Agenda for week 5: Quantized electromagnetic field

Learning goals for this week:

• Understanding what is quantized when the electromagnetic field is quantized
• The basic concept of coherent and squeezed states
• Origin of spontaneous emission resulting from the quantized em field
• Concept of hybridization, avoided crossing

Reading assignment:

Notes for week 5: Quantized electromagnetic field

More details: - Tuominen, chapter 10 (has much more detail than here)

Preliminary exercises,

Do these during/after reading the assignment work. Will be discussed in class April 13th.

1. Express the electric field in terms of the creation and annihilation operators of photons.

2. Calculate the expectation values of and in a coherent state .

3. Assume you know the wavefunctions and and energies of an excited state and the ground state of a hydrogen-like atom. Write down (but don't calculate) the expression that you need to calculate to obtain the lifetime of the excited state by one photon emission in the dipole approximation.

Homework exercises, week 5

Will be discussed in the tutorial session on Thursday April 15th. Return a scanned pdf with your solution by Friday April 16th, at 9 pm using the form below. Then check and grade your solution with the help of the model solutions and resubmit your graded solutions by Tuesday April 20th at 9 am.

Notes for week 5: quantized electromagnetic field

The first part of the QM II course contained a discussion of the interaction between a classical electromagnetic field with charged particles. Here we lift the assumption of the classical field and see what happens when the electromagnetic field is quantized. The idea is

1. Quantize the electromagnetic field.

2. Study the properties of the quantized field, and express the coherent field in terms of the quantum description of the field. Coherent field is the closest analogue to a classical field, and it is a useful starting point for describing physical fields within the quantum realm. In addition, the path integral theory for bosons is most conveniently described in terms of the coherent fields (however, we do not discuss it here).

3. Study the interaction of the quantized electromagnetic field and a two-level atom. For weak interaction, we find the spontaneous emission as the clearest quantum effect. For stronger coupling, the two systems hybridize, which can be seen in the coupled response.

The wider field describing the properties of the quantized electromagnetic field is called quantum optics. One exhaustive online source on quantum optics is for example D.A. Steck: Quantum and Atom Optics.

Prelude: First and second quantization

Let us first recall what we have done in Quantum Mechanics I and II for fermions, such as the electron.

1. First we notice that the electron has both particle- and wavelike properties. Thus it is useful to define the concept of the electron wave function, which satisfies a wave equation (the Schrödinger equation). Since the number of electrons is (in nonrelativistic quantum mechanics) conserved, we can in a meaningful way normalize these wave functions, so that they always describe one particle. This passage from a particle to a wave is referred to as first quantization (first, because it was discovered first, by Bohr, Schrödinger, Heisenberg et al).
2. Then we want to study systems of many fermions. They must be antisymmetric, so the wave functions soon become cumbersome. We move to the particle number (Fock state) representation. Now our Hilbert space of states describing the physical state is larger, it can contain a varying number of particles. In fact, for the BCS theory of superconductivity, we see that it can be convenient to describe the system as a state that does not have a fixed number of fermions, but is a superposition of states with different numbers.
3. Mathematically, in the Fock space we have fermion field operators in stead of wave functions. This passage from wave functions to field operators is referred to as second quantization

Now let us move from electrons to photons. The wave-particle duality applies to them as well as to electrons: photons are both particles and waves of the electromagnetic field. The wave equations (Maxwell's equations) are in some sense the Schrödinger equations for the "first quantized" photon field. In classical physics we do not, however, usually construct "normalized" solutions of the Maxwell equations that would describe a single photon. The main reason for this is that the number of photons is not a conserved quantity: photons can be absorbed or emitted. Having a single photon flying somewhere is possible, but in classical situations the photon tends to disappear when it interacts with something. Thus a single photon state is more rare in physical problems than a single electron. Because the number of photons is not conserved, "first quantization" of photons (normalized states satisfying the wave equation) is not very useful, although possible. Thus, in order to treat photons as quantum mechanical particles, we jump directly from classical electromagnetic fields to second quantization, i.e. electromagnetic field operators. These are completely analogous to the fermion field operators in the previous weeks. They allow us to express the electromagnetic field as a superposition of photon creation and annihilation operators, and address physical situations where we can have varying numbers of photons, but nevertheless must count photons, i.e. be sensitive to the fact that they are particles, not just waves of the classical electromagnetic field.

