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Hessian facilitated analysis of optimally controlled quantum dynamics of systems with coupled primary and secondary states Chuan-Cun Shu,ab Melanie Edwalds,a Alireza Shabani,ac Tak-San Hoa and Herschel Rabitz*a The efficacy of optimal control of quantum dynamics depends on the topology and associated local structure of the underlying control landscape defined as the objective as a function of the control field. A commonly studied control objective involves maximization of the transition probability for steering the quantum system from one state to another state. This paper invokes landscape Hessian analysis performed at an optimal solution to gain insight into the controlled dynamics, where the Hessian is the second-order functional derivative of the control objective with respect to the control field. Specifically, we consider a quantum system composed of coupled primary and secondary subspaces of energy levels with the initial and target states lying in the primary subspace. The primary and secondary subspaces may arise in various scenarios, for example, respectively, as sub-manifolds of ground and excited electronic states of a poly-atomic molecule, with each possessing a set of rotational–vibrational
Received 7th May 2015, Accepted 23rd June 2015
levels. The control field may engage the system through electric dipole transitions that occur either (I) only in the primary subspace, (II) between the two subspaces, or (III) only in the secondary subspace.
DOI: 10.1039/c5cp02660a
Important insights about the resultant dynamics in each case are revealed in the structural patterns of the corresponding Hessian. The Fourier spectrum of the Hessian is shown to often be complementary
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to mechanistic insights provided by the optimal control field and population dynamics.
I. Introduction Quantum dynamics control is generally concerned with finding optimal control fields to effectively manipulate dynamical processes. A frequently studied control process involves maximization of the transition probability for steering the quantum system from one state to another state. Quantum optimal control theory (OCT)1–5 has been applied to a wide variety of physical and chemical problems.6–10 The generally successful outcome from these OCT calculations on model systems is encouraging for wide ranging manipulation of quantum phenomena. A quantitative understanding of the topological properties of quantum control landscapes helps to explain why the underlying optimal control problems may be readily solved using a host of search algorithms. The dynamics of a quantum system in the presence of an applied field is often described in terms of a subset of energy levels that are directly engaged by the field. However, it is
a
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA. E-mail: [email protected] b School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia c Google Inc., Venice, California 90291, USA
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generally difficult to precisely know the number of engaged energy levels, for example, the rotational–vibrational–electronic (rovibronic) levels of a poly-atomic molecule, that are involved in the intervening transitions for achieving the control objective. Many quantum control studies consider an effective Hamiltonian that accounts for the neglected part of the energy levels of the quantum system.11–15 This work considers the state-to-state transition control problem and treats the optimally controlled quantum system as composed of two coupled subspaces of energy levels: one is the primary subspace that is a set of quantum states, generally of low energy levels, including both the initial and target states, and the other is the secondary subspace that may be a set of dynamically distinct internal states containing the energy levels above the primary subspace. The two subspaces are coupled by the field-free off-diagonal portion of the Hamiltonian. For example, this circumstance could arise from anharmonic coupling of a nominal set of multi-mode harmonic oscillator states. The aim of this paper is to explore the role of the primary and secondary subspaces when they are engaged with the control field in various distinct ways. The quantum system of field-free coupled primary and secondary quantum subspaces with additional dipole-mediated radiative coupling schemes can have distinct and complex controlled dynamics such that a complete mechanistic picture of the
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transitions utilized to reach the target state may not always be obtained by simply correlating the transition frequencies to those present in the optimal field.16 Often nascent frequencies may appear in the field that do not correspond to any known transition energy due to power shifting and broadening as well as field-free coupling within the Hamiltonian. In this context, a more detailed dynamical analysis can facilitate a better understanding of the underlying mechanism, and this paper applies the landscape Hessian analysis technique to explore optimally control dynamics of such quantum systems. A quantum control problem generally can be formulated as the maximization of an objective J = J(e()) by a suitable optimal control field e(t), t A [0, T], T 40. The objective J, as a functional of control field e(t), specifies the landscape of the quantum control problem, and may be expanded as ðT dJ J½eðÞ þ deðÞ ¼ J½eðÞ þ deðtÞdt deðtÞ 0 ð ð 1 T T d2 J deðt 0 ÞdeðtÞdt 0 dt þ . . . ; þ 2 0 0 deðt 0 ÞdeðtÞ (1) with respect to an arbitrary perturbation de(t). The topology of the dynamical landscape is defined by the properties of the Hessian d2J/de(t 0 )de(t), as the second derivative of the objective J with respect to the control field e(t) evaluated at a critical point, where the gradient dJ/de(t) = 0.17,18 The Hessian has been used to traverse the level sets on the top and bottom of quantum control landscapes and thereby find other solutions.19 The Hessian can also be analyzed at a particular optimal field solution to (i) gain insight into which field frequencies are most important to the controlled dynamics as well as (ii) identify the effective number of participating states in the quantum system revealed by the rank (the number of the non-zero eigenvalues) of the Hessian.20 The Hessian analysis may be used in conjunction with Fourier analysis of the optimal control field to gain deeper insight into the control mechanism. The ability to experimentally measure the Hessian has been demonstrated including in complex controlled molecular fragmentation experiments.21,22 Rich information was obtained by the analysis of the Hessian in these studies. The present work explores this prospect further by considering control through field entering the system in diverse ways in relation to the physical objective. Through numerical simulations, we illustrate the optimal quantum control of coupled primary and secondary quantum subspaces with various levels of complexity by considering three different dipole-mediated transition scenarios: (I) the dipole matrix elements are non-zero only in the primary subspace, (II) the dipole matrix elements are non-zero only between the primary and secondary subspaces, and (III) the dipole matrix elements are non-zero only in the secondary subspace. These circumstances could be realized in some cases when operating with particular bandwidth limited field resources. The Hessian analysis is used to aid in gaining a general mechanistic understanding of how these different dipole-mediated controls of two coupled quantum subspaces are achieved. To this end, the Hessian is expressed in a
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visual fashion to reveal information about the temporal and frequency response relationships between the control field and the target yield. The most complex cases show that combined information from the state population dynamics, the power spectrum of the field and the Hessian structure are needed to gain the fullest picture of the control mechanism. The remainder of this paper is organized as follows. Section II presents a general description of the quantum system and the three control scenarios. Section III discusses the results of the optimal control simulations and the patterns that are revealed through the Hessian analysis. Conclusions and potential future directions are discussed in Section IV.
