Entanglement between nitrogen vacancy spins in diamond controlled by a nanomechanical resonator

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by  L. Chotorlishvili

Entanglement between nitrogen vacancy spins in diamond controlled by a nano
mechanical resonator
L. Chotorlishvili1, D. Sander2, A. Sukhov1, V. Dugaev1,3, V. R. Vieira4, A. Komnik5, J. Berakdar1
1Institut fu¨r Physik, Martin-Luther Universita¨t Halle-Wittenberg, D-06120 Halle/Saale, Germany
2Max Planck Institute of Microstructure Physics, D-06120 Halle/Saale, Germany
3Department of Physics, Rzesz´ow University of Technology Al. Powstanc´ow Warszawy 6, 35-959 Rzesz´ow, Poland
4Department of Physics and CFIF, Instituto Superior T´ecnico,
Universidade T´ecnica de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
5Institut fu¨r Theoretische Physik, Universita¨t Heidelberg, Philosophenweg 19, D-69120
(Dated: January 21, 2013)
We suggest a new type of nano-electromechanical resonator, the functionality of which is based
3 on a magnetic field induced deflection of an appropriate cantilever that oscillates between nitrogen
1 vacancy (NV) spins in daimond. Specifically, we consider a Si(100) cantilever coated with a thin
0 magnetic Ni film. Magnetoelastic stress and magnetic-field induced torque are utilized to induce a
2
controlled cantileverdeflection. Itisshown that,dependingonthevalueofthesystemparameters,

n the induced asymmetry of the cantilever deflection substantially modifies the characteristics of the
a system. In particular, thecoupling strength between the NV spins and thedegree of entanglement
J can be controlled through magnetoelastic stress and magnetic-field induced torque effects. Our

7 theoretical proposal can be implemented experimentally with the potential of increasing several
1 times the coupling strength between the NV spins as compared to the maximal coupling strength
reported before in P.Rabl, et al. Phys. Rev. B 79, 041302(R) (2009).

]
h
p I. INTRODUCTION netic resonance force microscopy (MRFM) which was
-
t proposed to improve the detection resolution for the
n three-dimensional imaging of macromolecules27. Oper-
a Nano-electromechanical resonators (NEMs) are at-
u ation of the MRFM is based on the detection of the
tracting intense research efforts due to a number of fa-
q magnetic force between a ferromagnetic tip and spins.
vorable properties such as the high sensitivity and the
[ The possibility to achievestrong(up to 0.1MHz or even
swiftresponsetoanexternalforcewithalowpowercon-
stronger) coupling between the quantized motion of the
1 sumption. This makesNEMs attractivefor applications,
nano-resonatorand NV impurity spin was demonstrated
v
6 e.g. for microwave switches, nano-mechanical memory in Ref.29. However, an efficient control of the coupling
elements, and for single molecule sensing1–3. The role of
5 strengthandthedevelopmentofultrasensitivecantilever-
quantized mechanical motion coupled to other quantum
2 based force sensors is still a fundamental challenge. Yet
4 degrees of freedom as well as the influence of dissipation another application is based on using such a resonator
. andnoiseareimportantissuesinNEMsresearchwithnu-
1 in order to produce controllable entanglement between
merous findings and demonstrations of applications, e.g.
0 spins. Despite rather long decoherence time scalesit has
3 Refs.4–23 and further references therein. These studies not been demonstrated yet, to our knowledge.
1 also evidence the potential of NEMs for studying funda-
: mentalquestionsconcerningthequantum-classicalinter- In this paper we propose a new type of a nano-
v
relationandissuesrelatedtoentanglementandquantum electromechanical resonator system, the functionality of
i
X correlations. Particularlyinterestingforthepresentwork which is based on the NV spin controlled by a magnetic
r are spin states in nitrogen vacancy (NV) impurities in field. Thenano-resonatorisacantilever,whichiscovered
a diamond24–31 as utilized for quantum information stud- withamagneticfilm,suchthatwecanexploitmagnetoe-
ies. Theirquantummechanicalpropertiescanbemapped lastic stress or magnetic torque effects for a controllable
onto effective two-level systems which possess very long cantilever deflection. At the free end of the cantilever
decoherence times even at a room temperature. On the magnetic tips are mounted, as indicated in Fig. 1. The
otherhand,theyallowforahighdegreeoftunability via magnetic field allows for a full control of the cantilever
externalmagneticfieldswhichrenderspossibletheuseof configuration via the amplitude of the applied magnetic
NV impurity spins as sensors. For instance, a detection field. This is the major novelty of the current proposal.
