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Quantum-limited metrology with nonlinear detection schemes

Quantum-limited metrology with nonlinear detection schemes Complutense University of Madrid, Department of Optics, Faculty of Physical Sciences, 28040 Madrid, Spain alluis@fis.ucm.es Abstract. Quantum mechanics limit the resolution of detection schemes. Typical arrangements are based on linear processes, so that the corresponding quantum limits are usually understood as unsurpassable and ultimate. Recently it has been shown that nonlinear schemes allow signal detection and measurement with larger resolution than linear processes. In particular, this affects the quantum limits. We review the proposals introduced so far in this novel area of quantum metrology. C 2010 Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/6.0000007] Keywords: quantum metrology; uncertainty relations; nonclassical states; interferometry; quantum optics. Paper SR100110 received Apr. 30, 2010; accepted for publication Aug. 1, 2010; published online Sep. 23, 2010. 1 Introduction Quantum metrology investigates the ultimate limits that quantum physics places on the accuracy of the measurement of any magnitude, such as length, time, frequency, temperature, population, etc., referred to in general as the signal. Roughly speaking, the structure of detection schemes is quite universal. The signal modifies the state of a suitably prepared input probe state. This change is then detected by performing a measurement on the output probe state, whose outcome serves to estimate the value of the signal. By focusing on quantum limits, it is assumed that all sources of technical noise have been removed. In standard metrology the signal induces linear transformations, so that previously known quantum limits heavily depend on the assumption of linearity. Thus, a new frontier arises if we consider that the signal can be imprinted in the probe via nonlinear processes. The key point is that nonlinear schemes allow us to reach larger resolutions than the linear ones. This leads to new quantum limits, new experiments, and eventually new devices. The improvement holds even when using probes in classical states, which is very relevant concerning robustness against practical imperfections. Nonlinear transformations have played a distinguished role in quantum metrology, being extensively used for the preparation of nonclassical probe states (typically squeezed states), while signal transformation was always assumed to be linear. Nonlinear detection schemes change this perspective by showing that it is more advantageous to use nonlinear transformations to imprint the signal on the probe. The objective of this review is to examine the work exploring this possibility [1–25]. Also, we hope to provide a quick start on the subject for newly interested readers. Technical details are deferred to appendices in Secs. 6, 7, and 8. 2 Quantum Metrology Every quantum detection scheme contains five ingredients (see Fig. 1): the system, the input probe state, the transformation induced by the signal, the measurement performed on the output probe state, and the data analysis of the measurement outcomes. 1946-3251/2010/$25.00 SPIE Reviews 2010 SPIE 018006-1 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . estimation transformation probe signal Fig. 1 Detection schemes. measurement 2.1 System The first step is to identify elementary physical systems sensitive enough to changes induced by the signal. Two main possibilities have been considered in the bibliography: quantum light and atoms, usually in Bose-Einstein condensates (for more details, see the proposals reviewed in Sec. 4.1). Fortunately, the quantum theory allows us to describe them in a unified fashion. When the system is made of identical particles (light and Bose-Einstein condensates), we use bosonic annihilation operators aj representing the complex amplitude of the allowed modes (i.e., the degrees of freedom of the particles). Usually just two modes are considered, in which case the angular-momentum operators J are very useful (see Appendix A in Sec. 6). These are Stokes operators for light beams or collective spin operators for atoms. They allow us also to represent collections of distinguishable particles. For more details, see Appendix A in Sec. 6. 2.2 Input Probe State The system is prepared in a probe state |ψ (assumed pure for simplicity) that undergoes the transformation caused by the signal. The probe can be either in a classical or nonclassical state, usually entangled (see Appendix B in Sec. 7). The main advantage of nonclassical states is that they can provide better accuracy. The main advantage of classical states is that they are more robust against practical imperfections. Throughout this work probe realizations are assumed as statistically independent. 