II. Methods We consider an N-level quantum system composed of two coupled quantum subspaces: a primary subspace Hp of dimensions Np and a secondary subspace Hs of dimension Ns = N Np. The N-dimensional Hilbert space of the quantum system is written as a direct product H = Hp = " Hs. In order to consider a general circumstance, the subspaces Hp and Hs are assumed to be coupled (in an uncontrolled inherent fashion) even without a field being present. The control objective J in this work is to maximize the transition probability Pi-f, where |ii, | f i A Hp, with the aid of a control field e(t) t A [0, T] that may be coupled to the quantum system either (I) only within Hp, or (II) only between Hp and Hs, or (III) only within Hs. These cases cover broad scenarios of potential situations that may arise from specifically directed control resources. Employing the dipole approximation, the N N total Hamiltonian H(t) of the quantum system can be written as H(t) = H0 me(t)
(2)
where H0 is the field-free Hamiltonian and m is the transition dipole matrix. The N N field-free Hamiltonian H0 is expressed as the following block matrix: " # Hp V H0 ¼ ; (3) V y Hs where the Np Np Hermitian matrix Hp and Ns Ns Hermitian matrix Hs are, respectively, the uncoupled Hamiltonians for the primary and secondary subspace, and the Np Ns matrix V represents the field-free inherent coupling between the primary and secondary subspaces. The N N dipole matrix m can take on either of the following three expressions: 2 3 " # " # + mps mp + + + I II III 4 5 ;m ¼ m ¼ ;m ¼ ; (4) myps + + + + ms where + denotes a zero sub-block matrix. The evolution of the closed quantum system is described by the unitary propagator ¨dinger equation: U(t, 0) satisfying the time-dependent Schro @Uðt; 0Þ ¼ HðtÞUðt; 0Þ; i h @t
Uð0; 0Þ ¼ I;
(5)
such that the state |C(t)i at the time t is |C(t)i = U(t, 0)|C0i, with |C0i being the initial state |ii A Hp. The dipole couplings
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in eqn (4) consider three extreme forms as a basis of evaluating their dynamical consequences. In practice, all of the dipole components may be present thereby leaving the distinctions to lie in the spectral content of the field. Given the initial and target states |ii and | f i in the primary subspace Hp, the objective is to find an optimal control e(t) that maximizes the probability Pi-f of population transfer from the initial state |ii to the target state | f i at time T:19 Pi-f = |h f |U(T,0)|ii|2.
Within the dipole formulation, the gradient dPi-f /de(s, t) is given by19 N dPi!f 2X ¼ Re hijU y ðT; 0Þj f ih f jUðT; 0Þj ji Im½h jjmðtÞjii deðs; tÞ h jai
N 2X Im hijU y ðT; 0Þj f ih f jUðT; 0Þj ji Re½h jjmðtÞjii h jai
(11)
(6)
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†
Meeting this goal may be viewed as taking an excursion to the global maximum of the underlying quantum control landscape, defined by Pi-f in eqn (6) as a functional of the control field e(t). In the following, we assume that the system is controllable23 such that the desired unitary propagator U(T, 0) can always be generated by a suitably chosen field e(t) at a sufficiently large final time T. Satisfaction of this assumption may be difficult in any particular application with the Hamiltonian having the structure of H0 in eqn (3) and some of the cases in eqn (4). However, the simulations in Section III were successful, although the level search effort differed depending on the coupling circumstance. A full analysis of the landscape topology involves more than assuming controllability,18,24 but no hindrances were observed here in reaching the desired objective. In order to perform optimizations, it is convenient to parameterize the field e(s, t) by an index s Z 0 to track the progress to the top of the landscape, in which we have25 ðT @Pi!f dPi!f @eðs; tÞ ; (7) ¼ dt @s deðs; tÞ @s 0 where Pi-f is a function of s. As the objective is to maximize Pi-f by increasing s 4 0, it is sufficient to require that qPi-f /qs 4 0, which can be attained by forcing the field e(s, t) to satisfy the following partial differential equation26 @eðs; tÞ dPi!f ¼ ; @s deðs; tÞ
eð0; tÞ ¼ e0 ðtÞ;
(8)
¨dinger equation, eqn (5), and which is coupled to the Schro collectively corresponds to the gradient algorithm. It is easily seen that after substituting eqn (8) into eqn (7) we obtain the monotonically increasing relation ðT @Pi!f dPi!f 2 ¼ dt 0: (9) @s deðtÞ 0 The initial control at s = 0 is chosen as " # K 1 T 2 X t e0 ðtÞ ¼ F exp sinðok t þ fk Þ; 20 2 k¼1
(10)
where K is the number of frequency components, fk are random phases between 0 and 2p, and F is an arbitrary constant. The frequencies ok are chosen randomly on a predefined spectral band in each example. After choosing the initial field in eqn (10), the field was discretized into an adequate number of time points to assure an accurate solution for each application, and the field at each time point was treated as the control for optimization.
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where U(T, 0) and m(t) = U (T, 0)mU(T, 0) are implicitly functions of s. Note that the gradient dPi-f/de(s, t) is a linear superposition of up to 2N 2 independent functions Re(h j|m(t)|ii) and Im(h j|m(t)|ii), which are, respectively, the real and imaginary parts of the matrix element h j|m(t)|ii, with j a i and 1 r j r N. Eqn (8) was solved using a variable step size fourth-order Runge– Kutta method by MATLAB.27 The solution is achieved by successive steps of solving eqn (5) and (8) until Pi-f was maximized to a satisfactory degree (i.e. Pi-f 4 0.999). At the global maximum of the control landscape, Pi-f = 1 with dPi-f /de(s, t) = 0, 8t, and the corresponding Hessian can be written as:19 Hðt; t 0 Þ ¼
d2 Pif deðt 0 ÞdeðtÞ
¼
2 Refhij½mðtÞ hijmðtÞjii½mðt 0 Þ hijmðt 0 Þjiijiig 2 h
¼
N X 2 Re ½hijmðtÞj ji½h jjmðt 0 Þjii: 2 h jai
(12)
From eqn (12), we see the Hessian at the global maximum is the covariance of the corresponding time-dependent dipole moment operator m(t). Furthermore, the Hessian is negative semi-definite and at most of rank 2N 2, possessing all negative diagonal elements, i.e., H(t, t) r 08t A [0, T]. To facilitate the analysis, the Hessian can be further expressed in terms of its non-zero eigenvalues sj o 0 and associated eigenvectors uj (t)19 X
sj uj ðtÞuj ðt 0 Þ; Hðt; t 0 Þ ¼ (13) j2N2
which contains information about the second-order response of the state-to-state transition probability to an infinitesimal perturbation (noise) de(t) of the optimal control control e(t). If the number of significant non-zero negative eigenvalues is B(2M 2) where M r N, then the system may be considered as having an effective dimension of M, implying that there are effectively BM energy levels participating in the optimal control dynamics. The robustness of the controlled system to field noise is also reflected in the eigenvalues and eigenvectors of the Hessian. The eigenfunctions associated with the significant non-zero eigenvalues indicate which coordinated perturbations in the control field will reduce the value of Pi-f. Specifically, the presence of a small field perturbation, de(t), will cause a corresponding change 2 2M2 X ð T
s j
dPi!f deðtÞuj ðtÞdt (14) j¼1
0
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in the transition probability Pi-f. In turn, the Fourier transform of the Hessian in eqn (12) indicates which field frequencies are the most important for attaining the optimal yield with the Hessian spectrally represented as Hðo; o0 Þ ¼
2M2 X
sj hj ðoÞhj ðo0 Þ
(15)
j¼1
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where hj ðoÞ
ÐT 0
uj ðtÞeiot dt.
