with continuous measurement of observables that are linear combinations of the canonical variables [70].
From this point on we will only consider the simplified Gaussian representation (used in the next section) since it allows us to apply established techniques from estimation and control theory. The replacement of the quantum mechanical model with a classical noise model is discussed more fully in the appendix. Throughout this treatment, we keep in mind the constraints that the original model imposed. As before, we assumeF is large enough to maintain the Gaussian approximation and that time is small compared to the measurement induced damping rate, t 1/M. Also, the description of our original problem demands that h∆ ˆFz2i(0) = F/2 for a coherent state. 1 Hence our prior information for the initial value of the spin component will always be dictated by the structure of Hilbert space.
As stated in the appendix, we implicitly make the associations: ˜z(t) =hFˆzi(t) = Tr[ ˆρ(t) ˆFz] and ˜b(t) =R
p(b, t)b db, although no further mention of ˆρ(t) or p(b, t) will be made.
We assume the measurement induced damping of F to be negligible for short times (Fexp[−M t/2]≈F ift1/M) and approximate the dynamics as
Dynamics
dx(t) = Ax(t)dt+Bu(t)dt+
0
√σbF
dW1 (10.6)
A ≡
0 γF 0 −γb
B ≡
γF
0
Σ0 ≡
σz0 0 0 σb0
Σ1 ≡
0 0
0 σbF
where the initial value x(0) for each trial is drawn randomly from a Gaussian distribution of mean zero and covariance matrix Σ0. The initial field variance σb0 is considered to be due to classical uncertainty, whereas the initial spin variance σz0 is inherently nonzero due to the original quantum state description. Specifically, we impose σz0 = h∆ ˆFz2i(0). The Wiener increment dW1(t) has a Gaussian distribution with mean zero and variancedt. Σ1
represents the covariance matrix of the last vector in equation (10.6).
We have given ourselves a magnetic field control input, u(t), along the same axis, y, of the field to be measured, b(t). We have allowed b(t) to fluctuate via a damped diffusion (Ornstein-Uhlenbeck) process [52]
db(t) = −γbb(t)dt+√
σbF dW1. (10.7)
Theb(t) fluctuations are represented in this particular way because Gaussian noise processes are amenable to LQG analysis. The variance of the field at any particular time is given by the expectationσbF ree ≡E[b(t)2] =σbF/2γb. (Throughout the chapter we use the notation
E[x(t)] to represent the average of the generally stochastic variable x(t) at the same point in time, over many trajectories.) The bandwidth of the field is determined by the frequency γb alone. When considering the measurement of fluctuating fields, a valid choice of prior might be σb0 = σbF ree, but we choose to let σb0 remain independent. For constant fields, we set σbF ree= 0, but σb06= 0.
Note that only the small angle limit of the spin motion is considered. Otherwise we would have to consider different components of the spin vector rotating into each other. The small angle approximation would be invalid if a field caused the spins to rotate excessively, but using adequate control ensures this will not happen. Hence, we use control for essentially two reasons in this chapter: first to keep our small angle approximation valid, and, second, to make our estimation process robust to our ignorance of F. The latter point will be discussed in section 10.6.
Our measurement ofz is described by the process Measurement
y(t)dt = Cx(t)dt+√
σMdW2(t) (10.8)
C ≡ h 1 0
i Σ2 ≡ σM ≡1/4M η
where the measurement shotnoise is represented by the Wiener incrementdW2(t) of variance dt. Again,√
σM represents the sensitivity of the measurement,M is the measurement rate (with unspecified physical definition in terms of probe parameters), and η is the quantum efficiency of the measurement. The incrementsdW1 and dW2 are uncorrelated.
Following [49], the optimal estimator for mapping y(t) tom(t) takes the form
Estimator
dm(t) = Am(t)dt+Bu(t)dt
+KO(t)[y(t)−Cm(t)]dt (10.9)
m(0) =
0 0
KO(t) ≡ Σ(t)CTΣ−12 dΣ(t)
dt = Σ1+AΣ(t) +Σ(t)AT
−Σ(t)CTΣ−12 CΣ(t) (10.10) Σ(t) ≡
σzR(t) σcR(t) σcR(t) σbR(t)
(10.11)
Σ(0) = Σ0 ≡
σz0 0 0 σb0
. (10.12)
equation (10.9) is the Kalman filter that depends on the solution of the matrix Riccati equation (10.10). The Riccati equation gives the optimal observation gain KO(t) for the filter. The estimator is designed to minimize the average quadratic estimation error for each variable: E[(z(t)−z(t))˜ 2] and E[(b(t)−˜b(t))2]. If the model is correct, and we assume the observer chooses his prior informationΣ(0) to match the actual variance of the initial data Σ0, then we have the self-consistent result:
σzE(t) ≡ E[(z(t)−z(t))˜ 2] =σzR(t) σbE(t) ≡ E[(b(t)−˜b(t))2] =σbR(t).
Hence, the Riccati equation solution represents both the observer gain and the expected performance of an optimal filter using that same gain.
