Measure of phonon-number moments and motional quadratures through infinitesimal-time probing of trapped ions

0.18 MB·English

by  T. Bastin

Measure of phonon-number moments and motional
quadratures through infinitesimal-time probing of
trapped ions
6
0
T Bastin1, J von Zanthier2 and E Solano3
0
2 1InstitutdePhysiqueNucl´eaire,AtomiqueetdeSpectroscopie, Universit´ede
Li`egeauSartTilman,Baˆt.B15,B-4000Li`ege,Belgium
n 2Institutfu¨rOptik,InformationundPhotonik, Universit¨atErlangen-Nu¨rnberg,
a Staudtstrasse7/B2,D-91058Erlangen,Germany
J 3Max-Planck-Institutfu¨rQuantenoptik, Hans-Kopfermann-Strasse1,D-85748

0 Garching,Germany
1 Seccio´nF´isica,DepartamentodeCiencias,PontificiaUniversidadCato´licadel
Peru´,Apartado1761,Lima,Peru

2 E-mail: [email protected], [email protected],
v [email protected]
8
9
0 Abstract. A method for gaining information about the phonon-number
moments and the generalized nonlinear and linear quadratures in the motion
6
oftrappedions(inparticular,positionandmomentum) isproposed,validinside
0
andoutsidetheLamb-Dickeregime. Itisbasedonthemeasurementoffirsttime
5
derivatives of electronic populations, evaluated at the motion-probe interaction
0
time τ = 0. In contrast to other state-reconstruction proposals, based on
/
h measuring Rabi oscillations or dispersive interactions, the present scheme can
p be performed resonantly at infinitesimal short motion-probe interaction times,
- remainingthus insensitivetodecoherence processes.
t
n
a
u
PACSnumbers: 03.65.Wj,32.80.Lg,42.50.Ct
q
Keywords: trapped ions; quantum measurement
:
v
i
X 1. Introduction
r
a A single ion in a radio-frequency trap is an ideal system for the investigation of
the basic and intriguing features of quantum mechanics [1]. Recent advances in
the manipulation of the internal and external degrees of freedom of the trapped
particle allowed, for example, the realization of laser cooling to the ground state
for extended periods of time [2, 3], the creation of nonclassical motional states [4, 5],
entanglement between external and internal degrees of freedom of one ion or several
ions [6, 7, 8, 9, 10], quantum gates for quantum computation [11, 12, 13], and single
photon emission on demand [14].
An important aspect at this advanced level of quantum state engineering is the
ability to obtain as much information as possible, and with high efficiency, about the
motional state of the trapped atom. Moreover, since the Lamb-Dicke (LD) regime,
wherethe motionofthe particleis restrictedtoaregionsmallerthanawavelength,is
sometimes difficult to achieve in a trap, it is desirable to have measurement schemes
working also outside this regime [15].
Measure of phonon-number moments and motional quadrature 2
So far, several techniques have been proposed to gain information about the
motional quantum state of a trapped particle. The central idea in all these methods
is the mapping ofthe ionic externaldynamics to aninternaldegree offreedom,where
the latter can be read out, for example, with electron shelving techniques [16]. Some
of these procedures are applicable only inside the LD regime [17, 18, 19, 20], whereas
others work also beyond this limit [21, 22, 23, 24, 25, 26], allowing to derive in some
cases the complete motional density operator. However, a typical requirement of
conventionalschemesisthatthemeasurementshavetobeperformedduringrelatively
long interaction times, through several Rabi oscillations in the resonant case or slow
phaseshiftsinthedispersivecase,wheredecoherencemechanismscannotbeneglected.
Inthispaper,wepresentamethodthatallowstodeterminetheexpectationvalue
ofthephonon-numbermoments nˆp ofatrappedionforanypositiveintegerp,aswell
h i
as any generalized motional nonlinear quadrature 1 fˆ(nˆ)aˆe−iφ+aˆ†fˆ(nˆ)eiφ . Here,
2h i ‡
aˆ and aˆ† are the phonon annihilation and creation operators, respectively, nˆ = aˆ†aˆ,
fˆ(nˆ) is a function of nˆ, and φ is an arbitrary phase factor. In particular, we are
able to measure the expectation value of the linear quadratures, i.e., when fˆ(nˆ)= 1,
includingtheionicpositionandmomentum. OurtechniqueisnotrestrictedtotheLD
regime and, moreover,can be extended easily to all spatial dimensions, or to systems
containing more than one trapped ion. Furthermore, in contrast to the methods
presentedsofar,ourproposalallowstoobtaintherequiredinformationinanextremely
shortmotion-probeinteractiontime,whichisusefulinparticularinpresenceofstrong
decoherence processes.