III. Results and discussions Many particular quantum optimal control models were considered with the concepts and methodology in Section II. Here we present an illustrative case of a quantum system of N = 10, in which the primary and secondary subspaces are defined by Hp = |mi, m = 1, 2, 3, 4, 5 with diagonal elements Hp(m, m) = m(m 1)/4 in arbitrary dimensionless units, and Hs = |ni, n = 6, 7, 8, 9, 10 with diagonal elements Hs(n, n) = n(n 1)/4, respectively. All of the simulations were performed for the state-to-state probability transfer |1i - |5i, in which both the initial and target states lie entirely in the primary part, Hp, of the system. The elements throughout of the field-free intersubspace coupling V are chosen at random between 1 and 1. To illustrate the general principles involved in the optimal control of various classes of quantum systems, the chosen random numbers in V were then each multiplied by a random factor in range of [0.01, 1]. In the following simulations, the average magnitude of V in H0 is 0.05 (weak) for classes (I) and (II), and 0.4 (strong) for class (III). Although class (III) may still be controllable for weak V, we found that the simulations took an increasing number of iterations for weak V. In this regard, class (III) may naturally be viewed as the most difficult circumstance, as the control only acts in the secondary subspace, while the target state is in the primary subspace. The dipole matrix elements between the nearest neighbor states in each of the cases of eqn (4) is set to be 1.0, with each subsequent neighbor coupling decreasing by a factor of one-half, (i.e., 1 mab ¼ jabj1 , with a and b denoting the energy levels). All 2 diagonal elements are set to zero (maa = 0). Additionally, the dipole matrix element between the initial state |1i and the final state |5i is also set to zero (m15 = m51 = 0). A final time of T = 75 was selected and the time grid was discretized into 8192 points. The control field was optimized until the target objective value P1-5 = 0.999 was obtained in all cases. Fig. 1 shows the population evolution with the final optimal control fields. It was found that the population in Fig. 1a and b for class (I) are mainly confined to the primary subspace Hp, but the population in Fig. 1c and d for class (II) and in Fig. 1e and f for class (III) first enter into the secondary subspace Hs before reaching the target state |5i and back to Hp. The dominant transition for class (II) is reminiscent of a threelevel L system, in which the population on the initial state |1 i is transferred to the target state |5i through the intermediate state |6i. For class (III), despite lacking direct dipole coupling
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Fig. 1 The final optimal population dynamics as a function of time, for class (I) (a) and (b), class (II) (c) and (d), and class (III) (e) and (f). In all cases the goal is to maximize P1-5 at final time T = 75.
between the two subspaces, the strong field-free coupling V enables the indirect control of the field to funnel the quantum system through the secondary subspace back to the primary subspace to reach the desired target state |5i. The temporal dynamics reflected in the Hessian in eqn (12) can be complex, but very evident temporal response correlation patterns of the target yield were present. Fig. 2 displays the Hessian Hðt; t 0 Þ for the above three control scenarios, revealing that (1) all three Hessian diagonals are negative and (2) the three Hessian off-diagonal features behave quite differently at the global maximum of the state-to-state transition probability landscape. Fig. 2a shows that the target yield in class (I) is
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sensitive to the control perturbation at all times. Fig. 2b for class (II) shows no temporal response correlations for t and t 0 near the initial time t = 0, but very large response correlations at the bottom right quadrant. Fig. 2c for class (III) shows significant temporal response correlations for t E t 0 and |t t 0 | E 11. Important details of the state-to-state transition dynamics can be understood by further analyzing the individual structure of each Hessian. For class (I), distinct patterns are found in the Hessian for t, t 0 A [0, 38] and t, t 0 A [38, 75] associated with high probability of occupying states |1i and |5i, respectively. For class (II), a directional change is seen in the diagonal pattern of the Hessian for either t or t 0 greater than B70. The relationship
between the class (III) Hessian and the corresponding population evolution is not as clear as their counterparts in classes (I) and (II). Nevertheless, from comparing Fig. 1f and 2c it was found that the largest magnitude of Hessian at t = t 0 C 40 corresponds to the dominant transition occurring at t C 40 between the primary subspace and the secondary subspace due to the strong field-free coupling V between the two subspaces. To gain further insight, Fig. 3 shows the Fourier power spectrum of the Hessian |H(o, o 0 )|2 as well as for the optimal field for class (I). The Hessian power spectrum in Fig. 3a contains three diagonal major peaks at points (o, o 0 ) = (0.5, 0.5), (1.5, 1.5), and (2,2), corresponding to the frequencies needed to make the |1i - |2i transition, |1i - |3i or |3i |4i transitions, and the |4i - |5i transition, respectively. These transitions naturally play a significant role in the optimal control of this system, and there are also two significant offdiagonal peaks at (o, o 0 ) = (0.5, 2.5) and at (1.5, 3.5), indicating correlation between participating transitions. The frequency correlation between 0.5 and 2.5 implies that the transition pathway |1i - |2i - |4i plays a significant role in the corresponding optimal control process. Similarly, the peak at (1.5, 3.5) implies that the pathway |1i - |3i - |5i is significant. Thus, in addition to the implied stepwise control mechanism |1i - |2i - |3i - |4i - |5i in Fig. 1a and b, two other optimization mechanism contributions are revealed by the Fourier spectrum of the Hessian. This information is consistent with the state-to-state transition dynamics shown in Fig. 1a and b, but it can not be extracted by directly analyzing the Fourier spectrum of the optimal field from Fig. 3b, including the
Fig. 2 Distinctive patterns in the Hessian Hðt; t 0 Þ for the class (I) (a), class (II) (b), and class (III) (c).
