Generating arbitrary analytically solvable two-level systems

We present a new approach for generating arbitrary analytically solvable two-level systems. This method offers the ability to completely derive all analytically solvable Hamiltonians for any analytical evolutions of two-level systems. To demonstrate the effectiveness of this approach, we reconstruct the Rosen–Zener model and generate several new exact solutions. Using this approach, we present the exact evolution of the semi-classical Rabi model with new analytical properties. The parameters used to generate Hamiltonians have direct physical interpretations within the Bloch sphere, the quantum speed limit, and the geometric phase. As a result, the physical properties of the generated Hamiltonian are highly controllable, which plays a significant role in the fields of quantum control, quantum computing, and quantum information.

In general, it is challenging to solve the Schrödinger equation with time-dependent Hamiltonians.However, the inverse process, which involves generating the analytically solvable Hamiltonian from the corresponding time evolution operator, is more manageable.This inverse process can be implemented by using Lie algebras to represent the evolution operator [19][20][21][22][23][24][25] and is named the inverse technique by Fernández [23].Another approach in two-level systems is to directly parameterize the matrix elements of the evolution operator, as demonstrated in several previous works [26][27][28][29].The inverse technique can generate an unlimited number of analytically solvable two-level Hamiltonians.
In this paper, we present a new theoretical approach to generate arbitrary analytically solvable two-level quantum systems.The proposed approach here is capable of generating complete analytically solvable two-level systems, contrasting with existing methods that were only limited to specific models.By combining aspects of Lie algebra and direct representation of the evolution operator matrix, this new approach offers a direct and physically interpretable method.This approach parameterizes the time evolution operator by three Euler angles {ϕ, θ, γ} and establishes a connection between the time evolution operator and the Hamiltonian through a brief equation.There are two main advantages of this approach.The first advantage is the ability to generate all the analytically solvable two-level Hamiltonians and their evolutions.For example, we apply this approach to reconstruct the time-evolution of the Rosen-Zener model, derive the analytical time-evolution of the semi-classical Rabi model, and generate novel exact solutions.The second advantage is that the three parameters {ϕ, θ, γ} have direct physical interpretations.ϕ and θ represent the trajectory of the state in the Bloch representation, which allows using desired physical features of the evolution to generate analytically solvable Hamiltonians.This advantage, enabling the achievement of desired probabilities and relative phases of states, plays a significant role in quantum control and quantum computing.
The paper is structured as follows.In section 2 we introduce a new generating approach parameterized by Euler angles.In section 3 we demonstrate the completeness of our generating approach through several examples and generate new exact solutions.In section 4 we discuss the physical meanings of the parameters within the Bloch sphere, the quantum speed limit, and the geometric phase.Section 5 contains our conclusions.