In Sec. 2, we show for a single trapped ion how to obtain information about the
expectation values of the phonon-number moments (Sec. 2.1) and the generalized
motional nonlinear and linear quadratures (Sec. 2.2) through infinitesimal-time
motion-probe interactions. In Sec. 3, we extend the method to the N-ion case. In
Sec. 4, we summarize our central results.
2. Single ion case
2.1. Phonon-number moments
We consider a single two-level ion trapped in a harmonic potential. For the sake of
simplicity, we confine our treatment to one motional degree of freedom, though all
results are easily generalizable to three dimensions. The system is described by the
Hamiltonian
Hˆ0 =~ν(nˆ+1/2)+~ω0 e e, (1)
| ih |
where ν is the harmonic oscillator frequency, e and g are the electronic upper
| i | i
and lower states of the two-level ion, respectively, and ω0 is the associated transition
frequency.
The ion is excited by a travelling laser beam resonant with the carrier electronic
transition, leaving unchanged the motional populations. In the usual rotating-wave
approximation, and in the interaction picture, the Hamiltonian reads [15, 24]
1
Hˆint = ~ΩL(σˆ++σˆ−)fˆ0(nˆ;η), (2)
2
‡ Generalizednonlinearquadraturescanberelatedtononlinearcoherentstates,theoreticallystudied
in [27]. A method for generating nonlinear coherent states in trapped ions, but not for measuring
them,canbefoundin[28].
Measure of phonon-number moments and motional quadrature 3
where Ω is the Rabi frequency, σˆ± are the electronic two-level flip operators, η is
L
the LD parameter (η = kx x0 , x0 being the extension of the ground state of the
h i h i
motional mode and k the projection of the laser wave vector on the trap axis), and
x
fˆ0(nˆ;η) is given by
fˆ0(nˆ;η)=e−η2/2 ∞ (ilη!)22l (nˆnˆ!l)!. (3)
l=0 −
X
At any time t, the probability of finding the ion in the excited state e is given
| i
by
P (t)=Tr ρˆ(t)e e = e e , (4)
e
| ih | | ih |
h i D E
whereρˆ(t)isthesystemdensityoperator,describingtheinternalandexternaldegrees
of freedom of the trapped particle.
Considering that, for any operator Aˆ,
d 1 ∂Aˆ
Aˆ = [Aˆ,Hˆ] + , (5)
dth i i~ ∂t
D E D E
we get, for Aˆ= e e,
| ih |
dP 1
e = e e,Hˆ . (6)
dt i~ | ih |
Dh iE
In our case, we have e e,Hˆ0 =0, so that
| ih |
ddPte (cid:2)= i1~ |ei(cid:3)he|,Hˆint . (7)
Dh iE
Since, in the interaction picture,
~Ω
e e,Hˆint = L(σˆ+ σˆ−)fˆ0(nˆ,η), (8)
| ih | 2 −
we obtain, from(cid:2)(7) and (8),(cid:3)
dP Ω
e = LTr ρˆ(t)(σˆ+ σˆ−)fˆ0(nˆ,η) . (9)
dt 2i −
Next, we consider that th(cid:2)e ion is prepared initia(cid:3)lly in the state
ρˆ(0)= ρˆ , (10)
φ φ f
|± ih± |⊗
where
1
= (g eiφ e ), (11)
φ
|± i √2 | i± | i
andρˆ isthephononstateweaimtocharacterize. Then,astraightforwardcalculation
f
yields
Tr ρˆ(0)(σˆ+ σˆ−)fˆ0(nˆ,η) = isin(φ)Tr ρˆffˆ0(nˆ,η) , (12)
− ∓
and thus, using Eq(cid:2). (9), (cid:3) (cid:2) (cid:3)
dP±φ
e = sin(φ) fˆ0(nˆ,η) , (13)
dτ (cid:12) ∓ h i
(cid:12)τ=0
(cid:12)
where τ is the dimens(cid:12)ionless time ΩLt/2 and the token φ stands for the two
(cid:12) ±
parameters (sign and phase φ) defining state of Eq. (11).