Now consider the control problem, which is in many respects dual to the estimation problem. We would like to design a controller to mapy(t) tou(t) in a manner that minimizes the quadratic cost function
Minimized Cost
H = Z T
0
xT(t)Px(t) +u(t)Qu(t) dt
+xT(T)P1x(T) (10.13)
P ≡
p 0 0 0
Q ≡ q
where P1 is the end-point cost. Only the ratio p/q ever appears, of course, so we define the parameterλ≡p
p/q and use it to represent the cost of control. By settingλ→ ∞, as we often choose to do in the subsequent analysis to simplify results, we are putting no cost on our control output. This is unrealistic because, for example, makingλarbitrarily large implies that we can apply transfer functions with finite gain at arbitrarily high frequencies, which is not experimentally possible. Despite this, we will often consider the limitλ→ ∞ to set bounds on achievable estimation and control performance. The optimal controller for minimizing equation (10.13) is
Controller
u(t) = −KC(t)m(t) (10.14)
KC(t) ≡ Q−1BTV(T −t) dV(T)
dT = P+ATV(T) +V(T)A
−V(T)BQ−1BTV(T) (10.15) V(T = 0) ≡ P1.
HereV(T) is solved in reverse timeT, which can be interpreted as the time left to go until the stopping point. Thus if T → ∞, then we only need to use the steady-state of the V Riccati equation (10.15) to give the steady state controller gain KC for all times. In this case, we can ignore the (reverse) initial condition P1 because the controller is not designed to stop. Henceforth, we will makeKC equal to this constant steady-state value, such that the only time varying coefficients will come from KO(t).
In principle, the above results give the entire solution to the ideal estimation and control problem. However, in the nonideal case where our knowledge of the system is incomplete,
e.g.,F is unknown, our estimation performance will suffer. Notation is now introduced that produces trivial results in the ideal case, but is helpful otherwise. Our goal is to collect the above equations into a single structure that can be used to solve the nonideal problem. We define thetotal state of the system and estimator as
Total State
θ(t) ≡
x(t) m(t)
=
z(t) b(t)
˜ z(t)
˜b(t)
. (10.16)
Consider the general case where the observer assumes the plant contains spinF0, which may or may not be equal to the actualF. All design elements depending onF0 instead ofF are now labeled with a prime. Then it can be shown that the total state dynamics from the above estimator-controller architecture are a time-dependent Ornstein-Uhlenbeck process Total State Dynamics
dθ(t) = α(t)θ(t)dt+β(t)dW(t) (10.17) α(t) ≡
A −BK0C
K0O(t)C A0−B0K0C−K0O(t)C
β(t) ≡
0 0 0 0
0 √
σbF 0 0
0 0 √
σMKO10 (t) 0
0 0 √
σMKO20 (t) 0
where the covariance matrix ofdW isdt times the identity. Now the quantity of interest is the following covariance matrix
Total State Covariance
Θ(t) ≡ E[θ(t)θT(t)]
≡
σzz σzb σzz˜ σz˜b σzb σbb σb˜z σb˜b
σz˜z σb˜z σz˜z˜ σz˜˜b σz˜b σb˜b σz˜˜b σ˜b˜b
(10.18)
σzz ≡ E[z(t)2] σzb ≡ E[z(t)b(t)]
... ≡ ....
It can be shown that this total covariance matrix obeys the following deterministic equations of motion,
Total State Covariance Dynamics dΘ(t)
dt = α(t)Θ(t) +Θ(t)αT(t) +β(t)βT(t) (10.19) Θ(t) = exp
− Z t
0
α(t0)dt0
Θ0exp
− Z t
0
αT(t0)dt0
+ Z t
0
dt0exp
− Z t
t0
α(s)ds
β(t0)βT(t0)
×exp
− Z t
t0
αT(s)ds
(10.20)
Θ0 =
σz0 0 0 0 0 σb0 0 0
0 0 0 0
0 0 0 0
.
Equation (10.20) is the matrix form of the standard integrating factor solution for time- dependent scalar ordinary differential equations [52]. Whether we solve this problem nu- merically or analytically, the solution provides the following quantity that we ultimately care about,
Average Magnetometry Error
σbE(t) ≡ E[(˜b(t)−b(t))2]
= E[b2(t)] + E[˜b2(t)]−2E[b(t)˜b(t)]
= σbb(t) +σ˜b˜b(t)−2σb˜b(t). (10.21) When all parameters are known (andF0 =F), this total state description is unnecessary because σbE(t) =σbR(t). This equality is by design. However, when the wrong parameters are assumed (e.g.,F0 6=F) the equality does not holdσbE(t)6=σbR(t) and either equation (10.19) or equation (10.20) must be used to findσbE(t). Before addressing this problem, we consider in detail the performance in the ideal case, where all system parameters are known by the observer, includingF.
At this point, we have defined several variables. For clarity, let us review the meaning of several before continuing. Inputs to the problem include the field fluctuation strength σbF, equation (10.7), and the measurement sensitivity σM, equation (10.8). The prior information for the field is labeledσb0, equation (10.12). The solution to the Riccati equation isσbR(t), equation (10.11), and is equal to the estimation varianceσbE(t), equation (10.21), when the estimator model is correct. In the next section, we additionally useσbS, equation (10.24), andσbT(t), equation (10.25), to represent the steady-state and transient values of σbE(t) respectively.