2.3 Transformation Information about the signal χ is imprinted in the probe state by letting it experience a signaldependent transformation |ψ → |ψχ . For simplicity we assume that the transformation is unitary |ψχ = exp (iχ G) |ψ , (1) where the generator G is Hermitian. A key point of this work is to distinguish between linear and nonlinear transformations. 2.3.1 Linear transformations By linear transformations, we mean that the output complex amplitudes exp(−iχ G)aj exp(iχ G) †are linear functions of the input ones aj , maybe including their Hermitian conjugates aj . In such a case the generators G are at most quadratic polynomials of aj , ak . Typical examples are phase ˆ shifts, generated by the number operator n = a †a, and rotations of the angular-momentum (k) operators J, generated by a J component, say Jz . For two-level atoms this is Jz ∝ k σz , where SPIE Reviews 018006-2 Vol. 1, 2010 †Luis: Quantum-limited metrology with nonlinear detection schemes. . . detection linear nonlinear Fig. 2 Linear versus nonlinear schemes. k indexes atoms and σ (k) are the Pauli matrices acting on the two-level space of each particle. Linear transformations correspond to free evolution, light propagation through optically linear media, beamsplitting, phase shifting, the most common interferometers, and atoms driven by external fields. Thus, most detection processes in interferometry and spectroscopy are linear. 2.3.2 Nonlinear transformations By nonlinear transformations, we mean that the output complex amplitudes are not linear †functions of the input ones. Their generators contain combinations of aj , ak with powers above ˆ the second. The most simple examples are G = n2 and G = Jz2 . For light, this corresponds to propagation through optically nonlinear media (such as the Kerr effect). In terms of single-atom (k) ( ) operators, Jz2 ∝ k, σz σz includes two-body interactions (see Sec. 4.1). Roughly speaking, in linear transformations each particle experiences the same transformation independent of the presence of other particles, irrespectively of whether the particles are prepared in independent or collective states (see Fig. 2). This can be exemplified by the famous saying that each photon interferes with itself [26]. In nonlinear schemes, particles interact among themselves during the imprinting of the signal, so that they experience a collective transformation. 2.4 Measurement The signal information encoded in the output probe state |ψχ must be disclosed by a measurement M with statistics P (m|χ ) = m|ψχ detection particles particles (2) where P(m|χ ) is the conditional probability of obtaining the outcome m when the true but unknown value of the signal is χ , and we have assumed for simplicity projection on pure states |m , usually the eigenstates of the measured observable M|m = m|m . In practical ˆ terms this is always a counting of atoms or photons per mode; this is M = nj , Jz . This is usually done after a linear coupling so that output populations depend via interference on the signal-induced phase shifts. The only exceptions seem to be the measurement of field quadratures X ∝ a + a †[4, 5], Jz2 [to avoid the vanishing of Jz for strong SU(2) squeezed probes] [4,9], parity [8], and phase difference [9]. Nevertheless, more or less directly these are also always particle-counting measurements. 2.5 Inference The last goal is to obtain the best hypothesis χ about χ along with its uncertainty χ . Both ˜ ˜ should be derived from the statistics P(m|χ ) of the measurement. In this work we are mainly interested in χ, since this allows us to compare the performance of different schemes. The ˜ estimation of the uncertainty can be done in many different ways. The most popular are recalled SPIE Reviews 018006-3 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . in Appendix C in Sec. 8. This subject is not trivial, since different data analysis can lead to different, or even contradictory, conclusions. Nevertheless, for the purpose of this work it is enough to mention that most approaches lead essentially to a minimal signal uncertainty of the form 1 , χ= √ ˜ 2 ν G ( G)2 = ψ|G2 |ψ − ψ|G|ψ 2 , (3) and ν is the number of independent repetitions of the measurement. Leaving aside the statistical factor ν, this recalls a Heisenberg-kind of uncertainty relation χ G ≥ 1/2, although χ is ˜ ˜ not necessarily the variance of an operator. 