# Kaczmarz Projection Algorithms in Moving Window : Performance Improvement
via Extended Orthogonality & Forgetting

Alexander Stotsky
Department of Electrical Engineering
Chalmers University of Technology
Gothenburg SE - 412 96, Sweden
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

New Kaczmarz algorithms with rank two gain update, extended orthogonality
property and forgetting mechanism which includes both exponential and
instantaneous forgetting (implemented via a proper choice of the forgetting
factor and the window size) are introduced and associated in this report with
well-known Kaczmarz algorithms with rank one update.

_Keywords_ Kaczmarz Algorithms in Weighted Moving Window $\cdot$ Exponential &
Instantaneous Forgetting $\cdot$ Updating & Downdating, Rank Two Update Versus
Rank One Update $\cdot$ Extended Orthogonality Property

##  1 Previous & Related Work

Kaczmarz projection method, [1], [2] is robust and computationally efficient
algorithm (alternative to the least squares approach) for estimation of the
frequency contents of the oscillating signals in a number of signal processing
and system identification applications. In the Kaczmarz algorithm
$\displaystyle\theta_{k}=\theta_{k-1}-\frac{\varphi_{k}}{\varphi^{T}_{k}\leavevmode\nobreak\
\varphi_{k}}\leavevmode\nobreak\ [\varphi^{T}_{k}\leavevmode\nobreak\
\theta_{k-1}-y_{k}]$ the output of the model
$\varphi^{T}_{k}\leavevmode\nobreak\ \theta_{k}$ matches exactly the measured
signal $y_{k}=\varphi^{T}_{k}\leavevmode\nobreak\ \theta_{k}$ in each discrete
step $k$, which can be obtained by multiplying the parameter vector
$\theta_{k}$ by the regressor $\varphi^{T}_{k}$. The algorithm has the
remarkable orthogonality property $\varphi_{k}^{T}\leavevmode\nobreak\
\tilde{\theta}_{k}=0$, where the vector of the parameter mismatch
$\tilde{\theta}_{k}=\theta_{k}-\theta_{*}$ (difference between adjustable and
true parameters $\theta_{*}$) is orthogonal to the regressor vector. The
square of the length of the harmonic regressor (which consists of the
trigonometric functions at different frequencies),
$\varphi^{T}_{k}\leavevmode\nobreak\ \varphi_{k}$ is constant and depends on
the number of frequencies only. The Kaczmarz algorithms were successfully
tested in different estimation problems in automotive applications, [3].
Aiming for convergence rate improvement of the Kaczmarz projection method the
gain update algorithm was introduced in [4]. Further development was
associated with introduction of the forgetting factor which has significant
impact on the estimation performance, discounts exponentially old measurements
in the gain update and creates a virtual moving window, [5].
The choice of the forgetting factor is associated with the trade-off between
rapidity and accuracy of estimation. The gain matrix with forgetting factor is
associated in this paper with the window which is moving in time for further
improvements. Introduction of the forgetting factor in the sliding window,
[6],[7] creates extended forgetting mechanism that includes both exponential
and instantaneous forgetting and provides new opportunities for achievement of
the trade-off between rapidity and accuracy.
The movement of the window is associated with data updating and downdating
that results in recursive updates of the information matrix, which can occur
sequentially, [3] , [8] or simultaneously, [9]. Sequential updating and
downdating results in computationally complex algorithms. The gain update in
known Kaczmarz algorithms, [4], [5] is associated with recursive rank one
update, (similar to recursive least squares algorithms, [10], [11]) whereas
simultaneous updating/downdating is associated with computationally efficient
rank two update, [12].
New gain update which is explicitly associated with the movement of the
sliding window of the size $w$ with exponential forgetting is introduced in
Kaczmarz algorithms. The parameter update law extends the property of
orthogonality where the output of the model matches measured data in two end
points of the weighted window. The development is performed and systematically
associated with well-known Kaczmarz algorithms with rank one update, [5].

##  2 Kaczmarz Algorithms in the Moving Window with Exponential Forgetting

Suppose that the measured oscillating signal is presented in the following
form ${y}_{k}=\varphi_{k}^{T}\theta_{*}$, where $\varphi_{k}$ is the harmonic
regressor, $\varphi_{k}^{T}=[cos(q_{0}k)\leavevmode\nobreak\
sin(q_{0}k)\leavevmode\nobreak\ ...\leavevmode\nobreak\
cos(q_{h}k)\leavevmode\nobreak\ sin(q_{h}k)]$, $q_{0},...q_{h}$ are the
frequencies and $\theta_{*}$ is the vector of unknown parameters, $k=1,2,...$.
The window of the size $w$ which is moving in time can be presented as rank
two update of the information matrix (which is defined as the sum of the outer
products of the regressor vectors) where new observation is added (updating)
and old observation is deleted (downdating). The estimation performance is
highly influenced by the forgetting factor, $0<\lambda\leq 1$ which discounts
exponentially old measurements and creates a virtual window inside of the
moving window. In other words, the new data $\varphi_{k}$ (with the largest
forgetting factor which is equal to one) enter the window and the data with
the lowest priority
$\tilde{\varphi}_{k-w}=\sqrt{\lambda^{w}}\leavevmode\nobreak\ \varphi_{k-w}$
leave the window in step $k$ as follows, [12]:

$\displaystyle A_{k}$
$\displaystyle=\sum_{j=k-(w-1)}^{j=k}\lambda^{k-j}\varphi_{j}\leavevmode\nobreak\
\varphi_{j}^{T}=\lambda\leavevmode\nobreak\ A_{k-1}+Q_{k}\leavevmode\nobreak\
D\leavevmode\nobreak\ Q^{T}_{k}$ (1)

where $Q_{k}=[\varphi_{k}\leavevmode\nobreak\ \tilde{\varphi}_{k-w}]$,
$D=diag[1,-1]$, $k\geq w+1$.
The gain matrix (as inverse of $A_{k}$) which is derived by application of the
matrix inversion lemma, [15] to the identity (1) and the parameter update law
can be written in the following form, [12], [20]:

$\displaystyle\Gamma_{k}=\frac{1}{\lambda}\leavevmode\nobreak\
[\leavevmode\nobreak\ \Gamma_{k-1}-\Gamma_{k-1}\leavevmode\nobreak\
Q_{k}\leavevmode\nobreak\ S^{-1}\leavevmode\nobreak\
Q^{T}_{k}\leavevmode\nobreak\ \Gamma_{k-1}\leavevmode\nobreak\ ]$ (2)
$\displaystyle\theta_{k}=\theta_{k-1}-\Gamma_{k-1}\leavevmode\nobreak\
Q_{k}\leavevmode\nobreak\ [Q^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\leavevmode\nobreak\ Q_{k}]^{-1}\leavevmode\nobreak\
[Q^{T}_{k}\leavevmode\nobreak\ \theta_{k-1}-\tilde{y}_{k}]$ (3)

where $S=\lambda\leavevmode\nobreak\ D+Q^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\leavevmode\nobreak\ Q_{k}$,
$\tilde{y}^{T}_{k}=[y_{k}\leavevmode\nobreak\ \sqrt{\lambda^{w}}y_{k-w}]$ is
the synthetic output, provided that the matrix $Q^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\leavevmode\nobreak\ Q_{k}$ is invertible. The vector of
adjustable parameters $\theta_{k}$ in (3) is constructed so that the output of
the model matches exactly the measured signal $Q^{T}_{k}\leavevmode\nobreak\
\theta_{k}=\tilde{y}_{k}$ (see Section 1) and the following extended
orthogonality property $Q^{T}_{k}\leavevmode\nobreak\ \tilde{\theta}_{k}=0$ is
valid for the parameter mismatch $\tilde{\theta}_{k}=\theta_{k}-\theta_{*}$.

##  3 Kaczmarz Algorithms with Rank One Update

Introduction of the forgetting factor allows to establish relationship between
Kaczmarz algorithms with rank two and rank one updates. Notice that $Q_{k}$
gets the following form $Q_{k}=[\varphi_{k}\leavevmode\nobreak\ 0]$ without
downdating when $\lambda^{w}\to 0$. Straightforward substitution of $Q_{k}$
defined above in (2) yields to following well-known Kaczmarz algorithms,
[4],[5] :

$\displaystyle\Gamma_{k}$ $\displaystyle=\frac{1}{\lambda}\leavevmode\nobreak\
[\leavevmode\nobreak\ \Gamma_{k-1}-\frac{\Gamma_{k-1}\leavevmode\nobreak\
\varphi_{k}\leavevmode\nobreak\ \varphi^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\leavevmode\nobreak\ }{\lambda+\varphi^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\varphi_{k}}\leavevmode\nobreak\ ]$ (4) $\displaystyle\theta_{k}$
$\displaystyle=\theta_{k-1}-\frac{\Gamma_{k-1}\leavevmode\nobreak\
\varphi_{k}}{\varphi^{T}_{k}\leavevmode\nobreak\
\Gamma_{k-1}\varphi_{k}}\leavevmode\nobreak\ \leavevmode\nobreak\
[\varphi^{T}_{k}\theta_{k-1}-y_{k}]$ (5)

with the orthogonality property $\varphi^{T}_{k}(\theta_{k}-\theta_{*})=0$.

Detailed description and comparisons of the Kaczmarz algorithms with rank two
and rank one updates together with their convergence properties are presented
in [20] :

Stotsky A., Kaczmarz Projection Algorithms with Rank Two Gain Update, J Sign
Process Syst, 2024.
https://doi.org/10.1007/s11265-024-01915-w

where the performance of new algorithms is also examined in the problem of
estimation of the grid events in the presence of significant harmonic
emissions.

##  4 Conclusion

The gain matrix with rank two update which allows exponential weighting of the
data inside of the moving window, prioritizes recent measurements and improves
estimation performance for fast varying changes of the signal formed the basis
for new Kaczmarz algorithms. New Kaczmarz adjustment law has extended
orthogonality property in the sliding window. The enhancement of the
performance was achieved via a proper choice of two adjustable parameters
(small window size and forgetting factor) associated with fast forgetting.
The results are valid under the assumption associated with invertibility of
the matrix $Q^{T}_{k}\leavevmode\nobreak\ \Gamma_{k-1}\leavevmode\nobreak\
Q_{k}$ which can be seen as drawback of proposed algorithm. For the sake of
robustness the Kaczmarz algorithms with rank two update can be modified. The
modification can be associated with modification of well-known Kaczmarz
algorithm, [1].

## References

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https://staff.uz.zgora.pl/acegiels/Publications-Kaczmarz-method.pdf

  * [3] Stotsky A., Automotive Engines: Control, Estimation, Statistical Detection, Springer-Verlag, Berlin-Heidelberg, 2009.
  * [4] Avedyan, E. D. Modified Kaczmarz Algorithms for Estimating the Parameters of Linear Plants, Autom. Remote Control, 1978, vol. 39, N5, pp. 674–680.
  * [5] Stotsky A. A New Frequency Domain System Identification Method, Proc. IMechE Part I: Journal of Systems and Control Engineering, vol. 226, N 1,2012, pp. 111-124.
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  * [12] Stotsky A., Recursive Estimation in the Moving Window: Efficient Detection of the Distortions in the Grids with Desired Accuracy, Journal of Advances in Applied and Computational Mathematics, vol. 9, 2023, pp. 181-191.
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  * [14] Stotsky A., Simultaneous Frequency and Amplitude Estimation for Grid Quality Monitoring : New Partitioning with Memory Based Newton-Schulz Corrections 2022 ; IFAC PapersOnLine 55(9): 42-47.
  * [15] Hager W., Updating the Inverse of a Matrix, SIAM Review, vol. 31, N 2, 1989, pp. 221-239.
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  * [20] Stotsky A., Kaczmarz Projection Algorithms with Rank Two Gain Update, J Sign Process Syst, 2024.
https://doi.org/10.1007/s11265-024-01915-w


# Measurement-based Feedback Control of a Quantum System in a Harmonic
Potential

Amy Rouillard  School of Chemistry and Physics, University of KwaZulu-Natal,
Private Bag X54001, Durban 4000, South Africa  Anirudh Reddy  School of
Chemistry and Physics, University of KwaZulu-Natal, Private Bag X54001, Durban
4000, South Africa  Humairah Bassa  School of Chemistry and Physics,
University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
Shamik Maharaj  School of Chemistry and Physics, University of KwaZulu-Natal,
Private Bag X54001, Durban 4000, South Africa  Lajos Diósi  Wigner Research
Centre for Physics, H-1525 Budapest, Hungary  Department of Physics of Complex
Systems, Eötvös Loránd University, Pázmány Péter stny. 1/A, H-1117 Budapest,
Hungary  Thomas Konrad<EMAIL_ADDRESS>School of Chemistry and Physics,
University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
National Institute of Theoretical and Computational Sciences (NITheCS),
KwaZulu-Natal, South Africa

###### Abstract

We present a formulation of measurement-based feedback control of a single
quantum particle in one spatial dimension. An arbitrary linear combination of
the position and momentum of the particle is continuously monitored, and
feedback proportional to the measured signal is used to control the system. We
derive a feedback master equation and discuss a general approach to computing
the steady-state solutions for arbitrary potentials. For a quantum harmonic
oscillator or a free particle, we show that it is possible to cool and confine
the system using feedback that simultaneously damps the measured observable
and its conjugate momentum, as well as compensates for noise introduced by the
measurement. In addition, we demonstrate that appropriate feedback adds a
quadratic term in the measured observable to the Hamiltonian of the system.
For the particular case of the harmonic potential, we describe the exact
dynamics of the system that can be cooled to the ground state. Moreover, we
provide an argument for the possibility to cool systems with arbitrary
potentials provided that the measurement is strong enough to localise the
particle on an interval smaller than the characteristic length scale of the
potential.