of a single electronic spin by a classical cantilever was As will be demonstrated below, the field-induced deflec-
demonstrated in27. Magnetic tips attached on the free tion of the cantilever has a major impact on the inter-
end of a cantilever generate a magnetic field gradient action strength between the NV spins and the magnetic
during oscillations, which induces a magnetic coupling tips,aswellasonthestrengthofthe indirectinteraction
between the nanomechanical oscillator and the spin sys- between the spins mediated by the cantilever. We will
tem. Using the backaction due to this coupling one can show that the asymmetry of the nanomechanicalsystem
readout the cantilever motion24,25. with respect to the shape of the cantilever controlled by
One further fascinating application for the NV spin magnetic fields drastically changes the degree of entan-
based nano-electromechanical resonator system is mag- glement.
2
The paper is organized as follows. In the next section δ
+
we introduce our system as well as the model Hamilto- |+1>
~B S
nian and discuss the relevant experimental parameters. 0 z
SectionIIIdescribesthemagneticfieldinducedcantilever
Ω
deflection. The subsequent Section IV focuses on the ef- δ- +
fects ofthe inducedindirectinteractionofthe impurities ω0 |-1> Ω
mediated by the cantilever coupling. The influence of -
theasymmetrycouplingonthedegreeofentanglementis |0>
considered in Section V. Finally, Section VI summarizes
the results gives some perspectives for further progress. FIG. 2. Configuration of the energy levels. Ω± stand for the
Rabi frequencies between the ground and the excited levels
a) (| 0 >,| −1 >) and (| 0 >,| 1 >). δ± denote the detuning
Spin 1 between microwave frequency ω0 and transition frequencies.
Magnetic tip
h
1 anadjacentvacancy. Thetotalspinofthemany-electron
Nickel layer orbitalgroundstate ofthe NVcenteris describedby the
spin triplet S = 1, m = 1,0,1. States with differ-
S
ent m are separated by a−zero field splitting barrier24
h S
2 whic|his|of the orderω =2.88GHz. This splitting is an
0
intrinsicpropertyoftheNVspinsystemandoriginates31
Spin 2
fromtheeffectofthespin-orbitandspin-spininteractions
Z b) Microwave ltehaedtinhgeotroettihcealsdinegslcer-iapxtiisonspwineasnetis~ot=rop1y. TDhSez2r≈ole~ωo0f.thIne
X applied external magnetic field is twofold: First of all,
Spin 1 B the applied external magnetic field µBB0 < ~ω0, due to
Nickel layer h1 0 the Zeeman shift proportional to B0Sz, removes the de-
generacy of the levels 1 >, 1 >. Besides, as will be
|− |
shownbelow,theexternalmagneticfield,duetothethin
Magnetic tips
magnetic Ni film deposited on the cantilever, modifies
NAMR
h the shape of the cantilever. We will demonstrate that
2
theasymmetryofthecantileverpositionbetweenspins1
Microwave
Vibrational direction and2(Fig.1),inducedbyanexternalmagneticfield,has
Spin 2
importantconsequencesforthecouplingstrengthandthe
entanglement between the NV spins. Therefore, one can
FIG. 1. Schematics of the nanomechanical resonator system
controlthedegreeofentanglementandincreasetheinter-
discussed in the present project. a) The distances h1,2 be- action strength between the NV spins. This constitutes
tween the NV spins and magnetic tips can be adjusted by
an important advance in nanomechanics.
the field-induced deflection of the cantilever. The deflection
The Hamiltonianofthe singleNV spinsystemreads24
iscontrolledbytheappliedmagneticfieldB0. Intheabsence
of the external constant magnetic field B0 = 0 the system
Ω
becomes symmetric, and h1 = h2 = h0 ≈ 25 nm b). Mi- H = δ i i + i (0 i + i 0) . (1)
NV i
crowaves areusedtodrivespintransitions at theNVcenters − | ih | 2 | ih | | ih |
i=±1(cid:18) (cid:19)
of spin 1 and spin 2. b) Two magnetic tips are mounted on X
the free end of the cantilever, which mechanically oscillates In the case of a weak magnetic field µ B ~ω
B 0 0
in the z-axis direction. The length of the cantilever is of the ≪
( 30 mT) one can neglect level splitting and set δ− =
orderL∼3000 nm . Thecantileversurface iscoveredwith a ≪
10 nm thin Nifilm. SeeSection III for furtherdetails. δ+, Ω− = Ω+. In this case the Hamiltonian (1) cou-
ples the ground state 0> to the “bright” superposition
| 1
of the excited states b >= 1 > +1 > , while
| √2 |− |
1 (cid:0) (cid:1)
II. THEORETICAL MODEL the “dark” state d >= 1 > 1 > is readout
| √2 |− −|
and decoupled from the pro(cid:0)cess. Since only(cid:1)two states
The system of our interest is schematically shown in are involved, the NV spin triplet can be described via
Fig. 1. Magnetic tips are attached to the free ends of a S = 1/2 pseudo-spin model. If the external constant
the cantilever on both sides. The deflection of the can- magnetic field is strong enough, the splitting between
tilever and consequently the distance between magnetic the levels 1>, 1> is larger and the system becomes
tipsandspinsh iscontrolledviathemagneticfieldB~ identicalto|−thethr|eelevelgeneralizedJaynes-Cummings
1,2 0
appliedalongtheZ axis. Thenitrogenvacancycenterin modelinthesocalledlambdaconfiguration32,33(SeeFig.
