Two-level system - AbsoluteAstronomy.com

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, a two-state system (also known as a TLS or two-level system) is a system which has two possible states. More formally, the Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

of a two-state system has two degrees of freedom

Degrees of freedom (physics and chemistry)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

, so a complete basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

spanning the space must consist of two independent states. An example of a two-state system is the spin

Spin (physics)

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

of a spin-1/2 particle such as an electron, whose spin can have values +½ or -½ in units of ħ.

The physics of a quantum mechanical two-state system is trivial if both states are degenerate, that is, if the states have the same energy. However, if there is an energy difference between the two states, then nontrivial dynamics can occur.

Two-state dynamics

If the time-independent Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

, and we label the two levels as

and

with corresponding orthonormal energy eigenvalues

and

, then the dynamics of the system can be specified as follows:

At some time

, let the system be in an arbitrary (and completely general) state,

then after evolving under

, at time

,
the state will be

The physics of two state systems can be usefully applied to multi-state systems where the system is known to have only enough energy available to excite the lowest two states, thus effectively creating a two state system. In fact, in nature, it is difficult to identify any true two-state systems; merely systems where the energetics of the circumstances isolate two particular states.

The set of all states in a two-level system can be mapped onto a representation known as the Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

Free Precession

The interaction of a two state system with an external field results in the precession of the state vector. The ability to carefully control the position of the state vector on the Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

is central to the realization of the qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

. 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

Magnetic moment

The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...

is the magnetic field, and

is the vector of Pauli matrices

Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

. In analogy to related systems,

is a coupling constant, and

is an external field. Solving the time dependent Schrödinger equation

yields the solution

where

is the state vector at time

, and

. Physically, this corresponds to the Bloch vector precessing around

with angular frequency

. Without loss of generality, assume the field points in

, such that

The representation on the Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

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

its components on the Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

will simply be

. This is a unit vector that begins pointing in

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

plane making an angle

with the

-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.

NMR

NMR may refer to:Applications of Nuclear Magnetic Resonance:* Nuclear magnetic resonance* NMR spectroscopy* Solid-state nuclear magnetic resonance* Protein nuclear magnetic resonance spectroscopy* Proton NMR* Carbon-13 NMR...

is an important example in the dynamics of two-state systems because it is involves the exact solution to a time dependent Hamiltonian. Consider a spin-1/2 particle in a magnetic field

This corresponds physically to a situation where there is a time independent field of strength

pointing in

, and a field of strength

rotating in a right-handed manner with an angular frequency

. The dynamics of this situation are found as above by solving the time dependent Schrödinger equation

Which, after dividing through by

and expanding out the dot product is

where

. 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

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

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

Rotating wave approximation

The rotating wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic resonance,...

in deriving such results.

Relation to Bloch equations

The Bloch equations

Bloch equations

In physics and chemistry, specifically in NMR or MRI , or ESR the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = as a function of time when relaxation times T1 and T2 are present...

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

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

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

Magnetization

In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

is proportional to the Bloch vector

. All that is left to obtain the final form of the Bloch equations

Bloch equations

is the inclusion of the phenomenological relaxation

Relaxation (NMR)

In nuclear magnetic resonance spectroscopy and magnetic resonance imaging the term relaxation describes several processes by which nuclear magnetization prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins "forget" the...

terms.

As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator

Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry...

in the Heisenberg picture

Heisenberg picture

In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...

Which, when coupled with the fact that

, is the same equation as before.

Examples of two-state quantum systems

Spin-1/2 particles are two-state quantum systems when only the spin degree of freedom is considered.
The "inverting" degree of freedom in an ammonia
Ammonia
Ammonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...

molecule; the nitrogen at the vertex of an ammonia molecule exhibits two molecular states - "up" and "down", corresponding to opposite positions with respect to the plane of the three hydrogen atoms. In an electric field, these two states are non-degenerate.
Two-level systems are important in the field of quantum computing
Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...

as they are used to implement qubit
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

s.

Two-state dynamics

Free Precession

NMR

Relation to Bloch equations

Examples of two-state quantum systems

See also