Table Of ContentFormation and Dynamics of a Schr¨odinger-Cat State
in Continuous Quantum Measurement
G.P. Berman1, F. Borgonovi1,2, G. Chapline1,3, S.A. Gurvitz1,4, P.C. Hammel5,
D.V. Pelekhov5, A. Suter5, and V.I. Tsifrinovich6
1Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, NM 177545
2Dipartimento di Matematica e Fisica, Universit`a Cattolica, via Musei 41 , 25121 Brescia, Italy, and
I.N.F.M., Gruppo Collegato di Brescia, Italy, and I.N.F.N., sezione di Pavia , Italy
3Lawrence Livermore National Laboratory, Livermore, CA 94551
4Department of Particle Physics, Weizmann Institute of Sciences, Rehovot 76100, Israel
5MST-10, Los Alamos National Laboratory, MS K764, Los Alamos, NM 87545
6IDS Department, Polytechnic University, Six Metrotech Center, Brooklyn NY 11201
1
0 fect: the quantum dynamics is completely suppressed
0 Weconsidertheprocessofasingle-spinmeasurementusing by the measurement process and the quantum system
2 magneticresonanceforcemicroscopy(MRFM)asanexample remains in the same eigenstate of the measurement de-
of a truly continuous measurement in quantum mechanics.
n vice [2]. On the other hand, quantum dynamics is not
This technique is also important for different applications,
a completelysuppressedbyasequenceof“non-traditional”
J includingameasurementofaqubitstateinquantumcompu-
tation. Themeasurementtakesplacethroughtheinteraction non-destructive measurements [11,12]. These measure-
7
of a single spin with a quasi-classical cantilever, modeled by ments assume a weak interaction between the quantum
1 a quantum oscillator in a coherent state in a quasi-classical system and the classical measuring device.
v region of parameters. Theentiresystem is treated rigorously The other type of “non-traditional” non-destructive
5 within the framework of the Schr¨odinger equation, without measurements are “truly continuous” measurements
3 anyartificialassumptions. Computersimulationsofthespin- [13,14]. A “truly continuous” measurement of a quan-
0 cantileverdynamics,wherethespiniscontinuouslyrotatedby tum system means a continuous monitoring of the dy-
1 means of cyclic adiabatic inversion, show that the cantilever
namics of a macroscopic system caused by the dynamics
0 evolvesintoaSchr¨odinger-catstate: theprobabilitydistribu-
1 ofaquantumsystem. The resultofa“truly continuous”
tionforthecantileverpositiondevelopstwoasymmetricpeaks
0 measurement can be different from the output of a se-
that quasi-periodically appear and vanish. For a many-spin
/ quence of many repeated measurements [13]. Probably,
h systemourequationsreducetotheclassical equationsofmo-
p tion,andweaccurately describeconventionalMRFMexperi- thebestexampleofa“trulycontinuous”measurementis
- mentsinvolvingcyclicadiabatic inversionof thespinsystem. the veryattractiveidea ofa single-spinmeasurementus-
t
n We surmise that the interaction of the cantilever with the ing magnetic resonance force microscopy (MRFM) [15].
a environment would lead to a collapse of the wave function; The essence of the idea is the following. A single spin
u
however,weshow thatin such acasethespin doesnot jump (e.g., a nuclear spin) is placed on the tip of a cantilever.
q
into a spin eigenstate. An frequency-modulated radio-frequency magnetic field
:
v PACS: 03.67.Lx, 03.67.-a,76.60.-k induces a periodic change in the direction of the spin.
i
X Thiscanbeachieved,forexample,byusingawell-known
r method of the fast adiabatic inversion [16], in which the
a I. INTRODUCTION spin follows the effective magnetic field [17]. In the ro-
tating reference frame, this field periodically changes its
Continuous quantum measurement is a very challeng- direction [16]. The period of this motion must be much
ing problem for the modern quantum theory [1–4]. It greater than the period of precession about the effective
is well-known that a traditional measurement puts the magnetic field, but still less than the nuclear spin-lattice
quantum system into contact with its environment lead- relaxation time. If this spin is placed in a strong mag-
ingtocollapseofthecoherentwavefunctionofthequan- netic field gradient the rotation of the spin leads to a
tum system into one of the eigenstates of the measure- periodic force on the cantilever. If this period matches
ment device [4]. This phenomenon is sometimes referred the period of the cantilever the cantilever can be driven
toasaquantumjump. Eventhoughtraditionalmeasure- into oscillationwhoseamplitude increasesto suchextent
ments leadeventually to wavefunction collapse,initially that a single-spin detection may be possible [15].
