Exactly solvable two-level quantum systems and Landau-Zener interferometry
A simple algorithm is presented based on a type of partial reverse-engineering that generates an unlimited number of exact analytical solutions to the Schrödinger equation for a general time-dependent two-level Hamiltonian. I demonstrate this method by deriving new exact solutions corresponding to fast control pulses that contain arbitrarily many tunable parameters. It is shown that the formalism is naturally suited to generating analytical control protocols that perform precise non-adiabatic rapid passage and Landau-Zener interferometry near the quantum speed limit. A general, exact formula for Landau-Zener interference patterns is derived.
Although they have pervaded quantum physics since its inception, very few time-dependent two-level quantum systems are known to be analytically solvable. Among the most famous examples of exactly soluble two-level evolution is the Landau-Majorana-Stückelberg-Zener (LMSZ) problem Landau (1932); Zener (1932); Stückelberg (1932); Majorana (1932); Shevchenko et al. (2010), which remains a very active area of research due to numerous applications pertaining to quantum phase transitions Zurek et al. (2005), quantum control Rudner et al. (2008); Petta et al. (2010); Ribeiro et al. (2012); Quintana et al. (2013); Zhang et al. (2012) and quantum state preparation Wu et al. (2011); Brierley et al. (2012); Malossi et al. (2012). The hyperbolic secant pulse of Rosen and Zener Rosen and Zener (1932) has played an important role in self-induced transparency McCall and Hahn (1969) and qubit control Economou et al. (2006); Greilich et al. (2009); Poem et al. (2011), and it has since been found to belong to a larger family of analytical controls Bambini and Berman (1981); Bambini and Lindberg (1984); Hioe (1984); Zakrzewski (1985); Silver et al. (1985); Robinson (1985); Ishkhanyan (2000); Carmel and Mann (2000); Kyoseva and Vitanov (2005); Vitanov (2007); Hioe (2007). Several of these examples have proven very beneficial to the fields of quantum control and computation Economou et al. (2006); Greilich et al. (2009); Economou (2012); Motzoi et al. (2009); Chow et al. (2010); Gambetta et al. (2011), where analytical solutions are often central in the design of control fields that are fast, precise, and robust against noise. However, the rarity of such solutions has severely limited one’s options in developing an analytical approach to qubit gate design.
In a recent work Barnes and Das Sarma (2012), a systematic method for deriving arbitrarily many families of exactly solvable two-state systems was presented, vastly extending the number of known analytical solutions. This method allows one to input many of the basic features of the desired control field and then compute exactly the corresponding evolution of the system with the provided formulas. However, a limitation of this work is that it applies only to systems where the driving is along a single axis of the Bloch sphere, such as in the case of electrically-driven singlet-triplet qubits Petta et al. (2005); Foletti et al. (2009); Maune et al. (2012); Wang et al. (2012), making it inapplicable to the majority of driven two-level systems.
In this paper, I address this limitation by presenting a method to generate arbitrarily many families of solutions in the most general case where the two-level Hamiltonian has time-dependence along any set of axes of the Bloch sphere. Of course, one can easily generate exactly solvable Hamiltonians by first choosing the evolution operator and then differentiating to obtain the corresponding Hamiltonian, but it is challenging to arrive at a physically meaningful Hamiltonian in this way. In contrast, the method presented here allows one to specify the basic form and many features of the Hamiltonian whose evolution one wishes to solve before proceeding to compute the exact solution for this evolution. This method has important applications in a vast range of problems, including the development of quantum controls for essentially any quantum computing platform and control protocols for performing LMSZ interferometry and non-adiabatic rapid passage (NARP). I illustrate this by deriving new, exactly solvable LMSZ driving fields and control pulses that execute a desired evolution at speeds approaching the quantum speed limit (QSL) Mandelstam and Tamm (1945); Bhattacharyya (1983); Margolus and Levitin (1998); Giovannetti et al. (2003); Caneva et al. (2009); Bason et al. (2012); Hegerfeldt (2012). Attaining fast evolution times is especially crucial in quantum computing where quantum gates need to be performed on timescales much shorter than the decoherence time. In the case of periodic driving through a level anti-crossing, I show that the formalism allows one to easily derive analytical expressions for LMSZ interference patterns and conditions for coherent destruction of tunneling Grossmann et al. (1991); Stehlik et al. (2012).
