Coupling-assisted Landau-Majorana-Stuckelberg-Zener transition in two-interacting-qubit systems
aa r X i v : . [ qu a n t - ph ] M a y Coupling-assisted Landau-Majorana-St ¨uckelberg-Zener transition in two-interacting-qubit systems
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: May 21, 2019)We analyse a system of two interacting spin-qubits subjected to a Landau-Majorana-St¨uckelberg-Zener(LMSZ) ramp. We prove that LMSZ transitions of the two spin-qubits are possible without an external trans-verse static field since its role is played by the coupling between the spin-qubits. We show how such a physicaleffect could be exploited to estimate the strength of the interaction between the two spin-qubits and to generateentangled states of the system by appropriately setting the slope of the ramp. Moreover, the study of effects ofthe coupling parameters on the time-behaviour of the entanglement is reported. Finally, our symmetry-basedapproach allows us to discuss also effects stemming from the presence of a classical noise or non-Hermitiandephasing terms.
I. INTRODUCTION
The Landau-Majorana-St¨uckelberg-Zener (LMSZ)scenario and the Rabi one represent two milestonesamong exactly solvable time-dependent semi-classical mod-els for two-level systems. A common fundamental propertyof these two models is the possibility of realizing a fullpopulation inversion in a two-state quantum system. In theformer case through an adiabatic passage via a level crossing,in the second case thanks to the application of a resonant π -pulse.It is important to underline that the LMSZ scenario, dif-ferently from the Rabi case, is an ideal model. The word“ideal” refers to the fact that it consists in a process charac-terized by an infinite time duration resulting, then, practicallyunrealisable. This fact leads, indeed, to not physical prop-erties such as, for example, the fact that the energies of theadiabatic states diverge at initial ( − ∞ ) and final instant ( + ∞ ).As a consequence, both mathematical and physical problemsarise when amplitudes and not only probabilities are neces-sary, e.g. when initial states present coherences . In suchcases one can alternatively use either the exact solutions ofthe finite LMSZ scenario or the Allen-Eberly-Hioe model ,the Demkov-Kunike model or other models , where no di-vergency problems arise and the transition probability is rathersimple.However, despite this circumstance, it is a matter of factthat the LMSZ grasps peculiar dynamical aspects of a lot ofphysical systems . This relevant aspect has increased thepopularity of the LMSZ model and several efforts have beendone towards its generalization to the case of N -level quantumsystems and total crossing of bare energies . Moreover,its experimental feasibility gave it a basic role in the area ofquantum technology thanks also to the several sophisticatedtechniques developed for a precise local manipulation of thestate and the dynamics of a single qubit in a chain .In such an applicative scenario, as we know, several sourcesof incoherences can be present : incoherent (mixed)states, relaxation processes (e.g., spontaneous emission) or in-teraction with a surrounding environment (e.g., nuclear spin bath). They generate incoherent excitation leading to de-parture from a perfect (ideal) population transfer. There-fore, more realistic descriptions of quantum systems sub-jected to LMSZ scenario comprising such effects have beenproposed .In this respect, the most relevant influence in the dynam-ics of a spin-qubit primarily stems from the coupling with itsnearest neighbours. Recently the attention has been focusedon double interacting spin-qubit systems subjected to LMSZscenario . These papers investigate the coupling effects inthe two-spin system dynamics in view of possible experimen-tal techniques and protocols. Moreover, such systems, un-der specific conditions, behave effectively as a two-level sys-tem with relevant applicability in quantum information andcomputation sciences . In the references cited before, in-deed, generation of entangled states or the singlet-triplettransition in the two-qubit system under the LMSZ sce-nario have been studied.With the same objective in mind, that is to characterizephysical effects stemming from the coupling between twospin-qubits subjected to a LMSZ scenario, in this paper westudy a two-spin-1/2 system described by a C -symmetryHamiltonian model. We consider coupling terms compati-ble with the symmetry of the Hamiltonian, namely isotropicand anisotropic exchange interaction. The two spin-1/2’s aremoreover subjected only to a LMSZ ramp with no transversestatic field. We show that LMSZ transitions for the two spin-qubits are still possible thanks to the presence of the coupling,playing the role of an effective transverse field. Such an effect,we call coupling-assisted LMSZ transition , deserves particularattention for two reasons. Firstly, it can be exploited to esti-mate the presence and the relative weight of different couplingparameters determining the symmetry of the Hamiltonian andthen the dynamics of the two spins. Secondly, through suchan estimation, it is possible to set the slope of the field rampin such a way to generate asymptotic entangled states of thetwo qubits.The paper is organized as follows. In Sec. II we intro-duce the model and its symmetry properties on which