Home
Search
Collections
Journals
About
Contact us
My IOPscience
Sensitive detection of Majorana fermions based on a hybrid spin–microcantilever via enhanced spin resonance spectrum
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 195501 (http://iopscience.iop.org/0957-4484/26/19/195501) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 169.230.243.252 This content was downloaded on 21/04/2015 at 12:17
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 26 (2015) 195501 (8pp)
doi:10.1088/0957-4484/26/19/195501
Sensitive detection of Majorana fermions based on a hybrid spin–microcantilever via enhanced spin resonance spectrum Wen-Hao Wu1 and Ka-Di Zhu2 1
Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, Peopleʼs Republic of China 2 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, Peopleʼs Republic of China E-mail: [email protected] Received 7 December 2014, revised 10 March 2015 Accepted for publication 12 March 2015 Published 21 April 2015 Abstract
Motivated by recent experimental progress towards the detection and manipulation of Majorana fermions in ferromagnetic atomic chains on a superconductor, we present a novel proposal based on a single-crystal diamond (SCD) microcantilever with a single nitrogen-vacancy (NV) center spin embedded in ultrapure diamond substrate to probe Majorana fermions in an all-optical domain. With this scheme, a possible distinct Majorana signature is investigated via the electron spin resonance spectrum. In the proposal, the SCD microcantilever behaves as a phonon cavity and is robust for detecting of Majorana fermions, while the NV center spin can be considered as a sensitive probe. Further, the vibration of the microcantilever will enhance the coupling effect, which makes the Majorana fermions more sensitive to detection and the well-established optical NV spin readout technology will certainly promote the detection. This proposed method may provide a potential supplement for the detection of Majorana fermions. Keywords: Majorana fermions, hybrid spin–microcantilever, spin resonance spectrum (Some figures may appear in colour only in the online journal) 1. Introduction
[6–10] and many experimental attempts have also been devoted to identify them [11–18]. According to these attempts, the zero-bias conductance peak [19–21] as a representative property of Majorana modes is often considered as a Majorana signature. These zero-bias conductance peaks have already been observed in several experiments [13– 17]. However, it is fair to say that these experimental results cannot serve as definitive evidence to prove the existence of MFs in condensed matter systems because the zero-bias conductance peaks can appear due to many other mechanisms [22] such as zero-bias anomaly due to Kondo resonance [23] and disorder or band bending in the semiconductor nanowire (SNW) [24]. Therefore, to obtain definitive signatures of MFs, alternative setups or proposals are necessary. Due to the coherent long-range quantum effects of MFs, the understanding of genuine quantum effects on macroscopic length scales, i.e. nanoelectromechanical systems [25] and
Currently, the issue of Majorana fermions (MFs) as the most intriguing features of topological states of matter has received a great deal of attention. Besides the fundamental interest attached to the experimental discovery of Majorana fermions in nature, MFs as realized in one-dimensional topological superconductors (1DTSC) also have attractive features relating to various aspects of fundamental quantum physics. On one hand, the non-Abelian anyonic nature of MFs shows great promise for topological quantum information processing architecture [1–3]. On the other hand, the delocalized pair of MFs at the ends of a 1DTSC can be viewed as a single ordinary (spinless Dirac) fermionic zero mode leading to electron teleportation mechanisms [4, 5], i.e. coherent longrange quantum effects. By now, various systems that might host MFs in condensed matter systems have been proposed 0957-4484/15/195501+08$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
microcantilever with a magnetic tip attached to the end and a single NV spin center embedded in ultrapure diamond placed below [33, 43], in the simultaneous presence of a external magnetic field to detect MFs. By applying a strong pump microwave and a weak probe microwave to the NV center spin simultaneously, the possible Majorana signature could be probed via the spin resonance spectrum of the NV center spin, which could be detected through well-established [44] optical NV spin readout technology, which is maturely developed and vitally accurate. Moreover, the optical pump–probe technology has been demonstrated experimentally in several systems [45] and provides an effective way to investigate the light–matter interaction. Compared with electrical detection schemes, our optical scheme can avoid the heat effect and energy loss caused by the electric circuitry which will broaden the electrical response spectrum and finally affect the sensitivity of MF detection. Since the distance between the NV center spin and the MFs can be adjusted in several nanometers, the tunneling between the NV center spin and MFs can be ensured. The signal change in the spin resonance spectrum as a possible signature for the MFs is another potential piece of evidence in the ferromagnetic atomic chains. Further, the vibration of the microcantilever acting as a phonon cavity will enhance the spin resonance spectrum significantly and make the MFs more sensitive to detection. Figure 1. (a) Schematic diagram of the hybrid spin–microcantilever
system for optically detecting MFs at the ends of the ferromagnetic atomic chains on a superconductor. The inset is the energy level diagram of a NV center spin. [εi ]i = x, y, z are the diagonal components of the strain tensor defined in the NVʼs basis (note we have neglected shear). (b) An energy-level diagram of a NV center spin coupled to MFs and microcantilever.
2. Model and theory Figure 1(a) presents the schematic setup that will be studied in this work. Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin–orbit interaction. So, ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb) are the analyte at the ends of which an MF pair is expected to locate [17, 18]. To detect MFs, we employ a hybrid system consisting of a highQ SCD microcantilever with a magnetic tip attached to the end and a single NV spin center embedded in diamond placed below. By applying a strong pump microwave ωpu , a weak probe microwave ωpr and a external magnetic field Bz along the z axis to the NV center spin simultaneously, one could probe the MFs via optical pump–probe technique [45]. In such a hybrid system, the motion of the microcantilever along the z axis is quantized and described by the Hamiltonian Hr = ℏωr a+a , with ωr as the frequency of the fundamental bending mode and a and a+ as the corresponding annihilation and creation operators. On the other hand, for the NV center spin in the diamond, the schematic diagram of energy levels is shown in the inset of figure 1(a). The electronic ground state of the NV center is an S = 1 spin triplet, and we label states by ms, where m s = 0, ±1. Spin states with different values of ms are separated by a zero-field splitting of D0 ≈ 2.87 GHz. For moderate applied magnetic field Bz along the NV axis, the NV spin can be restricted to the twolevel subspace spanned by ∣ 0〉 and ∣ + 〉 and mixing of ∣ − 〉
nanooptomechanical systems [26], may be a helpful motivator. Quite excitingly, nitrogen-vacancy (NV) centers embedded in diamond are amongst the most promising implementations of quantum bits for quantum information processing [27] and nanoscale field sensors [28] because of their excellent coherence properties [29] and their ability to coherently couple to various external fields such as photons [30], nuclear spins [31], and mechanical resonators [32–35]. Promoted by recent demonstrations of high quality (Q) factor (exceeding 106 ) single-crystal diamond (SCD) mechanical resonators [36, 37] and the NV spinʼs long quantum coherence time T2 = 1−10 ms [31, 38–40], this hybrid spin– microcantilever system could be a promising alternative platform for the detection of MFs, thus paving the way for coupling MFs to nontopological qubits NV center spin embedded in diamond. Additionally, we notice that all the theoretical proposals and experimental schemes [6–17, 41, 42] have focused on electrical methods, while other effective methods such as optical schemes for detecting MFs have received less attention. In the present article, based on the recent experiment by Stevan et al [17], where a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin– orbit interaction (figure 1(a)), we will employ a high-Q SCD 2
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
and ∣ + 〉 can be neglected [34, 35]. Therefore, the Hamiltonian of the NV center spin can be described as Hs = ℏωs S z , where ωs 2π = D0 , the spin operator can be characterized by the spin operators S ± and S z . The schematic diagram of energy levels of the coupling between the NV spin and the cantileverʼs vibration is shown in figure 1(b). Therefore, the spin–microcantilever interaction can be described by [33, 35, 43] Hs − r = ℏλ (a + a+) S z and λ = gs μ B Gm a 0 is the coupling of spin to phonon, where gs ≃ 2, μ B is the Bohr magneton, Gm is the magnetic field gradient, and a 0 = ℏ 2mωr is the amplitude of zero-point fluctuations for a resonator m. Therefore, we obtain the total Hamiltonian of the coupled spin–microcantilever:
should also be noted that the third term of non-conservation for energy iℏg (S −f − S +f + ) is generally neglected. We have made the numerical calculations (not shown in the following figures) and shown that the effect of this term is too small to be considered in our theoretical treatment. So, the HMF here can be rewritten as: HMF = ℏω M (f + f − 1 2) + iℏg (S −f + − S +f ). In terms of this scheme, we apply the pump–probe scheme to the NV center spin simultaneously. We treat these microwave fields classically. The Hamiltonian of the NV center spin-transitions through microwaves is described as * e iωpu t )− μ [52] Hs − p = −μ B(S +Bpu e−iωpu t + S −Bpu B (S +Bpr e−iωpr t + S −Bpr* e iωpr t), where ωpu (ωpr ) is the frequency of the pump field (probe field), and Bpu (Bpr ) is the slowly varying envelope of the pump field (probe field). The driven spin system with one microwave field has been realized in the hybrid spin–resonator system [32]. Therefore, one can obtain the total Hamiltonian of the hybrid system as H = Hs − MC + HMF + Hs − p . In a rotating frame at the pump field frequency ωpu , the total Hamiltonian of the system reads as follows:
Hs − MC = Hr + Hs + Hs − r = ℏωr S z + ℏωs aa+ + ℏλ ( a + a+ ) S z .