Fig. 3 (a) the Fourier power spectrum of Hessian (b) the optimal field for class (I).
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presence of two broad peaks between 7.5 and 10 that do not correspond to any known transition frequencies. The sharp peaks at the low frequency band correspond to the transitions in the primary subspace, but their significance is only fully revealed by the many Hessian cross peaks in Fig. 3a. For class (II), the Fourier spectrum of the Hessian in Fig. 4a indicates that the frequency at 2.5 corresponding to the transition between states |6i and |5i is mainly responsible for attaining the optimization objective. On the other hand, the Fourier spectrum of the optimal field in Fig. 4b has a single dominant frequency at 7.5, corresponding to the transition |1i - |6i. A detailed examination of the power spectrum of the Hessian in Fig. 4a shows a cross pattern centered at 2.5 indicating that this frequency is also coupled to others for reaching the objective. Fig. 1c and d show that almost all of the population lies in states included in the pathway |1i - |6i |5i as the mechanism. Interestingly, the field and the Hessian alone do not give the full picture. The lack of the |6i - |5i frequency component in the field suggests that the control process relies on the coupling through V for this last step, which shows up with high sensitivity in the Hessian. This behavior is characteristic of class (II) Hessian’s processes. For class (III) in Fig. 1e and f, the population dynamics may be interpreted as involving the transitions |1i - |7i and then |7i - |5i along with |7i - |8i - |5i. The most important frequency in the Hessian in Fig. 5(a) is 3.4 corresponding to the gap between states |7i and |8i reflected in Hp and Hs consistent with the latter process. However, the Fourier spectrum of the optimal field in Fig. 5b shows three large peaks at 5.7, 9.1, and
Fig. 4 Fourier spectra of (a) the Hessian and (b) the optimal field for class (II).
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Fig. 5
Fourier spectra of (a) the Hessian and (b) the optimal field for class (II).
10.7, which correspond to the transitions |7i 2 |5i, |8i 2 |5i and |1i 2 |7i, respectively, identified from the fully diagonalized system Hamiltonian H0, where the energies are shifted due to the relatively large coupling V between the primary and secondary states. Since there are no non-zero dipole elements in mIII for these transitions, these shifted spectral locations and the strong coupling through V implies that these transitions essentially ‘‘borrow’’ radiative coupling by mixing of the directly allowed mIII couplings amongst others states. The full picture of the dynamical process is only revealed by examining the population evolution along with the power spectra of the Hessian and the field. The power spectra of the Hessian and the field in classes (I) and (II) could be linked to from the transition frequencies in and between Hp and Hs, as the coupling V was relatively weak. However, while the spectrum of the Hessian in class (III) can be well understood from the unperturbed transition frequencies between |7i and |8i described by the diagonal portion H0, the power spectrum of the field can only be understood in the eigenbasis of the field-free total Hamiltonian H0 in eqn (3). This circumstance illustrates the richness contained in optimally controlled dynamics, and that a single basis may not always be adequate to understand the processes enabling the best performance. Finally, the effective dynamic dimension M of the optimized system may be estimated based on the number (i.e. taken as equal to B(2M 2)) of significant negative eigenvalues of the Hessian in eqn (14), as shown in Fig. 6. Here we plot log(sj) and the number of significant eigenvalues is identified by the location of the first significant drop in value by a factor of B10.
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Fig. 6 Log scale plots of the Hessian eigenvalues for the representative class (I) (a), class (II) (b), and class (III) (c).
For class (I) in Fig. 6(a), we find there are 8 dominant negative eigenvalues (see the vertical dashed line), implying that the effective dimension of the system is 5, rather than the full 10 dimensions of H. Thus, the effective dimension corresponds to that of the primary subspace Hp, consistent with the analysis of Fig. 1a and b, 3 along with the weak coupling V between the two subspaces. In contrast to class (I), interpreting the effective dynamic dimension of class (II) system is less straightforward. Based on the dynamics in Fig. 1c and d and the analysis of Fig. 4, one might predict that there should be 4 significant negative eigenvalues, and thus the effective dimension of the system is 3, but Fig. 6b reveals two dominant Hessian eigenvalues, implying that the effective dimension of the system is 2. This behavior appears likely, due to the well separated sequence of |1i - |6i (pump) and |6i - |5i (dump) two-step transitions, along with the dynamics in the Hessian of Fig. 4 dominated by two state |6i - |5i transition. For class (III) in Fig. 6c, there exist 6 dominant Hessian eigenvalues and the corresponding effective dimension of 4, in agreement with the above mechanistic analysis based on Fig. 1e and f, 5.
IV. Conclusions As quantum dynamics of increasing complexity (i.e., in larger polyatomic molecules) are subject to control in the laboratory, a broad variety of dynamical scenarios can arise. These scenarios can include direct and indirect control along with inherent coupling present between the states. The model classes (I)–(III)
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illustrated here correspondingly show varied behaviour, and further richness could likely occur upon relaxing the strict classes of dipole coupling in eqn (4). The power spectrum of the field and population dynamics are traditionally employed as a means to give mechanistic insight into the controlled dynamics. This paper also introduced the analysis of the Hessian spectrum as an additional spectral source of mechanistic information. Importantly, the examples shown here demonstrated the complementary mechanistic insights offered by these sources of information including cases where all these sources are needed to attain the clearest picture of the dynamics. Recent experimental studies show the ability to measure the Hessian in the laboratory, and again rich information about control dynamics was revealed.21,22 The optimal control of a primary system coupled with a bath is a subject that is only beginning to be understood.11 The Hessian analysis techniques explored in this work can be applied to these types of simulations and may be useful in revealing hidden mechanistic details. Furthermore, the tools in this work could be applied to the so-called leakage problem for quantum computing28 as well as for superconducting qubits29 and may lead to further insights which can aid in the design of qubits based on complex (e.g. molecular vibrational and rotational) modes. A physical application for a model of coupled primary and secondary states is transmon qubits (a type of superconducting charge qubit) for which an accurate physical model requires considering a few levels above two lowest levels that defines a qubit.30
Acknowledgements We acknowledge Dr. Xi Xing for fruitful discussions. CCS, ME and TSH acknowledge partial support by the US Department of Energy grant DE-FG02-02ER15344, AS and HR by the National Science Foundation grant CHE-1058644, and CCS also by the Vice-Chancellor’s Postdoctoral Research Fellowship program of The University of New South Wales, Australia.