3 Quantum Limits The idea of precise detection is to have χ be as small as possible; this is G as large as ˜ possible. If no further restrictions are placed, nothing would prevent acquiring G as large as desired, so there would be no limit to accuracy [27]. Quantum limits can only emerge when we impose constraints. The most popular is to consider a fixed number of particles, although time limitations might also be considered [14,28] (see also Sec. 4.3). This is reasonable, since the number of available photons and atoms during the duration of the measurement seems to be always limited, for example, because of the maximum output power of lasers at hand. Nevertheless, this constraint is not trivial, since, for example, the quadrature coherent states have by definition an unbounded number of particles. Thus, there is the ambiguity of whether we refer to constraints in the total number or in the mean number. Because of this it can be advantageous to split the analysis for bounded and unbounded numbers of particles. 3.1 Bounded Number of Particles Let us say that we have just N particles. This is tantamount to say that the Hilbert space of the problem is of finite dimension, and thus G is bounded. In such a case the maximum G is reached by a 50% coherent superposition of the eigenstates of G, |Gmax,min , with maximum and minimum eigenvalues Gmax,min [28] G= 1 (Gmax − Gmin ) , 2 1 |ψ = √ (|Gmax + |Gmin ) . 2 (4) Let us particularize this to linear and nonlinear schemes. 3.1.1 Linear schemes To be more specific, let us consider G = Jz as the generator so that (see Appendix A in Sec. 6 for more details) Gmax = −Gmin = N , 2 G= N , 2 (5) where the optimum probe state (in the angular-momentum basis |j, m ) and the optimum resolution are 1 |ψ = √ (|j, j + |j, −j ) , 2 χ=√ ˜ 1 . νN (6) This can be compared with the maximum G and minimum χ achievable when the probe ˜ is in a classical state. This is an equatorial SU(2) coherent state with polar angle θ = π /2, SPIE Reviews 018006-4 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . leading to √ G= Let us draw some conclusions. 1. According to Eq. (6), the minimum uncertainty χ scales as the inverse of the number ˜ of particles N. This is usually referred to as the Heisenberg limit [28–40], considered the best that can be done with linear schemes. Nevertheless, other schemes where the probe state |ψ may be different for different repetitions can achieve a better use of resources, leading to uncertainties χ scaling as 1/(ν N) [40–43]. ˜ 2. The optimum resolution requires the probe to be prepared in a maximally entangled state (i.e., it cannot be expressed as a product of states in the corresponding modes) |ψ ∝ |j, j + |j, −j in the angular-momentum basis, equivalent to |ψ ∝ |N 1 |0 2 + |0 1 |N 2 in the particle-number basis. This is a coherent superposition of extreme distinguishable states, with all particles in one mode or the other (Schr¨ dinger o cat states or N00N states) [28,29,42–50]. 3. As an alternative to the N00N states, the Heisenberg limit can also be approached by probes prepared in SU(2) squeezed states [9,51–59]. 4. When using probes in coherent states (this is often regarded as equivalent to unentangled particles), the uncertainty in Eq. (7) scales as the inverse of the square root of the number of particles. This uncertainty is much larger than the Heisenberg limit in Eq. (6), referred to as the standard quantum limit [29,60]. 5. Some approaches to quantum linear metrology propose to reach the Heisenberg limit with classical states [14,28,42,43,61,62] by using multiround or sequential protocols, where the same probe experiences the same signal-induced transformation several times. As stated in Ref. 14, the number of interactions is a discrete version of the evolution time τ . Increasing evolution time generally improves sensitivity, since χ ∝ τ . As pointed out in Ref. 28, multiround protocols are an example of the frequent interplay between time and entanglement in quantum information. From a different perspective, this can be regarded as signal amplification with gain given by the number of rounds. Incidentally, nonlinear schemes might be regarded also as operating via signal amplification, as discussed in detail in Sec. 4.3. In any case, it seems that a proper comparison between different schemes (this is with different |ψ and G) should be done on an equal-time basis, considering the same number of applications of the transformation. When this is done it arises that for linear schemes, resolution beyond coherent states requires nonclassical probes [21]. N , 2 χ=√ ˜ 1 νN . (7) 3.1.2 Nonlinear schemes To be more specific, we consider G = Jz2 so that, for N Gmax = and G= N2 , 8 χ=√ ˜ 4 . νN 2 (9) N2 4 Gmin , 1, (8) Thus, with the same number of particles N, the uncertainty in nonlinear schemes can be much smaller than in linear schemes, so we can say that nonlinear schemes provide resolution SPIE Reviews 018006-5 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . beyond the Heisenberg limit. It might be argued that for different generators