(1)
Since several experiments [13–16] have reported the distinct signatures of MFs in the hybrid semiconductor/ superconductor heterostructure via electrical methods, and because Stevan et al [17, 18] give more clear observation of MFs in ferromagnetic atomic chains on a superconductor, we assure the reader that the MFs may exist in the ferromagnetic atomic chains on a superconductor under some appropriate conditions. According to these experiments, the coupling between the NV-spin and the MFs should be the tunnel coupling. It should be noted that the magnetic moment of the ferromagnetic iron chains may have an effect on the NV center spin. However, compared to the applied magnetic field H = 1 T which is the precondition for the observation of MFs in ferromagnetic atomic chains, the effect of the magnetic moment can be negligible. As each MF is its own antiparticle, + one can introduce a MF operator γMF such that γMF = γMF and 2 γMF = 1 to describe MFs. Supposing the NV center spin tunneling couples to γ2 , the Hamiltonian of the spin and MF system is [7, 41, 46–50] HMF = ℏω M γ1γ2 2 + iℏg ( S − − S + ) γ2.
H = ℏΔpu S z + ℏΔr aa+ + ℏλ ( a + a+ ) S z + ℏΔ M f + f
(
(
)
− μ B S +Bpr e−iδt + S −Bpr* eiδt ,
(4)
where Δpu = ωs − ωpu , Δr = ωr − ωpu , ΔM = ω M − ωpu is the detuning of the MF frequency and the pump frequency, Ω = μ B Bpr ℏ, is the pump microwave Rabi frequency, and δ = ωpr − ωpu is the detuning between the probe microwave field and the pump microwave field. As the diamond substrate is artificial ultrapure, any other magnetic impurity is virtually nonexistent. Actually, we have neglected the regular fermions (i.e. normal electrons) in the microcantilever that interact with the spin in the above discussion. To describe the interaction between the normal electrons and the spin, we use the tightbinding Hamiltonian of the whole wire as [9, 53] Hspin − e = ℏωs S z + ℏ∑k ω k ck+ck + ℏζ ∑k (ck+S − + ck S +), where ck and ck+ are the regular fermion annihilation and creation operators with energy ω k and momentum k obeying the anti-commutative relation and ζ is the coupling strength between the normal electrons and spin (here, for simplicity, we have neglected the k -dependence of ζ as in [54]). Applying the Heisenberg equation of motion for operators S z , S −, f and N and introducing the damping and noise terms phenomenologically, we obtain the corresponding quantum Langevin equations as follows: [55, 56]
(2)
To detect the existence of MFs, it is helpful to switch from the Majorana representation to the regular fermion one via the exact transformation γ1 = f + + f and γ2 = i (f + − f ). f + and f are the fermion annihilation and creation operators obeying the anti-commutative relation {f + , f } = 1. Accordingly, HMF can be rewritten as: HMF = ℏω M ( f + f − 1 2) + iℏg ( S −f + − S +f ) + iℏg ( S −f − S +f + ),
)
+ iℏg ( S −f + − S +f ) − ℏ ΩS + + Ω*S −
(3)
where the first term gives the energy of the MF at frequency ω M , and ℏω M = ε M ∼ el ξ ∼ 0 with the wire length l and the superconducting coherent length ξ. This term is so small that it can approach zero when the wire length is large enough. The second term describes the tunnel coupling between the nearby MFs and the NV center spin with the coupling strength g, where the coupling strength is related to the distance between the NV center spin and the ferromagnetic atomic chains. In the rotating wave approximation [51], it
dS z = −Γ1 ( S z + 1 2) + iΩ ( S + − S −) dt ⎛ iμ B Bpr ⎞ +⎜ ⎟ S + e−iδt − S −eiδt ⎝ ℏ ⎠
(
− g ( S −f + + S +f ),
3
)
(5)
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
dS − = −⎡⎣ i Δpu + λN + Γ2 ⎤⎦ S − dt − 2iΩS z − 2iμ B Bpr e−iδt S z ℏ
(
optical NV spin readout to perform optically detected electron spin resonance (ESR). Actually, this is the population of the spin states 〈S z〉 which has been measured in the majority of experiments [33–35]. The average population of the spin states can be given by
)
+ 2 (gf − iΩ) S z + Fˆn,
(6)
df = −( iΔ M + k M 2) f + gS − + ξˆ, dt
(7)
d2N dN + γr + ωr2 N = −2ωr λS z + ξˆ. dt dt 2
S+z =
(
h1h2 h 3 d1d*3 + h1h 3 d2 d*3 + h 3 d2 d*4 − iw0 h2 h1d*3 + d*4 d2 d*4
−
h1h2 d1d*3
),
(8)
(10)
where N = a+ + a is the position operator, Γ1 and Γ2 are the electronic spin relaxation rate and dephasing rate, kM is the decay rate of the MF, and γr = ωr Q is the intrinsic decay rate of the high-Q SCD cantilever. Fˆn is the δ-correlated Langevin noise operator, which has zero mean Fˆn = 0 and obeys the
where b1 = g (iΔM + k M 2 − iδ ), b 2 = g (iΔM + k M 2 + iδ ), b3 = 2λ 2 ωr (δ 2 + iδγr − ωr2 ), h1 = [−g (S0 b 2* + f0 ) + iΩ] (Γ1 − iδ ), h 2 = [−g (S0* b1 + f0* ) − iΩ] (Γ1 − iδ ), h 3 = iS0* (Γ1 − iδ ), d 2 = i(Δpu + λN0 − δ ) d1 = 2gf0 − i(λS0 b3 + 2Ω ), + Γ2 − gw0 b1− d1h 2 , d 3 = 2gf0 − i(λS0 b3* + 2Ω ),
+
following correlation functions Fˆn (t ) Fˆn (t′) ∼ δ (t − t′). The motion of the nanomechanical resonator is affected by a thermal bath of Brownian and non-Markovian stochastic processes [56]. The quantum effects on the resonator are only observed in the limit of very high quality factor, which obeys Q = ωr γr ≫ 1. The Brownian noise operator can be modeled as Markovian with the decay rate cr of the resonator mode. Therefore, the Brownian stochastic force has zero mean value that can be characterized as ξˆn = 0 ⎛ ℏω ⎞ ⎤ γr dω −iω (t − t′) ⎡ + ⎢ 1 + coth ⎜ ξˆ (t ) ξˆ (t′) = ⎟⎥ ∫ ωe ⎢⎣ ωr 2π ⎝ 2k B T ⎠ ⎥⎦ [57], where kB and T are the Boltzmann constant and the temperature of the reservoir of the coupled system, respectively. To go beyond weak coupling, the Heisenberg operator can be rewritten as the sum of its steady-state mean value and a small fluctuation with zero mean value: and S z = S0z + δS z, S − = S0− + δS −, f = f0 + δf N = N0 + δN . Since the driving fields are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [58]. Simultaneously, inserting these operators into the Langevin equations (equations (5)–(8)) and neglecting the nonlinear term, we can obtain two equation sets about the steady-state mean value and a small fluctuation. The steadystate equation set consisting of f0 , N0 and S0− is related to the population inversion (w0 = 2S0z ) of the exciton which is determined by
d 4 = i(Δpu + λN0 + δ )+ Γ2 − gw0 b 2 − d 3h1* (where O* indicates the conjugate of O). The quantum Langevin equations of the normal electrons coupled to the NV spin have the same form as MFs; therefore, we omit their derivation and only give the numerical results in the following.
