Ramsey Interferometry as Quantum Estimation

Ramsey interferometry estimates a parameter by preparing coherence, allowing a relative phase to accumulate, and converting that phase into a measured population. Its physical sequence is often summarized as

$$\text{prepare} \longrightarrow \text{encode }\phi \longrightarrow \text{choose analysis phase} \longrightarrow \text{measure}.$$

The population fringe is only the visible part of the protocol. A complete sensing statement must also specify the likelihood, working point, prior or capture range, interrogation and dead times, coherence loss, readout errors, estimator, uncertainty calibration, and comparison resource. Ramsey interferometry is therefore both a coherent-control primitive and a compact laboratory for quantum estimation theory.

Required background. Parameter estimation, Fisher information, and Cramér–Rao bounds supply the likelihood and local precision bounds; the standard quantum limit gives the independent-probe reference. Projective measurement explains the binary readout.

The AMO treatment of Ramsey interferometry derives the pulse dynamics and finite-pulse line shape. Here the focus is the information in the outcomes, frequency ambiguity, and the choice of interrogation time.

Ramsey phase encoding and readout

Consider a two-level probe with basis states $|0\rangle$ and $|1\rangle$. An ideal preparation pulse creates an equatorial state, which may be written

$$|+x\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}.$$

During the sensing interval, the unknown parameter produces a relative phase $\phi$. At the end of the interval the state is

$$|\psi_\phi\rangle = \frac{ |0\rangle+e^{i\phi}|1\rangle }{\sqrt2}.$$

An analysis pulse with controllable phase $\theta$ selects an equatorial measurement axis. With a convenient convention, the probability of the outcome $x=1$ is

$$p_\theta(1|\phi) = \frac12 \left[ 1+\sin(\phi-\theta) \right].$$

For example, on the encoded qubit this outcome corresponds to the effect

$$E_{\theta,1}=\frac{I-\sin\theta\,\sigma_x+\cos\theta\,\sigma_y}{2}, \qquad p_\theta(1|\phi)=\operatorname{Tr}(\rho_\phi E_{\theta,1}).$$

Changing pulse conventions can replace the sine by a cosine or reverse its sign. Nothing operational changes if the same convention is used for the state, analysis phase, and estimator. The parameter $\theta$ is not cosmetic: it moves the high-slope region of the fringe to the current estimate of $\phi$.

Loss of coherence reduces the fringe contrast. A common effective model is

$$\begin{aligned} p_\theta(1|\phi,C) &= \frac12 \left[ 1+C\sin(\phi-\theta) \right], \\ 0&\le C\le 1. \end{aligned}$$

Here $C$ can include dephasing, inhomogeneous broadening, imperfect preparation, and analysis-pulse errors. Those mechanisms should be separated when the scientific claim depends on their interpretation, even if they enter the simplest likelihood through the same observed contrast.

From Outcomes to a Likelihood

A single projective readout does not reveal a probability or a phase. It produces one bit. For $N$ independent probes exposed to the same phase in one shot, let $K$ be the number of outcomes equal to one. Conditional on fixed $\phi$, $C$, and $\theta$,

$$K \sim \operatorname{Binomial} \left( N,p_\theta(1|\phi,C) \right).$$

For settings indexed by $j$, the likelihood is

$$\mathcal L(\phi,C) = \prod_j {N_j\choose K_j} p_j^{K_j} (1-p_j)^{N_j-K_j},$$

with

$$p_j = p_{\theta_j}(1|\phi_j,C_j).$$

The index can represent repeated shots, different analysis phases, different interrogation times, or controlled calibration settings. Writing this likelihood before choosing an estimator exposes several issues that a fringe plot can hide:

• $\phi$ is periodic and may not be globally identifiable;
• contrast and offset can be nuisance parameters;
• outcomes within a shot may cease to be independent under common-mode noise;
• postselection changes the likelihood and accepted resource count;
• drift can make nominally repeated trials non-identical.

