Nonlinear Landau-Zener Processes in a Periodic Driving Field

Effects of a periodic driving field on Landau-Zener processes are studied using a nonlinear two-mode model that describes the mean-field dynamics of a many-body system. A variety of different dynamical phenomena in different parameter regimes of the driving field are discussed and analyzed. These include shifted, weakened, or enhanced phase dependence of nonlinear Landau-Zener processes, nonlinearity-enhanced population transfer in the adiabatic limit, and Hamiltonian chaos on the mean field level. The emphasis of this work is placed on how the impact of a periodic driving field on Landau-Zener processes with self-interaction differs from those without self-interaction. Aside from gaining understandings of driven nonlinear Landau-Zener processes, our findings can be used to gauge the strength of nonlinearity and for efficient manipulation of the mean-field dynamics of many-body systems.


I. INTRODUCTION
Landau-Zener (LZ) transition [1] forms a fundamental dynamical process relevant to a variety of contexts, such as quantum control and quantum information [2,3,4,5,6], quantum dots [7], molecular clusters [8], to name a few. If the external bias involved in a LZ process is tuned adiabatically, then the LZ tunneling probability goes to zero, yielding a robust scenario for realizing a complete quantum population inversion. If the external bias is varied non-adiabatically, then the LZ tunneling offers a diagnosis tool for understanding the quantum dynamics. Indeed, the paramount importance of LZ processes has motivated a large body of theoretical and experimental work [9,10].
As a very recent development, Wubs et al. [11] studied numerically and analytically how a LZ process might be further manipulated by considering a periodic driving field. Within the rotating-wave-approximation (RWA), Wubs et al. [11] showed that a LZ process in the presence of a driving field possesses two intriguing features. First, the driving field can induce interesting quantum interference effects between two well-separated LZ sub-processes, with the final quantum transition probability dependent upon the phase of the driving field.
Second, contrary to a standard LZ problem, the population inversion probability approaches zero (instead of unity) if the bias is tuned adiabatically. These findings may find novel applications of LZ processes, especially in identifying an unknown phase of a driving field or in studying decoherence time scales of a system coupled with an environment.
Generalizing LZ processes for single quantum systems to those associated with the meanfield dynamics of interacting many-body systems, one obtains nonlinear LZ (NLZ) processes.
Due to their implications for quantum control of the dynamics of Bose-Einstein condensates (BEC) [12], NLZ processes have also attracted considerable interests. Note that the meanfield dynamics of an interacting many-body system is necessarily nonlinear, and as a result simple physical intuitions based on linear LZ processes may become invalid in nonlinear cases. For example, the adiabatic following of a quantum state with an external bias may break down in an NLZ process, even when the external bias varies at an infinitely slow rate [13,14,15].
Extending the work by Wubs et al. [11] from the linear regime to the nonlinear regime, our interest here is in how a periodic driving field might affect NLZ processes. It is hoped that a complete picture of driven NLZ processes will help make the best use of a driving field in manipulating the mean-field dynamics of many-body systems. To reach that long-term goal, it is necessary to first examine how the impact of a periodic driving field on NLZ processes differs from that on conventional LZ processes.
In our previous work [16], NLZ processes subject to a high-frequency and large-amplitude driving field were studied. There, using a high-frequency approximation, we showed that the driven NLZ dynamics can be effectively described by a stationary Hamiltonian. The spectrum of the effective Hamiltonian can have new degenerate eigenstates and hence display new topological structures that are absent in the non-driven cases. The results offer a simple approach to the complete suppression of NLZ tunneling, even when the tunneling is doomed to happen in the absence of a driving field.