3. Results and discussions For illustration of the numerical results, we consider an experimentally realistic hybrid spin–microcantilever: [33] the magnetic tip attached to the end of the SCD microcantilever of dimensions (L,W,T) is placed above a single NV center spin and the ferromagnetic atomic chains on a superconductor [17], as shown in figure 1. For the NV center spin in the coupled hybrid system, the longitudinal spin relaxation of NV center can be T2 = 1 ms at T = 4 K [31, 40]. The physical parameters of SCD microcantilever are ωr = 5 MHz, the quality factor Q = 106 [36, 37] and the decay rate of the microcantilever is γr = ωr Q = 5 Hz. For MFs in the ferromagnetic atomic chains on a superconductor, there are no experimental values for the lifetime of the MFs and the coupling strength between the spin and MFs in recent literature. However, according to a few experimental reports [13–17], it is reasonable to assume that the lifetime of the MFs is k M = 0.1 MHz. As the tunnel coupling strength between the NV center spin and nearby MFs is related to their distance apart, we expect the tunnel coupling strength g = 0.3 MHz via adjusting the distance between the ferromagnetic atomic chains and the NV center spin. Firstly, we consider the case that there is no coupling between the spin and microcantilever (λ = 0 MHz), i.e. only a single spin is coupled to the ferromagnetic atomic chains on a superconductor. Figure 2 plots the average population of the spin states 〈S z〉 as a function of the probe detuning Δpr ( Δpu = 2 MHz, ΔM = −4 MHz ). As shown in figure 2, the black solid curve indicates the spin resonance spectrum without the spin–MF tunnel coupling, and the red one shows the result with the spin–MF tunnel coupling g = 0.3 MHz. It is obvious that when the MFs are presented at the ends of the
2 2 0 = Γ1 ( w0 + 1) ⎡⎣ Δ M + k M2 4 Δpu + Γ22 + w02 λ4 ωr2
(
2
)
− 2w0 Δpu λ ωr + g
2
)( (g
w02
2
− 2Δ M λ2 ωr
2 + 2Δ M Δpu − Γ2 k M ⎤⎦ + 4Ω 2w0 Δ M + k M2 4 .
)
(
)
(9)
For the equation set of the small fluctuation, we make the ansatz [52] δS z = S+z e−iδt + S −z eiδt , and δS − = S+e−iδt + S −eiδt , 〈δf 〉 = f+ e−iδt + f− eiδt −iδt iδt . Solving the equation set and 〈δN 〉 = N+e + N −e working to the lowest order in Bpr but to all orders in Bpu detect NV spin transitions through well-established [44] 4
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
Figure 2. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–MF coupling
strengths without the spin–microcantilever coupling. The inset shows the result for the normal electrons in the microcantilever that couple to the spin at the coupling strength ζ = 0.3 MHz . The parameters used are Γ1 = 2 kHz, Γ2 = 1 kHz, k M = 0.1 MHz, γr = ωr Q = 5 Hz, 2 Ωpu = 100 (kHz)2 , Δpu = 2 MHz, ΔM = −4 MHz, λ 2π = 0 MHz and ωr 2π = 5 MHz .
ferromagnetic atomic chains, the two sharp sideband peaks corresponding to ΔM (as shown in the left inset) will appear in the spin resonance spectrum of the NV center spin. As the diamond substrate is artificially ultrapure, any other magnetic impurity is virtually nonexistent. The physical origin of this result is due to the spin–MF tunnel coupling, which makes the resonant enhancement of the spin resonance effect in the NV venter spin. This result also implies that the sharp peaks in the spin resonance spectrum may be the signature of MFs at the ends of the ferromagnetic atomic chains. Because there are also normal electrons in the ferromagnetic atomic chains, in order to determine whether or not this signature (i.e. the sharp peaks) is caused by MFs indeed, we plot the inset of figure 2, which uses the tight binding Hamiltonian to describe the normal electrons. In the figure, the parameters of normal electrons are chosen to be the same as those of the MFs so that we can compare them with the case of the MFs. However, there is no sharp peak and only a nearly zero line in the spin resonance spectrum (see the green line in the inset). Therefore, the result demonstrates that the coupling between the spin and the normal electrons in the ferromagnetic atomic chains can be neglected in our theoretical treatment. In this case, we may utilize the spin resonance effect in the NV spin to detect the existence of MFs provided that the NV spin is close enough to the ends of the ferromagnetic atomic chains. Secondly, we turn on the coupling to the microcantilever (λ ≠ 0 ) and then plot the spin resonance spectrum as a
function of probe detuning for λ = 0.02 MHz as shown in figure 3. Taking the coupling between the NV spin and microcantilever into consideration, the other two sharp peaks located at ± ωr will also appear. The red and black curves correspond to the spin resonance spectrum with and without the spin–MF tunnel coupling, respectively. Without the spin– MF coupling, the two sharp peaks locate at the resonator frequency of microcantilever induced by its vibration, i.e. the two peaks are at Δpr = ±5 MHz as shown in figure 3. The physical origin of this result is due to mechanically induced coherent population oscillation (MICPO), which makes quantum interference between the resonator and the beat of the two optical fields via the spin when the probe–pump detuning is equal to the resonator frequency [59]. Turning on the spin–MF tunnel coupling, the other two sideband peaks induced by the spin–MF coupling appear at Δpr = ±4 MHz simultaneously. In order to demonstrate the function of the microcantilever that enhances the sensitivity for detecting MFs, we should consider the vibration of the microcantilever. Figure 4 presents the spin resonance spectrum of the probe field as a function of the probe detuning Δpr . We adjust the detuning ΔM = −4 MHz to ΔM = −5 MHz, ωr = 5 MHz. Figure 4 gives the result of the spin resonance spectrum as a function of probe detuning with or without the spin–microcantilever coupling for the spin–MF coupling g = 0.3 MHz. The black and red curves correspond to λ = 0 MHz and λ = 0.02 MHz, 5
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
Figure 3. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–MF
coupling strengths with the spin–microcantilever coupling λ 2π = 0.02 MHz. The other parameters used are the same as figure 2.
Figure 4. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–
microcantilever coupling strengths λ 2π = 0 MHz and λ 2π = 0.02 MHz with the spin–MFs coupling g 2π = 0.3 MHz, ΔM = −5 MHz. The other parameters used are the same as figure 2.
6
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
respectively. In this case, the location of the two sideband peaks induced by the spin–MF coupling coincides with the two sharp peaks induced by the vibration of microcantilever, so the microcantilever is resonant with the coupled spin–MF system and makes the tunneling interaction of spin–MF more strong. As shown in the inset, the blue and green curves correspond to only the spin–MF tunnel coupling and only the spin–microcantilever coupling, respectively. It is obvious that only when the spin–microcantilever coupling is turned on, a more significant characteristic peak (the red curve) can be observed in the spin resonance spectrum, and the role of microcantilever is to narrow and to increase the spin resonance effect. In this case, the microcantilever as a phonon cavity will enhance sensitivity for detecting MFs, which is analogous to the optical cavity-enhanced effect in quantum optics. In addtion, compared to the existing electrical measurement scheme for detection of MFs in the hybrid semiconductor/superconductor heterostructure [13–16], our alloptical system can avoid the heat effect and energy loss caused by the electric circuitry which will broaden the electrical response spectrum and finally affect the sensitivity of the MFs detection. Moreover, the spin resonance spectrum of the NV center spin embedded in diamond can be detected through well-established optical NV spin readout technology, which is maturely developed and vitally accurate.
