Landau-Majorana-Stuckelberg-Zener dynamics driven by coupling for two interacting qutrit systems
aa r X i v : . [ qu a n t - ph ] J un Landau-Majorana-St ¨uckelberg-Zener dynamics driven by coupling for two interacting qutritsystems
R. Grimaudo,
N. V. Vitanov, and A. Messina
2, 4 Dipartimento di Fisica e Chimica dell’Universit`a di Palermo, Via Archirafi, 36, I-90123 Palermo, Italy INFN, Sezione Catania, I-95123 Catania, Italy Department of Physics, St. Kliment Ohridski University of Sofia, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria Dipartimento di Matematica ed Informatica dell’Universit`a di Palermo, Via Archirafi, 34, I-90123 Palermo, Italy (Dated: June 11, 2019)A time-dependent two interacting spin-qutrit model is analysed and solved. The two interacting qutritsare subjected to a longitudinal field linearly varying over time as in the Landau-Majorana-St¨uckelberg-Zener(LMSZ) scenario. Although a transverse field is absent, we show the occurrence of LMSZ transitions assistedby the coupling between the two spin-qutrits. Such a physical effects permits to estimate experimentally thecoupling strength between the spins and allows the generation of entangled states of the two qutrits by appro-priately setting the slope of the ramp. Furthermore, the possibility of local and non-local control as well as theexistence of dark states of the two qutrits have been brought to light. Effects stemming from a noisy surround-ing environment are also taken into account by introducing a random fluctuating field component as well asnon-Hermitian terms in the Hamiltonian model.
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I. INTRODUCTION
Spin chains are the reference experimental scenario forquantum technology applications thanks to the possibility ofentanglement generation also over long distances . Entan-glement, indeed, is the key resource for quantum informationtasks and its manipulation by field application is of courseof fundamental importance.In this context, a growing interest in qutrits - three-statequantum systems - should be emphasized. Besides the obvi-ous exponential increase of their Hilbert space, qutrits, andqudits in general, offer several advantages over qubits. Forexample, among the most important applications of qutrit sys-tems we find: optimization of the Hilbert space dimensional-ity vs. control complexity , larger violations of nonlocality ,new types of quantum protocols and entanglement , moresecure quantum communication , Bell inequalities resis-tant to noise . Moreover, efficient protocols and methodshave been developed for the manipulation of qutrits andqudits .In this respect the possibility of realizing a local applicationof fields on a single qudit while it interacts with other ones isof basic interest to generate physical effects in the spin chainby manipulating the single spin dynamics. Through the Scan-ning Tunneling Microscopy (STM), for example, it is possibleto construct atom by atom a chain of interacting nanomagnetsand to manipulate the state of a single spin by applying a lo-cal magnetic field on atomic scale with a STM tip . Moreprecisely, the field created on the single spin is an effectivemagnetic field stemming from the tunable exchange interac-tion between the target spin we wish to manipulate and thespin present on the STM tip . Such an effective field maybe also time-dependent thanks to the possibility of varying thedistance between the tip spin and the one in the chain . It ispossible, for example, to create a field varying linearly in timeand changing its direction , as in the well known Landau-Majorana-St¨uckelberg-Zener (LMSZ) scenario . Thus, STM makes experimentally possible, by atomic manipulation, tocontrol the quantum state and the quantum dynamics of a sin-gle spin while the latter is interacting with other neighbouringspins and to generate, then, delocalized effects by local fieldapplication.The LMSZ scenario is one of the most famous and im-portant exactly solvable time-dependent single-spin modelsthanks to the fact that, though its unphysical nature (infinitetime duration of the physical process, implying divergence ofthe instantaneous energy separation as time goes on), it fur-nishes accurate predictions also for more realistic situations(finite times). However, the exact solutions of the LMSZ dy-namical problem exist and may be given in terms of paraboliccilinder functions . Its popularity is confirmed also by a lotof studies, both theoretical and experimental, which have beendeveloped aiming at generalizing the LMSZ scenario consid-ering N -level systems , total crossing of bare energies and the presence of classical and quantum noise stemmingfrom sources of incoherences : incoherent (mixed) states,relaxation processes (e.g., spontaneous emission) or inter-action with a surrounding environment (e.g., nuclear spinbath). Moreover, recently, the attention has been focusedon double interacting spin-qubit systems subjected to LMSZscenario with the scope of identifying the signatures ofthe coupling in the two-spin dynamics and their potentialityfor possible future applications .In this paper we analyse, instead, the quantum dynamics oftwo interacting qutrits subjected to a LMSZ ramp along thequantization axis (ˆ z ). Both the manipulation of the quantumdynamics of a single high spin-value magnetic atom or molec-ular magnet and the control of the interaction between qu-dits in a chain offer, indeed, the experimental backgroundin which such systems turn out to be actual powerful buildingblocks for quantum information and computation tasks.The physical interest of the work is twofold. Firstly, webring to light the existence of a physical effect consisting inthe possibility of generating LMSZ transitions in the two-qutrit system, though a transverse constant field is absent.This fact results possible thanks to the coupling existing be-tween the two spin-1’s which plays the role of an effectivetransverse field making possible avoided crossings and conse-quent LMSZ transitions of the two qutrit system. Secondly,we show how such an effect may be exploited for two relevantapplications: the estimation of the strength of the coupling pa-rameters and the possibility of generating asymptotically en-tangled states of the two qutrits by appropriately setting theslope of the ramp. Our symmetry-based analysis of the Hamil-tonian model, usefully used in several problems , allowsus to consider also effects stemming from the presence of anoisy field component.The structure of the paper is the following. The modeland the symmetry-based dynamical reduction are presentedin Sec. II. In Sec. III and IV the quantum dynamics of thetwo qutrits is investigated in the four- and five-dimensionaldynamically invariant subspace, respectively. In both sectionswe report the formal general solution of the dynamical prob-lem and the LMSZ transition probabilities when a linearlyvarying ramp is applied on just one spin as well as on boththe spins. Basing on such a result, we show the possibility oflocal a non-local control of the dynamics of one of the twoqutrits in the chain as well as physical effects related to theanisotropy of the coupling. We bring to light moreover theexistence of dark states, that is, not evolving states indepen-dently of the time-dependence of the applied fields. Finallywe discuss the modification of the LMSZ probabilities whena random fluctuating field component is present. In Sec. V thestudy of the Negativity as measure of Entanglement betweenthe two qutrits is developed and the possibility of generatingentangled states of the two spin-1’s through a LMSZ processis analysed. Finally, conclusive remarks and perspectives maybe found in the last section VI. II. THE MODEL
Let us consider the following model of two interactingqutrits subjected to local time-dependent fields H = ¯ h ω ˆ Σ z + ¯ h ω ˆ Σ z + γ x ˆ Σ x ˆ Σ x + γ y ˆ Σ y ˆ Σ y + γ z ˆ Σ z ˆ Σ z (1)where ω i ( i = ,
2) are the characteristic frequencies of the twoqutrits and γ s are the different energy contributions stemmingfrom the coupling between the two three-level systems. ThePauli operators ˆ Σ ki ( k = x , y , z ) for a spin-1 system are relatedwith the spin-1 operator components asˆ S xi = ¯ h √ Σ xi , ˆ S yi = ¯ h √ Σ yi , ˆ S zi = ¯ h ˆ Σ zi . (2)Our scope is to study a Landau-Majorana-St¨uckelberg-Zener(LMSZ) scenario for the two qutrits and analyse how the cou-pling between them and a noisy component of the magneticfield affect their dynamics.In Ref. it was shown that two dynamically invariantHilbert subspaces exist: one of dimension four spanned by {| i , | i , | − i , |− i} and the other one of dimension five spanned by {| i , | − i , | i , |− i , |− − i} . Theyare related to the two eigenvalues ( ±
1) of the constant of mo-tion ˆ K = cos ( π ˆ Σ z tot ) , (3)where ˆ Σ z tot = ˆ Σ z + ˆ Σ z is the total spin of the composed systemalong the z direction. It is worth to emphasize, at this point,that the Hamiltonian model keeps its symmetry also for twolarger spin systems, that is, for two interacting spins ˆJ and ˆJ . In such a case, it is always possible to decompose thedynamical problem into two sub-problems related to the twodynamically invariant subspaces linked to the two eigenvalues(1 and −
1) of the constant of motion cos [ π ( ˆ J z + ˆ J z )] . How-ever, for larger spin systems, the sub-dynamics could be verydifficult to solve due to the high degeneracy of both eigenval-ues.In this respect, in an important property of the two-qutritsystem was discovered, which is of basic importance for ouranalysis and to get exact solutions of the dynamical prob-lem. The Hamiltonian governing the two-qutrit dynamics inthe four-dimensional subspace may be written in terms of two non interacting qubits as follows H − = H ⊗ ˆ + ˆ ⊗ H , (4)with H = ¯ h Ω + σ z + γ − ˆ σ x , H = ¯ h Ω − σ z + γ + ˆ σ x (5)where ˆ σ k ( k = x , y , z ) are the standard Pauli matrices and weset Ω ± = ω ± ω and γ ± = γ x ± γ y . The mapping at the basisof such a rewriting is | i ↔ | ++ i , | i ↔ | + −i , | − i ↔ |− + i , |− i ↔ |−−i . (6)The Hamiltonian governing the five dimensional subspace,instead, under the following conditions γ z = , γ x = γ y = γ / , (7)is reduced to the following block-diagonal form H + = ¯ h Ω + h Ω − γ γ γ
00 0 γ − ¯ h Ω −
00 0 0 0 − ¯ h Ω + . (8)The three-dimensional middle block possesses an su(2) struc-ture and hence can be written in terms of spin variables of afictitious spin-1, namely H = γ ˆ Σ x + ¯ h Ω − ˆ Σ z . (9)We emphasize that the choice γ z = H and H in Eq.(5) do not depend on γ z .Now, we want to study the two interacting qutrits when theyare subjected to time-dependent fields, ω ( t ) and ω ( t ) . Tothis end we stress that the results and the analysis reportedbefore in Ref. are still valid also when we consider time-dependent fields and, more generally, when all the Hamilto-nian parameters depend on time. This is due to the fact thatthe Hamiltonian structurally commutes with the constants ofmotion independently of its time-dependence. In the follow-ing we show that we are able to construct formally the timeevolution operator for both four- and five-state subdynamics.In particular, we analyse the case in which the z -magneticfield is a ramp as in the LMSZ scenario. We are interestedin revealing intriguing dynamical effects stemming from thehomogeneity or heterogeneity of both the coupling parame-ters and the two fields. In addition, we want to exploit oursymmetry-based approach to take into account the influenceof a surrounding environment by considering a random fluc-tuating field component. III. FOUR-DIMENSIONAL SUBDYNAMICSA. General Solution
