Quantum Characterization of a Werner-like Mixture

8 Pages·0.36 MB·

by  Anna Felicetti

Quantum Characterization of a Werner-like Mixture
Anna Felicetti, Stefano Mancini and Paolo Tombesi
INFM, Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy
(February 1, 2008)
WeintroduceaWerner-likemixture[R.F.Werner,Phys. Rev. A40,4277(1989)]byconsidering
two correlated but different degrees of freedom, one with discrete variables and the other with
continuous variables. We evaluate the mixedness of this state, and its degree of entanglement
establishing its usefulness for quantum information processing like quantum teleportation. Then,
weprovideitstomographiccharacterization. Finally,weshowhowsuchamixturecanbegenerated
and measured in a trapped system like one electron in a Penningtrap.
Pacs No: 03.65.Wj, 03.65.Ud, 03.65.Ta
2
0
0 I. INTRODUCTION ous variable. Then we characterize such a state in terms
2
of mixedness and entanglement. Section III is devoted

n to the tomographic method employed for such a state
a Itisnowadayswellknownthatthenonlocalproperties reconstruction. In Section IV we present the results of
J
of Quantum Mechanics [1,2] enable striking processes in numerical simulations. Finally, in Section V we discuss
7 quantum information [3]. In these processes it is promi- a possible implementation and Section VI is devoted to
1
nent the role of maximally entangled states [4,5]. How- conclusions.

ever, very often, the decoherence effects due to the envi-
1
ronment transform the pure entangled state into a sta-
v
1 tistical mixture and degrade quantum entanglement in II. WERNER-LIKE MIXTURE
7 the real world [6]. Although purification schemes may
0 be applied to noisy channels [7], there exist some mix- In his pioneering paper, Bell proved that a local real-
1 ture states which maintain interesting properties. An istic interpretation of Quantum Mechanics is impossible
0
illuminating example is providedby the Werner mixture [2],andforthecaseofpurestatesitisknownthat,when
2
0 [8],whichis nota mixture ofproductstates,nonetheless measurements are performed on two quantum systems
/ not violating any Bell’s inequality [2], but still useful for separatedin space, their results are correlatedin a man-
h quantum information processing [9,10]. Such states be- ner which, in general, cannot be explained by a local
p
longto systemswith twodiscretedegreesoffreedomlike hidden variables model [20]. Since the only pure states
-
nt two spin-21. However, information processing may some- satisfying the Bell inequality are pure product states,
a timesinvolvehybridsystemswhereonedegreeoffreedom one might naively think that the only mixed states that
u hasdiscretevariablesandtheothercontinuousvariables. do not violate Bell’s inequality are mixtures of product
q It may happen, for instance, in trapped ions [11,12], or states. However, Werner [8] showed that this conjecture
v: in trapped electrons [13] or in cavity quantum electro- is false for the so-called Werner states
dynamics [14,15]. Thus, it will be the aim of this paper
i 1 1
X to consider a mixture, which resembles the Werner one, ρ= I I + Ψ Ψ , (1)
1 2 − −
8 ⊗ 2| ih |
r butoneofthetwosubsystemsisdescribedbycontinuous
a
variables. where I (i = 1,2) stands for the identity operator of
i
Onthe other hand, states and processesusedin quan- a single qubit ( ii + ii ) and Ψ− =
tuminformationtypicallyneedofawellcharacterization 1 ( | ↓i h↓) i|s the| s↑piinhs↑ing|let stat|e. i
√2 |↓i1|↑i2−|↑i1|↓i2
[3]. Thiscanbeaccomplishedbyusingtomographictech- A more general Werner mixture can be obtained by
niques[16]. Concerningthequantumstatemeasurement, consideringoneofthetwosubsystems,say2,asdescribed
after the seminal work by Vogel and Risken [16], a lot of by continuous variables. A way to encode qubit in con-
progress has been obtained and further techniques and tinuousvariablesystemscouldbetheuseofevenandodd
algorithms were developed [17]. We would just mention catstateswhichareorthogonal[21],thusresultinginthe
the possibility of state reconstruction, for a composite same situation of Eq.(1). Instead, the choice we are go-
system of discrete and continuous variables, by simply ing to make is more general and gives the possibility to
measuring the set of rotated spin projections and dis- explore a variety of situations.