[3] Alicea J, Oreg Y, Refael G, von Oppen F and Fisher M P A 2011 Non-Abelian statistics and topological quantum informaiton processing in 1D wire networks Nat. Phys. 7 412 [4] Semenoff G and Sodano P 2007 Stretched quantum states emerging from a Majorana medium J. Phys. B: At. Mol. Opt. Phys. 40 1479 [5] Fu L 2010 Electron teleportation via majorana bound states in a mesoscopic superconductor Phys. Rev. Lett. 104 056402 [6] Tanaka Y, Yokoyama T and Nagaosa N 2009 Manipulation of the majorana fermion, andreev reflection, and josephson current on topological insulators Phys. Rev. Lett. 103 107002 [7] Oreg Y, Refael G and von Oppen F 2010 Helical liquids and majorana bound states in quantum wires Phys. Rev. Lett. 105 177002 [8] Sau J D, Tewari S, Lutchyn R M, Stanescu T D and Das Sarma S 2010 Non-abelian quantum order in spin-orbitcoupled semiconductors: search for topological majorana particles in solid state systems Phys. Rev. B 82 214509 [9] Nadj-Perge S, Drozdov I K, Bernevig B A and Yazdani A 2013 Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor Phys. Rev. B 88 020407(R) [10] Klinovaja J, Stano P, Yazdani A and Loss D 2013 Topological superconductivity and majorana fermions in RKKY systems Phys. Rev. Lett. 111 186805 [11] Rokhinson L P, Liu X and Furdyna J K 2012 The fractional A C Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles Nat. Phys. 8 795 [12] Williams J R et al 2012 Unconventional Josephson effect in hybrid superconductor-topological insulator devices Phys. Rev. Lett. 109 056803 [13] Das A, Ronen Y, Most Y, Oreg Y, Heiblum M and Shtrikman H 2012 Zero-bias peaks and splitting in an AlInAs nanowire topological superconductor as a signature of Majorana fermions Nat. Phys. 8 887 [14] Mourik V et al 2012 Signatures of Majorana fermions in hybrid superconductorsemiconductor nanowire devices Science 336 1003 [15] Churchill H O H et al 2013 Superconductor-nanowire devices from tunneling to the multichannel regime zero-bias oscillations and magnetoconductance crossover Phys. Rev. B 87 241401(R) [16] Lee E J H et al 2014 Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures Nat. Nanotechnology 9 79 [17] Nadj-Perge S, Drozdov I K, Bernevig B A and Yazdani A 2014 Observation of Majorana fermions in ferromagnetic atomic Chains on a superconductor Science 10 1126 [18] Lee A P 2014 Seeking out Majorana under the microscope Scinece 346 545 [19] Flensberg K 2010 Tunneling characteristics of a chain of Majorana bound states Phys. Rev. B 82 180516 [20] Prada E, San-Jose P and Aguado R 2012 Transport spectroscopy of NS nanowire junctions with Majorana fermions Phys. Rev. B 86 180503(R) [21] Yamakage A, Yada K, Sato M and Tanaka Y 2012 Theory of tunneling conductance and surface-state transition in superconducting topological insulators Phys. Rev. B 85 180509(R) [22] Chang W, Manucharyan V E, Jespersen T S, Nygard J and Marcus C M 2013 Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction Phys. Rev. Lett. 110 217005 [23] Finck A D K, van Harlingen D J, Mohseni P K, Jung K and Li X 2013 Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device Phys. Rev. Lett. 110 126406
4. Conclusion We have proposed an all-optical method to detect the existence of Majorana fermions in ferromagnetic atomic chains on a superconductor via the hybrid spin–microcantilever. The spin resonance effect may provide another supplement for detecting Majorana fermions. Due to the microcantileverʼs vibration, the spin resonance effect becomes much more significant and then enhances the detectable sensitivity of Majorana fermions. Additionally, the well-established [44] optical NV spin readout technology will certainly promote detection. Finally, we hope that our proposed scheme can be realized experimentally in the future.
Acknowledgments This study was supported by the National Natural Science Foundation of China (Nos. 10974133 and 11274230) and the Basic Research Program of the Committee of Science and Technology of Shanghai (No. 14JC1491700).
References [1] Kitaev A 2001 Unpaired Majorana fermions in quantum wires Phys. Usp. 44 131 [2] Nayak C, Stern A, Freedman M and das Sarma S 2008 NonAbelian anyons and topological quantum computation Rev. Mod. Phys. 80 1083 7
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
[42] Walter S and Budich J C 2014 Teleportation-induced entanglement of two nanomechanical oscillators coupled to a topological superconductor Phys. Rev. B 89 155431 [43] Rabl P, Kolkowitz S J, Koppens F H L, Harris J G E, Zoller P and Lukin M D 2010 A quantum spin transducer based on nanoelectromechanical resonator arrays Nat. Phys. 6 602–8 [44] Gruber A et al 1997 Scanning confocal optical microscopy and magnetic resonance on single defect centers Science 276 2012 [45] Xu X et al 2007 Coherent optical spectroscopy of a strongly driven quantum dot Science 317 929 [46] Cao Y S, Wang P Y, Xiong G, Gong M and Li X Q 2012 Probing the existence and dynamics of Majorana fermion via transport through a quantum dot Phys. Rev. B 86 115311 [47] Dong E L and Harold U B 2011 Detecting a Majorana-fermion zero mode using a quantum dot Phys. Rev. B 84 201308(R) [48] Lutchyn R M, Jay D S and das Sarma S 2010 Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures Phys. Rev. Lett. 105 077001 [49] da Silva S J S, Seridonio A C, del Nero J and Souza F M 2014 Nonequilibrium transport in a quantum dot attached to a Majorana bound state arXiv:1403.3575 [50] Seridonio A C, Siqueira E C F, Dessotti A, Machado R S and Yoshida M 2014 Fano interference and a slight fluctuation of the Majorana hallmark J. Appl. Phys. 115 063706 [51] Ridolfo A, Stefano O D, Fina N, Saija R and Savasta S 2010 Quantum plasmonics with quantum dot-metal nanoparticle molecules: influence of the Fano effect on photon statistics Phys. Rev. Lett. 105 263601 [52] Boyd R W 1992 Nonlinear Optics (San Diego, CA: Academic) p 225 [53] Mahan G D 1992 Many-Partcle Physics (New York: Plenum Press) [54] Hewson A C 1993 The Kondo Problem to Heavy Fermions (New York: Cambridge University Press) [55] Scully M O and Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) [56] Gardiner C W and Zoller P 2004 Quantum Noise (New York: Springer) [57] Vitali D and Giovannetti V 2001 Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion Phys. Rev. A 63 023812 [58] Weis S, Riviére R, Deleglise S, Gavartin E, Arcizet O, Schliesser A and Kippenberg T J 2010 Optomechanically Induced Transparency Science 330 1520 [59] Li J J and Zhu K D 2013 All-optical mass sensing with coupled mechanical resonator systems Phys. Rep. 525 223–54
[24] Bagrets D and Altland A 2012 Class D spectral peak in majorana quantum wires Phys. Rev. Lett. 109 227005 [25] Poot M, Zan H S and van Dert J 2012 Mechanical systems in the quantum regime Phys. Rep. 511 273 [26] Aspelmeyer M, Kippenberg T J and Marquardt F 2014 Cavity Optomech. Rev. Mod. Phys. 86 1391 [27] Childress L and Hanson R 2013 Diamond N V centers for quantum computing and quantum networks MRS Bull 38 134–8 [28] Rondin L et al 2014 Magnetometry with nitrogen-vacancy defects in diamond Rep. Prog. Phys. 77 056503 [29] Balasubramanian G et al 2008 Nanoscale imaging magnetometry with diamond spins under ambient conditions Nature 455 648–51 [30] Togan E et al 2010 Quantum entanglement between an optical photon and a solid-state spin qubit Nature 466 730–4 [31] Childress L et al 2006 Coherent dynamics of coupled electron and nuclear spin qubits in diamond Science 314 281–5 [32] Kolkowitz S et al 2012 Coherent sensing of a mechanical resonator with a single-spin qubit Science 335 1603 [33] Rabl P et al 2009 Strong magnetic coupling between an electronic spin qubit and a mechanical oscillator Phys. Rev. B 79 041302(R) [34] Teissier J, Barfuss A, Appel P, Neu E and Maletinsky P 2014 Resolved sidebands in a straincoupled hybrid spin-oscillator system Phys. Rev. Lett. 113 020503 [35] Ovartchaiyapong P, Lee K W, Myers B A and Bleszynski Jayich A C 2014 Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator Nat. Commun. 5 4429 [36] Ovartchaiyapong P, Pascal L M A, Myers B A, Lauria P and Bleszynski Jayich A C 2012 High quality factor singlecrystal diamond mechanical resonators Appl. Phys. Lett. 101 163505 [37] Tao Y, Boss J M, Moores B and Degen C L 2014 Singlecrystal diamond nanomechanical resonators with quality factors exceeding one million Nat. Commun. 5 3638 [38] Jarmola A, Acosta V M, Jensen K, Chemerisov S and Budker D 2012 Temperature and magnetic field dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond Phys. Rev. Lett. 108 197601 [39] Acosta V M, Bauch E, Ledbetter M P, Waxman A, Bouchard L S and Budker D 2010 Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond Phys. Rev. Lett. 104 070801 [40] Balasubramanian G et al 2009 Ultralong spin coherence time in isotopically engineered diamond Nat. Mater. 8 383–7 [41] Walter S, Schmidt T L and Trauzettel B 2011 Detecting Majorana bound states by nanomechanics Phys. Rev. B 84 224510
8
Search
Collections
Journals
About
Contact us
My IOPscience
Sensitive detection of Majorana fermions based on a hybrid spin–microcantilever via enhanced spin resonance spectrum
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 195501 (http://iopscience.iop.org/0957-4484/26/19/195501) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 169.230.243.252 This content was downloaded on 21/04/2015 at 12:17
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 26 (2015) 195501 (8pp)
doi:10.1088/0957-4484/26/19/195501
Sensitive detection of Majorana fermions based on a hybrid spin–microcantilever via enhanced spin resonance spectrum Wen-Hao Wu1 and Ka-Di Zhu2 1
Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, Peopleʼs Republic of China 2 Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, Peopleʼs Republic of China E-mail: [email protected] Received 7 December 2014, revised 10 March 2015 Accepted for publication 12 March 2015 Published 21 April 2015 Abstract
