Derivation of the quantum-optical master equation based on coarse-graining of time

This is a derivation of the quantum-optical master equation using coarse-graining of time, which brings new insights into a decades old technique. My derivation is quite similar to derivations using quantum stochastic methods or Kraus operators, though I go through the derivation without explicitly invoking any of these concepts, so it may be easier to follow as an introduction. I also address the major pitfall of nearly all microscopic derivations of the master equation, namely that they assume the state of the system and bath factorize for all times. I show why this assumption actually holds for spontaneous emission, and coincidentally turns out to be correct.

If the waveguide is linearly coupled to a local quantum system, with the Hamiltonian H sys , then in the rotating wave approximation the interaction part of the Hamiltonian is The field mode operators obey the commutation relation and σ is a lowering operator in the Hilbert space of the system. This gives a total Hamiltonian The quantum-optical master equation describes the reduced dynamics of the system alone after having traced out the field modes ρ sys (t) = Tr field |Ψ(t) Ψ(t)| or ρ sys (t) = Tr field ρ(t) , depending on the initial conditions. It has been widely used in quantum optics and is responsible for many of the field's successes. Further, it has been derived in many forms, though in my opinion they all have numerous shortcomings: 1. The original microscopic derivation [2] has the seemingly unphysical assumption that the state of the bath and system factorize at all times, i.e. ρ(t) = ρ sys (t) ⊗ ρ field (t). This is obviously not true since it is easily possible to maximally entangle the system and field, e.g. in a spin-photon interface [3,4].
3. Methods based on stochastic evolution equations require significant knowledge of stochastic calculus which most quantum mechanics students do not possess [8].
4. Information-theoretic derivations based on the properties of dynamical semigroups [9,10] do not immediately reveal how physical models can be cast into the same form as their results.
In this tutorial, I aim to address these issues by borrowing techniques and concepts from measurement theory, while basing the calculations exclusively on a microscopic model. I hope this presentation will be helpful for those trying to understand the quantum-optical master equation.

II. INTERACTION-PICTURE HAMILTONIAN
Due to the coupling between the local system and all frequency modes of the waveguide, integrating Schrödinger's equation for this Hamiltonian is a bit unwieldy. The first step to getting a Hamiltonian that is easier to work with is to transform into an interaction picture to remove the free evolution of the waveguide H 0 . Specifically, I choose the state vector Then, Schrödinger's equation for |Ψ I (t) becomes with Here, I used the relationship which follows from the commutation in Eq. 3. Next, I define a new operator, which naturally appeared as a Fourier I then rewrite the interaction-picture Hamiltonian as where The operator b(t) obeys commutations similar to the frequency mode operators, but in time

III. COARSE-GRAINED TIME EVOLUTION OPERATOR
The Schrödinger equation can be rewritten by iteratively applying its integral form, yielding a Dyson series Note that the limits on the second time index t ′ are only from t 0 < t ′ < t and not t 0 < t ′ < t 1 . Since the arguments of each set of integrals are symmetric with respect to exchange of time indices, I can alternatively let t ′ vary until t 1 by applying the time-ordering operator T . On the second-order term it is defined by yielding the restatement is called a time-ordered exponential and is a shorthand for the expansion in Eq. 24. Now, I can proceed to finding an approximate dynamical map that takes the wavefunction from time t = k∆t → (k + 1)∆t, i.e. the map U [k + 1, k] where The goal of this map is to remove the time-ordering (which is intractable for this Hamiltonian), while maintaining unitarity (so the norm of the wavefuctnion is conserved) and being correct to O(∆t) (so that the error vanishes in the limit of ∆t → 0) [1]. Hence, I choose This map differs from the correct map via time-ordering where the limits of integration are only over the upper half of the coordinate plane for t < t ′ . Here, the operators H I (t) and H I (t ′ ) need to be reordered, which gives their commutator in Eq. 30. I can break up the commutation between the Hamiltonian at different times based on Eq. 12 The commutations between H sys (t) and V (t ′ ) are assigned an O(1/ √ ∆t) because the singular operators in V (t) (Eq. 13) obey the commutation Combining the orders of [H I (t), H I (t ′ )] in Eq. 30, I see that my choice of map is correct to O(∆t), with a leading error O(∆t 3/2 ) from the commutation of H sys (t) and V (t ′ ). This approximation amounts to a coarse-graining of the system-bath-interaction dynamics to a timescale of ∆t, which occur on a much faster timescale than the dynamics generated by the system evolution. Nevertheless, in the limit of ∆t → 0 the error vanishes and the map becomes exact.
Writing the coarse-grained map out explicitly where I defined the coarse-grained operator This operator obeys the commutation and in the limit (A very nice alternative to this derivation is based on using a wavelet expansion of b(t), which yields an approximate Hamiltonian directly rather than just a coarse-grained map [7].) Now, the accessible Hilbert space of the field is also coarse-grained. Specifically, the map can only create or remove excitations from the field in time bins of width ∆t. Owing to the commutation in Eq. 37, the relevant Hilbert space of the field is now a product of a bunch of harmonic oscillator spaces, each labeled by the time-bin number n The vacuum state with zero excitations is |0 = |0 ⊗ |0 ⊗ · · · , the field with a single excitation is in a state or with multiple excitations in the same j-th bin

IV. QUANTUM-OPTICAL MASTER EQUATION FOR SPONTANEOUS EMISSION
Consider the initial state of the local system to be a mixed state, with the field in the vacuum state To obtain the density matrix of the system at time t 1 = ∆t, I need to evolve the total system with the unitary map U The first step to evaluating this expression is to expand the trace as a series of partial traces over the field time bins, with the partial trace over the j-th bin as Thus, the creation and annihilation operators from partial traces over bins other than the 0-th one in Eq. 45 commute with the unitary evolution operators and evaluate to zero in their expectations. This leaves only the first partial trace and the zero elements from the other partial traces-hence Notably this map is of the operator-sum representation [5], i.e.
where K m [k] are the so-called Kraus operators (which are typically associated with quantum measurements, though I do not invoke any measurement theory in my derivation). In my specific scenario, the Kraus operators for the first time step ∆t in evolution are To evaluate these operators, I expand the discrete map to O(∆t) Then, I evaluate these Kraus operators for zero photon emissions into the field one photon emission into the field and two photon emissions into the field Operators representing more photon emissions are higher order in ∆t and hence are taken K m>2 [0] ≈ 0. Then, keeping all terms O(∆t) in the density matrix map Rearranging, I show this is in the standard Lindblad form It is further important to show that this map holds for all time steps. For example, consider the second time step, i.e. from t = 0 → ∆t → 2∆t.
Again, I have used Eq. 46 to reduce the complexity of the trace by commuting away operators that have no effect on the expectations. Next, I chronologically order and group all operators based on their action on the time bins and insert a field identity operator 1 field between the groups. Then, I note that all the elements of the identity with nonzero photon number can commute past the other operators, either to the left or right depending on their time bin, and annihilate to zero. Hence, I can equivalently insert |0 0| rather than the identity. The left expectation in Eq. 65 then becomes Applying to all of Eq. 65, I obtain Notably, the entire derivation holds for other system operators like σ † σ to yield a dephasing rate instead of spontaneous emission. See reference [5] for a detailed explanation of how thermal bath states drive the system, and their corresponding Lindblad superoperators. In summary, I derived the quantum-optical master equation from a microscopic model of linear system-bath interactions, without relying on the unsupported assumption that the system and bath factorize at all times. I hope the techniques presented here will be useful in helping students enter the field of open quantum systems.