Quantum quenches of ion Coulomb crystals across structural instabilities II: thermal effects

1 MB·English

by  Jens D. Baltrusch

Quantum quenches of ion Coulomb crystals across structural instabilities II:
thermal effects.
Jens D. Baltrusch1,2,∗ Cecilia Cormick1,3, and Giovanna Morigi1
1Theoretische Physik, Universita¨t des Saarlandes, D-66123 Saarbru¨cken, Germany
2Grup d’Optica, Departement de F´ısica, Universitat Auto`noma de Barcelona, E-08193 Bellaterra, Spain
3Institut fu¨r Theoretische Physik, Universita¨t Ulm, D-89069 Ulm, Germany
(Dated: January 17, 2013)
Wetheoreticallyanalyzetheefficiencyofaprotocolforcreatingmesoscopicsuperpositionsofion
chains, described in [Phys. Rev. A 84, 063821 (2011)], as a function of the temperature of the
crystal. The protocol makes use of spin-dependent forces, so that a coherent superposition of the
electronicstatesofoneionevolvesintoanentangledstatebetweenthechain’sinternalandexternal
degrees of freedom. Ion Coulomb crystals are well isolated from the external environment, and
should therefore experience a coherent, unitary evolution, which follows the quench and generates
3 structuralSchro¨dingercat-likestates. Thetemperatureofthechain,however,introducesastatistical
1 uncertainty in the final state. We characterize the quantum state of the crystal by means of the
0 visibility of Ramsey interferometry performed on one ion of the chain, and determine its decay as
2 afunctionofthecrystal’sinitialtemperature. Thisanalysisallowsonetodeterminetheconditions

n on the chain’s initial state in order to efficiently perform the protocol.
a
J PACSnumbers: 03.65.Ud,42.50.Dv

6
1
I. INTRODUCTION

]
h
p The quantum to classical transition is an intriguing
- problem of quantum physics [1] and a central issue of
t
n quantum-based technologies, where efforts are being in-
a vestedindevelopingprotocolsforimplementingquantum
u
dynamicsofsystemsofincreasingsize[2]. Increasingthe
q
size of a system is usually associated with loss of coher-
[ (a) (b)
ence: Even when the physical object undergoes unitary

1 evolution, the particles composing it can often be seen
FIG. 1: (Color online) Ramsey interferometry with an ion
v as a reservoir for each individual one [3]. As a result, a chain whose vibrations are at temperature T. A quench
6
coherent and localized excitation can dephase on a rate across the linear-zigzag instability is performed by exciting
4
which increases with the number of components. Such thecentralionwithalaserpulseinpresenceofspin-dependent
6
3 dephasingcanbeexaminedusingtheso-calledLoschmidt forces. In (a) the collective motion is initially in a thermal
. echo [4–6], which can be measured by means of the vis- stateofazigzagstructure,andthecentralionintheinternal
1 ibility of an interferometric measurement performed on state |g(cid:105). A√π/2 laser pulse prepares it in the superposition
0 (|g(cid:105)+|e(cid:105))/ 2. (b) The ion in state |e(cid:105) experiences a tighter
the system [7, 8].
3
state-dependentpotential. Thecorrespondingconditionaldy-
1 Incrystalsoftrappedionssuchaninterferometricmea- namics entangle the ions’ internal and external degrees of
v: surement can be performed on an internal transition of freedom. A subsequent laser pulse performs a −π/2 rotation
i one ion of the crystal. In Ref. [8] a protocol for imple- on the ion’s internal transition. The final occupation of the
X
menting Ramsey interferometry on the ion of the crystal ground state |g(cid:105) as a function of the time t elapsed between
r hasbeenproposed. Itwasshownthatthevisibilityofthe thetwopulsesallowsonetoextractinformationonquantum
a
interferometricsignal, herecorrespondingtotheoccupa- coherence and entanglement created by these dynamics.
tion of one electronic state of the ion, gives information
on the quantum state of the crystal when analysed as
a function of the time t elapsed between the two Ram- inaset-upwherethetrapfrequencydependsontheelec-
sey pulses. This protocol is at the basis of the proposal tronic state [9, 13]. In these settings, a first laser pulse
of Ref. [9] to create a superposition of two different crys- prepares the ion in a coherent superposition of the elec-
tallinestructuresacrossthelinear-zigzagstructuraltran- tronic states, which evolves into an entangled state be-
sition [10–12]: The mesoscopic superposition of a crystal tween the chain’s internal and external degrees of free-
in the linear and in the zigzag structure can be accessed dom as sketched in Fig. 1. The visibility of the Ramsey
bydrivingtheelectronictransitionofoneionofthechain signal does indeed decay fast with the time elapsed be-
tweenthepulses,evenwhenthechainispreparedatzero
temperature and the dynamics are purely unitary. How-
ever, for quenches close to the structural instability the
∗Email: [email protected] visibilityexhibitsquasiperiodicrevivalswhicharevisible
2
atlongertimest. Theserevivalsappearwithafrequency oflasercooling, suchthattheyarealignedalongastring
whichisdeterminedbythefrequencyofthezigzagmode andperformsmallvibrationsabouttheirequilibriumpo-
andpersistwhenthesizeofthecrystalisincreased. The sitions[18]. Inthisworkweassumethatafterlasercool-
analysis of the spectrum of the visibility’s temporal be- ing all ions are in the same stable electronic state, which
haviour shows features that can be associated with the is denoted by |g(cid:105). In absence of external perturbations,
presenceofentanglementgeneratedbythefirstquantum their motion is governed by the Hamiltonian [14, 20]
quench [14].