##  I Introduction

Recent developments in the fields of quantum computing and quantum meteorology
have resulted in a demand for protocols to control individual quantum
systems[1, 2, 3]. One paradigm of quantum control is measurement-based
feedback control in which the dynamics of a system are altered as information
received from continuous measurements is fed back into the system. For
example, in order to impede the movement of a particle a force proportional to
the measured velocity can be applied in the opposite direction to that of the
motion. Although a measurement inevitably changes the state of a quantum
system, feedback control can be achieved with weak or unsharp measurements
[4], which allow to gain information about a system without much disturbance.

In this paper we present the theory that describes the dynamics of a system
that is controlled by instantaneously feeding back the measurement signal at
each moment, which corresponds to the Markovian limit. This is done by
introducing a feedback Hamiltonian which depends on the stochastic measurement
signal. The possibility of continuous feedback control was studied
theoretically by Wiseman and Milburn[4] as well as experimentally in [5, 6,
7]. In particular, there are studies of the cooling of a single trapped ion
[5, 6], the cooling of an optically levitated nano-particle [7], and the
cooling of trapped atoms probed by off-resonant light [8].

Feedback presupposes a measurement step which necessarily introduces noise
into the process even if all classical noise sources are eliminated. In our
theory, this is manifest as a diffusion-like term in the master equation. The
beauty of direct quantum feedback is that it might be used to cancel the noise
which drives the mean values of the dynamical variables as shown in Section V.
This is possible because the noise in the measurement signal is the same noise
that drives the system. The result is that the steady-state of the feedback
master equation can have the same variances as the conditioned states since
all of the fluctuations in the mean values are overcome.

Here, we use the method developed by Diosi in [9] to express the state change
in terms of generators and in the ”co-moving” frame, that is the reference
frame in which the expectation values of both position and momentum are zero.
We show that this method can greatly reduce the complexity of computations
without additional approximation, as well as give insight into the type of
feedback necessary to achieve the desired control.

In order to illustrate the power of this method we consider the case that the
measured and feedback observables are an arbitrary linear combination of the
position and momentum. We derive a stochastic feedback master equation in
Section II and compute the corresponding steady-state in Section IV. In
addition, we show that appropriate feedback can be used to compensate a part
of the Hamiltonian of the system (Section V). In addition, for a harmonic
potential, we determine how feedback can confine the particle to the centre of
phase space in the laboratory frame, thereby reducing the energy of the
particle. This phenomenon is also discussed for other potentials.

##  II The Feedback Master Equation

Our feedback scheme relies on the use of information obtained from continuous
observation to manipulate the motion of the system, e.g. to cool or confine
it. Assuming the measured observable is associated with the operator
$\hat{M}$, the increment of the readout of the continuous measurement at a
certain instance is

$d\mathcal{M}=\langle\hat{M}\rangle dt+\tfrac{1}{\sqrt{\gamma}}dW$ (1)

where $\gamma$ is the measurement strength which determines the rate at which
information is extracted from the system. The first term is the expectation
value of the observable,
$\langle\hat{M}\rangle\equiv\mathrm{Tr}[\hat{M}\rho]$, defined with respect to
the density operator $\rho$ in the selective regime which represents the
current state of the system for a given measurement record. The second term
represents a contribution of Gaussian white noise ($dW$), modelling the
deviation of the measurement result from the expectation value of the
observable by a random Wiener process.

According to continuous measurement theory [4, 10], the state change of the
system, due to the measurement of the observable after a time step $dt$, is
given by the Ito stochastic differential equation

$\displaystyle d\rho=-$
$\displaystyle\tfrac{i}{\hbar}[\hat{H},\rho]dt-\tfrac{\gamma}{8}\left[\hat{M},\left[\hat{M},\rho\right]\right]dt$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2}\left\\{\hat{M}-\langle\hat{M}\rangle,\rho\right\\}\mathrm{d}W.$
(2)

The first term corresponds to the unitary evolution related to the Hamiltonian
$\hat{H}$ of the system. The second term refers to the dissipative dynamics
due to measurement and tends to diagonalize the density operator in the
eigenbasis of the measured observable, $\hat{M}$. The final term is
proportional to the Wiener increment, $dW$, and indicates an update of the
observer’s knowledge about the system according to a certain measurement
result, $d\mathcal{M}$. The Ito differential $dW$ satisfies the following
algebra:

$\displaystyle\langle dW\rangle_{st}$ $\displaystyle=0,$ $\displaystyle dWdW$
$\displaystyle=dt,$ $\displaystyle dW^{n}$ $\displaystyle=0\text{ for }n>2\,.$
(3)

The Wiener noise interment $dW$ at a given time $t$, points to the future and
is thus statistically independent of the state $\rho$ and the measurement
signal (1) at time $t$.

A simple Markovian feedback can be achieved by amplifying and feeding back the
measurement signal, via the Hamiltonian [11]

$\hat{H}_{\mathrm{fb}}(t)=\hat{F}\frac{d\mathcal{M}}{\delta t}\,,$ (4)

where $\hat{F}$ represents an hermitian operator, and $\delta t$ is a
sufficiently small time interval during which the feedback is implemented by
means of the unitary time evolution

$U=\exp\left[-\frac{i}{\hbar}\hat{F}d\mathcal{M}\right]\,.$ (5)

Although Markovian feedback is not as powerful as Bayesian feedback [12],
which uses the best estimate of the system state, it is theoretically and
experimentally simpler to implement [11, 13]. Realistic feedback is not
Markovian, but has some time delay. For the Markovian approximation to be
satisfied, the time delay between measurement and feedback should be small
compared to the time scale on which the system evolves. We point out that it
is possible to implement feedback without delay by incorporating it into the
interaction with a probe system, that also realises the measurement [14].

A system undergoing continuous observation and Hamiltonian evolution changes
as $\rho\rightarrow\rho+d\rho$, according to Eq. (2). Adding feedback,
described by the unitary operator given in Eq. (5), causes the system to
evolve as $\rho\rightarrow U(\rho+d\rho)U^{\dagger}$. The resulting stochastic
master equation for the density operator of the system is thus given by

$\displaystyle d\rho=$
$\displaystyle-\frac{i}{\hbar}\left[\hat{H},\rho\right]dt-\frac{\gamma}{8}\left[\hat{M},\left[\hat{M},\rho\right]\right]dt$
$\displaystyle-\frac{1}{2\hbar^{2}\gamma}\left[\hat{F},\left[\hat{F},\rho\right]\right]dt-\frac{i}{2\hbar}\left[\hat{F},\left\\{\hat{M},\rho\right\\}\right]dt$
$\displaystyle+\frac{\sqrt{\gamma}}{2}\left\\{\hat{M}-\langle\hat{M}\rangle,\rho\right\\}dW-\frac{i}{\hbar\sqrt{\gamma}}\left[\hat{F},\rho\right]dW\,,$
(6)

and has a Lindblad structure. The complete derivation is given in Appendix A.
This expression also appears in [4, 15]. The first, second and fifth terms of
this equation describes the free evolution and measurement back-action, while
the remaining terms include the effects of the feedback loop. The fourth term
causes the expected damping while the third term leads to diffusion in the
observable conjugate to the generator of feedback $\hat{F}$. The diffusion
occurs as a result of the noise in the measurement signal, which is also
amplified and fed back to the system.

##  III Determining the stationary state using the co-moving frame

Here we follow the method of Diosi [9] in order to derive the stationary-state
solution. Accordingly, we use the coordinate free representation and assume
the state of the system to be a pure state $\ket{\psi}$. We can thus write the
feedback equation for the state vector in the equivalent form

$\displaystyle d\ket{\psi}=$
$\displaystyle\Big{\\{}\Big{[}-\tfrac{i}{\hbar}\hat{H}-\tfrac{\gamma}{8}\left(\hat{M}-\langle\hat{M}\rangle\right)^{2}$
$\displaystyle\quad-\tfrac{1}{2\hbar^{2}\gamma}\hat{F}^{2}-\tfrac{i}{2\hbar}\hat{F}\left(\hat{M}+\langle\hat{M}\rangle\right)\Big{]}dt$
$\displaystyle+\left[\tfrac{\sqrt{\gamma}}{2}(\hat{M}-\langle\hat{M}\rangle)-\tfrac{i}{\hbar\sqrt{\gamma}}\hat{F}\right]dW\Big{\\}}\ket{\psi}\,.$
(7)

It is possible to formulate this non-deterministic state change in terms of a
generator as

$\displaystyle\ket{\psi}+d\ket{\psi}=\exp(\hat{G})\ket{\psi}\,,$ (8)

where

$\displaystyle\hat{G}:=$
$\displaystyle\Big{[}-\tfrac{i}{\hbar}\hat{H}-\tfrac{\gamma}{4}(\hat{M}-\langle\hat{M}\rangle)^{2}+\tfrac{i}{4\hbar}\left[\hat{M},\hat{F}\right]-\tfrac{i}{\hbar}\langle\hat{M}\rangle\hat{F}\Big{]}dt$
$\displaystyle\,+\Big{[}\tfrac{\sqrt{\gamma}}{2}(\hat{M}-\langle\hat{M}\rangle)-\tfrac{i}{\hbar\sqrt{\gamma}}\hat{F}\Big{]}dW\,.$
(9)

Up to this point we have not placed any restrictions on the Hamiltonian,
$\hat{H}$, the measured observable, $\hat{M}$, or the generator of feedback,
$\hat{F}$. Equation (8) is therefore valid for any system undergoing
continuous measurement of an observable $\hat{M}$ and feedback of the form (5)
with hermitian feedback operator $\hat{F}$.

As outlined by Diosi in [9], we now consider the dynamics of the system in the
co-moving frame, i.e. the reference frame in which the expectation values of
both position and momentum are zero. For this purpose, we transform the state
$\ket{\psi}+d\ket{\psi}$ at time $t+dt$ into the co-moving frame as follows

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}=e^{-\frac{i}{\hbar}(\langle\hat{p}\rangle+d\langle\hat{p}\rangle)\hat{q}}e^{\frac{i}{\hbar}(\langle\hat{q}\rangle+d\langle\hat{q}\rangle)\hat{p}}\left(\ket{\psi}+d\ket{\psi}\right)\,,$
(10)

where $d\langle\hat{q}\rangle=\mathrm{Tr}(\hat{q}d\rho)$ and
$d\langle\hat{p}\rangle=\mathrm{Tr}(\hat{p}d\rho)$. Transformation (10)
resembles feedback however, it is cannot be realised in a laboratory and
therefore does not represent standard feedback, rather a purely mathematical
operation. As a consequence of Eq. (10), the expectation value of position
with respect to the transformed state at time a $t+dt$ vanishes:

$\displaystyle(\bra{\tilde{\psi}}+d\bra{\tilde{\psi}})\hat{q}(\ket{\tilde{\psi}}+d\ket{\tilde{\psi}})$
$\displaystyle=(\bra{\psi}+d\bra{\psi})e^{-\tfrac{i}{\hbar}(\langle\hat{q}\rangle+d\langle\hat{q}\rangle)\hat{p}}\hat{q}e^{\tfrac{i}{\hbar}(\langle\hat{q}\rangle+d\langle\hat{q}\rangle)\hat{p}}(\ket{\psi}+d\ket{\psi})$
$\displaystyle=(\bra{\psi}+d\bra{\psi})\left[\hat{q}-(\langle\hat{q}\rangle+d\langle\hat{q}\rangle)\right](\ket{\psi}+d\ket{\psi})$
$\displaystyle=0\,.$

Similarly the expectation value of momentum,
$(\bra{\tilde{\psi}}+d\bra{\tilde{\psi}})\hat{p}(\ket{\tilde{\psi}}+d\ket{\tilde{\psi}})$
equals zero. The state $\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$ therefore
represents the state of the system viewed from the co-moving reference frame.