the system which is being observed may exist in a mys- Theresonantvibrationsofaclassicalcantileverdriven
terious Schr¨odinger-cat state – a superposition of classi- bya continuousoscillationsofa single-spinz-component
callydistinguishedstatesofamacroscopicsystem[5–10]. would seem to violate the traditional expectation that
If a traditional measurement is continuously repeated, coupling of a spin system to a measuring device would
and the time interval between two subsequent measure- cause quantum jumps of the z-components of the spin
ments approaches zero, one faces the quantum Zeno ef- [18]whichwouldpreventsuccessfulspindetectionbythis
1
method. The fundamental questions are the following:
What is the cause of quantum jumps in a “truly contin-
uous” measurement? What specific feature of the quan-
tum dynamics causes the collapse of the wave function
of a quantum system? The answers to these questions
have both a fundamental anda practicalimportance be- z
B
cause a single-spin detection is commonly recognized as 0
a significant application of the MRFM, with particular
importance in the context of quantum information pro-
cessing [15,17–19].
As a necessary step to approach the above problems
we perform a detailed quantum mechanical analysis of
the coupling of a single spin to a cantilever. We rig-
B
orously treat the measurement device (a quasi-classical M 1
cantilever) together with a single spin as an isolated
quantum system described by the Schr¨odinger equation,
without any additional assumptions. In Section 2, we
present the Hamiltonian and the equations of motion S
for a single spin-cantilever system in the Schr¨odinger
representation. The cantilever is prepared initially in
a coherent quantum state using parameters that place
it in a quasi-classical regime. In Section 3, we derive
the equations of motion in the Heisenberg representa-
tion;wedemonstratethatthisdescriptionreducestothe
classical equations of motion for a many spin system;
FIG.1. A schematic setup of the cantilever-spin system.
and we present the results of numerical simulations of B~0istheuniformpermanentmagneticfield;B~1istherotating
the classical spin-cantilever dynamics under the condi- magneticfield;S~isasinglespin(S =1/2);M~ isthemagnetic
tions of the cyclic adiabaticinversionofthe spin system. moment of theferromagnetic particle.
In Section 4, we consider the quantum dynamics of the
spin-cantilever system when the spin is rotated by cyclic The uniform magnetic field, B~0, oriented in the posi-
adiabatic inversion. Our computer simulations explic- tive z-direction,determines the groundstate of the spin.
itlydemonstratetheformationofaSchr¨odinger-catstate The rotating magnetic field, B~1, induces transitions be-
of the cantilever in the process of a “truly continuous” tweenthe groundandthe excitedstatesofthe spin. The
quantum measurement: an asymmetric two-peak prob- originischosentobetheequilibriumpositionofthecan-
ability distribution for the cantilever—a Schr¨odinger-cat tilever tip with no ferromagnetic particle. The rotating
state—is found. This Schr¨odinger cat quasi-periodically magnetic field can be represented as,
appears andvanishes with a periodthat matches that of
the cyclic adiabatic inversion of the spin. We show that Bx =B1cos(ωt+ϕ(t)), By = B1sin(ωt+ϕ(t)), (1)
−
the two peaks of the Schr¨odinger-cat state each involve
where ϕ(t) describes a smooth change in phase required
a superposition of both the stationary spin states. In
for a cyclic adiabatic inversion of the spin (dϕ/dt ω).
Section 5, we summarize our results and discuss the in- ≪
fluenceofthe environmentonthedynamicsofthe“truly In the reference frame rotating with B~1, the Hamilto-
nian of the system is,
continuous”quantummeasurement. Inparticular,wear-
gue that after the collapse of the wave function the spin
P2 m∗ω2Z2 dϕ
does not jump into one of its stationary states. = z + c c ¯h ω ω S (2)
H 2m∗c 2 − L− − dt! z−
II. THE HAMILTONIAN AND THE EQUATIONS
∂B
OF MOTION ¯hω1Sx gµ zZSz.
