Quantum stochastic calculus

An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a heat bath.

It is appropriate in many circumstances to model the heat bath as an assembly of harmonic oscillators.

One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following Hamiltonian:[3]: 42, 45 where

is a vector containing the system variables corresponding to a finite number of degrees of freedom,

For many purposes it is convenient to make approximations about the nature of the heat bath in order to achieve a white noise formalism.

[3]: 148  This model uses the rotating wave approximation and extends the lower limit of

in order to admit a mathematically simple white noise formalism.

The coupling strengths are also usually simplified to a constant in what is sometimes called the first Markov approximation:[4]: 3763 Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output.

allows the model to have a strict correspondence with a Markovian master equation.

[2]: 142 In the white noise setting described so far, the quantum Langevin equation for an arbitrary system operator

takes a simpler form:[4]: 3763 For the case most closely corresponding to classical white noise, the input to the system is described by a density operator giving the following expectation value:[3]: 154 In order to define quantum stochastic integration, it is important to define a quantum Wiener process:[3]: 155 [4]: 3765 This definition gives the quantum Wiener process the commutation relation

The property of the bath annihilation operators in (WN2) implies that the quantum Wiener process has an expectation value of: The quantum Wiener processes are also specified such that their quasiprobability distributions are Gaussian by defining the density operator: where

[3]: 159  In the context of the white noise formalism described earlier, the Itô QSDE can be defined as:[3]: 156 where the equation has been simplified using the Lindblad superoperator:[2]: 105 This differential equation is interpreted as defining the system operator

as the quantum Itô integral of the right hand side, and is equivalent to the Langevin equation (WN1).

Unlike the Itô formulation, the increments in the Stratonovich integral do not commute with the system operator, and it can be shown that:[3] The Stratonovich QSDE can be defined as:[3]: 158 This differential equation is interpreted as defining the system operator

as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation (WN1).

[4]: 3766–3767 The two definitions of quantum stochastic integrals relate to one another in the following way, assuming a bath with

A peculiarity in the quantum generalization is the necessity to define both Itô and Stratonovitch integration in order to prove that the Stratonovitch form preserves the rules of noncommuting calculus.

The unconditioned Markovian evolution of a quantum system (averaged over all possible measurement outcomes) is given by a Lindblad equation.

In order to describe the conditioned evolution in these cases, it is necessary to unravel the Lindblad equation by choosing a consistent QSDE.

In the case where the conditioned system state is always pure, the unraveling could be in the form of a stochastic Schrödinger equation (SSE).

If the state may become mixed, then it is necessary to use a stochastic master equation (SME).

[2]: 148 Consider the following Lindblad master equation for a system interacting with a vacuum bath:[2]: 145 This describes the evolution of the system state averaged over the outcomes of any particular measurement that might be made on the bath.

The following SME describes the evolution of the system conditioned on the results of a continuous photon-counting measurement performed on the bath: where are nonlinear superoperators and

Another type of measurement that could be made on the bath is homodyne detection, which results in quantum trajectories given by the following SME: where

is a Wiener increment satisfying:[2]: 161 Although these two SMEs look wildly different, calculating their expected evolution shows that they are both indeed unravelings of the same Lindlad master equation: One important application of quantum trajectories is reducing the computational resources required to simulate a master equation.

For a Hilbert space of dimension d, the amount of real numbers required to store the density matrix is of order d2, and the time required to compute the master equation evolution is of order d4.

Storing the state vector for a SSE, on the other hand, only requires an amount of real numbers of order d, and the time to compute trajectory evolution is only of order d2.

The master equation evolution can then be approximated by averaging over many individual trajectories simulated using the SSE, a technique sometimes referred to as the Monte Carlo wave-function approach.

Not only does this technique yield faster computation time, but it also allows for the simulation of master equations on machines that do not have enough memory to store the entire density matrix.

Plot of the evolution of the z-component of the Bloch vector of a two-level atom coupled to the electromagnetic field undergoing damped Rabi oscillations . The top plot shows the quantum trajectory for the atom for photon-counting measurements performed on the electromagnetic field, the middle plot shows the same for homodyne detection, and the bottom plot compares the previous two measurement choices (each averaged over 32 trajectories) with the unconditioned evolution given by the master equation.