Motivated by recent experimental progress towards the detection and manipulation of Majorana fermions in ferromagnetic atomic chains on a superconductor, we present a novel proposal based on a single-crystal diamond (SCD) microcantilever with a single nitrogen-vacancy (NV) center spin embedded in ultrapure diamond substrate to probe Majorana fermions in an all-optical domain. With this scheme, a possible distinct Majorana signature is investigated via the electron spin resonance spectrum. In the proposal, the SCD microcantilever behaves as a phonon cavity and is robust for detecting of Majorana fermions, while the NV center spin can be considered as a sensitive probe. Further, the vibration of the microcantilever will enhance the coupling effect, which makes the Majorana fermions more sensitive to detection and the well-established optical NV spin readout technology will certainly promote the detection. This proposed method may provide a potential supplement for the detection of Majorana fermions. Keywords: Majorana fermions, hybrid spin–microcantilever, spin resonance spectrum (Some figures may appear in colour only in the online journal) 1. Introduction
[6–10] and many experimental attempts have also been devoted to identify them [11–18]. According to these attempts, the zero-bias conductance peak [19–21] as a representative property of Majorana modes is often considered as a Majorana signature. These zero-bias conductance peaks have already been observed in several experiments [13– 17]. However, it is fair to say that these experimental results cannot serve as definitive evidence to prove the existence of MFs in condensed matter systems because the zero-bias conductance peaks can appear due to many other mechanisms [22] such as zero-bias anomaly due to Kondo resonance [23] and disorder or band bending in the semiconductor nanowire (SNW) [24]. Therefore, to obtain definitive signatures of MFs, alternative setups or proposals are necessary. Due to the coherent long-range quantum effects of MFs, the understanding of genuine quantum effects on macroscopic length scales, i.e. nanoelectromechanical systems [25] and
Currently, the issue of Majorana fermions (MFs) as the most intriguing features of topological states of matter has received a great deal of attention. Besides the fundamental interest attached to the experimental discovery of Majorana fermions in nature, MFs as realized in one-dimensional topological superconductors (1DTSC) also have attractive features relating to various aspects of fundamental quantum physics. On one hand, the non-Abelian anyonic nature of MFs shows great promise for topological quantum information processing architecture [1–3]. On the other hand, the delocalized pair of MFs at the ends of a 1DTSC can be viewed as a single ordinary (spinless Dirac) fermionic zero mode leading to electron teleportation mechanisms [4, 5], i.e. coherent longrange quantum effects. By now, various systems that might host MFs in condensed matter systems have been proposed 0957-4484/15/195501+08$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
microcantilever with a magnetic tip attached to the end and a single NV spin center embedded in ultrapure diamond placed below [33, 43], in the simultaneous presence of a external magnetic field to detect MFs. By applying a strong pump microwave and a weak probe microwave to the NV center spin simultaneously, the possible Majorana signature could be probed via the spin resonance spectrum of the NV center spin, which could be detected through well-established [44] optical NV spin readout technology, which is maturely developed and vitally accurate. Moreover, the optical pump–probe technology has been demonstrated experimentally in several systems [45] and provides an effective way to investigate the light–matter interaction. Compared with electrical detection schemes, our optical scheme can avoid the heat effect and energy loss caused by the electric circuitry which will broaden the electrical response spectrum and finally affect the sensitivity of MF detection. Since the distance between the NV center spin and the MFs can be adjusted in several nanometers, the tunneling between the NV center spin and MFs can be ensured. The signal change in the spin resonance spectrum as a possible signature for the MFs is another potential piece of evidence in the ferromagnetic atomic chains. Further, the vibration of the microcantilever acting as a phonon cavity will enhance the spin resonance spectrum significantly and make the MFs more sensitive to detection. Figure 1. (a) Schematic diagram of the hybrid spin–microcantilever
system for optically detecting MFs at the ends of the ferromagnetic atomic chains on a superconductor. The inset is the energy level diagram of a NV center spin. [εi ]i = x, y, z are the diagonal components of the strain tensor defined in the NVʼs basis (note we have neglected shear). (b) An energy-level diagram of a NV center spin coupled to MFs and microcantilever.
2. Model and theory Figure 1(a) presents the schematic setup that will be studied in this work. Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin–orbit interaction. So, ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb) are the analyte at the ends of which an MF pair is expected to locate [17, 18]. To detect MFs, we employ a hybrid system consisting of a highQ SCD microcantilever with a magnetic tip attached to the end and a single NV spin center embedded in diamond placed below. By applying a strong pump microwave ωpu , a weak probe microwave ωpr and a external magnetic field Bz along the z axis to the NV center spin simultaneously, one could probe the MFs via optical pump–probe technique [45]. In such a hybrid system, the motion of the microcantilever along the z axis is quantized and described by the Hamiltonian Hr = ℏωr a+a , with ωr as the frequency of the fundamental bending mode and a and a+ as the corresponding annihilation and creation operators. On the other hand, for the NV center spin in the diamond, the schematic diagram of energy levels is shown in the inset of figure 1(a). The electronic ground state of the NV center is an S = 1 spin triplet, and we label states by ms, where m s = 0, ±1. Spin states with different values of ms are separated by a zero-field splitting of D0 ≈ 2.87 GHz. For moderate applied magnetic field Bz along the NV axis, the NV spin can be restricted to the twolevel subspace spanned by ∣ 0〉 and ∣ + 〉 and mixing of ∣ − 〉
nanooptomechanical systems [26], may be a helpful motivator. Quite excitingly, nitrogen-vacancy (NV) centers embedded in diamond are amongst the most promising implementations of quantum bits for quantum information processing [27] and nanoscale field sensors [28] because of their excellent coherence properties [29] and their ability to coherently couple to various external fields such as photons [30], nuclear spins [31], and mechanical resonators [32–35]. Promoted by recent demonstrations of high quality (Q) factor (exceeding 106 ) single-crystal diamond (SCD) mechanical resonators [36, 37] and the NV spinʼs long quantum coherence time T2 = 1−10 ms [31, 38–40], this hybrid spin– microcantilever system could be a promising alternative platform for the detection of MFs, thus paving the way for coupling MFs to nontopological qubits NV center spin embedded in diamond. Additionally, we notice that all the theoretical proposals and experimental schemes [6–17, 41, 42] have focused on electrical methods, while other effective methods such as optical schemes for detecting MFs have received less attention. In the present article, based on the recent experiment by Stevan et al [17], where a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin– orbit interaction (figure 1(a)), we will employ a high-Q SCD 2
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
and ∣ + 〉 can be neglected [34, 35]. Therefore, the Hamiltonian of the NV center spin can be described as Hs = ℏωs S z , where ωs 2π = D0 , the spin operator can be characterized by the spin operators S ± and S z . The schematic diagram of energy levels of the coupling between the NV spin and the cantileverʼs vibration is shown in figure 1(b). Therefore, the spin–microcantilever interaction can be described by [33, 35, 43] Hs − r = ℏλ (a + a+) S z and λ = gs μ B Gm a 0 is the coupling of spin to phonon, where gs ≃ 2, μ B is the Bohr magneton, Gm is the magnetic field gradient, and a 0 = ℏ 2mωr is the amplitude of zero-point fluctuations for a resonator m. Therefore, we obtain the total Hamiltonian of the coupled spin–microcantilever:
should also be noted that the third term of non-conservation for energy iℏg (S −f − S +f + ) is generally neglected. We have made the numerical calculations (not shown in the following figures) and shown that the effect of this term is too small to be considered in our theoretical treatment. So, the HMF here can be rewritten as: HMF = ℏω M (f + f − 1 2) + iℏg (S −f + − S +f ). In terms of this scheme, we apply the pump–probe scheme to the NV center spin simultaneously. We treat these microwave fields classically. The Hamiltonian of the NV center spin-transitions through microwaves is described as * e iωpu t )− μ [52] Hs − p = −μ B(S +Bpu e−iωpu t + S −Bpu B (S +Bpr e−iωpr t + S −Bpr* e iωpr t), where ωpu (ωpr ) is the frequency of the pump field (probe field), and Bpu (Bpr ) is the slowly varying envelope of the pump field (probe field). The driven spin system with one microwave field has been realized in the hybrid spin–resonator system [32]. Therefore, one can obtain the total Hamiltonian of the hybrid system as H = Hs − MC + HMF + Hs − p . In a rotating frame at the pump field frequency ωpu , the total Hamiltonian of the system reads as follows:
Hs − MC = Hr + Hs + Hs − r = ℏωr S z + ℏωs aa+ + ℏλ ( a + a+ ) S z .
