Universality in Exact Quantum State Population Dynamics and Control
We consider an exact population transition, defined as the probability of finding a state at a final time being exactly equal to the probability of another state at the initial time. We prove that, given a Hamiltonian, there always exists a complete set of orthogonal states that can be employed as time-zero states for which this exact population transition occurs. The result is general: it holds for arbitrary systems, arbitrary pairs of initial and final states, and for any time interval. The proposition is illustrated with several analytic models. In particular we demonstrate that in some cases, by tuning the control parameters a complete transition might occur, where a target state, vacant at , is fully populated at time .
Introduction.— The central goal of quantum control is the transfer of population from an initial state to a final target state Bookrice (); BSbook (). Within the framework of coherent quantum control, focus has been primarily on designing specific laser-based scenarios that achieve this goal (for some bound state examples see, e.g., shapiro (); jiangbin (); Kbergmann ()), whereas within the framework of optimal control, focus has been on identifying control fields that achieve this goal, both computationally and experimentally.
Despite the enormous interest in this area there are very few analytic control results about realistic systems. These include theorems such as that of Huang-Tarn-Clark HTC1983 (), a theorem by Ramakrisnan . on the dimensionality of the Lie Algebra induced by the interaction between the system and the control field Rama1995 (), and a theorem by Shapiro and Brumer Brumer95 (), where control was shown to depend on the dimensionality of the controlled subspaces. As a consequence, any proven fundamental result adds considerably to the knowledge base (e.g., Openwu ()). In this paper we expose a universal feature of quantum dynamics that has significant implications for control. The focus here is in the proving this dynamical result; future studies will be directed to control applications.
Specifically, consider an initial state that evolves under a Hamiltonian to yield the state at time . Of interest is the probability of the system being initially in state undergoing a transition with probability to an orthogonal component at time . We focus on the possibility of an “exact quantum transition” between these states defined as
i.e., where the probability of observing state at final time equals the probability of observing the state initially.
In this paper we prove that there always exists, for arbitrary evolution operator and for an arbitrary time , a complete set of orthogonal states that undergo the exact state transition (1) from to . For a given Hamiltonian, the magnitude of the associated is determined by the choice of , and . As examples, we obtain the set for some analytical models, and furthermore provide instances of significant transfer, defined by .
While this universality might seem surprising, we show below that it simply stems from unitarity of quantum evolution. Based on unitary evolution, the universal existence of exact quantum state transmission between different subspaces was demonstrated in Wu09 (), and cyclic quantum evolution in the theory of geometric phase was established in Wu94 (). Unitarity is also at the heart of the no-cloning theorem, which is fundamental to quantum information science. In the present case an inclusive theorem in quantum dynamics based on unitarity is derived that is expected to be influential in quantum technologies. In particular, note that the dynamical principle is here established within the same Hilbert space, unlike Wu09 (), giving an approach that is propitious for a broad range of applications, e.g., for quantum computing Chuang (), coherent control of atomic and molecular processes BSbook (), and laser control of chemical reaction in molecules Bookrice ().
Universality of the exact population transition.— Consider an dimensional system ( can be infinite), spanned by the bases and described by density matrix . The equality in Eq. (1) becomes
We assume that the system is prepared in a pure state so that .
Proposition. There always exists a complete orthogonal set , which depends on , such that an exact population transition described by Eqs. (1) or (2) takes place if the initially prepared state is a member of this set.
Proof. Assuming that at time the state of the system is , the left side of Eq. (2) can be written as
where is the time evolution operator of the system, and we have introduced the exchange operator
satisfying , with . The exchange operator swaps the states and while keeping other states intact. It is easy to prove that . We have also defined the auxiliary density matrix , with and . This operator behaves similarly to the time-evolution operator. It is significant to note that the operator is unitary, satisfying
As a unitary operator can be diagonalized to yield a complete set of orthonormal eigenvectors and exponential eigenvalues . Any vector in the set thus obeys the eigenequation
Comparing Eq. (2) with Eq. (3), we note that if the state of the system at time zero is one of the ’s in Eq. (6), then the equality trtr in Eq. (2) holds. In other words, an exact population transition occurs between states and independent of the choice of these states other than that they are orthogonal. The result is also valid for the exchange operator , where and are real numbers. In this case the unitary condition (5) translates to , since may not be equal to .
