Coherent state


In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
These states, expressed as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e. g.
In the quantum theory of light and other bosonic quantum field theories, coherent states were introduced by the work of Roy J. Glauber in 1963 and are also known as Glauber states.
The concept of coherent states has been considerably abstracted; it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing. For this reason, the coherent states associated to the quantum harmonic oscillator are sometimes referred to as canonical coherent states, standard coherent states, Gaussian states, or oscillator states.

Coherent states in quantum optics

In quantum optics the coherent state refers to a state of the quantized electromagnetic field, etc. that describes a maximal kind of coherence and a classical kind of behavior. Erwin Schrödinger derived it as a "minimum uncertainty" Gaussian wavepacket in 1926, searching for solutions of the Schrödinger equation that satisfy the correspondence principle. It is a minimum uncertainty state, with the single free parameter chosen to make the relative dispersion equal for position and momentum, each being equally small at high energy.
Further, in contrast to the energy eigenstates of the system, the time evolution of a coherent state is concentrated along the classical trajectories. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems. They occur in the quantum theory of light and other bosonic quantum field theories.
While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention until Roy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field. In this respect, the concurrent contribution of E.C.G. Sudarshan should not be omitted,.
Glauber was prompted to do this to provide a description of the Hanbury-Brown & Twiss experiment which generated very wide baseline interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence.
In classical optics, light is thought of as electromagnetic waves radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are in phase. Actually, the picture of one photon being in-phase with another is not valid in quantum theory. Laser radiation is produced in a resonant cavity where the resonant frequency of the cavity is the same as the frequency associated with the atomic electron transitions providing energy flow into the field. As energy in the resonant mode builds up, the probability for stimulated emission, in that mode only, increases. That is a positive feedback loop in which the amplitude in the resonant mode increases exponentially until some non-linear effects limit it. As a counter-example, a light bulb radiates light into a continuum of modes, and there is nothing that selects any one mode over the other. The emission process is highly random in space and time. In a laser, however, light is emitted into a resonant mode, and that mode is highly coherent. Thus, laser light is idealized as a coherent state.
Besides describing lasers, coherent states also behave in a convenient manner when describing the quantum action of beam splitters: two coherent-state input beams will simply convert to two coherent-state beams at the output with new amplitudes given by classical electromagnetic wave formulas; such a simple behaviour does not occur for other input states, including number states. Likewise if a coherent-state light beam is partially absorbed, then the remainder is a pure coherent state with a smaller amplitude, whereas partial absorption of non-coherent-state light produces a more complicated statistical mixed state. Thermal light can be described as a statistical mixture of coherent states, and the typical way of defining nonclassical light is that it cannot be described as a simple statistical mixture of coherent states.
The energy eigenstates of the linear harmonic oscillator are fixed-number quantum states. The Fock state is the most particle-like state; it has a fixed number of particles, and phase is indeterminate. A coherent state distributes its quantum-mechanical uncertainty equally between the canonically conjugate coordinates, position and momentum, and the relative uncertainty in phase and amplitude are roughly equal—and small at high amplitude.

Quantum mechanical definition

Mathematically, a coherent state is defined to be the eigenstate of the annihilation operator with corresponding eigenvalue. Formally, this reads,
Since is not hermitian, is, in general, a complex number. Writing || and are called the amplitude and phase of the state.
The state is called a canonical coherent state in the literature, since there are many other types of coherent states, as can be seen in the companion article Coherent states in mathematical physics.
Physically, this formula means that a coherent state remains unchanged by the annihilation of field excitation or, say, a particle. An eigenstate of the annihilation operator has a Poissonian number distribution when expressed in a basis of energy eigenstates, as shown below. A Poisson distribution is a necessary and sufficient condition that all detections are statistically independent. Compare this to a single-particle state : once one particle is detected, there is zero probability of detecting another.
The derivation of this will make use of dimensionless operators, and, normally called field quadratures in quantum optics.
These operators are related to the position and momentum operators of a mass on a spring with constant,
the mean photon number is equal to the variance of the photon number distribution. Bars refer to theory, dots to experimental values.
For an optical field,
are the real and imaginary components of the mode of the electric field inside a cavity of volume.
With these operators, the Hamiltonian of either system becomes
Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally distributed between and satisfies the equation
or, equivalently,
and hence
Thus, given, Schrödinger found that the minimum uncertainty states for the linear harmonic oscillator are the eigenstates of.
Since â is, this is recognizable as a coherent state in the sense of the above definition.
Using the notation for multi-photon states, Glauber characterized the state of complete coherence to all orders in the electromagnetic field to be the eigenstate of the annihilation operator—formally, in a mathematical sense, the same state as found by Schrödinger. The name coherent state took hold after Glauber's work.
If the uncertainty is minimized, but not necessarily equally balanced between and, the state is called a squeezed coherent state.
The coherent state's location in the complex plane is centered at the position and momentum of a classical oscillator of the phase and amplitude |α| given by the eigenvalue α. As shown in Figure 5, the uncertainty, equally spread in all directions, is represented by a disk with diameter. As the phase varies, the coherent state circles around the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space.
.
Since the uncertainty stays constant at as the amplitude of the oscillation increases, the state behaves increasingly like a sinusoidal wave, as shown in Figure 1. Moreover, since the vacuum state is just the coherent state with =0, all coherent states have the same uncertainty as the vacuum. Therefore, one may interpret the quantum noise of a coherent state as being due to vacuum fluctuations.
The notation does not refer to a Fock state. For example, when =1, one should not mistake for the single-photon Fock state, which is also denoted in its own notation. The expression with =1 represents a Poisson distribution of number states with a mean photon number of unity.
The formal solution of the eigenvalue equation is the vacuum state displaced to a location in phase space, i.e., it is obtained by letting the unitary displacement operator operate on the vacuum,
where and.
This can be easily seen, as can virtually all results involving coherent states, using the representation of the coherent state in the basis of Fock states,
where are energy eigenvectors of the Hamiltonian
For the corresponding Poissonian distribution, the probability of detecting photons is
Similarly, the average photon number in a coherent state is
and the variance is
That is, the standard deviation of the number detected goes like the square root of the number detected. So in the limit of large, these detection statistics are equivalent to that of a classical stable wave.
These results apply to detection results at a single detector and thus relate to first order coherence. However, for measurements correlating detections at multiple detectors, higher-order coherence is involved. Glauber's definition of quantum coherence involves nth-order correlation functions for all. The perfect coherent state has all n-orders of correlation equal to 1. It is perfectly coherent to all orders.
Roy J. Glauber's work was prompted by the results of Hanbury-Brown and Twiss that produced long-range first-order interference patterns through the use of intensity fluctuations, with narrow band filters at each detector. Almost all of optics had been concerned with first order coherence. The Hanbury-Brown and Twiss results prompted Glauber to look at higher order coherence, and he came up with a complete quantum-theoretic description of coherence to all orders in the electromagnetic field. He coined the term coherent state and showed that they are produced when a classical electric current interacts with the electromagnetic field.
At , from Figure 5, simple geometry gives Δθ |α | = 1/2.
From this, it appears that there is a tradeoff between number uncertainty and phase uncertainty, Δθ Δn = 1/2, which is sometimes interpreted as a
number-phase uncertainty relation; but this is not a formal strict uncertainty relation: there is no uniquely defined phase operator in quantum mechanics.