− ∂Z
We consider the cantilever–spin system shown in Fig. In Eq. (2), Z is the coordinate of the oscillator which
1. A single spin (S = 1/2) is placed on the cantilever describes the dynamics of the quasi-classical cantilever
tip. The tip can oscillate only in the z-direction. The tip;P isitsmomentum,m∗andω aretheeffectivemass
z c c
ferromagnetic particle, whose magnetic moment points and the frequency of the cantilever; S and S are the
z x
in the positive z-direction,produces a non-uniformmag- z- and the x-components of the spin; ω is its Larmor
L
netic field at the spin. frequency; ω1 is the Rabi frequency (the frequency of
2
the spin precession about the field B1 at the resonance eigenfunctions, n , of the unperturbed oscillator Hamil-
condition: ω =ω , ϕ˙ =0); g and µ are the g-factor and tonian, (p2+z2|)/i2,
L z
the magneticmomentofthe spin. The parametersin(2)
∞ ∞
can be expressed in terms of the magnetic field and the
Ψ1(z,τ)= An(τ)n , Ψ2(z,τ)= Bn(τ)n , (11)
cantilever parameters: | i | i
n=0 n=0
X X
m∗c =mc/4, ωc =(kc/m∗c)1/2, ωL =γBz, ω1 =γB1,
(3) n =π1/42n/2(n!)1/2e−z2/2H (z),
n
| i
where m and k are the mass and the force constant
c c
ofthecantilever;Bz includestheuniformmagneticfield, where Hn(z) is the Hermitian polynomial. Substituting
B0,andthemagneticfieldproducedbytheferromagnetic (10) and (11) in (9), we derive the coupled system of
particle; γ =gµ/¯h is the gyromagnetic ratio of the spin. equationsforthecomplexamplitudes,An(τ),andBn(τ),
Next, we introduce the following “quants” of the os-
iA˙ =(n+1/2+ϕ˙/2)A (12)
cillator (cantilever): energy (Ec), force (Fc), amplitude n n−
(Z ), and momentum (P ),
c c
(η/√2)(√nAn−1+√n+1An+1) (ε/2)Bn,
−
E =h¯ω , F = k E , Z = E /k , P =h¯/Z . (4)
c c c c c c c c c c
Using these quanptities and settping ω = ωL, we rewrite iB˙n =(n+1/2+ϕ˙/2)Bn+
the Hamiltonian (2) in the dimensionless form,
H′ =H/¯hωc =(p2z +z2)/2+ϕ˙Sz −εSx−2ηzSz, (5) (η/√2)(√nBn−1+√n+1Bn+1)−(ε/2)An.
To derive Eqs (12), we used the well-known expressions
where,
for creation and annihilation operators,
pz =Pz/Pc, z =Z/Zc, ε=ω1/ωc, ϕ˙ =dϕ/dτ, τ =ωct, an =√nn 1 , a† n =√n+1n+1 , (13)
(6) | i | − i | i | i
η =gµ(∂B /∂Z)/2F . [(p2+z2)/2]n =(n+1/2)n ,
z c z | i | i
To estimate the “quants” in (4) and the dimensionless
z =(a†+a)/√2, p =i(a† a)/√2, [a,a†]=1.
parameters in (5), we use parameters from the MRFM z
−
measurement [17] of protons in ammonium nitrate,
ωc/2π =1.4 103 Hz, kc =10−3 N/m, B1 =1.2 10−3 T, III. THE HEISENBERG REPRESENTATION
× × AND CLASSICAL EQUATIONS OF MOTION
(7)
∂B /∂Z =600 T/m, γ/2π=4.3 107Hz/T. In this section, we will find the relation between the
z
×
Schr¨odinger equation and the classical equations of mo-
Using these values, we obtain, tion for the spin-cantilever system. For this, we present
the operator equations of motion in the Heisenberg rep-
E =9.2 10−31 J, F =3 10−17 N, Z =3 10−14 m,
c c c resentation,
× × ×
(8)
z˙(τ)=p (τ), (14)
z
P =3.5 10−21 kgm/s, ε=37, η =2.8 10−7.