The studies in Refs. [9, 14] assume unitary dynamics (cid:88)3N (cid:18) 1(cid:19)
of a chain initially at T = 0. The assumption of uni- Hg =E0g+ (cid:126)ωjg bgj†bgj + 2 . (1)
tary dynamics is reasonable for state-of-the-art ion-trap j=1
experiments [15–17], in which the coherence times are of
This Hamiltonian describes the harmonic vibrations
the order or larger than the typical time scales required
about the equilibrium positions along the string, where
for observing the dynamics predicted in Ref. [9]. Much
Eg is the energy of the classical ground state, ωg are
more stringent is, however, the condition that the chain 0 j
thenormalmodefrequenciesofthecrystalformingalin-
should be initially at temperature T = 0. A feasibility
ear (or a zigzag) chain and bg†,bg are the corresponding
analysis needs in fact a more quantitative statement on j j
bosonic creation and annihilation operators.
the temperatures required so that the protocol can be
Structural superpositions can be obtained by prepar-
successfully performed. The present work extends the
ing, for instance, one ion in a coherent superposition
analysis of Ref. [14] to the case in which the chain is ini-
of state |g(cid:105) and a second stable state |e(cid:105), in which the
tially at finite temperature T, considering temperatures
ion experiences an additional potential due to a dipole
that can be achieved by means of Doppler or ground-
trap [9, 22–24]. When this potential is sufficiently steep,
state cooling [18]. The behaviour of the visibility as a
the ion’s equilibrium position is displaced with respect
funtion of T is analysed for small quenches across the
to the case in which all ions are in state |g(cid:105). Then,
linear-zigzag phase transition. This allows us to deter-
the long-range Coulomb repulsion causes a distortion of
mine the experimental requirements on the temperature
the crystalline structure, such that the ground state and
of the chain in order to perform the protocol.
the normal mode spectrum are now different. Therefore,
The article is organized as follows: In Sec. II the pro-
when one ion is in |e(cid:105), its dynamics are governed by the
posal of Ref. [9] is summarized. The theoretical model is
Hamiltonian
presented in Sec. III, which includes the detailed evalu-
ation of the visibility signal as a function of the number
3N (cid:18) (cid:19)
of ions and of the initial temperature of the crystal. The H =Ee+(cid:88)(cid:126)ωe be†be+ 1 . (2)
behaviour of the visibility is analysed in Sec. IV for a e 0 j j j 2
j=1
chain of three ions and different temperatures, and the
conclusions are drawn in Sec. V. Theoretical details for Here, Ee is the energy of the corresponding classical
0
the derivation of the results in Sec. III are given in the ground state, ωe are the normal mode frequencies and
j
appendix. be†,bg are the corresponding bosonic creation and anni-
j j
hilation operators. By preparing the ion’s internal state
inalinearsuperpositionof|g(cid:105)and|e(cid:105),thedynamicsgen-
II. ION COULOMB CRYSTALS IN A erated by the state-dependent Hamiltonian entangle the
THERMAL STATE electronic and motional degrees of freedom [9]. A probe
ofquantumcoherenceandentanglementcanbeobtained
Inthissectionwebrieflyreviewthephysicalmodeland byperformingRamseyinterferometrywithasingleionof
the protocol proposed in Ref. [9]. These are the starting the crystal [8]. The scheme is sketched in the following.
pointsoftheanalysisperformedinthefollowingsections.
B. Ramsey Interferometry with Thermal States
A. Ion Coulomb Crystals in State-dependent Traps
InRef.[8]itwasproposedtouseRamseyinterferome-
The system we consider are atomic ions, which are tryasatooltoprobethedynamicsandthermodynamics
confined in an external anisotropic potential. The ions of ion chains close to the zigzag instability. In Ref. [9]
have been laser cooled to sufficiently low temperatures, it was shown that the visibility of the Ramsey signal in
so that they perform small vibrations about the equilib- chains of three ions presents features that can be asso-
rium positions determined by the competition between ciated with the creation of a superposition of motional
the external trap and the Coulomb repulsion [19, 20]. states corresponding to linear and zigzag structures. In
In the following we assume N identical ions with Ref. [14] it was shown that some of these features are
massmandchargeq,whichareconfinedbyalinearPaul found for chains of generic size N. In these articles, it
trap [11] or a Penning trap [21]. The ions are assumed was assumed that the chain was prepared in the vibra-
to have been prepared at low temperature T by means tional ground state, at T = 0. The scope of this paper
3
is to analyze how the coherence of the superposition is theprobabilitythatthecentralionisinthegroundstate
affectedwhentheinitialvibrationalstateisnotpurebut, after a second pulse has been applied which performs a
say, a thermal state, as it is the experimental situation −π/2-pulse. Theprobabilitythatthegroundstateofthe
afterlasercoolingthechain. Inthissubsectionwebriefly ion is occupied after the pulse reads
review the interferometric scheme and derive the expres-
sionforthevisibility,whichisgoingtobeexplicitlyeval- Pg(φ,t)=Tr{(cid:37)f|g(cid:105)j0(cid:104)g|}=Tr{ρf,gg(t)}, (7)
uated in the rest of this work.
Let the initial state of the ion chain be described by where(cid:37)f isthedensitymatrixafterthesecondpulseand
the density matrix (cid:37) =(cid:37)(t=0). This reads
0
1(cid:16)
ρ (t)= ρ (t)+R† ρ (t)R
(cid:37)0 =|g(cid:105)(cid:104)g|⊗ρ0, (3) f,gg 2 2,gg k(cid:48) 2,ee k(cid:48)
(cid:17)
+eiφρ (t)R +e−iφR† ρ (t) , (8)
where 2,ge k(cid:48) k(cid:48) 2,eg
(cid:18) (cid:19)
ρ = 1 exp − Hg , withk(cid:48)thewavevectorofthesecondpulse. UsingEq.(8)
0 Z k T in Eq. (7), the probability can be recast in the form
B
isthedensitymatrixfortheexternaldegreesoffreedom, P (φ,t)= 1(cid:0)1+Re(cid:2)eiφO(t)(cid:3)(cid:1) , (9)
with kB Boltzmann constant and Z = Tr{e−Hg/(kBT)} g 2
thepartitionfunction. InFig.1atheinitialstateistaken
to be a thermally excited zigzag structure. A laser pulse where O(t) measures the coherence between ground and
applied for a time ∆τ drives resonantly the transition excited state. It reads
|g(cid:105) → |e(cid:105) of the central ion, which we label by j0. As- (cid:110) (cid:111)
sumingthatthepulseareacorrespondstoaπ/2rotation O(t)=Tr Rk†(cid:48)UeRkρ0Ug† , (10)
ofthedipole,whileitsduration∆τ issufficientlyshort,so
that the chain motion can be neglected during ∆τ, then and determines the visibility V of the Ramsey signal
the density matrix immediately after the pulse takes the through the relation V = |O|. In Sec. III we determine
form O(t) as a function of the time elapsed between the two
pulses and of the initial temperature of the chain, and in
(cid:37)1 =U(cid:37)0U†, Sec. IV we discuss its behavior as a function of the tem-
√ perature T for a chain of three ions in a two-dimensional
whereU =(|e(cid:105)j0(cid:104)e|+|g(cid:105)j0(cid:104)g|+Rk|e(cid:105)j0(cid:104)g|+Rk† |g(cid:105)j0(cid:104)e|)/ 2 geometry.
