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In quantum mechanics, a two-state system (also known as a two-level system) is a 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.
Contents
- Representation of the Two state quantum system
- The Two state Hamiltonian
- Eigenvalues of the Hamiltonian Basis vectors and Time evolution
- Dynamics of the Two state System The Rabi formula
- Precession in a field
- Evolution in a Time dependent Field Nuclear magnetic resonance
- Relation to Bloch equations
- The Validity of the Two state formalism
- Some more examples and the significance of the Two state formalism
- References
Two-state systems are the simplest quantum systems that can exist, since the dynamics of a one-state system is trivial (i.e. there is no other state the system can exist in). 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.
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. Another example, frequently studied in atomic physics, is the transition of an atom to or from an excited state; here the two-state formalism is used to quantitatively explain stimulated and spontaneous emission of photons from excited atoms.
Representation of the Two-state quantum system
The state of a two-state quantum system can be described by a two-dimensional complex Hilbert space, this means every state vector
If the vectors are normalized,
All observable physical quantities associated with this systems are 2
The Two-state Hamiltonian
The most general form of the Hamiltonian of a two-state system is given
here,
Here,
The Hamiltonian can be written (in a slightly different vector form) as:
The vector
Eigenvalues of the Hamiltonian, Basis vectors and Time evolution
Let
the unitary time evolution operator
where,
It is to be noted that the
Dynamics of the Two-state System: The Rabi formula
If
the above vector is assumed to be normalized. The time evolution of the state
further eliminating an overall phase factor of
It is easy to infer that if the system was initially in one of the eigestates (
Where
where it has been assumed that
It can be seen that the probability of finding the system in its initial quantum state oscillates between
Precession in a field
Consider the case of a spin-1/2 particle in a magnetic field
where
where
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 however 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) 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
The components of
Evolution in a Time-dependent Field: Nuclear magnetic resonance
Nuclear magnetic resonance (NMR) 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 (the "holding field") 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
As in the free precession case, the Hamiltonian is
where
As per the previous section, the solution to this equation has the Bloch vector precessing around
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
Multiplying by a Pauli matrix
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,
where the fact that
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
As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator in the Heisenberg picture.
Which, when coupled with the fact that
The Validity of the Two-state formalism
Two-state systems are the simplest non-trivial quantum systems that occur in nature however it should be noted that the above mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be effectively treated as two-state system as long as a particular property of is being considered (which behaves as a two-state system) an example of this that of a spin-1/2 particle, which may have additional translational or even rotational degrees of freedom, however in the preceding analysis, the additional degrees freedom are ignored.
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 (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest.
Some more examples and the significance of the Two-state formalism
Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. The most fundamental quantum mechanical phenomenon such as the interference exhibited by particles of the polarization states of the photon. but also more complex phenomenon 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 phenomenon such as resonance stabilization and other level crossing related symmetries. Such phenomenon have a wide variety of application in chemistry. Phenomena with tremendous industrial applications such as Maser and laser can be explained using the two-state formalism.
The two-state formalism form 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.