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Quantum superposition, entanglement, and state teleportation of a microorganism

on an electromechanical oscillator

Tongcang Li1, 2, 3, ∗ and Zhang-Qi Yin4, †

1Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA2School of Electrical and Computer Engineering,

Purdue University, West Lafayette, IN 47907, USA3Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA

4Center for Quantum Information, Institute of Interdisciplinary

Information Sciences, Tsinghua University, Beijing, 100084, China

(Dated: September 15, 2015)

Schrodinger’s thought experiment to prepare a cat in a superposition of both alive and dead statesreveals profound consequences of quantum mechanics and has attracted enormous interests. Herewe propose a straightforward method to create quantum superposition states of a living microor-ganism by putting a small bacterium on top of an electromechanical oscillator. Our proposal isbased on recent developments that the center-of-mass oscillation of a 15-µm-diameter aluminiummembrane has been cooled to its quantum ground state [Nature 475, 359 (2011)], and entangledwith a microwave field [Science, 342, 710 (2013)]. A microorganism with a mass much smaller thanthe mass of the electromechanical membrane will not significantly affect the quality factor of themembrane and can be cooled to the quantum ground state together with the membrane. Quantumsuperposition and teleportation of its center-of-mass motion state can be realized with the help ofsuperconducting microwave circuits. More importantly, the internal states of a microorganism, suchas the electron spin of a glycine radical, can be prepared in a quantum superposition state and en-tangled with its center-of-mass motion. Our proposal can be realized with state-of-art technologies.The proposed setup is also a quantum-limited magnetic resonance force microscope (MRFM) thatnot only can detect the existence of an electron spin, but also can coherently manipulate and detectthe quantum state of the spin.

I. INTRODUCTION

In 1935, Erwin Schrodinger proposed a famous thoughtexperiment to prepare a cat in a superposition of bothalive and dead states [1]. He imagined that a cat, a smallradioactive source, a Geiger counter, a hammer and asmall bottle of poison were sealed in a chamber. If oneatom of the radioactive source decays, the counter willtrigger a device to release the poison. So the state of thecat will be entangled with the state of the radioactivesource. After a certain time, the cat will be in super-position of both alive and dead states. The possibilityof an organism to be in a superposition state is counter-intuitive and dramatically reveals the profound conse-quences of quantum mechanics. Besides their importancein studying foundations of quantum mechanics, quantumsuperposition and entangled states are key resources forquantum metrology and quantum information [2]. Fur-thermore, the explanation of how a “Schrodinger’s cat”state collapses is the touchstone of different interpreta-tions of quantum mechanics. While many-worlds inter-pretations propose that there is no wave function collapse[3, 4], objective collapse theories [5–8] propose that thewave function collapse is an intrinsic event that happensspontaneously.

∗ tcli@purdue.edu† yinzhangqi@mail.tsinghua.edu.cn

The “Schrodinger’s cat” thought experiment has at-tracted enormous interest since it was proposed eightyyears ago [2]. Many great efforts have been made to cre-ate large quantum superposition states. Superpositionstates of photons, electrons, atoms, and some moleculeshave been realized [9]. Wavelike energy transfer throughquantum coherence in photosynthetic systems has beenobserved [10, 11]. Recent developments in quantum op-tomechanics [12–15] and electromechanics [16–23] pro-vide new opportunities to create even larger superposi-tion states experimentally [24]. In 2009, Romero-Isart etal proposed to optically trap a living microorganism ina high-finesse optical cavity in vacuum to create a su-perposition state [25]. This paper increased our hope toexperimentally study the quantum nature of living or-ganisms [1, 26, 27]. Meanwhile, Li et al demonstratedoptical trapping of a pure silica microsphere in air andvacuum [28], and cooled its center-of-mass motion fromroom temperature to about 1.5 mK in high vacuum [29].Recently, parametric feedback cooling [30] and cavitycooling of pure dielectric nanoparticles [31–33] were alsodemonstrated. These are important steps towards quan-tum ground state cooling of a levitated dielectric particle.Optical trapping of an organism in vacuum, however, hasnot been realized experimentally. The main difficulty isthat the optical absorption coefficient [34] of organismsis much larger than that of a pure silica particle, whichcan lead to significant heating of an optically trapped mi-croorganism in vacuum. Recently, Fisher proposed thatthe nuclear spin of phosphorus can serve as the putative