Quantization of the electromagnetic field

In what follows, we introduce the common approach to quantize an electromagnetic field. The idea is first to define some eigenmodes of the field in some confined region, express the general field as a superposition of those eigenmodes, and identify the amplitude of the eigenmodes as a bosonic variable. We then discuss two alternative ways to describe the quantized field: first in terms of the Fock states of the bosonic field, and the second in terms of the coherent states, defined as the fields with a minimum uncertainty between different quadrature operators (corresponding to the generalized position and momentum of the harmonic oscillator fields). This notion can then be generalized to introduce squeezed states, which retain the minimum uncertainty, but make a difference between the two quadratures.

This discussion follows partially that in the book Quantum Optics by D.F. Walls and G.J. Milburn (2nd Edition, Springer 2010).

Maxwell's equation, radiation gauge

The most natural way to start describing an electromagnetic field is from Maxwell's equations describing a free radiation field, i.e., a field without source or sink terms. It satisfies where is the speed of light in free space.

We remember from th A-part of the course, that it is useful to think in terms of the scalar and vector potentials The 6 components of the electric and magnetic fields are not independent degrees of freedom, and we can reduce the number by switching to the potential. Also, we recall that the interaction of quantized charged particles with classical electromagnetic fields is expressed in terms of the potential.

Our aim here is to quantize the photon, i.e. electromagnetic radiation. As in the A-part of the course, it is convenient to do this in the radiation gauge where the electric and magnetic fields are

Electromagnetic waves

Substituting the gauge potentials into the fourth Maxwell equation, we find that satisfies the wave equation,

This wave equation has solutions of the form with some frequency . Due to the reality of the physical field, we have .

In the same way as when we calculate some physical observable in an electronic system, it is convenient to consider a finite region of space and impose some boundary conditions to the field. These boundary conditions can also describe real boundaries, such as in the case of optical cavities. Nevertheless, such a procedure leads to an orthonormal set of eigenmodes of the wave equation so that we can write Here denotes the wave vector of the eigenmode, denotes the polarization, and the coefficients are position independent coefficients. The mode functions satisfy the boundary conditions, the wave equation in Fourier space and the Coulomb gauge condition In the case of a cubic volume with side length , the modes are and is a unit vector perpendicular to due to the gauge condition. In 3D space, this can be satisfied with two different polarization directions, so denotes these two possibilities. Moreover, with

Fig: Eigenmodes of wave equation with infinite potential well boundary conditions ( and ).

Quantized field

We are now ready to quantize our electromagnetic field potential. This is done by changing the coefficients of the field modes into operators

Before doing this it is useful to go to appropriate dimensionless coefficients. We choose The physical unit of the vector potential is (from the defining equations) Teslameter or Newton/Ampere (unit of an electric field is N/As). A quick check for example with this shows that this results in a dimensionless .

We are now ready to quantize the field. This is done by writing the the vector potential in the free space but within the confined volume as with operators .

Note: now the operator has an explicit time dependence. Here the operators are time-independent, we have separated out the time depedence explicitly. Thus this is a Heisenberg picture (time dependent) field operator. In second quantization one goes from a time-dependent classical field to a time dependent field operators. The corresponding Fock states are time-independent. The Heisenberg picture of time dependence is much more natural here than the Schrödinger one.

Quantization: commutation relations

Since the polarization direction does not have any particular role in the quantization of the field, in what follows we suppress the index , but should remember to re-insert it when necessary.

Now we are ready to quantize the electromagnetic field. At this point it is straightforward: instead of treating as some complex numbers, we treat them as bosonic operators. In this case we must write instead of a complex conjugate. In other words, annihilates (creates) a photon with wave number .

These operators hence satisfy the bosonic commutation relations

Hamiltonian

Let us then derive the Hamiltonian describing the electromagnetic field. We can start from the classical Hamiltonian yielding the energy of the electromagnetic field, Using the above definition of the fields with respect to the vector potential, and the vector potential with respect to the bosonic operators , , you will show in the exercises that this can be expressed as In other words, we get a set of Hamiltonians for free harmonic oscillators, so that the total energy is the sum of the mode energies times the number of photons in each mode.

