Exactly Solvable Generalized Semi-Classical Rabi Systems
The exact quantum dynamics of a single spin-1/2 in a generic time-dependent classical magnetic field is investigated and compared with the quantum motion of a spin-1/2 studied by Rabi and Schwinger. The possibility of regarding the scenario studied in this paper as a generalization of that considered by Rabi and Schwinger is discussed and a notion of time-dependent resonance condition is introduced and carefully legitimated and analysed. Several examples help to disclose analogies and departures of the quantum motion induced in a generalized Rabi system with respect to that exhibited by the spin-1/2 in a magnetic field precessing around the -axis. We find that, under generalized resonance condition, the time evolution of the transition probability between the two eigenstates of may be dominated by a regime of distorted oscillations, or may even exhibit a monotonic behaviour. At the same time we succeed in predicting no oscillations or even oscillations of maximum amplitude in the behaviour of under general conditions. New scenarios of experimental interest originating a Landau-Zener transition is brought to light.
Keywords: Time-dependent Hamiltonians, Exactly solvable single spin models, SU(2) dynamical problems, Semi classical time-dependent models.
Spin language is transversal to every physical scenario wherein, by definition, the states of the specific system under scrutiny live in a finite-dimensional Hilbert space. When the generally time-dependent Hamiltonian of the system may in particular be expressed as linear combination of the three generators of su(2), the corresponding quantum dynamical problem coincides with that of a spin in a time-dependent magnetic field. It is well known that the time evolution of the spin in this case is fully recoverable from that of a spin subjected to the same magnetic field . On the other hand, however, a closed general form of the 2x2 unitary evolution operator is still unavailable. As a consequence, to bring to light new dynamical scenarios of a spin-1/2 represents a target of basic and applicative importance in many contexts such as quantum optics , quantum control [3, 4], quantum information, and quantum computing [5, 6, 7].
Rabi [8, 9] and Schwinger  exactly solved the quantum dynamics of a spin-1/2 in the now called Rabi scenario, that is, subjected to a static magnetic field along the -axis and an r.f. magnetic field rotating in the - plane with frequency , namely
and being fixed unit vectors in the laboratory frame. Their seminal papers show that the probability of transition between the two Zeeman states generated by is dominated by periodic oscillations reaching maximum amplitude under the so-called resonance condition . Here is the spin Larmor frequency. The exact treatment of this basic problem provides the robust platform for the NMR technology implementation . The recently published special issue on semiclassical and quantum Rabi models  witnesses the evergreen attractiveness of this problem.
Searching new exactly solvable time-dependent scenarios of a single spin-1/2 could be very interesting and worth both from basic and applicative points of view, expecially in the quantum control context. To pursue this target, over the last years, new methods have been developed to face the problem with an original strategy [14, 15, 16, 17, 18]. We stress that progresses along this direction might be of relevance even for treating time-dependent Hamiltonians describing interacting qubits or qudits [19, 20, 21]. In turn exact treatments of such scenarios might stimulate the interpretation of experimental results in fields from condensed matter physics [22, 23] to quantum information and quantum computing [24, 25, 27, 26].
In this paper we construct the exact time evolution of a spin-1/2 subjected to time-dependent magnetic fields never considered before. Our investigation introduces, in a very natural way, three different classes of Generalized Rabi Systems (GRSs) wherein Rabi oscillations of maximum amplitude still survive. We succeed indeed in identifying generalized resonance conditions which are, as in the Rabi scenario, at the origin of the complete population transfer between the two Zeeman levels of the spin. We bring to light that, even at resonance, these oscillations might loose its periodic character, significantly differing, thus, from the sinusoidal behaviour occurring in the Rabi scenario. In this paper we have also considered time-dependent magnetic fields giving rise to exactly solvable models not satisfying the resonance condition. In this way, we are able to write down a special link among all the time-dependent parameters appearing in the Hamiltonian of the system even if the resonance condition is not met, for which, however, the dynamical problem is exactly solvable. The departure of the time evolution of the transition probability out of generalized resonance from the corresponding behaviour in the Rabi scenario is illustrated with the help of two exemplary cases.
The paper is organized as follows: in Sec. 2 two sets of parametrized solutions of the dynamical problem of a single spin-1/2 in a time-dependent magnetic field are given. The generalized time-dependent resonance condition and the generalized out of resonance case are introduced and physically legitimated In Sec. 3. In the subsequent section (Sec. 4) several examples illustrate the occurrence of effects on the Rabi transition probability due to the magnetic field time-dependence. Finally, in the Conclusion section, a summary of the results with possible applications and future outlooks are briefly discussed.