The sufficient statistic is $K$ only under the independent, identical Bernoulli model. Time tags, detector channels, or individual-probe labels can carry additional information when noise or readout varies across the data.

Local Readout and Projection Noise

Suppose the controller keeps the interferometer near $\phi=\theta$, where the sine fringe crosses one half. If $f=K/N$ is the measured excitation fraction, a local linear estimator is

$$\widehat\phi \simeq \theta + \frac{2f-1}{C}.$$

At the working point, $p=1/2$ and

$$\operatorname{Var}(f) = \frac{1}{4N}.$$

Error propagation therefore gives

$$\operatorname{Var}(\widehat\phi) \simeq \frac{1}{NC^2}.$$

For $\nu$ statistically independent shots with stable settings,

$$\operatorname{Var}(\widehat\phi) \simeq \frac{1}{\nu NC^2}.$$

At $C=1$, this is the independent-probe standard quantum limit for phase. The word projection describes the Bernoulli population fluctuations of a coherent superposition; it does not mean that every measured variance is quantum projection noise. Atom-number fluctuations, detector noise, phase jitter, and technical state-preparation noise must be modeled or bounded separately.

The estimator above is locally unbiased exactly at the calibrated working point: its mean equals $\phi$ there and has derivative one. Away from that point its mean is $\theta+\sin(\phi-\theta)$, so fringe curvature creates bias; for sufficiently low $C$, it can also return values outside the chosen phase branch. A maximum-likelihood or Bayesian estimator using the full periodic likelihood is usually preferable when the phase uncertainty is not already small.

Fisher Information of the Fringe

For one binary outcome, the classical Fisher information for $\phi$ is

$$F_1(\phi;\theta,C) = \frac{ \left(\partial_\phi p_\theta\right)^2 }{ p_\theta(1-p_\theta) }.$$

Substituting the contrast-reduced sine fringe gives

$$F_1 = \frac{ C^2\cos^2(\phi-\theta) }{ 1-C^2\sin^2(\phi-\theta) }.$$

At the quadrature working point,

$$F_1(\theta;\theta,C) = C^2.$$

Independent probes and shots add information:

$$F_{\mathrm{tot}} = \nu N C^2$$

at that point. The local Cramér–Rao bound then reproduces the projection-noise result,

$$\operatorname{Var}(\widehat\phi) \ge \frac{1}{\nu NC^2}.$$

For imperfect contrast, the fringe extrema contain little local information because their slopes vanish. In the mathematical $C=1$ model the expression has a limiting value away from the extrema, but an exactly deterministic outcome is a nonregular operating point and is not robust to model error. Precision experiments deliberately use a high-slope discriminator rather than relying on that singular limit.

Quantum Fisher Information

If $C$ describes the encoded state before measurement and is independent of the unknown phase, an equatorial qubit with dephasing can be written

$$\rho_\phi = \frac12 \left[ I + C\left( \cos\phi\,\sigma_x + \sin\phi\,\sigma_y \right) \right].$$

Its quantum Fisher information for $\phi$ is

$$F_Q[\rho_\phi] = C^2.$$

The quadrature Ramsey measurement attains this value locally. This example is important because it shows what measurement optimization means in practice: the QFI is not merely an abstract bound; an analysis phase aligned to the current estimate implements an optimal local measurement for this model.

Three qualifications remain:

1. the optimal axis depends on the unknown phase, so a prior, pilot estimate, or adaptive controller is needed;
2. QFI is local and does not resolve the periodic phase aliases;
3. an achieved detector likelihood may contain SPAM errors or nuisance parameters absent from the state-only model.

The general distinctions between implemented Fisher information, SLD QFI, and attainable multiparameter bounds belong to Classical and Quantum Fisher Information.