The wavefunction of a coherent state

To find the wavefunction of the coherent state, the minimal uncertainty Schrödinger wave packet, it is easiest to start with the Heisenberg picture of the quantum harmonic oscillator for the coherent state. Note that
The coherent state is an eigenstate of the annihilation operator in the Heisenberg picture.
It is easy to see that, in the Schrödinger picture, the same eigenvalue
occurs,
In the coordinate representations resulting from operating by, this amounts to the differential equation,
which is easily solved to yield
where is a yet undetermined phase, to be fixed by demanding that the wavefunction satisfies the Schrödinger equation.
It follows that
so that is the initial phase of the eigenvalue.
The mean position and momentum of this "minimal Schrödinger wave packet" are thus oscillating just like a classical system,
The probability density remains a Gaussian centered on this oscillating mean,

Mathematical features of the canonical coherent states

The canonical coherent states described so far have three properties that are mutually equivalent, since each of them completely specifies the state, namely,
  1. They are eigenvectors of the annihilation operator: .
  2. They are obtained from the vacuum by application of a unitary displacement operator: .
  3. They are states of minimal uncertainty: .
Each of these properties may lead to generalizations, in general different from each other. We emphasize that coherent states have mathematical features that are very different from those
of a Fock state; for instance, two different coherent states are not orthogonal,
.
Thus, if the oscillator is in the quantum state it is also with nonzero probability in the other quantum state
. However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis, in which one can diagonally decompose any state. This is the premise for the Sudarshan-Glauber P representation.
This closure relation can be expressed by the resolution of the identity operator in the vector space of quantum states,
This resolution of the identity is intimately connected to the Segal–Bargmann transform.
Another peculiarity is that has no eigenket. The following equality is the closest formal substitute, and turns out to be useful for technical computations,
This last state is known as an "Agarwal state" or photon-added coherent state and denoted as
Normalized Agarwal states of order can be expressed as
The above resolution of the identity may be derived by taking matrix elements between eigenstates of position,, on both sides of the equation. On the right-hand side, this immediately gives. On the left-hand side, the same is obtained by inserting
from the previous section, then integrating over using the Fourier representation of the delta function, and then performing a Gaussian integral over.
In particular, the Gaussian Schroedinger wavepacket state follows from the explicit value
The resolution of the identity may also be expressed in terms of particle position and momentum.
For each coordinate dimension,
the closure relation of coherent states reads
This can be inserted in any quantum-mechanical expectation value, relating it to some
quasi-classical phase-space integral and explaining, in particular, the origin of
normalisation factors for classical
partition functions, consistent with quantum
mechanics.
In addition to being an exact eigenstate of annihilation operators, a coherent state is
an approximate common eigenstate of particle position and momentum. Restricting to
one dimension again,
The error in these approximations is measured by the uncertainties
of position and momentum,

Thermal coherent state

A single mode thermal coherent state is produced by displacing a thermal mixed state in phase space, in direct analogy to the displacement of the vacuum state in view of generating a coherent state. The density matrix of a coherent thermal state in operator representation reads
where is the displacement operator which generates the coherent state with complex amplitude, and . The partition function is equal to
Using the expansion of the unity operator in Fock states,, the density operator definition can be expressed in the following form
where stands for the displaced Fock state. We remark that if temperature goes to zero we have
which is the density matrix for a coherent state. The average number of photons in that state can be calculated as below
where for the last term we can write
As a result, we find
where is the average of the photon number calculated with respect to the thermal state. Here we have defined, for ease of notation,
and we write explicitly
In the limit we obtain, which is consistent with the expression for the density matrix operator at zero temperature. Likewise, the photon number variance can be evaluated as
with. We deduce that the second moment cannot be uncoupled to the thermal and the quantum distribution moments, unlike the average value. In that sense, the photon statistics of the displaced thermal state is not described by the sum of the Poisson statistics and the Boltzmann statistics. The distribution of the initial thermal state in phase space broadens as a result of the coherent displacement.

Coherent states of Bose–Einstein condensates