In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.
Two-state systems are the simplest quantum systems that are of interest, since the dynamics of a one-state system is trivial (as there are no other states in which the system can exist). The mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states.
A well known example of a two-state system is the spin of a spin-1/2 particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the reduced Planck constant.
The two-state system cannot be used as a description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two-state system are oscillatory.
Analytical solutions for stationary state energies and time-dependence
Representation Supposing the two available basis states of the system are and , in general the state can be written as a superposition of these two states with probability amplitudes ,
Since the basis states are orthonormal, where and is the Kronecker delta, so . These two complex numbers may be considered coordinates in a two-dimensional complex Hilbert space. Thus the state vector corresponding to the state is and the basis states correspond to the basis vectors, and
If the state is normalized, the norm of the state vector is unity, i.e. .
All observable physical quantities, such as energy, are associated with hermitian operators. In the case of energy and the corresponding Hamiltonian, , this means i.e. and are real, and . Thus, these four matrix elements produce a 2×2 hermitian matrix,
The time-independent Schrödinger equation states that ; substituting for in terms of the basis states from above, and multiplying both sides by or produces a system of two linear equations that can be written in matrix form, or which is a 2×2 matrix eigenvalues and eigenvectors problem. As mentioned above, this equation comes from plugging a general state into the time-independent Schrödinger equation. Remember that the time-independent Schrödinger equation is a restrictive condition used to specify the eigenstates. Therefore, when plugging a general state into it, we are seeing what form the general state must take to be an eigenstate. Doing so, and distributing, we get , which requires or to be zero ( cannot be equal to both and , the energies of the individual states, which are by definition different). Upon setting or to be 0, only one state remains, and is the energy of the surviving state. This result is a redundant reminder that the time-independent Schrödinger equation is only satisfied by eigenstates of H, which are (by definition of the state vector) the states where all except one coefficient are zero. Now, if we follow the same derivation, but before acting with the Hamiltonian on the individual states, we multiply both sides by or , we get a system of two linear equations that can be combined into the above matrix equation. Like before, this can only be satisfied if or is zero, and when this happens, the constant will be the energy of the remaining state. The above matrix equation should thus be interpreted as a restrictive condition on a general state vector to yield an eigenvector of , exactly analogous to the time-independent Schrödinger equation.
Of course, in general, commuting the matrix with a state vector will not result in the same vector multiplied by a constant E. For general validity, one has to write the equation in the form with the individual eigenstate energies still inside the product vector. In either case, the Hamiltonian matrix can be derived using the method specified above, or via the more traditional method of constructing a matrix using boundary conditions; specifically, by using the requirement that when it acts on either basis state, it must return that state multiplied by the energy of that state. (There are no boundary conditions on how it acts on a general state.) This results in a diagonal matrix with the diagonal elements being the energies of the eigenstates and the off-diagonal elements being zero. The form of the matrix above that uses bra-ket-enclosed Hamiltonians is a more generalized version of this matrix.
One might ask why it is necessary to write the Hamiltonian matrix in such a general form with bra-ket-enclosed Hamiltonians, since should always equal zero and should always equal . The reason is that, in some more complex problems, the state vectors may not be eigenstates of the Hamiltonian used in the matrix. One place where this occurs is in degenerate perturbation theory, where the off-diagonal elements are nonzero until the problem is solved by diagonalization.
Because of the hermiticity of the eigenvalues are real; or, rather, conversely, it is the requirement that the energies are real that implies the hermiticity of . The eigenvectors represent the stationary states, i.e., those for whom the absolute magnitude of the squares of the probability amplitudes do not change with time.
Eigenvalues of the Hamiltonian The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system is given by where and are real numbers with units of energy. The allowed energy levels of the system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way.
Equivalently, this matrix can be decomposed as, Here, and are real numbers. The matrix is the 2×2 identity matrix and the matrices with are the Pauli matrices. This decomposition simplifies the analysis of the system, especially in the time-independent case, where the values of and are constants.
The Hamiltonian can be further condensed as
The vector is given by and is given by . This representation simplifies the analysis of the time evolution of the system and is easier to use with other specialized representations such as the Bloch sphere.
If the two-state system's time-independent Hamiltonian is defined as above, then its eigenvalues are given by . Evidently, α is the average energy of the two levels, and the norm of is the splitting between them. The corresponding eigenvectors are denoted as and .
Time dependence We now assume that the probability amplitudes are time-dependent, though the basis states are not. The Time-dependent Schrödinger equation states , and proceeding as before (substituting for and premultiplying by again produces a pair of coupled linear equations, but this time they are first order partial differential equations: . If is time independent there are several approaches to find the time dependence of , such as normal modes. The result is that where is the statevector at . Here the exponential of a matrix may be found from the series expansion. The matrix is called the time evolution matrix (which comprises the matrix elements of the corresponding time evolution operator ). It is easily proved that is unitary, meaning that .
It can be shown that where
When one changes the basis to the eigenvectors of the Hamiltonian, in other words, if the basis states are chosen to be the eigenvectors, then and and so the Hamiltonian is diagonal, i.e. and is of the form,
Now, the unitary time evolution operator is easily seen to be given by:The factor merely contributes to the overall phase of the operator, and can usually be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any perturbation to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation the new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction.
Rabi formula for a static perturbation Suppose that the system starts in one of the basis states at , say so that , and we are interested in the probability of occupation of each of the basis states as a function of time when is the time-independent Hamiltonian.
The probability of occupation of state is . In the case of the starting state, , and from above, Hence,
Obviously, due to the initial condition. The frequency is called the generalised Rabi frequency, is called the Rabi frequency, and is called the detuning.
At zero detuning, , i.e., there is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1, etc., with frequency . As the detuning is increased away from zero, the frequency of the flopping increases (to ) and the amplitude of exciting the electron decreases to .
For time dependent Hamiltonians induced by light waves, see the articles on Rabi cycle and rotating wave approximation.
Some important two-state systems
Precession in a field
Consider the case of a spin-1/2 particle in a magnetic field . The interaction Hamiltonian for this system is where is the magnitude of the particle's magnetic moment and is the vector of Pauli matrices. Solving the time dependent Schrödinger equation yields where and . Physically, this corresponds to the Bloch vector precessing around with angular frequency . Without loss of generality, assume the field is uniform and points in , so that the time evolution operator is given as
It can be seen that such a time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of Larmor precession)
The above method can be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic field in the previous case) if the interaction is given by an appropriate coupling term that is analogous to the magnetic moment. The precession of the state vector (which need not be a physical spinning as in the previous case) can be viewed as the precession of the state vector on the Bloch sphere.
The representation on the Bloch sphere for a state vector will simply be the vector of expectation values . As an example, consider a state vector that is a normalized superposition of and , that is, a vector that can be represented in the basis as
The components of on the Bloch sphere will simply be . This is a unit vector that begins pointing along and precesses around in a left-handed manner. In general, by a rotation around , any state vector can be represented as with real coefficients and . Such a state vector corresponds to a Bloch vector in the xz-plane making an angle with the z-axis. This vector will proceed to precess around . In theory, by allowing the system to interact with the field of a particular direction and strength for precise durations, it is possible to obtain any orientation of the Bloch vector, which is equivalent to obtaining any complex superposition. This is the basis for numerous technologies including quantum computing and MRI.
Evolution in a time-dependent field: Nuclear magnetic resonance