the dy-namical reduction is based. In Sec. III the application of theLMSZ scenario on both the subdynamics (that is the two-qubitdynamics restricted to the invariant subspaces) is performed.Moreover, physical effects stemming from the (an)isotropy ofthe exchange interaction are brought to light. In the subse-quent Sec. IV, we emphasize the possibility of estimating thevalues of the coupling parameters. The generation of asymp-totic entangled states of the two spins through coupling-basedLMSZ transitions is reported instead in Sec. V. Some ef-fects of a possible interaction with a surrounding environment,providing for either a classical noisy field component or non-Hermitian terms in the Hamiltonian model, are taken into ac-count in Sec. VI. Finally, some conclusive comments and fur-ther remarks can be found in the last Sec. VII. II. THE MODEL
Let us consider the following model, describing two inter-acting spin-qubits: H = ¯ h ω ( t ) ˆ σ z + ¯ h ω ( t ) 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 ˆ σ xi , ˆ σ yi and ˆ σ zi ( i = ,
2) are the Pauli matrices and allthe parameters may be thought as time-dependent. The ma-trices are represented in the following ordered two-spin basis {| ++ i , | + −i , |− + i , |−−i} ( ˆ σ z |±i = ±|±i ) .The C -symmetry with respect to the z -direction, possessedby the Hamiltonian, causes the existence of two dynamicallyinvariant Hilbert subspaces related to the two eigenvalues ofthe constant of motion ˆ σ z ˆ σ z . Basing on such a symmetry,the time evolution operator, solution of the Schr¨odinger equa-tion i ¯ h ˙ U = HU , may be formally put in the following form U = a + ( t ) b + ( t ) a − ( t ) b − ( t ) − b ∗− ( t ) a ∗− ( t ) − b ∗ + ( t ) a ∗ + ( t ) . (2)The condition U ( ) = is satisfied by putting a ± ( ) = b ± ( ) =
0. It is worth noticing that a ± ( t ) and b ± ( t ) are thetime-dependent parameters of the two evolution operators U ± = e ∓ i γ z t / ¯ h (cid:18) a ± ( t ) b ± ( t ) − b ∗± ( t ) a ∗± ( t ) (cid:19) , (3)solutions of two independent dynamical problems of fictitioussingle spin-1/2, namely i ¯ h ˙ U ± = H ± U ± , U ± ( ) = ± , with H ± = (cid:18) ¯ h Ω ± ( t ) γ ± γ ± − ¯ h Ω ± ( t ) (cid:19) ± γ z ± = ¯ h Ω ± ( t ) ˆ σ z + γ ± ˆ σ x ± γ z ± , (4)where Ω ± ( t ) = [ ω ( t ) ± ω ( t )] , γ ± = ( γ x ∓ γ y ) , (5)and ± represent the identity operators within the two-dimensional subspaces. Thus, it means that the solution of thedynamical problem of the two interacting spin-1/2’s is tracedback to the solution of two independent problems, each one ofsingle (fictitious) spin-1/2 . The explicit expressions of a ± ( t ) and b ± ( t ) depend onthe specific time-dependences of ω ( t ) and ω ( t ) . It is wellknown that it is not possible to find the analytical solution ofthe Schr¨odinger equation for a spin-1/2 subjected to a generictime-dependent field. Therefore, specific exactly solvabletime-dependent scenarios for a single spin-1/2 might be ofgreat help to investigate the dynamics of the two interactingspin system under scrutiny . III. COUPLING-BASED LMSZ TRANSITION
In this section we investigate the case in which a LMSZramp is applied on either just one or both the spins. Our fol-lowing theoretical analysis is based on the possibility of ex-perimentally addressing at will the spin systems exploiting,for example, the Scanning Tunneling Microscopy (STM). Itappears hence appropriate to furnish a sketch of such a tech-nique.STM proved to be an excellent experimental technique incontrolling the dynamics of spin-qudit systems for two mainreasons: 1) the possibility of building atom by atom atomic-scale structures , such as spin chains and nano-magnets ; 2)the possibility of controlling the whole system by addressinga single element (qudit) while it interacts with the others ,succeeding in realizing, for example, logic operations . Themanipulation of a single qudit dynamics is performed throughthe exchange interaction between the atom on the tip of thescanning tunneling microscope and the target atom in thechain. It is possible to show that such an interaction is equiv-alent to a magnetic field applied on the atom we want tomanipulate . In this way, it is easy to guess that a time-dependent distance between the tip and the target atom gener-ates a time-dependent exchange coupling, giving rise, in turn,to a time-dependent effective magnetic field on the atom of thechain, as analysed in Ref. . Basing on such an observation,in Ref. the authors study the spin dynamics and entangle-ment generation in a spin chain of Co atoms on a surface ofCu N/Cu(110). Precisely, they consider a LMSZ ramp alongthe z direction produced in a time window of 20 ps and a shortGaussian pulse in the x direction (half-width: 10 ps ). A. Collective LMSZ Dynamics