is the evolution operator describing the dynamics due to
the laser pulse and the operator R (x)=eik·x describes
k
the mechanical effect on the crystal associated with the III. EVALUATION OF THE VISIBILITY OF
absorption of a laser photon. THE RAMSEY FRINGES
The crystal evolves then freely for a time t according
to the Hamiltonian H = |e(cid:105)(cid:104)e|H +|g(cid:105)(cid:104)g|H , such that
e g In this section we carry out the theoretical evaluation
the density matrix at time t reads (cid:37)(t)=(cid:37) , with
2 of the visibility of the Ramsey signal as a function of
the temperature T and of the time t elapsed between
(cid:88)
(cid:37)2 = ρ2,pq|p(cid:105)(cid:104)q|, the two Ramsey pulses. The calculation here reported
p,q=e,g extends the one presented in [14], which was performed
assuming that the ion crystal is initially prepared in the
and
vibrational ground state. We also include the possibility
1 ofamechanicaleffectassociatedwiththeabsorptionand
ρ (t)= U (t)R ρ R†U†(t), (4)
2,ee 2 e k 0 k e emissionofaphotonofthethepulses. Thefinalresultis
1 reported in Eq. (50). It is valid for a three-dimensional
ρ (t)= U (t)ρ U†(t), (5)
2,gg 2 g 0 g geometry,foranynumberofionsandforanyinitialtem-
perature T, as long as the assumption of that the ions
eiφ
ρ (t)=ρ† (t)= U (t)R ρ U†(t). (6) performharmonicvibrationsabouttheirequilibriumpo-
2,eg 2,ge 2 e k 0 g
sitions is valid.
There,U (t)=exp(−iH t/(cid:126)),whileφisaphase-shiftap-
p p
plied when the atom is in the excited state, which allows
onetoperforminterferometry. Atthisstage,atomicmo- A. Some useful relations
tionandinternaldegreesoffreedomareentangledbythe
state-dependent evolution. A graphical representation is In order to evaluate Eq. (10), we make use of the uni-
shown in Fig. 1b, illustrating a coherent superposition tarytransformationswhichrelatethestatesandnormal-
between zigzag and linear chain. Information about this mode operators of the two structures. These have been
structural superposition can be extracted by measuring derived in Ref. [14], and are reported in this section.
4
Wewritetheions’positionsr (j =1,...,3N)assmall Thevibrationalgroundstateofeachstructureisdenoted
j
excursions qg,qe away from the equilibrium positions of by|0(cid:105) ,withs=g,e. Theyaremappedintooneanother
j j s
the corresponding structures, rg,re, namely, by the transformation [14, 25]
j j
rj =rjg+qjg =rje+qje, (11) |0(cid:105)g =ZDe(β1e,...,β3eN)eA|0(cid:105)e . (19)
where the superscripts g and e indicate whether the
Here, A reads
central ion is in the ground state |g(cid:105) or in the excited
state |e(cid:105). We denote by dg = re −rg the equilibrium
j j j 1(cid:88)
displacements of ion j between the structures. The nor- A= 2 Ajkbej†bek† (20)
malmodesareobtainedbydiagonalizingHamiltonianHs jk
(s=g,e),afterithasbeenexpandedaroundtheequilib-
riumpositionsuptosecondorderinthedisplacementsqs. with Ajk a real and symmetric matrix with elements
j
Thus,theydependontheinternalstateofthecentralion
(cid:88)
and read Ajk = (u−1)jlvlk. (21)
Qs =(cid:88)Ms qs, (12) l
l kl k
k The scalar
wonhaelrizeinmgattrhiexhMarsmisonthicepoarrtthoogfotnhaelptroatnensftoiarml;adteiotanildsiaarge- Z =det(cid:104)(cid:0)1−A2(cid:1)1/4(cid:105) (22)
reported in Ref. [14]. Equations (11) and (12) give the
mapping connecting the normal modes in the two struc- is warranting the correct normalization, while the term
tures: D (βe,...,βe ) is a displacement operator of the 3N
e 1 3N
Qg =(cid:88)T Qe +Dg, (13) normalmodes,whenthestructureistheonecorrespond-
j jk k j ing to the central ion being excited. It is defined as
k
(cid:88)
Pg = T Pe, (14) D (βe)=⊗3N D(j)(βe), (23)
j jk k e j=1 e j
k
with Ps the momentum canonically conjugated to the with
j
displacement Qs and
l D(j)(βe)=exp{βeb†−βe∗b )}. (24)
(cid:88) (cid:88) s j j s j s
T = Mg Me , Dg = Mg dg. (15)
jl kj kl j kj k
It is useful to introduce the relation between displace-
k k
ment operators in the basis of normal modes of each
Correspondingly,theannihilationandcreationoperators
(cid:113) structure. For a generic displacement λg, here given for