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Hessian facilitated analysis of optimally controlled quantum dynamics of systems with coupled primary and secondary states Chuan-Cun Shu,ab Melanie Edwalds,a Alireza Shabani,ac Tak-San Hoa and Herschel Rabitz*a The efficacy of optimal control of quantum dynamics depends on the topology and associated local structure of the underlying control landscape defined as the objective as a function of the control field. A commonly studied control objective involves maximization of the transition probability for steering the quantum system from one state to another state. This paper invokes landscape Hessian analysis performed at an optimal solution to gain insight into the controlled dynamics, where the Hessian is the second-order functional derivative of the control objective with respect to the control field. Specifically, we consider a quantum system composed of coupled primary and secondary subspaces of energy levels with the initial and target states lying in the primary subspace. The primary and secondary subspaces may arise in various scenarios, for example, respectively, as sub-manifolds of ground and excited electronic states of a poly-atomic molecule, with each possessing a set of rotational–vibrational
Received 7th May 2015, Accepted 23rd June 2015
levels. The control field may engage the system through electric dipole transitions that occur either (I) only in the primary subspace, (II) between the two subspaces, or (III) only in the secondary subspace.
DOI: 10.1039/c5cp02660a
Important insights about the resultant dynamics in each case are revealed in the structural patterns of the corresponding Hessian. The Fourier spectrum of the Hessian is shown to often be complementary
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to mechanistic insights provided by the optimal control field and population dynamics.
I. Introduction Quantum dynamics control is generally concerned with finding optimal control fields to effectively manipulate dynamical processes. A frequently studied control process involves maximization of the transition probability for steering the quantum system from one state to another state. Quantum optimal control theory (OCT)1–5 has been applied to a wide variety of physical and chemical problems.6–10 The generally successful outcome from these OCT calculations on model systems is encouraging for wide ranging manipulation of quantum phenomena. A quantitative understanding of the topological properties of quantum control landscapes helps to explain why the underlying optimal control problems may be readily solved using a host of search algorithms. The dynamics of a quantum system in the presence of an applied field is often described in terms of a subset of energy levels that are directly engaged by the field. However, it is
a
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA. E-mail: [email protected] b School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia c Google Inc., Venice, California 90291, USA
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generally difficult to precisely know the number of engaged energy levels, for example, the rotational–vibrational–electronic (rovibronic) levels of a poly-atomic molecule, that are involved in the intervening transitions for achieving the control objective. Many quantum control studies consider an effective Hamiltonian that accounts for the neglected part of the energy levels of the quantum system.11–15 This work considers the state-to-state transition control problem and treats the optimally controlled quantum system as composed of two coupled subspaces of energy levels: one is the primary subspace that is a set of quantum states, generally of low energy levels, including both the initial and target states, and the other is the secondary subspace that may be a set of dynamically distinct internal states containing the energy levels above the primary subspace. The two subspaces are coupled by the field-free off-diagonal portion of the Hamiltonian. For example, this circumstance could arise from anharmonic coupling of a nominal set of multi-mode harmonic oscillator states. The aim of this paper is to explore the role of the primary and secondary subspaces when they are engaged with the control field in various distinct ways. The quantum system of field-free coupled primary and secondary quantum subspaces with additional dipole-mediated radiative coupling schemes can have distinct and complex controlled dynamics such that a complete mechanistic picture of the
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transitions utilized to reach the target state may not always be obtained by simply correlating the transition frequencies to those present in the optimal field.16 Often nascent frequencies may appear in the field that do not correspond to any known transition energy due to power shifting and broadening as well as field-free coupling within the Hamiltonian. In this context, a more detailed dynamical analysis can facilitate a better understanding of the underlying mechanism, and this paper applies the landscape Hessian analysis technique to explore optimally control dynamics of such quantum systems. A quantum control problem generally can be formulated as the maximization of an objective J = J(e()) by a suitable optimal control field e(t), t A [0, T], T 40. The objective J, as a functional of control field e(t), specifies the landscape of the quantum control problem, and may be expanded as ðT dJ J½eðÞ þ deðÞ ¼ J½eðÞ þ deðtÞdt deðtÞ 0 ð ð 1 T T d2 J deðt 0 ÞdeðtÞdt 0 dt þ . . . ; þ 2 0 0 deðt 0 ÞdeðtÞ (1) with respect to an arbitrary perturbation de(t). The topology of the dynamical landscape is defined by the properties of the Hessian d2J/de(t 0 )de(t), as the second derivative of the objective J with respect to the control field e(t) evaluated at a critical point, where the gradient dJ/de(t) = 0.17,18 The Hessian has been used to traverse the level sets on the top and bottom of quantum control landscapes and thereby find other solutions.19 The Hessian can also be analyzed at a particular optimal field solution to (i) gain insight into which field frequencies are most important to the controlled dynamics as well as (ii) identify the effective number of participating states in the quantum system revealed by the rank (the number of the non-zero eigenvalues) of the Hessian.20 The Hessian analysis may be used in conjunction with Fourier analysis of the optimal control field to gain deeper insight into the control mechanism. The ability to experimentally measure the Hessian has been demonstrated including in complex controlled molecular fragmentation experiments.21,22 Rich information was obtained by the analysis of the Hessian in these studies. The present work explores this prospect further by considering control through field entering the system in diverse ways in relation to the physical objective. Through numerical simulations, we illustrate the optimal quantum control of coupled primary and secondary quantum subspaces with various levels of complexity by considering three different dipole-mediated transition scenarios: (I) the dipole matrix elements are non-zero only in the primary subspace, (II) the dipole matrix elements are non-zero only between the primary and secondary subspaces, and (III) the dipole matrix elements are non-zero only in the secondary subspace. These circumstances could be realized in some cases when operating with particular bandwidth limited field resources. The Hessian analysis is used to aid in gaining a general mechanistic understanding of how these different dipole-mediated controls of two coupled quantum subspaces are achieved. To this end, the Hessian is expressed in a
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visual fashion to reveal information about the temporal and frequency response relationships between the control field and the target yield. The most complex cases show that combined information from the state population dynamics, the power spectrum of the field and the Hessian structure are needed to gain the fullest picture of the control mechanism. The remainder of this paper is organized as follows. Section II presents a general description of the quantum system and the three control scenarios. Section III discusses the results of the optimal control simulations and the patterns that are revealed through the Hessian analysis. Conclusions and potential future directions are discussed in Section IV.