We may formally write the time evolution operator U j ( j = ,
2) related to H j , solution of the Schr¨odinger equation i ¯ h ˙ U j = H j U j , as follows U j = (cid:18) a j b j − b ∗ j a ∗ j (cid:19) , (10)where a j and b j are time-dependent Cayley-Klein parameterssatisfying | a j | + | b j | =
1. The time evolution operator U − ,satisfying the Schr¨odinger equation i ¯ h ˙ U − = H − U − , then reads U − = U ⊗ U = a a a b b a b b − a b ∗ a a ∗ − b b ∗ b a ∗ − b ∗ a − b ∗ b a ∗ a a ∗ b b ∗ b ∗ − b ∗ a ∗ − a ∗ b ∗ a ∗ a ∗ . (11)The mathematical expressions of a j ( t ) and b j ( t ) depend onthe time-dependence of the two local magnetic fields ω ( t ) and ω ( t ) . B. STM Scenario
1. Local Dynamics
We firstly analyse the case of a single local z -magnetic field B z ( t ) applied on the first spin consisting in a LMSZ ramp,such that ¯ h ω ( t ) = α t , t ∈ ( − ∞ , ∞ ) , (12)where α is considered a positive real number and rules the adi-abaticity of the process since ˙ B z ∝ α . Let us consider the case of an excitation present in the system and localized in one ofthe two qutrits, say the second spin; in this case the initial stateof the two qutrits (fictitious qubits) is |− i ( |−−i ). In thisinstance, each fictitious spin-1/2 is subjected to a LMSZ sce-nario with ω ( t ) as longitudinal magnetic field and a constant(effective) transverse magnetic field determined by the cou-pling parameters [see Eq. (5)]. In this way, the first and secondfictitious spin-1/2 have the probability to make the transitionto the up-state, respectively P = − exp {− πβ − } , (13)and P = − exp {− πβ + } , (14)with β ± = γ ± / ¯ h α . Thus, the joint probability for the two ficti-tious spin-1/2’s to be found in the state | ++ i , | + −i and |− + i ,starting from |−−i , are respectively P P , P ( − P ) , ( − P ) P , (15)being nothing but the probability of finding the two qutrits inthe state | i , | i and | − i , respectively. We know thatin the standard LMSZ scenario applied on a single spin-qubit,the transverse field couples the two levels and is then respon-sible of the avoided crossing. It is worth noticing that, in ourcase, the transverse field role is played by the coupling exist-ing between the two qutrits, as it is clear by the two Hamil-tonians in Eq. (5). Hence, we may reproduce adiabatic con-ditions by appropriately setting the ratio between the longi-tudinal fields and the coupling parameters in order to have afull LMSZ transition of the two fictitious spin-1/2’s. The threeprobabilities in Eq. (15) are reported in Fig. 1 against the pa-rameter β = β + for β + / β − =
2. In this case we are realizinga local control of the dynamics of the first qutrit, leaving theother one unaltered. For a complete LMSZ transition, indeed,the first qutrit accomplishes the LMSZ transition |− i → | i ,while the second qutrit’s state does not change.Analogously, we may consider the excitation initially lo-calized in the first spin-1, so that the two qutrits start from thestate | − i . In this instance the two-qutrit system is asymp-totically driven to the state | i and the probability of the re-lated transition acquires the same expression as the previousone in Eq. (15). It is worth noticing that in this case we gener-ate a LMSZ transition from |− i to | i in the second spin, byapplying a local magnetic field only on the first qutrit which,instead, remains in its initial state. Such a circumstance, thus,may be identified as the achievement of a non-local control ofthe second qutrit.
2. State Transfer between the Qutrits
Another interesting effect to be highlighted is the possibil-ity of realizing a state transfer between the two qutrits. Indeed,if the two qutrits (fictitious qubits) are initialized in the state |− i ( |−−i ) and we assume γ x = γ y , the transition probabil-ity of the first fictitious spin-1/2 is forbidden, while the second Β Figure 1: (Color online) a) Asymptotic LMSZ probabilities[Eq. (15)] of finding the two qutrits in the state | i (bluedotted line), | i (magenta dot-dashed line), | − i (reddashed line) and |− i (green full line), when they start fromthe state |− i for γ x = γ y , β = β + and β + / β − = | + i with probability P = P . In this way, the twoqutrits (fictitious qubits) reach the state | − i ( |− + i ) havinginterchanged their initial state. The same effect is present ifthe two qutrits are initially prepared in | i passing to | i .In such a case, the transitions between the states of the twoqutrit system in the four dimensional subspace are differentsince the condition γ x = γ y introduces a further symmetry inthe model related to the commutation of H with ˆ Σ z tot . Thisfact generates, in the subspace under scrutiny, the existenceof other two dynamically invariant subspaces related to theeigenvalues of ˆ Σ z tot . It is easy to verify that, this time the twoqutrits starting from |− i ( | i ) can be asymptotically foundonly in the state | − i ( | i ).At the light of the STM scenario, the physical effects pre-viously discussed and analytically derived are of relevant in-terest. They show, indeed, that the presence of the couplingbetween the two qutrits allows us to manipulate the dynamicsof the whole two-qutrits chain by the application of a singlelocal magnetic field on one of the two spins, being exactly oneof the task of the application of the STM technique. More-over, the previous examples brought to light that, by studyingthe kind of transitions occurring in the two-qutrit system, wemay get information about the coupling parameters determin-ing the symmetries of the Hamiltonian.