2

quantum memory in brains [35].Here we propose to create quantum superposition and

entangled states of a living microorganism by puttinga small bacterium on top of an electromechanical os-cillator, such as a membrane embedded in a supercon-ducting microwave resonant circuit (Fig. 1). Our pro-posal also works for viruses. Since many biologists donot consider viruses as living organisms [27], we focus onsmall bacteria in this paper. Our proposal has severaladvantages. First, it avoids the laser heating problemas no laser is required in our scheme. Second, quan-tum ground state cooling and advanced state control ofan electromechanical oscillator integrated in a supercon-ducting circuit have been realized experimentally [16–21, 23]. Quantum teleportation based on superconduct-ing circuits has also been demonstrated [36]. In addition,most microorganisms can survive in the cryogenic envi-ronment that is required to achieve ground state coolingof an electromechanical oscillator. Although microorgan-isms are frozen in a cryogenic environment, they can bestill living and become active after thawing [37]. Cry-opreservation is a mature technology that has been usedclinically worldwide [37]. Most microorganisms can bepreserved for many years in cryogenic environments [38].Even some organs [39] can be preserved at cryogenic tem-peratures. More importantly, the internal states of a mi-croorganism, such as the electron spin of a glycine radicalNH+

3 CHCOO− [40, 41], can also be prepared in superpo-sition states and entangled with its center-of-mass motionin the cryogenic environment. This will be remarkablysimilar to Schrodinger’s initial thought experiment of en-tangling the state of an entire organism (‘alive’ or ‘dead’state of a cat) with the state of a microscopic particle (aradioactive atom).Our proposed setup not only can be used to study

macroscopic quantum mechanics, but also has applica-tions in quantum-limited magnetic resonance force mi-croscopy (MRFM) for sensitive magnetic resonance imag-ing (MRI) of biological samples [42]. This will providemore than structural information that can be obtainedby cryo-electron microscopy [43], which also requires thesample to be cooled to cryogenic temperatures. This sys-tem can coherently manipulate and detect the quantumstates of electron spins, which enables single electronspins that could not be read with optical or electricalmethods to be used as quantum memory.

II. THE MODEL

Recently, a 15-µm-diameter aluminum membrane witha thickness of 100 nm has been cooled to quantumground state by sideband cooling with a superconduct-ing inductor-capacitor (LC) resonator [17]. Coherentstate transfer between the membrane and a traveling mi-crowave field [19], as well as entangling the motion ofthe membrane and a microwave field [18] have been re-alized. The experiment in Ref. [17] was performed in a

MembraneMicroorganism

ΓΩ

(a) (b)

(c)

FIG. 1. (a) Scheme to create quantum superposition statesof a microorganism by putting a small bacterium or a viruswith a mass of m on top of an electromechanical membraneoscillator with a mass of Mmem. The membrane is the upperplate of a capacitor embedded in a superconductor inductor-capacitor (LC) resonator. The LC resonator is coupled to atransmission line. The membrane has an intrinsic mechanicaloscillation frequency of Ωm and a linewidth of Γm when themicroorganism is not on it. The LC resonator has a resonantfrequency of ωc and an energy decay rate of κ. Some mi-croorganisms have smooth surfaces (b) while some have pilion their surfaces (c).