The above result is a fundamental attribute of quantum dynamics and should serve as a basic building block in quantum control theory (see also Wu09 () and Wu94 ()). Given an arbitrary Hamiltonian at an arbitrary time , and an arbitrary pair of states and , can be numerically diagonalized to obtain its eigenvalue spectrum and eigenstates, giving the states and between which the ”exact transition” takes place. This can be readily done for small systems, and we illustrate below several simple eigenproblems where the spectrum of can be analytically obtained. However, we emphasize that, unlike specific control scenarios, this result is universal, arising only from the fact that a unitary operator possesses a complete set of orthogonal eigenvectors. Of particular interest in control scenarios are transitions, which we denote as significant, when . Ideally, in control scenarios, we seek exact transitions that are (what we term) complete, i.e. where and , so that an initial state, fully populated at time zero, transfers its population to a target state at time . Experience gained from individual sample cases below sheds light on the theorem and will allow one to assess future directions for control applications.
Example: A two level system.— We discuss three variants of the two-level-system (TLS) model. In the first static case the theorem holds in a trivial way, though there is no actual population transition. In the second case a weak time dependent perturbation leads to a significant population transfer. The last example demonstrates that in a delta-kicked TLS a complete transition might take place. The unperturbed TLS model is described by the states and of energies and respectively. Taking into account different types of interactions, we explore next the transition between to .
I. Static two-level system.— Assuming for simplicity that , we obtain the eigenstates of , with eigenvalues ; and . This case is trivial since there is no actual transition during the course of time. However, the dynamics is still Hamiltonian and hence (2) is valid.
II. Two-level system under a time dependent perturbation.— Consider next the two states and of energies and respectively, on-resonance with a periodic perturbation , where is a parameter characterizing the order of the perturbation expansion. Setting again , , we obtain the approximate eigenstates of to first order of ,
where ; and are real numbers. If we obtain the following inequality for the initial state
Since , a significant transfer is realized here. For the same inequality holds for . In both cases the initial and final probabilities satisfy
Figure 1 demonstrates an exact population transfer in the present model, computed without approximation for the evolution of the TLS under a harmonic perturbation. Panel (a) demonstrates a significant transition, while for a different set of parameters panel (b) shows that at specific times (or for a designed time dependent field) a complete transition might take place, even for weak perturbations.
The analytic calculation (7)-(9) exemplifies a “significant transition”, i.e. where . Having demonstrated that population transfer can be achieved, we further address the question, particularly relevant to control, of what is the maximum achievable significance, defined as . For the TLS case the significance equals , which we want to maximize under the conditions that (a) the state is pure, and (b) an exact quantum transition is achieved at time . It can be shown that the condition for exact transition can be rewritten as , where we define the vectors and ; are the Pauli matrices, respectively. If, for example, the component was equal to 1, the condition for an exact transition forces to be zero, and there is no initial state for which there is significant exact transfer. Assuming that , on the other hand, the significance becomes maximal for the initial state
Note then that generally, states that maximize the significance will not be eigenstates of . Thus, although in accord with the above theorem one achieves exact transitions, the ideal complete transition is obtained rarely. However, the proven theorem provides a new framework in within which to modify the Hamiltonian to achieve transitions with increasingly larger significance.
III. Kicked two-level system.— For the same model, again with and , consider a non-perturbative time-dependent Hamiltonian,
where , and are the Pauli matrices. Here one can generically write the operator , where in this case the parameters are
with . The two eigenstates of are with eigenvalue and with eigenvalue , where . Hence, adopting as the time-zero state, we obtain the results
As an example, if and a strong pulse kicks at such that , in the limit one gets that , and , which is a complete quantum transition from to , satisfying and . In Fig. 2 we show that by carefully tuning the interaction parameters, e.g. the delay time , one can achieve such a complete transition. We next demonstrate that a complete transition can take place in general in an adiabatically evolving system.