diamond consists of a substitutional nitrogen atom with 2).
3
ApeculiarityofthegeneralizedJaynes-Cummingsmodel by eq. (2)) reads
is the expectation of two different transition frequencies
1
bHeatmweiletnonsiatanteosf t(h|0e s>y,st|e−m1ho>ld)saanSdU(|(03)>sy,m|1m>e)t.ryTanhde Sz = 2[cosθσz+sinθ(σ++σ−)],
can be specified in terms of the Gell-Mann generators32. H = 1ωσ , ω = Ω2+δ2 1/2. (6)
S z
The three-level Jaynes-Cummings model is exactly solv- 2
able in the general case. If the rf-field contains only (cid:0) (cid:1)
Taking into account eqs. (4)-(6) we have for the Hamil-
one frequency resonant to the transition (0 >, 1 >)
| |− tonianofasingleNVspininteractingwiththecantilever
the system can be reduced to the S = 1/2 pseudo-spin
model which will describe transitions between the states 1 λ
Hˆ = ωσz +ω a+a+1/2 + a++a
1 >, 0 > only, since the transition between the lev- 2 r 2 ×
|− |
els 1 >, 0 > is off resonance and therefore forbidden. (cid:0)cosθσz +s(cid:1)inθ σ(cid:0)++σ−(cid:1) . (7)
| |
However, the Rabi frequency Ω− of the transition be-
tween the states 1 >, 0 > in this case is different Now we generalize th(cid:2)e model given (cid:0)by eq. (7(cid:1))(cid:3)for the
| − |
from the transition frequency between the ground state system of two spins (see Fig. 1). In the absence of an
0 > and the bright superposition of the excited states applied magnetic field the system is symmetric and the
| 1 distance between the spins and the cantilever is equal
b >= 1 > +1 > . Nevertheless, in both cases
| √2 |− | h1 = h2 = h. However, due to the magnetic field in-
our system(cid:0) can be describ(cid:1)ed via the effective two-level duced cantilever deflection, the external magnetic field
model with a different Rabi transition frequency. The leads to an asymmetry h = h . The imposed asymme-
1 2
6
Hamiltonian of the system in the frame rotating with try ∆h(B ) = 2h h /(h + h ), 0 < ∆h(B ) < 1
0 1 2 1 2 0
| − |
the frequency of the rf-field has the form can be quantified in terms of the amplitude of applied
magnetic field B and the geometric and material char-
0
Ω(B)
H = δ 1 1 + ( 1 0 + 0 1). (2) acteristics of the cantilever, as outline din Section III.
S
− |− ih− | 2 |− ih | | ih− | The amplitude of zero point oscillations of the can-
cHreorweaδve=freωq0u−encΩy(Ba)ndistthhee idnettruinnsiincgfbreeqtwueenecnythoef mthie- wtilheevreerims ofth6e ord1e0r−o1f7ak0g=is t~h/e(2rmesωorn)a≈to5r×m1a0s−s1a3mnd,
≈ × p
spins. The Rabi frequency of the transition between up ωr/(2π) 4 MHz is the first resonance frequency of the
≈
and down spin states Ω(B) depends on the amplitude of cantilever. Details on the cantilever are given in Sec-
themagneticfieldand,thus,includesbothlimitingcases. tion 3.
InthecaseofzeroexternalconstantmagneticfieldB =0 Thisoscillationamplitudeisdefinitelysmallerthanthe
andzerosplittingbetweenthelevels 1>, 1>theRabi asymmetry imposed by the external magnetic field, i.e.
frequency for the transition between|−the br|ight b> and ∆h(B0) a0. Nevertheless,oscillationsofthecantilever
the ground 0 > states is equal to Ω(0) = Ω ,|while in produce≫a varying magnetic field due to the attached
0
the caseofa|nonzeroexternalfieldandnonzerosplitting magnetic tips, which is proportional to the oscillation
for the transition between levels 1 >, 0 > the Rabi amplitude. The key consequence of the asymmetry is
frequency is equal to: |− | that constants of the interaction between the spins and
themagnetictipsλ aredifferentfordifferentspinsand
1,2
Ω(B)=Ω ∆Ω(B), ∆Ω(B)=µ (B +B ). (3) the Rabi transitionfrequencies Ω are different as well.
