron Spins of Donors in Silicon ()
1. Introduction
Most working scientists hold fast to the concepts of “realism” according to which an external reality exists independent of observation and “locality” which means that local events cannot be affected by actions in space-like separated regions [1]. The significance of these concepts goes far beyond science. Based on these deep-rooted reasonable assumptions, in their seminal 1935 paper, Einstein, Podolsky, and Rosen (EPR) advocated that quantum mechanics is incomplete [2]. The EPR arguments about the physical reality of quantum systems is shifted from the realm of philosophy to the domain of experimental physics since 1964 when Bell and others constructed mathematical inequalities—one of the profound scientific discoveries of the 20th century [3,4], which must be satisfied by any theory based on the joint assumption of realism and locality and be violated by quantum mechanics. Many experiments [1,5-12] have since been done that are consistent with quantum mechanics and inconsistent with local realism. So far, however, all these tests suffered from “loopholes” allowing a local-realistic explanation of the experimental results by exploiting either the low detector efficiency [13,14] or the timelike interval between the detection events [15,16]. The first loophole names the detection loophole allowing the possibility that the subensemble of detected events agrees with quantum mechanics even though the entire ensemble is consistent with Bell inequalities. So a fair-sampling hypothesis that the detected events represent the entire ensemble must be assumed. The second refers the locality or “lightcone” loophole allowing the correlations of apparently separate events resulting from unknown subluminal signals which propagate between space-like regions of the apparatus to take place.
Several schemes were proposed closing these loopholes based on entangled photon pairs [14,17], Hg atoms [18], Rydberg atoms [19], trapped ions [20], or non-Gaussian states of light and balanced homodyning [21,22], but all face a formidable experimental challenge. Here we discuss a scheme for the loophole-free Bell test based on the Kane Si:P architecture [23], in which two qubits are encoded onto two electron spins of donor atoms31P in nanoscale doped silicon electronic devices in two high Q cavities 300 m apart. Entanglement between the two qubits is created by using bright coherent-light pulses which interact with the donor atoms 31P through a weak dispersive light-matter interaction, respectively, via homodyne detections and postselections [24]. The qubits are then read out with perfect efficiency and high accuracy above 99% in about 0.6 µs using optically induced spin to charge transduction [25-27]. The realization of the setup is within the ability of the current semiconductor fabrication technology [28,29], and the read-out of the single donor electron spin can be realized through resonant spin-dependent charge transfer where the resulting electron current is measurable using radio-frequency single electron transistor (rf-SET) [23,25-27,30].
2. Bell Inequality Test
A Bell measurement of inequality of Clauser, Horne, Shimony, and Holt (CHSH) [4] comprises of three basic ingredients. First, a pair of particles entangled with each other is prepared in a repeatable starting configuration. Then a variable classical manipulation is applied independently and randomly to each particle; these manipulations are labeled as 
and
. At last, a classical property with two possible outcome values 1 and −1 is measured for each of the particles. The correlation is measured by repeating the experiment many times and can be expressed as
(1)
where 
and 
denote the numbers of measurements in which the measured results are the same or different, respectively. The CHSH form of Bell inequalities states that the correlations resulting from local realistic theories must satisfy:
(2)
where 
and 
(
and
) are specific values of 
(
). For a Bell measurement based on electron spins, we have
(3)
where 
with 
(i = A, B, and
) being the Pauli matrices, and 
(
) are unit vectors. The CHSH inequality (2) is maximally violated by quantum mechanics at certain sets of 
and
, one such set is that both of the 
(
) are in the xy-plane, and the polar angles of 
(
) are
, 
for
, and
, 
for
. For these phase angles and state
, the quantum mechanics gives
(4)
The architecture of the basic phosphorus 31P donor electron spin qubit in silicon with control gates and a resonant readout mechanism are shown in Figure 1. The donors serve to localize the electron spins in space which encode quantum information in the conventional fashion as 
and
, and to provide local qubit addressability through the electron nuclear hyperfine interaction [23]. Indirect spin detection involves transfer of the spin information to the charge degrees of freedom through a spin-dependent tunneling process, during which the resulting electron current can be detected by an ultrasensitive electrometer, rf-SET [25]. This concept depends on the application of a small dc electric field 
and an ac electric field 
with the amplitude 
resonant at the energy gap 
of the two states 
and 
to induce the tunneling of the qubit electron to a secondary (spin polarized) “SET-donor”. Here 
denotes the state with two electrons being bound to the same donor formed by the tunneling. The resulting charge re-distribution can be detected by a rf-SET. This detection is equivalent to the measurement of the qubit spin state, because the tunneling event 
will be Pauli blocked if the qubit and the SET-donor electron spins are parallel.
To generate entanglement between donor qubits at Alice’s and Bob’s sites 300 m apart, a bright coherent pulse sequentially interacts with the qubits, entangled qubit pairs will then be postselected conditioned upon the results of probe homodyne measurements. For a sufficient dispersive interaction between the donor electron and the light, the system should be placed in a cavity resonant with the light. For the cavity, weak coupling is
Figure 1. (Color online) Schematic picture of the device for the resonant spin-dependent charge transfer of a single electron of a donor 31P in silicon. The tunneling event D0D0 ® D+D– is Pauli blocked if the qubit donor and the SET-donor spins are parallel.
sufficient, but a high value of 
is required, where Q is the quality and V is the mode-volume of the cavity [24].
