Coherent state

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In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. 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 state, arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of the particle in a quadratic potential well. In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories they were introduced by the work of Roy J. Glauber in 1963. Here the coherent state of a field describes an oscillating field, the closest quantum state to a classical sinusoidal wave such as a continuous laser wave.

Figure 1: The electric field, measured by optical homodyne detection, as a function of phase for three coherent states emitted by a Nd:YAG laser. The amount of quantum noise in the electric field is completely independent of the phase. As the field strength, i.e. the oscillation amplitude α of the coherent state is increased, the quantum noise or uncertainty is constant at 1/2, and so becomes less and less significant. In the limit of large field the state becomes a good approximation of a noiseless stable classical wave. The average photon numbers of the three states from top to bottom are <n>=4.2, 25.2, 924.5 (source: link 1 and ref. 2)
Figure 1: The electric field, measured by optical homodyne detection, as a function of phase for three coherent states emitted by a Nd:YAG laser. The amount of quantum noise in the electric field is completely independent of the phase. As the field strength, i.e. the oscillation amplitude α of the coherent state is increased, the quantum noise or uncertainty is constant at 1/2, and so becomes less and less significant. In the limit of large field the state becomes a good approximation of a noiseless stable classical wave. The average photon numbers of the three states from top to bottom are <n>=4.2, 25.2, 924.5 (source: link 1 and ref. 2)
Figure 2: The oscillating wave packet corresponding to the second coherent state depicted in Figure 1. At each phase of the light field, the distribution is a Gaussian of constant width.
Figure 2: The oscillating wave packet corresponding to the second coherent state depicted in Figure 1. At each phase of the light field, the distribution is a Gaussian of constant width.
Figure 3: Wigner function of the coherent state depicted in Figure 2. The distribution is centered on state's amplitude α and is symmetric around this point. The ripples are due to experimental errors.
Figure 3: Wigner function of the coherent state depicted in Figure 2. The distribution is centered on state's amplitude α and is symmetric around this point. The ripples are due to experimental errors.
Figure 4: The probability of detecting n photons, the photon number distribution, of the coherent state in Figure 3. As is necessary for a Poissonian distribution the mean photon number is equal to the variance of the photon number distribution. Bars refer to theory, dots to experimental values.
Figure 4: The probability of detecting n photons, the photon number distribution, of the coherent state in Figure 3. As is necessary for a Poissonian distribution the mean photon number is equal to the variance of the photon number distribution. Bars refer to theory, dots to experimental values.
Figure 5: Phase space plot of a coherent state. This shows that the uncertainty in a coherent state is equally distributed in all directions. The horizontal and vertical axes are the X and P quadratures, respectively (see text). The red dots on the x-axis trace out the boundaries of the quantum noise in Figure 1.
Figure 5: Phase space plot of a coherent state. This shows that the uncertainty in a coherent state is equally distributed in all directions. The horizontal and vertical axes are the X and P quadratures, respectively (see text). The red dots on the x-axis trace out the boundaries of the quantum noise in Figure 1.

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[edit] Coherent states in quantum optics

In classical optics light is thought of as electromagnetic waves radiating from a source. Specifically, coherent light is thought of as light that is emitted by many such sources that are in phase. For instance a light bulb radiates light that is the result of waves being emitted at all the points along the filament. Such light is incoherent because the process is highly random in space and time (see thermal light). In a laser, however, light is emitted by a carefully controlled system in processes that are not random but interconnected by stimulation and the resulting light is highly ordered, or coherent. Therefore a coherent state corresponds closely to the quantum state of light emitted by an ideal laser. Semi-classically we describe such a state by an electric field oscillating as a stable wave.

Contrary to the coherent state, which is the most wave-like quantum state, the Fock state (e.g. a single photon) is the most particle-like state. It is indivisible and contains only one quantum of energy. These two states are examples of the opposite extremes in the concept of wave-particle duality. A coherent state distributes its quantum-mechanical uncertainty equally, which means that the phase and amplitude uncertainty are approximately equal. Conversely, in a single-particle state the phase is completely uncertain. Also, the phase is completely uncertain in a state with fixed number of particles.

[edit] Quantum mechanical definition

Mathematically, the coherent state |\alpha\rangle is defined to be the eigenstate of the annihilation operator \hat a. Formally, this reads:

\hat{a}|\alpha\rangle=\alpha|\alpha\rangle

Since \hat a is not hermitian, α is complex, and can be represented as

\alpha = |\alpha|e^{i\theta}~~~,

where ~\theta~ is real number. Here ~|\alpha|~ and ~\theta~ are called the amplitude and phase of the state.

Physically, this formula means that a coherent state is left unchanged by the detection (or annihilation) of a particle. Consequently, in a coherent state, one has exactly the same probability to detect a second particle. Note, this condition is necessary for the coherent state's Poissonian detection statistics, as discussed below. Compare this to a single-particle state (Fock state): once one particle is detected, we have zero probability of detecting another.

For the following discussion we need to define the dimensionless quadratures ~X~ and ~P~. For a harmonic oscillator, ~x=\left( \frac{m\omega}{2\hbar}\right) ^{-1/2} is the oscillating particle's position and p=\left( \frac{m\omega\hbar}{2}\right)^{1/2} is its momentum. For an optical field, ~E_{\rm R} = \left(\frac{\hbar\omega}{2\epsilon_0 V} \right)^{1/2} \cos(\theta) X~ and ~E_{\rm I} = \left(\frac{\hbar\omega}{2\epsilon_0 V}\right)^{1/2} \sin(\theta) X~

are the real and imaginary components of the mode of the electric field.