It is always possible to choose a special coordinate system where
$\langle\hat{q}\rangle=0=\langle\hat{p}\rangle$ at the given instant $t$ when
$\ket{\psi}$ is considered. At this instant $\ket{\psi}$ coincides with
$\ket{\tilde{\psi}}$ and it follows that the state change with respect to this
coordinate system reads

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}=$ $\displaystyle
e^{-\frac{i}{\hbar}d\langle\hat{p}\rangle\hat{q}}e^{\frac{i}{\hbar}d\langle\hat{q}\rangle\hat{p}}e^{\hat{G}}\ket{\tilde{\psi}}\,.$
(11)

Note, that we have not placed any restrictions on the Hamiltonian, $\hat{H}$,
measured observable, $\hat{M}$, or the generator of feedback, $\hat{F}$.
Therefore, by replacing $q$ and $p$ in Eq. (11) with any pair of conjugate
variables, we can shift the description of the system into a reference frame
with respect to which the expectation values of these conjugate variables
vanish simultaneously. Using the measured observable $M$ and its conjugate
momentum as the conjugate variables, greatly simplifies the calculation of the
stationary states as we shall see in Section IV. This simplification is the
essence of Diosi’s method, and the form invariance of Eq. (11) under canonical
transformations makes it applicable to compute the stationary states for a
large class of continuous measurements with feedback.

##  IV Stationary state of a particle in a harmonic potential

In order to illustrate the power of this generalised approach we consider the
specific case that the measured observable is an arbitrary linear combination
of position and momentum,

$\displaystyle\hat{M}=\alpha\hat{q}+\beta\hat{p}\,,$ (12)

where $\alpha$ and $\beta$ are real constants of units $1$ and
${\mathrm{kg}}^{-1}\text{\,}\mathrm{s}$, respectively. Furthermore, let the
generator of feedback be

$\displaystyle\hat{F}=\chi\hat{q}+\delta\hat{p}$ (13)

where $\chi$ and $\delta$ are real gain factors of units
$\mathrm{k}\mathrm{g}\mathrm{s}^{-2}$ and ${\mathrm{s}}^{-1}$, respectively.
In order to simplify the calculations that follow, we introduce the following
operators

$\displaystyle\hat{Q}$ $\displaystyle:=\hat{M}=\alpha\hat{q}+\beta\hat{p}$
(14) $\displaystyle\hat{P}$
$\displaystyle:=-\beta^{\prime}\hat{q}+\alpha^{\prime}\hat{p}$ (15)

that satisfy the canonical commutation relation, $[\hat{Q},\hat{P}]=i\hbar$,
where for simplicity we assume that the factor
${\alpha\alpha^{\prime}+\beta\beta^{\prime}}=1$. This assumption also implies
that $Q$ and $P$ are canonically conjugate. In addition, we can write

$\displaystyle\hat{F}=u\hat{Q}+v\hat{P}\,,$ (16)

where $u$ and $v$ are real factors of units
$\mathrm{k}\mathrm{g}\mathrm{s}^{-2}$ and ${\mathrm{s}}^{-1}$, respectively.
The transition to the co-moving frame can also be achieved by replacing
variables $q$ ($p$) in transformation (10) by the conjugate pair $Q$ and $P$
since the expectation values of position and momentum vanish if, and only if,
those of the generalised variables $Q$ and $P$ vanish.

As an example, let us consider a particle in a harmonic potential with
Hamiltonian

$\displaystyle\hat{H}=\frac{1}{2m}\hat{p}^{2}+\frac{m\omega^{2}}{2}\hat{q}^{2}\,.$
(17)

If $\alpha^{\prime}=\alpha$ and $\beta^{\prime}=m^{2}\omega^{2}\beta$, the
Hamiltonian of a harmonic oscillator is form invariant under linear
transformations of this type, so that

$\displaystyle\hat{H}=\frac{1}{2m}\hat{P}^{2}+\frac{m\omega^{2}}{2}\hat{Q}^{2}\,.$
(18)

We now show that the transformation (11) for a particle in a harmonic
potential leads to a unique solution in the stationary regime. Applying the
Baker-Campbell-Hausdorff formula and the Ito rule (3) to Eq. (11) results in a
state change in the co-moving frame given by

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}=$
$\displaystyle\exp\Big{\\{}\left[-\tfrac{i}{\hbar}\hat{{H}}-\tfrac{\gamma}{4}\left(\hat{Q}^{2}-\langle\hat{Q}^{2}\rangle\right)\right]dt$
$\displaystyle\hskip
18.0pt-\tfrac{i}{\hbar}\sqrt{\tfrac{\gamma}{2}}\Big{[}\left(i\hbar+\langle\\{\hat{Q},\hat{P}\\}\rangle\right)\hat{Q}-2\langle\hat{Q}^{2}\rangle\hat{P}\Big{]}dW\Big{\\}}\ket{\tilde{\psi}},$
(19)

where we ignored irrelevant phase factors and assumed Hamiltonian (18). The
full derivation and the general state change for arbitrary Hamiltonian are
provided in Appendix B.

We note that Eq. (19) no longer depends on feedback. This is to be expected as
feedback of form (5) corresponds to a shift of the Wigner function [16] in
phase space, and consequently its shape remains unchanged. Therefore, from the
point of view of the co-moving frame, feedback has no impact on the state of
the quantum particle. We note that if the generator of feedback $\hat{F}$
contains terms of higher orders of position and momentum, for example angular
momentum, then this is no longer true. However, this method still leads to a
simplified state change in the co-moving frame since all first-order terms in
Q and P are eliminated.

The stationary solutions are of the form
$\ket{\tilde{\psi}_{\infty}}\exp(-iEt/\hbar)$, where
$\ket{\tilde{\psi}_{\infty}}$ is the time-independent part and $E$ is a real
number with units of energy. Substitution of this ansatz into Eq. (19)
followed by a comparison of coefficients of the independent increments $dt$
and $dW$ yields

$\displaystyle\left[\hat{{H}}-i\tfrac{\hbar\gamma}{4}\left(\hat{Q}^{2}-\langle\hat{Q}^{2}\rangle_{\infty}\right)\right]\ket{\tilde{\psi}_{\infty}}=E\ket{\tilde{\psi}_{\infty}}\,,$
(20)

such that
$E=\braket{\tilde{\psi}_{\infty}}{\hat{H}}{\tilde{\psi}_{\infty}}\equiv\langle\hat{H}\rangle_{\infty}$,
and

$\displaystyle\Big{[}\left(i\hbar+\langle\\{\hat{Q},\hat{P}\\}\rangle_{\infty}\right)\hat{Q}-2\langle\hat{Q}^{2}\rangle_{\infty}\hat{P}\Big{]}\ket{\tilde{\psi}_{\infty}}=0\,.$
(21)

If we introduce the wave function in position representation
$\tilde{\psi}(q)$, Eq. (21) becomes a first-order linear ordinary differential
equation of the form $aq\tilde{\psi}(q)={d\tilde{\psi}(q)}/{dq}$, where $a$ is
a complex constant, namely

$\displaystyle\big{(}i\hbar\alpha+\langle\\{\hat{Q},\hat{p}\\}\rangle_{\infty}$
$\displaystyle\big{)}{q}\tilde{\psi}_{\infty}(q)$
$\displaystyle=i\hbar\big{(}i\hbar\beta-\langle\\{\hat{Q},\hat{q}\\}\rangle_{\infty}\big{)}\frac{d\tilde{\psi}_{\infty}(q)}{dq}\,.$
(22)

This differential equation is known to have a Gaussian solution and it follows
that the stationary solution in position representation is given by

$\displaystyle\tilde{\psi}_{\infty}(q)=$
$\displaystyle\left(\frac{\operatorname{Re}\\{\sigma^{2}\\}}{2\pi|\sigma^{2}|^{2}}\right)^{\frac{1}{4}}\exp\left(-\frac{q^{2}}{4\sigma^{2}}\right)\,,$
(23)

where

$\displaystyle\sigma^{2}$
$\displaystyle:=\frac{i\hbar}{2}\frac{\langle\\{\hat{q},\hat{Q}\\}\rangle_{\infty}-i\hbar\beta}{\langle\\{\hat{p},\hat{Q}\\}\rangle_{\infty}+i\hbar\alpha}\,,$
(24)

and similarly in momentum space

$\displaystyle\tilde{\psi}_{\infty}(p)=$
$\displaystyle\left(\frac{2\operatorname{Re}\\{\sigma^{2}\\}}{\pi\hbar^{2}}\right)^{\tfrac{1}{4}}\exp\left(-\frac{\sigma^{2}p^{2}}{\hbar^{2}}\right)\,.$
(25)

The eigenvalue equation (21), and equivalently Eq. (22), holds for arbitrary
Hamiltonian $\hat{H}$, see Eq. (82). Therefore, one could ask the question
whether arbitrary Hamiltonians allow the energy eigenvalue equation and
eigenvalue equation (21), that follow from Eq. (82), to be simultaneously
satisfied. For the free particle it is possible to use the results of [9] to
show that both eigenvalue equations hold. We briefly argue that this is also
the case for a general Hamiltonian however, the analysis is be beyond the
scope of this paper.

For the harmonic oscillator we can show explicitly that these Gaussian
stationary wave functions also satisfy the energy eigenvalue equation (20) for
all measurements strengths $\gamma$, see Appendix C. Proving this requires us
to express the stationary value of the variance of the measured observable in
the co-moving frame, $\langle\hat{Q}^{2}\rangle_{\infty}$, as a function of
the expectation value of the anti-commutator
$\langle\\{\hat{Q},\hat{P}\\}\rangle_{\infty}$ and the variance
$\langle\hat{P}^{2}\rangle_{\infty}$. The calculation of these terms is the
subject of the next section. It turns out that under certain circumstances the
quantities $\langle\hat{Q}^{2}\rangle$, $\langle\hat{P}^{2}\rangle$ and
$\langle\\{\hat{Q},\hat{P}\\}\rangle$ obey a set of coupled first-order
differential equations and thus evolve deterministically. Further, in Appendix
C it is shown that for a particle in a harmonic potential the energy
eigenvalue in Eq. (20) is given by

$\displaystyle E=$
$\displaystyle\frac{\hbar^{2}}{4m\langle\hat{Q}^{2}\rangle_{\infty}}\,.$ (26)

It also turns out that in the weak measurement limit, $\gamma\rightarrow 0$,
the energy with respect to the co-moving frame is minimal, i.e.,
$\langle\hat{Q}^{2}\rangle_{\infty}\rightarrow\frac{\hbar}{2m\omega}$. In
addition, the product
$\langle\hat{Q}^{2}\rangle_{\infty}\langle\hat{P}^{2}\rangle_{\infty}\rightarrow\frac{\hbar^{2}}{4}$
so that $E\rightarrow\langle\hat{P}^{2}\rangle_{\infty}/m$, which is twice the
expectation value of the kinetic energy in agreement with the Virial theorem.

###  IV.1 Calculation of stationary widths for a particle in a harmonic
potential

From the expressions of second-order moments of position and momentum a system
of differential equations can be derived that allows to calculate the width of
the asymptotic wave function,

$\displaystyle d\langle\Delta\hat{q}^{2}\rangle=$ $\displaystyle
d\langle\hat{q}^{2}\rangle-2\langle\hat{q}\rangle
d\langle\hat{q}\rangle-(d\langle\hat{q}\rangle)^{2}$ (27) $\displaystyle
d\langle\Delta\hat{p}^{2}\rangle=$ $\displaystyle
d\langle\hat{p}^{2}\rangle-2\langle\hat{p}\rangle
d\langle\hat{p}\rangle-(d\langle\hat{p}\rangle)^{2}$ (28) $\displaystyle
d\langle\\{\Delta\hat{q},\Delta\hat{p}\\}\rangle=$ $\displaystyle
d\langle\\{\hat{q},\hat{p}\\}\rangle-2\langle\hat{q}\rangle
d\langle\hat{p}\rangle-2\langle\hat{p}\rangle d\langle\hat{q}\rangle$
$\displaystyle-2d\langle\hat{q}\rangle d\langle\hat{p}\rangle\,.$ (29)

Here we have introduced the notation
$\Delta\hat{o}=\hat{o}-\langle\hat{o}\rangle$, such that the variance of
$\hat{o}$ is given by
$\langle\Delta\hat{o}^{2}\rangle=\langle\hat{o}\rangle^{2}-\langle\hat{o}^{2}\rangle$.
Note that the last terms in Eqs. (27)-(29) take into account the stochastic
contribution $dW$ in the increment of the expectation value, see equation
below, the square of which leads to a contribution of order $dt$. The
increment $d\langle\hat{o}\rangle$ of an arbitrary operator $\hat{o}$ can be
calculated using the master equation (II),

$\displaystyle d\langle\hat{o}\rangle=$
$\displaystyle\mathrm{Tr}(\hat{o}d\rho)$ $\displaystyle=$
$\displaystyle\Big{[}-\tfrac{i}{\hbar}\langle[\hat{o},\hat{H}]\rangle-\tfrac{\gamma}{8}\langle[\hat{Q},[\hat{Q},\hat{o}]]\rangle$
$\displaystyle\quad-\tfrac{1}{2\hbar^{2}\gamma}\langle[\hat{F},[\hat{F},\hat{o}]]\rangle-\tfrac{i}{2\hbar}\langle\\{\hat{Q},[\hat{o},\hat{F}]\\}\rangle\Big{]}dt$
$\displaystyle+\Big{[}\tfrac{\sqrt{\gamma}}{2}\langle\\{\hat{o},\Delta\hat{Q}\\}\rangle-\tfrac{i}{\hbar\sqrt{\gamma}}\langle[\hat{o},\hat{F}]\rangle\Big{]}dW\,.$
(30)

Equations (27)-(29) are form invariant for any pair of conjugate variables. We
can thus compute these total differentials for $\hat{Q}$ and $\hat{P}$.