bsj and bsj†, defined by relations bsj = mωjs/(2(cid:126))[Qs + the structure in which the central ion is in the ground
iP /(mωs)]anditsadjoint,arerelatedbytheBogoliubov state, they are related by the equation
s j
transformations
bg =(cid:88)u be −(cid:88)v be†+βg, (16a) Dg(λg)=eiϕ[λg]De(λe), (25)
j jk k jk k j
k k where
(cid:88) (cid:88)
bej = ukjbgk + vkjbgk†+βje, (16b) ϕ[λg]=2Im(cid:104)(cid:88)λgβg(cid:105), (26)
k k j j
j
with the real and dimensionless coefficients
T (cid:115)ωe (cid:115)ωg and
ujk = 2jk  ωjkg + ωjke , (17a) λej =(cid:88)(λglulj +λgl∗vlj). (27)
T (cid:115)ωe (cid:115)ωg l
vjk = 2jk  ωkg − ωje , (17b)
j k B. Evaluation of the visibility for any initial state
(cid:113)
and displacements βg = mωg/2(cid:126)Dg, such that
j j j We now evaluate Eq. (10), whose modulus is the visi-
(cid:115) bility for an arbitrary initial state. We use that R (x)=
βe =−(cid:88) ωjeT βg =−(cid:88)(u +v )βg, (18a) exp(ik · x) is a displacement operator for each nkormal
j ωg kj k kj kj k mode, such that R (x)=D (κ). Here,
k k k k e
(cid:115)
βg =−(cid:88) ωjgT βe =−(cid:88)(u −v )βe. (18b) (cid:115) (cid:126)
j k ωke jk k k jk jk k κj =i 2mωjeKj, (28)
5
(cid:16) (cid:17)
where K = k Me +k Me +k Me is the with
j x j0x,j y j0y,j z j0z,j
projection of the wave vector onto the normal mode j, (cid:104)(cid:88) (cid:88) (cid:105)
ϕ =Im (κ +βg)λe∗+ κ βe , (37a)
assuming that ion j is illuminated (j labels the α = θ j j j j j
0 0α
x,y,z displacement of the ion). j j
(cid:104)(cid:88) (cid:88) (cid:105)
It is convenient to use a coherent state basis for per- ϕ =Im (κ(cid:48) +βg)λe∗(t)+ κ(cid:48)βe , (37b)
θ(cid:48) j j j j j
forming the evaluation of Eq. (10). Therefore, we take
j j
thetraceinEq.(10)overthebasisofcoherentstates|α(cid:105)
e
of the harmonic oscillators corresponding to the normal and where κ(cid:48) is the displacement due to the emission
modes when the ion is in state e, where |α(cid:105) =⊗ |α (cid:105) , of a photon with wave vector k(cid:48). In Eq. (35) we have
e j j e
such that D(j)(α )|0(cid:105) =|α (cid:105) . Using the cyclic proper- introduced the quantities
e j e j e
ties of the trace, we recast Eq. (10) in the form
θ =κ +βe+λe, (38a)
j j j j
(cid:90) d6Nα θ(cid:48) =κ(cid:48) +βe+λe(t), (38b)
O(t)= (cid:104)α|R ρ U†R† U |α(cid:105) . (29) j j j j
π3N e k 0 g k(cid:48) e e
as well as the operators
The initial density matrix can be expressed in the form
1(cid:88)
A(θ)= A (be†−θ ∗)(be†−θ ∗). (39)
(cid:90) d6Nλg 2 jk j j k k
ρ0 = π3N P0(λg)|λg(cid:105)g(cid:104)λg| , (30) jk
Exchanging the order of the integrations and evaluating
where |λg(cid:105)g = ⊗j(cid:12)(cid:12)λgj(cid:11)g is the basis of coherent states the operators, Eq. (35) becomes
of the harmonic oscillators, corresponding to the nor-
mal modes when the ion is in state g, and P (λg) (cid:90) d6Nλg
0 O(t)= Z2eiϕP (λg)I (λg), (40)
is the Glauber-Sudarshan-P distribution containing the π3N 0 α
information over the initial state [26], with λg =
(λg,...,λg ). Using Eq. (30) in Eq. (29), we find with
1 3N
O(t) = (cid:90) dπ63NNα(cid:90) dπ6N3Nλg P0(λg) (31) Iα(λg)=(cid:90) dπ63NNαe(cid:104)α|eA(θ)|θ(cid:105)ee(cid:104)θ(cid:48)|eA†(θ(cid:48))|α(t)(cid:105)e
(cid:90) d6Nα
× e(cid:104)α|Rk|λg(cid:105)gg(cid:104)λg|Ug†Rk†(cid:48)Ue|α(cid:105)e. = π3N e(cid:104)α|θ(cid:105)ee(cid:104)θ(cid:48)|α(t)(cid:105)e e(cid:104)α|eA(θ)|α(cid:105)e
This expression contains two matrix elements. We write × (cid:104)α(t)|eA†(θ(cid:48))|α(t)(cid:105) (41)
e e
the first one as
The explicit evaluation of the integral in the variables α
(cid:104)α|R |λg(cid:105) =Zeiϕ[λg] (cid:104)α|D (κ)D (λe)D (βe)eA|0(cid:105) ,
e k g e e e e e is reported in Appendix A, and leads to the expression
(32)
ewlehmereenwteinutsheedr|iλggh(cid:105)tg-h=anDdgs(idλeg)o|f0(cid:105)Egq..T(3h1e)sceacnonbde rmewatrriitx- O(t)=(cid:90) dπ6N3Nλg P0(λg)√Zd2eetiϕΩeG∗(θ(cid:48))eG(θ)e41sTΩ−1s,
ten as (42)
where
(cid:104)λg|U†R† U |α(cid:105) = (cid:104)λg(t)|R† |α(t)(cid:105)
g g k(cid:48) e e g k(cid:48) e
=Ze−iϕ[λg(t)]e(cid:104)0|eA†De†(βe)De†(λe(t))De†(κ(cid:48))|α(t)(cid:105)e, G(γ)=(cid:88)A2jkγj∗γk∗−(cid:88)|γ2j|2 , (43)
(33)
jk j