cryostat at 15 mK. The mechanical oscillation frequencyof this membrane is about 10 MHz, with a mechanicalquality factor of 3.3 × 105. The mass of the membraneis 48 pg (2.9× 1013 Da). As shown in Table I, this massis about four orders larger than the mass of ultra-smallbacteria and even more orders larger than the mass ofviruses [44–50]. We can use this membrane oscillator tocreate quantum superposition states of a small microor-ganism. Among different cells, mycoplasma bacteria areparticularly suitable for performing this experiment be-cause they are ubiquitous and their sizes are small [48].In addition, they have been preserved at cryogenic tem-peratures [38]. It will also be interesting to create super-position states of prochlorococcus. Prochlorococcus ispresumably the most abundant photosynthetic organismon the earth, with an abundance of about 105 cells/mLin surface seawater [49, 51]. It can be used to studyquantum processes in photosynthesis.

As shown in Fig. 1a, we consider a bacterium or viruson top of an electromechanical membrane oscillator. Weassume the mass of the microorganismm is much smallerthan the mass of the membrane Mmem. Some microor-ganisms have smooth surfaces (Fig. 1b) while some othermicroorganisms have pili (hairlike structures) on theirsurfaces (Fig. 1c). For simplicity, it will be better to usemicroorganisms with smooth surfaces. One way to puta microorganism on top of an electromechanical mem-brane is to spray a small amount of microorganisms toair in a sealed glove box, and wait one microorganismto settle down on the electromechanical membrane. Theprocess can be monitored with an optical microscope.After being cooled down to a cryogenic temperature, the

3

Microorganism Typical mass m/Mmem

(pg) (Mmem = 48 pg)

Bacteriophage MS2 6× 10−6 10−7

Tobacco mosaic virus 7× 10−5 10−6

Influenza virus 3× 10−4 10−5

WWE3-OP11-OD1 0.01 10−4

ultra-small bacterium

Mycoplasma bacterium 0.02 10−4

Prochlorococcus 0.3 10−2

E. coli bacterium 1 10−2

TABLE I. Comparison of masses of some viruses and bacteriato the mass of a membrane oscillator (Mmem = 48 pg) thathas been cooled to the quantum ground state in Ref. [17].These mass values are obtained from references [44–50].

microorganism can be exposed to high vacuum withoutsublimation of the water ice. Techniques developed incryo-electron microscopy can be utilized here [43]. Thesublimation rate of water ice is only about 0.3 monolay-ers/hour at 128 K with a sublimation energy of 0.45 eV[52]. The sublimation of water ice can be safely ignoredat millikelvin temperatures.At millikelvin temperatures, a frozen microorganism

will be in a hard and brittle state like a glass. It willbe stuck on the membrane due to van der Waals forceor even stronger chemical bonds. The pull-off force be-tween a 1 µm sphere and a flat surface due to van derWaals force is on the order 100 nN [53], which is about107 times larger than the gravitational force on a 1 µmparticle. So the microorganism will move together withthe membrane as long as there is a good contact betweenthem. The oscillation frequency of the membrane oscilla-tor will change by roughly −Ωmm/(2Mmem), where Ωm

is the intrinsic oscillation frequency of the membrane.The exact frequency shift depends on the position of themicroorganism on the membrane. Nonetheless, this fre-quency shift will be small and will not significantly affectthe ground state cooling or other state control of themembrane when m/Mmem << 1.The change of the quality factor Q of the mem-

brane oscillator due to an attached small microorganism(m/Mmem << 1) will also be negligible. For a frozenbacterium with a smooth surface (Fig. 1b), the lowest in-ternal vibration frequency can be estimated by the speedof sound in ice divided by half of its size. So the inter-nal vibration modes of the main body will be larger than1 GHz for a bacterium smaller than 1 µm. This is muchlarger than the frequency of the center-of-mass motionof the electromechanical membrane which is about 10MHz. Thus these internal vibration modes of the mainbody of a bacterium will not couple to the center-of-massvibration of the membrane. The situation will be a lit-tle complex for bacteria that have pili on their surfaces

(Fig. 1c) [54]. For a very thin and long pilus, it is possiblethat its vibration frequency to be on the same order oreven smaller than the center-of-mass vibration frequencyof the electromechanical membrane. However, becausethe quality factors of both the pili and electromechani-cal membrane are expected to be very high at milikelvintemperatures, it is unlikely that their frequencies matcheswithin their linewidths to affect the ground state coolingof the center-of-mass mode of the membrane. One canalso avoid this problem by embedding the pili in waterice.

