Quantum trajectories is a method for studying the dynamics of open quantum systems. In particular, it predicts the possibility of particular time evolutions in the continuous measurement regime. If the measurement record is not kept and many trajectories averaged together master equation dynamics are recovered. In this case quantum trajectories leads to an stochastic numerical algorithm that has some memory usage advantages. In the case that the measurement record is kept, individual trajectories correspond to possible observed time evolutions.

The quantum trajectory method was first discussed by Jean Dalibard, Yvan Catsinm and Klaus Mølmer et al.^[1][2] The theory was developed around the same time by Howard Carmichael who presented the material in a series of lectures at the Université Libre de Bruxelles that were later published as a book^[3]. Early discussions can also be found in^[4][5]

Theory

The dynamics of open quantum systems are governed by the Lindblad master equation

${\displaystyle {\dot {\rho }}=-{\frac {i}{\hbar }}[H,\rho ]+{\mathcal {L}}[\rho ]}$

where ${\displaystyle \rho }$ is the density matrix representing the quantum state and ${\displaystyle H}$ is the Hamiltonian of the system. The Lindbladian takes the form

${\displaystyle {\mathcal {L}}[\rho ]=-\displaystyle \sum _{\mu }{\frac {i}{\hbar }}[-i\hbar L_{\mu }^{\dagger }L_{\mu },\rho ]_{+}+\displaystyle \sum _{\mu }L_{\mu }^{\dagger }\rho L_{\mu }}$

The operators ${\displaystyle L_{\mu }}$ are called jump operators and depend on the particular physical situation being considered. We can rewrite this equation as

${\displaystyle {\dot {\rho }}=-{\frac {i}{\hbar }}\left(H_{eff}\rho -\rho H_{eff}^{\dagger }\right)+L_{\mu }^{\dagger }\rho L_{\mu }}$

Where the effective Hamiltonian is given by

${\displaystyle H_{eff}=H-i\hbar \displaystyle \sum _{\mu }L_{\mu }^{\dagger }L_{\mu }}$

Over a very short time interval this implies that the change in the density operator is (to first order in ${\displaystyle dt}$)

${\displaystyle \rho (t+dt)=\left(\mathbf {1} -{\frac {i}{\hbar }}H_{eff}dt\right)\rho (t)\left(1+{\frac {i}{\hbar }}H_{eff}^{\dagger }dt\right)-i\hbar dt\displaystyle \sum _{\mu }L_{\mu }^{\dagger }\rho (t)L_{\mu }}$

If the system begins in a pure state, ${\displaystyle |\psi (t)\rangle }$ and a continuous measurement is performed to determine whether the system is in any of the states ${\displaystyle L_{\mu }|\psi \rangle }$, then this equation says that

${\displaystyle |\psi (t+dt)\rangle ={\frac {L_{\mu }|\psi (t)\rangle }{\sqrt {\langle \psi (t)|L_{\mu }^{\dagger }L_{\mu }|\psi (t)\rangle }}}}$

with probability

${\displaystyle P_{\mu }=\langle \psi (t)|L_{\mu }^{\dagger }L_{\mu }|\psi (t)\rangle dt}$

When this happens it is referred to as a jump. Alternatively, with probability

${\displaystyle P_{0}=1-\displaystyle \sum _{\mu }P_{\mu }}$

the state is

${\displaystyle |\psi (t+dt)\rangle ={\frac {e^{-{\frac {i}{\hbar }}H_{eff}dt}|\psi (t)\rangle }{{\sqrt {\langle \psi (t)|e^{{\frac {i}{\hbar }}H_{eff}^{\dagger }dt}e^{-{\frac {i}{\hbar }}H_{eff}dt}|\psi (t}}\rangle }}}$

since, to first order in ${\displaystyle dt}$

${\displaystyle \mathbf {1} -{\frac {i}{\hbar }}H_{eff}dt=e^{-{\frac {i}{\hbar }}H_{eff}dt}}$

The normalization factors are necessary because neither of the time evolution options are non-unitary. It may seem mysterious that the evolution in non-unitary even in the case where no jump occurs. This can be understood in terms of the information gained about the system in that time. Put simply, the lack of a jump ${\displaystyle L_{\mu }}$ provides information that leads the observer to assign a lower probability to the system being in a state that can be effected by ${\displaystyle L_{\mu }}$. The non-Hermitian part of ${\displaystyle H_{eff}}$ accomplishes this.

A possible combination of non-unitary time evolution operators and jump operators applied to the state is referred to as a trajectory. Averaging over many trajectories recovers the master equation dynamics. If a measurement with back action of the system corresponding to the ${\displaystyle L_{\mu }}$ operators is actually being preformed by the experimenters then the time of evolution of the quantum state does follow some trajectory although exactly which trajectory is a matter of probability.