II. Methods We consider an N-level quantum system composed of two coupled quantum subspaces: a primary subspace Hp of dimensions Np and a secondary subspace Hs of dimension Ns = N Np. The N-dimensional Hilbert space of the quantum system is written as a direct product H = Hp = " Hs. In order to consider a general circumstance, the subspaces Hp and Hs are assumed to be coupled (in an uncontrolled inherent fashion) even without a field being present. The control objective J in this work is to maximize the transition probability Pi-f, where |ii, | f i A Hp, with the aid of a control field e(t) t A [0, T] that may be coupled to the quantum system either (I) only within Hp, or (II) only between Hp and Hs, or (III) only within Hs. These cases cover broad scenarios of potential situations that may arise from specifically directed control resources. Employing the dipole approximation, the N N total Hamiltonian H(t) of the quantum system can be written as H(t) = H0 me(t)
(2)
where H0 is the field-free Hamiltonian and m is the transition dipole matrix. The N N field-free Hamiltonian H0 is expressed as the following block matrix: " # Hp V H0 ¼ ; (3) V y Hs where the Np Np Hermitian matrix Hp and Ns Ns Hermitian matrix Hs are, respectively, the uncoupled Hamiltonians for the primary and secondary subspace, and the Np Ns matrix V represents the field-free inherent coupling between the primary and secondary subspaces. The N N dipole matrix m can take on either of the following three expressions: 2 3 " # " # + mps mp + + + I II III 4 5 ;m ¼ m ¼ ;m ¼ ; (4) myps + + + + ms where + denotes a zero sub-block matrix. The evolution of the closed quantum system is described by the unitary propagator ¨dinger equation: U(t, 0) satisfying the time-dependent Schro @Uðt; 0Þ ¼ HðtÞUðt; 0Þ; i h @t
Uð0; 0Þ ¼ I;
(5)
such that the state |C(t)i at the time t is |C(t)i = U(t, 0)|C0i, with |C0i being the initial state |ii A Hp. The dipole couplings
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in eqn (4) consider three extreme forms as a basis of evaluating their dynamical consequences. In practice, all of the dipole components may be present thereby leaving the distinctions to lie in the spectral content of the field. Given the initial and target states |ii and | f i in the primary subspace Hp, the objective is to find an optimal control e(t) that maximizes the probability Pi-f of population transfer from the initial state |ii to the target state | f i at time T:19 Pi-f = |h f |U(T,0)|ii|2.
Within the dipole formulation, the gradient dPi-f /de(s, t) is given by19 N dPi!f 2X ¼ Re hijU y ðT; 0Þj f ih f jUðT; 0Þj ji Im½h jjmðtÞjii deðs; tÞ h jai
N 2X Im hijU y ðT; 0Þj f ih f jUðT; 0Þj ji Re½h jjmðtÞjii h jai
(11)
(6)
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†
Meeting this goal may be viewed as taking an excursion to the global maximum of the underlying quantum control landscape, defined by Pi-f in eqn (6) as a functional of the control field e(t). In the following, we assume that the system is controllable23 such that the desired unitary propagator U(T, 0) can always be generated by a suitably chosen field e(t) at a sufficiently large final time T. Satisfaction of this assumption may be difficult in any particular application with the Hamiltonian having the structure of H0 in eqn (3) and some of the cases in eqn (4). However, the simulations in Section III were successful, although the level search effort differed depending on the coupling circumstance. A full analysis of the landscape topology involves more than assuming controllability,18,24 but no hindrances were observed here in reaching the desired objective. In order to perform optimizations, it is convenient to parameterize the field e(s, t) by an index s Z 0 to track the progress to the top of the landscape, in which we have25 ðT @Pi!f dPi!f @eðs; tÞ ; (7) ¼ dt @s deðs; tÞ @s 0 where Pi-f is a function of s. As the objective is to maximize Pi-f by increasing s 4 0, it is sufficient to require that qPi-f /qs 4 0, which can be attained by forcing the field e(s, t) to satisfy the following partial differential equation26 @eðs; tÞ dPi!f ¼ ; @s deðs; tÞ
eð0; tÞ ¼ e0 ðtÞ;
(8)
¨dinger equation, eqn (5), and which is coupled to the Schro collectively corresponds to the gradient algorithm. It is easily seen that after substituting eqn (8) into eqn (7) we obtain the monotonically increasing relation ðT @Pi!f dPi!f 2 ¼ dt 0: (9) @s deðtÞ 0 The initial control at s = 0 is chosen as " # K 1 T 2 X t e0 ðtÞ ¼ F exp sinðok t þ fk Þ; 20 2 k¼1
(10)
where K is the number of frequency components, fk are random phases between 0 and 2p, and F is an arbitrary constant. The frequencies ok are chosen randomly on a predefined spectral band in each example. After choosing the initial field in eqn (10), the field was discretized into an adequate number of time points to assure an accurate solution for each application, and the field at each time point was treated as the control for optimization.
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where U(T, 0) and m(t) = U (T, 0)mU(T, 0) are implicitly functions of s. Note that the gradient dPi-f/de(s, t) is a linear superposition of up to 2N 2 independent functions Re(h j|m(t)|ii) and Im(h j|m(t)|ii), which are, respectively, the real and imaginary parts of the matrix element h j|m(t)|ii, with j a i and 1 r j r N. Eqn (8) was solved using a variable step size fourth-order Runge– Kutta method by MATLAB.27 The solution is achieved by successive steps of solving eqn (5) and (8) until Pi-f was maximized to a satisfactory degree (i.e. Pi-f 4 0.999). At the global maximum of the control landscape, Pi-f = 1 with dPi-f /de(s, t) = 0, 8t, and the corresponding Hessian can be written as:19 Hðt; t 0 Þ ¼
d2 Pif deðt 0 ÞdeðtÞ
¼
2 Refhij½mðtÞ hijmðtÞjii½mðt 0 Þ hijmðt 0 Þjiijiig 2 h
¼
N X 2 Re ½hijmðtÞj ji½h jjmðt 0 Þjii: 2 h jai
(12)
From eqn (12), we see the Hessian at the global maximum is the covariance of the corresponding time-dependent dipole moment operator m(t). Furthermore, the Hessian is negative semi-definite and at most of rank 2N 2, possessing all negative diagonal elements, i.e., H(t, t) r 08t A [0, T]. To facilitate the analysis, the Hessian can be further expressed in terms of its non-zero eigenvalues sj o 0 and associated eigenvectors uj (t)19 X
sj uj ðtÞuj ðt 0 Þ; Hðt; t 0 Þ ¼ (13) j2N2
which contains information about the second-order response of the state-to-state transition probability to an infinitesimal perturbation (noise) de(t) of the optimal control control e(t). If the number of significant non-zero negative eigenvalues is B(2M 2) where M r N, then the system may be considered as having an effective dimension of M, implying that there are effectively BM energy levels participating in the optimal control dynamics. The robustness of the controlled system to field noise is also reflected in the eigenvalues and eigenvectors of the Hessian. The eigenfunctions associated with the significant non-zero eigenvalues indicate which coordinated perturbations in the control field will reduce the value of Pi-f. Specifically, the presence of a small field perturbation, de(t), will cause a corresponding change 2 2M2 X ð T
s j
dPi!f deðtÞuj ðtÞdt (14) j¼1
0
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in the transition probability Pi-f. In turn, the Fourier transform of the Hessian in eqn (12) indicates which field frequencies are the most important for attaining the optimal yield with the Hessian spectrally represented as Hðo; o0 Þ ¼
2M2 X
sj hj ðoÞhj ðo0 Þ
(15)
j¼1
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where hj ðoÞ
ÐT 0
uj ðtÞeiot dt.