This work continues to study driven NLZ dynamics, but with the parameter regimes different from that in Ref. [16]. We place our emphasis on comparisons between driven (linear) LZ processes and driven NLZ processes, thus shedding more light on the nonlinear and mean-field nature of NLZ dynamics. For example, we shall show that nonlinearity can induce significant quantum population transfer that is completely absent in linear cases. We will also show that the extent of population transfer in driven NLZ processes can be, as compared with driven LZ processes, either much more or much less sensitive to the phase of the driving field. It is hoped that our detailed results below will motivate more efforts towards better understanding and better control of many-body systems. This paper is organized as follows. In Sec. II we briefly discuss our two-mode model of NLZ processes in a periodic driving field. In Sec. III we discuss the high-frequency approximation in treating NLZ processes in a high-frequency and large-amplitude driving field. In Sec. IV we consider a different regime of field parameters, where the RWA can be applied. In Sec. V we study another parameter regime, where neither the high-frequency approximation nor the RWA is valid. Section V concludes this work.

DIAGONAL" DRIVING FIELD
Motivated by the model of linear driven LZ processes considered in Ref. [11], we consider a two-mode model of driven NLZ processes as follows: Here, denotes a time-dependent bias between two modes of interest and is being varied at a rate α; |a| 2 and |b| 2 represent occupation probabilities on the two modes, with the normalization condition taken as |a| 2 + |b| 2 = 1. The terms containing c characterize the nonlinear selfinteraction of a BEC under the mean-field treatment, with the value of c proportional to the number of bosons and the s-wave scattering length [12,13,14,15]; ∆ C represents the static coupling between the two modes; and ∆ 0 is the amplitude of an external driving, with the frequency ω and the phase parameter β. Unlike previous models for driven LZ/NLZ processes [12,17,18] or for coherent destruction of tunneling [9,19] (for a recent beautiful experimental validation of the latter phenomenon see in Refs. [20,21]), the driving field here appears only in the off-diagonal terms of the above Hamiltonian and is hence an "offdiagonal" driving field. That is, the external field directly modulates the coupling between the two modes, rather than modulating their energy bias. The initial state of an NLZ process is always taken as a = 1 and b = 0, and the occupation probability |b| 2 in the end, denoted by |b(∞)| 2 , is the final quantum transition probability (or the final population transferred between the two modes). Note also that all the variables here should be understood as scaled dimensionless variables withh = 1. Throughout we use ∆ 0 to scale. Then, ∆ 0 = 1, ω is in units of ∆ 0 /h, α is in units of ∆ 2 0 /h, and c is in units of ∆ 0 . Within the BEC context, the above Hamiltonian may be experimentally realized in several ways. For example, one may consider a BEC in a double-well potential [22,23,24,25], with the barrier height periodically modulated, or by a BEC in an optical lattice occupying two bands [26], with the well-depth of the optical lattice periodically modulated. One may also regard the two modes as two internal states of a BEC, such as 87 Rb [27], with the energy bias γ effectively realized by the detuning of a coupling field from the resonance and the off-diagonal modulation achieved by modulating the intensity of the coupling field. Though we only refer to the BEC context below, it should be noted that this two-mode model of NLZ might be relevant to physics of Josephson junctions as well as nonlinear optics [28].
In the special case of ∆ 0 = 0, the above Hamiltonian reduces to the well-known model of non-driven NLZ processes [14]. Therein the eigen-spectrum diagram as a function of γ is known to display a loop structure at the tip of the lower (upper) level for c > ∆ C (c < −∆ C ).
Such a loop structure, absent in linear systems, directly leads to a nonzero LZ transition probability even when γ changes adiabatically. This makes it interesting to examine what happens if an NLZ process is subject to a periodic driving field as introduced above.
Without loss of generality we will restrict ourselves to the c > 0 cases (cases with c < 0 can be mapped to those with c > 0). This c > 0 assumption requires, for example, an attractive interaction for bosons in a double-well potential or a repulsive interaction for bosons in two energy bands of an optical lattice. In current non-modulated double-well BEC experiments, typical values of |c/∆ C | for 10 3 bosons range from ∼ 10 0 to ∼ 10 1 [12,25]. For the latter two-band realization of our two-mode model, the value of c (scaled by ∆ 0 ) can be easily tuned by controlling the maximal depth of an optical lattice potential [14].