Phase, Frequency, and a General Signal

For a constant detuning $\delta\omega$ accumulated over free-evolution time $T$,

$$\phi = \delta\omega T.$$

Parameter transformation gives

$$F_{\delta\omega} = T^2F_\phi.$$

Thus one independent probe at the optimal working point carries

$$F_{\delta\omega} = C(T)^2T^2.$$

For a signal parameter $\lambda$ coupled through an instantaneous angular frequency shift $\Omega(\lambda,t)$, ideal Ramsey evolution gives

$$\phi(\lambda) = \int_0^T \Omega(\lambda,t) \,dt.$$

More generally, control modulation introduces a sensitivity function $y(t)$:

$$\phi(\lambda) = \int_0^T y(t)\Omega(\lambda,t) \,dt.$$

For ordinary free precession, $y(t)=1$ between the two pulses. If the contrast is independent of $\lambda$ and readout is at quadrature, the parameter information is

$$F_\lambda = C^2 \left( \frac{\partial\phi}{\partial\lambda} \right)^2.$$

This formula separates three design ingredients: coherent response, contrast, and readout. It also makes units explicit. A phase sensitivity, a frequency sensitivity, and a magnetic-field sensitivity are not interchangeable without the response coefficient and interrogation time.

Information per Unit Time

Longer free evolution increases phase response, but a real experiment pays for state preparation, control, measurement, and reset. Let

$$T_c = T+t_{\mathrm d}$$

be the cycle time, where $t_{\mathrm d}$ collects dead time and overhead. For a large number of cycles in total averaging time $\tau$, the number of independent cycles is approximately $\nu=\tau/T_c$. For $N$ independent probes per cycle,

$$\operatorname{Var}(\widehat{\delta\omega}) \ge \frac{T_c}{ \tau N C(T)^2T^2 }.$$

The frequency information rate is therefore

$$\dot F_{\delta\omega} = \frac{NC(T)^2T^2}{T_c}.$$

For a constant response $\partial_\lambda\phi=\kappa T$, a convenient sensitivity coefficient is

$$\eta_\lambda \equiv \sqrt{\tau} \,\Delta\widehat\lambda \ge \frac{ \sqrt{T_c} }{ |\kappa| C(T)T\sqrt N }.$$

This is why the smallest one-shot phase variance need not identify the best sensor. A protocol can improve one shot while losing through slower cycles, lower duty factor, or reduced accepted-shot probability.

For a stretched-exponential Ramsey contrast

$$C(T) = \exp\left[-(T/T_2^*)^q\right],$$

with $q>0$ and negligible dead time, maximizing information per unit time gives

$$T_{\mathrm{opt}} = T_2^* \left( \frac{1}{2q} \right)^{1/q}.$$

With nonzero dead time, the optimum solves

$$\frac{2}{T} - \frac{1}{T+t_{\mathrm d}} - \frac{2qT^{q-1}}{(T_2^*)^q} = 0.$$

The optimal $T$ depends on the objective. Maximizing single-shot frequency information, information per wall-clock time, bandwidth, or tracking performance produces different answers.

Periodicity and Dynamic Range

A Ramsey fringe is periodic:

$$p_\theta(1|\phi+2\pi m,C) = p_\theta(1|\phi,C), \qquad m\in\mathbb Z.$$

For fixed $T$, frequency values separated by $2\pi/T$ are aliases. A local uncertainty such as $1/(CT\sqrt{N\nu})$ is meaningful only after the correct fringe branch has been identified. If the prior support is broader than one period, a single interrogation time cannot determine the frequency globally.

Two analysis phases can distinguish the reflection ambiguity of one sine fringe. For settings $\theta$ and $\theta+\pi/2$, define, with known $C>0$,

$$\widehat s=\frac{2f_\theta-1}{C},\qquad \widehat c=-\frac{2f_{\theta+\pi/2}-1}{C}.$$

The minus sign follows from the sine convention: $\sin(\phi-\theta-\pi/2)=-\cos(\phi-\theta)$. The circular phase estimate is

$$\widehat\phi = \theta + \operatorname{atan2} \left( \widehat s, \widehat c \right).$$

Quadrature readout distinguishes phase within a $2\pi$ interval but does not remove the integer alias. Low contrast also makes the angle estimator non-Gaussian; near the origin of the estimated quadrature plane, a symmetric confidence region is usually misleading.