3. Effects of Environment
We wish to show now that the mapping of the two-qutritdynamics into that of two decoupled spin-1/2’s in the four-dimensional subspace is useful not only to solve exactly theproblem in ideal conditions, but also to take into account pos-sible external influences due to the action of a surroundingenvironment, such as nuclear spin bath. In Ref. , for ex-ample, it is experimentally demonstrated that decoherence ef-fects in the dynamics of a NV center in diamond (consist-ing in a three-level system), subjected to a LSZ interferom-eter, comes from the dipolar interaction of the system withthe surrounding C nuclear spins random fluctuating at roomtemperature. Such external influences may be theoreticallyregarded, for example, as noise in the magnetic field com- ponent. In Ref. the authors study the dynamics of a spin S subjected to a noisy LMSZ scenario. The noisy time-dependent magnetic field η ( t ) is considered only in the z di-rection and characterized by a time correlation function of theform h η ( t ) η ( t ′ ) i = Γδ ( t − t ′ ) . Reference experimentallylegitimates such an assumption; in that case, indeed, the au-thors shows as the transverse fluctuations can be neglected. Insuch a way the noisy component cannot generates transitionsbetween the different states but it leads only to loss of coher-ence. In Ref. , the authors show how the LMSZ transitionprobability is affected by the presence of such a noisy mag-netic field in the case of a spin-1/2, a spin-1 and a spin-3/2.For a spin-1/2 and for large values of Γ we have asymptoti-cally P + − = − exp {− π g / ¯ h α } , (16)where g is the energy contribution due to the coupling of thespin-1/2 with the constant transverse magnetic field. We seethat the transition probability does not depend on the specificvalue of Γ , provided that Γ is large. Moreover, it is importantto note that the effect of the noise is to hinder the transition.Indeed, in the most convenient case, that is for g / ¯ h α ≫ γ x = γ y , the probability in Eq.(15) becomes P /
4, reasonably meaning that, under the ef-fect of noise, we reach an equally populated condition of thefour states involved in the subdynamics under scrutiny. Anal-ogously, if γ x = γ y , had the two qutrits started form |− i weget the probability P / | − i , reach-ing this time an equally populated condition between thesetwo states.Such observation is based on the fact that, adding the noisycomponent η ( t ) to the field applied to the first qutrit, nothingchanges in the dynamics-decoupling procedure. The Hamil-tonian transformation is completely unaffected since the onlydifference consists in a redefinition of the longitudinal field.In this way, what we obtain is an effective z -field for the twofictitious spin-1/2’s supplemented by a random field compo-nent. Thus, also in this case, we may reduce the two-qutritdynamical problem into the analysis of the quantum dynam-ics of two decoupled spin-1/2’s.In this respect, it is worth pointing out that the argumentpreviously exposed continues to be valid also when we con-sider the possibility that the exited states | i and | i of the twoqutrits decay irreversibly out of the system by some mech-anism. Let us suppose that the spontaneous emission fromthe exited states to the ground one is negligible and that thetwo decay rates for the state | i and | i are ˜ Γ ( ˜ Γ ′ ) and 2 ˜ Γ (2 ˜ Γ ′ ), respectively, for the first (second) qutrit. It is easy tosee that the analysis of such a scenario is equivalent, up to adda constant imaginary term, to phenomenologically introducethe non-Hermitian terms i ˜ Γ ˆ Σ z and i ˜ Γ ′ ˆ Σ z in our Hamiltonianmodel. Also this time we have a simple redefinition of the pa-rameters in front of the operators ˆ Σ z and ˆ Σ z without alteringthe symmetries possessed by the Hamiltonian H . Therefore,in such a case, within the four-dimensional subspace the two-qutrit dynamics may be described in terms of two decoupledtwo-level systems subjected to effective external fields andcharacterized by decaying states. Several results have beenreported for a single qubit with a decaying state subjected tothe LMSZ scenario . Precisely, it has been proved that,on the one hand, in the standard (ideal) LMSZ scenario, thedecay rate influences only its the time-history of the transitionprobality but not its asymptotic value ; on the other hand,in the more realistic LMSZ scenario characterized by a lim-ited time-window, the exited state population exhibits a de-pendence on the decay rate . We emphasize that even suchresults allow to make quantitative predictions on the LMSZtransition probabilities for the system under scrutiny. C. Local Fields
Now, we want to discuss the possibility of applying localfields on both the qutrits. Let us consider, firstly, the case ω ( t ) = ω ( t ) = α t / , (17)with t going from − ∞ to + ∞ .In this case, the Hamiltonians of the two fictitious spin-1/2’s, through which we describe effectively the dynamics ofthe two qutrits in the four dimensional subspace, read H = ¯ h Ω + ( t ) ˆ σ z + γ − ˆ σ x , H = γ + ˆ σ x , (18)with Ω + ( t ) = α t . We see that the second fictitious spin-1/2is subjected only to a magnetic field in the x -direction, whilethe first one is subjected to standard Landau-Zener scenario.As before, the role of the external transverse constant field iseffectively played by the coupling existing between the twospins.