To examine the driven NLZ dynamics (i.e., ∆ 0 = 0), we consider below three different parameter regimes. In the first high-frequency and large-amplitude regime , a high-frequency approximation can be used to obtain an effective Hamiltonian. The second regime is the RWA regime, where the driven dynamics is again understood in terms of some effective Hamiltonians. In the third regime, both the driving frequency ω and the amplitude ∆ 0 are comparable to, or smaller than, the nonlinear parameter c. Our task below is to examine the driven NLZ dynamics in these three regimes.

III. HIGH-FREQUENCY AND LARGE-AMPLITUDE DRIVING FIELD
The NLZ dynamics under the condition ω, ∆ 0 ≫ γ 0 , c, ∆ C , γ 0 (where γ 0 is the initial value of γ) was first studied by us in Ref. [16] using a high-frequency approximation. For completeness and for the sake of comparison with other regimes, here we briefly revisit this regime. To that end we first introduce another pair of wave function parameters (a ′ , b ′ ) Substituting these two relations into the Hamiltonian in Eq. (1), one finds the equations of motion for (a ′ , b ′ ): where θ ≡ ∆ 0 ω sin(ωt + β). For sufficiently large ω, the oscillation in θ is much faster than all other time scales of the system. As such, during each period of T = 2π/ω, the change in a ′ and b ′ is negligible and considering their averages over T will greatly reduce the equations of motion. Specifically, upon a time averaging the odd functions of t in Eq. (4) vanish and the effective equations of motion become where , and J 0 is the zeroth order Bessel function of the first kind. Apparently, these newly defined parameters reflect the action of the high-frequency driving field. Based on Eq. (5), we can define the following effective Hamiltonian: where we have replaced a ′ by a, b ′ by b, and so on. Comparing this effective Hamiltonian with the original one in Eq. (1) for ∆ 0 = 0, it is seen that the nonlinear parameter c Z can be regarded as a re-scaled parameter c, and the new nonlinear terms containing c Y arise as a surprise. In addition, the ratio of the two nonlinear parameters c Z and c Y is given by , which is easily adjustable by tuning the ratio ∆ 0 /ω.
In terms of the mean-field eigen-energies of H eff as a function γ, three typical level structures of H eff are depicted in Fig. 1. The structures represented by the solid lines in Fig. 1 are also typical in non-driven NLZ models [14], but the dashed lines (each of them is associated with double degenerate eigenstates) in Fig. 1 are entirely induced by the high-frequency driving field. For an NLZ process with γ increasing at a very small rate of α = 0.002 (hence an adiabatic NLZ process), the time-evolving expectation values of the mean-field energy associated with H eff are shown by the discrete points in Fig. 1. Clearly, for the two cases in Fig. 1(a) and 1(c), the evolution of the system perfectly follows the new eigenstates (dashed lines). The driving field can hence dramatically affect the dynamics of adiabatic following.
Indeed, in the case of Fig. 1(a), if there were no additional levels induced by the driving field, then the loop structure there will necessarily cause the adiabatic following to break down (this is analogous to what is observed in non-driven NLZ models [14] and will be also seen below). For other important findings in this high-frequency large-amplitude regime, please see Ref. [16].
The high-frequency approximation used here is also applicable to high-frequency and small-amplitude fields.
the effective Hamiltonian in Eq. (6) will essentially reduce to the Hamitonian of Eq. (1) with ∆ 0 = 0. That is, in cases of ∆ 0 /ω << 1, the present high-frequency approximation based on the (a ′ , b ′ ) representation does not directly give useful insights into the dynamics.
Fortunately, in these cases the RWA treatment becomes valid and more advantageous to use. This is discussed in detail in the next section.

A. Rotating-Wave Approximation
In the high-frequency and small-amplitude regime, we adopt the RWA to understand the effects of a driving field on NLZ processes. The off-diagonal coupling term in Eq. (1) is now understood as a superposition of three terms, where the first two terms represent the circular modulation with frequency ω and −ω, and the last constant term can be regarded as a circular modulation with zero frequency.