Dynamic range is not just the fringe period. It also depends on prior knowledge, control range, sampling schedule, update latency, drift between settings, and the allowed probability of a cycle slip. A sensitivity number without an unambiguous range or tracking model is incomplete when the signal is not already locked near quadrature.

Multitime and Adaptive Ramsey Estimation

Broad-range protocols combine short and long interrogation times. A short time has weak phase leverage but a wide frequency period; a long time has high local resolution but many aliases. A geometric schedule such as

$$T_k = 2^kT_0$$

can identify coarse frequency bits before refining the estimate. Repetitions at each scale suppress branch errors.

For a fixed unknown angular detuning $\lambda$, each duration encodes $\phi_j=\lambda T_j$. In a Bayesian implementation, the posterior after outcome $x_j$ is

$$\pi_j(\lambda) \propto p(x_j|\lambda T_j,\theta_j) \pi_{j-1}(\lambda).$$

The next setting may be chosen to minimize expected posterior loss, maximize expected information gain, avoid a cycle slip, or satisfy a latency and bandwidth constraint. Adaptation can change $\theta_j$, $T_j$, or both.

The posterior tracks the same physical detuning at every duration, rather than treating each accumulated phase as the same unknown. The chosen setting is held fixed when differentiating its conditional likelihood; its dependence on previous outcomes is handled through the joint record.

Longer coherent phase accumulation can replace some spatial entanglement with repeated temporal queries. Heisenberg Scaling explains the ideal query scaling and its global ambiguity. Preparation, measurement, feedback latency, and branch errors determine whether that gain survives in elapsed time.

Readout Errors and Nuisance Parameters

Let $\alpha$ be the probability of reporting one when the true state is zero, and let $\beta$ be the probability of reporting zero when the true state is one. If the ideal Ramsey probability is $p$, the observed probability is

$$q = \alpha(1-p) + (1-\beta)p.$$

For the sine fringe,

$$q = \frac12 \left[ 1+\alpha-\beta + (1-\alpha-\beta) C\sin(\phi-\theta) \right].$$

Readout asymmetry creates an offset; the fringe amplitude is multiplied by $1-\alpha-\beta$. Its magnitude cannot increase. If this factor is negative, relabeling the reported outcomes reverses the fringe. At quadrature the single-outcome Fisher information is

$$F_1 = \frac{ (1-\alpha-\beta)^2C^2 }{ 1-(\alpha-\beta)^2 }.$$

For symmetric error $\alpha=\beta=\epsilon$, this becomes

$$F_1 = (1-2\epsilon)^2C^2.$$

Correcting a fitted contrast by a calibrated readout fidelity can estimate the premeasurement state, but it does not restore information lost by the actual detector. A state-characterization claim and an achieved sensor-sensitivity claim therefore answer different physical questions.

Contrast $C$, offset $b$, analysis-phase drift, and coupling calibration are often inferred jointly with the signal. For parameters $\boldsymbol\vartheta=(\phi,C,b,\ldots)$, the Fisher matrix is

$$[F(\boldsymbol\vartheta)]_{ab} = \sum_x p(x|\boldsymbol\vartheta) \, \partial_a\ln p \, \partial_b\ln p.$$

When the joint Fisher matrix is invertible and the estimator is locally unbiased with respect to both signal and nuisance directions, the bound is

$$\operatorname{Var}(\widehat\phi) \ge [F^{-1}]_{\phi\phi},$$

not $1/F_{\phi\phi}$. A single binary setting supplies a rank-one Fisher matrix and cannot identify several unconstrained parameters. Additional analysis phases or calibration measurements must provide the missing information; their shots and elapsed time also count.