c
× ×
The dimensionless Schr¨odinger equation can be written p˙z(τ)= z(τ)+2ηSz(τ),
−
in the form,
S˙ (τ)= ϕ˙S (τ)+2ηz(τ)S (τ),
iΨ˙ = ′Ψ, (9) x − y y
H
where, S˙ (τ)=ϕ˙S (τ)+εS (τ) 2ηz(τ)S (τ),
y x z x
−
Ψ1(z,τ)
Ψ(z,τ)= , (10)
Ψ2(z,τ) S˙z(τ)= εSy(τ).
(cid:18) (cid:19) −
is a dimensionless spinor, and Ψ˙ =∂Ψ/∂τ. Next, we ex- In Eqs (14), the time-dependent Heisenberg operators
pandthefunctions,Ψ1(z,τ)andΨ2(z,τ),intermsofthe are related to the Schr¨odinger operators in the stan-
dard way, e.g., z(τ) = U†(τ)zU(τ), where U(τ) =
3
Tˆexp( i τ ′(τ′)dτ′),whereTˆ isthe time-orderingop- dPz dBz
− 0 H = kcZ+ z ,
erator. dt − M dZ
R
To deriveclassicalequationswe firstpresentthe equa-
tions of motionforaveragesofthe Heisenbergoperators. d ~
We have from Eqs (14), M =[ ~ B~ ],
e
dt M×
z˙(τ) = p (τ) , (15)
z
h i h i
1dϕ ∂B
z
Bex =B1, Bez = + Z.
p˙ (τ) = z(τ) +2η S (τ) , −γ dt ∂Z
z z
h i −h i h i
To estimate the amplitude of the cantilever vibra-
hS˙x(τ)i=−ϕ˙hSy(τ)i+2ηhz(τ)ihSy(τ)i+2ηR1, tions for the experimental parameters (7), we set Sz =
(1/2)cosτ in (18). Then, the driven oscillations of the
cantilever are given by the expression,
S˙y(τ) =ϕ˙ Sx(τ) +ε Sz(τ) 2η z(τ) Sx(τ) 2ηR2,
h i h i h i− h ih i−
1
z = ∆Nητsinτ. (21)
S˙ (τ) = ε S (τ) , 2
z y
h i − h i
Next, to estimate the amplitude of the stationary vibra-
where R1 and R2 are quantum correlationfunctions,
tions of the cantilever within the Hamiltonian approach,
R1 = zSy z Sy , R2 = zSx z Sx . (16) weputτ =Qc,whereQc isthequalityfactorofthecan-
h i−h ih i h i−h ih i tilever. (The value τ = Q corresponds to time t = t ,
c c
NowconsiderN spinsinteractingwithacantilever. At where t = Q /ω is the the time constant of the can-
c c c
τ =0,someofthesespinsareintheirgroundstates,and tilever.) Taking parameters from the experiment [17],
othersareintheirexcitedstates. We introduceadimen-
sionless “thermal” magnetic moment, khS~ki. Neglect- ∆N =2.9×109, Qc ≈103, (22)
ing quantum correlations under the conditions,
P we obtain for the stationary amplitude of the cantilever,
z S z S z S , (17)
|h x,yi−h i h x,yi|≪|h ih x,yi| z =∆NηQ 8.1 105.