Consistent with the high-frequency assumption, the effects of the three circular modulation terms can be analyzed separately. To illustrate this, we depict in Fig. 2 a typical numerical result for this parameter regime. Evidently, the transitions primarily occur in three small time windows around γ ∼ −ω, 0, ω, each of them is associated with one of the three circular modulation terms in Eq. (7). For the sake of a direct comparison between driven NLZ processes and the driven LZ processes studied in Ref. [11], below we set ∆ C = 0. We then only need to consider two circular modulation terms. The effects of the first circular modulation term ∆ 0 2 exp(−iωt−iβ) become obvious in a rotating frame. We hence introduce another pair of wavefunction Using this representation and neglecting the counter-rotating terms, one obtains where the RWA effective HamiltonianH + (t) is given bỹ with γ + = γ + ω. This Hamiltonian is essentially identical with that for a non-driven NLZ model, with the bias parameter γ shifted by ω and with the effective static coupling term determined by the amplitude of the periodic driving field. This explains the observation from Fig. 2 that one main transition window is roughly at γ ∼ −ω. Because |b| 2 = |b| 2 , this RWA treatment also indicates that for a fixed initial state, the quantum population at the end of such a transition window should not depend on β. Similar treatments apply to the other circulation modulation term ∆ 0 2 exp(iωt + iβ), yielding the following effective HamiltonianH Here the effective bias is γ − = γ − ω, and as a result the main transition window will be close to γ ∼ ω.
Because the RWA treatment here is seen to be independent of the nonlinear parameter c, it applies as well to those driven (linear) LZ processes considered in Ref. [11]. Hence, for either NLZ or LZ processes, a periodic driving field is expected to shift the main quantum transition window at γ ∼ 0 to two new windows close to γ ∼ ∓ω (how the exact locations of the transition windows depend on c will be discussed later).

B. Phase-dependence weakened by strong nonlinearity
The above RWA treatment makes it clear that the NLZ dynamics in this regime can be described by effective Hamiltonians analogous to that for non-driven cases. However, because the final state of the transition window around γ ∼ −ω becomes the initial state of the second transition window around γ ∼ ω, the driven dynamics should be much richer than in non-driven cases. Indeed, in the linear case with c = 0, the analytical result from Ref. [11] predicts that the final transition probability is given by where β is the phase of the driving field defined above. Because the transition probability of the first LZ sub-process does not depend upon β, the factor cos 2 (β) in Eq. (12) reflects how the net result associated with the second transition window is affected by the first one.
Alternatively, this cos 2 (β) factor can be interpreted as a quantum interference between two LZ sub-processes at two time windows [11]. This intriguing phase dependence might be The weakened β-dependence due to strong nonlinearity requires a qualitative explanation.
Consider the state after the first transition window, i.e., the initial state for the second NLZ sub-process. As β varies, the occupation populations associated with this intermediate state can be hardly changed (as is evident from the above RWA treatment), but the relative phase between the amplitudesã ′ andb ′ will be affected. With this understanding, we explain below the observed weakened β-dependence of |b(∞)| 2 in terms of the increasing insensitivity of the second NLZ sub-process on the quantum phase of its initial state.
Let us map exactly the mean-field dynamics of the effective Hamiltonian in Eq. (11) to that of a classical Hamiltonian system, with the canonical variables S = |b ′ | 2 − |ã ′ | 2 and φ [13,16,18], with φ the relative phase between the amplitudesã ′ andb ′ . The mapped classical Hamiltonian, denoted byH c , is then given by [13]: With this mapping, the phase of the initial state for the second NLZ sub-process apparently becomes the initial value of the coordinate φ.   Indeed, in the large c limit, one may only keep the −cS 2 /2 term ofH c .H c then becomes the Hamiltonian for a "free-particle" system, with a trivial phase space structure filled with straight lines perpendicular to the S-axis. As such, the β-dependence of |b(∞) 2 | in the large c limit will disappear altogether. This completes our qualitative explanation of the weakened β-dependence due to increasing nonlinearity.