Generating approach parameterized by Euler angles
Consider a time-dependent two-level system described by a general form Hamiltonian H = 1 2 ω(t) • σ, where ω(t) = (ω x , ω y , ω z ) T , σ = (σ x , σ y , σ z ), and σ x , σ y , σ z are Pauli matrices.The corresponding unitary evolution operator can be parameterized as where ϕ (t), θ (t) and γ (t) are real functions of time t.This evolution (1) can be expressed more concisely as Using the Bloch sphere representation, we denote the density matrix of the state as ρ(t) = 1 2 (I + r(t) • σ), where r(t) is the normalized Bloch vector and I is the 2 × 2 identity matrix.The vector r(t) represents a point on the Bloch sphere corresponding to the state.For example, the Bloch vector of the state |0⟩ is (0, 0, 1) T (the north pole), where |0⟩ ≡ | ↑⟩ and |1⟩ ≡ | ↓⟩.
The evolution U(ϕ, θ, γ) acting on the state ρ 0 can be interpreted as a rotation applied to the initial Bloch vector r 0 .This rotation is expressed as r = R z (ϕ)R y (θ)R z (γ)r 0 , where R n (α) represents a counterclockwise rotation of angle α along the n-direction in the Bloch representation.The three parameters ϕ, θ and γ are the three Euler angles in the ZYZ convention, which plays a pivotal role in describing the system's physical properties.Set the initial time t = t 0 , then construct the time evolution operator as U(t, t 0 ) = U(t)U † (t 0 ), allowing that U(t 0 ) can be arbitrary and does not have to be the identity matrix I.For simplicity, we can still choose the initial unitary operator U(t 0 ) as I, and set the initial state at time t 0 as |ψ(t 0 )⟩ = |0⟩.When applying the time-evolution operator U(t, t 0 ) = U(t) = U(ϕ, θ, γ), the corresponding state's Bloch vector evolves into r = (sin θ cos ϕ, sin θ sin ϕ, cos θ) T and the corresponding state is e −i γ/2 |ϕ, θ⟩, where the factor e −i γ/2 can be omitted.The parameters ϕ and θ possess a direct physical interpretation as they indicate the direction of the final state in the Bloch sphere representation.For an arbitrary initial state |ψ(t 0 )⟩ = U 0 |0⟩, we can choose U(t 0 ) = U 0 and the evolving state can be expressed as |ψ(t)⟩ = |ϕ, θ⟩.Under this condition, the parameters ϕ and θ are connected to the geometric properties of the state |ψ(t)⟩ = |ϕ, θ⟩, and the parameter γ represents a freely changing intrinsic rotation angle associated with the total phase.
According to the Schrödinger equation, the evolution ( 2) satisfies H = i UU † (set ℏ = 1).Then one can derive three differential equations as which is equivalent to the vector form with the unit vector k in the z-direction.The nonlinearity of these differential equations, arising from the non-commutativity of the Pauli matrices, often presents a significant challenge in finding solutions.Fortunately, the inverse process is simple.By choosing a set of three arbitrary real functions {ϕ, θ, γ}, the analytically solvable Hamiltonian H(ϕ, θ, γ) can be determined by (4).
The main result of this paper is the approach to generate arbitrary analytically solvable two-level systems.The three parameters {ϕ, θ, γ} in this approach can be considered almost arbitrary, as they are only required to be continuous and differentiable without any additional constraints.Although different parametric methods have been proposed in a series of articles [23][24][25][26][27][28][29], our approach in this paper carries direct physical significance and can generate all analytically solvable two-level systems.We can represent an arbitrary unitary evolution U(ϕ, θ, γ) as a trajectory r(t) on the Bloch sphere.The reverse process is also true: all trajectories on the Bloch sphere can be represented by a set of {ϕ, θ}.As a result, all possible unitary evolutions for two-level systems can be generated, and the corresponding Hamiltonians are also determined.If the solution of the state is |ψ(t)⟩ = a|0⟩ + b|1⟩, the evolution U(t) can be expressed more briefly as U 11 = a and U 21 = b.
Our generating approach can be categorized into three different types.The first type, as demonstrated above, involves selecting three arbitrary real functions {ϕ, θ, γ}.With this selection and (4), one can easily obtain the related Hamiltonian.An example of a shamrock-pattern trajectory is presented in appendix C, illustrating the flexibility of the first type of the generating approach.The second type is selecting two out of the set {ϕ, θ, γ} and one constraint function for ω.For instance, we can choose ϕ(t), θ(t) and ω z (t) to generate the Hamiltonian by γ = ) ) This choice is natural since the functions ϕ(t) and θ(t) are associated with the evolution trajectory, whereas the function γ(t) is not.However (3c) is solvable for all different choices of the set {ϕ, θ, γ}.Generally, we can select f = ω • n 1 as the Hamiltonian constraint, where n 1 is the normalized vector of the controlled direction.With the notation n 2 = (− sin ϕ, cos ϕ, 0) T , we can obtain γ = (f − θn 2 • n 1 − φk • n 1 )/r • n 1 , and determine the Hamiltonian by (5a)-(5c).From another point of view, we can just denote n 1 as a new z-direction.Then the trajectory can be represented with the transformed functions ϕ n1 (t) and θ n1 (t), while the Hamiltonian can be still obtained by (5a)-(5c).
The third type of our generating approach is selecting only one function from the set {ϕ, θ} and two constraint functions for ω.Due to the symmetry of the Bloch sphere, we can always choose the constraint functions in the xy-plane as ω x = Ω cos Φ and ω y = Ω sin Φ.Then (3a) and (3b) can be reformulated as Using (6a), one can obtain the function ϕ (or θ) from θ (or ϕ), then γ can be derived by (6b) and the Hamiltonian is subsequently determined.