0 B 0 ms 1,2
−
Therefore,theHamiltonianofthesystemoftwoNVspins
Here B is the amplitude of the external constant mag-
0 interacting with the deformed cantilever reads
netic field applied on the system, B is the magnetic
ms
field produced by the magnetic film on the cantilever at Hˆ =Hˆ +Vˆ,
0
thepositionofthespinsSpin1and2,anditisB0 >Bms. Hˆ =1~ω σz + 1~ω σz +~ω a+a+1/2 , (8)
Theeigenbasisofthe Hamiltonian(2)isgivenbythefol- 0 2 1 1 2 2 2 r
lowing states25 Vˆ =λ a++a 1~ cosθ σz +(cid:0)sinθ σ+(cid:1)+σ− +
g =cos(θ/2) 1 +sin(θ/2) 0 , 1 2 1 1 1 1 1
|ei= sin(θ/2|)− i1 +cos(θ/2|)i0 , (4) λ (cid:0)a++a(cid:1)1~(cid:2)cosθ σz +sinθ (cid:0)σ++σ−(cid:1)(cid:3).
| i − |− i | i 2 2 2 2 2 2 2
Ω
tanθ =−δ. Here the t(cid:0)erm ~ω(cid:1)r(a+(cid:2)a + 1/2) describe(cid:0)s quantiz(cid:1)e(cid:3)d os-
cillations of the cantilever, tanθ = Ω1,2 and ω =
In the basis (4) the components of the pseudo-spinoper- 1,2 − δ 1,2
ator have the form Ω2 +δ2, thereby denotes Ω the Rabi frequency of
1,2 1,2
tqhe transition between up and down spin states and δ is
σz = e e g g , σ+ = e g , σ− = g e (5)
| ih |−| ih | | ih | | ih | the detuning between the microwave frequency and the
1 intrinsic frequency of the spins. In the symmetric case
whiletheZ componentofthespinS = 0><0
z 2 | |−|− h1 =h2 =h,ω1 =ω2 =ω andλ1 =λ2 =λtheHamilto-
1>< 1 andtheHamiltonianofthe syst(cid:0)emHS (given nian (1) recoversthe previously studied model24,25. The
− |
(cid:1)
4
coupling constants between the spins and the cantilever a valid scenario of stress-induced free two-dimensional
has the formλ =g µ Gm a ,where g 2, µ is the bending due to the large length-to-width ratio43.
1,2 S B 1,2 0 S ≈ B
Bohr magneton and Gm = 1 B~ is the magnetic field We assume that the cantilever is fabricated out of
gradientproducedbyt1h,e2 mazˆg|nettiipc|tips. Note thatifthe Si(100). Weusethecorresponding44YoungmodulusY =
asymmetry of the deformationof the cantilever is strong 130 GPa and Poisson ratio ν = 0.279, and density ρ =
∆h(B ) 1, the distance between the magnetic tip and 2.33 103 kg/m−3. Such a cantileverhas a mass of m=
the ne0are∼st to the cantilever adjacent NV spin becomes 6.29× 10−17 kg. Its first three resonance frequencies45
very small z = h1 leading to the very large magnetic are c×alculated from fres = tsβ2(Y/(3ρ))0.5/(4πL2) with
field gradient Gm = 1 B~ and to the large interac- β = (1.8751,4.6941,7.8548)as 4.02, 25.2 and 70.6 MHz.
1 h1| tip| This cantilever has a negligible deflection at its end due
tionconstantλ g µ Gm a /h ,whilethecouplingto
the second spin1b≈ecoSmeBs w1,e2ak0λ21≈gSµBGm1,2a0/h2 and itsomitsoroewtnhwaneigshixtoofrd3eρrLs4/o(f2mYat2sg)ni=tu2d.e4×sm1a0l−le1r5mth,awnhtihche
therefore the relation λ1 = h2 1 holds. This means
λ2 h1 ≫ field induced deflection, as described next.