placednumberoperators,[18,19]. Then,weshallprovide That is, we now replace the states , of the
2 2
| ↑i | ↓i
the tomographic characterization of a Werner-like mix- second qubit with α and α , where the latter are
2 2
| i |− i
ture by generalizing that method. coherent states of amplitude α and α respectively (we
The outline ofthe paperis the following: InSectionII shall consider α R throughout the−paper for the sake
∈
wediscusstheWernermixtureandweextendtheconcept of simplicity). Therefore, a Werner-like mixture would
by considering one of the two subsystems with continu- be
1
1
We can see that also for α= 0 the state is a mixture,
ρ= α α
11 22
8{|↓i h↓|⊗|− i h− | then, itbecomesmoreandmoremixedby increasingthe
+ 11 α 22 α value of α, but never reaching the maximum (a com-
|↑i h↑|⊗|− i h− |
+ α α pletely mixed density operator in a d-dimensional space
11 22
|↓i h↓|⊗| i h | hasentropylog d). Itisalsoworthnotingthatforα(κ)
+ α α 2
11 22
|↑i h↑|⊗| i h |} arbitrary, Eq.(3) is not a mixture of Bell’s states as it is
1
+ [ α α ][ α α] . for the Werner mixture (1), i.e., for α 1.
4 |↓i1| i2−|↑i1|− i2 1h↓|2h |−1h↑|2h− | ≫
Letus nowconsiderthe measureofentanglement. For
(2)
a two spin-1 system the state separability is related to
2
thepartialtranspositionoperation[4,24]. Thematrixel-
Since α α =exp( 2α2)=0, the above state doesn’t ements of partial transpositionρT2 of a state ρ are given
h |− i − 6
describe a real two qubit system, but rather a two qubit by ρT2 ρ where ρ = m µρν n
system with nonorthogonal states [22]. Of course, for withmmµ,,nnν,µ≡,ν =mν,,nµ. Adensitmyµm,naνtrixρ1hfor|2ahtw| o| sip2i|n-i11
α 1 Eq.(2) behaves like the state (1), but we want ↑ ↓ 2
≫ system is inseparable if and only if its partial transpose,
to study its characteristics for a generic value of α. To ρT2, has any negative eigenvalue [4,24]. Then, a suitable
this end, we map the above state in the two spin-1
2 measure of entanglement can be defined as [25]
Hilbert spaces by introducing for the subsystem 2 a vec-
tor ψ =κ +√1 κ2 and considering the non
2 2 2
orth|ogionals|ta↓ties|↓i2,−|ψi2|i↑nisteadof|−αi2,|αi2,with E(ρ)=−2 λ−i , (5)
ψ = κ exp( 2α2). Then, Eq.(2) can be rewritten Xi
h | ↓i ≡ −
as
whereλ−i isanegativeeigenvalueofρT2. Itisworthnot-
1 ing that this measure satisfies the necessary conditions
ρ=
11 22
8{|↓i h↓|⊗|↓i h↓| required for every measure of entanglement [5]. Then,
+ we have plotted in Fig.2 the quantity E calculated by
11 22
|↑i h↑|⊗|↓i h↓|
+ ψ ψ exploiting again the matrix representation of Eq.(3) in
11 22
|↓i h↓|⊗| i h | the basis .