$\displaystyle d\langle(\Delta\hat{Q})^{2}\rangle=$
$\displaystyle-\gamma\langle(\Delta\hat{Q})^{2}\rangle^{2}dt$
$\displaystyle-\frac{i}{\hbar}\langle\\{\Delta\hat{Q},\Delta[\hat{Q},\hat{H}]\\}\rangle
dt$ $\displaystyle+{\sqrt{\gamma}}\langle(\Delta\hat{Q})^{3}\rangle dW$ (31)
$\displaystyle d\langle(\Delta\hat{P})^{2}\rangle=$
$\displaystyle\frac{\hbar^{2}\gamma}{4}dt-\frac{\gamma}{4}\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle^{2}dt$
$\displaystyle-\frac{i}{\hbar}\langle\\{\Delta\hat{P},\Delta[\hat{P},\hat{H}]\\}\rangle
dt$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2}\langle\\{(\Delta\hat{P})^{2},\Delta\hat{Q}\\}\rangle
dW$ (32) $\displaystyle d\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle=$
$\displaystyle-\frac{i}{\hbar}\langle\\{\Delta\hat{Q},\Delta[\hat{P},\hat{H}]\\}\rangle
dt$
$\displaystyle-\frac{i}{\hbar}\langle\\{\Delta\hat{P},\Delta[\hat{Q},\hat{H}]\\}\rangle
dt$
$\displaystyle-\gamma\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle\langle(\Delta\hat{Q})^{2}\rangle
dt$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2}\langle\\{\\{\Delta\hat{Q},\Delta\hat{P}\\},\Delta\hat{Q}\\}\rangle
dW\,.$ (33)

This set of coupled equations holds for all times $t$ and for an arbitrary
Hamiltonian, $H$.

For a particle in a harmonic potential and for large times
$t>>\frac{m\omega}{\hbar\gamma}$, we can use the fact that the stationary
states have a Gaussian form, see Section IV. This also applies for a free
particle and it can also be assumed in good approximation whenever the
particle is localised due to the measurement on length scales on which the
potential does not vary much [17]. In this case higher order moments are zero,
in particular

$\displaystyle\langle\Delta\hat{Q}^{3}\rangle_{\infty}$
$\displaystyle=\langle\\{(\Delta\hat{P})^{2},\Delta\hat{Q}\\}\rangle_{\infty}$
$\displaystyle=\langle\\{\\{\Delta\hat{Q},\Delta\hat{P}\\},\Delta\hat{Q}\\}\rangle_{\infty}=0\,.$
(34)

Assuming that the Hamiltonian $\hat{H}$ is that of the harmonic oscillator
(18), we note that $[\hat{Q},\hat{H}]=\tfrac{i\hbar}{m}\hat{P}$ and
$[\hat{P},\hat{H}]=-i\hbar m\omega^{2}\hat{Q}$ so that Eqs. (31)-(33) are
reduced to a system of linear first-order coupled differential equations which
can be written as

$\displaystyle\dot{x}$ $\displaystyle=-2\kappa x^{2}+z$ (35)
$\displaystyle\dot{y}$ $\displaystyle=\frac{\kappa}{2}\left(1-z^{2}\right)-z$
(36) $\displaystyle\dot{z}$ $\displaystyle=2\left(y-x\right)-2\kappa xz\,,$
(37)

where

$\displaystyle x:=$
$\displaystyle\frac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle$ (38)
$\displaystyle y:=$ $\displaystyle\frac{1}{\hbar
m\omega}\langle(\Delta\hat{P})^{2}\rangle$ (39) $\displaystyle z:=$
$\displaystyle\frac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle$
(40)

are the unitless versions of the variances of interest and

$\displaystyle\kappa$ $\displaystyle:=\frac{\hbar\gamma}{2m\omega^{2}}$ (41)

is the measurement strength relative to the characteristic dynamics of the
particle. The steps taken to arrive at this result are shown in Appendix D.
The coupled differential equations (35)-(37) can be solved numerically, with
an example of the numerically obtained solutions for specific initial
conditions shown in Figure 1. In addition, we note that Eqs. (35)-(37) imply

$\displaystyle\langle(\Delta\hat{Q})^{2}\rangle\langle(\Delta\hat{P})^{2}\rangle=\frac{\hbar^{2}}{4}+\frac{1}{4}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle^{2}\,,$
(42)

so that $\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle$ increases the
minimal uncertainty [18].

Figure 1: Time dependant solutions to the coupled differential equations (35),
(36) and (37) for $\kappa=1/4$. The initial conditions were
${\frac{m\omega}{\hbar}}\langle\hat{Q}^{2}\rangle=1/\sqrt{2}$ and
$\frac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle=1/2$, with
${\frac{1}{\hbar m\omega}}\langle\hat{P}^{2}\rangle=5/(8\sqrt{2})$ according
to Eq. (42).

In the stationary regime the time derivatives
$d\langle(\Delta\hat{Q})^{2}\rangle/dt$,
$d\langle(\Delta\hat{P})^{2}\rangle/dt$ and
$d\langle\\{\Delta\hat{Q},\hat{P}\\}\rangle/dt$ (equivalently $\dot{x}$,
$\dot{y}$ and $\dot{z}$) are zero, so that Eqs. (31)-(33) (Eqs. (35)-(37))
yield

$\displaystyle\frac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle_{\infty}$
$\displaystyle=\frac{1}{\kappa}\left(-1+\sqrt{1+\kappa^{2}}\right)$ (43)
$\displaystyle\frac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle_{\infty}$
$\displaystyle=\frac{1}{\sqrt{2}\kappa}\sqrt{-1+\sqrt{1+\kappa^{2}}}$ (44)
$\displaystyle\frac{1}{\hbar
m\omega}\langle(\Delta\hat{P})^{2}\rangle_{\infty}$
$\displaystyle=\frac{\sqrt{1+\kappa^{2}}}{\sqrt{2}\kappa}\sqrt{-1+\sqrt{1+\kappa^{2}}}\,.$
(45)

The above solutions are valid for all values of $\gamma$, $\alpha$, $\beta$,
$m$ and $\omega$ provided that the assumption
$\alpha^{2}+m^{2}\omega^{2}\beta^{2}=1$ holds. We note that the asymptotic
values depend on $m\omega$ and the dimensionless relative measurement strength
$\kappa$, and not on $m$, $\omega$ and $\gamma$ independently.

The relative measurement strength $\kappa$ is the product of the measurement
strength $\gamma$ and the two quantities that characterise the dimensions of
the system. These are the time scale on which the system oscillates
$1/\omega$, and the square length scale given by the variance
$\frac{\hbar}{2m\omega}$ of the position of the ground state of the harmonic
oscillator. If $\kappa>>1$, measurement is the dominant feature and the
particle follows a random walk. If, on the other hand, $\kappa<<1$ the
measurement is weak compared to the unitary dynamics of the system and the
harmonic motion of the particle is dominant. The asymptotic second-order
moments (43)-(45) are plotted as functions of the relative measurement
strength $\kappa$ in Figure 2, which shows the transition that occurs in the
regime $1<\kappa<10$ from a state of minimal uncertainty to a state of maximal
uncertainty that the system can assume.

In the weak measurement limit $\gamma\rightarrow 0$, i.e. $\kappa\rightarrow
0$, the variance of the measured observable approaches the variance of the
ground state,

$\displaystyle\langle(\Delta\hat{Q})^{2}\rangle_{\infty}\rightarrow\frac{\hbar}{2m\omega}\,.$
(46)

Therefore, according to Eq. (26), the energy in the co-moving frame converges
to the ground state energy $\hbar\omega/2$, as alluded to in the discussion of
Eq. (26). Values of $\kappa<1$ are experimentally feasible for systems such as
trapped ions and optical oscillators.

Figure 2: Plot showing the stationary value of the second-order variances as a
function of $\kappa$ according to Eqs. (43)-(45). The dashed grey line
represents the asymptotic uncertainty as a function of $\kappa$. A lower bound
for the asymptotic energy $E_{\text{LB}}$ is also shown and is determined by
the expectation value of the Hamiltonian with respect to the stationary state
in the co-moving frame (26). Dots indicate the stationary values associated
with $\kappa=0.25$ for comparison with Figure 1.

##  V Feedback for cooling and confinement

The results of the previous section are dictated by measurement only, since
the feedback vanishes from the equation of motion in the co-moving frame. Here
we consider the effect of feedback on the motion of the quantum particle
viewed from the lab frame and how it can be used to cool and confine the
particle.

At each moment in time, feedback changes the position and momentum of the
particle by an amount proportional to the measurement signal and according to
coefficients $\chi$ and $\delta$ in Eq. (13). By tuning these coefficients, we
can alter the motion of the particle in such a way as to mimic the action of a
quadratic potential. In general, this allows for the Hamiltonian of the system
to be altered by a term quadratic in the measured observable.

Feedback can also be used for the purpose of cooling and confinement.
Intuitively, information gained from measurement can be used to gradually
reduce the expectation value of the measured observable $Q$ by shifting the
system towards a smaller value of this observable. This is achieved by
continuously applying feedback generated by the conjugate momentum $P$ with a
negative scaling. At the same time, it is possible to choose this scaling such
that the random changes in $Q$ introduced by the measurement diminish over
time. The same can be achieved for the conjugate momentum $P$. This is
realised by introducing friction generated by $Q$ which simultaneously
compensates for random changes of the conjugate momentum $P$ generated by the
measurement of $Q$. The combined feedback thus leads to damping of both the
measured observable $Q$ and its conjugate momentum $P$. To determine the
values of the scaling coefficients $u$ and $v$ in Eq. (16) that lead to
optimal cooling and confinement, it is necessary to consider the equations
that govern the motion of the particle.

The stochastic master equation (II) in terms of the operators $\hat{Q}$ and
$\hat{P}$ reads

$\displaystyle d\rho=$
$\displaystyle-\tfrac{i}{\hbar}\left[\hat{H}+\tfrac{u}{2}\hat{Q}^{2},\rho\right]dt$
$\displaystyle-\tfrac{\gamma}{8}\left(1+\left(\tfrac{2u}{\hbar\gamma}\right)^{2}\right)\left[\hat{Q},\left[\hat{Q},\rho\right]\right]dt$
$\displaystyle-\tfrac{v^{2}}{2\hbar^{2}\gamma}\left[\hat{P},\left[\hat{P},\rho\right]\right]dt-\tfrac{iv}{2\hbar}\left[\hat{P},\left\\{\hat{Q},\rho\right\\}\right]dt$
$\displaystyle-\tfrac{uv}{2\hbar^{2}\gamma}\left(\left[\hat{Q},\left[\hat{P},\rho\right]\right]+\left[\hat{P},\left[\hat{Q},\rho\right]\right]\right)dt$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2}\left\\{\hat{Q}-\langle\hat{Q}\rangle,\rho\right\\}dW-\tfrac{i}{\hbar\sqrt{\gamma}}\left[u\hat{Q}+v\hat{P},\rho\right]dW\,,$
(47)

where we inserted $\hat{F}=u\hat{Q}+v\hat{P}$, and used the identity
$[\hat{Q},\\{\hat{Q},\rho\\}]=[\hat{Q}^{2},\rho]$ to obtain the first term in
the this equation. Eq. (V) shows that by choosing $u\neq 0$, the Hamiltonian
of the system can be altered by a quadratic term in the measured observable
$Q$. This is discussed in Section V.1.

