Quantum nondemolition measurement

An ideal measurement preserving the measured eigenspaces. In the projective setting, the selective instrument $\rho \mapsto P_j\rho P_j$ preserves every vector within outcome $j$, including coherence inside a degenerate eigenspace. Repeated measurements agree in the absence of intervening dynamics changing the measured quantity. Measuring a finer observable can destroy degeneracy coherence and need not realize the same Lüders rule instrument.

A physical measurement interaction must correlate the system with a meter; in Haroche cavity photon nondemolition experiment that meter is a chemical atom. For a measured observable $A$, a useful sufficient ideal condition is $[A,H_{\mathrm{int}}]=0$ together with $[A,H_0]=0$, where $H_{\mathrm{int}}$ is the interaction Hamiltonian operator and $H_0$ the free Hamiltonian operator. These conditions protect its eigenvalues during probing and between probes. They do not protect every other observable, nor prevent conditional quantum state changes. In particular, preserving photon number can disturb the field's quantum phase.

In the 2007 cavity quantum electrodynamics experiment of Christine Guerlin and collaborators, including Sébastien Gleyzes, Michel Brune, Jean-Michel Raimond and Serge Haroche, an injected coherent state occupied a 51.1 GHz microwave cavity with niobium-coated mirrors exhibiting superconductivity. Velocity-selected rubidium chemical atoms were prepared as circular Rydberg atoms in levels with principal quantum numbers 50 and 51. Repeated atomic readouts progressively narrowed the inferred photon number to a Fock state; later records revealed jumps. The cavity damping time was $T_c=0.130\mathrm{s}$. Empty-cavity Ramsey interferometry scans fixed the analysis-phase origin; weak injected fields refined the response calibration. Finite fringe contrast required calibrated likelihoods; photon losses could interrupt convergence.^[ref]

The functional sequence is preparation, first Ramsey interferometry zone $R_1$, dispersive cavity $C$, second zone $R_2$, and state-selective detection. The first $\pi /2$ microwave pulse prepares an atomic quantum superposition; dispersive atom-cavity coupling adds a photon number-dependent relative quantum phase while suppressing energy exchange. The second pulse converts that quantum phase into atomic populations; state-selective field ionization reads $e$ or $g$. Successive chemical atoms carry the meter records away while the same field remains available.^[ref]

This arrangement makes quantum nondemolition measurement useful for following a field repeatedly and preparing a Fock state by conditioning. The photon number is inferred from a sequence of physical records; it is not a directly visible label attached to a photon. Each atomic probe makes a weak, partial measurement in quantum mechanics of photon number, and the accumulated records can approach a resolved count. A single binary meter need not identify one among many possible Fock states, as perfect discrimination of pure states requires orthogonality.

Consider an original balanced-readout model, rather than a fit to the experiment. Let $n\in \{0,\dots ,N_{\max }\}$, with a normalized prior probability $p_0(n)\ge 0$ and ${\sum }_n{p}_0(n)=1$. A detected chemical atom has outcome $s\in \{e,g\}$. Define the conditional probabilitiesHere $\chi$ is the accumulated quantum phase per photon, $\varphi$ the controlled analysis quantum phase, and $C$ the fringe contrast. The two probabilities sum to one for every $n$. Actual calibration may also require an offset and asymmetric detection errors; $C$ alone does not specify a quantum instrument.

Assume no photon number change during the record and fresh independently prepared atomic meters. For observed outcomes $s_1,\dots ,s_k$ at recorded settings ${\varphi }_1,\dots ,{\varphi }_k$, the Born rule and Bayes' theorem giveThe posterior probability is defined only for $Z_k>0$. Summing its numerator over $n$ proves normalization. This is also the update of diagonal elements of a field density operator under a measurement in quantum mechanics; it does not assume that an initial quantum superposition was an unknown pre-existing integer. The observed settings are part of the record, especially when selected adaptively.

For an exact finite example take $p_0=(1,1,1,1)/4$, $C=4/5$, and $\chi =\pi /2$. An $e$ outcome at $\varphi =0$ has likelihood $(9,5,1,5)/10$, giving $p_1=(9,5,1,5)/20$. Another $e$ at $\varphi =-\pi /2$ gives $p_2=(9,9,1,1)/20$. A subsequent $g$ at $\varphi =0$ givesThus three atomic records raise the posterior probability of $n=1$ from $1/4$ to $45/68$, while leaving finite uncertainty. No one of these outcomes identifies $n$ with certainty.