where withγ =θ ,θ(cid:48). Here,Ωisacomplexsymmetric6N-by-
j j j
λej(t)=(cid:88)k (cid:16)λgke−iωkgtukj +λgk∗e+iωkgtvkj(cid:17) . (34) 6N matrix(cid:18), wΩh+i+chΩre+a−d(cid:19)s (cid:18)1−A+ −iA−(cid:19)
Ω= = (44)
Ω−+ Ω−− −iA− 1+A+
Using these results, Eq. (29) can be cast in the form
with
(cid:90) d6Nα(cid:90) d6Nλg
O(t)= π3N π3N Z2eiϕP0(λg) A±jk = 21(cid:0)Ajk(e−i(ωje+ωke)t±1)(cid:1). (45)
× (cid:104)α|eA(θ)|θ(cid:105) (cid:104)θ(cid:48)|eA†(θ(cid:48))|α(t)(cid:105) , (35)
e ee e
Moreover, s is a 6N-dimensional vector given by
where
(cid:18) S+(cid:19)
s= , (46)
ϕ=ϕ[λg]−ϕ[λg(t)]+ϕ −ϕ , (36) −iS−
θ θ(cid:48)
6
with The first term on the right-hand side is given by
Sj±[Sθ,[θγ(cid:48)]]==S(cid:88)j[θA]±γS∗j∗[−θ(cid:48)γ]e−.iωjet, ((4477ba)) (cid:18)IIjj12(cid:19)=(cid:32)IIjj12((ζζ∗∗))++IIjj21((ζζ(cid:48)(cid:48)))ee−+iiωωjjggtt(cid:33) (53)
j jk k j
k where
Equation(42)givesthevisibilityasafunctionofanarbi- I1(ζ∗)=(cid:88)v A ζ∗− 1(cid:88)(cid:0)v ζ +u ζ∗(cid:1) , (54a)
l lj jk k 2 lj j lj j
traryinitialstate,foranarbitrarynumberofionsN and jk j
aacbcsoourpnttiionngafonrdtehmeimsseiocnhaonficaalpheffoteocnt aosfstohceialtaesderwpituhlsteh.e Il2(ζ∗)=(cid:88)uljAjkζk∗− 12(cid:88)(cid:0)uljζj +vljζj∗(cid:1) . (54b)
jk j
The second term can be written as
C. Visibility for an initial thermal state (cid:18)Jk1(cid:19)=(cid:32)βjg(1−e−iωjgt)+ 12(cid:0)Jk+(κ)−Jk+(κ(cid:48))e−iωkgt(cid:1)(cid:33)
We now evaluate the visibility when the chain is ini- Jk2 βjg(e+iωjgt−1)+ 12(cid:0)Jk−(κ)−Jk−(κ(cid:48))e+iωkgt(cid:1)
tially in a thermal state, as in Eq. (3); we need to inte- (55)
grate in Eq. (42) over the variables λg taking the distri- with
bution P (λg)=(cid:81) P (λg), such that [26] (cid:88)(cid:16) (cid:17)
0 j 0 j J±(κ)= κ (u +v )±βe(u −v ) (56)
k j kj kj j kj kj
1 (cid:34) (cid:12)(cid:12)λg(cid:12)(cid:12)2(cid:35) j
P (λg)= exp − j , (48) The third term reads
0 j π(cid:10)ngj(cid:11) (cid:10)ngj(cid:11) (cid:18)Kk1(cid:19)=(cid:88)(cid:88)(cid:34)(cid:32)Yjl[Ω−1]αjkβSkβ[κ,κ(cid:48)](cid:33)+
with K2 Yα[Ω−1]αβSβ[κ,κ(cid:48)]
k αβ jk jl jk k
(cid:10)ng(cid:11)=(cid:10)bg†bg(cid:11)= e−(cid:126)ωjg/kBT , (49) + (cid:32)Sjα[κ,κ(cid:48)][Ω−1]αjkβYkl(cid:33)(cid:35) , (57)
j j j 1−e−(cid:126)ωjg/kBT Sjα[κ,κ(cid:48)][Ω−1]αjkβYkβl
themeanvibrationalnumberofmodebg. Theintegralin where
j
the variable λg is a Gaussian integral and the resulting (cid:88)
Y = A v −u , (58)
jl jk lk lj
visibility reads:
k
O(t)= Z2eiϕ˜eC ex√p(cid:8)14LTX−1L(cid:9). (50) Yj±l =±Yjle−i(ωje−ωlg)t. (59)
(cid:104)n1(cid:105)···(cid:104)n3N(cid:105) detΩdetX The matrix X in Eq. (50) is given by the following
expression,
ThisexpressionisvalidforanyinitialtemperatureT and
any number of ions, as long as the harmonic approxima- (cid:18)Xl1m1 Xl1m2(cid:19)=(cid:18) 0 0(cid:19)+(cid:18)Yl0m −12e−i(ωlg−ωmg)t(cid:19)
tion at the basis of our model is valid. In Eq. (50) we Xl2m1 Xl2m2 Tlm 0 −12 Yl0me+i(ωlg+ωmg)t
haacvoeminptarcotdfuocrmed.aTsheersieesqoufaqnutiatnietsitaierseignivoerndehrerteoipnroovrdideer +(cid:88)(cid:88)(cid:18)Yjl 0 (cid:19)(cid:32)[Ω−1]αjkβ [Ω−1]αjkβ(cid:33)(cid:18)Ykm 0 (cid:19),
to make the presentation self-consistent. αβ jk 0 Yjαl [Ω−1]αjkβ [Ω−1]αjkβ 0 Ykβm
The prefactors contain two exponentials, whose expo- (60)
nents take the form
with
ϕ˜=(ϕ[κ]−ϕ[κ(cid:48)])/2, 1(cid:88)
Y0 = v Y , (61)
lm 2 lj jm
and j
and the thermal excitation,
C =G(ζ)+G∗(ζ(cid:48))+(cid:88)3N (cid:88) Sjα[κ,κ(cid:48)][Ω−1]αjkβSkβ[κ,κ(cid:48)], T =δ (cid:104)ng(cid:105)−1 . (62)
4 lm lm l
j,k=1α,β=± The integration in λg is facilitated by changing to real
where andimaginarypartsofλgj =xj+iyj,therebyintroducing
(cid:18)Xxx Xxy(cid:19) (cid:18)1 1(cid:19)(cid:18)X11 X12(cid:19)(cid:18)1 i(cid:19)
ζj =κj +βje, ζj(cid:48) =κ(cid:48)j +βje. (51) Xlymx Xlymy = i −i Xl2m1 Yl2m2 1 −i (63)
lm lm lm lm
The vector L is conveniently decomposed into three and
parts, (cid:18)Lx(cid:19) (cid:18)L1+L2(cid:19)
j = j j . (64)
Ly L1−L2
L=I+J +K. (52) j j j
7
g -0.1 -0.005 0 0.02 the mechanical effect is relevant and analyse its effect
νy/(2π) (MHz) 1.470 1.545 1.549 1.565 over the visibility signal.