In addition, there is the vacuum energy of from each mode. Note that since the sum over modes is not limited from above, this vacuum energy is in fact infinite! This as such does not have practical consequences, since no process can extract the vacuum energy directly. However, it is a relevant concept when considering for example (finite) changes in this energy due to changing boundary conditions as in the static or dynamic Casimir effect. These phenomena are however not considered here.

Now you should be able to go back and do Question 1 in the preliminary exercises

Multi-photon states

The most natural way to characterize the states of the quantized electromagnetic field are the eigenstates of the number operator , which then are also eigenstates of the Hamiltonian. In other words, we would hence describe the states via where denotes the number of photons in the momentum state . Note that since these are number states of bosons, the order of the states in the ket vector does not matter.

Description of the field in terms of number states is somewhat convenient, because they form an orthonormal set of vectors, and the algebra with them is somewhat straightforward. However, in practice it is very hard to bring the field into some specific Fock state except the vacuum state.

In fact, generating pure single- or two-photon states of electromagnetic field was achieved only quite recently, see Varcoe, et al., Nature 403, 743 (2000) and more general pure Fock states in M. Hofheinz, et al., Nature 454, 310 (2008).

More often, the electromagnetic radiation fields are in superpositions (pure state) or a mixture of number states (mixed state). Next, we discuss a useful way of describing a rather generic set of superposition states, those of coherent states.

Coherent states

The name 'coherent state' refers to the concept of optical coherence which is related with photon correlations. Historically, Roy Glauber constructed the theory of coherence (Phys. Rev. 130, 2529 (1963)) and, in doing so, investigated the coherent states (Phys. Rev. 131, 2766 (1963)).

There are three equivalent definitions of the coherent states and multiple ways to derive them. Let us take the one mentioned in the introduction: coherent states as minimum uncertainty states. In the following, we drop all the subscripts to simplify notation, consider a single mode, and denote this bosonic mode by . Then, we define the dimensionless quadrature operators which have the commutation relations . For the simple 1-dimensional harmonic oscillator is basically the coordinate and the momentum (up to a rescaling by some power of and ), which you can check from the definitions of the operators in that case. We are taking here.

The uncertainty relation in its general form follows from the Cauchy-Schwarz inequality which reads in terms of the operators and The Heisenberg uncertainty principle, the latter inequality, follows by writing the operator as a sum of the commutator and the anticommutator between and and then noting that the anticommutator term gives only a positive contribution. However, if we suppose that the lower bound of the Cauchy-Schwarz inequality is satisfied then necessarily where is a constant. We can solve by using the commutator relation The quantity is the variance of which is why we denote it by . If we now set we find that . Rearranging the equation and using the definitions of the quadrature operators, we find the eigenvalue equation This tells us that The minimum uncertainty states are, in fact, the eigenstates of the annihilation operator . These are called coherent states

The value of can be found even if . These solutions correspond to the squeezed states, which we discuss in detail below. Note that it is possible to squeeze the uncertainty in one quadrature (with the expense of the other quadrature) even below the value of vacuum uncertainty, e.g. .

Fig. Graph showing where the coherent states (red circle) and the squeezed states (black line) lie in -plane. The black line also represents the lower limit of Heisenberg uncertainty relation, i.e., . The white region is thus forbidden, while the blue region represents states for which one quadrature is squeezed below zero-point fluctuations.

As in the case of the harmonic oscillator and Fock states, we may now relabel our states based on the eigenvalue. Here, we denote and

Explicit expression in terms of Fock states

How does look in the Fock space? To answer this, we operate with from the left which gives From this recursion relation we find that Therefore The prefactor is fixed by demanding that the state is normalized giving in the end The probability to observe photons in the coherent state is thus . The probability distribution is thus Poissonian with the mean value .

Displacement operator

Alternatively, we may generate the coherent states with a displacement operator To show that this coincides with the earlier definition, one should use a simplified version of the Baker-Campbell-Hausdorff formula which holds when .