Xk Xk Xk c ≈ ×
we derive the classical equations for the spin-cantilever
system,
1 1 2
z˙ =p , (18) a) b) c)
z
0.5 0.5
p˙z =−z+2η∆NSz, 6z/10 0 6p/10z 0 11E/1001
−0.5 −0.5
S~˙ =[S~ ~b ], −1 −1 0
e 0 500 1000 0 500 1000 0 500 1000
× τ τ τ
wheretheangularbracketsareomitted;~b istheeffective 1 1 1
e d) e) f)
dimensionless magnetic field with components,
0.5 0.5 0.5
bex =ε, bez =−ϕ˙ +2ηz. (19) Sx 0 Sy0 Sz0
The thermal dimensionless magnetic moment, S~ , −0.5 −0.5 −0.5
kh ki
is represented as ∆N S , where ∆N is the difference in −1 −1 −1
h i P 0 500 1000 0 500 1000 0 500 1000
the population of the ground state and the excited state τ τ τ
of the spin system (i.e. the effective number of spins at
FIG. 2. Numerical simulations of the dynamics of a
given temperature) at time τ =0. classical spin-cantilever system (equations (18)); (a) the
Thesecondtermintheexpressionforbez describesthe cantilever coordinate, z; (b) the cantilever momen-
nonlineareffectsinthedynamicsoftheclassicalmagnetic tum, p; (c) the cantilever energy, E0 = (p2z + z2)/2;
moment. In terms of the dimensional quantities, Z, P , (d,e,f) x,y and z-components of the spin, S~. The values of
z
and ~ =γ¯h∆NS~,theclassicalequationsofmotionhave parametersare: ε=37,∆N =2.9×109,η=2.8×10−7. The
the fMorm, initial conditions are: z(0)=pz(0)=6.7×104, Sz(0)=1/2,
Sx,y(0) = 0. The initial conditions, z(0) and pz(0), corre-
dZ Pz spond to the root-mean-square values for z and pz at room
= , (20)
dt m∗ temperature.
c
4
The correspondingdimensionalvalue ofthe amplitude at four points in time, τ. Near τ = 40 the probabil-
is, Z 24nm. The experimental value in [17] is 16nm, ity distribution (26) splits into two peaks; after this the
≈
which is close to the estimated value. To estimate the separationbetweenthesetwopeaksvariesperiodicallyin
importance of nonlinear effects, we should compare the time. For the largest spatial separation, the ratio of the
effective “nonlinear field”, 2η z 0.45, with the trans- peak amplitudes is about 100 (hence we show amplitude
| | ≈
versefield,ε=37. Itfollowsthatforexperimentalcondi- on a logarithmic scale).
tions[17],the nonlineareffectsaresmall: 2η z /ε 0.01.
| | ≈
In our simulations of the classical dynamics we used 100 100
the following time dependence for ϕ˙, τ=336.8 τ=376.4
10−10 10−10
600+30τ, if τ 20, P(z)10−20 10−20
ϕ˙ = − ≤ (23)
100sin(τ 20), if τ >20.
(cid:26) − 10−30 10−30
−35 0 35 −35 0 35
This time-dependence of ϕ˙ produces cyclic adiabatic in- 100 100
version of the spin system [17]. The standard condition τ=416 τ=455.6
ϕf¨or aεfa2s,twahdiicahbiasticcleianrvleyrssiaotnis,fi|dedB~ein/dEt|q≪s (2γ3B)1.2,Fbiegc.o2m(eas-: P(z)1100−−2100 1100−−2100
≪
f) shows the results of our numerical simulations; these
10−30 10−30
show steady growth of the cantilever vibration ampli- −35 z0 35 −35 z0 35
tude, momentum and energy with time, as well as the FIG.3. Theprobabilitydistributionforthecantileverpo-
oscillations of the average spin, S~. sition, P(z) = |Ψ1(z)|2+|Ψ2(z)|2, for four instants of time.
The values of parameters are: the dimensionless Rabi fre-
quency, ε = 40; the dimensionless magnetic force, η = 0.03.
IV. QUANTUM DYNAMICS FOR A SINGLE Thefunction,ϕ˙(τ),istakenintheform(23). Theinitialcon-
SPIN-CANTILEVER SYSTEM ditions: thevalueofαinEq. (24)correspondstotheaverage
values, hz(0)i=−20, hpz(0)i=0.