Decoherence and Correlated Noise

The contrast function is an averaged coherence,

$$C(T) = \left| \left\langle e^{i\delta\phi(T)} \right\rangle \right|,$$

where $\delta\phi$ is random phase accumulated from uncontrolled fields or frequency fluctuations. Exponential contrast often indicates approximately Markovian dephasing, while Gaussian decay commonly arises from quasistatic or inhomogeneous detuning. The envelope alone rarely identifies a unique microscopic noise process.

For a stationary classical frequency noise $\delta\omega(t)$, the phase variance can be expressed through a filter function. With

$$Y(\omega) = \int_0^T y(t)e^{i\omega t} \,dt,$$

Take the two-sided angular-frequency spectrum $S_{\delta\omega}(\omega)=\int_{-\infty}^{\infty}dt\, e^{i\omega t}\langle\delta\omega(t)\delta\omega(0)\rangle$. For zero-mean stationary Gaussian noise and a deterministic $y(t)$,

$$C(T) = \exp\left[ -\frac12 \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} S_{\delta\omega}(\omega) |Y(\omega)|^2 \right].$$

Ordinary Ramsey evolution has

$$|Y(\omega)|^2 = \frac{ 4\sin^2(\omega T/2) }{ \omega^2 },$$

so it is strongly sensitive to quasistatic and low-frequency detuning. Echo and dynamical-decoupling sequences alter $y(t)$ and shift sensitivity toward selected nonzero frequencies; they answer a different sensing question.

Common-mode phase noise also invalidates naive independent-probe scaling. If all $N$ probes share an unobserved phase $\delta$, then

$$P(K|\phi) = \int d\delta\, P(\delta) \operatorname{Bin} \left[ K;N,p(\phi+\delta) \right].$$

Outcomes are conditionally independent at fixed $\delta$ but correlated after $\delta$ is marginalized. Increasing $N$ can suppress projection noise while leaving a common oscillator or environmental floor unchanged. Likewise, dead-time aliasing can fold oscillator noise into a clock measurement even when each Ramsey shot is otherwise ideal.

Correlated and Entangled Probes

Ramsey interrogation is not restricted to independent qubits. For a collective spin, the first operation prepares a transverse mean spin, the signal rotates it, and the final analysis maps a transverse component to a population difference. A spin-squeezed input can reduce readout noise while retaining signal slope. Its local phase variance is

$$(\Delta\phi)^2 = \frac{\xi_R^2}{N}$$

for an aligned ideal readout. The contrast, squeezed variance, cycle time, and detector noise must all refer to the same operating sequence. Spin Squeezing develops the collective covariance, preparation, and entanglement criteria.

An ideal $N$-qubit GHZ state,

$$|\mathrm{GHZ}_N\rangle = \frac{ |0\rangle^{\otimes N} + |1\rangle^{\otimes N} }{\sqrt2},$$

accumulates the amplified phase $N\phi$. A parity-like readout can have

$$p(1|\phi) = \frac12 \left[ 1+C_N\sin(N\phi-\theta) \right],$$

and local Fisher information

$$F_\phi = C_N^2N^2$$

at quadrature. The same factor $N$ reduces the state-phase period to $2\pi/N$; two analysis phases are still needed to resolve the reflection ambiguity of a single sine fringe within that period. Independent dephasing can make $C_N$ decay rapidly with $N$ and $T$. Under independent Markovian dephasing with no preparation dead time, maximally entangled probes need not improve the asymptotic frequency resolution over optimized uncorrelated Ramsey spectroscopy.

Entanglement can still improve constants, finite-resource performance, or operation under structured noise. The defensible question is not whether the input is entangled, but whether the complete protocol improves the declared loss under the same time, probe, prior, bandwidth, and success conditions.

Representative Sensing Mappings

The Ramsey information structure is portable, but each platform supplies a different phase response and noise model.