At the light of the STM experimental scenario, we take intoaccount firstly the case of a LMSZ ramp applied on the firstspin such that¯ h ω ( t ) = α t / , ¯ h ω ( t ) = , t ∈ ( − ∞ , ∞ ) , (6)where α is related to the velocity of variation of the field,˙ B z ∝ α , and it is considered a positive real number withoutloss of generality. Let us consider, moreover, the two spinsinitialized in the state |−−i . In this instance, the subdynam-ics governed by H + is characterized by a LMSZ scenariowhere the longitudinal ( z ) magnetic field produces the stan-dard LMSZ ramp ¯ h Ω + ( t ) = ¯ h ω ( t ) = α t / x -direction is given by γ − .It is well-known that the dynamical problem for such a time-dependent scenario can be analytically solved. The transitionprobability of finding the two-spin system in the state | ++ i coincides with the probability of finding the fictitious spin-1/2subjected to H + in its state | + i starting from |−i and reads P + = |h + + | U + ( ∞ ) | − −i| = − exp {− πγ + / ¯ h α } . (7)If we now, instead, consider the two spins initially preparedin |− + i , the probability for each spin-1/2 of undergoing aLMSZ transition, that is the probability of finding the two-spin system in the state | + −i , results P − = |h + − | U − ( ∞ ) | − + i| = − exp {− πγ − / ¯ h α } . (8)This time the transition probability is governed by the ficti-tious magnetic field given by γ − . The effective longitudi-nal magnetic field, instead, is the same, namely ¯ h Ω − ( t ) = ¯ h ω ( t ) = α t /
2. We see that in both cases, though a constanttransverse magnetic field is absent, the LMSZ transition ofboth the spins is possible thanks to the presence of the cou-pling between them. It is important to stress that, for the casesconsidered before, if γ x = γ y (as it often happens experimen-tally) we cannot have transition in the first case, that is in thesubdynamics involving | ++ i and |−−i . In this instance, in-deed, P + happens to be 0 at any time. B. Isotropy Effects: Local LMSZ Transition by nonlocalControl and State Transfer
The symmetry-based dynamical decomposition and the ap-plication of the STM LMSZ scenario in each subdynamicsallow us to bring to light peculiar evolutions of physical in-terest. For example, if we consider γ x = γ y and the followinginitial condition | + i ⊗ | + i + |−i√ , (9)the two states | ++ i and |−−i evolve independently and ap-plying the LMSZ ramp we have the probability P = P + P − tofind asimptotically the two-spin system in the state |−i ⊗ | + i + |−i√ . (10)We see, that such a dynamics leaves unaffected the secondspin, while it produces a LMSZ transition only on the firstspin. It is also relevant the dynamical evolution of the sym-metric initial condition | + i + |−i√ ⊗ | + i . (11)This time, we get the same probability P = P + P − of findingasymptotically the two-spin system in | + i + |−i√ ⊗ |−i . (12) This case results less intuitive even if we are reproducing thesame dynamics but with interchanged roles of the two spins.In this instance, in fact, we generate a LMSZ transition onlyon the second spin by locally applying the field on the firstone. This shows that the coupling between the two spins playsa key role to achieve a non-local control of the second spin bylocally manipulating the first ancilla qubit.If we consider, instead, γ x = γ y = γ / |−−i ↔ | ++ i is suppressed. This means that if weconsider as initial conditions the states in Eqs. (9) and (11),we get asymptotically, this time, the states | + i + |−i√ ⊗ | + i , (13a) | + i ⊗ | + i + |−i√ , (13b)respectively, with probability P = − exp {− πγ / ¯ h α } . Wesee that the isotropy properties of the exchange interactionconsistently change the dynamics of the system. When theexchange interaction is isotropic, indeed, the asymptotic statesreached by the initial conditions (9) and (11) radically change.In these cases, the resulting physical effect is a state transfer ora state exchange between the two spin-qubits. Therefore, thedifferent state transitions from the state (9) [(11)] to the state(10) or (13a) [(12) or (13b)] can reveal the level of isotropy ofthe exchange interaction. IV. COUPLING PARAMETER ESTIMATION
It is interesting noticing that the coupling-based LMSZtransition could be used to estimate the coupling parameters.By measuring P + and P − (Eqs. (7) and (8), respectively) in aphysical scenario describable by the Hamiltonian model (1),we get an estimation of γ + and γ − and then of the two cou-pling parameters γ x and γ y . Supposing to know P + and P − , wehave indeed γ x = r ¯ h α π "s log (cid:18) − P − (cid:19) + s log (cid:18) − P + (cid:19) , γ y = r ¯ h α π "s log (cid:18) − P − (cid:19) − s log (cid:18) − P + (cid:19) . (14)We wish to emphasize that we may estimate the coupling pa-rameters also through the Rabi oscillations occurring in thetwo subspaces. Applying, indeed, a constant field ω on thefirst spin, the two probabilities P + and P − become P + = γ + ¯ h ω + γ + sin (cid:18)q ω + γ + / ¯ h t (cid:19) , P − = γ − ¯ h ω + γ − sin (cid:18)q ω + γ − / ¯ h t (cid:19) . (15)So, by measuring the frequency and the amplitude of the os-cillations in the two cases we may get information about thethe relative weights of the coupling parameters. V. ENTANGLEMENT
A precise estimation of the coupling parameters is use-ful also to generate entangled states of the two spins. Bythe knowledge of them, indeed, we may set the parameter α in order to get asymptotically P ± = /
2, generating soan entangled state. If the two spins start from state |−−i or |− + i , being the dynamics unitary, they reach asymptoti-cally the pure state ( | ++ i + e i φ |−−i ) / √ ( | + −i + e i φ |− + i ) / √ ), in fact, when the two-spin system is initialized in |−−i or |− + i , read respectively C = | c ++ c −− | = p P + ( − P + ) = q ( − e − πβ + ) e − πβ + (16a) C = | c + − c − + | = p P − ( − P − ) = q ( − e − πβ − ) e − πβ − (16b)and they exhibit a maximum for β + = β − = log ( ) / π ≈ .