1. Determination of γ s We study now the instance in which only one excitation ispresent in the system, equally shared by the two qutrits. Weconsider, then, the entangled state ( |− i + | − i ) / √ |−i ⊗ | + i + |−i√ . (19)It is easy to see that the second spin does not change itsstate in time since the latter is an eigenvalue of H . The firstspin, instead, evolves according to the LMSZ dynamics, sothat the probability to find it in the opposite state | + i at verylarge time instants ( t → ∞ ) is P . Of course, it expresses toothe probability of the two spin-1/2’s to be found in the state | + i ⊗ | + i + |−i√ . The relevant point is that, in view of Eq. (6),it provides the probability for the two qutrits of reaching thestate | i + | i√ . (20)Thus, if β − ≫
1, through the linear ramp we have createdan excitation in the system. It is important to underline thatsuch a transition depends strongly on the coupling parametersbetween the two qutrits, since their difference constitute theeffective transverse magnetic field entering in the expressionof the LMSZ parameter β − . Indeed, if the two parametersare equal or very close, the transition is forbidden, while, ifthey are opposite, the transition probability reaches its maxi-mum efficiency. This suggests us that, choosing at will α andstudying the characteristic time of the transition, we may getinformation about the value of γ − .If we now consider ω ( t ) = − ω ( t ) = α t / ( |− i + | − i ) / √
2, we get a specular dynamics. Thatis, the first fictitious spin-1/2, subjected only to a static x -magnetic field ( H = γ − ˆ σ x ), does not evolve, while thesecond fictitious spin-1/2 makes a transition from |−i to | + i (being H = ¯ h α t ˆ σ z + γ + ˆ σ x ). Studying such a transition,this time, we get information about γ + since it rules thecharacteristic time of such a transition. Finally, by comparingthe two values of γ + and γ − we may estimate the originalcoupling parameters of the two qutrits γ x and γ y .
2. Dark States
We emphasize that, under the conditions γ x = γ y = γ / ω ( t ) = ω ( t ) = ω ( t ) / | ψ / i = | i ± | i√ , | ψ / i = |− i ± | − i√ ( | + i ± |−i ) / √
2] of its constant Hamiltonian H = γ ˆ σ x and evolves trivially, only acquiring the phase factorexp {− i γ t / ¯ h } ; the first fictitious spin-1/2, instead, (being in thestate |±i ) keeps only the phase factor exp {− i R t ω ( t ) dt } sinceits Hamiltonian H = ¯ h ω ( t ) ˆ σ z does not mix the two standardbasis states. This means that for these four states we have( j = . . . H ( t ) | ψ j i = E j ( t ) | ψ j i , E / ( t ) = ω ( t ) ± γ , E / ( t ) = − E / ( t ) (23)implying | ψ j ( t ) i = exp (cid:26) − i Z t dt ′ E ( t ′ ) / ¯ h (cid:27) | ψ j i . (24)It is easy to see that, considering the time-independent case,such states result to be the eigenstates of the Hamiltonian .So, this model, in this specific case, presents a peculiar char-acteristic consisting in maintaining its steady states also whenthe Hamiltonian parameters are time-dependent. A remark-able consequence of this circumstance is that the followingclass of states ρ = ∑ j p j | ψ j ih ψ j | ( ∑ j p j = p j = exp {− E j / k B T } , k B and T beingthe Boltzman constant and the Temperature, respectively), donot evolve in time, that is ρ ( t ) = ∑ j p j | ψ j ( t ) ih ψ j ( t ) | = ∑ j p j | ψ j ih ψ j | = ρ . (25)Therefore, any physical observable calculated for such classof states exhibit a constant value in time. We can call suchstates ‘dark states’ since, under the conditions written before,they are unaffected by both the coupling and the longitudinaltime-dependent field, also when the latter presents a randomfluctuating behaviour.Analogously, if we have γ x = − γ y and ω ( t ) = − ω ( t ) thefour dark states are | i ± | − i√ , | i ± |− i√ . (26)Finally, we emphasize that the previous results are not re-stricted to the LMSZ scenario, but they are valid whatever thetime-dependence of the field is. IV. FIVE-DIMENSIONAL SUBDYNAMICSA. General Solution
In the second section we saw that the central block of H + in Eq. (8) has an su(2) structure and then it is interpretable asthe Hamiltonian of a (fictitious) spin-1 subjected to (fictitiousas well) magnetic fields (see Eq. (9)). It is well known thatthe time evolution operator related to a 3x3 su(2) Hamiltonianmay be put in the following form U = a √ a b b −√ a b ∗ | a | − | b | √ a ∗ b b ∗ −√ a ∗ b ∗ a ∗ , (27)where a and b are two time-dependent parameters, solutionof the analogous dynamical problem for a single spin-1/2. Inother words, a and b may be found by solving the dynam-ical problem of a single spin-1/2 subjected to the same mag-netic field acting upon the fictitious spin-1. Thus, we mayformally write the time evolution operator U + , solution of theSchr¨odinger equation i ¯ h ˙ U + = H + U + , as follows U + = e − i ¯ h R Ω + a √ a b b −√ a b ∗ | a | − | b | √ a ∗ b b ∗ −√ a ∗ b ∗ a ∗
00 0 0 0 e i ¯ h R Ω + . (28) B. Dark States
First of all, it is important to underline that also for the five-dimensional subdynamics we have dark states. Indeed, if thetwo qutrits are initially prepared in | i or |− − i , inde-pendently of the time-dependence of the z -magnetic field, thetwo-qutrit system remains in its initial state, also if the mag-netic field component randomly fluctuates remaining alongthe z -direction. Moreover, if we consider the case ω ( t ) = ω ( t ) , also a generic state belonging to the three-dimensionalsubspace, namely c | − i + c | i + c |− i , (29)is completely unaffected by the presence of time-dependentmagnetic fields, since in this instance Ω − ( t ) = H = γ ˆ σ x . Such states, then, evolves only under theaction of the coupling between the two qutrits. It means thenthat the three eigenstates of ˆ Σ x rewritten in terms of two-qutritstates | ψ i = | − i + √ | i + |− i , | ψ i = | − i − |− i√ | ψ i = | − i − √ | i + |− i ω ( t ) = ω ( t ) and γ x = γ y ), we may conclude that, in this scenario, the ther-mal state of the system and, more in general, every mixtureinvolving the steady states | i , |− − i and the ones in Eqs.(25) and (30), namely ρ = k | ih | + ∑ j = p j | ψ j ih ψ j | + k |− − ih− − | , (31)such that k + k + ∑ j p j =
1, is a stationary state of the two-qutrit system.