φ
± |± i
Measure of phonon-number moments and motional quadrature 4
Eq. (13) shows that the mean value of the nonlinear operator fˆ0(nˆ;η) is
determined by the time derivative of the population of the excited state at the initial
interactiontimeτ =0. Thisallowstogaininformationabouttheionicmotionalstate,
in particular,as willbe shownbelow, aboutthe phonon-number moments. Moreover,
since the time derivative of the excitation probability P±φ is evaluated at τ =0, this
e
informationcanbeobtainedinaveryshortmotion-probeinteractiontime,evenbefore
decoherencemechanismscanaffecttheinitialcoherentevolution. Remarkfurtherthat
the“contrast”inthemeasurementof fˆ0(nˆ,η) ,asseeninEq.(13),canbetunedwith
h i
the proper choice of the phase φ.
For small LD parameters, a series expansion of the nonlinear operator fˆ0(nˆ;η)
leads to the expression
η4
fˆ0(nˆ;η) 1 η2 nˆ + nˆ2 . (14)
h i≃ − h i 4 h i
Thus, if we repeat the measurements with two known LD parameters, η1 and η2
(varying, e.g., the angle between the laser beam and the trap axis), we can derive nˆ
and nˆ2 by solving the linear system h i
h i
hfˆ0(nˆ,η1)i =1−η12hnˆi+ η414hnˆ2i, (15)

 hfˆ0(nˆ,η2)i =1−η22hnˆi+ η424hnˆ2i,
from which, for example, the Fano-Mandel parameter [29]
nˆ2 nˆ 2
Q= h i−h i (16)
nˆ
h i
can be extracted. It is known that the Fano-Mandel parameter allows to determine
the “classicality” of a given motional state, since quantum states with Q > 1 are
consideredasclassicalstates,andthosewithQ<1asnonclassicalstates,duetotheir
sub-Poissonianphonon-number distribution.
This low order moment determination scheme canbe generalizedto higher order
vibronicmoments nˆp (p>0). Indeed,forhigherLDparameters,theseriesexpansion
h i
offˆ0(nˆ,η)inEq.(3)willextendbeyondthe nˆ2 term,uptoacertainmoment nˆN .
h i h i§
We could repeat then the measurement technique, as in Eq. (15), for N known LD
parameters,yieldingalinearsystemfromwhichtheN firstvibronicmomentsbecome
accessible. Each nˆp , 0 < p N, could also be determined by measuring only once
theinitialtimedehrivaitiveofP≤±φ. Toshowthis,werequirethattheionissubmittedto
e
N simultaneouslaserinteractions,eachofthemresonantwiththeelectronictransition
and leaving the motional state unchanged. In the interaction picture, they give rise
to the simultaneous action of Hamiltonians
1
Hˆijnt = 2~Ωjσˆ+fˆ0(nˆ;ηj)+H.c., j =1,...,N, (17)
whereΩ andη are,respectively,theelectronicRabifrequencyandtheLDparameter
j j
of laser j. The total Hamiltonian will be given in this case by
N ~Ω
Hˆint = Hˆijnt = 2L(σˆ++σˆ−)Fˆ0(nˆ), (18)
j=1
X
§ The series of Eq. (3) is convergent for all η. When the sum of this series is truncated at l = N,
the coefficient of the nˆN term is given by (−1)Ne−η2/2η2N/N!2. For any η value, this coefficient
canbemadearbitrarilysmallbyaproperchoiceofN andtheseriesmaybetruncated withingood
approximationuptothisterm.