The donor electron spin system in the cavity is treated as a 
system with two stable and metastable ground states 
and
, and an excited state 
provided by the bound-exciton state. For the coherent pulses, the transition between 
and 
is suppressed due to a prohibitive selection rule and only 
and 
participate in the interaction with the cavity mode [31]. Assuming that the state of the qubits in Alice’s and Bob’s sites are initially prepared in the states 
and
, respectively, and the two SET 31P are in 
state that makes them not to take part in the interaction with the bright coherent pulses. The coherent light is sufficiently detuned from the transition between 
and the excited state to allow for a strictly dispersive light-matter interaction. When the probe pulse in coherent state 
reflects from the cavity at the Alice’s sitethe total output state may be described by [24]
(5)
The probe beam is then sent to the cavity at the Bob’s site and interacts with the qubit donor in the same way. Applying a further linear phase shift of θ to the pulse after it leaves the cavity will yield the total state
(6)
where 
with the conventional denotation 
and 
hereafter. In the presence of channel loss, we may model the photon loss by considering a beam splitter in the channel that transmits only a part of the probe pulse with transmission 
[24]. Tracing over the lost photons introduces decoherence and the total state can be described by density matrix ρ, with the following diagonal part for the pulse states
(7)
where
(8)
Here 
is the decoherence factor arising from the dispersive light-matter interaction in the cavities,
, and an extra phase
can be set to be naught, since it is independent of the measurement results and can be locally removed via static phase shifters.
With the balanced homodyne detection [32], the success probability of generating entanglement between two qubits at Alice’s and Bob’s sites is found to be [24]
(9)
where
, 
, and 
is the selection window of the homodyne measurements. The desired entangled output state is
, so the average fidelity after postselection has the form [24]
(10)
For the state obtained through the postselection with the configuration for 
(
) aforesaid, the violation of the CHSH inequalities reads
(11)
After the pulse leaves the cavity at the Bob’s site, Alice and Bob randomly and dependently manipulate the electron spins of the SET 31P from the initial state 
to the state 
corresponding to 
or the state 
and to state
or the state
, respectively. These manipulations on the SET 31P electron spins equivalent to the actions on the qubit donor spins can be finished in 0.1 µs [33]. Then Alice and Bob read out the electron spins of the qubit 31P using optically induced spin to charge transduction detected by the rf-SET. If the qubit and SET-donor electron spins are parallel, the tunneling event 
will be Pauli blocked, thus the rf-SET will detect nothing, which will be assigned a value +1. Otherwise, the rf-SET will detect a current signal, and the outcome will be assigned –1. The outcomes of every experiment can be used to compute the correlation function
, so that fair-sampling hypothesis is not required. In this case, the issue of the detection efficiency is replaced by the detection accuracy
. The main origin of the inaccuracy in the detection of electron spins comes from the imperfection in the spin to charge transduction, i.e., the tunneling event 
may not happen even if the qubit and SET-donor electron spins are antiparallel. This leads to an experimental result that the probability of the qubit spin parallel to the SET-donor spin will be larger than that obtained with the perfect detection accuracy. The read-out of the qubit spin with the detection accuracy 
may be possible according to [27,30]. Considering the detection accuracy 
and the detection error rate
, the violation of the CHSH inequalities B in Equation (11) may be rewritten as
(12)
Assuming the telecom fiber and wavelength where losses are about 1 dB/km [34], the transmission parameter of 300 m is
. For 31P donor impurity in silicon, distinguishability of 
corresponding to 
and 
is achievable [31]. Assuming that the decoherence factor 
and the detection accuracy
, from Equations (9), (10) and (12), we obtain the results shown in Figure 2. Assuming that the selection window 
and other parameters unchanged, we get the relation between the violation of the CHSH inequalities B and the detection accuracy 
as shown in Figure 3. When 
and
, we have
, 
and
. Even the detection accuracy is so low that
, we still have 
for
. With a repetition rate of 500 kHz and
, the number of data samples would be 150,000 per second, thus the whole Bell test experiments would be finished in less than one second.
Building the setup shown in Figure 1 is within the reach of the current technology. The two donors of distance 20 - 30 nm can be placed through random doping techniques, though only a small percentage of such devices will work properly [23]. The three gates with lateral dimensions and separation ~10 nm can be patterned on the surface through technologies such as self-assembly and the use of extreme ultraviolet radiation, x-rays, and electron beams [28,35]. A workable transistor with a gate length of 6 nm has already been realized in 2002 [28]. The two qubit states have an energy split 
meV for an applied 8.6 T magnetic field. The exited state 
is about 1.09 eV above the ground state [36]. The probe coherent pulses of wavelength about 1650 nm (
eV) (Figure 4) are far detuned from the transition
, but on resonant with the cavity with frequency
, of which the device of dimensions about 50 nm × 30 nm × 30 nm is at the antinode [31].
Figure 3. The violation of the CHSH inequalities
versus the detection accuracy 
with
. See the text for the values of other parameters.
3. Conclusions
As a summary, we present a scheme for the loopholefree test of the Bell inequalities. The detection efficiency of donor electron spins is unity using the optically induced spin to charge transfer detected by a rf-SET, and the fair sampling assumption is not required, thereby the detection loophole in this scheme is closed. The two qubit donors are 300 m apart, and the time of the random and independent measurement of the two qubits by Alice and Bob, respectively, is within 0.7 µs, thus the lightcone loophole may be closed too. The experimental realization of this scheme is within the reach of the current technology. Large violation of the CHSH inequality 
for the detection accuracy 
is achievable. Even if the detection accuracy is so low that
, we may still have
. This scheme may open a promising avenue towards a complete experimental Bell test which has a profound significance far beyond science.
4. Acknowledgements
This work was supported by the National Nature Science Foundation of China (Grant No.10874071, 50672088, 11072218, 11005031, and 60571029), by Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6110314), by Scientific Research Fund of Zhejiang Provincial Education Department (Grant No.Y200909693), and by Science Foundation of Zhejiang Sci-Tech University (ZSTU) under Grant No.0913842-Y.