Certified quantum non-demolition measurement of material systems

An extensive debate on quantum non-demolition (QND) measurement, reviewed in Grangier et al. [Nature, {\bf 396}, 537 (1998)], finds that true QND measurements must have both non-classical state-preparation capability and non-classical information-damage tradeoff. Existing figures of merit for these non-classicality criteria require direct measurement of the signal variable and are thus difficult to apply to optically-probed material systems. Here we describe a method to demonstrate both criteria without need for to direct signal measurements. Using a covariance matrix formalism and a general noise model, we compute meter observables for QND measurement triples, which suffice to compute all QND figures of merit. The result will allow certified QND measurement of atomic spin ensembles using existing techniques.


I. INTRODUCTION
A quantum non-demolition (QND) measurement is one which provides information about a quantum variable while leaving it unchanged and accessible for future measurements.
The approach was originally suggested as a means to avoid measurement back-action in gravitational wave detection [1][2][3][4][5]. QND measurements of optical fields both provided the first demonstrations and led to a considerable refinement of the understanding of QND measurements in practice [6]. More recently, QND measurements have been employed to prepare spin-squeezed atomic states [7][8][9][10][11] and with nano-mechanical systems [12].
In a generic QND measurement, a 'meter' and a 'system' variable interact via a selected Hamiltonian. The meter can then be directly measured to gain indirect information about the system. In the context of optical QND measurements, the question of when a measurement should be considered QND has been much discussed (see [6] for references). Two distinct non-classicality criteria emerge: A state preparation criterion requires small uncertainty in the system variable after the measurement while a second criterion describes the information-damage tradeoff in the measurement. While some operations such as filtering or optimal cloning can be non-classical in one or the other criterion, a true QND measurement is non-classical in both [6].
With the aid of figures of merit [13][14][15] describing the quantum-classical boundary, optical QND measurements satisfying both criteria have been demonstrated [14,[16][17][18][19][20][21][22][23][24]. These figures of merit make use of the fact that the optical signal beam, after the QND measurement, can be verified by a direct, i.e., destructive, measurement with quantum-noise-limited sensitivity. Typically such a direct measurement is not available in atomic QND. Rather, repeated QND measurement has been used to show the state preparation criterion [8][9][10][11]25] by conditional variance measurements. Here we show how repeated QND measurements can also be used to test the information-damage tradeoff, and thus to certify full QND performance without direct access to the system variable.

II. MODEL
As in the pioneering work by Kuzmich, et al. [7,26], we consider the collective spin of an atomic ensemble, described by the vector angular momentum operator J. We note that a variety of other physical situations are described in the same way, e.g. by using a pseudo-spin to describe a clock transition [9]. The optical polarization of any probe pulse is described by a vector Stokes operator S i = x, y, z where σ i are the Pauli matrices, a ≡ {a + , a − } T and a ± are annihilation operators for circular-plus and circular-minus polarizations.
We define Stokes operators P, Q for the first and second pulses, respectively. The operators J, P, Q each obey the angular momentum commutation relation [L x , L y ] = iL z and cyclic permutations (for simplicity, we takeh = 1). For notational convenience, we define the combined optical variables C ≡ P ⊕ Q and the total variable T ≡ J ⊕ C. We will be interested in the average values of these operators, which we write asJ ≡ J and similar, and the covariance matrices, which we write as and similar. Our approach follows that of Madsen and Mølmer [27,31].
We assume that the input probe pulses are polarized asP (in) x =Q (in) x

=S (in)
x and that the other average components are zero. We take the initial covariance matrix for the system to beT 0 =J ⊕C This form of the covariance matrix allows for arbitrary prior correlations (including correlated technical noise) among the two optical pulses, but no prior correlations between the atoms and either optical pulse.

The interaction is described by an effective Hamiltonian
where g is a constant [28]. This QND interaction, to lowest order in gτ , where τ is the interaction time of the pulse and atoms, produces a rotation of the state, This has the effect of imprinting information about J z on the light without changing J z itself: Here κ ′ = gτ and S is P or Q depending on which pulse-atom interaction is being described.
The rotation can be described by a linear transformation T (out) = M P T (in) (and thusT (out) = M PT (in) M T P ) where M P is equal to the identity matrix, apart from the elements (M P ) 2,6 = κ ′J (in) x , and (M P ) 5, x . The effect of the second pulse is described by the matrix exchanges the roles of P and Q, and I 3 is the 3 × 3 identity matrix.

III. REDUCTION OF UNCERTAINTY BY QND MEASUREMENT
We first consider the case in which the interaction does not introduce additional noise (although both the input atomic and optical states may be noisy). After interaction with the first pulse, but before the arrival of the second pulse, the state is described byT P ≡ M PT0 M T P . A component P y of the first pulse is measured. Formally, this corresponds to projection along the axis m P ≡ {0, 0, 0, 0, 1, 0, 0, 0, 0} T , andT P is reduced tõ where Π Q ≡ m P ∧ m P is the projector describing the measurement and () MP indicates the Moore-Penrose pseudo-inverse.
We can directly calculate the resulting variance of J z , This has a natural interpretation: The variance of the detected projection P y has two contributions: κ 2J 3,3 from the atomic signal andC 2,2 from the pre-existing optical noise.J 3,3 is reduced by the factor 1/(1 + SNR) where SNR is the signal-to-noise ratio of the measurement. A similar result is found in reference [27]. This post-measurement variance of the signal variable describes the state-preparation capability of the QND measurement. Absent the ability to directly measure J z , we must look for observables which contain this same information.
We note that for κ = 0, e.g. if the atoms are removed, the values are var NA (Q y ) =C 5,5 cov NA (P y , Q y ) =C 2,5 .
We see that the state preparation capability can be expressed in terms of measurable quantities as which uses the variance of the two measurements to determine the SNR. Another formulation, expresses the residual variance in terms of the atomic contribution to the correlation between first and second pulses.
These simple expressions are only valid for noise-free interactions, however. In a real experiment, other effects are present which introduce both noise and losses in the atomic and optical variables. We now account for these other effects.