+ 11 ψ 22 ψ B
|↑i h↑|⊗| i h |}
1
+ [ 1 ψ 2 1 2][1 2 ψ 1 2 ] . E 1
4 |↓i | i −|↑i |↓i h↓| h |− h↑| h↓|
(3)
0.8
Thedegreeofmixednessofthestate(3)canbeevaluated
0.6
by using the Von Neumann entropy [23]
0.4
S(ρ)= Tr(ρlog ρ)= λ log λ , (4)
− 2 − i 2 i
Xi 0.2
whereλ aretheeigenvaluesofthematrixrepresentation
i
of ρ. We have calculated such eigenvalues in the basis 0.2 0.4 0.6 0.8 1 1.2 1.4
, , , and we a
1 2 1 2 1 2 1 2
B ≡ {| ↓i | ↓i | ↓i | ↑i | ↑i | ↓i | ↑i | ↑i }
have plotted the entropy S in Fig.1.
FIG. 2. Degree of entanglement E as function of
α≡ −(1/2)lnκ.
S
2
p
1.75 WecanseethattheWerner-likemixtureisfactorisable
1.5 for α = 0, then, by increasing α, its degree of entangle-
ment increases saturating at the value 1/4 characteris-
1.25
tic of the true Werner mixture (1). This latter value of
1
entanglement is known to be sufficient to improve the
0.75 teleportation performances over classical limit once the
0.5 sender (Alice) and the receiver (Bob) initially share the
state (1) [9]. Thus, it is straightforward to ask to what
0.25
extent a Werner-like mixture can be used for the same
0.2 0.4 0.6 0.8 1 1.2 1.4 goal. Toestablishathresholdvalueforαwearegoingto
a consider the teleportation fidelity. To this end, we first
write the state ρ through the Hilbert-Schmidt decompo-
FIG.1. EntropyS as function of α≡ −(1/2)lnκ. sition
p
2
1
ρ= I I +rσ(1) I +I sσ(2) which lead to rotated (by angles θ and ϕ) spin projec-
1 2 2 1
4 ⊗ · ⊗ ⊗ · tion in the subsystem 1 [28], and to displaced number
h 3 state (by a complex amount β) in the subsystem 2 [29].
+ tn,mσn(1)⊗σm(2)# , (6) Then, it is possible to consider the following measurable
n,m=1 marginal distributions
X
where σn n=1,2,3 are the standard Pauli matrices, r, s w(s,n;θ,ϕ,β)=Tr ρD(β)U(θ,ϕ)
acoreeffivecci{etonrtss}itnn,Rm3=anTdr[rρ·σσn(1=)⊗Pσm3i(=2)1]rfoiσrmi. Fthuertrheearlmmoarter,ixthTe |si{11hs|⊗|ni22hn|U†(θ,ϕ)D†(β)(cid:9)(1,0)
describingthecorrelationsbetweenthetwoqubits. Thus,
the teleportation capabilities will depend on the specific having as variables the eigenvalues s, n of σ and a a
3 †
form of T. In particular it is shown in Ref. [26] that the as variables and parametrically depending on θ, ϕ and
teleportation fidelity amounts to β. Thus, measuring the state ρ would mean the possi-
bility to express ρ as a functional operator of w, i.e. to
F = 1 1+ 1Tr√T T . (7) invertexpression(10). Thisalsomeansthe possibilityto
2 3 † sample the density matrix elements (in some basis) from
(cid:20) (cid:21)
the quantity measured by spanning the whole space of
Then, in Fig.3 we have plotted the quantity F versus α parameters.
compared with the classical fidelity 2/3. In reality, we shall see that it is not necessary to con-
siderallpossiblevaluesofparameters. Asmatteroffact,
F we write the density operator (2) as
0.75
ρ ρ
0.7 ρ= ↑↑ ↑↓ , ρ =[ρ ] , (11)
ρ ρ ↓↑ ↑↓ †
↓↑ ↓↓
(cid:20) (cid:21)
0.65
where each operator ρ , ρ , ρ , ρ , can in turn be
↑↑ ↑↓ ↓↑ ↓↓
represented in the Fock basis of the subsystem 2.
