View source for Unraveling quantum open system dynamics

Open quantum system dynamics are traditionally studied with the master
equation, a differential equation for the density matrix describing
the system.  An alternative, and equivalent approach, utilizing
pure-state wavefunctions and stochastic evolution, is presented in
this section.  This approach, known variously as the "quantum
monte-carlo wavefunction" technique, or the method of quantum jumps
or quantum trajectories, is based on the fact that density matrices
can be represented as stochastic combinations of pure states.  We
apply this technique to modeling an atom coupled to a vacuum, and show
how the monte-carlo approach generates average evolution equivalent to
the optical Bloch equations.  The approach offers new ways to
physically interpret open quantum system evolution, but must be used
with caution, since there are an infinite number of equivalent
interpretations, many of which may appear to conflict with each other.

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=== Quantum jumps ===
Much of traditional atomic physics has focused on ensembles of many
atoms; correspondingly, the density matrix formalism we have employed
in studying an atom coupled to the vacuum provides an ensemble
description of quantum dynamics.  In fact, the concept of ensembles is
vital to the foundations of quantum mechanics; only ensemble averages
can be measured, and one might even say that in a sense, only ensemble
averages are "real."  Schr&ouml;dinger, in 1952, wrote (and we quote
here from Gerry and Knight, "Introductory Quantum Optics," Cambridge
University Press, 2005):

: ... we never experiment with just one electron or atom or (small) molecule.  In thought experiments, we sometimes assume that we do; this invariably entails ridiculous consequences\ldots In the first place, it is fair to state that we are not experimenting with single particles, any more than we can raise ichthysauria in the zoo.


Despite this rather extreme opinion, modern atomic physics experiments
are now broadly moving in the direction of coherently addressing and
manipulating single atoms and single molecules in the laboratory.
Experiments have been successful with both (hot and cold) neutral
atoms and ions, and with a variety of single (hot) molecules.  These
date back to Dehmelt's 1975 proposal to trap single ions to provide an
atomic time standard.

Such experiments with single quantum systems motivate a description of
open quantum system dynamics in terms of statistical averages over
single realizations, as illustrated by the following experiment.
Consider a single three-level ion, such as strontium, with a <math>8</math> ns
lifetime <math>422</math> nm S-P transition, and a <math>\sim 400</math>ms lifetime <math>674</math> nm
S-D transition:
::[[Image:Unraveling_quantum_open_system_dynamics-sr-levels.png|thumb|400px|none|]]
Experimental observation of the blue fluorescence from this atom, when
strongly excited at <math>422</math> nm, and weakly excited at <math>674</math> nm, shows a
very interesting signature: quantum jumps.  When the atom is in the
metastable D state, the <math>422</math> nm fluorescence ceases, because no
photons can be scattered from the strong transition until it
transitions back to the ground S state.  A typical observation record
might look like this:
::[[Image:Unraveling_quantum_open_system_dynamics-sr-jumps.png|thumb|408px|none|]]
Naturally, the average fluorescence rate will be consistent with
that expected by the relative strengths of the red and blue
excitations, but this would be just a constant value.  Such quantum
jumps, however, are ubiquitous in experiments involving single atoms
and other single quanta, and it would be nice to have a theoretical
prescription for describing such observations.  In addition, it would
be nice to have available possible interpretations of what how
the individual systems might be evolving to give some observed
ensemble behavior, such as an exponential decay.

=== Quantum Monte-Carlo wavefunction technique for atom + vacuum ===

Consider a single two-level atom evolving under the Hamiltonian <math>H_0</math>,
while also undergoing spontaneous emission.  This is the same scenario
we studied to formulate the optical Bloch equations.  It can be
modeled using the quantum Monte-Carlo wavefunction (QMCWF) technique,
using the following steps.