Finally, we discuss another interesting observation made from Fig. 3 system reaches the right "edge" of the loop, thus postponing the first transition window. In addition, because the loop structure is bigger for larger c, the first transition window will be postponed more as c increases.
Interestingly, after the first NLZ sub-process, the system will in general occupy both upper and lower levels considerably. Then, as γ continues to increase in the positive regime, those populations on the upper branch will make significant transitions at γ ∼ ω, whereas those on the lower branch will delay their transitions again. The overall result is that the precise location of the second transition window becomes less definitive and more dependent on the details of the dynamics. With this recognition we schematically place the second transition window in Fig. 5 between γ ∼ ω and the right edge of the right loop structure.

C. Nonlinearity-induced Transition Probabilities
Equation (12) contains another intriguing and stimulating result for driven linear LZ processes. That is, in the adiabatic limit α → 0, the final transition probability |b(∞)| 2 becomes zero for any β. The associated physical picture is as follows. After the first transition window of a linear driven LZ process, the adiabatic population transfer is already complete. But then the system approaches the second transition window, which transfers all the population back to the initial mode. In other words, for linear driven LZ processes, the transitions associated with the two time windows γ ∼ ∓ω exactly cancel each other in the adiabatic limit. Such a strong result for linear systems motivates us to examine driven NLZ processes in the same adiabatic limit.
For two values of the nonlinear parameter c, Fig. 6 shows the time-evolving occupation probability |b| 2 , using a sufficiently small value of α to simulate the dynamics in the adiabatic  one also sees that the larger c is, the more delayed the first transition window will be. This observation is analogous to what is found in Fig. 3.
To further motivate interests in the nonlinearity-induced transition probabilities in driven NLZ processes, we calculate |b(∞)| 2 in many adiabatic cases that cover a wide range of c values. A typical result, obtained with ω = 40 and α = 0.003, is depicted in Fig. 7. The non-monotonic dependence of |b(∞)| 2 upon c is apparent. In particular, for any c < 0.5, the transition probability right after the first transition window is close to unity and becomes essentially zero after the second transition window. This behavior is the same as in linear driven LZ processes in the adiabatic limit. Hence, for the RWA regime studied here, the threshold value for observing nonlinearity-induced population transfer at the end of the adiabatic process is around c ∼ 0.5. of c , and (iv) the nonlinearity-induced transition probabilities in the adiabatic limit, one may also design strict tests to check the validity of a two-mode description of a driven many-body quantum system.

V. LOW-FREQUENCY DRIVING FIELDS
If the driving frequency ω is comparable to, or smaller than, c and ∆ 0 , we call it a lowfrequency case. In this regime the previous high-frequency approximation or RWA does not apply. As it turns out, complications arise here due to the emergence of Hamiltonian chaos on the mean-field level.
Using the same approach in Sec. IV-B, let us first map the driven NLZ dynamics described by Eq. (1) (with ∆ C = 0) to that of a classical Hamiltonian, where S is given by (2|b| 2 − 1), and φ is now the relative phase between the amplitudes a and b.
Because ω is small, it is impossible to divide the driven NLZ process into two sub-processes. But this also implies that, roughly speaking, the phase space structure at γ ∼ 0 will be most important in understanding the mean-field dynamics. As such, we show in Fig. 8  values of c, the dynamics is primarily regular.
In Fig. 9 we plot |b(∞)| 2 as a function of β, with the values of ω same as in Fig. 8.