The Rosen-Zener model
The exact hyperbolic secant pulse solution of a two-level system was initially discovered by Rosen and Zener [3], and later extended to three models including the Hioe chirped pulse, the Bambini-Berman asymmetric pulse and the Zakrzewski general pulse [5][6][7].The Zakrzewski solution serves as a comprehensive and general form that encompasses the other three models.The general form Hamiltonian [7] can be expressed as where 2 F 1 denotes the hypergeometric function, c Here the notation (q) n is the (rising) Pochhammer symbol defined by (q) n = q(q + 1) • • • (q + n − 1), n⟩0 and (q) n=0 = 1.The elements u 11 and u 21 of the unitary evolution operator U(t) are respectively equal to the complex amplitudes of the states |0⟩ and |1⟩, which is expressed as u 11 = α 0 , u 21 = α 1 .These models can be regenerated using the first type of the generating approach described in (4) with three parameter functions as: New analytically solvable Hamiltonians of similar kinds can be generated by the third type of the generating approach in (6a) and (6b).By using the rotating frame R = e i(g+∆t)σz/2 , the effective Hamiltonian of H can be derived as , where g z = ġ + ∆ is a function of t.Specifically, for the Rosen-Zener and Bambini-Berman models, g z = ∆ is constant.We can choose ω y = 0 and ω x as constraints, and subsequently utilize (6a) and (6b) to obtain the Hamiltonian by ϕ or θ as which are consistent with the results in [26][27][28][29].If we choose ω y = 0 and ω z as constraints, the algorithm for deriving the Hamiltonian differs slightly.Solving the differential equations (3a)-(3c), we can determine the function θ(t) by sin θ = sec ϕ e − ´ωz tan ϕ dt /ηkm, where η = ±1.
Here, k ⩾ 1 is a constant related to the initial condition, and m is defined as the supremum of e − ´ωz tan ϕ dt | sec ϕ |.Next, the x-direction field ω x can be expressed as As an example, let us consider the choice of ϕ = arctan(−te −t 2 /σ 2 ) and a constant ω z = c.This selection leads to a Gaussian-like pulse with three tunable parameters (k, σ, c), which can be expressed as where h = e −t 2 /σ 2 .The impact of the standard deviation σ on the pulse width along the x direction can be observed in figure 1.Additionally, figure 2 visually depicts the evolution U(t) under various parameter configurations.It's important to emphasize that the curve in figure 2 can represent the trajectory of the state in the Bloch representation only when the initial state is U(t 0 )|0⟩, denoted as the black point in figure 2. If the initial state differs from U(t 0 )|0⟩, the trajectory will be different as well.The closed curves in figure 2 exemplify the final state's return to the initial state as t ranges from −∞ to +∞.