The displacement operator has the following properties The two latter equations can be shown to be true with the help of the formula where and . In other words, is a nested commutator with amount of 's.

Some properties of the coherent states

Non-orthogonality:

First, we derive the multiplication property of displacement operators Thus, the operation of the displacement operator to the state displaces the state in the complex plane from to . Consequently, which means that the coherent states are not orthogonal, unlike the Fock states.

Completeness:

The field coherent states form a complete set. They are in fact overcomplete since the label of coherent states is continuous and uncountable, although the underlying number state basis is countable. This means that the completeness relation is not unique. One useful expression of the completeness relation is The integration measure is defined as and the integration is over the whole complex plane. This can be proven by brute force, by inserting the Fock state representation of the coherent states and calculating the resulting integrals in polar coordinates.

Classicality:

The coherent states are in a sense 'classical states' of quantum harmonic oscillator which is described by a Hamiltonian . If the initial state is a coherent state , the state at a later time is given by the Schrödinger equation The state remains as a coherent state during time evolution. Hopefully this clarifies what is meant by 'a classical state': coherent state follows a classical trajectory with and . Besides the trajectory, the state is as localized (both in space and in momentum) as possible and this property does not change over time.

Summary of coherent states

• Minimum uncertainty states with
• Eigenvalues of the annihilation operator: with
• "Displaced vacuum states",
• Form an overcomplete set

Squeezed states

How to generate states that have ? We can guess that there must be a squeezing operator in the same way that there is a displacement operator . Indeed, we have the squeezing operator which defines the squeezed coherent states That is, we first squeeze the vacuum and then displace the resulting state. The order of operations matters here, but this is the usual convention.

By utilizing the formula, we find where . Here is the squeeze factor which determines how much a quadrature is squeezed. The phase determines the axis of squeezing which does not have to be either or .

We then look at the squeezed coherent states and their properties; especially we want to see the squeezing in effect.

Thus, we work our way back to and . We have Now, we can plug in these results to get If we now set the squeezing angle , then and . For non-zero then which implies squeezing. In a general case, we find the same exponential squeezing along some other quadratures and defined by a relation . This is illustrated below.

Fig. The effect of squeezing. The shapes represent uncertainties, or variances, in -plane. The solid gray circle shows the uncertainty of a coherent state and the ellipse outlines the uncertainty of a squeezed state . Note that the squeezing axes are and which are obtained by a rotation of .

Similar to coherent states, squeezed states remain as such in a free electromagnetic field (i.e. ). This can be shown either by finding the Fock state representation of a squeezed coherent state or by developing the theory of phase-space distributions. Both of these approaches are beyond this course, so we simply state the result: the time-evolution simply rotates the above figure around the origing, giving . The code below allows you to see how this affects, for instance, the expectation value of (or after suitable rotations a component of the electic field ).

Now you should be able to go back and do Question 2 in the preliminary exercises

Interaction of radiation with atomic system

Now, we can generalize the description of an atom in a radiation field to a quantum electromagnetic field. This leads to

• spontaneous emission and stimulated emission
• hybridization, avoided crossing.

In fact, only the first one is a 'true' quantum effect, the latter results from assuming the electromagnetic field to be constrained on a finite region of space. However, the details of the hybridized state spectrum can only be explained with the quantum model.

We start by deriving the Hamiltonian of this interaction, where many aspects of the derivation are similar to the case of a classical electromagnetic field in part A of the course.

We will do this derivation in two ways.

1. We second-quantize the electric field, but treat the states of the atom in the first-quantized formalism. This is not a good way to understand the time dependence, but is the easiest way to get to a point where one can calculate atomic transition rates.
2. We also treat the states of the atom in second quantization. This is the more complete way for nontrivial applications.

Hamiltonian with first-quantized electron

A full treatment of QED is beyond this course, but, fortunately, only the low energy (non-relativistic) theory is needed to explain many experiments in quantum optics. Thus, we may fix the gauge as and we can start from the minimal coupling Hamiltonian where now is the quantized vector potential and is the Hamiltonian of the free radiation field.