2 Resolution of 2x2 SU(2) quantum dynamical problems
The problem of a single spin-1/2 subjected to a generic time-dependent magnetic field is investigated by assuming the SU(2) Hamiltonian model
Here is the magnetic moment associated to the spin-1/2, and being the appropriate Landé factor and the Bohr magneton, respectively. The entries and of the unitary time evolution operator
generated by , must satisfy the Cauchy-Liouville problem , , which originates the following system of linear differential equations
It is possible to demonstrate  that if is a complex-valued function of satisfying the nonlinear integral-differential Cauchy problem
then the solutions of the Cauchy problem (5) can be represented as follows
Generally speaking, solving Eq. (6) is a difficult task. This equation however may be exploited in a different way, giving rise to a strategy  aimed at singling out exactly solvable dynamical problems represented by Eq. (5). Fixing, indeed, at will the function in Eqs. (6), that is, regarded now as a parameter (function) rather than an unknown, determines a link between and under which the corresponding dynamical problem may be exactly solved in view of Eqs. (7). We emphasize that if we knew the solution of the Cauchy problem given in Eq. (6), whatever , and are, then we would be in condition to solve in general the corresponding Cauchy dynamical problem expressed by Eqs. (5).
Another useful way of parametrizing the expressions of and is
being an arbitrary real number and having put, without loss of generality, . In this case, it is possible to check that they solve the system (5) if the following condition holds
It is stressed that this last equation does only express the condition under which, whatever is, the representations (9) satisfy the Cauchy problem (5). This means that the real number plays in this case the role of parameter. When Eq. (11) cannot be satisfied for any , of course the solution of the dynamical problem exists but cannot be represented using Eqs. (9). In this case there certainly exists a function enabling the representation of the solutions by using Eqs. (7). Finally, it is interesting to underline that Eq. (6a) turns into the simpler condition (11) on under an appropriate choice of the parameter function .
3 Generalized Resonance Condition and out of Resonance Cases
The experimental set-up considered by Rabi, as described in the introduction, leads to the Hamiltonian model (2) where
Then it is characterized by the three time-independent parameters: , and . In this paper we generalize this Rabi scenario by making some out of or all these parameters time-dependent: , and .
Firstly, we rewrite the general Hamiltonian (2) as follows
with and being Pauli matrices. Generalizing the approach in Ref. , we pass from the laboratory frame to the time-dependent one tuned with , where the time-dependent Schrödinger equation for the transformed state, , is governed by the following effective time-dependent transformed Hamiltonian
It is worth noticing its strict similarity with the analogous one got in Ref.  where the unitary transformation is indeed a uniform rotation around the -axis. In fact, it is enough to make , and time-independent in ( stands for Generalized Rabi) to immediately recover the transformed Hamiltonian got by Rabi . On the basis of this observation it then appears natural to refer to the following condition
as a generalized resonance condition, in accordance with the corresponding static resonance condition brought to light by Rabi in Ref. . We underline that the generalized resonance condition does not lead to a time-independent transformed dynamical problem (as it happens in the Rabi scenario), but, whatever is, it easily enables the explicit construction of the time evolution operator describing the quantum motion of the spin in the laboratory frame. In view of Eq. (11), the entries of such an operator are indeed exactly given by Eqs. (9) in the limit , namely
By definition, we say to be in generalized out of resonance when the left hand side of Eq. (15) is non-vanishing, namely
where is an arbitrary energy-dimensioned well-behaved function of time. Generally speaking, to find the exact quantum dynamics of the spin in the generic out of resonance case, is a very complicated mathematical problem. Let us observe that, on the basis of the structure of in Eq. (14), when is proportional to , the dynamical problem may be exactly solved. Indeed, this condition coincides with that expressed by Eq. (11) which in turn enables one to write down exact solutions of the Cauchy problem (5) in the form given by Eqs. (9). In this paper we report the exact solutions of special non trivial out of resonance dynamical problems. Our aim is to illustrate the occurrence of analogies and differences in the time behaviour on the Rabi transition probability
(), when the time evolution of the magnetic field acting upon the spin cannot be described as a perfect precession around the -axis.
4 Examples of Generalized Rabi models
This section is aimed at showing how the Rabi transition probability is only slightly as well as strongly affected by different choices of the time-dependent magnetic fields under general conditions. The following examples are reported to illustrate such behaviour.