Let us assume the atom begins in some state <math>|\psi{\rangle}</math>, and let the
atomic levels be <math>|e{\rangle}</math> and <math>|g{\rangle}</math>.  Let <math>\Gamma</math> be the decay rate of
the excited state, as determined, for example, from Fermi's golden
rule.  The QMCWF technique introduces a new non-Hermitian
"Hamiltonian"
:<math> 
	H = H_0 - i\frac{\Gamma}{2} |e \rangle  \langle e|
\,,
</math>
which governs evolution of the state, according to the following.
For each differential time step <math>dt</math>, the state <math>|\psi{\rangle}</math> evolves
as:

<blockquote>
<table border=1>
<tr><td>
* 1. Compute <math>dp = \Gamma dt | \langle e| \psi{\rangle}|^2</math>
* 2. Let <math>0\leq \epsilon \leq 1</math> be a uniformly distributed random number
* 3. If <math>\epsilon<dp</math> then <math>|\psi \rangle  \leftarrow |g{\rangle}</math>
* 4. If <math>\epsilon\ge dp</math> then <math>|\psi \rangle  \leftarrow
		  \left[    { e^{-i H dt}/\sqrt{1-dp}} \right]  |\psi{\rangle}</math>
* 5. Go to 1
</td></tr>
</table>
</blockquote>

Physically, <math>dp</math> is the probability of the atom jumping from <math>|e{\rangle}</math> to
<math>|g{\rangle}</math>, that is, spontaneously emitting a photon, during the time
interval <math>dt</math>.  Note that typically <math>dp\ll 1</math>, and
:<math> 
	dp = 1-{\langle}\psi| e^{iH^\dagger dt}e^{-iH dt}|\psi{\rangle}
\,,
</math>
so <math>1-dp</math> is the normalization of the state after <math>dt</math> of evolution
under the non-Hermitian Hamiltonian <math>H</math>.  Essentially, the QMCWF
scheme models an imaginary observer watching for a spontaneously
emitted photon coming from the atom.  If a photon is emitted, then the
atom transitions immediately into <math>|g{\rangle}</math>; this is the rule given in
step 3, and such a transition is known as a quantum jump.
Otherwise, if no photon is emitted, the state changes nevertheless,
albeit by only a small amount.  Because the observer saw no photon,
and thus the probability of the atom being in <math>|e{\rangle}</math> is slightly
diminished.  Specifically, if <math>H_0=0</math>, and <math>|\psi{\rangle}=\alpha
|g{\rangle}+\beta|e{\rangle}</math>, then according to the rule in step 4,
:<math> 
	|\psi \rangle  \leftarrow
		\frac{\alpha|g \rangle  + \beta e^{-\Gamma dt/2} |e{\rangle}}
			{\sqrt{|\alpha|^2 + |\beta|^2 e^{-\Gamma dt}}}
\,.
</math>
This is exactly the same physics we have seen earlier, in the
beamspliter model of the optical Bloch equations.

==== Equivalence of QMCWF and the optical Bloch equations ====

The equivalence of QMCWF and the optical Bloch equations (OBE) is
shown by demonstrating that the evolution of the density matrix
:<math> 
	\rho(t) = \overline{|\psi(t){\rangle}{\langle}\psi(t)|}
</math>
satisfies the OBE, where the average (denoted by the overline) is
taken over instances of running the QMCWF procedure.  This follows
from computing the density matrix for the state after one QMCWF
procedure step:
:<math>\begin{array}{rcl}  
\rho(t+dt) &=& dp \, |g \rangle  \langle g| + (1-dp)
	\frac{e^{-i H dt} |\psi{\rangle}{\langle}\psi|e^{+i H^\dagger dt}}{{1-dp}}
\\
&\approx& dp \, |g \rangle  \langle g| + (1-i H dt) \rho(t) (1+i H^\dagger  dt)
\\
&\approx& dp \, |g \rangle  \langle g| + \rho(t) - i (H\rho(t) - \rho(t)  H^\dagger ) dt
\\
&=& \Gamma dt  \langle e|  \rho(t) |e \rangle  \, |g \rangle  \langle g| + \rho(t)
	- i (H\rho(t) - \rho(t)  H^\dagger ) dt
\\
&=& \Gamma dt |g \rangle  \langle e|  \rho(t) |e \rangle  \langle g| + \rho(t)
	- i \left [H_0,\rho(t) \right ] -\frac{\Gamma}{2}  \left(    |e \rangle  \langle e| \rho(t) +
	\rho(t) |e \rangle  \langle e| \right)
\,.
\end{array}</math>
Writing this as a coarse-grained differential equation, taking the
limit of small <math>dt</math>, we find
:<math>\begin{array}{rcl}  
\frac{d}{dt} \rho(t) &\approx& \frac{\rho(t+dt)-\rho(t)}{dt}
\\
  &=& - i \left [H_0,\rho(t) \right] -\frac{\Gamma}{2}  \left(    |e \rangle  \langle e| \rho(t) +
	\rho(t)|e \rangle  \langle e| \right)  + \Gamma dt |g \rangle  \langle e|  \rho(t) |e \rangle  \langle g|
\,.
\end{array}</math>
This is the optical Bloch equation.