Furthermore, according to Eq. (IV.1) the increments in the measured observable
and its conjugate momentum are given by

$\displaystyle d\langle\hat{Q}\rangle=$
$\displaystyle-\tfrac{i}{\hbar}\langle[\hat{Q},\hat{H}]\rangle
dt+v\langle\hat{Q}\rangle dt$
$\displaystyle+\sqrt{\tfrac{2\hbar\kappa}{m}}\left(\tfrac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle+\tfrac{1}{2\kappa}\tfrac{v}{\omega}\right)dW$
(48)

and

$\displaystyle d\langle\hat{P}\rangle=$
$\displaystyle-\tfrac{i}{\hbar}\langle[\hat{P},\hat{H}]\rangle dt-u\langle
Q\rangle dt$ $\displaystyle+\sqrt{\tfrac{\hbar
m\omega^{2}\kappa}{2}}\left(\tfrac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle-\tfrac{1}{\kappa}\tfrac{u}{m\omega^{2}}\right)dW\,,$
(49)

respectively. These equations indicate how to simultaneously eliminate the
stochastic terms. As discussed in Section V.2.2, the corresponding choice of
feedback turns out to be sufficient for cooling and confinement of a particle
in a harmonic potential. Furthermore, it is possible to drive the particle
into the ground state by combining this choice of feedback with weak
measurement.

###  V.1 Manipulation of the Hamiltonian

First we consider the implications on the system dynamics if we choose
feedback in order to alter the Hamiltonian of the system. If the generator of
feedback is a linear combination of position and momentum, see Eq. (13) and
Eq. (16), then feedback proportional to the measured observable $Q$ allows us
to simulate an additional quadratic potential in $Q$. This is implied by the
form of the first term on the right-hand side of Eq. (V).

In the case of a harmonic potential (18), we consider how feedback might be
used to compensate for a portion of the Hamiltonian. Let us assume that
$\hat{F}=-m\omega^{2}\hat{Q}$, i.e. $u=-m\omega^{2}$ and $v=0$. The master
equation (V) becomes

$\displaystyle d\rho=$
$\displaystyle-\tfrac{i}{\hbar}\left[\hat{H}-\tfrac{m\omega^{2}}{2}\hat{Q}^{2},\rho\right]dt$
$\displaystyle-\tfrac{\gamma}{8}\left(1+\left(\tfrac{2m\omega^{2}}{\hbar\gamma}\right)^{2}\right)\left[\hat{Q},\left[\hat{Q},\rho\right]\right]dt$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2}\left\\{\hat{Q}-\langle\hat{Q}\rangle,\rho\right\\}dW+\tfrac{im\omega^{2}}{\hbar\sqrt{\gamma}}\left[\hat{Q},\rho\right]dW\,.$
(50)

Here feedback cancels a portion of the Hamiltonian so that the effective
Hamiltonian $\hat{H}_{\text{eff}}$ is given by

$\displaystyle\hat{H}_{\text{eff}}$
$\displaystyle:=\hat{H}-\tfrac{m\omega^{2}}{2}\hat{Q}^{2}$
$\displaystyle=\tfrac{1}{2m}\hat{P}^{2}$ (51)

The master equation (V.1) resembles that of a free particle under continuous
position measurement, with an amplified decoherence term, as well as a random
Hamiltonian (cp. last term in the equation).

To take a particular example, let us continuously measure position, with
$\hat{Q}=\hat{q}$ and $\hat{P}=\hat{p}$ (i.e. $\alpha=1$ and $\beta=0$), so
that $\hat{H}_{\text{eff}}$ is the Hamiltonian of a free particle. According
to Eq. (V) and in the non-selective regime of measurement, the expectation
value of the position $q$ grows linearly proportional to the expectation value
of the conjugate momentum, while the expectation value of the momentum ${p}$
remains constant, according to Eq. (V). A method to continuously infer the
position of the ion through spontaneous light scattering into mirror modes is
provided in [6].

Alternatively, let us continuously monitor momentum, with
$\hat{Q}=(m\omega)^{-1}\hat{p}$ and $\hat{P}=-m\omega\hat{q}$, i.e. $\alpha=0$
and $\beta=(m\omega)^{-1}$ in order to satisfy the condition
$\alpha^{2}+m^{2}\omega^{2}\beta^{2}=1$. The measurement of the momentum of
the ion can be realised via the measurement of the phase changes of a probe
beam using homodyning[19]. For this choice of feedback the effective
Hamiltonian $\hat{H}_{\text{eff}}$ is given by
$\frac{m\omega^{2}}{2}\hat{q}^{2}$. For this choice of feedback and in the
non-selective regime of measurement, the momentum $p$ grows linearly
proportional to the position $q$, while the position remains constant,
according to Eq. (V) and Eq. (V) respectively. Feedback continuously shifts
the quantum particle back to its original position, while it simultaneously
gains momentum as a result of the force applied by the harmonic potential.

###  V.2 Cooling and confinement of a particle in a harmonic potential

In order to investigate the thermalising effect of the feedback, we first
compare the master equation (V) with that of an optical harmonic oscillator
coupled to a heat bath. We consider the prototypical example of an optical
oscillator in the rotating wave approximation and in the non-selective regime
of measurement, allowing us to compare our results with the typical equations
obtained for such a system.

In the second part of this section we considering the equations governing the
increment of the energy, as well as the expectation values of the measured
observable and the conjugate momentum, outside of the limits of the
aforementioned approximations. This allows us to derive a generalised approach
to cooling that facilitates the simultaneous reduction of the expectation
values of both position and momentum at an exponential rate.

####  V.2.1 Optical oscillator coupled to a heat bath

The position and momentum operators, $\hat{q}$ and $\hat{p}$, can be expressed
in terms of creation and annihilation operators as,

$\hat{q}=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}+\hat{a}^{\dagger})\;\;\;\mbox{and}\;\;\;\hat{p}=i\sqrt{\frac{\hbar
m\omega}{2}}(\hat{a}^{\dagger}-\hat{a}).$ (52)

In Appendix E the master equation (II) written in terms of creation and
annihilation operators is given in Eq. (E). We consider this result in the
rotating wave approximation and in the non-selective regime of measurement,
such that the state of an optical harmonic oscillator obeys

$\displaystyle d\rho^{\prime}=$
$\displaystyle-\tfrac{i}{\hbar}\left[\hat{H},\rho\right]dt-\tfrac{iu}{4m\omega}[\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\rho]dt$
$\displaystyle+\left(c-v\right)(\hat{a}\rho\hat{a}^{\dagger}-\tfrac{1}{2}\\{\hat{a}^{\dagger}\hat{a},\rho\\})dt$
$\displaystyle+c(\hat{a}^{\dagger}\rho\hat{a}-\tfrac{1}{2}\\{\hat{a}\hat{a}^{\dagger},\rho\\})dt\,,$
(53)

where the real constant $c$ is given by

$\displaystyle
c=\tfrac{\kappa\omega}{4}+\tfrac{1}{4\kappa\omega}(\tfrac{u^{2}}{m^{2}\omega^{2}}+v^{2})+\tfrac{v}{2}\,.$
(54)

For a particular class of feedback, namely $\hat{F}\propto\hat{P}$ with $u=0$,
Eq. (V.2.1) has the same form as the equation governing a harmonic oscillator
coupled to a heat bath of harmonic oscillators at a certain temperature[4],
given by

$\displaystyle d\rho=$ $\displaystyle-\tfrac{i}{\hbar}[H,\rho]dt$
$\displaystyle+\gamma^{\prime}(N+1)(\hat{a}\rho\hat{a}^{\dagger}-\tfrac{1}{2}\\{\hat{a}^{\dagger}\hat{a},\rho\\})dt$
$\displaystyle+\gamma^{\prime}\ N\
(\hat{a}^{\dagger}\rho\hat{a}-\tfrac{1}{2}\\{\hat{a}\hat{a}^{\dagger},\rho\\})dt\,,$
(55)

where $\gamma^{\prime}$ is the decay rate and
$N=1/(\exp(\tfrac{\hbar\omega}{k_{B}T})-1)$ is the mean excitation of the heat
bath of harmonic oscillators, with $T$ being the temperature of the bath. The
expression for $N$ can be reformulated as

$T=\frac{\hbar\omega}{k_{B}\ln((N+1)/N)}\,.$ (56)

A system in thermal contact with a heat bath as described by Eq. (V.2.1) will
eventually be thermalised at temperature $T$. Because Eq. (V.2.1) also
represents the dynamics of our controlled harmonic oscillator, appropriate
measurement and feedback induce a thermalising effect similar to the coupling
to an external heat bath. By comparing Eqs. (V.2.1) and (V.2.1), the
conditions for thermalisation in terms of the measurement strength and
feedback can be determined.

From the difference between the coefficients of the pumping and the
dissipation terms, we obtain the rate of thermalisation, i.e.
$\gamma^{\prime}=-v$. If $\gamma^{\prime}$ is greater than zero, then the
system loses more energy via dissipation than it gains by pumping, which is
equivalent to cooling. We can therefore conclude that if the proportionality
constant $v$ is negative then energy is reduced, the system is cooled.

That the generator of feedback should be chosen proportional to the conjugate
momentum $P$ with a negative scaling factor is intuitively correct. Doing so
correspond to performing a shift of the measured observable towards a smaller
value, based on information obtained from measuring the system, as discussed
earlier. In Section V.2.2, we shall see that in the selective regime of
measurement the situation is more subtle, none-the-less the results that
follow will be useful for comparison.

The effective temperature of the system can be calculated using the ratio of
the coefficients of the pumping and the dissipation terms in Eq. (V.2.1). By
substituting this ratio into the Eq. (56), the effective temperature of the
system can be written as

$\displaystyle T$
$\displaystyle=\frac{\hbar\omega}{2k_{B}\ln\left(\frac{c-v}{c}\right)}$
$\displaystyle=\frac{\hbar\omega}{2k_{B}\ln\left(\frac{v-\kappa\omega}{v+\kappa\omega}\right)}\,.$
(57)

Minimal temperature is therefore achieved if $v=-\kappa\omega$, i.e.
$\hat{F}=-\kappa\omega\hat{P}$, with decay rate
$\gamma^{\prime}=\kappa\omega=\tfrac{\hbar\gamma}{2m\omega}$. As
$\gamma/\gamma^{\prime}\rightarrow 2m\omega/\hbar$, temperature $T\rightarrow
0\,K$. Assuming the control over $\gamma/\gamma^{\prime}$ to a fourth-order
degree i.e,
$\gamma/\gamma^{\prime}=[\tfrac{2m\omega}{\hbar}(1-10^{-4}),\tfrac{2m\omega}{\hbar}(1+10^{-4})]$,
for a mechanical system with frequency, $\omega$ of the order of magnitude
$10^{6}$, the temperature of the system is cooled down to the order of
$10^{-7}K$. Whereas for an optical system of frequency of the order $10^{12}$,
the system is cooled down to a temperature of the order $10^{-1}K$.

The dynamics of the mean excitation of the system can be calculated using
$\tfrac{d\langle n\rangle}{dt}=-\gamma^{\prime}d\langle
n\rangle+\gamma^{\prime}N$ [4]. This would give

$\langle n\rangle=N+(n_{i}-N)e^{-\gamma^{\prime}t},$ (58)

where $n_{i}$ is the initial mean excitation of the system and $N$ is the
asymptotic mean excitation of the system determined by $\gamma$ and
$\gamma^{\prime}$. The condition for the lowest phonon count can be achieved
by choosing $\gamma^{\prime}=X_{0}^{2}\gamma$, where
$X_{0}^{2}=\tfrac{\hbar}{2m\omega}$ is the variance of the ground state of the
harmonic oscillator.