4.1 Examples of GRSs Dynamics under Generalized Resonance Condition
Let us consider, firstly, the generalized resonance condition in Eq. (15). We know that, in this instance, the time evolution operator is characterized by the time behaviour of its two entries given in Eq. (16), so that the transition probability reads
It is immediately evident that could or could not be periodic. Indeed, e.g., setting , we get
resulting in a Landau-Zener-like transition , that is an asymptotic aperiodic inversion of population.
However, of course, it is easy to understand that it is possible to make choices either resulting in a oscillating but not periodic transition probability or exhibiting a periodic behaviour, even if not coincident with that characterizing the Rabi scenario. If we consider, for example,
reproducible by engineering the transverse magnetic field as
the resulting transition probability yields
with . We point out that, for the sake of simplicity, in Eqs. (22) we have put , even if, in general, the expression of the probability in Eq. (23) holds whatever is, provided that Eq. (15) is satisfied. In Figs. a and b we report the time dependence of the two components of the transverse magnetic field, with respect to , accomplishing the time dependence of given in Eq. (21). Figure c, instead, shows the time behaviour of the magnetic field in the - plane, against the dimensionless parameter , when and .
We recognize the existence of a transient wherein exhibits aperiodic oscillations of maximum amplitude which, after a finite interval of time, turn into a monotonic increase that asymptotically approaches 1. We emphasize that the number of complete oscillations, preceding the asymptotic behaviour of as well as itself, are -dependent. Equation (23), indeed, predicts
which immediately leads to
As our second example, we consider the following modulation of
realizable by engineering the transverse magnetic field as
Here , and with . The transverse field is represented in Fig. 3 against the adimensional time parameter , once more supposing for simplicity .
In this case, the Rabi’s transition probability results
Differently from the previous example, we see that, in this case, the characteristic sinusoidal behaviour of the Rabi transition probability turns into a periodic population transfer, still of maximum amplitude, between the two energy levels of the spin. We emphasize that, in view of Eq. (19), different time evolutions of require different choices of only, then regardless of and which are constrained by the generalized resonance condition (15) only. We stress, however, that distinct realizations of the resonance condition, keeping the same , introduce significant changes in the dynamical behaviour of the GRS with respect to the Rabi system. It is enough to consider, for example, that
depend on both and , in view of Eqs. (16).
4.2 Examples of GRSs Dynamics in Generalized out of Resonance Cases
In this subsection we analyse the generalized out of resonance case, defined in Eq. (17). Since it appears hopeless to have an exact closed treatment of the Cauchy problem in Eq. (6) with an arbitrary , we confine ourselves to the following specific forms
where and are non-negative adimensional functions. The first form coincides with the condition in Eq. (11) with . In this case, the solutions and of the system in Eq. (5) may be cast as reported in Eqs. (9) so that
In the limit we recover Eq. (19) from this equation. Thus, we may compare in the resonant and this non-resonant cases when is fixed in the same way. It is easy to convince oneself that the main effect of a positive value of the parameter on is nothing but a scale effect determined by the ratio .
We wish now to bring to light and to discuss some exactly solvable scenarios of generalized, out of resonance, Rabi problems wherein . To this end, it is useful to observe that postulating as function of through
we would get, by Eq. (6), the desired form of , by construction. We stress however that the corresponding function would be functionally-dependent on , that is determined by the knowledge of . We emphasize that this aspect, however, does not spoil of interest such a particular procedure. In the following examples we indeed report two applications of the general strategy  exposed after Eq. (8) in Sec. (2), where those s that make Eqs. (6) solvable are fixed.
Assuming the solution of the Cauchy problem (6) as
it is straightforward to show that
Equation (6a) immediately yields
From this point on, we are ready to specialize Eqs. (7) getting
with . It is interesting to consider a simple case in which . In this instance we have such a situation that goes from 0 (1), at , to , when , as it can be appreciated by their plots in Fig. 6: full blue and dot-dashed red lines, respectively. In Fig. a we may appreciate the time behaviour of related to this specific physical scenario. This specific out of resonance time-dependent scenario, then, asymptotically evolves the initial state towards an equal-weighted superposition of the two eigenstates of . One can convince oneself that this circumstance is intimately related to the fact that, in this case, the “detuning” vanishes asymptotically (see Fig. a), getting then established a dynamical reply of the system as if it were under the generalized resonance condition.