Task	Encoded phase	Essential calibration	Common limiting noise
Frequency comparison	$\phi=(\omega_0-\omega_{\mathrm{LO}})T$	interrogation time and oscillator phase	local-oscillator noise, dead time, collisions
DC magnetometry	$\phi=\gamma BT$	gyromagnetic ratio and field projection	$T_2^*$ dephasing, field drift, readout noise
Electric-field sensing	$\phi=\int\Delta\omega_E(t)dt$	Stark response and field geometry	electric-field noise, level mixing, calibration drift
Inertial phase readout	platform-dependent action phase	scale factor, timing, trajectory	vibration, laser phase, wave-front error
Qubit calibration	$\phi=\delta\omega T+\phi_c$	pulse phase and timing	control drift, leakage, correlated phase noise

For a spin with Hamiltonian

$$H_B = \frac{\hbar\gamma B}{2} \sigma_z,$$

the phase is $\phi=\gamma BT$. The projection-noise-limited field sensitivity per square root bandwidth is

$$\eta_B \ge \frac{ \sqrt{T_c} }{ |\gamma| C(T)T\sqrt N }.$$

This formula is useful only after specifying whether $N$ counts simultaneous probes, repeated probes, detected probes, or accepted probes. Spatial resolution, standoff distance, sensor volume, and invasiveness can be as important as field sensitivity in an application.

Exercises

1. Shared phase noise differs from independent dephasing

Let every probe in a shot experience the same $\delta\sim\mathcal N(0,\sigma^2)$ and let $p(1|\phi,\delta)=[1+\sin(\phi+\delta)]/2$. Find $\operatorname{Var}(K)$ at $\phi=0$ and the variance of the locally unbiased estimator $(2K/N-1)/C$, where $C=e^{-\sigma^2/2}$.

Solution

The marginal mean probability is $1/2$. Its conditional variation has

$$\operatorname{Var}_\delta(p) =\frac14\mathbb E[\sin^2\delta] =\frac{1-e^{-2\sigma^2}}8.$$

By total variance for $K|\delta\sim\operatorname{Binomial}(N,p)$,

$$\operatorname{Var}(K) =\frac N4+N(N-1)\frac{1-e^{-2\sigma^2}}8.$$

Consequently,

$$\operatorname{Var}(\widehat\phi) =\frac{e^{\sigma^2}}{N} +\left(1-\frac1N\right)\sinh(\sigma^2).$$

The large-$N$ variance floor is $\sinh(\sigma^2)\simeq\sigma^2$ for small phase noise. Independent phase errors with the same marginal contrast would instead give $e^{\sigma^2}/N$. A contrast measurement alone cannot distinguish these joint noise models.

2. Projection-noise estimator

$N$ independent probes are operated at quadrature with known contrast $C$. Let $f=K/N$. Show that

$$\widehat\phi = \frac{2f-1}{C}$$

is locally unbiased at $\phi=0$ and find its variance there.

Solution

Near zero,

$$\mathbb E[f] = \frac12(1+C\sin\phi) \simeq \frac12(1+C\phi).$$

Therefore

$$\mathbb E[\widehat\phi] \simeq \phi.$$

At $\phi=0$, $p=1/2$ and

$$\operatorname{Var}(f) = \frac{p(1-p)}{N} = \frac{1}{4N}.$$

Thus

$$\operatorname{Var}(\widehat\phi) = \frac{4}{C^2} \operatorname{Var}(f) = \frac{1}{NC^2}.$$

3. Optimal time under exponential dephasing

Assume negligible dead time and

$$C(T)=e^{-T/T_2^*}.$$

Find the interrogation time that maximizes frequency Fisher information per unit time.

Solution

The information per cycle is proportional to $C(T)^2T^2$. With cycle time $T$, the information rate is proportional to

$$R(T) = Te^{-2T/T_2^*}.$$

Its logarithmic derivative is

$$\frac{d}{dT}\ln R = \frac1T - \frac{2}{T_2^*}.$$

The maximum occurs at

$$T_{\mathrm{opt}} = \frac{T_2^*}{2}.$$

Maximizing single-shot information instead would give $T=T_2^*$, showing why the time objective must be stated.