that one can easily control the interaction between the
magnetictipsandtheNVspinssimplybytuningtheam-
plitudeoftheexternalmagneticfieldandthuscontrolling
1. Magnetoelastic-stress-induced cantilever deflection
thedeflectionofthecantilever∆h(B ). Inthesymmetric
0
casez =h =h,∆h=0,Gm =Gm =G 106[T/m]
and realist1i,c2 values of the pa1ramet2ers arem: ≈h 25 nm, Magnetoelastic stress is responsible for the change of
λ/(2π) 0.1 [MHz], ω /(2π) 5 MHz. In wh≈at follows length of a bulk sample upon magnetization, and the re-
r
the inte≈ractionconstantλ and≈the detuning between the sultingstrainisknownasmagnetostriction38,46. Infilms,
cantilever frequency and the spin splitting ∆ = ω ω, a change of length of the film upon magnetization is not
r
∆ 2λ defines the time scale of the problem. A−de- possibleduetothebondingtothesubstrate,andthefilm
viat≈ion of the values of the constants from the values develops a magnetoelastic stress. This stress induces a
corresponding to the symmetric case reads curvatureofthesubstrate,whichweexploittodeflectthe
end of the cantilever. The role of the external magnetic
λ =λ(h )=λ ∆λ, ∆ =∆(h )=∆ ∆ω, field is to induce a reorientation of the magnetization of
1,2 1,2 1,2 1,2
± ±
Ω2 the film from an in-plane (external magnetic field off)
ω1,2= (Ω ∆Ω)2+δ2 =ω ∆ω, ∆ω = (∆h),(9) to an out-of-plane direction (external magnetic field on
± ± ω
q along z-direction). Such a reorientation of the magne-
∆λ=λ∆h, ∆Ω=Ω∆h.
tization direction of the film with thickness t induces a
f
correspondingchangeofmagnetoelasticstress. Itinduces
Estimates of the asymmetry parameter ∆h(B ) describ-
0 a deflection of the free cantilever end given by:38
ingadeformationofthecantilever,forrealisticmaterials
and magnetic fields, follow in the next section.
3L2t (1+ν)B
f 1
defl = (10)
me Yt2
III. MAGNETIC-FIELD-INDUCED s
CANTILEVER DEFLECTION
For a magnetization reversal along the axes of a cu-
bic system the magnetoelastic coupling coefficient B
1
We propose to deposit a magnetic film on the can- enters38. For definiteness, we assume a Ni film of thick-
tilever to explore alternatively magnetoelastic stress or ness 10 nm and we take B of bulk Ni, 9.38 MJ/m3.
1
magnetic-fieldinducedtorquetoinduceacontrolledcan- We note that the effective magnetoelastic coupling in
tileverdeflectionduetoanexternalmagneticfieldB0. In thin films may deviate from its bulk value38,39,47, but
thefirstcaseweexploitthetendencyofafilmtodevelop we stick to the bulk value for this proof of principle case
magnetoelasticstressuponmagnetization,andthisstress study. With these assumptions we get a deflection of
induces a curvature of the thin cantileversubstrate38–40. defl = 27.7 nm. A Ni film with bulk properties de-
me
In the second case we exploit the magnetic torque41 positedonthetopsurfaceofthecantileverhasatendency
T~ =m~ B~0,wherem~ isthetotalmagneticmomentofthe to contract along the magnetization direction. Thus,
×
film,whichispreparedtobeorientedalongthecantilever the cantileverwouldbe curvedupwardsforzeroexternal
length(x-direction),andB~ istheappliedmagneticfield, field(magnetizationin-planealongx),anditwouldcurve
0
oriented perpendicularly to the cantilever long axis (z- downwardfor magneticfieldon(magnetizationalongz).
direction). In this geometry a cantileverdeflection along This is irrespective of the sign (along +z or along z)
−
the z-direction results. of the external field. For AC magnetic fields (ω ) the
mag
For a quantitative determination of the resulting can- deflection will change with 2 ωmag.
tilever deflection we specify the cantilever dimensions as The required deflection can be adjusted by varying
follows: length L = 3000 nm, width w = 300 nm, thick- the experimental parameters t ,t ,L, accordingly. Note
f s
nesst =30nm. Theseparametersarewellsuitedforthe that the magnitude of the external magnetic field does
s
nano-fabrication of Si cantilevers42. They also represent not enter. The only requirement is that it is large
5
enough to induce a magnetization reorientation. The ef- 2h h /(h +h ). We calculate the magnetic tip–NV
1 2 1 2
| − |
fective magnetic anisotropy of the magnetic film should spindistancesh ,h ash =h deflandh =h +defl,
1 2 1 0 2 0
−
befairlysmalltoachievethis. Itcanbetunedbyadjust- whereh describesthesymmetriccase,i.e. thecantilever
0
ing film thickness, morphology, interface modifications, end is at the center position between both NV spins,
multilayer structures, and film composition to achieve which is realized for zero magnetic field for the torque-
this goal. The maximum field is given by the require- induced deflection. This gives the asymmetry parameter
ment µ B < ~ω , as pointed out above. This gives
B 0 0
B <32.7 mT.