=== Quantum Monte-Carlo wavefunction technique: general case ===

The method of the quantum Monte-Carlo wavefunction technique can be
applied not just to the atom + vacuum scenario, but also to model any
open quantum system dynamics.  The relaxation part of a master
equation may be described by a Lindblad operator
:<math> 
	{ \mathcal  L}(\rho) = -\frac{1}{2} \sum_k  C^\dagger _k C_k \rho + \rho
	 C^\dagger _k C_k
		+ \sum_k C_k \rho  C^\dagger _k
\,,
</math>
where <math>C_k</math> are known as "quantum jump" operators.  The
corresponding QMCWF procedure employs the non-Hermitian Hamiltonian
:<math> 
	H = H_0 - \frac{i\hbar}{2} \sum_k  C^\dagger _k C_k
\,,
</math>
and the steps

<blockquote>
<table border=1>
<tr><td>
* 1. Compute <math>dp = \sum_k dp_k</math>, and <math>dp_k = dt
			{\langle}\psi| C^\dagger _k C_k|\psi{\rangle}</math>
* 2. Let <math>0\leq \epsilon \leq 1</math> be a uniformly distributed random number
* 3. If <math>\epsilon<dp</math> then <math>|\psi \rangle  \leftarrow C_k|\psi \rangle  / \sqrt{dp_k/dt}</math>	with <math>k</math> randomly chosen with probability <math>dp_k/dp</math>.
* 4. If <math>\epsilon\ge dp</math> then <math>|\psi \rangle  \leftarrow
		  \left[    { e^{-i H dt}/\sqrt{1-dp}} \right]  |\psi{\rangle}</math>
* 5. Go to 1
</td></tr>
</table>
</blockquote>

There are some computational advantages to the QMCWF approach, over
numerical solution of the differential equations normally obtained
with master equations.  In particular, an <math>N</math> dimensional Hilbert
space described by a density matrix requires <math>O(N^2)</math> variables,
whereas a pure-state wavefunction requires only <math>O(N)</math> variables.  Of
course, calculating expectation values means that the stochastic
evolution of the QMCWF technique must be repeated <math>\sim N</math> times, so
the overall effect is a tradeoff between storage space and
computational time.  However, the QMCWF technique is immediately
parallelizable, and often (but not always) desired observables
converge quickly.

For more about this, see, for example, the nice article <refbase>5239</refbase>
<!-- by Molmer,
Castin, and Dalibard, {\em Monte Carlo wave-function method in quantum
optics}, J. Opt. Soc. Am. B, v10, p524, 1993. -->

=== Example: spontaneous emission ===

The dynamics of an atom interacting with the vacuum can be described
not just by the usual optical Bloch equations, but also, by an
infinite variety of alternate pictures.  This can be seen through a
simple example using the quantum monte-carlo wavefunction technique.