The periodic likelihood function imposes an identifiability limit. If $\chi =2\pi /q$ for an integer $q>0$, then $L_s(n+q;\varphi )=L_s(n;\varphi )$ for every setting: the posterior odds between $n$ and $n+q$ never change. Varying $\varphi$ can resolve a cosine reflection ambiguity within a period, but cannot remove this exact period. Changing $\chi$ can supply additional information. A finite cutoff is therefore a physical assumption to justify, not evidence that excluded high photon numbers were absent. At $C=0$ no outcome changes the prior probability; at $0<C<1$ neither binary outcome has an exactly zero likelihood.

The no-loss assumption also has a time limit. In an original zero-temperature loss model, let $\kappa =1/T_c$. A Fock state $|n⟩$ loses its first photon at rate $n\kappa$, so its survival probability is $\exp (-n\kappa t)$ and its mean lifetime is $T_c/n$ for $n>0$. Several records must be acquired within that lifetime to identify a plateau. When losses matter, first predict $p_k^{-}(n)={\sum }_m{T}_{nm}(\Delta t)p_{k-1}(m)$ using a normalized transition matrix, then apply the atomic Bayesian inference update to $p_k^{-}$. Thermal excitation requires upward transitions too. A delayed or broadened inference after a jump need not indicate that an atom caused the jump.

An explicit original ideal meter model explains how photon number survives while field quantum coherence changes. Let $\hat{N}$ be the number operator and prepare an atomic qubit in $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. The dispersive unitary operatorcommutes with $\hat{N}\otimes I$. For a field quantum superposition ${\sum }_n{c}_n|n⟩$, it produces ${\sum }_n{c}_n|n⟩|a_n⟩$, where $|a_n⟩=(e^{-in\chi /2}|e⟩+e^{in\chi /2}|g⟩)/\sqrt{2}$. The atomic meter states encode $n$ through relative quantum phase and can become entangled with the field. The second Ramsey interferometry pulse and detector turn those correlations into the outcome probabilities above when $C=1$.

Tracing out the atom multiplies a field density matrix entry ${\rho }_{nm}$ by $⟨a_m|a_n⟩=\cos ((n-m)\chi /2)$. Hence the nonselective quantum measurement preserves every photon number population ${\rho }_{nn}$ while changing off-diagonal quantum coherence. For $\chi =\pi /2$, coherence between $|0⟩$ and $|1⟩$ is multiplied by $1/\sqrt{2}$ per fresh meter, even with no photons absorbed. Field quantum phase information can therefore be disturbed by an energy-preserving probe. No universal self-adjoint phase operator or naive canonical number–phase commutator is needed for this conclusion.

One possible finite-contrast quantum instrument makes the distinction sharper. Define diagonal Kraus operators $K_e=\cos ((\varphi +\chi \hat{N})/2)$ and $K_g=\sin ((\varphi +\chi \hat{N})/2)$. After their ideal readout, flip the reported label with probability $r=(1-C)/2$. The unnormalized selective maps areTheir traces reproduce $L_e,L_g$ on a Fock state, and their sum reproduces the displayed dephasing quantum channel, independently of $r$. At $C=0$ the recorded labels carry no information although the physical meter interaction can still cause measurement backaction. This is an illustrative instrument, not a reconstruction of all laboratory imperfections; equal likelihoods alone do not determine the disturbance.

Choose a shared random axis uniformly from $x,y,z$, perform its entanglement-assisted nondemolition parity measurement with one Bell pair, and accept the anticorrelated parity. The spin singlet state always passes unchanged. The averaged acceptance operator is $\Omega ={\sum }_j(I-{\sigma }_j\otimes {\sigma }_j)/6=P_s+(I-P_s)/3$, so an orthogonal triplet state passes with probability one third. This is a statistical verifier with a nonzero acceptance gap, rather than an exact single-shot singlet-versus-triplet decision. The Bell-pair cost of exact nondemolition singlet verification concerns the stronger zero-error task.

Exact singlet verification accepts the spin singlet state with certainty, rejects every orthogonal state, and preserves the singlet on acceptance. Each nonzero accepting Kraus operator is then a scalar multiple of the singlet projector, whose operator Schmidt rank is four. The local Kraus rank bound from an entangled resource excludes a single shared Bell pair, which has Schmidt rank two. Two shared Bell pairs suffice by a Bell-state nondemolition measurement followed by reporting whether its label is the singlet. This protocol may disturb triplet coherence. The rank bound concerns the resource Schmidt rank, and is not an entropy lower bound of two bits for arbitrary resource states.

Share a Bell state meter pair, apply local system-to-meter CNOT gates, and measure each meter in the computational basis. The parity of the local meter records measures system $Z_A{Z}_B$ while preserving all coherence within either parity sector. The complete tuple of local records has Kraus operator $P_{u\oplus v}/\sqrt{2}$. Each local record is uniform, and the nonlocal result requires local operations and classical communication to compare records. A local Hadamard gate conjugation supplies the corresponding $X_A{X}_B$ measurement.