Themagneticforcebetweenthecantileverandasingle
Fig. 4showstheprobabilitydistribution,P(z),forthe
spinis extremelysmall. To simulatethe dynamicsofthe
sameinitialconditionsasinFig. 3,butwehaveincreased
cantileverdrivenbyasinglespin,onreasonabletimes,we
take η 3 10−2. Such a value canbe already achieved η and ε by a factor of 10: η = 0.3 ε = 400; the cyclic
≈ × adiabatic inversionparameters are:
inthepresentdayexperiments[17]bymeasuringasingle
electronspin. To describe the cantilever as a sub-system
6000+300τ, if τ 20
close to the classical limit, we choose the initial wave ϕ˙ = − ≤ (27)
1000sin(τ 20), if τ >20
functionofthecantileverinthecoherentstate, α ,inthe (cid:26) −
quasi-classical region of parameters (α2 1)|. Niamely,
| | ≫
the initial wave function of the cantilever was taken in
the form (10), where: 100 100 100
τ=0 τ=20 τ=104
) 10−10 10−10 10−10
z
∞ (
P 10−20 10−20 10−20
Ψ1(z,0)= An(0)n , Ψ2(z,0)=0, (24)
| i 10−30 10−30 10−30
n=0 −50 0 50 −50 0 50 −50 0 50
X
100 100 100
τ=126.4 τ=160 τ=182.4
A (0)=(αn/√n!)exp( α2/2). ) 10−10 10−10 10−10
n −| | (z
P 10−20 10−20 10−20
The initial averages of z and p can be represented as,
z 10−30 10−30 10−30
−50 0 50 −50 0 50 −50 0 50
z = 1 (α∗+α), p = i (α∗ α). (25) 100 100 100
h i √2 h zi √2 − ) 10−10 τ=199.2 10−10 τ=210.4 10−10 τ=221.6
z
(
The numerical simulations of the quantum dynamics P 10−20 10−20 10−20
using Eqs (12) reveal the formation of the asymmet- 10−30 10−30 10−30
−50 z0 50 −50 z0 50 −50 z0 50
ricquasi-periodicSchr¨odinger-catstate ofthe cantilever.
(The dimensionless period, ∆τ =2π, corresponds to the FIG. 4. Probability distribution of the cantilever coordi-
dimensionalperiod,∆t=2π/ω .) Fig. 3(a-c)showsthe nate, z, for ε = 400 and η = 0.3. The initial conditions are
c
probability distribution, thesame as in Fig. 3.
P(z,τ)= Ψ1(z,τ)2+ Ψ2(z,τ)2, (26) As shown in Fig. 4, the two peaks of the Schr¨odinger-
| | | |
5
catstatearemoreclearlyseparated. Whentheprobabil- One might expect that the two peaks are associated
itydistributionsplitsintotwopeaks,thedistance,d,be- with the functions P (z,τ) = Ψ (z,τ)2, n = 1,2. In
n n
| |
tween them initially increases. Then, d decreases. Then, fact the situation is more subtle: each function, P (z,τ)
n
the two peaks overlapand the Schr¨odinger-catstate dis- splits into two peaks. Fig. 5, shows these two functions
appears. After this, the probability distribution splits fornineinstantsintime: τ =92.08+0.8k,k =0,1,...,8.
k
again so that the position of the minor peak is on the One can see the splitting of both P1(z,τ) and P2(z,τ);
opposite side of the major peak. Again, the distance, d, each peak of the function P1(z,τ) has the same position
first increases, then decreases until the Schr¨odinger-cat as the two peaks of P2(z,τ), but the amplitudes of these
state disappears. This cycle repeats for as long as the peaks differ. For instance for k = 1 (τ = 92.88) the
simulations are run. left-handpeakisdominantlycomposedofP1(z,τ),while
right hand peak is mainly composed of P2(z,τ).
100
a) b) c)
50
10−2
10−4
25
10−1 d) e) f)
)
τ
P(z,10−3 (τ)> 0
10−5 <z
g) h) i)
10−3 −25
10−5
−30 −10 10 −30 −10 10 30 −10 10 30
z −50
0 100 200 300
τ
FIG. 5. Probability distributions, P1(z,τ) = |Ψ1(z,τ)|2
(slid curves), and P2(z,τ) = |Ψ2(z,τ)|2 (dashed curves) for FIG.7. The dependence hz(τ)i for the same values of pa-
nineinstants of time: τk =92.08+0.8k, k=0,1,...,8. rameters and initial conditions as in Figs 4-6.