V. GENERAL NOISE AND LOSS
We now consider noise produced in the atom-light interaction itself, as well as losses. The noise model we employ is very general. The interaction of the first pulse with the atoms is described byT We assume that the coherent part of the interaction is M P ≡ r A I 3 ⊕ r L I 3 ⊕ I 3 apart from the elements (M P ) 2,6 = κJ (in) x , and (M P ) 5,3 = −κS (in) x . Here r A , r L describe the fraction of atoms and photons, respectively, that remain after the interaction. Thus M P includes both the effect of H eff and linear losses. We leave N P completely general, except that it does not affect Q: N P ≡ N ⊕ 0I 3 , where N is a six-by-six symmetric matrix.
Similarly, we describe interaction with the second pulse as where M Q = XM P X and N Q = XN P X.
Note that we assume that both the interaction M and the noise N are the same for the first and second pulses (but act on different variables, naturally). This implies that optical characteristics of the pulses such as detuning from resonance are the same, a condition that can be achieved in experiments. It also assumes that the noise generated by the interaction is incoherent and state-independent, as opposed to a more general, state-dependent noise N(J, S). Nevertheless, in many situations J and S are nearly constant (only small quantum components change appreciably), so that any reasonable N(J, S) would be effectively constant.
Equation (13) still holds for the case with no atoms. We define where the r L factors are included to account for atom-induced optical losses. It is then simple to check that We note that the QND measurement reduces the variance of J z if the quantity in parentheses is negative, i.e., if δcov 2 (Q y , P y ) > var(P y )[δvar(Q y ) − δvar(P y )].
Again, there is an intuitive explanation: δcov(Q y , P y ), which arises from the fact that both pulses measure the same atomic variable J z , is a measure of the atom-light coupling.
[δvar(Q y ) − δvar(P y )] expresses the difference in atom-induced noise between the first and second pulses. This difference indicates a change in the atomic state, namely an increase in var(J z ). The condition of Equation (28) compares these two effects and can be tested knowing the statistics of the various measurements on S y and the optical transmission r L . The factors κ 2 ,J 3,3 in equation (27) must be determined by independent means. For example, κ can be found by measuring the rotation of a state with known J z = 0 andJ 3,3 from the number of atoms, or the observed noise scaling of a known state [29,30].

VI. THREE-PULSE EXPERIMENTS
The above description of two-pulse experiments can be extended straightforwardly to three or more pulses [31]. While a two-pulse experiment, plus prior knowledge of κ and J 3,3 , gives sufficient information to find the post-measurement variance, and thus test the state-preparation property, a three-pulse experiment is required to find the other quantities used to characterize QND measurements.
If R denotes the Stokes vector of the third probe pulse, then statistics such as var(R y ) and cov(P y , R y ) can be determined, and these in turn provide enough constraints to determine the loss and noise. Expanding our system to T ≡ J ⊕ P ⊕ Q ⊕ R, and defining interaction and noise operators M R , N R in the obvious way, a direct calculation finds several useful relations r A = δcov(P y , R y ) δcov(P y , Q y )

VII. MEASURES OF QND PERFORMANCE
To quantify QND performance, Holland et al. use the degree of correlation between various combinations of the input and output system variable X = J z and meter variable Y = S y variables [13]. They define three figures of merit, each of which is unity for an ideal QND measurement. These describe the measurement quality, the preservation of the initial value, and the state preparation capability, respectively: VIII. NON-CLASSICALITY CRITERIA Roch, et al. [14] and Grangier et al. [15] define non-classicality criteria using the conditional variance ∆X 2 s|m , as in Eq. (27), and the quantities ∆X 2 m , the measurement noise referred to the input and ∆X 2 s , the excess noise introduced into the system variable. All are normalized by the intrinsic quantum noise of the system variable, a quantity which may depend on the system or the application. For example, in a spin-squeezing context the natural noise scale isJ 0 = | J x |/2 =J 3,3 , the J z variance of the input x-polarized coherent spin state, i.e., the projection noise. Here we choose to normalize ∆X 2 m byJ 0 , and ∆X 2 s|m , ∆X 2 s by r AJ0 , reflecting the reduction in size of the spin due to losses in the measurement process.

IX. CONCLUSIONS
Using the covariance matrix formalism and a general noise model, we have shown that full certification of QND measurements is possible without direct access to the system variable under study. We find that repeated probing of the same system gives statistical information sufficient to quantify both the state preparation capability and the information-damage tradeoff. The results enable certification of true quantum non-demolition measurement of material systems, and are directly applicable to ongoing experiments using QND measurements for quantum information [28] and quantum-enhanced metrology [30,32,33].

X. ADDITIONAL MATERIAL
The calculations described in this article can be performed in Mathematica using the notebook "ThreePulseCMCalculator," available as an ancillary file.