Ii Analytically solvable Hamiltonians
The Hamiltonian we consider has the general form
where the are real functions and the are Pauli matrices. This Hamiltonian describes any time-dependent two-level system, with the functions interpreted as either driving fields or time-dependent energy splittings. Alternatively, we may parametrize the Hamiltonian in terms of rotating-frame fields , , and , where and . In the appendix, it is shown that one can systematically find analytical solutions for the evolution operator generated by Hamiltonian (1) with and either or chosen as desired; although one cannot choose both and at will (were this the case, all two-state problems could be solved analytically), one still has a large amount of control over the features of the second, unspecified function. For concreteness, we suppose that one wishes to fix at the outset (the formalism can easily be modified to fix instead). While we cannot then find analytical solutions for arbitrary , there exists a different parametrization of the Hamiltonian in which is replaced with a new function, , such that one can systematically generate an analytical expression for the evolution operator for arbitrary choices of , and . Parametrizing the lab-frame evolution operator as
the explicit , and driving fields are (see the appendix)
The initial conditions , imply , and ensures that is finite, where . Eqs. (3),(4) embody one of the main results of this paper, as they constitute a general analytic solution of the evolution generated by the Hamiltonian of Eq. (1). The task of finding analytical solutions has been reduced to first choosing , by picking and at will. One then selects to produce a desired via Eq. (4), fixing the Hamiltonian. Once these choices are made, an analytical expression for the evolution operator follows from Eq. (3). A simple case is and , where Eq. (4) gives that , corresponding to an -rotation for any . Another simple example arises when , for which Eq. (3) yields a -rotation for any .
Iii Quantum speed limit
In Eqs. (3),(4), it is clear that proper solutions necessarily satisfy . The physical origin of this constraint lies in the notion of the quantum speed limit Mandelstam and Tamm (1945); Bhattacharyya (1983); Margolus and Levitin (1998); Giovannetti et al. (2003); Caneva et al. (2009); Bason et al. (2012); Hegerfeldt (2012), which refers to the minimum time it takes a quantum state to evolve to a different state in the Hilbert space due to energy-time uncertainty. Indeed, implies that the fastest possible evolution from to a desired final value is obtained by choosing , with the shortest time given by substituting in this expression and solving for in terms of and whatever parameters might appear in . For constant , we immediately obtain , which is the QSL time for states evolving under an arbitrary time-independent Hamiltonian in the “Heisenberg regime” Giovannetti et al. (2003). We refer to as the QSL constraint. The present work leads to a general definition of , , for arbitrary time-dependent two-level systems. This definition is consistent with that used in Ref. Caneva et al. (2009) for a certain class of time-dependent Hamiltonians. Notice that the QSL evolution coincides with , suggesting that the fastest quantum operations are those which tend to minimize , a tendency that is borne out in the examples given below.
The fact that the QSL appears as a simple condition on makes the formalism of Eqs. (3),(4) very effective for designing quantum controls that operate near the QSL. To see how this works for a general , note that a simple way to construct a function which obeys the QSL constraint is to first find a function which satisfies the constraint in the case where is a constant. Denoting this latter function by and defining , if we choose , then automatically follows. Note that all the single-axis driving examples of Barnes and Das Sarma (2012), where the notation there is related to the present by , can be extended to multi-axis solutions using this trick. Furthermore, if the control field corresponding to operates near the QSL, this will also tend to be the case for the one generated by . Focusing then on the case , we can construct controls that operate near the QSL by choosing a which contains parameters that can be tuned to values where . An important feature of solutions generated from a whose time dependence arises only through is that the evolution operator is an ordinary function of and , namely
where . This fact greatly facilitates the design of a desired evolution since one can directly control the values of and by adjusting and .