Erwin Schrödinger was searching for the most classical-like states when he first introduced coherent states. He described them as the quantum state of the harmonic oscillator which minimizes the uncertainty relation with uncertainty equally distributed in both X and P quadratures (i.e., ~\Delta X=\Delta Y=1/2~). From the generalized uncertainty relation, it is shown that such a state ~|\alpha \rangle~ must obey the equation

(P-\langle P\rangle)|\alpha\rangle=i(X-\langle X\rangle)|\alpha\rangle

In the general case, if the uncertainty is not equally distributed in the X and P component, the state is called a squeezed coherent state.

If this formula is written back in terms of \hat a and \hat a^\dagger, it becomes:

\hat a|{\alpha}\rangle=(\langle X\rangle+i\langle P\rangle)|{\alpha}\rangle

The coherent state's location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the same phase θ and amplitude (or the same complex electric field value for an electromagnetic wave). As shown in Figure 2, the uncertainty, equally spread in all directions, is represented by a disk with diameter 1/2. As the phase increases the coherent state circles 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 (and hence measurement noise) stays constant at 1/2 as the amplitude of the oscillation increases, the state behaves more and more like a sinusoidal wave, as shown in Figure 1. Conversely, since the vacuum state |0\rangle is just the coherent state with α = 0, all coherent states have the same uncertainty as the vacuum. Therefore one can interpret the quantum noise of a coherent state as being due to the vacuum fluctuations.

It should be noted that the notation ~|\alpha\rangle~ implies certain ambiguity. For example, at α = 1, one cannot recognize ~|1\rangle~ as coherent state, and, perhaps, interpret ~|1\rangle~ as a single-photon state ~\hat a^\dagger|0\rangle~.

Furthermore, it is sometimes useful to define a coherent state simply as the vacuum state displaced to a location α in phase space. Mathematically this is done by the action of the displacement operator D(α):

|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle

This can be easily obtained, as can virtually all results involving coherent states, using the representation of the coherent state in the basis of Fock states:

|\alpha\rangle=e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle.

where |n\rangle are eigenvectors of the Hamiltonian. A stable classical wave has a constant intensity. Consequently, the probability of detecting n photons in a given amount of time is constant with time. This condition ensures there will be shot noise in our detection. Specificially, the probability of detecting ~n~ photons is Poissonian:

P(n)=e^{-\langle n \rangle}\frac{\langle n \rangle^n}{n!}

Similarly, the average photon number in coherent state ~\langle n \rangle = \hat a^\dagger \hat a=|\alpha|^2~ and the variance ~(\Delta n)^2={\rm Var}\left(\hat a^\dagger \hat a\right)= |\alpha|^2~, identical to the variance of the Poissonian distribution. Not only does a coherent state go to a classical sinusoidal wave in the limit of large α but the detection statistics of it are equal to that of a classical stable wave for all values of ~\alpha~. This also follows from the fact that for the prediction of the detection results at a single detector (and time) any state of light can always be modelled as a collection of classical waves (see degree of coherence). However, for the prediction of higher-order measurement like intensity correlations (which measure the degree of nth-order coherence) this is not true. The coherent state is unique in the fact that all n-orders of coherence are equal to 1. It is perfectly coherent to all orders.

There are other reasons why a coherent state can be considered the most classical state. Roy J. Glauber coined the term "coherent state" and proved they are produced when a classical electrical current interacts with the electromagnetic field. In the process he introduced the coherent state to quantum optics. In general when a quantum state of light is split at a beamsplitter, the two output modes are entangled. Aharonov proved that coherent states are the only pure states of light that remain unentangled (and thus classical) when split into two states.

At ~\alpha \gg 1~, from Figure 5, simple geometry gives \Delta\theta|\alpha|=\frac{1}{2} From this we can see that there is a tradeoff between number uncertainty and phase uncertainty \Delta\theta~\Delta n=1/2, which sometimes can be interpreted as the number-phase uncertainty relation. This is not a formal uncertainty relation: there is no uniquely defined phase operator in quantum mechanics.

[edit] Mathematical characteristics

The coherent state does not display all the nice mathematical features of a Fock state; for instance two different coherent states are not orthogonal:

\langle\beta|\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha)}\neq\delta(\alpha-\beta)

so that if the oscillator is in the quantum state |α> it is also with nonzero probability in the other quantum state |β> (but the farther apart the states are situated in phase space, the lower the probability is). 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.

Another difficulty is that a† has no eigenket (and a has no eigenbra). The following formal equality is the closest substitute and turns out to be very useful for technical computations:

a^{\dagger}|\alpha\rangle=\left({\partial\over\partial\alpha}+{\alpha^*\over 2}\right)|\alpha\rangle

The last state is known as Agarwal state denoted as |\alpha,1\rangle. Agarwal states for order n can be expressed as |\alpha,n\rangle=\hat{a}^{\dagger~n}|\alpha\rangle

[edit] Coherent states of Bose–Einstein condensates

  • A Bose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. An approximate theoretical description of its properties can be derived by assuming the BEC is in a coherent state. However, unlike photons atoms interact with each other so it now appears that it more likely to be one of the squeezed coherent states mentioned above.

[edit] Coherent electron states in superconductivity

  • Electrons are Fermions, but when they pair up into Cooper pairs they act as bosons, and so can collectively form a coherent state at low temperatures. Such coherent states are part of the explanation of effects such as the Quantum Hall effect in low-temperature superconducting semiconductors.

[edit] Generalizations

  • In one-dimensional many-body quantum systems with fermionic degrees of freedom, low energy excited states can be approximated as coherent states of a bosonic field operator that creates particle-hole excitations. This approach is called bosonization.

[edit] See also

[edit] External links

[edit] References

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