4. Readout confusion

Take symmetric state-classification error $\alpha=\beta=\epsilon=0.05$ and intrinsic contrast $C=0.8$. Find the observed contrast and one-outcome Fisher information at quadrature.

Solution

Symmetric confusion multiplies contrast by $1-2\epsilon$, so

$$C_{\mathrm{obs}} = (1-2\epsilon)C = 0.9\times0.8 = 0.72.$$

At quadrature,

$$F_1 = C_{\mathrm{obs}}^2 = 0.5184.$$

Correcting the contrast can infer the state before readout, but the detector still extracts only this smaller amount of information per outcome.

5. Frequency aliases

A Ramsey experiment uses $T=20\,\mathrm{ms}$. By how much can angular frequency change while producing the same ideal fringe? Give the corresponding ordinary-frequency spacing.

Solution

The fringe is unchanged when

$$\delta\omega T \longrightarrow \delta\omega T+2\pi m.$$

Adjacent angular-frequency aliases are separated by

$$\Delta\omega_{\mathrm{alias}} = \frac{2\pi}{T} = 100\pi\,\mathrm{rad\,s^{-1}}.$$

The ordinary-frequency spacing is

$$\Delta f_{\mathrm{alias}} = \frac{1}{T} = 50\,\mathrm{Hz}.$$

6. Duty-cycle penalty

Two protocols have the same $N$, contrast, and interrogation time $T$. Protocol A has no dead time, while protocol B has $t_{\mathrm d}=3T$. Compare their frequency standard deviations after the same wall-clock averaging time.

Solution

The variance scales with cycle time $T_c=T+t_{\mathrm d}$. Hence

$$\frac{ \Delta\widehat{\delta\omega}_B }{ \Delta\widehat{\delta\omega}_A } = \sqrt{ \frac{T_{c,B}}{T_{c,A}} } = \sqrt{ \frac{4T}{T} } = 2.$$

Protocol B has twice the standard deviation after equal wall-clock time, despite identical one-shot statistics.

7. GHZ resolution and range

Compare an ideal $N$-qubit GHZ Ramsey fringe with an independent-probe fringe. What happens to local Fisher information and the state-phase period?

Solution

At quadrature and unit contrast, the GHZ fringe depends on $N\phi$, so

$$F_{\phi,\mathrm{GHZ}} = N^2.$$

$N$ independent probes give total information $N$. The GHZ state therefore has an ideal factor-$N$ local information gain for the same simultaneous probe count. Its fringe period is

$$\frac{2\pi}{N},$$

instead of $2\pi$. The local gain is accompanied by a factor-$N$ reduction in phase period, before including GHZ preparation cost or faster decoherence. A single sine measurement also has a reflection ambiguity within its period; a second analysis phase resolves that ambiguity.

8. A magnetic-field sensitivity

An ensemble of $N=10^4$ independent spins has $\gamma=2\pi\times28\,\mathrm{GHz\,T^{-1}}$, $T=1\,\mu\mathrm{s}$, $C=0.7$, and $T_c=5\,\mu\mathrm{s}$. Estimate the projection-noise-limited $\eta_B$.

Solution

Use

$$\eta_B = \frac{\sqrt{T_c}}{\gamma CT\sqrt N}.$$

Substitution gives

$$\begin{aligned} \eta_B &= \frac{ \sqrt{5\times10^{-6}\,\mathrm s} }{ (2\pi)(28\times10^9\,\mathrm{s^{-1}T^{-1}}) (0.7)(10^{-6}\,\mathrm s)(100) } \\ &\simeq 1.8\times10^{-10} \,\mathrm{T\,Hz^{-1/2}}. \end{aligned}$$

This is about $180\,\mathrm{pT\,Hz^{-1/2}}$. It is a model benchmark, not a complete instrument sensitivity: calibration noise, spatial response, technical phase noise, and long-term drift are absent.

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