0 defl 16t µ B L3
torque f B 0
∆h(B )=2 = . (12)
0 h h ρ Yt3
0 0 atomic s
2. Magnetic-torque-induced cantilever deflection
IV. EFFECTIVE HAMILTONIAN: INDIRECT
The deposition of a magnetic film on the cantilever INTERACTION BETWEEN SPINS
opens also the possibility to exploit the torque induced
by the external magnetic field along the z-direction on
Interaction of the NV spins with the magnetic tips
the magnetic moments oriented along the x-direction of
thefilmtoinduceacantileverdeflection48. Themagnetic leads to an indirect interaction between the NV spins.
The Hamiltonian of the indirect interaction between the
torqueT~ =m~ B~ isproportionaltoboththetotalmag-
× 0 NV spins can be evaluated using the Fr¨ohlich method35
netic moment of the film m and the external magnetic
field B0. Here, we assume that the magnetic anisotropy 0
of the film is large enough to ensure an in-plane mag- Hˆ = iλ2 dt′[V (t′),V (0)], Vˆ(t)=e−iHˆ0tVˆeiHˆ0t
eff
neticmomentinpresenceoftheout-of-planefield. Thus, 2
−Z∞
the requirement on the magnetic anisotropy of the film
(13)
differs fromthat discussedabove for the magnetoelastic-
Taking into account eqs. (8) and (13) in the rotating
stress-induced deflection, as here an effective magnetic
wave approximationwe deduce
anisotropy favoring in-plane magnetization is required.
peFrofirlmdefiantiotmen,esasnwdewaessuremfeeratmoatghneetaitcommoicmevnotluomfe2µoBf Hˆeff =−41 sin2θ1∆λ21 (2n+1)σ1z +sin2θ2∆λ22 (2n+1)σ2z
bulk Ni, ρ = 1.096 10−29 m−3 to calculate the (cid:26) 1 2 (cid:27)
atomic
total number of Ni atoms×in the film. We obtain for the 1sinθ sinθ λ1λ2 + λ1λ2 σ+σ−+σ−σ+ ,
torque-induced deflection41 defltorque = 4TL2/(Ywt3), −4 1 2(cid:18) ∆1 ∆2 (cid:19) 1 2 1 2
and from this we find n= a+a , ∆ =ω ω , ∆ =ω(cid:0) ω . (cid:1)(14)
1 r 1 2 r 2
hh ii − −
Takingintoaccountthatsinθ Ω 1 δ2∆h , δ2
4tf2µBB0L3 1,2 ≈ ω ± ω2 ω2 ≪
defltorque = ρatomicYt3s . (11) 1 and, therefore, sinθ1,2 ≈ Ωω = sinθ(cid:16)we can rew(cid:17)rite the
interaction constant in terms of the asymmetry parame-
We obtain a deflection of 5.2 nm for the parameters ter
quoted above at a film thickness of t = 10 nm for an
f 2
externalfieldB0 =10mT.Notethatherethedeflectionis λ λ sin2θ∆1+∆2 2λ2Ω2 1−(∆h) =
proportionalto the externalmagnetic field, whichallows 1 2 ∆1∆2 ≈ ω2∆ · 1 (cid:16) Ω2 2(∆(cid:17)h)2
acontinuouscontrolofthedeflection. Themagneticfilm − ω∆
couldbe depositedonboth sidesof the cantilever,which 2λ2 Ω2 1+ Ω2 2 1 (∆(cid:0)h)2(cid:1) . (15)
woulddouble the deflection in a givenexternalmagnetic
field. Larger deflections at a given field are obtained by ω (cid:18)ω∆(cid:19)" (cid:18)ω∆(cid:19) − ! #
e.g. increasing the film thickness. From eq. (15) we see that the asymmetry can enhance
We foresee that the exploitationof the magnetoelastic the interaction between the spins if
stress for achieving a well defined cantilever deflection
is experimentally more challenging as compared to the Ω4
>1. (16)
exploitation of torque. One reason is the required exact ω2∆2
tuning of the magnetic anisotropy. The torque approach
In the opposite case,
is muchmorerobustinthataspect,as onlyasufficiently
large energy barrier against out-of-plane magnetization Ω4
is needed. Already the shape anisotropy of out-of-plane <1, (17)
ω2∆2
magnetization fulfills this requirement41. Therefore we
focus now on the torque-induced deflection. the asymmetry lowers the interaction strength. Since
To quantify the indirect interaction between the NV the variance of parameters is relatively large24,25 ωr
2π ≈
spins mediated by the interaction with the magnetic tip 1 5 MHz, Ω 0.1 10 MHz, δ 0.01 0.1 MHz, both
÷ ≈ ÷ ≈ ÷
we consider again the asymmetry parameter ∆h(B ) = cases given by eqs. (16) and (17) can be realized.