11. In the previous expressions we put β + = γ + / ¯ h α and β − = γ − / ¯ h α , while c ++ and c −− ( c + − and c − + ) are theasymptotic amplitudes of the states | ++ i and |−−i ( | + −i and |− + i ), respectively. Therefore, log ( ) / π is exactly thevalue the LMSZ parameters β + ad β − must have to realize thegeneration of the entangle states ( | ++ i + e i φ |−−i ) / √ ( | + −i + e i φ |− + i ) / √ |−−i or |− + i , respectively. Figure 1a reports the two curves for β − / = β + = β . Β (a) - - -
10 0 10 20 30 Τ (b) Figure 1: (Color online) a) The two curves of theconcurrence in Eq. (16a) (full blue line) and Eq. (16b) (reddashed line) for β − / = β + = β ; b) Time behaviour ofconcurrence for the initial condition |−−i and β + = . τ = p α / ¯ h t .We may verify this fact by investigating the behaviour ofthe concurrence in time. To this end, the exact solutions ofthe two time-dependent parameters determining the two timeevolution operators U + and U − in Eq. (3), related to each subdynamics, are necessary and they reads, namely 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 )] . (17) Γ f is the Gamma function, while D ν ( z ) are the paraboliccylinder functions and τ = p α / ¯ h t is a time dimensionlessparameter; τ i identify the initial time instant. If the systemstarts, e.g., from the state |−−i the amplitudes result c ++ = b + , c −− = a ∗ + , c + − = c − + = , (18)and the related time-behaviour of the concurrence C = | b + || a + | for β + = . β + determines not only the asymptotic value of the concur-rence but also its time behaviour. This fact is confirmed andcan be appreciated by Figs. 2a and Fig. 2b reporting the con-currence against the dimensionless parameter τ for β + = / β + =
2, respectively. The physical meaning of the asymp-totic vanishing of C in Fig. 2b is that for the specific value of β + the system evolves quite adiabatically towards the factor-ized states |−−i . On the contrary, in Figs. 2a the slope of theramp induces a non adiabatic evolution towards a coherent notfactorizable superposition of | ++ i and |−−i . - - -
10 0 10 20 30 Τ (a) - - -
10 0 10 20 30 Τ (b) Figure 2: (Color online) Time behaviour of the concurrenceagainst the dimensionless parameter τ = p α / ¯ h t during aLMSZ process when the system starts from the state |−−i for a) β + = / β + =
2. The upper straight curvecorresponds to C ( τ ) = |− + i . Inthis case, only the states |− + i and | + −i would be involvedand the LMSZ parameter determining the different concur-rence regimes would be β − . For such initial conditions, then,the ratio β + / β − , imposing precise relationships between thecoupling parameters γ x and γ y , does not matter.Such a ratio, conversely, results determinant for other initialconditions, e.g. the one considered in Eq. (10). In this casethe amplitudes read c ++ = a + , c −− = − b ∗ + , c + − = a − , c − + = − b ∗− . (19)In Figs. 3a-3f we may appreciate the influence of both the ra-tio β − / β + and the free parameter β + ; the former influencesonly qualitatively the behaviour of the concurrence, while thelatter both qualitatively and quantitatively. This time too, theconcurrence vanishes for high values of β + witnessing anasymptotic factorized state. For small values of β + , instead,positive values of entanglement even for large times indicatea superposition of the four standard basis states. - - -
10 10 20 30 Τ (a) - - -
10 10 20 30 Τ (b) - - -
10 10 20 30 Τ (c) - - -
10 10 20 30 Τ (d) - - -
10 10 20 30 Τ (e) - - -
10 10 20 30 Τ (f) Figure 3: (Color online) Time behaviour of the concurrenceagainst the dimensionless parameter τ = p α / ¯ h t during aLMSZ process when the system starts from the state( | ++ i + | + −i ) / √ β − / β + = / β + = /
2, b) β + = β − / β + = β + = .