∆ 0.005 0.01 0.015 0.02 0.025 Before we start, some considerations on the choice of
νdip/(2π) (kHz 110 155 190 219 245 the parameter ∆ are in order. We first note that the
modelweconsider, acrystallinestructurewheretheions
TABLE I: Conversion table for the dimensionless quantities
perform harmonic vibrations about the equilibrium con-
to actual frequencies used for three ions with an axial trap
ditions, require that anharmonicities are not relevant for
frequencyofν =2π×1MHz. Thecriticalfrequencyisν =
x c
(cid:112) the dynamics we investigate. This sets in general an up-
12/5ν =2π×1.549MHz.
x
per bound to the choice of the quench’s amplitude ∆. In
addition, anharmonic corrections are naturally relevant
veryclosetothelinear-zigzaginstability[12],sothatthe
IV. RESULTS
initial and final state should be sufficiently distant from
the critical point. Hence, this sets a lower bound to ∆
We now analyze the visibility of the Ramsey fringes
when the quench is performed across the linear-zigzag
when the central ion is subject to a sequence of two
instability, such that the initial state is, say, a zigzag
Ramsey pulses in presence of a state-dependent poten-
structure and the excited state is a linear array. The pa-
tial. The results we present are obtained by evaluating
rameters we choose are chosen in accordance with these
explicitly the visibility in Eq. (50) for a given set of pa-
conditions.
rameters, assuming that the vibrations along the direc-
tion perpendicular to the plane of the zigzag are frozen
out,namely,themotioniseffectivelyconfinedtothex−y
A. Initial thermal excitation
plane. Wewillfocusonachaincomposedbythreeionsin
alineartrapwithaxialfrequencyν andtransversesecu-
x
lar frequency ν . In the following we will consider values We assume that the initial state of the crystal is a
y
of ν close to the critical value ν , separating the linear thermal state of the corresponding equilibrium structure
y c
fromthezigzagphase[12,20], andusethedimensionless at a given temperature T. Table II reports the mean
parameter vibrational number of each normal mode for the values
of g and T we consider in this section. In the following
ν2−ν2 wewillseethatonenormalmodewillbecomeimportant
g = y c (65)
ν2 in our discussion. For the crystal being in the linear
c
structure, this mode is (for the parameters considered)
in order to indicate whether the ions form a linear array the zigzag mode [8, 12]. Its frequency and its motional
(g > 0), or a zigzag chain (g < 0). The instability is pattern are displayed in Table II in the top row in the
at g = 0. The effect of the quench on the chain, due to upper block. The frequency of the zigzag mode goes to
the internal excitation of the central ion, is represented zero when approaching the linear-zigzag transition, and
by a shift of the trapping frequency that the central ion when the mode crosses the transition it becomes mixed
experiences,denotedbyν . Thestrengthofthequench with a second normal mode. The motional pattern is
dip
is here described by the dimensionless parameter displayed for two values in the zigzag in the top row of
the lower two blocks of Table II. For convenience we will
ν2 name in the following also this mode as the zigzag mode
∆= dip , (66)
ν2 when the crystal is below the transition. We also will
c
also use the term soft mode for this mode.
thatisheretakentobepositive,∆>0. Hence,whenthe ThecorrespondingvisibilityisdisplayedinFig.2when
centralionisintheexcitedstate,thetrappingpotentialit the strength of the quench is ∆ = 0.025. Panel (a) dis-
experiences is steeper. Table I reports the experimental playsthevisibilityatdifferenttemperaturesforg =0.02,
parameters corresponding to the values of g and ∆ we namely, when the initial and final state of the quench
consider in this section. correspond to excitations of a linear structure. The visi-
Theplotswepresentdisplaythevisibility,namely,the bilityforT =0isgivenbytheblacklineanditoscillates
absolute value of the overlap O(t) in Eq. (50), as a func- between unity and a value above 0.95. The oscillation
tion of the time t elapsed between the two pulses and is at the frequency of the zigzag eigenmode, which is ex-
of the temperature. The plots are evaluated for a chain citedbythequench[14]. Asthetemperatureisincreased
composedbythree9Be+ ions,atdifferentvaluesofg and thevisibilitydecays,itstillexhibitsamodulation,which
∆ and at different initial temperatures T of the chain. is markedly at a smaller frequency but at a larger am-
In the first part of this section we discard possible me- plitude than in the case at T = 0. The correspond-
chanical effects of the laser pulse; this situation can be ing maxima are at a time-scale which is independent
realised with suitably tailored excitation schemes, for in- of the temperature and exhibit a double-peak structure,
stancebytakingcopropragatinglaserbeamsinaRaman whichbecomesevidentatsufficientlylargetemperatures.
