Two-state quantum system


In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent 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 can exist, since the dynamics of a one-state system is trivial. 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 very 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, then 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 statevector 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 schrodinger equation states that, and substituting for in terms of the basis states from above, and premultiplying 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. Because of the hermiticity of the eigenvalues are real, or rather vice versa 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.
Alternatively, this matrix can be decomposed as,
Here, and are real numbers. The matrix is the 2 2 identity matrix and the matrices 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 written even more compactly 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 is the average energy of the two levels, and the norm of is the splitting between them. The corresponding eigenvectors are denoted 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. 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 only 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 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 and the amplitude decreases to.
See also Rabi cycle and Rotating wave approximation for time dependent Hamiltonians induced by light waves.

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 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
The above method can be applied to the analysis of any generic two-state system that is interacting with some field if the interaction is given by an appropriate coupling term that is analogous to the magnetic moment. The precession of the state vector 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

is an important example in the dynamics of two-state systems because it is involves the exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field B0 and then applying a weak, transverse field B1 that oscillates at some radiofrequency ωr. Explicitly, consider a spin-1/2 particle in a holding field and a transverse rf field B1 rotating in the xy-plane in a right-handed fashion around B0:
As in the free precession case, the Hamiltonian is, and the evolution of a state vector is found by solving the time-dependent Schrödinger equation. After some manipulation, it can be shown that the Schrödinger equation becomes
where and.
As per the previous section, the solution to this equation has the Bloch vector precessing around with a frequency that is twice the magnitude of the vector. If is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that, in the rotating frame the state vector will precess around with frequency, and will thus flip from down to up releasing energy in the form of detectable photons. This is the fundamental basis for NMR, and in practice is accomplished by scanning until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use is made of the rotating wave approximation in deriving such results.
Derivation of above expression for the NMR Schrödinger equation
Here the Schrödinger equation reads
Expanding the dot product and dividing by yields
To remove the time dependence from the problem, the wave function is transformed according to. The time dependent Schrödinger equation becomes
which after some rearrangement yields
Evaluating each term on the right hand side of the equation
The equation now reads
which by Euler's identity becomes

Relation to Bloch equations

The optical Bloch equations for a collection of spin-1/2 particles can be derived from the time dependent Schrödinger equation for a two level system. Starting with the previously stated Hamiltonian, it can be written in summation notation after some rearrangement as
Multiplying by a Pauli matrix and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields
Adding this equation to its own conjugate transpose yields a left hand side of the form
And a right hand side of the form
As previously mentioned, the expectation value of each Pauli matrix is a component of the Bloch vector,. Equating the left and right hand sides, and noting that is the gyromagnetic ratio, yields another form for the equations of motion of the Bloch vector
where the fact that has been used. In vector form these three equations can be expressed in terms of a cross product
Classically, this equation describes the dynamics of a spin in a magnetic field. An ideal magnet consists of a collection of identical spins behaving independently, and thus the total magnetization is proportional to the Bloch vector. All that is left to obtain the final form of the optical Bloch equations is the inclusion of the phenomenological relaxation terms.
As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator in the Heisenberg picture.
When coupled with the fact that, this equation is the same equation as before.

Validity

Two-state systems are the simplest non-trivial quantum systems that occur in nature, but the above-mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be treated as a two-state system as long as the observable one is interested in has two eigenvalues. For example, a spin-1/2 particle may in reality have additional translational or even rotational degrees of freedom, but those degrees of freedom are irrelevant to the preceding analysis. Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues.
Another case where the effective two-state formalism is valid is when the system under consideration has two levels that are effectively decoupled from the system. This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of charge qubits. In this case it should be kept in mind that the perturbations are in the right range and do not cause transitions to states other than the ones of interest.

Significance and other examples

Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. It can be used to illustrate fundamental quantum mechanical phenomena such as the interference exhibited by particles of the polarization states of the photon, but also more complex phenomena such as neutrino oscillation or the neutral K-meson oscillation.
Two-state formalism can be used to describe simple mixing of states, which leads to phenomena such as resonance stabilization and other level crossing related symmetries. Such phenomena have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as the maser and laser can be explained using the two-state formalism.
The two-state formalism also forms the basis of quantum computing. Qubits, which are the building blocks of a quantum computer, are nothing but two-state systems. Any quantum computational operation is a unitary operation that rotates the state vector on the Bloch sphere.
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