Complete transitions: Adiabatic evolution.— Consider a time-dependent Hamiltonian . If it is varied sufficiently slowly, the evolution of the system is adiabatic, and the system occupies an (instantaneous) eigenstate of the Hamiltonian , provided the time-zero state is an eigenstate of . If the state of the system at time , , is orthogonal to the time-zero state, one can obtain a complete transition by setting and , both eigenstates of . As an example, consider the magnetic Zeeman effect where a magnetic field splits the atomic (or molecular) degenerate levels, characterized by the magnetic quantum numbers . The Hamiltonian is effectively given by
where the second term is responsible to quantum transitions between different values of . The time dependent modulation is controlled by the parameter , and we manipulate the magnetic field such that  is an even [odd] function of , and . We now choose the initial state as , the lowest eigenstate of . If we control the evolution of adiabatically from time to , the state of the system at time becomes , which is the lowest eigenstate of . Since and are orthogonal, the quantum transition (2) is complete. The results of Ref. Nori06 () may be an example of this scenario when .
Superposition: Eigenstates in a three-level system.— Finally, we address the following conceptual question: can one achieve an exact population transition (1) with a superposition of the eigenstates of as the time-zero state of the system? We explain next the conditions for this transfer by considering a three-level Hamiltonian . If one chooses the initial and final states to be and , the exchange operator commutes with comment (). One can easily obtain the eigenstates and eigenvalues of ,
are functions of and but their exact form is not important for the discussion below. Examining (Universality in Exact Quantum State Population Dynamics and Control), it is obvious that there is no significant transition if the time-zero state is any of these three eigenstates, since for . We show next that under some strict conditions, a superposition state can yield a complete transition. Since the set is a complete orthogonal set, a general time-zero state can be expanded as , not necessarily an eigenstate of ,
An exact transition can still take place at a specific time , obeying the eigenvalue equation
This equality is satisfied if . Thus, for the contributing coefficients, , we obtain a set of conditions , where are arbitrary integers. This is a very restrictive condition when there are many nonzero coefficients. However, it may be still satisfied for particular systems with special symmetry. We exemplify this within the three-level model presented above [See Eq. (Universality in Exact Quantum State Population Dynamics and Control)]. Assuming the following superposition state at time zero, , we obtain
which is generally not an eigenstate of . However, Eq. (17) is satisfied at the specific time leading to
manifesting a significant transition and a population transfer
The transition can be made complete, depending on the superposition preparation coefficients and .
Conclusion.— We have proved the universality of exact quantum transitions, demonstrating that for a given Hamiltonian and a pair of states and in the associated Hilbert space, there always exists a complete set of orthogonal states that when employed as the time-zero state of the system, lead to an exact population transition between the pair and . This universal proposition is a fundamental feature of quantum dynamics and a promising building block in quantum control. We have demonstrated the result analytically on the time-modulated two-level-system model, showing that in some cases a complete population transfer can be obtained. We have also shown that an adiabatic evolution can lead to a complete transition. Finally, we have analyzed exact population transitions in a superposition of states. Applications to specific control studies are the subject of future work.
Acknowledgments. LAW has been supported by the Ikerbasque Foundation. DS acknowledges support from the University of Toronto Start-up Fund, and PB was supported by NSERC.
- (1) S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, New York, 2000).
- (2) M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003).
- (3) E.g., P. Kral, I. Thanopulos, and M. Shapiro, Rev. Mod. Phys. 79, 53 (2007) and E. A. Shapiro, V. Milner, and M. Shapiro, Phys. Rev. A 79, 023422 (2009) and references therein.
- (4) E.g., J. B. Gong and S. A. Rice, J. Chem. Phys. 120, 5117 (2004); J. B. Gong and S. A. Rice, J. Chem. Phys. 120, 9984 (2004).
- (5) U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J. Chem. Phys. 92, 5363 (1990); K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998).
- (6) G. M. Huang, T. J. Tarn, and J. W. Clark, J. Math. Phys. 24, 2608 (1983).
- (7) V. Ramakrishna et al., Phys. Rev. A 51, 960 (1995).
- (8) M. Shapiro and P. Brumer, J. Chem. Phys. 103, 487 (1995).
- (9) For an extension of this result to open quantum systems see, L.-A. Wu, A. Bharioke, and P. Brumer, J. Chem. Phys. 129, 041105 (2008).
- (10) L.-A. Wu, Y.-x. Liu, and F. Nori, Phys. Rev. A 80, 042315 (2009).
- (11) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge: Cambridge University Press 2000).
- (12) L.-A. Wu, Phys. Rev. A 50, 5317 (1994).
- (13) S. Ashhab, J. R. Johansson, and F. Nori, Phys. Rev. A 74, 052330 (2006).
- (14) does not commute in general with the Hamiltonian. For example, if , then .