scheme[15],orbyusingradiofrequencyfields[27]. Inthe Panel(b)displaysthevisibilitywhenthechainisinitially
last part we then consider a pulsed excitation in which azigzagstructureandthequenchisperformedacrossthe
8
ωg/2π mode T(µK)
j (a) 1
(MHz) 5 10 50 100
0.8
g=0.02
01..20109010 [→[↓→↑↓→] ] 00..10309011 00..50307813 40..26722086 01..06129335 bility0.6
1.2033 [↓·↑] 0.0000 0.0031 0.4600 1.2794 Visi0.4
1.5646 [↑↑↑] 0.0000 0.0005 0.2866 0.8937 0.2
1.7321 [→·←] 0.0000 0.0002 0.2341 0.7715
2.4083 [→←→] 0.0000 0.0000 0.1100 0.4594 00 5 10 15 20
g=−0.005 t (µs)
0.1593 [⇓⇑⇓]+[→·←] 0.2974 0.9187 6.3014 13.0844 (b) 10.4
1.0000 [→→→] 0.0001 0.0083 0.6206 1.6235
1.1674 [⇓·⇑] [→←→] 0.0000 0.0037 0.4839 1.3313 0.8
+
0.2
112...573449572382 [[⇒⇒⇐·⇒⇐[↓↓]]↓++][[↑↑↓·↑↓]] 000...000000000000 000...000000000620 000...221921391579 000...974066914626 Visibility00..46 05.6 5.8 6 6.2 6.4
g=−0.1 0.2
0.6102 [⇓⇑⇓] [→·←] 0.0029 0.0565 1.2559 2.9391
+
0
0.8873 [⇓·⇑]+[→←→] 0.0002 0.0143 0.7443 1.8837 0 5 10 15 20
1.0000 [→→→] 0.0001 0.0083 0.6206 1.6235 t (µs)
1.4697 [↓↓↓] 0.0000 0.0009 0.3227 0.9760
(c) 1
1.9313 [⇒·⇐] [↑↓↑] 0.0000 0.0001 0.1857 0.6550
+
2.1425 [⇒⇐⇒]+[↑·↓] 0.0000 0.0000 0.1467 0.5567 0.8 0.4
TABLE II: Mean vibrational number for each normal mode bility0.6 0.2
of the different initial structures, determined by the choice Visi0.4 0 5.2 5.4 5.6
of g. The corresponding temperatures are given in µK. In
0.2
the second row, the vibrations of the ions for each mode are
sketched,andaredisplayedassumsoftheeigenmodesofthe 0
0 5 10 15 20
linear configuration. The modes of the zigzag chain are com-
t (µs)
posed of two normal modes of the linear chain, one of which
(denoted by the thicker arrows) has the main contribution.
FIG. 2: (Color online) Visibility signal as a function of the
time t elapsed between the two Ramsey pulses. The sig-
nal is evaluated for temperatures 0µK (black line), 5µK
linear-zigzag instability. The feature characterising the
(brown/dark gray line), 10µK (green/medium gray line),
behaviour at T =0 is the rapid decay of the visibility to
50µK (blue/medium light gray line) and 100µK (light
zero, and then the appearance of revivals the period of pink/light gray line). The parameters are ∆ = 0.025 and
the soft mode. This feature is independent of the num- (a) g = 0.02, (b) g = −0.005, (c) g = −0.1. The insets dis-
ber of ions [14]. Increasing the temperature leads to a play a zoom of (b) the first double-peak and of (c) the third
decrease of the amplitude of the revivals, as also visible peak.
in the inset: The amplitude significantly drops already
at T = 5µK. Panel (c) displays the visibility when the
quench connects two different zigzag structures. Also in
the corresponding peaks are broadened. Moreover, ad-
this case, at T =0 the visibility rapidly decays and then
ditional peaks appear that are located at the beat fre-
exhibits periodic revivals. Thermal effects lead to a de- quency ω =|ωe−ωg| between the zigzag eigenmodes
cay of the amplitude of the revivals. Here, however, the beat 1 1
of the two structures, namely, the equilibrium structure
signal is not significantly altered at temperatures as low
when the ion is in the ground state and the one in which
as T =10µK, as one can observe in the inset.
the ion is in the excited state. The appearance of a peak
These features can be better understood by analysing
at this beating frequency is due to the fact that the cor-
the spectrum of the signal. In particular, we choose to
respondingmodeintheinitialconfigurationisthermally
study the spectrum of the logarithmic visibility [14], de-
excited. Thenumberofpeaksincreaseswiththetemper-
fined by
ature; they appear at multiples of ω . This behaviour
beat
1 (cid:90) T shows that the eigenmodes which most relevantly con-
S (ω )= dt ln[V(t)]e−iωnt. (67) tribute to the overlap integral, and thus to the visibility,
ln n T
0 are the soft modes of the initial and excited structures,
while the contribution of the other modes is marginal.
Figure 3 displays the spectra of the logarithmic visibil-
Note that the soft modes are, for the parameters here
ity corresponding to the curves in Fig. 2. Let us first
considered, the ones which are at lowest frequency and
recall the behaviour of the spectra at T = 0. These
significantly occupied, as one can see from Table II.
exhibit well defined peaks at the frequency, or at mul-
tiples, of the soft mode. As the temperature increases We now compare the signals obtained for different
9
100 (a) 1
(a) 100
0.8
10−2 bility0.6
ω) 0 0.05 0.1 0.15 0.2 Visi0.4
S(ln10−4 0.2
0
0 10 20 30 40 50 60
t (µs)
10−6
(b) 1
1
0 1 2 3 4 0.8
Frequency (MHz)
(b) 102 102 Visibility00..46 0.05
101 100 0.2 0 1 2 3 4
10−2 0
ω) 100 0 0.05 0.1 0.15 0.2 0 10 20 t (3µ0s) 40 50 60
(n
Sl
10−1 FIG. 4: (Color online) Visibility as a function of the time t
elapsed between the two Ramsey pulses for T =100µK and
10−2 ∆=0.005 (light pink), ∆=0.010 (blue), ∆=0.015 (green),
∆ = 0.020 (brown), ∆ = 0.025 (black line). In a scale of
gray, as ∆ increases the line becomes darker. The quench is
0 1 2 3 4
Frequency (MHz) performed for (a) g = 0.02 and (b) g = −0.1, such that the
initial and quenched equilibrium strcutures are either both
(c) 102 linear or zigzag chains.
100
100
ated timescale, which is the period of the beating, corre-
0 0.05 0.1 0.15 0.2
ω) spondingly longer. Panel (b) shows the behaviour when
S(ln10−2 the initial and the quenched structures are both zigzag.
Here, the visibility rapidly decays to zero, and then ex-
hibits some revivals whose height also damps down to
10−4 zero. This latter decay is slower for weaker quenches,
i.e., for smaller values of ∆.