The second scenario is based on the following assumption
which, notwithstanding its apparent similarity with the previous case given in Eq. (34), leads however to a remarkable different temporal behaviour of the correspondent generalized Rabi system. This time it results in
so that the solutions of (5) read
Finally, the special form of underlying this specific scenario is
In this case, it is easy to see that if , () goes from 0 (1) to 1 (0) asymptotically. These behaviours, evoking the transition probabilities in the Landau-Zener scenario , are illustrated by green dashed and brown dotted lines, respectively, in Fig. 6.
In this case, the time behaviour of the “detuning” is characterized by an asymptotic linear dependence on , as shown in Fig. b.
As in the resonant scenario, even here different time-dependences of the magnetic field may give rise to qualitatively different time evolutions of with respect to the Rabi scenario. It is worth to note the possibility of occurrence of an oscillating behaviour of the transition probability, by making appropriate choices for . By the way, we note that and lead to an oscillating time evolution of which, in view of Eq. (7b), does have maximum amplitude. This aspect seems to be peculiar of out of resonance generalized Rabi systems in view of the fact that always oscillates with reduced amplitude for an out of resonance Rabi system. In such a case, however, it could be very hard to get analytical expressions for and , in Eq. (7a) and (7b), respectively. We would be, instead, always able to find accordingly the expressions of and . Nevertheless, such a bottleneck does not influence our capability to predict the Rabi transition probability and the expression of in order to know how to engineer the magnetic fields to get the desired time evolution. We stress that, as a consequence, when , and are chosen in such a way to generate the same detuning , the related different physical scenarios share the same analytical expressions of and and then of all physical observables depending only on these quantities, e.g. or
The Rabi scenario consists in a spin-1/2 subjected to a time-dependent magnetic field precessing around the quantized axis ()  and is characterized by three time-independent parameters: , and . Rabi shows that when the transverse magnetic field acts as a probe of the energy separation due to the longitudinal field alone. The measurable physical quantity revealing is the transition probability which, at resonance, oscillates between 0 and 1 with frequency now referred to as Rabi frequency.
In this work we generalize this Rabi scenario by assuming an SU(2) general time-dependent Hamiltonian model where then , and are now replaced with time-dependent counterparts. Along the lines of the Rabi approach , we firstly show that, in the rotating frame with the time-dependent angular frequency , the condition plays the same role of the Rabi resonance condition in the Rabi scenario. Such an occurrence makes of basic interest a direct comparison between the Rabi scenario and its generalized version on both time-dependent resonance and out of resonance () cases. To bring to light the occurrence of analogies and differences, we have focussed our attention on the study of the transition probability between the two eigenstates of .
We show that, on resonance, depends only on the integral of . Our examples illustrate that this circumstance determines a transition probability characterized by three possible different regimes: oscillatory (the only one dominating the Rabi scenario), monotonic and mixed which means an initial oscillatory transient followed by an asymptotic monotonic behaviour.
To capture significant dynamical consequences stemming from the detuning time dependence, we have constructed  exactly solvable problems and analysed the corresponding quantum dynamics of the spin-1/2. We have thus highlighted that when is proportional to , the main effect emerging in the time behaviour of is a scale effect both in amplitude and in frequency (like in the Rabi scenario). However, we have even shown that, in the generalized out of resonance case, oscillations of maximum amplitude are also possible, contrary to what happens in the Rabi scenario. We have further investigated two specific exactly solvable scenarios of experimental interest for which varies over time. One of them predicts a Landau-Zener transition, while the other an equal weighted superposition of the two states of the system. It is important to underline that our examples illustrate exactly solvable cases for which, then, the corresponding system dynamics may be fully disclosed. We highlighted, however, that when one is interested in the Rabi transition probability only, it is enough the knowledge of and . We point out that this circumstance leads us to wider and richer classes of physical scenarios, since it gets us rid of possible analytical difficulties stemming from Eq. (8).
As a final comment, we emphasize that the knowledge of the exactly solvable problems reported in this paper provide stimulating ideas for technological applications with single qubit devices. In addition it furnishes ready-to-use tools for interacting qudits systems [19, 20, 21], being of relevance in several fields, from condensed matter physics [22, 23] to quantum information and quantum computing [24, 25, 27, 26].
ASM de Castro acknowledges the Brazilian agency CNPq financial support Grant No. 453835/2014-7. RG acknowledges for economical support by research funds in memory of Francesca Palumbo.
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