0.6
Now, we set θ = ϕ = 0 and we suppose to retain
only the measurement results s= ; then, expanding the
0.55
↑
density operator ρ in the Fock basis, and defining N
↑↑ c
as an appropriate estimate of the maximum number of
0.2 0.4 0.6 0.8 1
excitations (cut-off), we have
a
FIG. 3. Teleportation fidelity as Nc
w( ,n;0,0,β)= n,β k k ρ m mn,β . (12)
function of α ≡ −(1/2)lnκ. The dashed line represents ↑↑
↑ h | ih | | ih | i
theclassical boundof 2/3. k,m=0
X
p
The projectionof the displaced number state n,β onto
We immediately recognize the presence of a threshold | i
theFockstate m canbeobtainedgeneralizingtheresult
value (α=0.2476)below which the Werner-like mixture | i
derived in Ref. [30].
becomes useless for quantum teleportation.
Let us now consider, for a given value of β , w as a
| |
function of φ = arg[β] [31] and calculate the coefficients
of the Fourier expansion
III. STATE MEASUREMENT
1 2π
w( ,n;0,0,r)= dφw( ,n;0,0,φ)eirφ , (13)
Wenowdiscussthepossibilityofacompletecharacter- ↑ 2π ↑
Z0
ization of the Werner-like mixture through tomographic
techniques. In particular, we generalize the method pre- for r =0,1,2,.... Combining Eqs. (12) and (13), we get
sented in Refs. [18,19] to non pure states.
Accordinglytothestatereconstructionprincipledevel- Nc−r
w( ,n;0,0,r)= G(r) (β ) m+rρ m , (14)
oped inRef. [27]we choose anobservable,here σ3 a†a, ↑ n,m | | h | ↑↑| i
then we apply suitable unitary transformations t⊗o get mX=0
a set of observables giving the whole state information where the explicit expression of the matrices G is given
upon measurements. In our case the transformations in Ref. [18].
would be We may now notice that if the distribution w(
↑
,n;0,0,β) is measured for n [0,N] with N N , then
c
U(θ,ϕ)=exp[ iθ(σ cosϕ+σ sinϕ)] , (8) ∈ ≥
1 2 Eq.(14)represents,foreachvalueofr,asystemofN+1
−
D(β)=exp βa β a , (9) linear equations between the N +1 measured quantities
† ∗
−
(cid:2) (cid:3)
3
and the N +1 r unknown density matrix elements. (a) (b)
c
−
Therefore, in order to obtain the latter, we only need to
invert the system
0.3 0.3
0.2 0.2
N 0.1 0.1
m+rρ m = M(r) (β )w( ,n;0,0,r), (15) 0 0
h | ↑↑| i m,n | | ↑ -0.1 m -0.1 m
n=0
X
n n
where the matrices M are given by M =(GTG) 1GT.
−
The procedure can be repeated with w( ,n;0,0,β) in (c) (d)
↓
order to get the matrix elements of ρ . Then, chang-
↓↓
ing the parameters so that θ = π/4 and ϕ = π/2 we 0.001 0.1
can get the real part of the matrix elements of−ρ . In- 0.0008 0.08
↑↓
0.0006 0.06
stead, with θ =π/4 and ϕ=0 we can get the imaginary
0.0004 0.04
part of the matrix elements of ρ↑↓, thus concluding the 0.0002 0.02
reconstruction procedure of the state (11). 0 m 0 m
n n
FIG. 4. Density matrix elements of ρ↑↑ for the state (2)
with α = 0.7. In the computer simulation N = 96 phases
φ
IV. NUMERICAL RESULTS are considered and N = 104 measurement events at each
φ
phase are assumed to be recorded using Nc =31. The other
parameters are |β| = 0.6 and η = 0.9. (a) Exact density
matrix; (b) reconstructed density matrix; (c) statistical er-
As an example of the proposed method, we show in rors;(d)absolutedifferencebetweenreconstructedandexact
Figs. 4, 5, 6 the results of numerical Monte-Carlo simu- density matrix elements.
lationsofthereconstructionofthestate(2)oncewritten
in the form (11). In this simulation we have used the
value α=0.7 which make the state different from a true (a) (b)
Werner mixture, but still having nonclassical features as
discussed in Section II.