III. CENTER-OF-MASS MOTION: COOLING,

SUPERPOSITION AND TELEPORTATION

We assume the frequency of the center-of-mass motionof the microorganism and the membrane together to beωm, which is almost the same as Ωm. The motion of themembrane alters the frequency ωc of the superconductingLC resonator. The frequency ωc can be approximatedwith ωc(x) = ω0 + Gx, where x is the displacement ofmembrane, andG = ∂ωc/∂x. The parametric interactionHamiltonian has the form HI = ~Ga†ax = ~Gnx0(am +a†m), where a (am) and a† (a†m) are the creation andannihilation operators for LC (mechanical) resonator, n

is the photon number operator, and x0 =√

~/2Mmemωm

is the zero point fluctuation for mechanical mode. Wedenote the single-photon coupling constant g0 = Gx0.In order to enhance the effective coupling between

LC resonator and mechanical oscillator, the LC res-onator is strongly driven with frequency ωd and Rabi fre-quency Ωd.The corresponding Hamiltonian reads Hd =Ωd

2e−iωdta+ h.c.. The detuning between the driving mi-

crowave source and the LC resonator is ∆ = ωd − ω0.The steady state amplitude can be approximated asα = Ωd/(2∆ + iκ), where κ is the decay rate of the LCresonator. We assume that the steady state amplitude αis much larger than 1. We expand the Hamiltonian witha− α, and the total Hamiltonian reads

H = ~∆a†a+ ~ωma†mam + ~g(a† + a)(a†m + am), (1)

where g = αg0. The detuning can be freely chosen to sat-isfy the requirements of different applications. In order tocool the membrane resonator, we choose ∆ = −ωm. Byusing rotating wave approximation under the conditionthat ωm ≫ g, we get the effective equation as [55]

Heff = ~ga†am + ~gaa†m. (2)

In order to cool the membrane oscillation to the groundstate, the system should fulfill the sideband limit ωm ≫κ, γm, where γm is the decay rate of mechanical mode ofthe whole system [56, 57]. γm will be almost the sameas the decay rate Γm of the mechanical mode of the elec-tromechanical membrane alone when the frequencies ofinternal modes of the microorganism do not match the

4

frequency of the electromechanical membrane. The mas-ter equation for the system is

ρ = −i[Heff , ρ]/~+ Laρ+ Lamρ, (3)

where the generators are defined by Laρ = κDaρ,Lam

ρ = (1 + nm)γmDamρ + nmγmDa

†m

ρ, and the nota-

tion of the generators has Lindblad form Dxρ = 2xρx†−x†xρ − ρx†x. In the limit that κ ≫ g, we can adiabati-cally eliminate the a mode, and get the effective masterequation

ρ = Lamρ+ L′

amρ, (4)

where L′am

= κ′Dam, and κ′ = g2/κ. We define the total

generator L = Lam+L′

am= (1+ n′

m)γ′Dam+ n′

mγ′D

a†m

,

where γ′ = γ + κ′, and n′m = nmγ/γ

′. As long as κ′ >γ, the steady state mean phonon number of mechanicalresonator n′

m is less than 1, which is in the quantumregime. As a bacterium or virus are attached on thetop of membrane, it is also cooled down to the quantumregime.Once the mechanical mode is cooled down to the quan-

tum regime, we can prepare the mechanical superpositionstate by the method of quantum state transfer betweenmechanical and LC resonators. For example, we can firstgenerate the superposition state |φ0〉 = (|0〉+ |1〉)/