5, d) β + = β − / β + = β + = .
1, f) β + =
10. The upper straight curvecorresponds to C ( τ ) = the authors considered a system of two spin-1/2’s interactingonly through the term ˆ σ z ˆ σ z and subjected to the same mag-netic field consisting in a Gaussian pulse uniformly rotatingin the x − y plane and a LMSZ ramp in the z direction. Theyshowed that the coupling between the two spins enhances sig-nificantly the probability to drive adiabatically the two-spinsystem from the separate state |−−i to the entangled state ( | + −i + |− + i ) / √
2. In this case the procedure to generatean entangled state is different from the scenario consideredhere because of the different symmetries of the Hamiltoniansruling the two-spin dynamics. Indeed, in Ref. the Hamil-tonian commutes with ˆ S and consequently two dynamicallyinvariant Hilbert subspaces exist: one of dimension three andthe other of dimension one. The three-dimensional subspaceis spanned by the states | ++ i , ( | + −i + |− + i ) √ |−−i ,making possible the preparation of the entangled state of thetwo spin-1/2’s by an adiabatic passage when they start fromthe separate state |−−i . In our case, instead, ˆ S is not con-stant while the integral of motion is ˆ σ z ˆ σ z . The symmetries ofthe Hamiltonian, thus, generate two two-dimensional dynam-ically invariant Hilbert subspaces: one spanned by | ++ i and |−−i and the other by | + −i and |− + i . Then, in our case, thetransition between the states considered in the other work isimpossible since such states belong to different invariant sub-spaces. VI. EFFECTS OF CLASSICAL NOISE
In experimental physical contexts involving atoms, ionsand molecules investigated and manipulated by application oflasers and fields, the presence of noise in the system stem-ming from the coupling with a surrounding environment isunavoidable. Though a lot of technological progresses and ex-perimental expedients have been developed, it is necessary tointroduce such decoherence effects in the theoretical modelsfor a better understanding and closer description of the experi-mental scenarios. There exist different approaches to treat theinfluence of a thermal bath; one is to consider the presenceof classical noisy fields stemming, e.g., from the presenceand the influence of a surrounding nuclear spin bath .In the last reference the authors study a noisy LMSZscenario for a N -level system. They take into account atime-dependent magnetic field η ( t ) only in the z -directionand characterized by the following time correlation function h η ( t ) η ( t ′ ) i = G δ ( t − t ′ ) . The authors show that the LMSZtransition probability P + − for a spin-1/2 to be found in the state | + i starting from |−i , in case of large values of G , changes as P + − = − exp {− π g / ¯ h α } , (20)where g is the energy contribution due to the coupling of thespin-1/2 with the constant transverse magnetic field and α isthe ramp of the longitudinal magnetic field. We see that thevalue of G , provided that it is large, does not influence thetransition probability. The unique effect of the noisy compo-nent is the loss of coherence. The field indeed cannot generatetransitions between the two diabatic states, being only in thesame direction of the quantization axis. In this way the tran-sition probability, as reasonable, results hindered by the pres-ence of the noise, since, for g / α ≫
1, the system reaches atmost the maximally mixed state.This result is of particular interest in our case since the ad-dition of the noisy component η ( t ) leaves completely unaf-fected the symmetry-based Hamiltonian transformation andthe validity of the dynamics-decoupling procedure. Thus, alsoin this case, the dynamical problem of the two-qubit systemmay be converted into two independent spin-1/2 problems af-fected by a random fluctuating z -field. Thus, we may writeeasily the transition probabilities when the two spins are sub-jected to a unique homogeneous field influenced by the noisycomponent considered before. We have precisely P + = − exp {− πγ + / ¯ h α } , ω ( t ) = ω ( t ) = [ α t + η ( t )] / , (21)We underline that the transition probability P − vanishes incase of an unique homogeneous magnetic field. In the relatedsubdynamics, indeed, the effective field ruling the two-spindynamics is zero, namely Ω − ( t ) =
0. Moreover, for γ x = γ y we would have no physical effects, since, in such a case, also P + would result zero.Another way to face with the problem of open quantumsystems is to use non-Hermitian Hamiltonians effectively in-corporating the information of the fact that the system theydescribe is interacting with a surrounding environment .We may suppose, for example, that the spontaneous emis-sion from the up-state to the down-one is negligible and thatsome mechanism makes the up-state | + i irreversibly decay-ing out of the system with rate ξ and ξ ′ for the first and sec-ond spin-1/2, respectively. It is well known that we can phe-nomenologically describe such a scenario by introducing thenon-Hermitian terms i ξ ˆ σ z / i ξ ′ ˆ σ z / and reread them in terms of the two-spin-1/2language. We know that the decaying rate affects only thetime-history of the transition probability but not, surprisingly,its asymptotic value . However, this result is valid for theideal LMSZ scenario; considering the more realistic case ofa limited time window, it has been demonstrated, indeed, thata decaying rate-dependence for the population of the up-statearises . VII. CONCLUSIVE REMARKS