0.3 0.3
In order to account for experimental conditions, we
0.2 0.2
have also considered the effects of a non-unit quantum
efficiency η in the counting of the number of excitations. 0.1 0.1
Whenη <1,theactuallymeasureddistributionisrelated 0 m 0 m
to the ideal distribution by a binomial convolution [32].
n n
Statistical errors are accounted for as well by consid-
ering an estimation of the marginal distributions given
by west(n;φ) = (n)/ , where (n) is the number
Nφ Nφ Nφ (c) (d)
of events with n counts at phase φ, while is the total
φ
N
numberofeventsatthe samephase. Then,followingthe
arguments given in Refs. [31,33], the quantities (n) 0.001 0.2
Nφ 0.0008
can approximately be regarded as independent Poisso- 0.15
0.0006
nian random variables, whose means and variances are 0.0004 0.1
given by w(n;φ) west(n;φ) . The variance of 0.0002 0.05
west(n;φ) may thNenφ b≈e approximaNteφd by west(n;φ)/ , 0 m 0 m
φ
so that the variances of the real and imaginary partNs of n n
the density matrix canbe easily estimatedusing Eq.(15) FIG. 5. Density matrix elements of ρ↓↓ for the state (2)
(and similarly for the other density operators). This with α = 0.7. In the computer simulation N = 96 phases
φ
4
meansthattheerrorscanbeestimatedinrealtimeasthe are considered and Nev = 10 measurement events at each
experimentruns, simultaneouslywith the reconstruction phase are assumed to be recorded using Nc =31. The other
of the density matrix elements. parameters are |β| = 0.6 and η = 0.9. (a) Exact density
matrix; (b) reconstructed density matrix; (c) statistical er-
Other error sources leading to discrepancies between
rors;(d)absolutedifferencebetweenreconstructedandexact
trueandreconstructeddensitymatricescanbeidentified
density matrix elements.
inthechoiceof β ,N andN (numberofphases). How-
c φ
| |
ever,asitcanbeseenfromFigs.4,5,6thereconstructed
density matrices turn out to be quite faithful.
4
(a) (b) W W
0.1 0.1 0.1 0.1
0.05 2 0.05 2
0 0 0 0Im g 0 0 Im g
-2 -2
-0.1 -0.1 Re0 g 2 -2 Re0 g 2 -2
m m
n n
W W
(c) (d)
0.1 0.1
0.05 2 0.05 2
00..0000018 00..125 0 -2 0 -2 0 Im g 0 -2 0 -2 0 Im g
0.0006 0.1 Re g 2 Re g 2
0.0004
FIG. 7. Top: Wigner function corresponding to the den-
0.0002 0.05
0 0 sitymatrixofFig.4(trueontheleftandreconstructedonthe
m m
right). Bottom: Wigner function corresponding to the den-
n n
sitymatrixofFig.5(trueontheleftandreconstructedonthe
FIG. 6. Density matrix elements of ρ↑↓ for the state (2) right).
with α = 0.7. In the computer simulation N = 96 phases
φ
4
are considered and Nev = 10 measurement events at each
phase are assumed to be recorded using Nc =31. The other
parameters are |β| = 0.6 and η = 0.9. (a) Exact density Re W Re W
matrix; (b) reconstructed density matrix; (c) statistical er-
rors; (d)absolutedifferencebetweenreconstructedandexact 0 0
density matrix elements. -0.04 2 -0.05 2
-0.08 0 Im g 0Im g
-2 -2
Re0 g 2 -2 Re0 g 2 -2
Inadditiontothepresentcasewehaveperformedother
simulationswithdifferentvaluesofαandseveralvaluesof
the parameters, which confirm that the present method
Im W Im W
is quite stable and accurate.
