Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions

The quantum noise of the vacuum limits the achievable sensitivity of quantum sensors. In non-classical measurement schemes the noise can be reduced to overcome this limitation. However, schemes based on squeezed or Schrödinger cat states require alignment of the relative phase between the measured interaction and the non-classical quantum state. Here we present two measurement schemes on a trapped ion prepared in a motional Fock state for displacement and frequency metrology that are insensitive to this phase. The achieved statistical uncertainty is below the standard quantum limit set by quantum vacuum fluctuations, enabling applications in spectroscopy and mass measurements.


Supplementary Note 1 Estimation of the achievable force sensitivity
The amplitude F of a resonant oscillating force required to get a displacement of α after time t F is given by where z 0 = /2mω z is the ground state wave packet extent for an atom with mass m, trapped in a harmonic potential with trap frequency ω z . Therefore, the statistical uncertainty for a force estimation can be written as with F, the Fisher information for the α-estimation (measured result shown in Fig. 2 a) and N the number of experiments. Rewriting this expression with N = τ /t cycle , and introducing R F = t cycle /t F as the ratio of cycle time and t F we obtain where ∆ c = 1/t cycle is the measurement bandwidth and τ the total measurement time. For 25 Mg + with ω z = 2π × 1.89 MHz, t F = 10 µs, t cycle = 15 ms, F = 5 and τ = t cycle the resulting force sensitivity is 112 yN/ √ Hz. It can be enhanced by one to two orders of magnitude by applying a smaller force for a longer time (see Eq. 2). For comparison, Biercuk et al. achieved a sensitivity of around 390 yN/ √ Hz [1] with 100 ions. Shaniv and Ozeri demonstrated a sensitivity of 28 zN/ √ Hz [2]. Both schemes rely on Doppler velocimetry. In contrast to that, the scheme presented here relies on a projective measurement, which is ideally capable of saturating the quantum Fisher information.
In a more general framework that is used in the context of heating rate measurements in ion traps, fluctuating electric fields exert forces onto the trapped ions [3]. The electric field is characterized by the single-sided electric field spectral density According to [4] the electric field spectral density can be related to the rate of motional excitation from the motional ground state to the first excited motional Fock state Γ by where q is the charge of the ion. It can be seen, that the change in motional excitation only depends on the electric field spectral density at the trapping frequency ω z . The root-mean-square force F rms can be inferred from the electric field spectral density by Consequently, in general it is not possible to infer a force from a measurement of the motional excitation only, but additional knowledge about the spectrum of the oscillating field is required.
The scenario treated in the manuscript corresponds to an electric field spectral density that is narrower than the Fourier-limited bandwith of the measurement. Therefore, we can assume a Fourierlimited, nearly monochromatic oscillating electric field E(t) = E 0 cos(ωt) applied for a time t F with Inserting the electric field spectral density into Supplementary Eq. 6, we find For a coherent drive, as considered in the manuscript, we can use | 0|α | 2 = e −|α| 2 ≈ 1 − |α| 2 assuming that the displacement is small. Therefore the rate of transfering population from motional Fock state |0 to |1 is given by Γt F = |α| 2 which gives us Supplementary Eq. 1. This formalism can also be used to derive a lower bound on detectable forces in the presence of heating with a rateṅ, which was done before by Maiwald et al. [5]. To distinguish a motional excitation n coh = |α| 2 due to a force signal from unintended motional heating n h =ṅt F , the inequality n coh ≥ n h must hold and therefore It should be noted, that this estimation does not consider the fact that stochastic forces such as motional heating and a coherent drive lead to different motionally excited states, which requires a more detailed treatment that also considers the particular measurement scheme. Stochastic forces with an electric field spectral density that is much broader than the Fourier-limited bandwidth are another extreme case usually investigated in the context of heating rate measurements. Modelling the electric field spectral density responsible for anomalous heating in ion traps is still subject of investigations [3]. For an order-of-magnitude estimate, we can assume a box-like spectrum centered around ω z with width ∆ h , i.e.
Under this assumption we can relate a root-mean-square force F rms to the measured heating rate with Considering the lowest reported electric field spectral densities from heating rate measurements in the S h E ∼ 10 −15 V 2 m 2 Hz range [6,7] we associate a force of F h rms ≈ 2 Hz . Note that ∆ h is not the Fourier-limited bandwidth ∆ c in this case but the intrinsic spectral width of the electric field noise that causes the heating, which certainly can exceed the MHz regime.

Supplementary Note 2 Error analysis for Fisher information measurement
Below we will use the shortened notation P |↓ (α i ) ≡ P i and s(α i ) ≡ s i .