In this work we considered a physical system of two inter-acting spin-1/2’s whose coupling comprises the terms stem-ming from the anisotropic exchange interaction. Moreover,each of them is subjected to a local field linearly varying overtime. The C -symmetry (with respect to the quantization axisˆ z ) possessed by the Hamiltonian allowed us to identify twoindependent single spin-1/2 sub-problems nested in the quan-tum dynamics of the two spin-qubits. This fact gave us thepossibility of decomposing the dynamical problem of the twospin-1/2’s into two independent problems of single spin-1/2.In this way, our two-spin-qubit system may be regarded as afour-level system presenting an avoided crossing for each pairof instantaneous eigenenergies related to the two dynamically invariant subspaces. This aspect turned out to be the key tosolve easily and exactly the dynamical problem, bringing tolight several physically relevant aspects.In case of time-dependent Hamiltonian models, such asymmetry-based approach and the reduction to independentproblems of single spin-1/2 has been used also in othercases . This fact permits a deep understanding of thequantum dynamics of the spin systems with consequent po-tential applications in quantum information and computation.We underline, in addition, that the dynamical reduction ex-posed in Sec. II is independent of the time-dependence ofthe fields. Thus, we may consider also different exactly solv-able time-dependent scenarios for the two subdynamics,resulting, of course, in different two-spin dynamics and phys-ical effects.In this paper, we showed that, although the absence of atransverse chirp or constant field, LMSZ transitions are stillpossible, precisely from |−−i to | ++ i and from |− + i to | + −i (the two couples of states spanning the two dynamicallyinvariant Hilbert spaces related to the symmetry Hamiltonian).Such transitions occur thanks to the presence of the couplingbetween the spins which plays as effective static transversefield in each subdynamics.It is worth noticing that, in our model, the two LMSZ sub-dynamics are ruled either by different combinations of the ex-ternally applied fields (when the local fields are different) orby the same field (under the STM scenario, that is when onelocal field is applied on just one spin). In the latter case weshowed the possibility of 1) a non-local control, that is to ma-nipulate the dynamics of one spin by applying the field on theother one and 2) a state exchange/transfer between the twospins. We brought to light how such effects are two differentreplies of the system depending on the isotropy properties ofthe exchange interaction.Concerning the interaction terms, each subdynamics ischaracterized by different combinations of the coupling pa-rameters. This aspect has relevant physical consequencessince, as showed, by studying the LMSZ transition probabil-ity in the two subspaces, it is possible both to evaluate thepresence of different interaction terms and to estimate theirweights in ruling the dynamics of the two-spin system. Webrought to light how the estimation of the coupling parame-ters could be of relevant interest since, through this knowl-edge, we may set the slope of variation of the LMSZ rampas to generate asymptotically entangled states of the two spin-1/2’s. Moreover, we reported the exact time-behaviour of theentanglement for different initial conditions and we analysedhow the coupling parameters can determine different entan-glement regimes and asymptotic values.Finally, we emphasized how our symmetry-based analy-sis has proved to be useful also to get exact results when aclassical random field component or non-Hermitian terms areconsidered to take into account the presence of a surround-ing environment interacting with the system. In this case,the dynamics decomposition is unaffected by the presenceof the noise or the dephasing terms and then we may applythe results previously reported for a two-level system andreread them in terms of the two spin-1/2’s.We wish to underline, in addition, that our results are validnot only within the STM scenario, but they are applicable toother physical platforms. Indeed, the local LMSZ model fora spin-qubit interacting with another neighbouring spin-qubitmay be reproduced also in laser-driven cold atoms in opticallattices where highly-selective individual addressing has beenexperimentally demonstrated . Another prominent exampleis laser-driven ions in a Paul trap where spatial individual ad-dressing of single ions in an ion chain has been routinely usedfor many years . Yet another example is microwave-driventrapped ions in a magnetic-field gradient where