√2 for

LC mode a with assistant of a superconducting qubit.Here |0〉 and |1〉 are vacuum and Fock state with 1 photonof mode a. Then, we turn on the beam-splitter Hamil-tonian Eq. (2) between a and am. After the interactiontime t = π/g, the mechanical mode will be in the su-perposition state |φ0〉. Here we suppose that the strongcoupling condition g > nmγ, κ fulfills. Therefore, thecoherence of the superposition state |φ0〉 can maintainduring the state transfer.The quantum state of the center-of-mass motion of

a microorganism can also be teleported to another mi-croorganism using a superconducting circuit. Quantumteleportation based on superconducting circuits has beendemonstrated recently [36]. We consider two remote mi-croorganisms, which are attached to two separate me-chanical resonators integrated with LC resonators. Theyare connected by a superconducting circuit as demon-strated in Ref. [36]. They are initially cooled down tothe motional ground states. We use the ground state andthe first Fock state |1〉 of both mechanical and LC res-onators as the qubit states. The mechanical mode am1 ofthe first microorganism and mechanical resonator is pre-pared to a superposition state |ψ1〉 = α|0〉m1 + β|1〉m1,where α and β are arbitrary and fulfill |α|2 + |β|2 = 1.The LC resonator modes a1 and a2 are prepared to theentangled state (|0〉1|1〉2 + |1〉1|0〉2)/

√2, through quan-

tum state transfer, or post selection [55]. Then by trans-ferring the state a2 to the mechanical mode am2 of thesecond microorganism and mechanical oscillator, the LCmode a1 entangles with mechanical mode am2. This en-tanglement can be used as a resource for teleporting the

state in mechanical mode am1 to the mechanical modeam2 [58].To do this task, we need to perform Bell measurements

on mode a1 and am1, which can be accomplished by aCPHASE gate between a1 and am1, and Hadamard gateson a1 and am1 [36]. In order to realize the CPHASE gate,we tune the driving detuning ∆ − ωm = δ ≪ ωm. Theeffective Hamiltonian between a1 and am1 becomes

H ′eff = ~δa†1a1 + ~ga†am + ~gaa†m (5)

In the limit δ ≫ g, we get an effective photon-phononcoupling Hamiltonian by adopting second order pertur-bation method,

Hpp = ~g2

δa†1a1a

†m1am1. (6)

By turning on the Hamiltonian (6) for time t =πδ/g2, the CPHASE gate between a1 and am1 ac-complishes. The state of the system becomes |Ψ〉 =(α|0〉m1|0〉1|1〉m2 + α|0〉m1|1〉1|0〉m2 + β|1〉m1|0〉1|1〉m2 −β|1〉m1|1〉1|0〉m2)

√2. Then we perform two Hadamard

gates on a1 and m1 modes, and project measurementson the basis |0〉 and |1〉. There are four different out-put 00, 01, 10, 11, which are one-to-one mapping to thefour Bell measurement basis. Based on the output, wecan perform the specific local operation on the mode am2

and recover the original state α|0〉+ β|1〉.

IV. INTERNAL ELECTRON SPINS:

ENTANGLEMENT AND DETECTION

The internal states of a microorganism can also beprepared in superposition states and entangled withthe center-of-mass motion of the microorganism at mil-likelvin temperatures. A good internal state of a microor-ganism is the electron spin of a radical or transition metalion in the microorganism. Radicals are produced duringmetabolism or by radiation damage. Some proteins inmicroorganisms contain transition metal ions, which alsohave non-zero electron spins can be used to create quan-tum superposition states. The smallest amino acid inproteins is glycine. The electron spin of a glycine radicalNH+