Errors on slope determination
Quantum projection noise The slope is determined by measuring the population at two neighboring points. The error associated with this measurement is ∆P i−1 = where N is the number of independent measurements and is propagated to the slope error due to quantum projection noise via Fit error for displacement determination To determine the implemented displacement, we fitted the expected theoretical curve to the measured data for the motional ground state. The fit error ∆α also gives an error on the denominator in the difference quotient. The resulting slope error reads Error due to finite step size The difference quotient only gives an approximation of the true slope of the signal. For a symmetric derivative the first order error to this approximation is given by and in the asymmetric case by For the error estimation we used the theoretically calculated derivative of P = exp(−|α| 2 )L n (|α| 2 ) 2 , where n denotes the number of excitations in the Fock state, without additional parameters accounting for the reduced contrast observed in the experiment.
Total slope error The total error on the estimation of the signal slope s then reads

Errors on QPN determination
Since we use a binary data set to determine the expectation value for the spin measurement, the value for the variance is exact. However, for finite N statistical fluctuation will give rise to an uncertainty given by ∆QP

Total error
The total estimated error on the measurement of the Fisher information is given by (18) The most significant violation of the standard quantum limit has been observed for point i = 5 in the Fock state n = 1 data. The Fisher Information for this measurement was F 5 = 5.37(63) and the different uncertainties are summarized in the error budget in Supplementary Table 1 Supplementary Note 3 Ramsey pattern line shape The individual displacement pulses start at t 0 and are applied for a duration t F . The oscillating force is detuned by δ from the axial trap frequency ω z , resulting in the time dependent interaction Hamiltonian where γ(t) = iΩe i(δ·t+φ LO ) and Ω = −qE 0 z 0 /(2 ) (see also Methods' section Trap modulation to implement displacement operator ). The dynamics is given by the unitary evolution [9] where the interaction starts at t 0 and has duration t. The displacement and phase are: For the total sequence the evolution is then where subscripts represent a phase change of the local oscillator to undo the initial displacement. Up to a global phase this results in a displacement So the residual displacement at the end of the sequence is This residual displacement can be detected by the overlap measurement described in the main text and gives the final result for the line shape of the Ramsey pattern | n|D(α)|n | 2 = exp(−|α| 2 ) L n (|α| 2 ) 2 . The dashed line indicates the standard quantum limit. Another value for n rescales the y-axis. It can be seen that the average value for the Fisher information only depends on the mean phonon excitation for states with X = 0 and P = 0. In the case of displacement sensing with an unknown phase, a suitable figure of merit is the sensitivity minimized over all phases. From the graph, it can be seen that Fock states (min φ F Q ( n = 1) = 12) outperform Schrödinger cat states (min φ F Q ( n = 1) = 4) and squeezed states (min φ F Q ( n = 1) ≈ 0.69).

Supplementary Note 4 Optimal estimation of a displacement amplitude without phase information
We consider the metrological task of estimating the amplitude of a displacement. The phase of the displacement is unknown at the time of the state preparation. To optimize the sensitivity of the estimation, the 'detector' shall be prepared in an optimal quantum state. The unitary process which generates the phase shift is given by the displacement where we defined real parameters θ and φ LO , such that α = θe −iφ LO /2 andR(φ LO ) = (sin(φ LO )X + cos(φ LO )P )/ √ 2. Furthermore, we usedX = (â +â † )/ √ 2 andP = i(â † −â)/ √ 2. Our goal is to estimate the parameter θ = 2|α|. To distinguish the Fisher Information with respect to |α|, F, from the Fisher information with respect to θ, the latter is denoted by a Gothic type F. The connection between them is The reason for introducing F is to normalize the SQL to one, which is a widely used convention in the literature [10], whereas in the main manuscript the classical limit is at 4, but the estimated parameter is |α| from the standard definition of the displacment operator. According to Eq. (7) of the main text, the sensitivity of the estimation of θ is bounded by the quantum Fisher information F Q [ρ,R(φ LO )], which depends on the phase φ LO via the generatorR(φ LO ). In a "worst-case" scenario the sensitivity may be reduced to In order to prepare the detector such as to render it most sensitive, even in this worst-case scenario, we need to maximize the figure of merit F min [ρ]. An alternative strategy consists in optimizing the average performance, as quantified by the figure of merit Below, we derive the limits on these two figures of merit as a function of the number of excitations.
We will see that a Fock state maximizes the sensitivity in both cases.