individual ad-dressing with extremely small cross-talk has been achieved infrequency space .We point out that the results obtained in this paper aredeeply different from the ones reported in other Refs. where systems of two spin-1/2’s in a LMSZ framework havebeen investigated on the basis of an approximate treatment. Inthese papers, indeed, the two spin-qubits are not directly cou-pled, but they interact through a common nuclear spin bathwhich they are coupled to. Such a composite system behavesas a two-level system under several assumptions and to de-rive the effective single spin-1/2 Hamiltonian requires severalapproximations. In Ref. , in particular, the effective Hamil-tonian describes the coupling between the two-level systemand a longitudinal time-dependent field which is not a pureLMSZ ramp, presenting a complicated functional dependenceon the original Hamiltonian parameters. There is, in addi-tion, a time-dependent effective interaction between the twostates possessing a complicated functional dependence on theconfinement energy as well as the tunneling and Coulomb en-ergies. Although such an effective Hamiltonian goes beyondthe standard LMSZ scenario, it may be considered similar tothe LMSZ one since both Hamiltonians describe an adiabaticpassage through an anticrossing.In our case, instead, the two spin-1/2’s are directly cou-pled, besides to be subjected to a random field stemming from the presence of a spin bath. Furthermore, the effective two-state Hamiltonians governing the two-qubit dynamics in thetwo invariant subspaces are easily got without involving anyassumption and/or approximation. The two two-level Hamil-tonians, indeed, are derived only on the basis of a transpar-ent mathematical mapping between the two-qubit states ineach subspace and the states of a fictitious spin-1/2. More-over, they describe exactly a LMSZ scenario with a standardavoided crossing where the transverse constant field is effec-tively reproduced by the coupling existing between the twoqubits. The treatment at the basis of this work remarkablyenables us to explore peculiar dynamical aspects of the sys-tem described by Eq. (1), leading, for example, to the exactevolution of the entanglement get established between the twospins.We underline, moreover, that our study is not a specialcase of the one considered in Ref. , where a Lipkin-Meskow-Glick (LMG) interaction model for N spin-qubits subjected toa LMSZ ramp is considered. The numerical results reportedin Ref. are, indeed, based on the mean field approximation.In addition, there is no possibility of considering in the LMGmodel effects stemming from the anisotropy between x and y interaction terms.Finally, two challenging problems naturally extending theinvestigation here reported are 1) that considering the inter-action of two qutrits in place of two qubits and 2) that tak-ing into account the coupling of the two spins with a quan-tum baths in place of the interaction with a classical randomfield. ACKNOWLEDGEMENTS
NVV acknowledges support from the EU Horison-2020project 820314 (MicroQC). RG acknowledges economicalsupports by research funds difc 3100050001d08+, Universityof Palermo. L. D. Landau, Phys. Z. Sowjetunion , 46 (1932); E. Majorana,Nuovo Cimento , 43 (1932); E. C. G. St¨uckelberg, Helv. Phys.Acta , 369 (1932); C. Zener, Proc. R. Soc. London, Ser. A ,696 (1932). I. I. Rabi, Phys. Rev.
652 (1937). G. S. Vasilev and S. S. Ivanov and N. V. Vitanov, Phys. Rev. A , 013417 (2007). B. T. Torosov and N. V. Vitanov, Phys. Rev. A , 063411 (2011). N. V. Vitanov and B. M. Garraway, Phys. Rev. A , 6 (1996). L. Allen and J. H. Eberly, Optical Resonance and Two-LevelAtoms (Dover, New York, 1987); F. T. Hioe, Phys. Rev. A ,2100 (1984). Y. N. Demkov and M. Kunike, Vestn. Leningr. Univ. Fiz. Khim. , 39 (1969). G. S. Vasilev and N. V. Vitanov, Phys. Rev. A , 023416 (2006). J. M. S. Lehto and K.-A. Suominen, Phys. Rev. A , 013404(2016). S.N. Shevchenko, S. Ashhab, Franco Nori, Phys. Rep. , 1(2010). S. S. Ivanov and N. V. Vitanov, Phys. Rev. A , 023406 (2008). N. A. Sinitsyn, Phys. Rev. A , 032701 (2013). B. D. Militello and N. V. Vitanov, Phys. Rev. A , 053402(2015). S. Thiele, F. Balestro, R. Ballou, S. Klyatskaya, M. Ruben, W.Wernsdorfer1, Science D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus, M. P. Hanson,A. C. Gossard, Science , 817 (2008). P. Huang, J. Zhou, F. Fang, X. Kong, X. Xu, C. Ju, and J. Du,Phys. Rev. X , 011003 (2011). J. Randall, A. M. Lawrence, S. C. Webster, S. Weidt, N. V. Vi-tanov, and W. K. Hensinger1, Phys. Rev. A , 043414 (2018). R. Wieser, V. Caciuc, C. Lazo, H. H¨olscher, E. Y. Vedmedenkoand R. Wiesendanger, New J. Phys. I. N. Sivkov, D. I. Bazhanov and V. S. Stepanyuk, Sc. Rep. ,2759 (2017). J. R. Petta et al. , Science (5744) 2180-2184 (2005). M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D.Phillips, and J. V. Porto Nature (7152) 452-456 (2007). H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Uman-sky and A. Yacoby Nat. Phys.
109 (2011). Xin Wang, L. S. Bishop, J. P. Kestner, E. Barnes, Kai Sun and S.Das Sarma Nat. Comm.