the Heisenberg limit should be different [24]. Leaving aside terminology, we stress that the key point is the different dependence on the number of particles N in Eqs. (6) and (9). Also, for nonlinear schemes optimum resolution requires entangled probes, perhaps with the only exception of the scheme proposed in Ref. 12. The optimum uncertainty can be approached also using SU(2) squeezed states [9]. When considering probes prepared in SU(2) coherent states, the maximum variance for Jz2 is obtained for θ = π/4, leading to [10] G= N 3/2 , 4 2 χ = √ 3/2 . ˜ νN (10) √ This uncertainty is larger than Eq. (9), although this is still N times smaller than the Heisenberg limit in Eq. (6). We emphasize that this is perhaps the key point of nonlinear detection schemes: improvement of resolution by robust classical states. 3.2 Unbounded Number of Particles When there is no restriction on the total number of particles, we have that G is usually unbounded, so there is no way to establish an upperbound on G, even for a fixed mean number of particles [27,38–40]. As a very simple example let us consider the variance of the number ˆ operator G = n in the probe state (in the photon-number basis) |ψ = 1 − p|0 + √ p|n/p , (11) where 1 ≥ p > 0. It should be understood that n/p is an integer. The mean number of photons is ˆ n = n for all p, while its uncertainty is ˆ n=n 1−p . p (12) ˆ Therefore, when p → 0, we have n → ∞. It seems that we would have arbitrarily high ˆ resolution χ → 0 with any fixed mean number of photons n = n, in contradiction with the ˜ usual idea of quantum limit. In a similar approach, Ref. 63 claims to beat the Heisenberg limit in a linear scheme with G = Jx if the probe is prepared in the mixed state (in the photon-number basis) ρ = (1 − p)|0, 0 0, 0| + p|n/p, n/p n/p, n/p|, where |n1 , n2 = |n1 1 |n2 (13) and n/p is an integer, leading to 1 2 n2 +n , p ˆ N = 2n, (14) ( Jx )2 = SPIE Reviews 018006-6 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . rather than to the Heisenberg limit to emphasize the dependence with the number of particles, rather than claim the existence of an absolute lower bound. Leaving aside these details, most analyses assume that the same limits of the bounded case ˆ ˆ hold simply by replacing the total number N by the mean number of particles N or n . ˆ Concerning nonlinear schemes, the most simple example in the single-mode case is G = nk . ˆ Variance scaling as G ∝ n k can be reached by quadrature squeezed states, while quadrature ˆ coherent states lead to G ∝ n k−1/2 [7,10,16]. Within this same framework, it has been shown that in nonlinear schemes, mixed classical states can lead to larger resolution than quadrature coherent states [21], which are pure states, contrary to what happens in linear schemes. This is revealed by the following probe state with density matrix √ √ ρ = (1 − p)|0 0| + p|α/ p α/ p|, (15) √ where |α/ p is a coherent state, |0 is the vacuum, and 1 ≥ p > 0, which includes the coherent ˆ ˆ state |α for p = 1. The mean number of photons is the same n = |α|2 for every p. For G = n2 and |α| 1, it holds that G=2 ˆ n 3/2 , p p χ= √ ˜ ˆ 4 ν n , (16) 3/2 4 Nonlinear Detection Schemes Next we provide a more detailed account of the material published on quantum limits of nonlinear detection schemes, commenting on some specific items. 4.1 Practical Implementations Essentially three basic schemes have been considered: atom-light interaction, light-light coupling via nonlinear optics, and atom-atom interactions, typically in Bose-Einstein condensates. Only the proposal in Ref. 14 dealing with nanomechanical resonators is somewhat out of this scope, although it can be clearly pictured as a nonlinear interferometer. 4.1.1 Atom-light interaction Most proposals based on atom-light interactions are nonresonant, with a typical generator ˆ G = nJz , where Jz refers to a component of the collective atomic spin [1–3]. This is essentially the same transformations arising in Faraday rotation [80–82] and the basis of signal amplification further discussed in Sec. 4.3. In Ref. 3 there is a slight variation in the form G = X Jz , where ˆ X ∝ a + a †is a field quadrature, that can be obtained from G = nJz by a large displacement of the field complex amplitude a. Reference 12 proposes a generalized resonant Jaynes-Cummings model, where a light mode of frequency Nω interacts with a molecule made of N two-level atoms of internal energies ω, SPIE Reviews 018006-7 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . via a coupling of the form (k) (k) G ∝ a ⊗ σx + iσy + HC, k=1 (k) N (17) where σx,y are the Pauli matrices for the k’th atom, and HC means Hermitian conjugated. This coupling produces the following evolution when the initial atomic state is |j, m = j (i.e., all