Measure of phonon-number moments and motional quadrature 5
where Ω =max(Ω ) and
L j
N
Ω
Fˆ0(nˆ)= jfˆ0(nˆ;ηj). (19)
Ω
j=1 L
X
Ifthesystemisagaininitiallypreparedinthestate ρˆ ,weget,similarly
φ φ f
|± ih± |⊗
to Eq. (13),
dP±φ
e = sin(φ) Fˆ0(nˆ) . (20)
dτ (cid:12) ∓ h i
(cid:12)τ=0
(cid:12)
It hasbeenshownb(cid:12)(cid:12)yde Matos Filho andVogel[24]thatFˆ0(nˆ)may be rewritten
in the form of a Taylor series
∞
Fˆ0(nˆ)= cpnˆp, (21)
p=0
X
where the Taylor coefficients c are given by
p
N
e−ηj2/2Ωj , if p=0,
Ω
cp = ( 1)p N e−Xj=ηj21/2Ωj ∞L amηj2m , if p=0, (22)
− j=1 ΩL m=p p m!2! 6
with  X X
1, if p=m,
am = m−1 (23)
p j j ...j if p<m.
 i1 i2 im−p
ji1<ji2<X...<jim−p=1
ThismeansthatforgivenvaluesoftheN LDparametersηj (fixedbythegeometry
of the laser beams regardingto the trapaxis up to the maximumvalue ηmax =k x0 ,
h i
k being the wave vector of the lasers), the c coefficients are linear combinationof all
p
RabifrequenciesΩ . Inthisway,theuseofN lasersallowstofixatwillN coefficients
j
of the Taylor series. In particular, if the N first coefficients are set to 0, we obtain
η2N
Fˆ0(nˆ)=O( Nm!a2xnˆN). (24)
NotethatitisalwayspossibletochoosetheN valuesuchthat (ηm2NaxnˆN)isnegligible,
O N!2
even outside the LD regime where ηmax can be greater than 1. The mean value of
(ηm2NaxnˆN) can be verified experimentally by measuring the time derivative of the
O N!2
population of the excited state at interaction time τ = 0. According to Eq. (20),
the outcome of this measurement yields (ηm2NaxnˆN) so that we can check if it is
hO N!2 i
indeed negligible for the phonon distribution we aim to characterize(in this case it is
recommendable to set φ= π/2).
±
The Rabi frequencies of the N lasers could also be chosen in such a manner that
only a single coefficient c (p < N) is equal to 1, the others remaining equal to 0.
p
In this case, according to Eq. (21) and considering that all terms nˆr, r > N, are
negligible,
Fˆ0(nˆ)=nˆp, (25)
Measure of phonon-number moments and motional quadrature 6
and the measurement of dPe±φ yields directly the vibronic moment nˆp . This
dτ h i
(cid:12)τ=0
step can be reproduced for any(cid:12)p<N.
(cid:12)
It is noteworthy to mention(cid:12) that the knowledge of the phonon-number moments
nˆp for all positive integers p allows to derive the complete phonon distribution
h i
p(n) nρˆ n ,asknownfromclassicalstatistics[30]. Also,thegeneralizationtothe
f
3D ca≡sehp|rov|idies measurement schemes for nˆ , nˆ , nˆ , nˆ2 , nˆ2 , nˆ2 , nˆ nˆ ,
h xi h yi h zi h xi h yi h zi h x yi
nˆ nˆ , nˆ nˆ , nˆ nˆ nˆ , and so on. Recently, the relevance of measuring different
x z y z x y z
h i h i h i
photon-number moments, through quantum field homodyning, for determining the
“classicality” of arbitrary quantum states has been considered [31].
2.2. Generalized nonlinear quadratures
Next, we show that by using a red or blue sideband excitation, expectation values
of generalized nonlinear quadrature moments, 1 fˆ(nˆ)aˆe−iφ + aˆ†fˆ(nˆ)eiφ , can be
2h i
determined using similar techniques. In particular, we have access to the expectation
values of the generalized linear quadrature moments, among them position xˆ and
h i
momentum pˆ . As will be shown below, these expectation values can be obtained
h i k
inside and outside of the LD regime.