III. Results and discussions Many particular quantum optimal control models were considered with the concepts and methodology in Section II. Here we present an illustrative case of a quantum system of N = 10, in which the primary and secondary subspaces are defined by Hp = |mi, m = 1, 2, 3, 4, 5 with diagonal elements Hp(m, m) = m(m 1)/4 in arbitrary dimensionless units, and Hs = |ni, n = 6, 7, 8, 9, 10 with diagonal elements Hs(n, n) = n(n 1)/4, respectively. All of the simulations were performed for the state-to-state probability transfer |1i - |5i, in which both the initial and target states lie entirely in the primary part, Hp, of the system. The elements throughout of the field-free intersubspace coupling V are chosen at random between 1 and 1. To illustrate the general principles involved in the optimal control of various classes of quantum systems, the chosen random numbers in V were then each multiplied by a random factor in range of [0.01, 1]. In the following simulations, the average magnitude of V in H0 is 0.05 (weak) for classes (I) and (II), and 0.4 (strong) for class (III). Although class (III) may still be controllable for weak V, we found that the simulations took an increasing number of iterations for weak V. In this regard, class (III) may naturally be viewed as the most difficult circumstance, as the control only acts in the secondary subspace, while the target state is in the primary subspace. The dipole matrix elements between the nearest neighbor states in each of the cases of eqn (4) is set to be 1.0, with each subsequent neighbor coupling decreasing by a factor of one-half, (i.e., 1 mab ¼ jabj1 , with a and b denoting the energy levels). All 2 diagonal elements are set to zero (maa = 0). Additionally, the dipole matrix element between the initial state |1i and the final state |5i is also set to zero (m15 = m51 = 0). A final time of T = 75 was selected and the time grid was discretized into 8192 points. The control field was optimized until the target objective value P1-5 = 0.999 was obtained in all cases. Fig. 1 shows the population evolution with the final optimal control fields. It was found that the population in Fig. 1a and b for class (I) are mainly confined to the primary subspace Hp, but the population in Fig. 1c and d for class (II) and in Fig. 1e and f for class (III) first enter into the secondary subspace Hs before reaching the target state |5i and back to Hp. The dominant transition for class (II) is reminiscent of a threelevel L system, in which the population on the initial state |1 i is transferred to the target state |5i through the intermediate state |6i. For class (III), despite lacking direct dipole coupling
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Fig. 1 The final optimal population dynamics as a function of time, for class (I) (a) and (b), class (II) (c) and (d), and class (III) (e) and (f). In all cases the goal is to maximize P1-5 at final time T = 75.
between the two subspaces, the strong field-free coupling V enables the indirect control of the field to funnel the quantum system through the secondary subspace back to the primary subspace to reach the desired target state |5i. The temporal dynamics reflected in the Hessian in eqn (12) can be complex, but very evident temporal response correlation patterns of the target yield were present. Fig. 2 displays the Hessian Hðt; t 0 Þ for the above three control scenarios, revealing that (1) all three Hessian diagonals are negative and (2) the three Hessian off-diagonal features behave quite differently at the global maximum of the state-to-state transition probability landscape. Fig. 2a shows that the target yield in class (I) is
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sensitive to the control perturbation at all times. Fig. 2b for class (II) shows no temporal response correlations for t and t 0 near the initial time t = 0, but very large response correlations at the bottom right quadrant. Fig. 2c for class (III) shows significant temporal response correlations for t E t 0 and |t t 0 | E 11. Important details of the state-to-state transition dynamics can be understood by further analyzing the individual structure of each Hessian. For class (I), distinct patterns are found in the Hessian for t, t 0 A [0, 38] and t, t 0 A [38, 75] associated with high probability of occupying states |1i and |5i, respectively. For class (II), a directional change is seen in the diagonal pattern of the Hessian for either t or t 0 greater than B70. The relationship
between the class (III) Hessian and the corresponding population evolution is not as clear as their counterparts in classes (I) and (II). Nevertheless, from comparing Fig. 1f and 2c it was found that the largest magnitude of Hessian at t = t 0 C 40 corresponds to the dominant transition occurring at t C 40 between the primary subspace and the secondary subspace due to the strong field-free coupling V between the two subspaces. To gain further insight, Fig. 3 shows the Fourier power spectrum of the Hessian |H(o, o 0 )|2 as well as for the optimal field for class (I). The Hessian power spectrum in Fig. 3a contains three diagonal major peaks at points (o, o 0 ) = (0.5, 0.5), (1.5, 1.5), and (2,2), corresponding to the frequencies needed to make the |1i - |2i transition, |1i - |3i or |3i |4i transitions, and the |4i - |5i transition, respectively. These transitions naturally play a significant role in the optimal control of this system, and there are also two significant offdiagonal peaks at (o, o 0 ) = (0.5, 2.5) and at (1.5, 3.5), indicating correlation between participating transitions. The frequency correlation between 0.5 and 2.5 implies that the transition pathway |1i - |2i - |4i plays a significant role in the corresponding optimal control process. Similarly, the peak at (1.5, 3.5) implies that the pathway |1i - |3i - |5i is significant. Thus, in addition to the implied stepwise control mechanism |1i - |2i - |3i - |4i - |5i in Fig. 1a and b, two other optimization mechanism contributions are revealed by the Fourier spectrum of the Hessian. This information is consistent with the state-to-state transition dynamics shown in Fig. 1a and b, but it can not be extracted by directly analyzing the Fourier spectrum of the optimal field from Fig. 3b, including the
Fig. 2 Distinctive patterns in the Hessian Hðt; t 0 Þ for the class (I) (a), class (II) (b), and class (III) (c).