C. STM Scenario and LMSZ Transition Probabilities
We investigate now the STM experimental scenario char-acterized by a single local magnetic field on the first spin-1,namely ω ( t ) = α t , and the two qutrits initialized in the state | − i . In this case the two-qutrit system behaves effectivelylike a three-level system (spin-1) subjected to a LMSZ rampwith an effective constant transverse magnetic field related tothe coupling constant γ . For such a time-dependent scenario,the transition probabilities, from | − i to the other two states | i and |− i , may be found analytically. Indeed, at thelight of the spin-1 - spin-1/2 transition probability relationshipbased on the SU(2) group structure, for large time instants, wehave P + − = P , P − = P ( − P ) , P − − = ( − P ) , (32)where P = ( − e − πβ ′ ) and β ′ = γ / ¯ h α . Also in this case,we appreciate how the coupling between the two qutrits is re-sponsible of an avoided crossing and a consequent full adia-batic LMSZ transition for the fictitious spin-1. In the previ-ous expressions we have labelled with -1, 0 and 1 the states | − i , | i and |− i , respectively. The plots of the asymp-totic probabilities are reported in Fig. 2 against the coupling-dependent LMSZ parameter β ′ . We see that the interplay be- Β '0.20.40.60.81.0P Figure 2: (Color online) Asymptotic LMSZ probabilities[Eq. (32)] of finding the two qutrits in the state | − i (bluedot-dashed line), | i (red dashed line) and |− i (green fullline), when they start from the state |− i for γ x = γ y .tween the coupling parameter γ and the ramp of the magneticfield α , defining β ′ , deeply influences the transition probabil-ity. For high values of the parameter β ′ we get a completeLMSZ transition of both the spins, getting, also this time, astate transfer between the two qutrits. This means that, mea-suring the state of the system and varying the ramp α , we mayestimate the parameter γ determining the strength of couplingbetween the two qutrits. D. Noise Effects
We consider now the field along the z axis affected by therandom fluctuating contribution we saw in the previous sec-tion. We may exploit again the results reported in Ref. where the authors solved the dynamical problem of a noisyramp in a LMSZ scenario also for a spin-1. In such a case,the transition probabilities affected by a noisy field compo-nent along the z -axis and characterized by the following time-correlation function h η ( t ) η ( t ′ ) i = Γδ ( t − t ′ ) , become P + − = ( + e − πβ ′ − e − πβ ′ ) , P − = ( − e − πβ ′ ) , P − − = ( + e − πβ ′ + e − πβ ′ ) . (33) Also these expressions, valid for large values of Γ , are inde-pendent of the value of the same Γ . We see that, also this time,the main effect of the noise is to hinder the transition gener-ating at most equally populated states when β ′ ≫
1. In thisway, we brought to light how the symmetry-based analysis ofthe model reported in the second sections plays a key role fordisclosing the exact quantum dynamics of the two interactingqutrits subjected to time-dependent magnetic fields, both inideal and more realistic conditions.
V. ENTANGLEMENT
The negativity, introduced by G. Vidal and R. F. Wernerin , of a two-qutrit system described by the density matrix ρ reads N ρ = || ρ T B || − , (34)where ρ T B is the partial transpose of the matrix ρ with respectto the subsystem B . The symbol || · || is the trace norm which,for a hermitian matrix, results in the sum of the absolute val-ues of the negative eigenvalues of ρ T B which is hermitian andsuch that Tr { ρ T B } =
1. The range of values of N ρ is [ , ] and its calculation is independent of the factorized orthonor-mal basis in which the matrix ρ is represented as well as of thesubsystem with respect to which we calculate the partial trans-pose, since ( ρ T A ) T = ρ T B and || X || = || X T || for any operator X . A. Four-Dimensional Sub-Dynamics
For our two-qutrit system, it has been proved that the neg-ativity for a generic pure as well as mixed state belongingto the four dimensional subspace possesses the upper bound N = /
2. In case of a generic pure state | Ψ i = w | i + w | i + w | − i + w |− i , the Negativity acquires indeedthe simple form N = p x ( − x ) , x = | w | + | w | . (35)If we consider as initial condition the two-qutrit state |− i , through the exact form of the time evolution operatorin Eq. (28), it is easy to verify that x ( t ) = | w ( t ) | + | w ( t ) | = | a | | a | + | b | | b | (36)At infinite time so we have x ( ∞ ) = P P + ( − P )( − P ) , (37)where the expressions of P and P are reported in Eq. (13) and(14), respectively. If we put the expression in Eq. (37) intoEq. (35), we get the asymptotic expression of the Negativity.In Fig. 3a such an expression of the negativity is reportedagainst the LMSZ parameter β = β + , for β − / β + = /
2. Wesee that two maxima are present and they correspond to thevalues log ( ) / π ≈ .
11 and log ( ) / π ≈ .