(1)
Since several experiments [13–16] have reported the distinct signatures of MFs in the hybrid semiconductor/ superconductor heterostructure via electrical methods, and because Stevan et al [17, 18] give more clear observation of MFs in ferromagnetic atomic chains on a superconductor, we assure the reader that the MFs may exist in the ferromagnetic atomic chains on a superconductor under some appropriate conditions. According to these experiments, the coupling between the NV-spin and the MFs should be the tunnel coupling. It should be noted that the magnetic moment of the ferromagnetic iron chains may have an effect on the NV center spin. However, compared to the applied magnetic field H = 1 T which is the precondition for the observation of MFs in ferromagnetic atomic chains, the effect of the magnetic moment can be negligible. As each MF is its own antiparticle, + one can introduce a MF operator γMF such that γMF = γMF and 2 γMF = 1 to describe MFs. Supposing the NV center spin tunneling couples to γ2 , the Hamiltonian of the spin and MF system is [7, 41, 46–50] HMF = ℏω M γ1γ2 2 + iℏg ( S − − S + ) γ2.
H = ℏΔpu S z + ℏΔr aa+ + ℏλ ( a + a+ ) S z + ℏΔ M f + f
(
(
)
− μ B S +Bpr e−iδt + S −Bpr* eiδt ,
(4)
where Δpu = ωs − ωpu , Δr = ωr − ωpu , ΔM = ω M − ωpu is the detuning of the MF frequency and the pump frequency, Ω = μ B Bpr ℏ, is the pump microwave Rabi frequency, and δ = ωpr − ωpu is the detuning between the probe microwave field and the pump microwave field. As the diamond substrate is artificial ultrapure, any other magnetic impurity is virtually nonexistent. Actually, we have neglected the regular fermions (i.e. normal electrons) in the microcantilever that interact with the spin in the above discussion. To describe the interaction between the normal electrons and the spin, we use the tightbinding Hamiltonian of the whole wire as [9, 53] Hspin − e = ℏωs S z + ℏ∑k ω k ck+ck + ℏζ ∑k (ck+S − + ck S +), where ck and ck+ are the regular fermion annihilation and creation operators with energy ω k and momentum k obeying the anti-commutative relation and ζ is the coupling strength between the normal electrons and spin (here, for simplicity, we have neglected the k -dependence of ζ as in [54]). Applying the Heisenberg equation of motion for operators S z , S −, f and N and introducing the damping and noise terms phenomenologically, we obtain the corresponding quantum Langevin equations as follows: [55, 56]
(2)
To detect the existence of MFs, it is helpful to switch from the Majorana representation to the regular fermion one via the exact transformation γ1 = f + + f and γ2 = i (f + − f ). f + and f are the fermion annihilation and creation operators obeying the anti-commutative relation {f + , f } = 1. Accordingly, HMF can be rewritten as: HMF = ℏω M ( f + f − 1 2) + iℏg ( S −f + − S +f ) + iℏg ( S −f − S +f + ),
)
+ iℏg ( S −f + − S +f ) − ℏ ΩS + + Ω*S −
(3)
where the first term gives the energy of the MF at frequency ω M , and ℏω M = ε M ∼ el ξ ∼ 0 with the wire length l and the superconducting coherent length ξ. This term is so small that it can approach zero when the wire length is large enough. The second term describes the tunnel coupling between the nearby MFs and the NV center spin with the coupling strength g, where the coupling strength is related to the distance between the NV center spin and the ferromagnetic atomic chains. In the rotating wave approximation [51], it
dS z = −Γ1 ( S z + 1 2) + iΩ ( S + − S −) dt ⎛ iμ B Bpr ⎞ +⎜ ⎟ S + e−iδt − S −eiδt ⎝ ℏ ⎠
(
− g ( S −f + + S +f ),
3
)
(5)
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
dS − = −⎡⎣ i Δpu + λN + Γ2 ⎤⎦ S − dt − 2iΩS z − 2iμ B Bpr e−iδt S z ℏ
(
optical NV spin readout to perform optically detected electron spin resonance (ESR). Actually, this is the population of the spin states 〈S z〉 which has been measured in the majority of experiments [33–35]. The average population of the spin states can be given by
)
+ 2 (gf − iΩ) S z + Fˆn,
(6)
df = −( iΔ M + k M 2) f + gS − + ξˆ, dt
(7)
d2N dN + γr + ωr2 N = −2ωr λS z + ξˆ. dt dt 2
S+z =
(
h1h2 h 3 d1d*3 + h1h 3 d2 d*3 + h 3 d2 d*4 − iw0 h2 h1d*3 + d*4 d2 d*4
−
h1h2 d1d*3
),
(8)
(10)
where N = a+ + a is the position operator, Γ1 and Γ2 are the electronic spin relaxation rate and dephasing rate, kM is the decay rate of the MF, and γr = ωr Q is the intrinsic decay rate of the high-Q SCD cantilever. Fˆn is the δ-correlated Langevin noise operator, which has zero mean Fˆn = 0 and obeys the
where b1 = g (iΔM + k M 2 − iδ ), b 2 = g (iΔM + k M 2 + iδ ), b3 = 2λ 2 ωr (δ 2 + iδγr − ωr2 ), h1 = [−g (S0 b 2* + f0 ) + iΩ] (Γ1 − iδ ), h 2 = [−g (S0* b1 + f0* ) − iΩ] (Γ1 − iδ ), h 3 = iS0* (Γ1 − iδ ), d 2 = i(Δpu + λN0 − δ ) d1 = 2gf0 − i(λS0 b3 + 2Ω ), + Γ2 − gw0 b1− d1h 2 , d 3 = 2gf0 − i(λS0 b3* + 2Ω ),
+
following correlation functions Fˆn (t ) Fˆn (t′) ∼ δ (t − t′). The motion of the nanomechanical resonator is affected by a thermal bath of Brownian and non-Markovian stochastic processes [56]. The quantum effects on the resonator are only observed in the limit of very high quality factor, which obeys Q = ωr γr ≫ 1. The Brownian noise operator can be modeled as Markovian with the decay rate cr of the resonator mode. Therefore, the Brownian stochastic force has zero mean value that can be characterized as ξˆn = 0 ⎛ ℏω ⎞ ⎤ γr dω −iω (t − t′) ⎡ + ⎢ 1 + coth ⎜ ξˆ (t ) ξˆ (t′) = ⎟⎥ ∫ ωe ⎢⎣ ωr 2π ⎝ 2k B T ⎠ ⎥⎦ [57], where kB and T are the Boltzmann constant and the temperature of the reservoir of the coupled system, respectively. To go beyond weak coupling, the Heisenberg operator can be rewritten as the sum of its steady-state mean value and a small fluctuation with zero mean value: and S z = S0z + δS z, S − = S0− + δS −, f = f0 + δf N = N0 + δN . Since the driving fields are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [58]. Simultaneously, inserting these operators into the Langevin equations (equations (5)–(8)) and neglecting the nonlinear term, we can obtain two equation sets about the steady-state mean value and a small fluctuation. The steadystate equation set consisting of f0 , N0 and S0− is related to the population inversion (w0 = 2S0z ) of the exciton which is determined by
d 4 = i(Δpu + λN0 + δ )+ Γ2 − gw0 b 2 − d 3h1* (where O* indicates the conjugate of O). The quantum Langevin equations of the normal electrons coupled to the NV spin have the same form as MFs; therefore, we omit their derivation and only give the numerical results in the following.