Fig. 3 (a) the Fourier power spectrum of Hessian (b) the optimal field for class (I).
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presence of two broad peaks between 7.5 and 10 that do not correspond to any known transition frequencies. The sharp peaks at the low frequency band correspond to the transitions in the primary subspace, but their significance is only fully revealed by the many Hessian cross peaks in Fig. 3a. For class (II), the Fourier spectrum of the Hessian in Fig. 4a indicates that the frequency at 2.5 corresponding to the transition between states |6i and |5i is mainly responsible for attaining the optimization objective. On the other hand, the Fourier spectrum of the optimal field in Fig. 4b has a single dominant frequency at 7.5, corresponding to the transition |1i - |6i. A detailed examination of the power spectrum of the Hessian in Fig. 4a shows a cross pattern centered at 2.5 indicating that this frequency is also coupled to others for reaching the objective. Fig. 1c and d show that almost all of the population lies in states included in the pathway |1i - |6i |5i as the mechanism. Interestingly, the field and the Hessian alone do not give the full picture. The lack of the |6i - |5i frequency component in the field suggests that the control process relies on the coupling through V for this last step, which shows up with high sensitivity in the Hessian. This behavior is characteristic of class (II) Hessian’s processes. For class (III) in Fig. 1e and f, the population dynamics may be interpreted as involving the transitions |1i - |7i and then |7i - |5i along with |7i - |8i - |5i. The most important frequency in the Hessian in Fig. 5(a) is 3.4 corresponding to the gap between states |7i and |8i reflected in Hp and Hs consistent with the latter process. However, the Fourier spectrum of the optimal field in Fig. 5b shows three large peaks at 5.7, 9.1, and
Fig. 4 Fourier spectra of (a) the Hessian and (b) the optimal field for class (II).
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Fig. 5
Fourier spectra of (a) the Hessian and (b) the optimal field for class (II).
10.7, which correspond to the transitions |7i 2 |5i, |8i 2 |5i and |1i 2 |7i, respectively, identified from the fully diagonalized system Hamiltonian H0, where the energies are shifted due to the relatively large coupling V between the primary and secondary states. Since there are no non-zero dipole elements in mIII for these transitions, these shifted spectral locations and the strong coupling through V implies that these transitions essentially ‘‘borrow’’ radiative coupling by mixing of the directly allowed mIII couplings amongst others states. The full picture of the dynamical process is only revealed by examining the population evolution along with the power spectra of the Hessian and the field. The power spectra of the Hessian and the field in classes (I) and (II) could be linked to from the transition frequencies in and between Hp and Hs, as the coupling V was relatively weak. However, while the spectrum of the Hessian in class (III) can be well understood from the unperturbed transition frequencies between |7i and |8i described by the diagonal portion H0, the power spectrum of the field can only be understood in the eigenbasis of the field-free total Hamiltonian H0 in eqn (3). This circumstance illustrates the richness contained in optimally controlled dynamics, and that a single basis may not always be adequate to understand the processes enabling the best performance. Finally, the effective dynamic dimension M of the optimized system may be estimated based on the number (i.e. taken as equal to B(2M 2)) of significant negative eigenvalues of the Hessian in eqn (14), as shown in Fig. 6. Here we plot log(sj) and the number of significant eigenvalues is identified by the location of the first significant drop in value by a factor of B10.
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Fig. 6 Log scale plots of the Hessian eigenvalues for the representative class (I) (a), class (II) (b), and class (III) (c).
For class (I) in Fig. 6(a), we find there are 8 dominant negative eigenvalues (see the vertical dashed line), implying that the effective dimension of the system is 5, rather than the full 10 dimensions of H. Thus, the effective dimension corresponds to that of the primary subspace Hp, consistent with the analysis of Fig. 1a and b, 3 along with the weak coupling V between the two subspaces. In contrast to class (I), interpreting the effective dynamic dimension of class (II) system is less straightforward. Based on the dynamics in Fig. 1c and d and the analysis of Fig. 4, one might predict that there should be 4 significant negative eigenvalues, and thus the effective dimension of the system is 3, but Fig. 6b reveals two dominant Hessian eigenvalues, implying that the effective dimension of the system is 2. This behavior appears likely, due to the well separated sequence of |1i - |6i (pump) and |6i - |5i (dump) two-step transitions, along with the dynamics in the Hessian of Fig. 4 dominated by two state |6i - |5i transition. For class (III) in Fig. 6c, there exist 6 dominant Hessian eigenvalues and the corresponding effective dimension of 4, in agreement with the above mechanistic analysis based on Fig. 1e and f, 5.
IV. Conclusions As quantum dynamics of increasing complexity (i.e., in larger polyatomic molecules) are subject to control in the laboratory, a broad variety of dynamical scenarios can arise. These scenarios can include direct and indirect control along with inherent coupling present between the states. The model classes (I)–(III)
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illustrated here correspondingly show varied behaviour, and further richness could likely occur upon relaxing the strict classes of dipole coupling in eqn (4). The power spectrum of the field and population dynamics are traditionally employed as a means to give mechanistic insight into the controlled dynamics. This paper also introduced the analysis of the Hessian spectrum as an additional spectral source of mechanistic information. Importantly, the examples shown here demonstrated the complementary mechanistic insights offered by these sources of information including cases where all these sources are needed to attain the clearest picture of the dynamics. Recent experimental studies show the ability to measure the Hessian in the laboratory, and again rich information about control dynamics was revealed.21,22 The optimal control of a primary system coupled with a bath is a subject that is only beginning to be understood.11 The Hessian analysis techniques explored in this work can be applied to these types of simulations and may be useful in revealing hidden mechanistic details. Furthermore, the tools in this work could be applied to the so-called leakage problem for quantum computing28 as well as for superconducting qubits29 and may lead to further insights which can aid in the design of qubits based on complex (e.g. molecular vibrational and rotational) modes. A physical application for a model of coupled primary and secondary states is transmon qubits (a type of superconducting charge qubit) for which an accurate physical model requires considering a few levels above two lowest levels that defines a qubit.30
Acknowledgements We acknowledge Dr. Xi Xing for fruitful discussions. CCS, ME and TSH acknowledge partial support by the US Department of Energy grant DE-FG02-02ER15344, AS and HR by the National Science Foundation grant CHE-1058644, and CCS also by the Vice-Chancellor’s Postdoctoral Research Fellowship program of The University of New South Wales, Australia.
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