Optimizing the minimum sensitivity
Let us first focus on the quantity given by Supplementary Eq. (31).
, we find the following upper bound where we introduced the 2 × 2 covariance matrix with Cov(X,P )ρ = 1 2 and a unit vector n = (sin(φ LO ), cos(φ LO )). The minimum in Supplementary Eq. (33) is given by the smallest eigenvalue λ min of the matrix Γρ. This eigenvalue can again be bounded from above: where n = Tr{ρâ †â } determines the number of excitations. Hence, the minimal sensitivity is generally bounded by

Optimal states must be quantum non-Gaussian
States which reach the upper bound (37) for n > 0 must necessarily be quantum non-Gaussian, i.e., they cannot be written as a mixture of Gaussian states (see e.g. Supplementary References [12,13]). To see this, notice that in (36), equality is reached if and only if the conditions X ρ = P ρ = Cov(X,P )ρ = 0 and ∆Xρ = ∆Pρ are satisfied, or equivalently, â ρ = â † ρ = ââ ρ = â †â † ρ = 0. These conditions can only be satisfied by a Gaussian state when n = 0, i.e., the vacuum state |0 . The above statement then follows together with the convexity of the quantum Fisher information.
All pure states lead to equality in Supplementary Equation (33). In the case of a pure state, non-Gaussianity is equivalent to a negative Wigner function [14].
Hence, mixtures of Gaussian states will always perform sub-optimally for the estimation of a displacement amplitude in a "worst-case" scenario.

Average sensitivity bound
The same bound also holds for the average of Supplementary Eq. (32). We obtain and equality is reached only by states that are not displaced: X ρ = P ρ = 0.
In summary, for a fixed energy (given by n), the Fock state provides the optimal precision for displacement detections with unknown phase. This is true for both strategies, i.e., preparing for the worst-case scenario (31) or optimizing the mean performance (32).

Supplementary Note 5 Classical limit
Based on the Glauber-Sudarshan P-representation, we define classical statesρ cl as those for which P (α) describes a probability distribution. Here |α =D(α)|0 is a coherent state. The classical limit is then given as the maximum quantum Fisher information, taken over all classical statesρ c . Since the Fisher information is convex, the maximum is attained by a pure coherent state |α .
which is independent of φ and α and corresponds to the sensitivity of the vacuum state. Thus, any observation of Supplementary Figure 3: Analogy with a two-mode interferometer. The presented measurements can be understood in terms of a two-mode interferometer, with a coherent state and a Fock state at the input ports |ψ in = |n |α . The output state is |Ψ out =Û (ϕ) |Ψ in , withÛ given in Eq. 49 reveals that the stateρ is non-classical according to the above definition [15]. This form of nonclassicality is a necessary resource to overcome the classical limit eq. (43). The resulting upper bound on the Fisher information for amplitude measurements using classical states is given by The corresponding bound for the phase measurement can be infered from Assuming that the detuning δ is small compared to the displacement Ω rate, we get and for the Quantum Fisher information which results in the SQL limit given in eq. (3) in the main text.

Supplementary Note 6 Analogy with a two-mode interferometer
Fock-state metrology can be understood in the wider context of a two-mode Mach-Zehnder interferometer [16] as shown in Supplementary Fig. 3. The evolution in a two-mode interferometer (with modeŝ a andb) is described by,Û whereĴ y = (â †b −b †â )/2i and ϕ is the phase shift. For the two-mode transformation (49) one obtains the classical limit, i.e., the maximum sensitivity for two-mode classical states, where n a + n b is the total number of excitations (n a = Tr{ρâ †â } and n b = Tr{ρb †b }). This bound is known as the shot-noise limit and coincides with the sensitivity bound for states that are separable among particles [17]. For a Fock state |n in input modeâ, the quantum Fisher information reads which yields a quantum-enhanced sensitivity for any n > 0 [16], in agreement with the results from the previous section. The specific case of the displacement discussed in Supplementary Note 4 is recovered in the homodyne limit, in which a classical, highly populated coherent state is inserted in one of the input ports. We thus assume that input modeb is prepared in a coherent stateρ b = |α 0 α 0 | with |α 0 | 2 = n 0 1 particles, while the quantum stateρ a of modeâ input is arbitrary. Neglecting quantum fluctuations, by making the replacementb → α andb † → α * in Eq. (49), we obtain an effective transformation of modeâ, described byÛ The evolution of modeâ is effectively given by the displacement (29)  For a Fock state in the second input port, we obtain a precision of F Q [||n ,Ĥ(φ LO )] = 2n 0 n + n 0 , independently of the phase of α 0 , which allows for sub-shot-noise sensitivity for n > 0, in agreement with our previous considerations (see Supplementary Note 4).
As discussed above, the classical bound and the Fock-state sensitivity can be equivalently obtained from the two-mode results (50) and (51), remembering that in the homodyne limit considered here, the contribution of n a or n to the total number of particles is negligible due to n b = n 0 n a , n.