particles in the same mode), eiχG |j, j → cos 2N−1 χ |j, j − i sin 2N−1 χ |j, −j . (18) As pointed out in Ref. 6, in this case the generator depends on N, so that when the number of atom changes, the interaction changes, which is somewhat peculiar. This might be the reason why this scheme reaches a resolution scaling exponentially χ = 1/2N instead of the most ˜ typical polynomial form. Within this same framework we can place the proposals of the magnetometer in Refs. 17 and 18, where a far-detuned laser beam passes twice through an atomic sample as it undergoes the Larmor precession, continuously observing the polarization of the light beam using techniques of quantum filtering. As stated by the authors, this double-pass scheme mimics a nonlinear magnetic interaction, although it is not trivial to explain the emergence of nonlinearity from effective linear couplings and formulate it resembling more typical nonlinear or multibody interactions. Finally, the approach in Refs. 22 and 25 considers a near-resonant atom-light interaction 2 leading to coupling terms of the form G ∝ S Jk + Sm Jn , where S are field Stokes operators and Jk are components of the collective atomic spin, with the possibility of tuning the linear and nonlinear parts. Within this scheme, Ref. 25 presents an experimental demonstration of the sensitivity scaling in Eq. (10) obtained by letting pulses of polarized light interact with an ensemble of 106 cold 87 Rb atoms in a dipole trap, and probing one component of the collective angular momentum via measurements of the polarization changes in the light. 4.1.2 Nonlinear optics Typical effective generators for light propagation in nonresonant nonlinear media are given by ˆ powers and products of photon-number operators of the form G = nk in a single-mode approach [4,5,7,8,16,24], G = Jz2 in a two-mode scheme [4], G = ⊗k nj in a multimode configuration [7], jˆ ˆj or even G = j nk [23]. 4.1.3 Bose-Einstein condensates Most of the proposals based on atomic systems focus on two-mode Bose-Einstein condensates [6,9–11,13,15,19], where each mode corresponds to a different internal atomic state, typically two hyperfine levels. The most common generator is of the form G = Jz2 , arising from two-body collisions. Nevertheless, under some specific circumstances nonlinear terms of the form Jz2 can ˆ be negligible, and the relevant leading term can be of the form G = N Jz [13,15,19], which is extremely interesting concerning the role of entanglement and amplification, as shown later. The signal χ is the strength of the two-body interaction being proportional to the scattering lengths. They can be tuned by varying an external magnetic field, and depend on the electron-to-proton mass ratio [9]. In Refs. 13,15, and 19, there are complete analyses of potential problems and challenges of such implementation, including the dependence of the nonlinear coupling with the number of atoms. In Refs. 6 and 10, there are very general analyses in terms of generators involving symmetric k-body terms. In Ref. 9, the implementation via fermionic atoms in an optical lattice loaded with one atom per site is discussed, with the advantage that losses are suppressed. SPIE Reviews 018006-8 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 4.2 Role of Entanglement Entanglement enters at two stages: entanglement carried by the probe and entanglement caused by the nonlinear transformation during detection. 4.2.1 Entangled probes As we have shown in Sec. 3.1, optimum probes are usually entangled [28,29,83,84]. The only exception is in the proposal in Ref. 12, where the optimum resolution is seemingly reached by a separable probe. For linear schemes, improved resolution beyond the one provided by coherent states requires first and foremost nonclassicality of the probe [21,85] (the situation is different for nonlinear schemes, as shown in Sec. 3.2 [21]). In turn, for distinguishable two-level particles, nonclassicality is equivalent to entanglement, because all states in 2-D spaces are classical [59]. For identical particles such as photons or Bose-Einstein condensates, the role of entanglement is much more obscure. On the one hand, entanglement plays no role in single-mode schemes. On the other hand, for two-mode schemes there are mode-factorized nonclassical states |ψ1 |ψ2 , where |ψj is in the mode with complex amplitude aj (such as the product of quadrature squeezed states) that reach Heisenberg scaling [86]. Thus, it seems that the key point for improved resolution is nonclassicality rather than entanglement. From a more practical perspective, most analyses focus on the resolution achievable with classical or separable probes. Actually, there are few works considering in some detail nonclassical or entangled probes. In