22. It means that,by appropriately setting the parameter β , when the two-qutritsystem start from the state |− i , through the LMSZ processwe may generate asymptotically an entangled state of the twospin-qutrits with the maximum level of entanglement possiblein such a subspace. This fact is confirmed by Fig. 3b wherethe time behaviour of the Negativity is reported against thedimensionless parameter τ = p α / ¯ h t for β = .
11. In thiscase, we used the expression of x ( t ) in Eq. (36) with the exactsolution of the LMSZ dynamical problem which read a / = Γ f ( − i β ± ) √ π × [ D i β ± ( √ e − i π / τ ) ∗ D − + i β ± ( √ e i π / τ i )+ D i β ± ( √ e i π / τ ) ∗ D − + i β ± ( √ e − i π / τ i )] , b / = Γ f ( − i β ± ) p πβ e i π / × [ − D i β ± ( √ e − i π / τ ) ∗ D − + i β ± ( √ e i π / τ i )+ D i β ± ( √ e i π / τ ) ∗ D − + i β ± ( √ e − i π / τ i )] . (38) Γ f is the gamma function, D ν ( z ) are the parabolic cylinderfunctions and τ i identifies the initial time instant. We em-phasize that the parameter β , besides the asymptotic value,deeply influences the trend in time of the Negativity curve, asit can be appreciated by Figs. 3c and 3d, related to β = . β =
2, respectively. Β (a) - - -
10 10 20 30 Τ (b) - - -
10 10 20 30 Τ (c) - - -
10 10 20 30 Τ (d) Figure 3: (Color online) a) β -dependence of the asymptoticNegativity of the two qutrits [Eqs. (35) and (37)] for theinitial condition |− i . Time behaviour of the Negativityagainst the dimensionless parameter τ = p α / ¯ h t during aLMSZ process when the two-qutrit system starts from thestate |− i for 2 β − = β + and b) β + = .
11, c) β + = / β + =
2. The upper straight curve represents N = . ( |− i + | − i ) / √ ω ( t ) = ω ( t ) , we have takeninto account in Sec. III C, exhibits a constant maximum level of entanglement (1/2) during the evolution. Such a peculiarfeature is independent of the specific time-dependence of thefield and it may be understood at the light of the analysis re-ported in . There the authors analyse the same two-qutritmodel but with time-independent fields. They have brought tolight the existence of eight states with such a feature which isrelated to the symmetry property of the Hamiltonian. Beingsuch property unaffected by a general time-dependence of theapplied field as we showed before, we find of course the samefeature here too.We stress that it is not possible to get physical informa-tion about the entanglement get established between the twoqutrits by studying correlations emerging between the twofictitious qubits. Indeed, by the mapping in Eq. (6), it iseasy to see that entangled states of the two qutrits, such as ( | i + | i ) / √
2, correspond to separable states of the twoqubits, ( | ++ i + | + −i ) / √
2, and, vice versa , separable statesof the qutrits ( | i + |− i ) / √ ( | ++ i + |−−i ) / √
2. Such a featurestems from the non-locality of the mapping established be-tween the two systems. This observation implies that, withinthe four-dimensional subspace, we cannot use the Concur-rence, but we are obliged to consider another Entanglementmeasure. This is why we use Negativity to quantify the En-tanglement get established between the two qutrits.
B. Three-Dimensional Sub-Dynamics
In the three-dimensional subspace the Negativity for thegeneral state in Eq. (29) reads N = | c || c | + | c || c | + | c || c | . (39)Its time evolution related to the initial condition |− i results N ( t ) = | a || b | [ √ + | a || b | ] , (40)and then asymptotically we get N ( ∞ ) = P ( − P ) + p P ( − P ) , (41)where P is defined after Eqs. (32). This quantity reaches itsmaximum value for P = / β ′ = log ( ) / π ≈ .
11 (see Fig. 4a). This means that, for such a value of theparameter β ′ , the LMSZ process generates asymptotically anentangled state of the two qutrits with the maximum availablevalue of Negativity for the initial condition under scrutiny, asconfirmed by Fig. 4b. We got the latter figure by putting in Eq.(40) the expressions of a + and b + (or, equivalently, a − and b − ) in Eqs. (38), replacing β + ( β − ) with β ′ . In the same waywe have analysed the time behaviour of the Negativity for thesame initial condition for other two values of the parameter β ′ , namely β ′ = / β ′ = Β '0.20.40.60.81.0N (a) - - -
10 10 20 30 Τ (b) - - -
10 10 20 30 Τ (c) - - -
10 10 20 30 Τ (d) Figure 4: (Color online) a) β ′ -dependence of the asymptoticNegativity of the two qutrits [Eqs. (40)] for the initialcondition |− i . Time behaviour of the Negativity againstthe dimensionless parameter τ = p α / ¯ h t during a LMSZprocess when the two-qutrit system starts from the state |− i for b) β ′ = .