Discussion
The velocity of the evolution can be regarded as the classical speed of the state point r(t) on the Bloch sphere, and the expression can be derived [46,[56][57][58][59][60][61][62] as where ω(t) is the angular velocity determined by 1  2 ω • σ ≡ H.This equation can also be derived by the quantum dynamics expressed in (3a)-(3c) (see appendix B.6), which can be viewed as the Euler kinematic equations [58][59][60][61][62]. From another perspective, if we can solve the vector differential equation ( 15) for a classical ω(t), the quantum dynamics will also be obtained by (1) or (2).Therefore ( 1) and (3a)-(3c) establish a brief link between classical and quantum physics, which is similar to [62] but more concise.If the Hamiltonian H is bounded by |ω| ⩽ Q, the classical speed is also bounded by |v| ⩽ Q.Then the fastest evolution follows the geodesic line on the Bloch sphere with the shortest time τ .An inequality for the quantum speed limit [63] can be constructed as t ⩾ τ = θ Q , where θ is the angle between the two Bloch vectors of two states.
The three parameters {ϕ, θ, γ} in this generating approach have a direct connection to the quantum geometric phase, which was originally studied by Berry [64] for adiabatic cyclic motion and later explored by Aharonov and Anandan [65] for any cyclic evolution.The evolution can be rewritten as U(ϕ, θ, γ) = e − 1 2 iθ(cos ϕσy−sin ϕσx) e − 1 2 i(γ+ϕ)σz .Assuming the initial state is |ψ(t 0 )⟩ = |0⟩, we can easily get the final state as |ψ(t)⟩ = e − 1 2 i(γ+ϕ) (cos θ 2 |0⟩ + sin θ 2 e iϕ |1⟩), where the total phase is γ T = − 1 2 (γ + ϕ).For a cyclic evolution [64][65][66][67][68], we can obtain the geometric phase and the dynamic phase where Ω(C) represents the solid angle enclosed on the Bloch sphere by the curve C. Geometric rotation on the Bloch sphere leads to changes in the dynamic phase without altering the total phase, which implies that the geometric phase is inherently related to such geometric rotations.This generating approach can be extended to Hamiltonians governed by arbitrary finitedimensional Lie algebras A. The modified evolution operator takes the form [19] U = e −iα1A1 e −iα2A2 . . .e −iαnAn , where A i is the generator of A. By selecting a set of α i , an analytically solvable Hamiltonian can be obtained by H = i UU † .The generating approach within the SU(2) algebra is simply the same as the two-level systems approach, since the spin-s does not alter the parametric evolution [60].For instance, our generating approach is applicable to a system of N noninteracting two-level atoms [34][35][36] which can be regarded as an SU(2) system.However, whether arbitrary evolution can be generated through a specific set of α i values requires further investigation in future studies.

Conclusion
In summary, we present a new parametric approach to completely generate all analytically solvable two-level systems.The key parameters in this approach are directly linked to the Euler angles, which are related to the geometric phase and carry clear physical meanings.This approach holds the advantage of enabling the design of analytically solvable Hamiltonians based on desired probabilities and relative phases of states.This feature is particularly relevant in the realms of quantum control and quantum computing.To demonstrate our approach, we regenerate several well-known models and present new exact solutions.A central contribution of our approach is its ability to establish a concise link between all analytically solvable Hamiltonians and their corresponding evolutions.This connection significantly enhances our understanding of quantum mechanics and provides a valuable tool for exploring the dynamics of quantum systems.