Letusconsiderthattheionisexcitedwithared-sidebanddetunedlaser(theblue-
sidebandcasecanbetreatedequally,leadingtosimilarresults). Intheusualrotating-
wave approximation, and in the interaction picture, the Hamiltonian reads [15, 24]
i
Hˆint = ~ΩLησˆ+fˆ1(nˆ;η)aˆ+H.c., (26)
2
where
fˆ1(nˆ;η)=e−η2/2 ∞ (iη)2l nˆ! . (27)
l!(l+1)!(nˆ l)!
l=0 −
X
If the ion system is initially prepared in the state ρˆ , we obtain
φ φ f
|± ih± | ⊗
straightforwardly from Eq. (6)
dP±φ 1
e = fˆ1(nˆ;η)aˆe−iφ+aˆ†fˆ1(nˆ;η)eiφ , (28)
dτ (cid:12) ±2
(cid:12)τ=0 D E
(cid:12)
whereτ isthedimension(cid:12)(cid:12)lesstimeηΩLt/2. IntheLDregime,fˆ1(nˆ;η)=1andEq.(28)
reduces to the generalized quadrature Xˆ (aˆe−iφ+aˆ†eiφ)/2,
φ
≡
dP±φ 1 xˆ pˆ
e = Xˆ = cos(φ) h i +sin(φ) h i , (29)
φ
dτ (cid:12) ±h i ±2 x0 p0
(cid:12)τ=0 (cid:18) h i h i(cid:19)
(cid:12)
where xˆ and pˆ are the(cid:12)trapped ion position and momentum, respectively, and x0
(cid:12) h i
and p0 are the spreads of these quantities in the ground state of the trap potential.
Eq. (h29)i shows that any quadrature moment Xˆ can be experimentally determined
φ
from the measurement of the initial time derivative of the population of the excited
state, whereas, in particular, xˆ and pˆ can be obtained by choosing φ = 0,π and
h i h i
φ= π/2, respectively.
±
k In this limit our results for trapped ions are analogous to the ones derived for measuring field
quadratures incavityQEDsetups: see[32].
Measure of phonon-number moments and motional quadrature 7
OutsidetheLDregime,asimilarinformationisgainedusingN simultaneousred-
sidebanddetunedlasers. Intheinteractionpicture,theygiverisetotheHamiltonians
i
Hˆijnt = 2~Ωjηjσˆ+fˆ1(nˆ;ηj)aˆ+H.c., j =1,...,N, (30)
whereΩ andη are,respectively,theelectronicRabifrequencyandtheLDparameter
j j
of laser j. The total Hamiltonian is given in this case by
N
i
Hˆint = Hˆijnt = 2~ΩLσˆ+Fˆ1(nˆ)aˆ+H.c., (31)
j=1
X
where Ω =max(Ω ) and
L j
N
Ω η
Fˆ1(nˆ)= j jfˆ1(nˆ;ηj). (32)
Ω
j=1 L
X
If the ion is initially in the state ρˆ , we obtain, similarly to Eq. (28),
φ φ f
|± ih± |⊗
with τ =Ω t/2,
L
dP±φ 1
e = Fˆ1(nˆ)aˆe−iφ+aˆ†Fˆ1(nˆ)eiφ . (33)
dτ (cid:12) ±2
(cid:12)τ=0 D E
(cid:12)
Accordingto de Ma(cid:12)(cid:12)tosFilho andVogel[24], Fˆ1(nˆ)maybe againrewritteninthe
form of a Taylor series
∞
Fˆ1(nˆ)= cpnˆp, (34)
p=0
X
where the c coefficients are linear combination of the N laser Rabi frequencies Ω .
p j
Similarly to the case of a resonant laser described in the preceding section, we can
choose the N value in such a way that (c nˆN) is negligible and fix next the N
N
O
Rabi frequencies to set all coefficients cp (p < N) to 0, except c0 to 1. In this case,
Fˆ1(nˆ) 1, Eq. (33) reduces to Eq. (29) and the quadrature moments (and more
≃
specifically xˆ and pˆ )aredeterminedasdoneinthe caseofthe LDregime. Clearly,
byfollowinghaisimilahriprocedure,wecouldalsoengineerFˆ1(nˆ)todescribeanarbitrary
polynomial.
3. N-ion case
We now briefly discuss how our proposalcan be generalizedto a chain of N identical
two-level ions in a linear Paul trap. This system is described by the Hamiltonian
N N
Hˆ0 = ~νj(nˆj +1/2)+~ω0 e k e, (35)
| i h |
j=1 k=1
X X
where ν arethe frequencies associatedwith the collective motionalmodes,nˆ =aˆ†aˆ
j j j j
are the respective phonon-number operators, e and g are the electronic states of
k k
| i | i
the two-level ion k, and ω0 is the electronic transition frequency of each ion.