With increasing time the relative contribution of each
spin state to the two peaks varies. Fig. 6 shows the rel-
ative contribution of each spin state to the spatially in-
1
tegratedprobabilitydistributions: P11(τ)= P1(z,τ)dz
and P22(τ) = P2(z,τ)dz, as “truly continuous” func-
R
tions of time, τ. (Vertical arrows show the time in-
R
stants, τ .) It is important to note that the formation
k
P (τ) of the Schr¨odinger-cat state does not suppress, in the
11 frameworks of the Scro¨dinger equation, the increase of
P22(τ) the average amplitude, z(τ) , of the cantilever oscilla-
0.5 h i
tions. Fig. 7, demonstrates the dependence z(τ) for
h i
the same values of parameters and initial conditions as
in Figs 4-6. We explored the affect of varying the initial
state of the cantilever, including the effect of an initial
state which is a superposition for the spin system, and
found no significant affect on the Schr¨odinger-cat state
dynamics. The value α cannot be significantly reduced
0 | |
92 93 94 95 τ 96 97 98 99 if we are going to simulate a quasi-classical cantilever,
and increasing α increases number of states n involved
| i
making simulation of the quantum dynamics more diffi-
FIG. 6. Integrated probability distributions of the spin
cult. Ourcurrentbasisincludes2000stateswhichallows
z-components (diagonal components of the spin density ma-
accurate simulations.
trix): P11(τ),forSz =1/2(•);andP22(τ),forSz =−1/2(◦),
as functions of time. Vertical arrows show the time instants,
τ =92.08+0.8k, k=0,1,...,8 depicted in Fig. 5.
k
6
V. SUMMARY thelife-timeoftheSchr¨odinger-catstatetakingintocon-
sideration the interaction with the environment. Then,
We considerthe problemof a “truly continuous”mea- one can choose a specific sequence of the life-times, τdk,
surementinquantumphysicsusingtheexampleofsingle- of the order τd and a specific sequence of the wave func-
spin detection with a MRFM. Our investigation of the tion collapses into the major and the minor peaks. (The
dynamicsofa purequantumspin1/2systeminteracting probability of a collapse in any peak is proportional to
with a quasi-classicalcantilever has revealed the genera- the integrated area of a peak.) After this, one can con-
tionofaquasi-periodicasymmetricSchr¨odinger-catstate sider quantum dynamics using Eqs (12) which generates
for the measurement device. theSchr¨odinger-catstateandinterruptedbythecollapse
In our example of the MRFM we considered the dy- intooneofthepeaks. Thisratherphenomenologicalcom-
namics of a spin-cantilever system within the frame- putational program repeated many times with different
work of the Schr¨odinger equation without any artificial sequencesofjumpscouldprovideanadequatedescription
assumptions. For a large number of spins, this equa- of a possible experimental realization of a “truly contin-
tion describes the classical spin-cantilever dynamics, in uous” quantum measurement which takes into account
good agreement with previous experimental results [17]. the interaction with the environment.
For single-spin detection, the Schr¨odinger equation de- A movie demonstrating a Schr¨odinger-cat state dy-
scribes the formation of a Schr¨odinger-cat state for the namics can be found on the WEB:
quasi-classical cantilever. In a realistic situation, the in- www.dmf.bs.unicatt.it/˜borgonov/4cats.gif
teraction with the environment may quickly destroy a
Schr¨odinger-catstateofamacroscopicoscillator(see,for
ACKNOWLEDGMENTS
example,Refs.[20,21]). Inourcasewewouldexpectina
similarwaythatinteractionwiththe environmentwould
cause the wave function for the coupled spin-cantilever We thank G.D. Doolen for discussions. This workwas
systemtocollapse,leadingtoaseriesofquantumjumps. supported by the Department of Energy under contract
Finally, we discuss briefly the possible influence of W-7405-ENG-36. The work of GPB and VIT was sup-
the wave function collapse on the quantum dynamics of ported by the National Security Agency.
the spin-cantilever system. This collapse causes a sud-
den transformation of a two-peak probability distribu-
tion into a one peak probability distribution. It should
be notedthatthe disappearanceofthe secondpeak does
not produce a definite value of the z-component of the
spinbecausebothP1(z,t)andP2(z,t)contributetoboth [1] A. Peres, Continuous Monitoring of Quantum Systems,
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|↑i |↓i
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7
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