Alternative Unravelings of the Master Equation

The choice of jump operators is not unique. For example, the master equation is invariant under the substitutions^[6]

${\displaystyle L_{\mu }\rightarrow L_{\mu }+\gamma _{\mu }}$

${\displaystyle H\rightarrow H-i{\frac {\hbar }{2}}\displaystyle \sum _{\mu }\left(\gamma _{\mu }^{*}L_{\mu }-\gamma _{\mu }L_{\mu }^{\dagger }\right)}$

where ${\displaystyle \gamma \in \mathbb {C} }$. The quantum trajectories produced by using these new jump operators is referred to as a different unraveling of the master equation. If many trajectories are averaged then the various possible unravelings are all equivalent, i.e. they all reproduce the master equation dynamics. However, if the measurement record is going to be kept then the jump operators which correspond to the particular measurement being preformed must be used to obtain accurate predictions about possible trajectories.

The Quantum Jump Method

This perspective leads to a numerical algorithm that can be used to simulate quantum dynamics. This algorithm is sometimes referred to as the quantum jump method, sometimes as Monte Carlo wavefunctions, and sometimes simply as quantum trajectories. A summary of the algorithm is as follows. If at a time ${\displaystyle t}$ the system is in the state ${\displaystyle |\psi (t)\rangle }$ then the time step from ${\displaystyle t}$ to ${\displaystyle dt}$ occurs in five steps. First, the probably of each type of collapse is calculated as described in the theory section. Second, The interval ${\displaystyle [0,1]}$ is divided up as follows

${\displaystyle I_{1}=[0,P_{1}]}$

${\displaystyle I_{2}=(P_{1},P_{1}+P_{2}]}$

${\displaystyle I_{j}=\left(\displaystyle \sum _{\mu =1}^{j-1}P_{\mu },\displaystyle \sum _{\mu =1}^{j}P_{\mu }\right]}$

Third, a random number ${\displaystyle r}$ in the interval ${\displaystyle [0,1]}$ is drawn. Fourth, if

${\displaystyle r\in I_{j}}$

then the jump operator ${\displaystyle L_{j}}$ is applied. If

${\displaystyle r\notin \displaystyle \cup _{\mu }I_{\mu }}$

Then time evolution occurs according to the non-unitary time evolution operator. Finally, the state must be normalized since in general the jump operators are also non-unitary.

In addition to physical insight about the behavior of quantum systems in the continuous measurement regime, this algorithm offers some efficiency advantages in the case that the state may always be represented by a pure state. In that case, only ${\displaystyle n}$ complex numbers must be stored in contrast to the ${\displaystyle n^{2}}$ that must be stored in the entire density matrix in going to be kept track of.

Example: The Jaynes-Cummings Model with Photon Counting

As a simple example lets consider the dynamics of the Jaynes-Cummings model. The Jaynes-Cummings model describes the interaction of a two level atom with a single cavity mode. The Hamiltonian is given by

${\displaystyle H=\hbar \omega _{c}a^{\dagger }a+\hbar \omega {\frac {\sigma _{z}}{2}}+\hbar g(\sigma _{-}a+a^{\dagger }\sigma _{+})}$

where ${\displaystyle \omega }$ is the atomic resonance frequency, ${\displaystyle \omega _{c}}$ is the resonance frequency of the cavity, ${\displaystyle g}$ is the coupling between the cavity and the atom, ${\displaystyle a}$ is the photon annihilation operator, and ${\displaystyle \sigma _{-}}$ is the atomic lower operator. The jump operators used in this example, which correspond to photon being observed outside the cavity, are given by

${\displaystyle L_{1}={\sqrt {\Gamma }}\sigma _{-}}$

${\displaystyle L_{2}={\sqrt {\kappa }}a}$

where ${\displaystyle \Gamma }$ is the atomic decay rate out of the cavity and ${\displaystyle \kappa }$ is that cavity decay rate. The Figure displays the photon number and atomic state expectation values as a function of time for (a) master equation evolution, (b) trajectory evolution averaged over many trajectories, and (c) a single trajectory. First, notice that (a) and (b) are basically identical. The single trajectory on the other hand appears quite different. The first two time evolutions are damped oscillations. In the case of the single trajectory the oscillations appear virtually undamped and then suddenly the oscillations stop completely. This end of the oscillations corresponds to a photon being detected outside of the cavity. We should also note that in the single trajectory case the oscillations actually are slightly damped due to the non-Hermitian nature of the effective Hamiltonian but it is not noticeable on this time scale.

Trajectories in Experiments

The idea of a quantum jump dates back to Bohr's 1913 proposal^[9] to explain atomic spectra and as discussed above the dynamics of open quantum systems can always be understood in terms of an average over many trajectories. In a modern context, Nagourney et al.^[10] observed quantum jumps in a trapped ${\displaystyle {\ce {Ba^{+}}}}$ ion in 1986. This was done by coupling the ${\displaystyle {\ce {6^{2}S_{{\frac {1}{2}}}}}}$ state strongly to the ${\displaystyle {\ce {5^{2}D_{{\frac {3}{2}}}}}}$ state and weakly to the ${\displaystyle {\ce {5^{2}D_{{\frac {5}{2}}}}}}$ state. They then observed the florescence from the ion. The florescence jump in an apparently discontinuous fashion from being more or less constant to being essentially zero. This corresponded to the ion jumping between a bright state in which it was Rabi flopping between the two strongly coupled states and a dark state in which it was in the weakly coupled ${\displaystyle {\ce {5^{2}D_{{\frac {5}{2}}}}}}$ state.

References