Consider first Fig. 9(a) for the c = 0 case. The β-dependence is seen to be rather weak, in contrast to the cos 2 (β) result in the RWA regime. In addition, |b(∞)| 2 is small for the entire range of β. Next, in Fig. 9(b) for c = 0.5, the β dependence remains weak, but the final transition probability |b(∞)| 2 becomes very significant in general: it can even reach almost 100% for certain values of β. We hence observe nonlinearity-enhanced population transfer again, but now outside the RWA regime. As c is increased to c = 1.25 in Fig. 9(c), drastic and full-range oscillations of |b(∞)| 2 are observed. Because Fig. 8(c) shows that this case corresponds to full Hamiltonian chaos, the full-range oscillation in Fig. 9(c) is clearly related to the underlying complete chaos on the mean-field level. As the nonlinearity parameter c further increases, the oscillation range of |b(∞)| 2 decreases, consistent with the observation from Fig. 8 that the chaotic layer becomes smaller and smaller. In the case of c = 75 in Fig.   9(f), there is essentially no transition probability for any value of β. We have also done more detailed analysis of how mean-field trajectories should move as γ is ramped, confirming that the size of the chaotic layer determines the oscillation range of |b(∞)| 2 . Comparing results in Fig. 9(c) and 9(d) with those in Fig. 9(a) and 9(b), one finds another interesting aspect of driven NLZ dynamics. That is, contrary to our observation in the RWA regime, nonlinearity may also enhance the β-dependence of |b(∞)| 2 . Note, however, that the strong β-dependence observed here is due to the mean-field chaos. Because chaos brings about strong instabilities and hence large fluctuations in the quantum system, results directly associated with the mean-field chaos should be applied with caution. For example, the validity of the mean-field treatment itself in the chaotic cases might be questionable when the number of particles in a BEC is not extremely large. This being so, a low-frequency driving field generating the mean-field chaos could be used to study effects of quantum fluctuations via the β-dependence of |b(∞)| 2 .

VI. CONCLUSION
Nonlinear extensions of LZ processes, and in particular, nonlinear LZ processes in a periodic driving field, are of importance in realizing the control of interacting many-body systems. By considering different parameter regimes of a periodic driving field, we have exposed a number of interesting dynamical phenomena in driven NLZ processes. Many of these dynamical phenomena can be used for the control of NLZ processes by a driving field.
In the high-frequency and large-amplitude regime, a driving field can effectively generate new self-interaction terms that are absent in non-driven two-mode NLZ models. The field can hence induce new energy-level structures and can be useful in simulating new systems not considered before. A driving-field-based scenario for the suppression of undesired NLZ tunneling probabilities also becomes possible.
In the high-frequency and small-amplitude regime, driven NLZ processes are also much different from driven LZ processes in at least three aspects. First, for small nonlinearity strength the final transition probability shows a shifted phase dependence on the driving field. Second, for strong nonlinearity strength the transition probability shows a much weakened phase dependence upon the driving field. Third, in the adiabatic limit, significant transition probabilities become a purely nonlinear effect, because they were precisely zero in the absence of the self-interaction. These differences between driven NLZ and driven LZ processes suggest that a driving field may be used to directly expose and measure nonlinear effects. Furthermore, the adiabatic population transfer based entirely on the nonlinear nature of the system can be almost complete for certain amplitudes of the driving field, thus leading to an interesting approach to the improvement of adiabatic population transfer in the presence of nonlinear Landau-Zener tunneling.
In the third low-frequency regime, nonlinearity may also enhance the final transition probability as well as its dependence on the phase of the driving field. The enhanced phase dependence occurs for intermediate nonlinear strength, where the NLZ dynamics displays Hamiltonian chaos. A driving field in this regime can therefore be used for studies of quantum fluctuations beyond the mean-field level, as well as studies of manifestations of mean-field chaos in the final transition probabilities.
To conclude, we have exposed many interesting aspects of driven nonlinear Landau-Zener processes, by considering different parameter regimes of a single-frequency and constantamplitude driving field. It is plausible that the driven nonlinear Landau-Zener processes are even much richer in the presence of more complex driving fields, such as those with multiple frequency components that have time-evolving amplitudes and phases. It should be also interesting to extend our nonlinear Landau-Zener treatments to other many-body generalizations of Landau-Zener processes that can go beyond the mean-field level [29,30,31].