3 CHCOO− has a relaxation time T1 = 0.31s and aphase coherent time TM = 6µs at 4.2 K [40]. As shown inFig. 4 of Ref. [40], the phase coherent time increases dra-matically when the temperature decreases below 10 K.So it will be much longer at millikelvin temperatures.Moreover, universal dynamic decoupling can be used toincrease the coherent time TM by several orders, even-tually limited by the relaxation time T1 [59]. Thus weexpect the coherent time of the electron spin of a glycineradical to be much longer than 1 ms at millikevin tem-peratures. The electron spin of some other radicals ordefects in a frozen microorganism may have even longercoherence time at millikelvin temperatures.As shown in Fig. 2, our scheme to couple the spin state

and the center-of-mass motion of a microorganism with

5

MicroorganismMagne c p

Rigid can lever

ΓΩSpin

FIG. 2. Scheme to couple the internal states of a microorgan-ism to the center-of-mass motion of the microorganism witha magnetic gradient. A ferromagnetic tip mounted on a rigidcantilever produces a strong magnetic gradient. A RF currentiRF passing through a superconducting microwire generatesa BRF that excites an electron spin in the microorganism.Different from a typical MRFM apparatus, the cantilever isrigid, while the substrate (membrane) of the microorganismis flexible to oscillate in this setup.

a magnetic tip is similar to the scheme used in MRFM.Recently, single electron spin detection with a MRFM[60], and nanoscale magnetic resonance imaging of the1H nuclear spin of tobacco mosaic viruses [42] have beenrealized. A MRFM using a superconducting quantuminterference device (SQUID)-based cantilever detectionat 30 mK has also been demonstrated [61].As shown in Fig. 2, in order to couple the internal spins

states of a microorganism to the center of mass motion ofthe microorganism, a magnetic field gradient is applied.Above the microorganism, there is a ferromagnetic tipmounted on a rigid cantilever, which produces a mag-netic field B with a large gradient. In a microorganism,there are usually more than one radical that has unpairedelectrons. Because the magnetic field is inhomogeneous,the energy splitting between electron spin states dependson the relative position between an electron and the ferro-magnetic tip. The Hamiltonian for N unpaired electronspins reads

He =

N∑

i

~gsµBSi(~xi) ·B(~xi) (7)

where gs ≃ 2, µB is Bohr magneton, ~xi is the position ofthe ith electrons, Si is the spin operator for ith electronsspins. To make sure that the electron spin of interestis initially in the ground state at 10 mK, we supposethat the electron spin level spacings are larger than 500MHz, which requires a magnetic field larger than 18 mTat the position of the radical. Because the finite size of

the microorganism, the magnetic field at the position ofmembrane can be smaller than 10 mT, which is the criti-cal magnetic field of aluminium. So the membrane can bemade of aluminium [17] if necessary. However, supercon-ducting materials (e.g. Nb) with a relative large criticalmagnetic field will be better for making the membraneand the RF wire [20, 61].The oscillation of membrane induces a time-varying

magnetic field on electrons in the microorganism. Wedefine the single phonon induced frequency shift λ =gsµB|Gm|x′o/~, where x′0 is the zero field fluctuation ofmicroorganism, and Gm = ∂B(~x1)/∂~x1. Here x′0 is dif-ferent from x0 of membrane. Usually, x′0 is two timeslarger than x0 when the microorganism is at the centerof the membrane. Here we assume that the magneticgradient is (un)parallel to both the magnetic field B(~x1)and the mechanical oscillation. The z axis is definedalong the direction ofB(~x1). We apply a microwave driv-ing with frequency ω′

d, which is close to the electron 1’slevel spacing ω1 = gµBB(~x1). The Rabi frequency is Ω′

d,which is tuned to near the mechanical mode frequencyωm. In all electrons, the level spacing ω2 = gµBB(~x2)of the 2nd electron is closest to the 1st one. Underthe condition that gsµb|B(~x1) − B(~x2)| ≫ |Ω|, we have|Ω| ≪ |ω1 − ω2|. Therefore, only electron 1 is driven bythe microwave. We can neglect the effects of all otherfree electrons in microorganism. The Hamiltonian con-tains electron 1 and the motion of the membrane and themicroorganism reads [62]