Let <math>X</math>, <math>Y</math>, and <math>Z</math> denote the Pauli matrices, as usual.  Recall
that the optical Bloch equations
:<math> 
    \dot{\rho} = -\frac{i}{\hbar} [H,\rho] + \left[ { L \rho L^\dagger  - \frac{1}{2}( L^\dagger  L \rho + \rho L^\dagger  L ) } \right]
 
</math>
can be written as
:<math> 
    \dot{\vec{r}} = \left[ \begin{array}{ccc} {-\Gamma/2}& {0} & {0} \\ {0} & -\Gamma/2 & 0 \\ 0 & 0 & -\Gamma \end{array} \right] \vec{r} - \left[ \begin{array}{c} 0 \\ 0 \\ \Gamma \end{array} \right]
\,,
</math>
using
:<math> 
    \rho = \frac{I+\vec{r}\cdot{\vec{\sigma}}}{2} \,.
</math>
In the "standard" unraveling of spontaneous emission into quantum
jumps, in the absence of a classical driving field, there is no
Hamiltonian evolution (<math>H=I</math>) and one jump operator,
<math>L=\sqrt{\Gamma}\sigma_- = \sqrt{\Gamma} ( X-i Y)/2</math>.  This gives exactly the standard
optical Bloch equations, as was seen above.  Unraveling the dynamics with the quantum monte carlo wavefunction technique, we obtain evolution trajectories which look like this:

<blockquote>
<jwplayer width="700" height="820" repeat="true" displayheight="800" 
image="http://feynman.mit.edu/8.422/simse5a.png"
autostart="false">http://feynman.mit.edu/8.422/simse5a.flv</jwplayer>
</blockquote>

The average over many such evolutions leads to the exponential decay expected from the optical Bloch equations:
<blockquote>
<jwplayer width="700" height="440" repeat="true" displayheight="420" 
image="http://feynman.mit.edu/8.422/simse4a.png"
autostart="false">http://feynman.mit.edu/8.422/simse4a.flv</jwplayer>
</blockquote>

==== An alternate unraveling of spontaneous emission ====

However, it turns out that identical dynamics are obtained, on average,
with the same form of master equation, but with <math>H=-\Gamma Y/4</math> and
<math>L=\sqrt{\Gamma}(I + X-iY)/2</math>, instead.  

<blockquote>
<jwplayer width="700" height="820" repeat="true" displayheight="800" 
image="http://feynman.mit.edu/8.422/simse5.png"
autostart="false">http://feynman.mit.edu/8.422/simse5.flv</jwplayer>
</blockquote>

The average over many such evolutions also leads to the exponential decay expected from the optical Bloch equations:
<blockquote>
<jwplayer width="700" height="440" repeat="true" displayheight="420" 
image="http://feynman.mit.edu/8.422/simse4.png"
autostart="false">http://feynman.mit.edu/8.422/simse4.flv</jwplayer>
</blockquote>
Analytically, the exact agreement can be proven by writing
the master equation as a set of differential equations for <math>\vec{r}</math>,
with the alternate choice of <math>H</math> and <math>L</math>.

The origin of this alternate unraveling of spontaneous emission arises
in the lack of knowledge about what basis the environment is observing
the system in, as can be seen in the discussion about another
representation of open quantum system dynamics, the operator sum representation.

=== Example: phase and amplitude damping ===
Spontaneous emission, which is mainly what we have studied so far,
with the OBE and QMCWF, is described by the density matrix evolution
:<math> 
	\rho(t=0) =  \left[	\begin{array}{cc}{1-c}&{b^*}\\{b}&{c}\end{array}\right]  \longrightarrow
	\rho(t) = \left[\begin{array}{cc}{1-c e^{-\Gamma t}}&{b^* e^{-\Gamma t/2}}\\
		{b e^{-\Gamma t/2}}&{c e^{-\Gamma t}}\end{array}\right]
\,.
</math>
Here, <math>\Gamma</math> is the spontaneous emission rate of the atom,
parameterizing the loss of energy from the atom (the decay of the
diagonal elements of <math>\rho</math> to <math>|g \rangle  \langle g|</math>), and the simultaneous loss
of phase coherence of the atom (the decay of the off-diagonal elements
of <math>\rho</math> to zero).  The quantum jump operator for spontaneous
emission, in a two-level atom, is <math>C = \sqrt{\Gamma} \sigma^-</math>.
More generally, however, we can have two parameters which describe
separate decay of the diagonal and off-diagonal elements,
:<math> 
	\rho(t) = \left[ \begin{array}{cc}{1-c e^{-t/T_1}}&{b^* e^{-t/T_2}}\\
		{b e^{-t/T_2}}&{c e^{-t/T_1}}\end{array}\right]
\,,
</math>
where <math>T_1</math> is the energy loss rate, and <math>T_2</math> is the rate of loss of
quantum coherence, also known as the phase damping rate.  Both of
these effects are known as decoherence.