0.05 0.05
The phase-space description corresponding to (11) is 0 2 0 2
given by the Wigner-function matrix [34] -0.05 0 Im g -0.05 0Im g
-2 -2
Re0 g 2 -2 Re0 g 2 -2
FIG.8. Wigner function corresponding to thedensity ma-
trixofFig.6(trueontheleftandreconstructedontheright).
W (γ) W (γ)
W = ↑↑ ↑↓ ThereconstructedWignerfunctionsaswellturnoutto
W (γ) W (γ)
(cid:20) ↓↑ ↓↓ (cid:21) be quite faithful. We would like to emphasize their par-
= Tr2[ρ↑↑δ(γ−aˆ)] Tr2[ρ↑↓δ(γ−aˆ)] , (16) ticular shape. In W↑↑ there are two hills, one centered
Tr2[ρ↓↑δ(γ aˆ)] Tr2[ρ↓↓δ(γ aˆ)] in Reγ = α and the other in Reγ = α coming from
(cid:20) − − (cid:21) −
therandompartofthestate(2),instead,thepseudosin-
gletpartofthestatecontributesonlytothebumpin α,
−
thusproducingtheasymmetriceffect. Theoppositehap-
pens for W . The shape of W is due to the quantum
↓↓
interference given by the entan↑g↓lement between the two
degrees of freedom: in fact, in absence of entanglement
where δ(γ aˆ) is the Fourier transform of the displace-
− ρ would just be a replica of the diagonalpartsρ and
ment operator [30]. ↑↓ ↑↑
ρ .
↓↓
Then, the Wigner functions correspondingto the den-
sity matrices of Figs.4, 5 and 6, are shown in Figs.7 and
8.
5
V. PHYSICAL REALIZATION and α , starting from the typical initial state
1 2
| ↓i |± i
0 .
1 2
|↑i | i
WenowbrieflydiscussasystemwheretheWerner-like Instead,forwhatconcernsthegenerationoftheentan-
mixture could be synthesized and even measured. It is a gled fraction of the mixture (2), we recall the procedure
singleelectrontrappedinaPenningtrap[35],wheretwo developed in Ref. [18]. That is, we consider the possi-
different degrees of freedom of the same particle can be bility of introducing pulsed standing waves through the
entangled and also measured. microwaveinlet [35] so that ǫ, ζ become time dependent
AnelectroninaPenningtrapisconfinedbythecombi- andǫ˜,ζ˜indicatethepulsearea(thedurationofthepulse
nationofahomogeneousmagneticfieldalongthepositive isassumedtobemuchshorterthanthecharacteristicax-
z axisandanelectrostaticquadrupolepotentialinthexy ial period, which is of the order of microseconds). Then,
plane [35]. The spatial part of the electronic wave func- nonclassical cyclotron states can be entangled with the
tion consists of three degrees of freedom, neglecting the spin states through the following steps [18].
slow magnetron motion (whose characteristic frequency
lies in the kHz region), here we only consider the axial First, we consider χ = 0, Ω = ωs, and a pulsed
•
and cyclotron motions, which are two harmonic oscilla- standing wave lasting ∆t1 =t1 t0 =t1;
−
torsradiatinginthe MHz andGHz regions,respectively.
Second, we allow a free evolution for a time ∆t =
On the other hand, the spin dynamics results from the • 2
t t =π/(2ω );
interactionbetweenthemagneticmomentoftheelectron 2− 1 z
and the static magnetic field, so that the free Hamilto- Third, we consider the action of another pulsed
nian reads as [35] • standing wave with χ = π/2, Ω = ω , for a time
c
−
∆t =t t .