3. Results and discussions For illustration of the numerical results, we consider an experimentally realistic hybrid spin–microcantilever: [33] the magnetic tip attached to the end of the SCD microcantilever of dimensions (L,W,T) is placed above a single NV center spin and the ferromagnetic atomic chains on a superconductor [17], as shown in figure 1. For the NV center spin in the coupled hybrid system, the longitudinal spin relaxation of NV center can be T2 = 1 ms at T = 4 K [31, 40]. The physical parameters of SCD microcantilever are ωr = 5 MHz, the quality factor Q = 106 [36, 37] and the decay rate of the microcantilever is γr = ωr Q = 5 Hz. For MFs in the ferromagnetic atomic chains on a superconductor, there are no experimental values for the lifetime of the MFs and the coupling strength between the spin and MFs in recent literature. However, according to a few experimental reports [13–17], it is reasonable to assume that the lifetime of the MFs is k M = 0.1 MHz. As the tunnel coupling strength between the NV center spin and nearby MFs is related to their distance apart, we expect the tunnel coupling strength g = 0.3 MHz via adjusting the distance between the ferromagnetic atomic chains and the NV center spin. Firstly, we consider the case that there is no coupling between the spin and microcantilever (λ = 0 MHz), i.e. only a single spin is coupled to the ferromagnetic atomic chains on a superconductor. Figure 2 plots the average population of the spin states 〈S z〉 as a function of the probe detuning Δpr ( Δpu = 2 MHz, ΔM = −4 MHz ). As shown in figure 2, the black solid curve indicates the spin resonance spectrum without the spin–MF tunnel coupling, and the red one shows the result with the spin–MF tunnel coupling g = 0.3 MHz. It is obvious that when the MFs are presented at the ends of the
2 2 0 = Γ1 ( w0 + 1) ⎡⎣ Δ M + k M2 4 Δpu + Γ22 + w02 λ4 ωr2
(
2
)
− 2w0 Δpu λ ωr + g
2
)( (g
w02
2
− 2Δ M λ2 ωr
2 + 2Δ M Δpu − Γ2 k M ⎤⎦ + 4Ω 2w0 Δ M + k M2 4 .
)
(
)
(9)
For the equation set of the small fluctuation, we make the ansatz [52] δS z = S+z e−iδt + S −z eiδt , and δS − = S+e−iδt + S −eiδt , 〈δf 〉 = f+ e−iδt + f− eiδt −iδt iδt . Solving the equation set and 〈δN 〉 = N+e + N −e working to the lowest order in Bpr but to all orders in Bpu detect NV spin transitions through well-established [44] 4
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
Figure 2. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–MF coupling
strengths without the spin–microcantilever coupling. The inset shows the result for the normal electrons in the microcantilever that couple to the spin at the coupling strength ζ = 0.3 MHz . The parameters used are Γ1 = 2 kHz, Γ2 = 1 kHz, k M = 0.1 MHz, γr = ωr Q = 5 Hz, 2 Ωpu = 100 (kHz)2 , Δpu = 2 MHz, ΔM = −4 MHz, λ 2π = 0 MHz and ωr 2π = 5 MHz .
ferromagnetic atomic chains, the two sharp sideband peaks corresponding to ΔM (as shown in the left inset) will appear in the spin resonance spectrum of the NV center spin. As the diamond substrate is artificially ultrapure, any other magnetic impurity is virtually nonexistent. The physical origin of this result is due to the spin–MF tunnel coupling, which makes the resonant enhancement of the spin resonance effect in the NV venter spin. This result also implies that the sharp peaks in the spin resonance spectrum may be the signature of MFs at the ends of the ferromagnetic atomic chains. Because there are also normal electrons in the ferromagnetic atomic chains, in order to determine whether or not this signature (i.e. the sharp peaks) is caused by MFs indeed, we plot the inset of figure 2, which uses the tight binding Hamiltonian to describe the normal electrons. In the figure, the parameters of normal electrons are chosen to be the same as those of the MFs so that we can compare them with the case of the MFs. However, there is no sharp peak and only a nearly zero line in the spin resonance spectrum (see the green line in the inset). Therefore, the result demonstrates that the coupling between the spin and the normal electrons in the ferromagnetic atomic chains can be neglected in our theoretical treatment. In this case, we may utilize the spin resonance effect in the NV spin to detect the existence of MFs provided that the NV spin is close enough to the ends of the ferromagnetic atomic chains. Secondly, we turn on the coupling to the microcantilever (λ ≠ 0 ) and then plot the spin resonance spectrum as a
function of probe detuning for λ = 0.02 MHz as shown in figure 3. Taking the coupling between the NV spin and microcantilever into consideration, the other two sharp peaks located at ± ωr will also appear. The red and black curves correspond to the spin resonance spectrum with and without the spin–MF tunnel coupling, respectively. Without the spin– MF coupling, the two sharp peaks locate at the resonator frequency of microcantilever induced by its vibration, i.e. the two peaks are at Δpr = ±5 MHz as shown in figure 3. The physical origin of this result is due to mechanically induced coherent population oscillation (MICPO), which makes quantum interference between the resonator and the beat of the two optical fields via the spin when the probe–pump detuning is equal to the resonator frequency [59]. Turning on the spin–MF tunnel coupling, the other two sideband peaks induced by the spin–MF coupling appear at Δpr = ±4 MHz simultaneously. In order to demonstrate the function of the microcantilever that enhances the sensitivity for detecting MFs, we should consider the vibration of the microcantilever. Figure 4 presents the spin resonance spectrum of the probe field as a function of the probe detuning Δpr . We adjust the detuning ΔM = −4 MHz to ΔM = −5 MHz, ωr = 5 MHz. Figure 4 gives the result of the spin resonance spectrum as a function of probe detuning with or without the spin–microcantilever coupling for the spin–MF coupling g = 0.3 MHz. The black and red curves correspond to λ = 0 MHz and λ = 0.02 MHz, 5
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
Figure 3. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–MF
coupling strengths with the spin–microcantilever coupling λ 2π = 0.02 MHz. The other parameters used are the same as figure 2.
Figure 4. Average population of the spin 〈S+z 〉 (spin resonance spectrum) as function of probe detuning Δpr for two different spin–
microcantilever coupling strengths λ 2π = 0 MHz and λ 2π = 0.02 MHz with the spin–MFs coupling g 2π = 0.3 MHz, ΔM = −5 MHz. The other parameters used are the same as figure 2.
6
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
respectively. In this case, the location of the two sideband peaks induced by the spin–MF coupling coincides with the two sharp peaks induced by the vibration of microcantilever, so the microcantilever is resonant with the coupled spin–MF system and makes the tunneling interaction of spin–MF more strong. As shown in the inset, the blue and green curves correspond to only the spin–MF tunnel coupling and only the spin–microcantilever coupling, respectively. It is obvious that only when the spin–microcantilever coupling is turned on, a more significant characteristic peak (the red curve) can be observed in the spin resonance spectrum, and the role of microcantilever is to narrow and to increase the spin resonance effect. In this case, the microcantilever as a phonon cavity will enhance sensitivity for detecting MFs, which is analogous to the optical cavity-enhanced effect in quantum optics. In addtion, compared to the existing electrical measurement scheme for detection of MFs in the hybrid semiconductor/superconductor heterostructure [13–16], our alloptical system can avoid the heat effect and energy loss caused by the electric circuitry which will broaden the electrical response spectrum and finally affect the sensitivity of the MFs detection. Moreover, the spin resonance spectrum of the NV center spin embedded in diamond can be detected through well-established optical NV spin readout technology, which is maturely developed and vitally accurate.