####  V.2.2 General approach to optimising feedback for a particle in a
harmonic potential

With the intuition gained from the previous calculation, let us consider the
expected increments in energy computed according to Eq. (IV.1) without
approximation:

$\displaystyle\tfrac{1}{\hbar\omega^{2}}$ $\displaystyle
d\langle\hat{H}\rangle=$ $\displaystyle\Big{\\{}$
$\displaystyle\tfrac{\kappa}{4}\left[1-\left(\tfrac{u}{\kappa
m\omega^{2}}\right)^{2}-\left(\tfrac{v}{\kappa\omega}\right)^{2}\right]$
$\displaystyle+\tfrac{v}{\omega}\left[\tfrac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle+\tfrac{1}{2\kappa}\tfrac{v}{\omega}+\tfrac{m\omega}{\hbar}\langle\hat{Q}\rangle^{2}\right]$
$\displaystyle-\tfrac{1}{2}\tfrac{u}{m\omega^{2}}\left[\tfrac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle-\tfrac{1}{\kappa}\tfrac{u}{m\omega^{2}}+\tfrac{2}{\hbar}\langle\hat{Q}\rangle\langle\hat{P}\rangle\right]\Big{\\}}dt$
$\displaystyle+$
$\displaystyle\sqrt{\tfrac{\kappa}{2\omega}}\Big{\\{}2\sqrt{\tfrac{m\omega}{\hbar}}\langle\hat{Q}\rangle\left[\tfrac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle+\tfrac{1}{2\kappa}\tfrac{v}{\omega}\right]$
$\displaystyle\quad\quad\quad+\tfrac{1}{\sqrt{\hbar
m\omega}}\langle\hat{P}\rangle\left[\tfrac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle-\tfrac{1}{\kappa}\tfrac{u}{m\omega^{2}}\right]$
$\displaystyle\quad\quad\quad+\left(\tfrac{m\omega}{\hbar}\right)^{\frac{3}{2}}\langle(\Delta\hat{Q})^{3}\rangle$
$\displaystyle\quad\quad\quad+\tfrac{1}{2\sqrt{\hbar^{3}m\omega}}\langle\\{(\Delta\hat{P})^{2},\Delta\hat{Q}\\}\rangle\Big{\\}}dW\,.$
(59)

Here we note that the terms
$\tfrac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle+\tfrac{1}{2\kappa}\tfrac{v}{\omega}$
and
$\tfrac{1}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle-\tfrac{1}{\kappa}\tfrac{u}{m\omega^{2}}$
also appear in Eqs. (V) and (V) where they govern the stochastic portion of
the increments of the expectation values of the measured observable $\hat{Q}$
and its conjugate momentum $\hat{P}$, respectively.

By choosing the generator of feedback as follows

$\displaystyle u$
$\displaystyle=\tfrac{m\omega^{2}\kappa}{\hbar}\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle_{\infty}=m\omega^{2}\left(-1+\sqrt{1+\kappa^{2}}\right)$
(60) $\displaystyle v$
$\displaystyle=-\tfrac{2m\omega^{2}\kappa}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle_{\infty}=-\sqrt{2}\omega\sqrt{-1+\sqrt{1+\kappa^{2}}}\,,$
(61)

the stochastic terms in Eq. (V.2.2), Eq. (V) and Eq. (V) are reduced over
time, where $\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle_{\infty}$ and
$\langle(\Delta\hat{Q})^{2}\rangle_{\infty}$ are given by Eq. (43) and Eq.
(44), respectively. We note that for weak measurement, $\kappa\ll 1$, this
choice of feedback coincides with our earlier result obtained for an optical
oscillator, namely $u=0$ and $v=-\kappa\omega$.

After some time $t\gg\tfrac{1}{\omega\kappa}$ the stochastic terms in the
increments of the expectation value of the measured observable (V) and its
conjugate momentum (V) are negligible, so that

$\displaystyle\frac{d\langle\hat{Q}\rangle}{dt}=$
$\displaystyle\tfrac{1}{m}\langle\hat{P}\rangle-\sqrt{2}\omega\sqrt{-1+\sqrt{1+\kappa^{2}}}\langle\hat{Q}\rangle$
(62) $\displaystyle\frac{d\langle\hat{P}\rangle}{dt}=$
$\displaystyle-m\omega^{2}\sqrt{1+\kappa^{2}}\langle Q\rangle\,.$ (63)

The above equations are equivalent to the equations of motion of a damped
harmonic oscillator and solving them yields a decay rate for the expectation
values of $Q$ and $P$ of

$\displaystyle\frac{\omega}{\sqrt{2}}\sqrt{-1+\sqrt{1+\kappa^{2}}}\,.$ (64)

For $\kappa<<1$, the decay rate is approximately $\kappa\omega/2$. In Figure
3(a) the results of a numerical simulation of the time evolution of the
expectation values of the measured observable and its conjugate momentum are
shown for $\kappa=0.25$, under the assumption that the initial wavefunction of
the system is Gaussian.

(a) Time evolution of the unitless expectation values of the measured
observable $M=Q$ and the conjugate momentum $P$.

(b) Time evolution of the unitless version of the expected energy.

Figure 3: The following results were obtains by numerical simulation using the
second-order weak scheme [20] with feedback chosen according to Eqs. (60) and
(61) and constant $\kappa=0.25$. The initial conditions were
$\langle\hat{Q}\rangle=0$ and $\langle\hat{P}\rangle=1$ and those of the
second-order variances are given in Figure 1. Grey lines indicates individual
simulations while the black line indicates the mean values taken over 100
simulations.

For times $t>>\frac{1}{\omega\kappa}$, the increment in the expectation value
of the energy (V.2.2) is given by

$\displaystyle\tfrac{1}{\hbar\omega^{2}}d\langle\hat{H}\rangle=$
$\displaystyle\left(\tfrac{v}{\omega}\tfrac{m\omega}{\hbar}\langle\hat{Q}\rangle^{2}-\tfrac{u}{m\omega^{2}}\tfrac{1}{\hbar}\langle\hat{Q}\rangle\langle\hat{P}\rangle\right)dt\,.$
(65)

The energy, as well as the increment in the energy, decays to its asymptotic
value at a rate twice that of the decay of the measured observable $Q$ and the
conjugate momentum $P$. For weak measurement $\kappa<<1$, this decay rate
coincides with the decay rate $\gamma^{\prime}=\omega\kappa$ found for an
optical oscillator in the rotating wave approximation and the non-selective
regime of measurement. In Figure 3(b) a numerical simulation of the evolution
of the energy over time illustrates this exponential decay.

For feedback determined by Eqs. (60) and (61),
$\langle\hat{Q}\rangle_{\infty}$ and $\langle\hat{P}\rangle_{\infty}$ are
negligible at times $t>>\tfrac{1}{\omega\kappa}$, and the particle is
approximately stationary in the centre of the potential with an asymptotic
energy given by

$\displaystyle E_{\infty}$
$\displaystyle=\frac{\hbar\omega}{2\sqrt{2}}\frac{\kappa}{\sqrt{-1+\sqrt{1+\kappa^{2}}}}\,.$
(66)

As discussed in Section IV, the energy of the particle converges to the ground
state energy with respect to the co-moving frame in the weak measurement
limit. We can therefore combine weak measurement and feedback to achieve
confinement, resulting in a convergence to the ground state in the lab frame.

##  VI Discussion

In this paper we investigated continuous measurement and feedback on a
particle in a harmonic potential, where the observable and the generator of
feedback are arbitrary linear combinations of position and momentum. We have
shown that the asymptotic state, assumed by the particle after a certain
convergence time determined by the measurement strength, is represented by a
Gaussian wave function. Its width does not depend on the feedback but only on
the product of the measurement strength, the variance of the ground state and
the time period of the oscillator. On the other hand, the location of the
Gaussian in phase space depends on the feedback. This is not surprising, since
the feedback merely shifts the wave function in position and/or momentum
space.

It is possible to localise the wave function in the centre of the harmonic
potential with vanishing momentum expectation value. The harmonic oscillator
wave function for the different scenarios (with and without measurement and
feedback) is depicted in Figure 4. Feedback enables localisation in the ground
state, measurement allows the wave function to climb up the potential mountain
without being squeezed as is the case in the absence of measurement. This
surprising effect can be understood in the Heisenberg Picture where the
observable is rotated periodically into its conjugate momentum, resulting in a
homogeneous localisation of the particle.

(a) Evolution of the wave function in $Q$-space in the absence of measurement.

(b) Measurement introduces noise into the dynamics of the system. In addition,
the width of the wave function stabilises over time.

(c) By measurement and appropriate feedback, the wave function is localised in
the centre of the potential with a stable width.

Figure 4: Comparison of dynamics in the absence of measurement (a), with
measurement (b) and with measurement and feedback (c). The initial state of
the wave function is shown in white, while the evolved wave function is
indicated in increasingly darker shades of grey. A dotted line represents the
shape of the potential $V(Q)=\tfrac{m\omega^{2}}{2}Q^{2}$, with unitless
values shown on the right axis. The probability $P(Q)$ per interval of length
$dQ$ is given on the left axis.

Our analysis shows that, remarkably, for a free particle or a particle in a
harmonic potential, the feedback can be employed to cancel the measurement
noise completely, cp. Eqs. (V) and (V). This can be achieved by adjusting the
feedback to the values of the variances of the measured observable and its
conjugate momentum. These values can be obtained analytically if the initial
state is a known Gaussian state. For arbitrary initial states the values are
known in the asymptotic regime as a function of the relative measurement
strength. They can also be eventually inferred from the measurement results
(cp. state monitoring [21]). By combining this confining feedback with
sufficiently weak measurement, it is possible to cool the system to the ground
state.

We conjecture that cooling is also possible for arbitrary potentials. Feedback
compensating the noise that contributes to the change of the expectation value
of the measured observable simultaneously damps the motion of the particle.
For sufficiently strong measurement, the asymptotic wave function can be
localised on a length scale such that the potential can be approximated by a
polynomial of second-order[17]. Thus the dynamics is reduced to the case of
the harmonic oscillator as discussed above.

On the other hand, it is also possible to control the motion of the particle
by means of feedback which adds an extra term quadratic in the measured
observable $Q$ to the Hamiltonian. This allows for modifying or even
compensating the kinetic or potential energy in the Hamiltonian. However, such
dynamic control contributes in general to the generation of noise, since it is
not compatible with the compensation of noise.

## Acknowledgements

Amy R thanks the National Research Foundation of South Africa for funding.
Anirudh R, SM and TK acknowledge funding by the South African Quantum
Technology Initiative grant from the Department of Science and Innovation of
South Africa.

##  Appendix A Ito Equations

In this appendix we derive the feedback master equation for an applied
feedback corresponding to the action of the unitary operator given in Eq. (5).
The unitary that describes the feedback is given by

$\displaystyle U$
$\displaystyle=\exp\left[-\frac{i}{\hbar}\hat{F}d\mathcal{M}\right]$
$\displaystyle=\mathbb{I}-\frac{i}{\hbar}\hat{F}d\mathcal{M}-\frac{1}{2\hbar^{2}}\hat{F}^{2}d\mathcal{M}^{2}$
$\displaystyle=\mathbb{I}-\frac{i}{\hbar}\hat{F}d\mathcal{M}-\frac{1}{2\hbar^{2}\gamma}\hat{F}^{2}dt\,,$
(67)

where the Ito rules, Eq. (3) are used and all terms $\mathcal{O}(dt^{2})$ are
neglected. Application of the unitary feedback to the state of the system
after measurement leads to

$\displaystyle U$ $\displaystyle(\rho+d\rho)U^{\dagger}$ $\displaystyle=$
$\displaystyle\rho+d\rho-\frac{i}{\hbar}\left[\hat{F},\rho\right]d\mathcal{M}-\frac{i}{\hbar}\left[\hat{F},d\rho\right]d\mathcal{M}$
$\displaystyle-\frac{1}{2\hbar^{2}\gamma}\left[\hat{F},\left[\hat{F},\rho\right]\right]dt$
$\displaystyle=$
$\displaystyle\rho+d\rho-\frac{i}{\hbar}\left[\hat{F},\rho\right]\left(\langle\hat{M}\rangle
dt+\tfrac{1}{\sqrt{\gamma}}dW\right)$
$\displaystyle-\frac{i}{\hbar\sqrt{\gamma}}\left[\hat{F},d\rho\right]dW-\frac{1}{2\hbar^{2}\gamma}\left[\hat{F},\left[\hat{F},\rho\right]\right]dt\,.$
(68)

We can now substitute the expression for $d\rho$ given by the Ito differential
equation (2), into the third term of the Eq. (68) such that

$\displaystyle\frac{i}{\sqrt{\gamma}}\left[\hat{F},d\rho\right]dW=$
$\displaystyle\frac{i}{\sqrt{\gamma}}\left[\hat{F},\tfrac{\sqrt{\gamma}}{2}\left\\{\hat{M}-\langle\hat{M}\rangle,\rho\right\\}\right]dt$
$\displaystyle=$
$\displaystyle\frac{i}{2}\left[\hat{F},\left\\{\hat{M},\rho\right\\}-\left\\{\langle\hat{M}\rangle,\rho\right\\}\right]dt$
$\displaystyle=$
$\displaystyle\frac{i}{2}\left[\hat{F},\left\\{\hat{M},\rho\right\\}\right]dt+i\langle\hat{M}\rangle\left[\hat{F},\rho\right]dt\,.$
(69)