Our full Hamiltonian is now where the electron (first quantized), the photon and the interaction contributions are respectively. Unless dealing with very intense fields, the second order term may be neglected from the interaction; this is what we now assume. Also, the anticommutative structure may be simplified using the gauge condition as Here we have inserted a "dummy" wave function that this operator operates on. In fact, this relation should be seen as an operator relation between and .

Interaction term, Schrödinger picture time dependence

We can now insert the quantized electromagnetic field in terms of the photon creation and annihilation operators from above. We need to change one thing, however. There the field was written in the Heisenberg picture, i.e. with time dependence. Here we are working in the Schrödinger picture, so we must remove the oscillating time dependent factors. We also replace the mode functions with explicit plane waves

The interaction term becomes Schrödinger picture interaction term

This is already a form that you can use for the simplest practical calculations using Fermi's golden rule. This is a Schrödinger picture interaction Hamiltonian and what appears in the expression for Fermi's golden rule are matrix elements of the Schrodinger picture interaction Hamiltonian. You can now calculate matrix elements of this operator between states with - a single electron in the initial state (described by some wave function) - a single electron in the final state (described by some other wave function) - an arbitrary number of photons in the initial state, and photons in the final state.

At lowest order in perturbation theory, you just have one matrix element of this operator: you can only emit or absorb one photon.

Reminder: Fermi's golden rule. Transition rate lifetime = Remember: in practice a sum over a continuum of final states.

Spontaneous and stimulated emission

We will discuss below the fermion part of the matrix element. For the photons, the interaction operator a has a photon creation and annihilation operator. Concerning the photon states with initial and final photons the matrix elements are of the form From these one observes the basic facts that are discussed in more detail below. We have

1. Spontaneous emission: the matrix element is nonzero
2. Stimulated emission: if there are already photons in the initial state, the emission probability is enhanced because
3. There is also absorption: : the atom can be excited by absorbing a photon.

Dipole approximation

Let us briefly recall the "dipole approximation" from part A of the course. Let us assume that you want to use the form of the interaction term above to calculate some atomic transition. You are in principle calculating (for the electron) matrix elements like where is some fixed polarization vector of the photon and is a momentum operator acting on the electron wavefunctions. The dipole approximation consists of two steps:

1. (This is the actual approximation) If we consider the electronic system to be small, e.g. an atom, the extent of wavefunctions is much smaller than the optical wavelength. We can then treat the atom as point-like, located at , and replace the oscillatory term inside the integral.
2. Then, we would like to do something to inside the integral. We can use the Schrödinger equation to simplify the integral; to this end we employ the relation which you can easily check. Thus, we have effectively changed to inside the integral. This allows us to write the light-matter coupling in terms of a matrix element of the electric dipole moment operator .

Now you should be able to go back and do Question 3 in the preliminary exercises

Interaction picture, second quantized fermion

Now let us do the same thing again, also treating the electron in second quantization. Our Hamiltonian is

We are now using the interaction picture (Dirac picture) of time dependence. This means that the free parts of our Hamiltonian do not have a time dependence. These free parts for the electron and photons are Note that here we have included a potential for the electrons in the "free" term in the sense of the interaction picture of time evolution (which requires separating the Hamiltonian into free and interaction terms).

The interaction part of the Hamiltonian is, using the Coulomb gauge condition and neglecting the -term: Now our time dependence follows the interaction picture, and thus the interaction part of the Hamiltonian has a time dependence, which is given by the free part of the Hamiltonian. Since the position eigenstates for the fermion are not eigenstates of the free Hamiltonian, we cannot write out this time dependence explicitly yet.

Single particle eigenstates

Next, we rewrite the field operators in another basis, which represents the single-particle eigenstates of the Hamiltonian without radiation, with and . The interaction term can be rearranged to where is the light-matter coupling constant. Note the indices of ; the Hamiltonian must be Hermitian. Note also that here the polarization vector plays a role, and that the we can use the dipole approximation to evaluate

Here, since we are working in the interaction picture, the field operators in the interaction part of the Hamiltonian have a time dependence given by the free part of the Hamiltonian (the single particle energies), they appear in the combinations:

Two state atomic system

Fig: Electron system and transitions due to absorption and emission of photons.