B.2. The derivation of the effective Hamiltonian
The Hamiltonian given in ( 8) can be transformed into a rotating frame using the rotation operator R = e i 1 2 (g+∆t)σz .The resulting effective Hamiltonian can be obtained as , where g z = ġ + ∆.
We can obtain several intermediate results first.We can get φ as φ = − , where h = e − t 2 σ 2 .Then, we obtain cos ϕ = 1 √ h 2 t 2 +1 and ´tan ϕ dt = hσ 2  2 + C 0 .Next, we substitute ϕ = arctan(−te −t 2 /σ 2 ) and ω z = c into (B.9) and can obtain which matches the expression given by (11).For the parameters in figures 1 and 2, the number B.4.The derivation of (13) Considering the initial state is |0⟩, the final state can be expressed as where z = sin 2 t 2 , η = 1 + 2⌊ t−π T ⌋ with the floor function ⌊x⌋.We can generate the evolution operator U 1 by setting its elements as (U 1 ) 11 = a 1 and (U 1 ) 21 = b 1 .The three parameter functions of the first type of our approach are selected as follows: . By combining U 1 and U 2 , the evolution of half a period is attainable as ), which can be expressed in the matrix form as which can be expressed as ) )] )] where Then, the elements of U T 2 can be derived as 2 iΘσy e − 1 2 iΓσz .Then we can get the evolution over a complete period as , where σ n = sin Φσ x + cos Φσ y .Consequently, the total time evolution operator can be expressed as where We can compare the exact solution with the numerical and approximate results in [36], as illustrated in figure B1.The energy E g (t) is expressed as E g (t) = ⟨ψ g |U † (t + t 0 , t 0 )σ x U(t + t 0 , t 0 )|ψ g ⟩.Given the driven function is A √ 2 cos(ωt), we set t 0 = T/4.The remaining parameters are defined as g = √ 2A/ω, β = ∆/ω, ω 0 = ω, t = ωτ .In figure B1, we use A = 3∆ and ω = 1.2∆, with ∆ = 1 for simplicity.Then the exact expression of E g (t) for − T 4 < t < 3T 4 can be derived as ) )] (B.28) Comparison between the exact Eg (left subfigure) and the numerical and approximate Eg in [36] (right subfigure).Reproduced from [36].© 2020 The Author(s).
Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.CC BY 4.0. where . The exact function of energy E g (τ ) = E g (t = ωτ ) matches the numerical result, as shown in figure B1.
The parameters in figure B3 are ω 0 = 2, g = 2, β = 0.02, which are the same as those in the second figure in [45] which is obtained to fifth order in β.However, the results in figure B3 are the exact solution for the amplitude |b| 2 , which can give more details for the evolution.The  first order in β is now expressed as Ω( g ω0 ) β ω0 = β ω0 π J 0 ( g ω0 ).We can obtain the quasi-period T a of the amplitude |b| 2 (half of the quasi-period of b), which is ) . (B.34) When ω 0 = 2, g = 2, β = 0.02, one can obtain T a = 100π J0(1) = 410.56,which is in agreement with figure B3.The second subfigure in figure B3 represents the amplitude |b| 2 varying over the first four period 4T, which shows that this amplitude variation exhibits a slightly oscillatory behavior.And the third subfigure in figure B3 shows that the amplitude |b| 2 does not exactly decrease down to zero at the quasi-period T a .

Note that ρ
which can be simplified as Taking the time derivative of both sides of (B.36), the left side becomes ṙ(t) • σ while the right side becomes d dt For a cyclic evolution [64][65][66][67][68] as ϕ(T) = ϕ(0) (mod 2π) and θ(T) = θ(0), we can obtain the geometric phase and the dynamic phase where Ω(C) represents the solid angle enclosed on the Bloch sphere by the curve C.

Appendix C. Generating the Hamiltonian with a shamrock-pattern
The three generating parameters are ϕ = ∆t, θ = kf (ϕ) and γ = 0, where k = 1 and f (ϕ) has two choices as f 1 (ϕ) = cos(nϕ) and f 2 (ϕ) = (cos(nϕ) + 1)/2 with n ∈ Z. Then the generated Hamiltonians can be expressed as H 1 = 1 2 φ(n sin(nϕ) sin ϕσ x − n sin(nϕ) cos ϕσ y + σ z ) and H 2 = 1 4 φ(n sin(nϕ) sin ϕσ x − n sin(nϕ) cos ϕσ y + 2σ z ).These two Hamiltonians can be simplified in a rotating frame R = e i 2 ϕσz as H 1,eff = − 1 2 n sin(nϕ)σ y and H 2,eff = − 1 4 n sin(nϕ)σ y .Figure B4 shows the trajectory representation of the evolution with function f 1 (ϕ), while  figure B5 shows the trajectory representation of the evolution with function f 2 (ϕ).In the second subfigure of figure B4, the blue shamrock-pattern with the parameter n = 3 is the trajectory of the evolution and the yellow line with φ = 2  3 represents the projection of the Hamiltonian H 1 = sin(3ϕ) sin ϕσ x − sin(3ϕ) cos ϕσ y + 1 3 σ z onto the xy-plane , which is also a shamrockpattern.As we can see from these figures, the integer parameter n controls the number of the leaves, and when n = 3 both the two functions f 1 (ϕ) and f 2 (ϕ) generate the shamrock-pattern trajectory.Figure C1 shows the trajectory of the evolution on the Bloch sphere.

Figure 2 .
Figure 2. Evolution corresponding to the Hamiltonian from (11) with σ = 0.3 (blue solid line), σ = 0.5 (yellow dot-dashed line), and σ = 1 (green dashed line).The other parameters are consistent with those in figure 1.The purple dotted line is a circle with a radius of θ = π/4.