In particular, we define decoherence as ''any process which can turn
pure states into statistical mixtures''.  A density
matrix <math>\rho</math> is pure if either <math>{\rm Tr}(\rho^2)=1</math>, or <math>\rho =
|\psi{\rangle}{\langle}\psi|</math> for some state <math>|\psi{\rangle}</math>, or the entropy <math>S(\rho) =
-{\rm Tr}(\rho \log \rho)=0</math>.  Otherwise, it is mixed.

Phase damping, which gives rise to <math>T_2</math>, is in a sense the most
"quantum" kind of noise; it comes along with spontaneous emission,
but can also exist by itself.  Historically, the term "decoherence"
has sometimes been identified exclusively with phase damping, because
of its important role in the emergence of classical behavior from
quantum systems, but today, decoherence is used as a much more general
term, because there are so many ways to loose coherence from a quantum
system other than just phase damping.

=== Open system dynamics have infinitely many equivalent unravelings ===
The various physical origins of phase damping are interesting to
consider, because they can teach us an important fact about models of
decoherence for single quantum systems: just as <math>\rho</math> can be ''unraveled'' in infinitely many ways as <math>\sum_k p_k |\psi_k{\rangle}{\langle}\psi_k|</math>,
decoherence processes may also be unraveled in an infinite
number of ways, each equally equivalent and equally physically
meaningful.

We consider three different physical models of phase damping, on a
two-level atom.

==== Random phase noise ====

A two-level atom of frequency <math>\omega_0</math> excited by far off-resonance
light, <math>\omega\gg\omega_0</math>, experiences an AC stark shift of amount
proportional to the light intensity.  If this intensity fluctuates,
then the atom's phase is randomly modulated, causing the evolution
{{Eq
|math=<math> 
	|e \rangle  \rightarrow e^{i\theta}|e{\rangle}
\,,
</math>
|num=6.2.1
}}
where <math>\theta</math> is the phase imparted by the AC stark shift.  Suppose
<math>\theta</math> is modeled as a Gaussian distributed random variable, with
mean zero and variance <math>2\lambda t</math>, such that
:<math> 
	{\rm prob}(\theta) = \frac{1}{\sqrt{4\pi\lambda t}}
	e^{-\theta^2/4\lambda t}
\,.
</math>
If the initial state of the atom is <math>\psi \rangle  = a|g \rangle  + b|e{\rangle}</math>, then
after time <math>t</math>, it evolves into the average state described by the
density matrix
:<math> 
	\rho(t) = \int_{-\infty}^{+\infty}
	 \left[	\begin{array}{cc}{|a|^2}&{ab^* e^{i\theta}}\\{a^*b e^{-i\theta}}&{|b|^2}\end{array}\right] 
		{\rm prob}(\theta) \,d\theta
\,,
</math>
which is found to be
{{Eq
|math=<math> 
	\rho(t) =
		\left[ \begin{array}{cc}
          {|a|^2}& {ab^* e^{-\lambda t}} \\
          {a^*b e^{-\lambda t}}& {|b|^2} \end{array}\right]
\,.
</math>
|num=6.2.2
}}
If the atomic Hamiltonian is <math>H_0 = \Omega |e \rangle  \langle e|</math>, and the atom's
initial state is <math>|\psi \rangle  = (|g{\rangle}+|e \rangle )/\sqrt{2}</math>, then the dipole
moment of the atom shows a simple Rabi oscillation:
::[[Image:Unraveling_quantum_open_system_dynamics-sim0.png|thumb|306px|none|]]
When random phase noise is imposed on the atom, then its dipole moment
decays with time.  The evolution of a single atom, according to
the "trajectory" described by the random walk of
Eq.(6.2.1), is a noisy Rabi oscillation:

::[[Image:Unraveling_quantum_open_system_dynamics-sim1.png|thumb|306px|none|]]

Here is an animation of seven sample trajectories, which makes it clear that the average of all these evolutions produces an exponential decay, due to the random phase jitter between the trajectories:
<blockquote>
<jwplayer width="700" height="820" repeat="true" displayheight="800" 
image="http://feynman.mit.edu/8.422/sim1b.png"
autostart="false">http://feynman.mit.edu/8.422/sim1b.flv</jwplayer>
</blockquote>

==== Elastic collisions ====
Another physical origin for phase damping is elastic collisions.
Assume the two-level atom bounces along a waveguide, interacting with
the walls without loosing kinetic energy, but changing its trajectory
slightly at each bounce, in a manner depending on the state of the
atom.  This can be modeled by a Hamiltonian interacting the atom with
a single mode environment,
:<math> 
	H_{SE} = |e \rangle  \langle e| \otimes   \left[    { \gamma |0 \rangle  \langle 1| + \gamma^*
			|1 \rangle  \langle 0| } \right] 
\,,
</math>
with coupling constant <math>\gamma</math>.  During a small differential timestep
<math>dt</math>, an initial atomic state <math>a|g \rangle  + b|e{\rangle}</math> coupled to an environment
<math>|0{\rangle}</math> evolves to become
{{Eq
|math=<math>\begin{array}{rcl}  
	(a|g \rangle  + b|e \rangle ) \otimes |0{\rangle}
	&\rightarrow& a|g{\rangle}|0 \rangle  + b|e \rangle  (\cos\theta|0 \rangle  + \sin\theta|1 \rangle )
\\ &=&   \left[    { a|g \rangle  + b\cos\theta |e \rangle  } \right]  |0{\rangle}
	+   \left[    { b\sin\theta|e \rangle  } \right]  |1{\rangle}
\,, 
\end{array}</math>
|num=6.2.3
}}
where <math>e^{-\lambda dt} = \cos\theta</math>.  This expression is very similar
to that obtained for the gedankenexperiment used in the QMCWF model of
spontaneous emission; the difference is that when a photon is observed
in the environment, the atom does ''not'' collapse into <math>|g{\rangle}</math>, but
rather, into <math>|e{\rangle}</math>.  In other words, it does not loose energy; it
only looses information about what state it was in, prior to the
collapse.

Just as in the proof of the equivalence of QMCWF to the OBE, we can
compute the density matrix evolution which this model gives rise to,
by writing down an expression for <math>\rho(t)</math>, based on
Eq.(6.2.3),
:<math>\begin{array}{rcl}  
	\rho(t) &=&   \left[    { a|g \rangle  + b\cos\theta |e \rangle  } \right] 
			  \left[    {  \langle g|a^* +  \langle e| b^*\cos\theta } \right] 
		+ |b|^2 \sin^2 \theta |e \rangle  \langle e|
\\
&=&  \left[	\begin{array}{cc}{|a|^2}&{ab^* e^{-\lambda t}}\\{a^*b e^{-\lambda t}}&{|b|^2}\end{array}\right] 
\,.
\end{array}</math>
Note that this is exactly the same evolution as we obtained for the
random phase noise model, Eq.(6.2.2).

Despite the density matrix evolution being identical to that of the
random phase model, the elastic collision model implies a different
single-particle evolution trajectory.  In contrast to the noisy Rabi
oscillations previously seen, for the elastic collisions, the atomic
state initially decays, then jumps into the <math>|e{\rangle}</math> state at some
random time; this is illustrated by this sample trajectory:
::[[Image:Unraveling_quantum_open_system_dynamics-sim2.png|thumb|306px|none|]]

Again, here is an animation of seven sample trajectories:
<blockquote>
<jwplayer width="700" height="820" repeat="true" displayheight="800" 
image="http://feynman.mit.edu/8.422/sim2b.png"
autostart="false">http://feynman.mit.edu/8.422/sim2b.flv</jwplayer>
</blockquote>