The ions are illuminated by a laser beam resonant with their electronic
transitionwhileleavingunchangedthephononpopulation,realizinganonlinearcarrier
Measure of phonon-number moments and motional quadrature 8
excitation. In the usual rotating-wave approximation and in the interaction picture,
the Hamiltonian reads [15, 24]
N N
1
Hˆint = 2~ΩL (σˆk++σˆk−) fˆ0(nˆj;ηj), (36)
k=1 j=1
X Y
whereΩ istheelectronicRabifrequency,σˆ±aretheelectronictwo-levelflipoperators
L k
ofatomk,ηj istheLDparameterrelatedtothecollectivemodej (ηj =kx x0 j, x0 j
h i h i
being the extension of the ground state of mode j and k the projection of the laser
x
wave vector on the trap axis), and
fˆ0(nˆj;ηj)=e−ηj2/2 ∞ (iηl!j2)2l (nˆnˆj!l)!. (37)
l=0 j −
X
In the following, we denote the product Nj=1fˆ0(nˆj;ηj) by Fˆ0. The generalized
relation
Q
d(P ) 1
e k = e e,Hˆ , (38)
dt i~ | ikh |
Dh iE
can be easily obtained, as in Eq. (6), where (P ) is the probability of finding ion k
e k
in its excited state. As e k e commutes with Hˆ0 and any operator associated with
| i h |
other ions k′, we get immediately
d(P ) 1 1
dte k = i~ |eikhe|,2~ΩL(σˆk++σˆk−)Fˆ0 , (39)
Dh iE
and thus
d(P ) Ω
dte k = 2Li Tr ρˆ(t)(σˆk+−σˆk−)Fˆ0 , (40)
where ρˆ(t) is the N-ion system d(cid:2)ensity operator. (cid:3)
Let us consider the following initial state
ρˆ(0)=ρˆ ρˆ ρˆ , (41)
k A f
⊗ ⊗
where
ρˆ = (42)
k φ k φ
|± i h± |
is the electronic density operator of the k-th ion, with
1
= (g eiφ e ), (43)
φ k k k
|± i √2 | i ± | i
ρˆ is an arbitrary electronic density operator of the remaining ions, and ρˆ is the
A f
collective motional density operator associated with the N eigenmodes. Following
similar steps as in the previous sections, we obtain
Tr ρˆ(0)(σˆk+−σˆk−)Fˆ0 =∓isin(φ)Tr ρˆfFˆ0 , (44)
similar to Eq. (12),(cid:2)and thus, using E(cid:3)q. (40), (cid:2) (cid:3)
d(P )
e k = sin(φ) ˆ0 , (45)
dτ ∓ hF i
(cid:12)τ=0
(cid:12)
where τ is the dimension(cid:12)less time τ =ΩLt/2. Eq. (45) tells us that by measuring the
(cid:12)
time derivative of the population of the excited state of one ion at τ =0, we are able
to gain information about the collective nonlinear operator ˆ0.
F
Measure of phonon-number moments and motional quadrature 9
4. Summary
In conclusion, we have proposed a method that allows to determine the phonon-
number moments and the motional nonlinear and linear quadratures of a trapped
ion, in particular the ionic position and momentum. Our method makes use of the
nonlinear behavior of the ion-laser interaction in harmonic traps. The measurement
of the phonon-number moments nˆp requires resonant carrier interaction, with no
h i
phonon gain or loss, while the measurement of the nonlinear quadrature moments
1 fˆ(nˆ)aˆe−iφ +aˆ†fˆ(nˆ)eiφ demands the use of red or blue sideband excitations. In
2h i
contrast to methods presented so far, our proposal is designed for measurements
realized in very short probe laser interaction times, thus preventing the noisy action
of decoherence processes. In addition, we have shown that our scheme works inside
and outside the LD regime, and that it can be generalized to the case of N ions. In
the latter case, informationabout a collective property is gainedby the measurement
of the excited state population of a single ion.
Acknowledgments
TB acknowledges support from the Belgian Institut Interuniversitaire des Sciences
Nucl´eaires (IISN), and thanks JVZ and ES for the hospitality at the University of
Erlangen, Erlangen, and Max-Planck-Institut fu¨r Quantenoptik, Garching, Germany.
ES acknowledges support from EU through RESQ project.
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