3 3 2
H =h¯ω a a +h¯ω a a +h¯ω σ /2, (17) −
free z †z z c †c c s 3
Finally,ifweconsidertheinitialaxialstateasaGaus-
where the indices z, c, and s refer to the axial, cyclotron sian state with momentum width much smaller than ζ˜
and spin motions, respectively.
(which is easily obtained in the case of the ground state
Here, in addition to the usual trapping fields, we con-
oftheaxialoscillator),weendupwithanevolutionoper-
sideranexternalradiationfieldasastandingwavealong
the z direction and rotating, i.e. circularly polarized, in atoroftheformD(ασ1),whereαisrelatedtoǫ˜,ζ˜andϕ.
It is then immediate to see that the initial state 0
the xy plane with frequency Ω [36]. To be more specific, |↑i1| i2
may evolve with the aid of D(ασ ) and a spin rotation
weconsiderastandingwavewithinthecylindricalcavity 1
into
configuration [37] with the (dimensionless) wave vector
k. Then, the interaction Hamiltonian reads [36] 1
( α α ) . (19)
1 2 1 2
√2 |↓i | i −|↑i |− i
Hint =h¯ǫ acei(Ωt+ϕ)+a†ce−i(Ωt+ϕ) cos(kz+χ)
ThisstatehasbeenalreadydiscussedinRefs.[12,15]and
h i
+h¯ζ σ ei(Ωt+ϕ)+σ+e−i(Ωt+ϕ) sin(kz+χ), (18) constitutes the pseudo singlet component of the mixture
−
(2).
h i
where σ =(σ iσ )/2, and z =a +a . The phase χ Thus, at each run of the experiment the desired com-
definest±heposi1ti±ono2fthecenterofthzeax†zialmotionwith ponent of the Werner-like mixture can be synthesized,
respect to the wave. Depending onits value the electron thus allowing the generation of the state (2) on average
can be positioned in any place between a node (χ = 0) ensemble.
and an antinode (χ = π/2). Instead, the phase ϕ is For what concerns the measurement, the addition of a
±
related to the initial direction of the electric (magnetic) particular inhomogeneous magnetic field (known as the
fieldinthexyplane,ortothephaseofanyreferencefield. magnetic bottle field [35]) to that already present in the
The two coupling constants ǫ and ζ are proportional to trap, allows to perform a simultaneous measurement of
the amplitude of the applied radiation field. Depending boththespinandthecyclotronexcitationsnumber. The
on Ω and χ, the interaction Hamiltonian (18) gives rise usefulinteractionHamiltonianforthe measurementpro-
to different contributions at leading order in the Taylor cess is
expansion of sin(kz+χ) and cos(kz+χ). g
We immediately recognize the possibility of imple- Hbottle =h¯ωb a†cac+ 2σ3 z2, (20)
menting the transformations (8), (9) on the spin and h i
cyclotrondegreesof freedom by appropriatelyexploiting where the angular frequency ω is directly related to the
b
the Hamiltonian (18). For instance, U can be realized strength of the magnetic bottle field.
by setting Ω = ω , χ = 0, and then adjusting ϕ and Equation(20)describesthefactthattheaxialangular
s
θ =ζt. Differently, D can be realized by setting Ω=ω , frequencyisaffectedbothbythenumberofcyclotronex-
c
χ= π/2, and then adjusting β =ǫte iϕ. citations a a andby the eigenvalueof σ . The modified
− − †c c 3
Thesetransformationseasilyallowtogeneratethedis- (shifted)axialfrequencycanbeexperimentallymeasured
entangledcomponentsofthemixture(2),i.e., α [35]aftertheapplicationoftheinhomogeneousmagnetic
1 2
|↑i |± i
6
bottle field. One immediately sees that it assumes dif- [10] C. Bennett, H. J. Bernstein, S. Popescu and B. Schu-
ferent values for every pair of eigenvalues of a a and σ macher, Phys. Rev. A 53, 2046 (1996); C. H. Bennett,
†c c 3
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