Ref. 9, the probe is prepared in an SU(2) squeezed state reaching maximum resolution for G = Jz2 , while in Ref. 16, quadrature squeezed probes are considered ˆ in a single-mode configuration with G = nk , showing that they can reach maximum resolution with the same amount of squeezing for all k. 4.2.2 Signal-induced entanglement It has been debated whether the improved resolution provided by nonlinear schemes is due to entanglement caused by the signal-induced nonlinear transformation during the detection process, as more or less directly suggested in Refs. 6,9, and 12. In fact, the same nonlinear generators considered here for the signal transformation were proposed previously to produce the nonclassical probes reaching optimum resolution in linear schemes. However, there are some conclusive arguments showing that entanglement plays no role, and even might be counterproductive [7,10,11,13,15,23]. A very first proof is provided by the singlemode schemes in quantum optics [1–5,8,16], since for single-mode photons there is no room for entanglement. Moreover, in Ref. 7 there is a simple comparison between single-mode and multimode configurations showing that, ceteris paribus, single-mode transformations provide better performance than multimode entangling ones. A similar conclusion has been reached in Ref. 23, showing that for nonlinear generators the best strategy is to concentrate all resources in a single mode instead of splitting them over many modes. Further conclusive proofs are provided in Refs. 10 and 13. In Ref. 10 there is a very illustrative model showing that signal-generated entanglement becomes negligible (see the linearization in Sec. 4.3). An even more convincing argument is presented in Ref. 13, where a ˆ ˆ generator G = N Jz is suitably engineered, where N is the total number operator, so that the signal-induced transformation does not produce entanglement at all. Moreover, it is argued in Refs. 11,13, and 15 that entanglement may lead to phase dispersion that, far from being an aid, prevents reaching optimum resolution. Next we discuss that built-in signal amplification, rather than entanglement, serves to account for the improved performance of nonlinear schemes. SPIE Reviews 018006-9 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 4.3 Phase Amplification Here we present an intuitive picture of the enhancement of resolution caused by nonlinear ˆ processes. Let us consider the simplest single-mode situation, comparing the linear G = n and ˆ nonlinear schemes G = n2 , with a probe in a coherent state |α in both cases. In the linear arrangement, the mean value of the complex amplitude in the transformed state is ψχ |a|ψχ = αeiχ , |ψχ = exp(iχ G)|α , (19) ˆ where |ψχ = exp(iχ n)|α = |αeiχ is a coherent state with complex amplitude α exp(i χ ). On ˆ the other hand, in the nonlinear case with G = n2 , we have ψχ |a|ψχ = αeiχ α|αei2χ = αeiχ e−2|α| , 1 (20) ˆ ˆ where we have used af (n) = f (n + 1)a. This can be simplified if χ ψχ |a|ψχ ˆ αei2 n χ , 1 and |α|χ (21) ˆ with n = |α|2 . Comparing Eqs. (19) and (21), we see that in the nonlinear case there is an ˆ effective and noiseless amplification of the signal χ → 2 n χ [1,2,13], which is referred to as increased or enhanced rotation in Refs. 10 and 15. The signal amplification is also very clear in Ref. 12 [see Eq. (18)]. Moreover, in Refs. 17 and 18, it is explicitly stated that the improvement arises from amplification of the Larmor precession. A different approximation leading to an equivalent conclusion for our purposes considers, ˆ prior to performing any transformation, the linearization of n2 around its mean value (roughly ˆ ˆ speaking valid provided that n n ), ˆ n2 ˆ ˆ ˆ ˆ ˆ ˆ n 2 + 2 n (n − n ) → 2 n n, (22) ˆ where the constant terms n 2 can be ignored, since they lead to global phases that play no essential role. This is the uniform-fringe approximation Jz2 → 2 Jz Jz considered in Refs. 10 and 15. Within this limit both phase dispersion and signal-induced entanglement disappear, and ˆ |α exp(i2 n χ ) . an initial coherent probe state remains always coherent |ψχ 4.3.1 Phase ambiguity Phase-shift amplification unavoidably implies signal ambiguity, since in the previous example ˆ in Eq. (21) we have that χ and χ + π/ n produce the same effects. This parallels the free spectral range in spectroscopic measurements using Fabry-Perot interferometers or diffraction gratings. It can be avoided by suitably distributing the measurement into several steps of different precision [10,40–43,61,62]. Otherwise, the signal must be known with a prior uncertainty of the ˆ order of 1/ n . The posterior resolution after ν repetitions of the measurement with the probe in √ ˆ a coherent state is of the order of 1/( ν n 3/2 ), so even in such a case there is an improvement ˆ of resolution by a factor ν n from the prior to posterior situations. 