Iv Pulse examples
iv.1 Gaussian-like pulses
To illustrate this method of obtaining multi-axis solutions from single-axis ones, consider the example where and , and the QSL constraint is satisfied. Using the method described above, we can extend this to a solution for any :
which yields the following driving terms:
where and the heights, widths and centers of these pulses can be controlled by tuning the parameters and . These pulses and the evolution they produce are shown in Fig. 1, where it is clear that these pulses have a simple, smooth shape.
iv.2 Pulses with arbitrarily many parameters
For a near-QSL pulse example, consider the case
where and the are arbitrary constants, is an even integer, and , . The QSL constraint is satisfied regardless of how large is, so that this yields an exact solution with arbitrarily many parameters . We can make the corresponding control field a pulse by setting , so that and as . The initial value of the pulse is set by : . Examples of these pulses are shown in Fig. 2a. The duration of the pulse approaches in the limit , , as can be seen by observing that in this limit. The substantial amount of tunability in this solution already makes it very attractive for applications in quantum computation such as dynamically corrected gates Motzoi et al. (2009); Gambetta et al. (2011); Wang et al. (2012), where the shape of the pulse is tuned to perform a specific quantum operation while simultaneously suppressing errors.
Using the prescription outlined above, we can extend this solution to the case of non-constant :
This class of pulses can be used to implement quantum operations by tuning for a given choice of and . We demonstrate this by designing a fast pulse that, together with , implements a Hadamard gate, a quantum operation that is ubiquitous in the field of quantum information processing and which is equivalent to a -rotation about . First choose , which ensures that . Supposing , if we let the system evolve for a time such that , then the phases of and will also attain their Hadamard values. Such a pulse is shown in Fig. 2b for an oscillatory . From Fig. 2b, it is evident that quickly sets the magnitudes of and , while the remainder of the evolution with allows their phases to accumulate. As before, the duration of the pulse approaches as , . This example illustrates how one can use this formalism to design analytical controls near the QSL in the presence of additional driving fields.
V LMSZ interferometry
The present formalism is also natural for designing driving fields that perform controlled LMSZ interferometry and non-adiabatic rapid passage, phenomena which have many applications in quantum control Petta et al. (2010); Ribeiro et al. (2012); Quintana et al. (2013); Zhang et al. (2012), state preparation Wu et al. (2011); Brierley et al. (2012) and qubit readout Petta et al. (2005); Foletti et al. (2009); Maune et al. (2012). (See Refs. Berry (1990); Lim and Berry (1991); Vitanov and Suominen (1999); Berry (2009); Malossi et al. (2012); Ruschhaupt et al. (2012) and references therein for previous analytical approaches for the LMSZ problem.) The LMSZ problem is generally setup as follows. Define the eigenstates of to be and and set so that ; when , these states are approximate energy eigenstates. A nonzero produces an anti-crossing with an energy gap of (see Fig. 3) which may be time-dependent. Now suppose that we drive through the anti-crossing, starting from some large negative value at up to a large positive value at . Assuming that the system is initially prepared in state at time , the probability for the system to be in state at time is
The fact that this depends only on demonstrates the suitability of the present formalism to the LMSZ problem. If we choose such that and , then we achieve a perfect LMSZ transition: the system is driven through the anti-crossing and returns to state with probability 1. On the other hand, we may choose , in which case the system undergoes NARP and ends up in state after being driven through the anti-crossing. Other values of lead to superpositions of and . We may also consider LMSZ interferometry, where the system is driven through the anti-crossing periodically, and the resulting time-averaged probabilities to be in state or after many periods is again largely determined by , as we will see. In choosing a for the LMSZ problem, we must impose appropriate initial conditions. For simplicity, we focus on the case , , for which we need , ; the analysis can be extended straightforwardly to the case where is finite at .
For constant , a simple example which satisfies these boundary conditions and the QSL constraint is
where choosing ensures that the from Eq. (3) is finite in the interval , and . This saturates the QSL constraint when , implying that will implement near-QSL evolution for small . To achieve a target , set in Eq. (12) and solve for : . Plugging this into Eqs. (12) and (4) yields a family of driving fields parametrized by that achieve the desired evolution for any ; some of these are shown in Fig. 4 along with the corresponding NARP probabilities. The restrictions on impose bounds on : ; the upper bound is particular to Eq. (12), while the lower bound is the familiar, universal QSL and gives rise to the step-like curves in Figs. 4a and 4c. These curves reveal that the desired LMSZ transition is achieved as quickly as possible by first driving to zero very rapidly, allowing the system to evolve for a time , and then driving quickly up to its final value (see also Ref. Hegerfeldt (2012)). In addition to NARP, these near-QSL driving fields could be important for LMSZ-based generation of entanglement in superconducting qubits Quintana et al. (2013), where fidelities are often limited by short relaxation times.
