Lecture 4: Quantum fluctuations, the quantum harmonic oscillator, and coherent states#
Expected prior knowledge
Before the start of this lecture, you should be able to:
write down the energy spectrum of the quantum harmonic oscillator
recall that the state of a quantum object is described by the wavefunction
calculate the expectation value of an observable given a wavefunction
write down and apply the Heisenberg x-p and the generalised uncertainty relations
Learning goals
After this lecture you will be able to:
derive the quantum fluctuations for a given wavefunction
calculate expectation values of coherent states
show that the number of photons in a coherent state follows a Poisson distribution
So far, we have been reviewing concepts from classical noise, and discovering fun things like the fluctuation-dissipation theorem by accident. This week, we will start to explore fluctuations in quantum mechanics, using the harmonic oscillator as the core example. But first, we will review the uncertainty principle.
Review: The Heisenberg Uncertainty principle and quantum fluctuations#
One of important concepts in quantum mechanics is the Heisenberg uncertainty principle.
In its more precise formulation, the uncertainty principle describes the relation between the uncertainties in position
and momentum of a particle that arise from the Schroedinger wave equation.
The uncertainties we are talking about in this case are defined by expectations values calculated using the wavefuntion
The expecation values above are calculated using the wave function in the usual way, for example:
Quantum fluctuations
If one were to repeatedly prepare a particle in state
Quantum fluctuations
Finally, the irregularity that quantum fluctuations introduce into a set of measurements can be referred to as quantum noise, and in general, the presence of non-zero quantum fluctuations / noise / uncertainty can have important physics consequences even if one does not collapse the wavefunction by performing a measurement.
Note: although quantum noise is frequently referred to as “fluctuations”, keep in mind that there is nothing associated with the quantum states
that has to be changing in time. Quantum fluctuations do not “fluctuate in time”. We will see an explicit example of this below: in particular, the stationary states (energy eigenstates) of the Harmonic oscillator are absolutely “stationary” in every sense; the expectation value of all operators in quantum mechanics do not change in time for a stationary state. Quantum fluctuations only “fluctuate” when a measurement apparatus breaks the evolution of the Schroedinger equation and collapses the wave function.
The (original) Heisenberg uncertainty principle places a lower limit on product of the quantum fluctuations of position and momentum:
The can also be understood intuitively from the Schroedinger wave equation: making a smaller wave packet in space
The same concept is linked to “confinement energy”: if I confine a particle to a fixed region of space of length
This lower bound on
We can then see that the particle must have some (expectation value of its) kinetic energy, since the kinetic energy operator
More review: Beyond and : The generalised uncertainty principle#
The “traditional”
In the most general case, the uncertainty principle for two observables
For any two “compatible” observables whose operators
Note also that the expression above includes an expectation value:
So in general, the exact limit that uncertainty principle imposes will also depend on the specific quantum state
The
which means that we can put it out of our braket above, being left with
Quantum noise in a harmonic oscillator#
Here, we will explore in more detail the types of quantum noise states can haveWe will start with a problem you have seen before: the quantum harmonic oscillator (i.e., a quantum mass on a spring).
Why is the quantum harmonic oscillator such an important concept?
for small displacements from a minimum, all potentials are approximately harmonic (i.e., parabolic, via Taylor expansion)
Electrical circuits can be harmonic oscillators
light and lattice vibrations are descirbed by harmonic oscillators
The last one is particularly interesting, and we can gain a deeper understanding of what “photons” are by looking at how light, and photons, are related to harmonic oscillators.
Imagine an empty box that contains electromagnetic radiation. Classically, each mode of the box is a standing wave that bounces up and down in time:
mode
:mode
:mode
: Associated with every electric field , there is a magnetic field that oscillates out of phase with respect to the electric field (this follows from Maxwell’s Equations). The total energy is given by:
For a given mode, this will be proportional to:
This is looking a lot like a harmonic oscillator
where the electric and magnetic fields play the roles of position and momentum, respectively.
Each mode of the box can be mapped to an individual harmonic oscillator. Each such oscillator can be quantized, and each mode can be excited by an arbitrary number of photons.
Having established the importance of the harmonic oscillator, let’s look at the properties of some of its states.
The ground state of the oscillator is the lowest-energy state allowed in quantum mechanics. Unlike in classical mechanics, the ground does not sit at the bottom of the potential. Instead, it has a zero-point energy of
, where is the eigenfrequency of the oscilator.The ground-state wavefunction is “smeared out” in space, so there are zero-point fluctuations in the position of the oscillator. The standard deviation of the position is
.
What does it mean for the wavefunction to be “smeared out”? Since we interpret
Let’s take a step back, and pretend that we don’t perform any measurements. Does any “property” of the particle in the ground state of the quantum harmonic oscillator change in time?
Under the evolution of the Schrödinger equation, we have
So the total wavefunction
So, if the oscillator is stationary, why does it have kinetic energy?
The first term is energy associated with classical motion, and the second comes from quantum fluctuations.
So if all of the oscillator eigenstates are stationary, how can we get them to “move”? Suppose we prepare a superposition of two such states, like
The relative phase between the two states will cause the observable
Coherent states#
Another very important state of the quantum harmonic oscillator is the “coherent state” (see problem 3.35 from Griffiths). Mathematically, the coherent state is defined as
where
Properties of coherent states
The coherent state has the following properties:
The coherent state is not an eigenstate of the Hamiltonian. Therefore, generally, it will “move” (i.e.,
and oscillate in time).Instead, the coherent state is an eigenstate of the annihilation operator
.
Creation and Annihilation operators
Recall that the creation operator
The last operator,
we can derive
There are an infinite number of possible coherent states, since
can vary continuously: .All coherent states are Heisenberg-limited minimum uncertainty wavepackets that satisfy
(see exercise).As a function of time, a coherent state evolves into a new coherent state with the same amplitude but a different phase:
.The ground state
is also a coherent state, with and .By expressing the operators for position and momentum in terms of creation and annihilation operators, it follows that a coherent state
has:
Using the fact that
, we can see that the above expectation values oscillate in time.In quantum optics,
and are called quadratures.The photon number operator
is (i.e. ), and the Hamiltonian then becomes . Using this, we find that
It turns out that the variance of the number of photons in a coherent state,
, is . The property that the variance is equal to the mean tells us that coherent states are Poissonian. If you recall that lasers are also described by Poissonian statistics, then we have found that lasers are actually just coherent states!
Coherent states have maximal “classical” energy, and even though they are (infinitely) large quantum superpositons, in some sense they are the most classical quantum states possible because they reach the limit of the smallest amount of quantum fluctuations allowed by the Heisenberg limit.
Conclusions
The expectation values of observables of stationary states do not depend on time.
However, there are quantum fluctuations. These are related to the spread/curvature of the wavefunction.
The coherent state is not a stationary state. Its time dynamics resemble that of a classical particle.
The coherent state is an eigenstate of the annihilation operator.
The expectation values for position and momentum of the coherent state
can be expressed in terms of .