4.4 Robustness Against Practical Imperfections A relevant feature of nonlinear detection is the possibility of reducing uncertainty with classical probe states. This is an interesting property, because nonclassical states tend to be fragile against practical imperfections [87–90]. Thus, unlike other applications of quantum theory, quantum metrology might not face formidable problems fighting decoherence [12]. The robustness of nonlinear schemes against photon and atom losses, inefficient detectors, damping, and atomic decoherence has been addressed in most of the proposals [2,3,5,9,10,14–16]. The conclusion is that for classical probes, practical imperfections lead to a reduction of sensitivity without affecting scaling with the number of particles [10]. In this context, there are SPIE Reviews 018006-10 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 5 Conclusions We hope to have included all proposals introduced so far in the new area of quantum nonlinear metrology. These works confirm that nonlinear schemes allow us to put quantum limits farther than linear schemes. Much remains to be done, especially concerning experimental confirmations and applications, and extending its usefulness to a wider range of signals. To this end, a key point is the possibility of using classical probes because of their robustness against practical imperfections. In any case, we think that this issue provides a deeper understanding of quantum metrology. 6 Appendix A: Atom-Photon Equivalence Bose-Einstein condensates and photons can be described alike in terms of annihilation operators †or complex amplitude operators ak with commutation relation [ak , a ] = δk, . The index k typically represents internal energies, minima of trapping potentials, spin or polarization states, frequency, or spatial distributions, such as input, internal, or output arms of interferometers. For two-mode situations, the following operators are especially useful [91,92] ††ˆ N = a1 a1 + a2 a2 , Jx = 1 2 1 2 a2 a1 + a1 a2 , a1 a1 − a2 a2 , ††††Jy = i 2 a2 a1 − a1 a2 , Jz = ††(23) n=x,y,z k, ,n Jn , ˆ [N, J] = 0, (24) with J2 = ˆ N 2 ˆ N +1 , 2 (25) ˆ where k, ,n is the fully antisymmetric tensor with x,y,z = 1. The operator N is the total number of particles, while Jz is proportional to the difference of the number of particles in the two modes. There is the following useful correspondence |j, m = |n1 = j + m |n2 = j − m , (26) between the standard angular-momentum basis |j, m of simultaneous eigenvectors of Jz and J 2 , i. e., Jz | j, m = m| j, m , J 2 | j, m = j (j + 1)| j, m , and the product of number states in †the two modes |n1 1 |n2 2 with aj aj |nj j = nj |nj j . For N two-level atoms, we can consider also the collective spin picture ˆ N= k=1 σ0 , (k) J= σ (k) , k=1 (27) where σ (k) are the Pauli matrices acting on the two-level space of the k’th particle spanned by | ↑, ↓ k σ0 = | ↑ (k) σy (k) k ↑|+|↓ ↑|−|↑ ↓ |, σx = ↓ (k) |) , σz (k) i 2 (| ↓ (| ↓ (| ↑ k k ↑|+|↑ ↑|−|↓ k k ↓ |) , ↓ |) . (28) SPIE Reviews 018006-11 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 7 Appendix B: Classical and Nonclassical States Nonclassicality is a relevant issue in quantum metrology. The common wisdom is that the ultimate quantum limits can be approached exclusively by nonclassical or entangled states, this being a paradigm of practical applications of quantum physics. 7.1 Classical States The most famous classical states, and the only ones that are pure states, are quadrature coherent states |α , which are eigenstates of the complex amplitude operator a|α = α|α [93–96]. In the number basis read |α = e −|α|2 /2 ∞ n=0 αn √ |n . n! (29) The most general classical state is given by a random distribution of coherent states |α , where the random amplitude α follows any real, positive, and normalized distribution P(α). Nevertheless, the classical/nonclassical frontier is somewhat misty [97–103]. Two-mode coherent states |α1 |α2 define the SU(2) coherent states |N, as [104,105] |α1 |α2 = e−r with (up to a global phase) /2 r N eiNδ | j = N/2, √ N! N=0 ∞ (30) ⎛ ⎝ 2j j +m ⎞1/2 ⎠sin θ 2 | j, m=−j j +m cos θ 2 j −m e−imφ |j, m , (31) where represents the pair of state parameters θ, φ. The quadrature and SU(2) state parameters are connected by the relations θ α1 = r sin eiδ e−iφ , 2 θ α2 = r cos eiδ . 