997 (2012). V. M. Akulin and W. P. Schleich, Phys. Rev. A , 7 (1992). N. V. Vitanov and S. Stenholm, Phys. Rev. A , 4 (1997). V. L. Pokrovsky and N. A. Sinitsyn, Phys Rev. B , 144303(2003). P. A. Ivanov and N. V. Vitanov, Phys. Rev. A , 063407 (2005). M. Scala, B. Militello, A. Messina and N. V. Vitanov, Phys. Rev.A , 023416 (2011). A. V. Dodonov, B. Militello, A. Napoli, and A. Messina, Phys.Rev. A , 052505 (2016). R. G. Unanyan, N.V. Vitanov, and K. Bergmann, Phys. Rev. Lett. , 137902 (2001). H. Ribeiro and G. Burkard, Phys. Rev. Lett. , 216802 (2009). H. Ribeiro, J. R. Petta, and G. Burkard, Phys. Rev. B , 235318(2013). M. J. Ranci´c, D. Stepanenko, Phys. Rev. B , 241301(R) (2016). J. Larson, Eur. Phys. Lett. , 30007 (2014). S. N. Shevchenko, A. I. Ryzhov, and Franco Nori, Phys. Rev. B , 195434 (2018). S. Mhel, Phys. Rev. B , 035430 (2015). R. Grimaudo, A. Messina, H. Nakazato, Phys. Rev. A , 022108(2016). A. A. Khajetoorians, J. Wiebe, B. Chilian, R. Wiesendanger, Sci-ence (6033) 1062-1064 (2011). S. Yan, D.-J. Choi, J. A. J. Burgess, S. Rolf-Pissarczyk and S.Loth, Nature Nanotechnology , 4045 (2015). B. Bryant, A. Spinelli, J. J. T. Wagenaar, M. Gerrits, and A. F.Otte, Phys. Rev. Lett. , 127203 (2013). Kun Tao, V. S. Stepanyuk, W. Hergert, I. Rungger, S. Sanvito, andP. Bruno, Phys. Rev. Lett. , 057202 (2009). W. K. Wootters, Phys. Rev. Lett. , 2245 (1998). M. Abramowitz and I. A. Stegun,
Handbook of MathematicalFunctions (Dover, New York, 1964). C. Benedetti and M . Paris, Phys. Lett. A H. Feshbach, Ann. Phys. , 357 (1958); H. Feshbach, Ann. Phys. , 287 (1962). N. Moiseyev, Non-Hermitian Quantum Mechanics (CambridgeUniv. Press, Cambridge, 2011). I. Rotter and J. P. Bird, Rep. Prog. Phys. , 114001, (2015); I.Rotter, J. Phys. A: Math. Theor. , 153001, (2009). L. S. Simeonov and N. V. Vitanov, Phys. Rev. A , 012123(2016). B. T. Torosov and N. V. Vitanov, Phys Rev. A , 013845 (2017). R. Grimaudo, A. S. M. de Castro, M. Ku´s, and A. Messina, Phys.Rev A , 033835 (2018). R. Grimaudo, A. Messina, P. A. Ivanov and N. V. Vitanov J. Phys.A R. Grimaudo,Y. Belousov, H. Nakazato and A. Messina, Ann.Phys. (NY) , 242 (2017). R. Grimaudo, L. Lamata, E. Solano, A. Messina, Phys. Rev. A ,042330 (2018). V. G. Bagrov, D. M. Gitman, M. C. Baldiotti and A. D. Levin,Ann. Phys. (Berlin) (11) 764 (2005). M. Kuna and J. Naudts Rep. Math. Phys. (1) 77 (2010). E. Barnes and S. Das Sarma Phys. Rev. Lett. A. Messina and H. Nakazato J. Phys. A: Math. Theor. L. A. Markovich, R. Grimaudo, A. Messina and H. NakazatoAnn. Phys. (NY)
522 (2017). R. Grimaudo, A. S. M. de Castro, H. Nakazato and A. Messina,Ann. Phys. (Berlin) , 12 1800198 (2018). T. Suzuki, H. Nakazato, Roberto Grimaudo and AntoninoMessina, Sc. Rep. , 17433 (2018). C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauss,T. Fukuhara, I. Bloch, and S. Kuhr, Nature , 319 (2011). T. Monz, D. Nigg, E. A. Martinez, M. F. Brandl, P. Schindler,R. Rines, S. X. Wang, I. L. Chuang, R. Blatt, Science , 1068(2016). A. Bermudez, X. Xu, R. Nigmatullin, J. OGorman, V. Negnevit-sky, P. Schindler, T. Monz, U.G. Poschinger, C. Hempel, J. Home,F. Schmidt-Kaler, M. Biercuk, R. Blatt, S. Benjamin, and M.M¨uller, Phys. Rev. X , 041061 (2017). C. Piltz, T. Sriarunothai, A. F. Varon, and C. Wunderlich, NatureCommun. , 4679, (2014). B. Lekitsch, S. Weidt, A. G. Fowler, K. Mølmer, S. J. Devitt, C.Wunderlich, and W. K. Hensinger, Science Advances , e1601540(2017). R. Grimaudo, N. V. Vitanov, A. Messina, arXiv:1901.00322v1(2019). A. Sergi, G. Hanna, R. Grimaudo and A. Messina, Symmetry,10