HeM = ~ωma†mam+

∆e

2~σz+

Ω′d

2~σx+

1

2~λ(am+a†m)σz ,

(8)where ∆e = ω′

d − ω1. We rotate the electron spin axisto diagonalize it, which has the effective level splittingωeff =

√

∆2e + Ω′2

d .We first need to identify the position and frequency of

an electron spin in the microorganism. We can search itby scanning the magnetic tip above the microorganism[60]. The microwaves driving the LC resonator and theelectron spins are turned on with Rabi frequencies Ωd

and Ω′d, and frequencies ωd and ω′

d. Once the electronspin resonance (ESR) reaches ωeff = ωm, the excitationof the electron spin could be efficiently transferred to theLC mode a, mediated with the mechanical mode am,and decay finally. We will observe the ESR signal onthe output spectrum of the LC mode. Then we want totune the ∆e to be zero and maximize the spin-phononcoupling strength. First, we scan the frequency of themicrowave on the electron spins, under the specific Rabifrequency Ω′

d. The ESR peaks appear in pair with thecenter frequency ω1. Then we tune the frequency of themicrowave drive on electron ω′

d = ω1, the ESR shouldhave only one peak. In this way, we find the resonantfrequency ω1 of the electron 1. The magnetic field B

near the electron can be got from the Eq. (7). Then wecan identify the relative location between the electron 1and the magnetic tip. This setup can also detect nuclearspins with lower sensitivity, which has broad applications

6

[35, 42, 63].Let’s suppose that ∆e has been tuned to be zero, and

denote the operators σ± = σz ± iσy, The qubit is definedon the eigenstates of σx. In the limit λ ≪ ωm, |Ω′

d|, weset Ω′

d = ωm and the effective interaction Hamiltonianbetween electrons and the motion of microorganism is

HI = ~λσ+a+ h.c.. (9)

Here the rotating wave approximation is used. We mayalso set Ω′

d = −ωm, and the effective Hamiltonian hasthe form

H ′I = ~λσ+a

† + h.c.. (10)

If we use the parameters in Ref. [17], ωm = 2π × 10MHz, Mmem = 48 pg, we have x0 = 4.2 × 10−15 m,and x′0 ≃ 2x0 = 8.4 × 10−15 m. Under the magneticgradient |Gm| = 107 T/m, the single phonon inducedfrequency shift λ = gsµB|Gm|x′o/~ = 14.8 kHz. Themechanical decay rate γm = 2π × 32 Hz. The effectivemechanical decay under temperature 10 mK is nmγm ≃20 × 2π × 32Hz = 4.0 kHz, which is much less than λ.

Recently, a mechanical decay rate as low as γm = 2π×9.2Hz was realized in a similar electromechanical membraneoscillator [64], which will be even better. The electronspin decay and dephase can be both less than kHz [40,59], which is also much less than λ. Therefore, strongcoupling condition is fulfilled. We can generate entangledstate and transfer quantum states between electron spin 1and the mechanical mode am with either Hamiltonian (9)or (10) [65, 66]. Thus this setup not only can detect theexistence of single electron spins like conventional MRFM[60], but also can coherently manipulate and detect thequantum states of electron spins. It enables some isolatedelectron spins that could not be read out with optical orelectrical methods to be used as quantum memory forquantum information.

TL would like to thank the support from PurdueUniversity and helpful discussions with G. Csathy, F.Robicheaux, C. Greene, and V. Shalaev. ZQY isfunded by the NBRPC (973 Program) 2011CBA00300(2011CBA00302), NNSFC NO. 11105136, NO. 61435007.ZQY would like to thank the useful discussion with LeiZhang.

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