11, c) β ′ = / β ′ =
2. The upperstraight curve represents N = . VI. CONCLUSIVE REMARKS
This paper investigates the quantum dynamics of two in-teracting qutrits subjected to local time-dependent fields. Wehave taken into account the anisotropic as well as isotropicHeisenberg interaction. The field applied on just one of thetwo qutrits or on both the two spin-1’s has been consid-ered linearly varying on time (LMSZ ramp) along the quan-tization z -axis. Atomic species with three metastable levelsmay be used in a linear ion crystal to realize the interactingspin-1 model under scrutiny through the application of laserfields . Moreover, a broad range of physical situations maybe covered by such a model: two spin-1’s in a double well op-tical lattice , interacting spin-1 nanomagnets and effectiveinteraction between two separated nitrogen-vacancy centres indiamond .The dynamical problem has been solved thank to the reduc-tion to two easier problems: one of two non-interacting ficti-tious spin-1/2’s and the other of a fictitious three-level system.Such a reduction relies on the symmetry-based analysis of theHamiltonian model reported in Ref. which is unaffected bythe time-dependences of the applied fields and, more gener-ally, by the time-dependences of all Hamiltonian parameters.This means that the same analysis may be developed consid-ering other possible time-dependences of the field leading toexactly solvable problems .The main result of the paper is the physical effect we calledcoupling-driven LMSZ transition. It consists in the fact that,though a transverse constant field is absent, LMSZ transitionsbetween two-qutrit states are still possible thanks to the pres-ence of the coupling between the two spin-1’s. Indeed, the fictitious dynamics of the two decoupled qubits and the oneof a fictitious spin-1 are characterized by a LMSZ longitu-dinal field and a fictitious constant transverse field stemmingfrom the coupling existing between the spin-qutrits. This factimplies that, avoided crossings in the two qutrit system arepossible thanks to the presence of such an interaction. Aremarkable consequence of this circumstance consists in thefact that an appropriate ratio between the applied fields andthe coupling parameters may result favourable for perform-ing adiabatic dynamics with consequent full LMSZ transitionsof the two spin-1 system. The knowledge of such a phys-ical effect makes it possible to have control on the dynam-ics of the system under scrutiny as well as to get informationabout the interaction characterizing the same system. We havebrought to light, moreover, how the LMSZ transition proba-bilities change according to the (an)isotropy of the couplingterms.We have showed that the physical relevance of thecoupling-driven LMSZ transitions is twofold. Firstly, by theknowledge of the transition probabilities we may estimate thecoupling parameters of the two-qutrit model. Secondly, bas-ing on such an estimation, we illustrated that an appropriateand specific choice of the slope of the LMSZ ramp can gen-erate asymptotically entangled states of the two qutrits. Wehave analysed the level of entanglement by studying both theasymptotic Negativity against the LMSZ parameters and itstime evolution. In the latter case, we have used the exact so-lutions of the LMSZ dynamical problem and we have in-vestigated the effects of the coupling determining the LMSZparameter. We reported how such a parameter, depending onthe ratio of the squared coupling and the slope of the ramp,determines not only the asymptotic value, but also the trendof the Negativity.Finally, we have discussed also how the LMSZ transitionprobabilities are modified by the presence of a noisy fieldcomponent stemming from the interaction of the the two-qutrit system with a surrounding environment. Such an anal-ysis is based on the fact that the dynamical reduction is unaf-fected by the presence of the noise and so, also in this case, wemay reduce the two-spin-1 problem to easier problems whosesolutions are known in literature. Following the same phi-losophy, we have exposed the possibility of treating exactlythe problem also by introducing the environment effects withnon-Hermitian terms in the Hamiltonian model.We note that the parameters of the applied magnetic field,including the magnetic field gradient, can be controlled invery wide ranges. For example, the magnetic field gradientcan reach values as large as 150-200 T/m in a microfabricatedion trap , which is far beyond what is needed here. The mostimportant parameter for the feasibility of our scheme is thespin-spin coupling constant γ . In nuclear magnetic resonance,its values typically vary from 10 Hz to 300 Hz depending onthe molecule , which implies that entanglement can be cre-ated on the millisecond scale. A very interesting physical plat-form, which allows the tuning of the spin-spin coupling in abroad range, is provided by microwave-driven trapped ionsin the presence of a static magnetic-field gradient . Theeffective spin-spin coupling is proportional to the magnetic-0field gradient and can reach the kHz range. A third example isprovided by Rydberg atoms and ions where, due to the hugeelectric-dipole moments of the Rydberg states, the effectivespin-spin coupling can reach a few MHz . This impliesentanglement creation on the sub-microsecond scale.We do emphasize that the results achieved in this paper arenot simple generalizations of the ones reported in where theHamiltonian model (1) has been investigated for two inter-acting qubits. Consider indeed that the symmetry-based ex-istence of two dynamically invariant subspaces regardless ofthe values of the two spins, does not represent the success-ful key of our approach in its own. It is in fact possible topersuade oneself that the effective quantum dynamics of twointeracting qudits, restricted to one of the two dynamicallyinvariant subspaces, turns out to be in general a challengingproblem whose difficulty grows with increasing the values ofthe qudits. This paper shows that, in the quantum dynamicsof two qutrits restricted to the two invariant subspaces, thesedifficulties can successfully overcome by establishing a directlink with su(2) problems. Summing up, the route followedin this paper has the merit of explicitly showing that the res-olution of the related dynamical problems cannot be derivedsimply generalizing technical aspects characterizing the anal- ogous dynamical problem of two qubits .It is worth noticing that the ideas and tools which our ap-proach hinges upon may be useful to investigate other evenmore complex physical scenarios, as for example done inRef.s . We feel that our results might stimulate pos-sible experimental investigations, for example within STMscenarios .Finally, the main perspective of this work is to take intoaccount also quantum degrees of freedom of the bath. In thiscase, the basic and fundamental symmetry-based dynamicalreduction might be joined with recent approaches to reacha deeper understanding of the dynamics of two-qutrit systemsin more realistic experimental situations. VII. ACKNOWLEDGEMENTS
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