0
6
In particular from eq.(15) we see that the interaction with the following notations
strength between spins depends on the asymmetry ∆h
and the dimensionless parameter α = Ω2. Considering λ2 Ω2 Ω2
ω∆ A= (2n+1) 1 1 (∆h)2 ,
standard values of the parameters24,25 ω Ω, λ = ∆ 2ω ω∆ − ω∆ −
≈ 2 (cid:20) (cid:18) (cid:19) (cid:21)
from eq. (15) for the interaction strength between spins λ2Ω2 Ω2
2
we get: λ 1+ α2 1 ∆h 2 . We see that the depen- B=−2ω2∆ 1− ω∆ −1 (∆h) ,
dence of the intera−ction strength between the NV spins (cid:20) (cid:18) (cid:19) (cid:21)
(cid:2) (cid:0) (cid:1)(cid:0) (cid:1) (cid:3) λ2 Ω2 Ω2
ontheasymmetryparameter∆hisnottrivial. Forsmall D= (2n+1) 1 ∆h,
−2ω ω∆ ω∆ −
values of the parameter α < 1 the asymmetry arising (cid:18) (cid:19)
fromthe deformationofcantileverleadstothe reduction F= B2+D2. (20)
of interaction strength, while for α > 1 the asymme-
p
try increases the interaction strength. In particular, for
Taking into account eqs. (18), (19) we can quantify the
the values α = 3, ∆h = 0.5 the interaction between NV
entanglement via the following expression
spinsisthreetimeslargerthantheinteractionstrengthin
the symmetric case 3λ. Stronger interaction between ∗
spins means larger e∼ntanglement. Therefore controlling C(|ψ(t)i)=|hψ (t)|σy ⊗σy|ψ(t)i|. (21)
the spincouplingstrengthwecaninfluence the degreeof
As a result, after straightforward but laborious calcula-
entanglement as well. From the physical point of view
smallvaluesoftheparameterα Ω <1corresponds tions for concurrence we deduce
≈ ωr−Ω
to a large detuning between oscillation frequency of the
cantileverωr andthe Rabifrequency Ωs,while for α>1 |C(t)|= C12(0)−C22(0) cos2At
we have the opposite case. 2
+e−2iFt (cid:12)(cid:12)(cid:0)[−DC3(0)+(F(cid:1)+B)C4(0)]
4F2
!
D2C (0) D(F +B)C (0) 2
e−2iFt 3 − 4
− (cid:2) 4F2(F +B)2 (cid:3) !
V. INFLUENCE OF ASYMMETRY ON THE
2
ENTANGLEMENT DEGREE +e2iFt [DC3(0)+(F −B)C4(0)] (22)
4F2
!
Inordertostudytheinfluenceoftheasymmetryonthe e2iFt D2C3(0)+D(F −B)C4(0) 2
degreeofentanglement,wedirectlysolvetheSchr¨odinger − (cid:2) 4F2(F B)2 (cid:3) !
equation corresponding to the Hamiltonian (14) −
[ DC (0)+(F +B)C (0)][DC (0)+(F B)C (0)]
3 4 3 4
+ − −
2F2 −
idd|ψti =Hˆeff|ψi, (18) [DC3(0)−D(F +B)2CF4(20()F][D2BC3)(20)+D(F −B)C4(0)](cid:12).
ψ =C (t) Φ+ +C (t) Φ− +C (t) ψ+ +C (t) ψ− . − (cid:12)(cid:12)
| i 1 2 3 4 (cid:12)
(cid:12)
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12) (cid:12)
VI. RESULTS
Here Φ± and ψ± are Bell states35–37. Taking into
| i | i
account eqs. (14), (18) for the resolution coefficients we
Following eq. (22) the concurrence is plotted in Figs.
obtain
3, 4 and 5 for a variation of two parameters α and ∆h.