Hence the state of the system after the application of feedback is given by

$\displaystyle U(\rho+d\rho)U^{\dagger}=$
$\displaystyle\rho+d\rho-\frac{i}{2\hbar}\left[\hat{F},\left\\{\hat{M},\rho\right\\}\right]dt$
$\displaystyle-\frac{1}{2\hbar\gamma}\left[\hat{F},\left[\hat{F},\rho\right]\right]dt-\frac{i}{\hbar\sqrt{\gamma}}\left[\hat{F},\rho\right]dW$
$\displaystyle=$ $\displaystyle\rho+d\rho^{\prime}\,,$ (70)

where

$\displaystyle d\rho^{\prime}:=$
$\displaystyle-\frac{i}{\hbar}\left[\hat{H},\rho\right]dt-\frac{\gamma}{8}\left[\hat{M},\left[\hat{M},\rho\right]\right]dt$
$\displaystyle-\frac{1}{2\hbar^{2}\gamma}\left[\hat{F},\left[\hat{F},\rho\right]\right]dt-\frac{i}{2\hbar}\left[\hat{F},\left\\{\hat{M},\rho\right\\}\right]dt$
$\displaystyle+\frac{\sqrt{\gamma}}{2}\left\\{\hat{M}-\langle\hat{M}\rangle,\rho\right\\}dW-\frac{i}{\hbar\sqrt{\gamma}}\left[\hat{F},\rho\right]dW\,.$
(71)

##  Appendix B Equation of motion of a pure state in the co-moving reference
frame

In order to determine the state $\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$ the
displacement operators in Eq. (10) are first combined into a single
exponential,

$\displaystyle\exp\big{\\{}-\tfrac{i}{\hbar}d\langle\hat{p}\rangle\hat{q}\big{\\}}$
$\displaystyle\exp\big{\\{}\tfrac{i}{\hbar}d\langle\hat{q}\rangle\hat{p}\big{\\}}$
$\displaystyle\quad=\exp\big{\\{}-\tfrac{i}{\hbar}\left(d\langle\hat{p}\rangle\hat{q}-d\langle\hat{q}\rangle\hat{p}\right)\big{\\}}\,,$
(72)

where we ignore the irrelevant phase factor which results from applying the
Baker–Campbell–Hausdorff formula. Eq. (11) can now be written as

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$
$\displaystyle=e^{-\tfrac{i}{\hbar}\left(d\langle\hat{p}\rangle\hat{q}-d\langle\hat{q}\rangle\hat{p}\right)}e^{\hat{G}}(\ket{{\psi}}+d\ket{{\psi}})\,,$
(73)

where $\hat{G}$ is given in Eq. (9). As noted previously $\hat{q}$ and
$\hat{p}$ can be replaced with any pair of conjugate variables. We therefore
replace them with $\hat{Q}$ and $\hat{P}$, defined in Eqs. (14) and (15), so
that

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$
$\displaystyle=e^{-\tfrac{i}{\hbar}\left(d\langle\hat{P}\rangle\hat{Q}-d\langle\hat{Q}\rangle\hat{P}\right)}e^{\hat{G}}(\ket{{\psi}}+d\ket{{\psi}})\,,$
(74)

Assuming that $\langle\hat{Q}\rangle=0=\langle\hat{P}\rangle$ at time $t$ when
$\ket{\psi}$ is considered, we have

$\displaystyle\hat{G}=$
$\displaystyle\Big{[}-\tfrac{i}{\hbar}\hat{H}-\tfrac{\gamma}{4}\hat{Q}^{2}+\tfrac{i}{4\hbar}[\hat{Q},\hat{F}]\Big{]}dt$
$\displaystyle+\Big{[}\tfrac{\sqrt{\gamma}}{2}\hat{Q}-\tfrac{i}{\hbar\sqrt{\gamma}}\hat{F}\Big{]}dW\,.$
(75)

and

$\displaystyle d\langle Q\rangle$
$\displaystyle=\operatorname{Tr}[{\hat{Q}d\rho}]$
$\displaystyle=-\tfrac{i}{\hbar}\langle[\hat{Q},\hat{H}]\rangle
dt-\tfrac{1}{2\hbar^{2}\gamma}\langle[\hat{F},[\hat{F},\hat{Q}]]\rangle dt$
$\displaystyle\quad-\frac{i}{2\hbar}\langle\\{\hat{Q},[\hat{Q},\hat{F}]\\}\rangle
dt$
$\displaystyle\quad+\frac{\sqrt{\gamma}}{2}\langle\\{\hat{Q},\hat{Q}\\}\rangle
dW+\frac{i}{\hbar\sqrt{\gamma}}\langle[\hat{F},\hat{Q}]\rangle dW\,.$ (76)

Given that feedback is a linear combination of position and momentum,
$[\hat{F},\hat{Q}]$ and $[\hat{F},\hat{P}]$ are constant, so that

$\displaystyle d\langle Q\rangle=$
$\displaystyle-\tfrac{i}{\hbar}\langle[\hat{Q},\hat{H}]\rangle dt$
$\displaystyle+\frac{\sqrt{\gamma}}{2}\langle\\{\hat{Q},\hat{Q}\\}\rangle
dW+\frac{i}{\hbar\sqrt{\gamma}}\langle[\hat{F},\hat{Q}]\rangle dW\,.$ (77)

Similarly,

$\displaystyle d\langle P\rangle$
$\displaystyle=\operatorname{Tr}[{\hat{P}d\rho}]$
$\displaystyle=-\tfrac{i}{\hbar}\langle[\hat{P},\hat{H}]\rangle
dt-\frac{1}{2\hbar^{2}\gamma}\langle[\hat{F},[\hat{F},\hat{P}]]\rangle dt$
$\displaystyle\quad-\frac{i}{2\hbar}\langle\\{\hat{Q},[\hat{P},\hat{F}]\\}\rangle
dt$
$\displaystyle\quad+\frac{\sqrt{\gamma}}{2}\langle\\{\hat{Q},\hat{P}\\}\rangle
dW+\frac{i}{\hbar\sqrt{\gamma}}\langle[\hat{F},\hat{P}]\rangle dW$
$\displaystyle=-\tfrac{i}{\hbar}\langle[\hat{P},\hat{H}]\rangle dt$
$\displaystyle\quad+\frac{\sqrt{\gamma}}{2}\langle\\{\hat{Q},\hat{P}\\}\rangle
dW+\frac{i}{\hbar\sqrt{\gamma}}\langle[\hat{F},\hat{P}]\rangle dW\,.$ (78)

The Baker–Campbell–Hausdorff formula can now be applied to Eq. (74). Hence,

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$
$\displaystyle=\exp\Bigg{\\{}-\tfrac{i}{\hbar}\left(\hat{{H}}-\tfrac{i}{\hbar}\langle[\hat{P},\hat{H}]\rangle\hat{Q}+\tfrac{i}{\hbar}\langle[\hat{Q},\hat{H}]\rangle\hat{P}\right)dt$
$\displaystyle\hskip
40.0pt-\tfrac{\gamma}{4}\hat{Q}^{2}dt+\frac{i}{4\hbar}[\hat{Q},\hat{F}]dt+\hat{A}dt$
$\displaystyle\hskip
17.0pt-\tfrac{i}{\hbar}\sqrt{\tfrac{\gamma}{2}}\left[\left(i\hbar+\langle\\{\hat{Q},\hat{P}\\}\rangle\right)\hat{Q}-2\langle\hat{Q}^{2}\rangle\hat{P}\right]dW\Bigg{\\}}\ket{\tilde{\psi}}\,,$
(79)

where we use the fact that
$\langle[\hat{F},\hat{P}]\rangle\hat{Q}-\langle[\hat{F},\hat{Q}]\rangle\hat{P}=i\hbar\hat{F}$.
$\hat{A}$ is the term which arises from the commutation relation between the
arguments of the two exponential terms in Eq. (74). Computing the term
$\hat{A}dt$ yields

$\displaystyle\hat{A}dt=$
$\displaystyle-\tfrac{i}{2\hbar}\left[d\langle\hat{P}\rangle\hat{Q}-d\langle\hat{Q}\rangle\hat{P},\hat{G}\right]$
$\displaystyle=$
$\displaystyle\tfrac{i}{2\hbar}\left[-d\langle\hat{P}\rangle\hat{Q}+d\langle\hat{Q}\rangle\hat{P},\tfrac{\sqrt{\gamma}}{2}\hat{Q}-\tfrac{i}{\hbar\sqrt{\gamma}}\hat{F}\right]dW$
$\displaystyle=$ $\displaystyle\tfrac{\sqrt{\gamma}}{4}d\langle\hat{Q}\rangle
dW-\tfrac{1}{2\hbar^{2}}\left(d\langle\hat{P}\rangle[\hat{Q},\hat{F}]-d\langle\hat{Q}\rangle[\hat{P},\hat{F}]\right)dW$
$\displaystyle=$ $\displaystyle\tfrac{\sqrt{\gamma}}{4}d\langle\hat{Q}\rangle
dW-\tfrac{i\sqrt{\gamma}}{4\hbar^{2}}\langle\\{\hat{Q},\hat{F}\\}\rangle
dt\,,$ (80)

where we have used the fact that $\hat{F}$ is a linear combination of
$\hat{Q}$ and $\hat{P}$ to obtain the last result. Substituting Eq. (77) into
the above gives

$\displaystyle\hat{A}=$
$\displaystyle\tfrac{\gamma}{4}\langle\hat{Q}^{2}\rangle+\tfrac{i}{4\hbar}\langle[\hat{F},\hat{Q}]\rangle+\tfrac{i}{4\hbar}\langle\\{\hat{Q},\hat{F}\\}\rangle\,.$
(81)

Substituting the above into Eq. (79) while ignoring the phase factor
associated with the last term in Eq. (81), leads to the expression

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}=\exp\Big{\\{}\Big{[}-\tfrac{i}{\hbar}\left(\hat{{H}}-\tfrac{i}{\hbar}\langle[\hat{P},\hat{H}]\rangle\hat{Q}+\tfrac{i}{\hbar}\langle[\hat{Q},\hat{H}]\rangle\hat{P}\right)-\tfrac{\gamma}{4}\left(\hat{Q}^{2}-\langle\hat{Q}^{2}\rangle\right)\Big{]}$
$\displaystyle dt$
$\displaystyle-\tfrac{i}{\hbar}\sqrt{\tfrac{\gamma}{2}}\Big{[}\left(i\hbar+\langle\\{\hat{Q},\hat{P}\\}\rangle\right)\hat{Q}-2\langle\hat{Q}^{2}\rangle\hat{P}\Big{]}$
$\displaystyle dW\Big{\\}}\ket{\tilde{\psi}}\,.$ (82)

If $\hat{H}$ is the Hamiltonian of a harmonic oscillator (18) then
$\langle[\hat{Q},\hat{H}]\rangle=\langle[\hat{P},\hat{H}]\rangle=0$, since
$\langle\hat{Q}\rangle=\langle\hat{P}\rangle=0$, and

$\displaystyle\ket{\tilde{\psi}}+d\ket{\tilde{\psi}}$
$\displaystyle=\exp\Big{\\{}\Big{[}-\tfrac{i}{\hbar}\hat{{H}}-\tfrac{\gamma}{4}\left(\hat{Q}^{2}-\langle\hat{Q}^{2}\rangle\right)\Big{]}dt$
$\displaystyle\hskip
25.0pt-\tfrac{i}{\hbar}\sqrt{\tfrac{\gamma}{2}}\Big{[}\left(i\hbar+\langle\\{\hat{Q},\hat{P}\\}\rangle\right)\hat{Q}-2\langle\hat{Q}^{2}\rangle\hat{P}\Big{]}dW\Big{\\}}\ket{\tilde{\psi}}\,.$
(83)

##  Appendix C Proof that the Gaussian steady-state solutions satisfy the
energy eigenvalue equation

This appendix will show that the stationary solution to the eigenvalue
equation (21) also satisfies the energy eigenvalue equation (20). The
eigenvalue equation (21), where $[\hat{Q},\hat{P}]=i\hbar$, has in “$Q$”
representation the solution

$\displaystyle\psi_{\infty}(Q)=\left(\frac{s}{\pi}\right)^{\frac{1}{4}}\exp\left({-\frac{s}{2}u^{2}}\right)$
(84)

where

$\displaystyle
s:=\frac{1}{2\langle(\hat{Q})^{2}\rangle_{\infty}}\left(1-\frac{i}{\hbar}\langle\\{\hat{Q},\hat{P}\\}\rangle_{\infty}\right)\,.$
(85)

The energy eigenvalue equation (20) can also be expressed in terms of
$\hat{Q}$ and $\hat{P}$ as

$\displaystyle\left[\frac{1}{2m}\hat{P}^{2}+\frac{m\omega^{2}}{2}\hat{Q}^{2}-i\tfrac{\hbar\gamma}{4}\left(\hat{Q}^{2}-\langle\hat{Q}^{2}\rangle_{\infty}\right)\right]$
$\displaystyle\ket{\tilde{\psi}_{\infty}}$ $\displaystyle=E$
$\displaystyle\ket{\tilde{\psi}_{\infty}}\,.$ (86)