In the context of LMSZ interferometry, the formalism of Eqs. (3),(4) yields an exact formula for LMSZ interference patterns. To show this, we begin by constructing a periodic driving field of period where determines the first half of one period and the second half, corresponding to retracing its path. Using Eq. (3), we find the evolution after one full period:
where and we have assumed that for simplicity. From this expression, it is straightforward to compute the time-averaged probability of being in the excited state after many periods:
Thus, we see that the present formalism readily produces a general, exact, analytic formula for , whereas analytic expressions for this important quantity typically require several approximations Shevchenko et al. (2010). This function takes values in the range , where for , while for . This is to be expected since corresponds to a 50-50 “beam splitter”, while ensures the system remains in the ground state after every sweep through the anti-crossing. In the context of charge qubits where states and correspond to an electron being in either the left or right dot of a tunnel-coupled double quantum dot, the case can be interpreted as coherent destruction of tunneling Grossmann et al. (1991); Stehlik et al. (2012) since the electron becomes localized in one dot. For more generic values of , is modulated by the phase , which can produce interference fringes as control parameters are adjusted, as shown in Fig. 5 for the example from Eq. (12) with . As is varied, an interference pattern emerges in which the peaks of the pattern sharpen and eventually disappear as approaches . Interestingly, Fig. 5 reveals a peak at the QSL time ; this is generally the case since at the QSL, , so that . This leads to the surprising conclusion that if we choose in order to trap the system in state , then driving very close to the QSL may not be ideal since small deviations away from will produce the peak at and, hence, large uncertainty in the state of the system.
In conclusion, a general formalism for deriving exactly solvable time-dependent two-level quantum systems has been presented. This formalism can vastly increase the number of known exact solutions for physical Hamiltonians, as has been demonstrated with explicit examples. These examples show that this theory can be a powerful tool in the design of control pulses both for quantum computation and for precise Landau-Majorana-Stückelberg-Zener interferometry near the quantum speed limit.
I thank Michael Berry, Lev Bishop, Łukasz Cywiński, and Sophia Economou for helpful discussions. I also thank Gerhard Hegerfeldt for useful comments. This work is supported by LPS-NSA and IARPA.
In this appendix, we derive Eqs. (3) and (4) of the main text. The general form of the Hamiltonian and its corresponding evolution operator are given in Eqs. (1),(2). The evolution operator obeys a Schrödinger equation whose form can be made compact by transforming to a rotating frame: , . Defining and , we have
Here, it is manifest that the evolution operator in the rotating frame depends on only two real functions, and . We must further specify to return to the lab frame, but this choice can be made after the evolution operator is computed in the rotating frame. In what follows, I will show that one can systematically find analytical solutions with either or chosen as desired; although one cannot choose both and at will (if this were the case, all two-state problems would be analytically solvable), we will see that one still has a large amount of control over the features of the second, unspecified function.
For concreteness, we will suppose that is chosen at the outset (the formalism can easily be modified to fix instead). While we cannot then find analytical solutions for arbitrary , there exists a different parametrization of the Hamiltonian in which is replaced with a new function, , such that one can systematically generate an analytical expression for the evolution operator for arbitrary choices of and . To see this, first express the rotating-frame evolution operator in terms of some complex :
with . This choice of parametrization is motivated by observing that we can combine the two equations in (15) to obtain , which generally implies , for some complex . The fact that this is true in general can be seen by noting that for , any complex function can be expressed as for some complex function , so that we may therefore express in this way. then implies . This argument does not hold when , however in this case we find , which is consistent with Eq. (15). This analysis is thus completely general and applies to any solution of the Schrödinger equation. Consistency between Eqs. (16) and (15) requires
which should be interpreted as follows: For any choice of a complex and real such that the computed from Eq. (17) is real, the evolution operator obtained from Eq. (16) is the exact solution for this and . Writing and imposing determines in terms of : , with
leading to an expression for that is real for any :
While this parametrization has the nice feature that can be chosen freely, one drawback is that one must then perform the integration in Eq. (18), making it harder to relate the features of to the driving field . We can avoid this by specifying directly, but at the expense of now having to choose functions that obey the quantum speed limit constraint, , which arises directly from Eq. (18). Solving Eq. (18) for in terms of , it is straightforward to turn the above expressions for the evolution operator into expressions which depend on rather than . The resulting lab-frame evolution operator and driving fields are given in Eqs. (3) and (4).
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