The contour plot of C(t) is presented in Fig. 6.
| |
FromFig. 4weseethatforsmallvaluesofthe param-
C (t)=C (0)cos(At) C (0)sin(At),
1 1 − 2 eterα=0,25,theincreaseoftheasymmetry∆hleadsto
C2(t)=C2(0)cos(At) C1(0)sin(At), asmallerconcurrence. The explanationis thatforα<1
−
D2C (0) D(F +B)C (0) the increase of the asymmetry ∆h reduces the coupling
C (t)= 3 − 4 eiFt (19)
3 strengthbetweenspinsseeeq. (12),while asweseefrom
2(F +B)F
Figs. 3, 5 and6, for largeα>1 concurrenceis increased
D2C (0)+D(F B)C (0)
+ 3 − 4 e−iFt, with asymmetry. In particular, the contour plot of Fig.
2(F B)F 6 defines domains of maximal and minimal concurrence
−
DC (0)+(F +B)C (0) asafunctionofparameters∆h, α. Forα<0,5concur-
C (t)=− 3 4 eiFt
4 2F rence is maximal for small asymmetry ∆h<0,25, while
DC (0)+(F B)C (0) for α<0,5 concurrence is maximal for large asymmetry
+ 3 − 4 e−iFt, ∆h>0,3.
2F
7
reduces the interaction strength. In addition we found
ÈCHtLÈ Α=0.25,Dh=0.4
0.5 ÈCHtLÈ Α=1.5,Dh=0.35
0.5
0.4 Α=0.50,Dh=0.4
0.4 Α=1.5,Dh=0.45
0.3
Α=2.00,Dh=0.4 0.3
0.2
Α=1.5,Dh=0.55
0.2
0.1
0.1
0.0 Timet
0 5 10 15 20 25 30
0.0 Timet
0 5 10 15 20 25 30
FIG. 3. Time evolution of the concurrence |C(t)| for the
given values of α and ∆h. Other parameters read: n = 1, FIG. 5. Time evolution of the concurrence |C(t)| for the
C1 = C2 = C3 = C4 = 1/2. Time-scale corresponds to the given values of α and ∆h. Other parameters read: n = 1,
microsecond. C1 = C2 = C3 = C4 = 1/2. Time-scale corresponds to the
microsecond.
ÈCHtLÈ Α=0.25,Dh=0.1 0.5 Dh,@d.u.D
0.5 0.35
0.4 Α=0.25,Dh=0.3
0.30
0.3
Α=0.25,Dh=0.5
0.2
0.25
0.1
0.0 Timet 0.20
0 5 10 15 20 25 30
0
FIG. 4. Time evolution of the concurrence |C(t)| for the 0.15
given values of α and ∆h. Other parameters read: n = 1, Α,@d.u.D
C1 = C2 = C3 = C4 = 1/2. Time-scale corresponds to the 0.10
microsecond. 0.5 1.0 1.5 2.0
FIG. 6. Contour plot of the concurrence |C(t)| for t = 13.3.
VII. CONCLUSIONS Otherparameters are: n=1, C1=C2 =C3 =C4 =1/2.
One of the main challenges for NV spin-based nano- that for α>1 entanglement is maximum for the case of
electromechanicalresonatorhas been the achievementof alargeasymmetry∆h,whileforα<1theentanglement
a high controlled degree of entanglement and a strong is maximal in the small asymmetry case (See Fig. 6).
coupling between NV spins. With this in mind, we pro- The values ofthe parameterαcanbe changedefficiently
posedin this worka new type ofnano-electromechanical viathechangeofthedetuningbetweentheoscillationfre-
resonator,the functionality of which is based on the NV quency of the cantilever and the spin splitting frequency
spincontrolledbyanexternalmagneticfield. Inparticu- ∆ = ω ω. This can be used as an effective tool for a
r
−
lar,we suggestto deposit a thin magnetic Ni film on the practical implementation of our theoretical proposal for
Si(100) cantilever to exploit alternatively magnetoelas- controllingtheentanglementandtheinteractionstrength
tic stressor magnetic-fieldinduced torquefor inducing a between NV spins in the experiment.
controlled cantilever deflection upon acting with an ex-
ternalmagneticfield. Wehaveshownthat,dependingon
the values of parameter α= Ω4 , the induced asymme- VIII. ACKNOWLEDGMENTS
ω2∆2
try ofthe cantileverdeflection substantially modifies the
characteristics of the system. In particular we demon- The financial support by the Deutsche Forschungs-
strated that is if α > 1 the asymmetry enhances the gemeinschaft (DFG) through SFB 762, contract BE
strengthofthe interactionbetweenthe NVspinsatleast 2161/5-1,15GrantNo. KO-2235/3isgratefullyacknowl-
three times 3λ, where λ = 100kHz is the maximal edged. This work is partly supported by the National
∼
coupling strength between NV spins for the symmetric Science Center in Poland as a research project in years
model reported in24. However, for α<1 the asymmetry 2011 - 2014.
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