0 1 2 3 4 The signal at ∆=0.005 is singled out in Fig. 5, where
Frequency (MHz)
it is plotted for a larger interval of elapsed times t. In
FIG. 3: (Color online) Spectra for the logarithmic signals, Fig. 5a, where the initial and quenched structures are
Eq.(67),forthevisibilitycurvesinFig.2. Theverticaldash- linear, the signal shows a slow modulation and a certain
dotted(green)linesshowthezigzageigenfrequencyωe (in(a) regularity. In Fig. 5b, where both structures are zigzag,
1
the line is at 2ωe). The dashed (orange) line shows the lo- themainpeaksofthesignalarelessregularlydistributed
1
cation of frequency ωbeat = |ω1e −ω1g|. The insets display and exhibit a fast quasi-periodic modulation (see inset).
a zoom of the low frequency part of the corresponding spec- In both situations one observes that at large times the
trum,highlightingthepeakstructureofthespectrumatmul-
visibility can be significantly above zero, showing that
tiples of ω .
beat coherence persists over time scales of the order of mil-
liseconds.
Therelevanttimescalesassociatedwiththesefeatures
strength ∆ of the quench at finite temperature. Fig- become evident by studying the spectrum of the loga-
ure 4 displays the visibility as a function of the elapsed rithm of the visibility. Figure 6a displays the spectrum
time t evaluated for different values of ∆ and when the corresponding to the signal in Fig. 5a. The additional
chain is initially at temperature T=100µK. The case in curves,fromtoptobottom,correspondtodecreasingval-
whichthetwostructuresarelinearisshowninpanel(a). ues of the temperature. The black solid line reports the
Here, the peaks arising from initial thermal occupation case T = 0, which is here plotted for comparison. This
wander to later times for smaller values of ∆. This curvedisplaysclearpeaksatvaluesofthefrequencycor-
can be understood by recalling that these peaks are de- responding to normal modes or to sum or difference of
termined by the beating between the zigzag modes of normal mode frequencies. The highest peak here corre-
the initial and of the quenched structure: For weaker sponds to the frequency difference |ωe −ωg|, which for
4 4
quenches, the difference becomes smaller and the associ- thelinearchainwhenallionsareinthegroundstatecor-
10
(a) 1 100
(a)
0.8 100
bility0.6 10−2
Visi00..24 ωS()ln10−4 0 0.05 0.1 0.15 0.2
0
0 200 400t (µs)600 800 1000 10−6
(b) 1
0.8 0.04 10−80 1 2 3 4
Frequency (MHz)
bility0.6 0.02 100
Visi0.4 080 100 120 140 160 (b) 100
0.2 10−1
0
0 200 400 600 800 1000 0 0.05 0.1 0.15 0.2
t (µs) ω()n10−2
Sl
FIG. 5: (Color online) Visibility as a function of the elapsed
time t for 100µK, ∆ = 0.005, and (a) g = 0.02 and (b) g = 10−3
−0.1. Theverticallinesindicatethelocationofω (dashed
beat
orange) and of the beating frequency between the transverse
center-of-mass frequencies (dash-dotted green). The inset in 10−4
0 1 2 3 4
(b) shows a zoom for the interval centered about ω . Frequency (MHz)
beat
FIG. 6: (Color online) Spectrum of the logarithm of the
responds to center-of-mass oscillations in the transverse visibility, Eq. (67), for ∆ = 0.005 and (a) g = 0.02 (lin-
ear to linear) and (b) g = −0.1 (zigzag to zigzag). The
direction. This peak is still present at finite tempera-
curves correspond to the temperatures T = 0 (black line),
tures but becomes less prominent. On the other hand,
T = 5µK (brown), T = 10µK (green), T = 50µK (blue)
atfinitetemperaturesoneobservestheappearanceofthe
andT =100µK(lightpink),correspondingtoascaleofgrey
peakatω . Moreover,resonancesatmultiplesofω
beat beat from dark to light. The inset is a zoom in the low-frequency
appearandtheirnumberincreaseswiththetemperature,
part. Theverticaldash-dotted(green)lineshowsthelocation
as is evident by inspecting the inset. of the zigzag eigenfrequency in panel (b) and of double the
The spectrum in Fig. 6b corresponds to the case in zigzag frequency in panel (a). The vertical dashed (orange)
which the quench connects two zigzag configurations. line shows the location of ω .
beat
Here,oneobservesthatthepeakscharacterizingthespec-
trum at T = 0 correspond to the frequencies of the nor-
mal modes; they are also present at finite temperatures, time when the zigzag mode has been cooled to 10µK
eventhoughtheybecomebroader. AtfiniteT apeakap- while the other modes are at T = 100µK (see the blue
pears at ωbeat, while the number of harmonics increases line). Bycomparingthisbehaviourwiththevisibilityfor
with T, as visible in the inset. the chain in the thermal state at T = 10µK and 100µK
we observe in (a) and (b) that for elapsed times of the
order of tens of microseconds the visibility qualitatively
B. Discussion reproduces the behaviour found by cooling all modes at
10µK. Thus, for quenches connecting two linear struc-
Theevaluatedvisibilityshowsthatfinitetemperatures tures (case a) or connecting a linear and a zigzag struc-
lead to the appearance of various features, which emerge ture(caseb)theinitialexcitationofthezigzagmodede-
because of coherence between the two states created by terminesthevisibilitybehaviouruptotimesoftheorder
the quench. The fact that the initial state is a statistical of10µs. Thisisalsoconfirmedwhencomparingwiththe
mixture leads to an overall decrease of the entanglement opposite case, in which the whole chain has been cooled
created by means of the quench. In particular, already to 10µK except for the zigzag mode, whose vibrational
at T = 100µK several features of the behaviour at zero excitations follow a thermal distribution corresponding
temperature have disappeared. to 100µK. The visibility in this case behaves similarly
One signature of thermal excitation is the appearance to the one where all modes are at T = 100µK. A dif-
ofpeaksattheharmonicoffrequencyω . Thesecanbe ferent situation is found when the quench connects two
beat
suppressed by cooling the zigzag mode, which is majorly zigzag structures and is displayed in panel (c). Here, all
excited by the quench, to a lower temperature. Figure 7 eigenfrequencies contribute in determining the dynamics
displays the visibility signal as a function of the elapsed at low temperatures.