Rotating wave approximation

The form we arrived at looks fairly simple but is, in fact, difficult to solve. Even recently, articles that try to provide insight to this problem are published, e.g., Phys. Rev. Lett. 123, 133603 (2019).

The interaction picture Hamiltonian for the case of two states becomes, writing down explicitly the sums over the states : with . If we look at the exponentials after taking the product, we have all the possible combinations of signs . If we were using the Fermi rule, the energy must be conserved, and the only term that contributes is where oscillating phase sums to zero, e.g. the combination with etc. Overall (even without the golden rule, i.e. the asymptotically large time limit) one can argue that the dominant ones should be the ones with a slow oscillation. Thus we will proceed to neglect the rapidly oscillating terms, and only keep the slow ones, and assume that their frequency is zero This assumption is called the rotating wave approximation.

The resulting interaction term is (with since we consider the case ): The neglected terms are called counter-rotating terms. As implicitly presented here, the rotating wave approximation is often done together with the dipole approximation, even though they are not connected in any way.

Transition rates

The total rate of transitions from the excited state to the ground state is We can make the corresponding calculation for the rate of transitions from the ground state to the excited states. In that case the matrix element is where the second term vanishes because . In this case the final state contains one less photon than in the initial state and we have to calculate . The total rate becomes

Comparing these results to what was found in the A part of the course, we can identify that the amplitude of the electromagnetic field, encoded by , is here characterized by the average number of photons in the field. However, there is a difference between the rates of relaxation and excitation : the previous is finite even when , i.e., when the field amplitude vanishes. The terms and in the relaxation are called

1. The term is spontaneous emission. This is an effect purely due to the quantum mechanical nature of the photon field. An atom in an excited state can decay into the ground state by emitting a photon, even in the absence of any external electromagnetic field
2. The term is stimulated emission. This is "less quantum mechanical" in the sense that a classical electromagnetic field has the same effect. This means that the decay rate of an excited atomic state by photon emission grows when there are already photons in the state. Lasers (laser=Light Amplification by Stimulated Emission of Radiation) are based on this effect. In a laser, one organizes a situation where there are many atoms in an excited state. When one of these states decays, the emitting photon increases the rate of the other decays into precisely the same photon mode (i.e. same direction of momentum, same phase). This causes a cascade where all the excited states decay simultaneously into the same photon state, producing a coherent collimated beam of light. In kinetic theory this term is called the "Bose enhancement" of a scattering rate, it is due to the bosonic nature of the photon. (Correspondingly there is a Pauli blocking for fermions, suppressing transitions to already occupied states).

The absorption rate is , i.e. proportional to the number of photons in the system; here the classical intuition is clear.

Now let us assume we can tune the cavity frequency, for example by moving one of the mirrors. Assume that the cavity field is coupled to the atom (or an ensemble of them). What happens to the absorption dip when becomes resonant with the atomic transition?

The states of the system can be characterized by , i.e., ground or excited state of the atom and states in the field. It is straightforward to show that is the ground state of the system, since the coupling terms evaluate to zero on this state. The energy of this state is , and hence the energies of the excited states become energy differences to the ground state. Now we can compute the action of the Hamiltonian on an arbitrary state except the ground state. For example, These two states are hence pairwise coupled. Note that the total number of excitations is the same between these two states. Expressed in terms of the matrix elements between these states, the Hamiltonian thus goes to a block diagonal form, where the blocks with different are not coupled. Diagonalizing the Hamiltonian thus becomes a question of diagonalizing each block separately. For the 'th block we hence have where . The corresponding normalized eigenstates of the 'th block are hybrid states of atoms and electromagnetic field, where . These are called dressed states. They have the eigenenergies We can hence plot the eigenenergy spectrum:

Let us return back to the initial experiment and consider the case of a fixed number of photons in the driving pulse (for example, a state with most is also possible for a low-amplitude coherent state). As the cavity frequency crosses the atomic spectrum, the total spectrum of the system shows avoided crossings of this form:

This is hence a way to study the coupling. Such an avoided crossing as such is a property of strongly coupled systems, and not a genuine quantum effect. On the other hand, the dependence of the spectrum on the number of photons is a quantum effect.