==== Phase flips ====
A third physical model for phase damping is based on quantum jumps.
The Lindblad operator for phase damping is evidently
:<math> 
	{ \mathcal  L}(\rho) = -\frac{1}{T_2}   \left[    { |e \rangle  \langle e| \rho |g \rangle  \langle g| +
		|g \rangle  \langle g| \rho |e \rangle  \langle e| } \right] 
\,.
</math>
Equivalently, it may be rewritten in standard form as
:<math> 
	{ \mathcal  L}(\rho) = -\frac{1}{2}   \left[    {  C^\dagger  C \rho + \rho
	 C^\dagger  C } \right] 
		+ C \rho  C^\dagger 
\,,
</math>
where
:<math> 
	C = \frac{1}{\sqrt{2 T_2}}   \left[    { |g \rangle  \langle g| - |e \rangle  \langle e| } \right] 
\,.
</math>
Note that <math>C^\dagger C</math> is proportional to the identity, and thus
<math>H=H_0</math>, such that no evolution occurs due to this relaxation process
except when a quantum jump occurs.  Moreover, the effect of a quantum
jump is to flip the phase of the atom by <math>-1</math>, changing <math>|e{\rangle}
\rightarrow -|e{\rangle}</math>.
If such a flip happens with probability <math>(1-e^{-\lambda t})/2</math> at each
moment in time, then the density matrix for this evolution is thus
:<math>\begin{array}{rcl}  
	\rho(t) &=& \frac{1+e^{-\lambda t}}{2}
		  \left[    { a|g \rangle  + b|e \rangle  } \right] 
			  \left[    {  \langle g|a^* +  \langle e| b^* } \right] 
		+ \frac{1-e^{-\lambda t}}{2}
		  \left[    { a|g \rangle  - b|e \rangle  } \right] 
			  \left[    {  \langle g|a^* -  \langle e| b^* } \right] 
\\
&=&  \left[	\begin{array}{cc}{|a|^2}&{ab^* e^{-\lambda t}}\\{a^*b e^{-\lambda t}}&{|b|^2}\end{array}\right] 
\,.
\end{array}</math>
This is again the same density matrix dynamics as previously obtained
for the random phase noise and elastic collision models.  However, the
trajectories of individual evolutions is different; at each moment in
time, the two-level atom either remains completely unchanged, or
its excited state flips sign, inverting its dipole moment:

::[[Image:Unraveling_quantum_open_system_dynamics-sim3.png|thumb|306px|none|]]

Again, here is an animation of seven sample trajectories:
<blockquote>
<jwplayer width="700" height="820" repeat="true" displayheight="800" 
image="http://feynman.mit.edu/8.422/sim3b.png"
autostart="false">http://feynman.mit.edu/8.422/sim3b.flv</jwplayer>
</blockquote>

==== Discussion ====
We have seen three models of phase damping, all of which produce the
same density matrix evolution, but each of which has very different
microscopic trajectories for individually evolving systems.
Which is correct?

The answer is that all of them are correct, and yet none are.  Any of
the three can be used for physical intuition and interpretation, but
only as long as the only conclusions drawn depend on statistical
averages.  In fact, in the absence of control over the environment,
no experiment can distinguish between phase damping processes
described by these three models, even in principle.  This strong
statement arises from the fact that there are an infinite number of
ways a (mixed) density matrix can be written as a statistical mixture
of pure states; correspondingly, there are an infinite number of
"unravelings" of density matrix time evolutions, into statistical
evolutions of pure state wavefunctions.

The freedom of interpretation which arises in studying
decoherence processes arises from a unitary degree of freedom.  For
example, in the QMCWF model, a gedankenexperiment is performed, in
which the atom is allowed to decay and the emitted photon is captured.
The evolution of the atom cannot depend on what measurement basis is
used for the photodetection.  This basis choice is a unitary transform
which can be chosen arbitrarily in the gedankenexperiment, and
different choices lead to the different unravelings of the master
equation into trajectories.

=== References ===

[[Category:Optical Bloch Equations]]

* <refbase>5239</refbase>
* <refbase>5249</refbase>
* <refbase>5250</refbase>
* <refbase>5251</refbase>