[3] Alicea J, Oreg Y, Refael G, von Oppen F and Fisher M P A 2011 Non-Abelian statistics and topological quantum informaiton processing in 1D wire networks Nat. Phys. 7 412 [4] Semenoff G and Sodano P 2007 Stretched quantum states emerging from a Majorana medium J. Phys. B: At. Mol. Opt. Phys. 40 1479 [5] Fu L 2010 Electron teleportation via majorana bound states in a mesoscopic superconductor Phys. Rev. Lett. 104 056402 [6] Tanaka Y, Yokoyama T and Nagaosa N 2009 Manipulation of the majorana fermion, andreev reflection, and josephson current on topological insulators Phys. Rev. Lett. 103 107002 [7] Oreg Y, Refael G and von Oppen F 2010 Helical liquids and majorana bound states in quantum wires Phys. Rev. Lett. 105 177002 [8] Sau J D, Tewari S, Lutchyn R M, Stanescu T D and Das Sarma S 2010 Non-abelian quantum order in spin-orbitcoupled semiconductors: search for topological majorana particles in solid state systems Phys. Rev. B 82 214509 [9] Nadj-Perge S, Drozdov I K, Bernevig B A and Yazdani A 2013 Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor Phys. Rev. B 88 020407(R) [10] Klinovaja J, Stano P, Yazdani A and Loss D 2013 Topological superconductivity and majorana fermions in RKKY systems Phys. Rev. Lett. 111 186805 [11] Rokhinson L P, Liu X and Furdyna J K 2012 The fractional A C Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles Nat. Phys. 8 795 [12] Williams J R et al 2012 Unconventional Josephson effect in hybrid superconductor-topological insulator devices Phys. Rev. Lett. 109 056803 [13] Das A, Ronen Y, Most Y, Oreg Y, Heiblum M and Shtrikman H 2012 Zero-bias peaks and splitting in an AlInAs nanowire topological superconductor as a signature of Majorana fermions Nat. Phys. 8 887 [14] Mourik V et al 2012 Signatures of Majorana fermions in hybrid superconductorsemiconductor nanowire devices Science 336 1003 [15] Churchill H O H et al 2013 Superconductor-nanowire devices from tunneling to the multichannel regime zero-bias oscillations and magnetoconductance crossover Phys. Rev. B 87 241401(R) [16] Lee E J H et al 2014 Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures Nat. Nanotechnology 9 79 [17] Nadj-Perge S, Drozdov I K, Bernevig B A and Yazdani A 2014 Observation of Majorana fermions in ferromagnetic atomic Chains on a superconductor Science 10 1126 [18] Lee A P 2014 Seeking out Majorana under the microscope Scinece 346 545 [19] Flensberg K 2010 Tunneling characteristics of a chain of Majorana bound states Phys. Rev. B 82 180516 [20] Prada E, San-Jose P and Aguado R 2012 Transport spectroscopy of NS nanowire junctions with Majorana fermions Phys. Rev. B 86 180503(R) [21] Yamakage A, Yada K, Sato M and Tanaka Y 2012 Theory of tunneling conductance and surface-state transition in superconducting topological insulators Phys. Rev. B 85 180509(R) [22] Chang W, Manucharyan V E, Jespersen T S, Nygard J and Marcus C M 2013 Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction Phys. Rev. Lett. 110 217005 [23] Finck A D K, van Harlingen D J, Mohseni P K, Jung K and Li X 2013 Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device Phys. Rev. Lett. 110 126406
4. Conclusion We have proposed an all-optical method to detect the existence of Majorana fermions in ferromagnetic atomic chains on a superconductor via the hybrid spin–microcantilever. The spin resonance effect may provide another supplement for detecting Majorana fermions. Due to the microcantileverʼs vibration, the spin resonance effect becomes much more significant and then enhances the detectable sensitivity of Majorana fermions. Additionally, the well-established [44] optical NV spin readout technology will certainly promote detection. Finally, we hope that our proposed scheme can be realized experimentally in the future.
Acknowledgments This study was supported by the National Natural Science Foundation of China (Nos. 10974133 and 11274230) and the Basic Research Program of the Committee of Science and Technology of Shanghai (No. 14JC1491700).
References [1] Kitaev A 2001 Unpaired Majorana fermions in quantum wires Phys. Usp. 44 131 [2] Nayak C, Stern A, Freedman M and das Sarma S 2008 NonAbelian anyons and topological quantum computation Rev. Mod. Phys. 80 1083 7
Nanotechnology 26 (2015) 195501
W-H Wu and K-D Zhu
[42] Walter S and Budich J C 2014 Teleportation-induced entanglement of two nanomechanical oscillators coupled to a topological superconductor Phys. Rev. B 89 155431 [43] Rabl P, Kolkowitz S J, Koppens F H L, Harris J G E, Zoller P and Lukin M D 2010 A quantum spin transducer based on nanoelectromechanical resonator arrays Nat. Phys. 6 602–8 [44] Gruber A et al 1997 Scanning confocal optical microscopy and magnetic resonance on single defect centers Science 276 2012 [45] Xu X et al 2007 Coherent optical spectroscopy of a strongly driven quantum dot Science 317 929 [46] Cao Y S, Wang P Y, Xiong G, Gong M and Li X Q 2012 Probing the existence and dynamics of Majorana fermion via transport through a quantum dot Phys. Rev. B 86 115311 [47] Dong E L and Harold U B 2011 Detecting a Majorana-fermion zero mode using a quantum dot Phys. Rev. B 84 201308(R) [48] Lutchyn R M, Jay D S and das Sarma S 2010 Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures Phys. Rev. Lett. 105 077001 [49] da Silva S J S, Seridonio A C, del Nero J and Souza F M 2014 Nonequilibrium transport in a quantum dot attached to a Majorana bound state arXiv:1403.3575 [50] Seridonio A C, Siqueira E C F, Dessotti A, Machado R S and Yoshida M 2014 Fano interference and a slight fluctuation of the Majorana hallmark J. Appl. Phys. 115 063706 [51] Ridolfo A, Stefano O D, Fina N, Saija R and Savasta S 2010 Quantum plasmonics with quantum dot-metal nanoparticle molecules: influence of the Fano effect on photon statistics Phys. Rev. Lett. 105 263601 [52] Boyd R W 1992 Nonlinear Optics (San Diego, CA: Academic) p 225 [53] Mahan G D 1992 Many-Partcle Physics (New York: Plenum Press) [54] Hewson A C 1993 The Kondo Problem to Heavy Fermions (New York: Cambridge University Press) [55] Scully M O and Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) [56] Gardiner C W and Zoller P 2004 Quantum Noise (New York: Springer) [57] Vitali D and Giovannetti V 2001 Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion Phys. Rev. A 63 023812 [58] Weis S, Riviére R, Deleglise S, Gavartin E, Arcizet O, Schliesser A and Kippenberg T J 2010 Optomechanically Induced Transparency Science 330 1520 [59] Li J J and Zhu K D 2013 All-optical mass sensing with coupled mechanical resonator systems Phys. Rep. 525 223–54
[24] Bagrets D and Altland A 2012 Class D spectral peak in majorana quantum wires Phys. Rev. Lett. 109 227005 [25] Poot M, Zan H S and van Dert J 2012 Mechanical systems in the quantum regime Phys. Rep. 511 273 [26] Aspelmeyer M, Kippenberg T J and Marquardt F 2014 Cavity Optomech. Rev. Mod. Phys. 86 1391 [27] Childress L and Hanson R 2013 Diamond N V centers for quantum computing and quantum networks MRS Bull 38 134–8 [28] Rondin L et al 2014 Magnetometry with nitrogen-vacancy defects in diamond Rep. Prog. Phys. 77 056503 [29] Balasubramanian G et al 2008 Nanoscale imaging magnetometry with diamond spins under ambient conditions Nature 455 648–51 [30] Togan E et al 2010 Quantum entanglement between an optical photon and a solid-state spin qubit Nature 466 730–4 [31] Childress L et al 2006 Coherent dynamics of coupled electron and nuclear spin qubits in diamond Science 314 281–5 [32] Kolkowitz S et al 2012 Coherent sensing of a mechanical resonator with a single-spin qubit Science 335 1603 [33] Rabl P et al 2009 Strong magnetic coupling between an electronic spin qubit and a mechanical oscillator Phys. Rev. B 79 041302(R) [34] Teissier J, Barfuss A, Appel P, Neu E and Maletinsky P 2014 Resolved sidebands in a straincoupled hybrid spin-oscillator system Phys. Rev. Lett. 113 020503 [35] Ovartchaiyapong P, Lee K W, Myers B A and Bleszynski Jayich A C 2014 Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator Nat. Commun. 5 4429 [36] Ovartchaiyapong P, Pascal L M A, Myers B A, Lauria P and Bleszynski Jayich A C 2012 High quality factor singlecrystal diamond mechanical resonators Appl. Phys. Lett. 101 163505 [37] Tao Y, Boss J M, Moores B and Degen C L 2014 Singlecrystal diamond nanomechanical resonators with quality factors exceeding one million Nat. Commun. 5 3638 [38] Jarmola A, Acosta V M, Jensen K, Chemerisov S and Budker D 2012 Temperature and magnetic field dependent longitudinal spin relaxation in nitrogen-vacancy ensembles in diamond Phys. Rev. Lett. 108 197601 [39] Acosta V M, Bauch E, Ledbetter M P, Waxman A, Bouchard L S and Budker D 2010 Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond Phys. Rev. Lett. 104 070801 [40] Balasubramanian G et al 2009 Ultralong spin coherence time in isotopically engineered diamond Nat. Mater. 8 383–7 [41] Walter S, Schmidt T L and Trauzettel B 2011 Detecting Majorana bound states by nanomechanics Phys. Rev. B 84 224510
8