Using Eq. (84) the energy eigenvalues equation (86) yields

$\displaystyle E=$
$\displaystyle\frac{\hbar^{2}s}{2m}+i\frac{\hbar\gamma}{4}\langle(\hat{Q})^{2}\rangle_{\infty}$
$\displaystyle+\frac{m\omega^{2}}{2}\left(1-\left(\frac{\hbar
s}{m\omega}\right)^{2}-i\frac{\hbar\gamma}{2m\omega^{2}}\right)Q^{2}\,.$ (87)

It is possible to evaluate the term proportional to $Q^{2}$ using the result
given in Eqs. (43) and (44). A straight forward calculation gives

$\displaystyle\left(\frac{\hbar s}{m\omega}\right)^{2}$
$\displaystyle=1-i\kappa=1-i\frac{\hbar\gamma}{2m\omega^{2}}\,,$ (88)

where $\kappa$ is defined in Eqn. (41). The $Q^{2}$ term therefore vanishes
and we can conclude from Eq. (87) that

$\displaystyle E$
$\displaystyle=\frac{\hbar^{2}}{4m}\left(2s+i\frac{m\gamma}{\hbar}\langle(\hat{Q})^{2}\rangle_{\infty}\right)$
$\displaystyle=\frac{\hbar^{2}}{4m\langle(\hat{Q})^{2}\rangle_{\infty}}$ (89)
$\displaystyle=\frac{\hbar\omega}{2\sqrt{2}}\frac{\kappa}{\sqrt{-1+\sqrt{1+\kappa^{2}}}}\,.$
(90)

For for a free particle, i.e. $\omega\rightarrow 0$, this solution equivalent
to the energy determined by Diósi in [9]. In addition, for $\kappa>>1$,

$\displaystyle E\approx\frac{\hbar\omega}{2\sqrt{2}}\sqrt{\kappa}\,,$ (91)

and for $\kappa<<1$,

$\displaystyle
E\approx\frac{\hbar\omega}{2}\left(1+\frac{\kappa^{2}}{8}\right)\,.$ (92)

In the weak measurement limit, $\kappa\rightarrow 0$, the energy approaches
the value associated with the ground state energy of the harmonic oscillator,
$\hbar\omega/2$.

##  Appendix D Dimensionless coupled differential equations

In this section we show that by dividing Eqs. (31)-(33) by the appropriate
constants of matching dimension it is possible to greatly simplify these
coupled differential equations, as shown below.

$\displaystyle\tfrac{m\omega}{\hbar}d\langle(\Delta\hat{Q})^{2}\rangle=$
$\displaystyle-2\kappa\left(\tfrac{m\omega}{\hbar}\right)^{2}\langle(\Delta\hat{Q})^{2}\rangle^{2}\omega
dt+\tfrac{1}{\hbar}\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle\omega
dt+\sqrt{2\kappa}\left(\tfrac{m\omega}{\hbar}\right)^{\frac{3}{2}}\langle(\Delta\hat{Q})^{3}\rangle\sqrt{\omega}dW$
(93) $\displaystyle\tfrac{1}{\hbar
m\omega}d\langle(\Delta\hat{P})^{2}\rangle=$
$\displaystyle\tfrac{1}{2}\kappa\left(1-\tfrac{1}{\hbar^{2}}\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle^{2}\right)\omega
dt-\tfrac{1}{\hbar}\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle\omega
dt+\sqrt{\tfrac{\kappa}{2}}\tfrac{1}{\sqrt{\hbar^{3}m\omega}}\langle\\{(\Delta\hat{P})^{2},\Delta\hat{Q}\\}\rangle\sqrt{\omega}dW$
(94)
$\displaystyle\tfrac{1}{\hbar}d\langle\\{\Delta\hat{Q},\Delta\hat{P}\\}\rangle=$
$\displaystyle 2\left(\tfrac{1}{\hbar
m\omega}\langle(\Delta\hat{P})^{2}\rangle-\tfrac{m\omega}{\hbar}\langle(\Delta\hat{Q})^{2}\rangle\right)\omega
dt-2\kappa\tfrac{m\omega}{\hbar^{2}}\langle\\{\Delta\hat{P},\Delta\hat{Q}\\}\rangle\langle(\Delta\hat{Q})^{2}\rangle\omega
dt$
$\displaystyle+\sqrt{\tfrac{\kappa}{2}}\sqrt{\tfrac{m\omega}{\hbar^{3}}}\langle\\{\\{\Delta\hat{Q},\Delta\hat{P}\\},\Delta\hat{Q}\\}\rangle\sqrt{\omega}dW\,.$
(95)

For a free particle or a particle in a harmonic potential and for $\tau=\omega
t>>{1}/{\kappa\omega}$, higher order moments can be ignored. By substitution
the variables defined in Eqs. (38)-(40) we arrive at the simple equations

$\displaystyle\frac{dx}{d\tau}$ $\displaystyle=-2\kappa x^{2}+z$ (96)
$\displaystyle\frac{dy}{d\tau}$
$\displaystyle=\frac{\kappa}{2}\left(1-z^{2}\right)-z$ (97)
$\displaystyle\frac{dz}{d\tau}$ $\displaystyle=2\left(y-x\right)-2\kappa
xz\,.$ (98)

##  Appendix E General feedback equation for Harmonic oscillator in terms of
creation and annihilation operators

According to the definitions of the creation and annihilation operators given
in Eq. (52), the measurement and feedback operators can be expressed,
respectively, as

$\displaystyle\hat{M}$ $\displaystyle=c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger}$
(99) $\displaystyle\hat{F}$
$\displaystyle=c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger}$ (100)

where

$\displaystyle c_{1}$
$\displaystyle=\alpha\sqrt{\tfrac{\hbar}{m\omega}}+i\beta\sqrt{\hbar m\omega}$
(101) $\displaystyle c_{2}$
$\displaystyle=\chi\sqrt{\tfrac{\hbar}{m\omega}}+i\delta\sqrt{\hbar
m\omega}\,.$ (102)

Substituting Eqs. (99) and (100) into Eq. (II) we obtain

$\displaystyle d\rho^{\prime}=$ $\displaystyle-\tfrac{i}{\hbar}[\hat{H},\rho]$
$\displaystyle-\tfrac{\gamma}{16}\left[c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger},\left[c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger},\rho\right]\right]dt$
$\displaystyle-\tfrac{1}{4\gamma\hbar^{2}}[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\left[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\rho\right]]dt$
$\displaystyle-\tfrac{i}{4\hbar}\left[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\left\\{c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger},\rho\right\\}\right]dt$
$\displaystyle-\tfrac{i}{\sqrt{2\gamma}\hbar}\left[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\rho\right]dW$
$\displaystyle+\tfrac{\sqrt{\gamma}}{2\sqrt{2}}\left\\{c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger}-\langle
c_{1}^{*}\hat{a}+c_{1}^{*}\hat{a}^{\dagger}\rangle,\rho\right\\}dW\,.$ (103)

The double commutator
$[c_{i}^{*}\hat{a}+c_{i}\hat{a}^{\dagger},\left[c_{i}^{*}\hat{a}+c_{i}\hat{a}^{\dagger},\rho\right]]$
can be written as

$\displaystyle
c_{i}^{*2}\left(\\{\hat{a}^{2},\rho\\}-2\hat{a}\rho\hat{a}\right)+c_{i}^{2}\big{(}\\{\hat{a}^{\dagger^{2}},\rho\\}-2\hat{a}^{\dagger}\rho\hat{a}^{\dagger}\big{)}$
$\displaystyle+|c_{i}|^{2}(\\{\hat{a}^{\dagger}\hat{a},\rho\\}-2\hat{a}^{\dagger}\rho\hat{a}+\\{\hat{a}\hat{a}^{\dagger},\rho\\}-2\hat{a}\rho\hat{a}^{\dagger})\,.$
(104)

Similarly, the mixed term
$\left[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\left\\{c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger},\rho\right\\}\right]$
can be written as

$\displaystyle c_{1}c_{2}\left[\hat{a}^{\dagger
2},\rho\right]+c_{1}^{*}c_{2}^{*}\left[\hat{a}^{2},\rho\right]$
$\displaystyle+c_{1}c_{2}^{*}\left(\hat{a}\hat{a}^{\dagger}\rho-\rho\hat{a}^{\dagger}\hat{a}\right)+c_{1}^{*}c_{2}\left(\hat{a}^{\dagger}\hat{a}\rho-\rho\hat{a}\hat{a}^{\dagger}\right)$
$\displaystyle+\left(c_{1}c_{2}^{*}-c_{1}^{*}c_{2}\right)\left(\hat{a}\rho\hat{a}^{\dagger}-\hat{a}^{\dagger}\rho\hat{a}\right)\,.$
(105)

The above can be expressed as

$\displaystyle c_{1}c_{2}\left[\hat{a}^{\dagger
2},\rho\right]+c_{1}^{*}c_{2}^{*}\left[\hat{a}^{2},\rho\right]-i\tfrac{1}{m\omega}\left[\hat{F},\hat{P}\right]\left[\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\rho\right]$
$\displaystyle-\left[\hat{F},\hat{Q}\right]\left[\\{\hat{a}^{\dagger}\hat{a},\rho\\}-2\hat{a}^{\dagger}\rho\hat{a}-\left(\\{\hat{a}\hat{a}^{\dagger},\rho\\}-2\hat{a}\rho\hat{a}^{\dagger}\right)\right]\,.$
(106)

Applying Eqs. (E) and (E) to Eq. (E) we obtain

$\displaystyle d\rho^{\prime}=$
$\displaystyle-\tfrac{i}{\hbar}[\hat{H},\rho]dt-\left(\tfrac{\gamma}{16}c_{1}^{*2}+\tfrac{1}{4\gamma\hbar^{2}}c_{2}^{*2}\right)\left(\\{\hat{a}^{2},\rho\\}-2\hat{a}\rho\hat{a}\right)dt-\left(\tfrac{\gamma}{16}c_{1}^{2}+\tfrac{1}{4\gamma\hbar^{2}}c_{2}^{2}\right)\big{(}\\{\hat{a}^{\dagger^{2}},\rho\\}-2\hat{a}^{\dagger}\rho\hat{a}^{\dagger}\big{)}dt$
$\displaystyle-\tfrac{i}{4\hbar}c_{1}c_{2}\left[\hat{a}^{\dagger
2},\rho\right]-\tfrac{i}{4\hbar}c_{1}^{*}c_{2}^{*}\left[\hat{a}^{2},\rho\right]dt-\tfrac{1}{4\hbar
m\omega}\left[\hat{F},\hat{P}\right]\left[\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\rho\right]dt-\tfrac{1}{2}\left(c-\tfrac{i}{\hbar}\left[\hat{F},\hat{Q}\right]\right)(\\{\hat{a}^{\dagger}\hat{a},\rho\\}-2\hat{a}^{\dagger}\rho\hat{a})dt$
$\displaystyle-\tfrac{1}{2}c(\\{\hat{a}\hat{a}^{\dagger},\rho\\}-2\hat{a}\rho\hat{a}^{\dagger})dt+\tfrac{i}{\sqrt{2\gamma}\hbar}\left[c_{2}^{*}\hat{a}+c_{2}\hat{a}^{\dagger},\rho\right]dW+\tfrac{\sqrt{\gamma}}{2\sqrt{2}}\left\\{c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger}-\langle
c_{1}^{*}\hat{a}+c_{1}\hat{a}^{\dagger}\rangle,\rho\right\\}dW\,,$ (107)

where

$\displaystyle c$
$\displaystyle:=\tfrac{\gamma}{8}|c_{1}|^{2}+\tfrac{1}{2\gamma\hbar^{2}}|c_{2}|^{2}+\tfrac{i}{2\hbar}\left[\hat{F},\hat{Q}\right]$
$\displaystyle=\tfrac{\hbar\gamma}{8m\omega}+\tfrac{1}{2\gamma\hbar
m\omega}(u^{2}+m^{2}\omega^{2}v^{2})-\tfrac{i}{2\hbar}\left[\hat{Q},\hat{F}\right]\,.$
(108)

In the rotating wave approximation fast oscillating terms quadratic in
$\hat{a}$ or $\hat{a}^{\dagger}$ are ignored. This leads to

$\displaystyle d\rho^{\prime}=$
$\displaystyle-\tfrac{i}{\hbar}\left[\hat{H},\rho\right]dt-\tfrac{1}{4\hbar
m\omega}\left[\hat{F},\hat{P}\right][\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\rho]dt$
$\displaystyle+\left(c+i\tfrac{1}{\hbar}\left[\hat{Q},\hat{F}\right]\right)(\hat{a}\rho\hat{a}^{\dagger}-\tfrac{1}{2}\\{\hat{a}^{\dagger}\hat{a},\rho\\})dt$
$\displaystyle+c(\hat{a}^{\dagger}\rho\hat{a}-\tfrac{1}{2}\\{\hat{a}\hat{a}^{\dagger},\rho\\})dt\,.$
(109)

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