2 (32) SU(2) coherent states are considered classical concerning just angular-momentum variables [106], but they are nonclassical from a wider two-mode perspective. For example, the polar SU(2) coherent states θ = 0, π, are the product of number states |n 1 |0 2 , |0 1 |n 2 , and therefore nonclassical. For distinguishable particles, the SU(2) coherent states factorize as the product of identical single-particle states | j = N/2, where | j = N/2, = ⊗ |j = 1/2, k=1 k, (33) are SU(2) coherent states for j = 1/2 θ θ = sin e−iφ/2 | ↑ k + cos eiφ/2 | ↓ k , 2 2 (34) | j = 1/2, and it can be appreciated that for j = 1/2, every pure state is an SU(2) coherent state. This relation supports the general understanding of coherent states as unentangled, and accordingly, nonclassical states as entangled. In this regard, note that SU(2) coherent states are in general SPIE Reviews 018006-12 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . mode-entangled in the sense that | j, corresponding mode aj . = |ψ1 1 |ψ2 2 , where |ψj are arbitrary states in the 7.2 Nonclassical States Most famous nonclassical states related to precision metrology are as follows. 1. Two-mode number states |n1 1 |n2 2 , with n1 n2 = 0, this is to say |j, m = ±j , especially cases with the same number of photons in each mode |n 1 |n 2 = | j = n, m = 0 [107– 109], which can be regarded as the limit of SU(2) squeezed states [36,37,51–58]. 2. Quadrature squeezed states: these are states with fluctuations of a rotated quadrature Xϕ = eiϕ a + e−iϕ a †smaller than for quadrature coherent states [93–95]. 3. SU(2) squeezed states: there are several definitions of SU(2) or spin squeezing [36,37, 51–58]. The most meaningful criteria are based on the angular-momentum components J⊥ orthogonal to the average vector J , so that by definition J⊥ = 0. The general idea is that J⊥ , or a suitable function of J⊥ and | J |, should be smaller than its value for SU(2) coherent states with J⊥ = 0. From the single-particle perspective discussed before, SU(2) squeezing would imply entanglement. 4. Schr¨ dinger cat states or N00N states [44–50]: these are coherent superpositions of o distinguishable states of the form (in the number and spin bases) 1 1 √ (|n 1 |0 2 + |0 1 |n 2 ) = √ (|j, j + |j, −j ) . 2 2 (35) 5. Phase states [92]: deep down, interferometry is related with phase-difference statistics. The states that represent this variable in the quantum domain are the phase-difference states 1 e−imφ |j, m . |j, φ = √ 2j + 1 m=−j (36) There are no simple procedures of generating or measuring these states, so that sometimes they are replaced by equatorial SU(2) coherent states | j, with θ = π/2 [110–112]. 8 Appendix C: Data Analysis In our context, our main interest is placed in the estimation of the signal uncertainty χ , since ˜ this allows us to compare schemes with different generators and determine optimum probes and measurements. Throughout we assume that the signal χ is small enough to simplify some expressions by series expansions on powers of χ retaining just the lower one. This is justified, since quantum limits are usually concerned with the detection of very weak signals. 8.1 Signal-to-Noise Ratio A convenient performance measure is the signal-to-noise ratio (see Fig. 3) √ M S = ν N SPIE Reviews χ − M M √ χ ν ∂ M ∂χ χ χ=0 (37) Vol. 1, 2010 018006-13 Luis: Quantum-limited metrology with nonlinear detection schemes. . . P(m 0) P(m ) Fig. 3 Signal-induced shift χ of the statistics of the observable M. where M is the variance of M in the probe state |ψ , ν is the number of repetitions of the measurement (assumed independent), and we considered the weak-signal approximation M being ∂ M ∂χ χ χ=0 χ = ψχ |M|ψχ +χ ∂ M ∂χ χ χ=0 (38) = | ψ|[M, G]|ψ |. (39) The minimum signal that can be detected (referred to as leading to unit signal-to-noise ratio S =1→ N χ=√ ˜ ν M ∂ M ∂χ χ χ ) can be estimated as the signal ˜ 1 ≥ √ , 2 ν G χ=0 (40) where the Heisenberg uncertainty relation has been used, G M≥ 1 ∂ M 1 | [M, G] | = 2 2 ∂χ χ (41) Equation (40) is essentially an error propagation in the inversion of Eq. (38), χ= ˜ M χ − M χ ∂ M ∂χ (42) χ=0 In principle, from Eqs. (40) and (41) it seems that optimum resolution requires both equality in the uncertainty relation in Eq. (41) and maximum G. However, these two conditions may be not compatible [16]. The main advantage of the signal-to-noise ratio is simplicity, allowing meaningful conclusions with very simple calculus. Nevertheless, relevant situations are excluded. For example, it may happen that M χ = 0 for all χ . This holds in linear schemes for relevant probes such as |j, 0 and |j, j + | j, −j . This might be avoided by considering that M k χ = 0 for a suitable integer k (otherwise, the measurement would be useless). Moreover, M χ may be a periodic function of χ , so that no single χ can be inferred from Eqs. (38) or (42), for example (see Sec. ˜ 4.3). SPIE Reviews 018006-14 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 8.2 Fisher Information A more powerful analysis considers a Bayesian strategy to get a probability distribution representing our knowledge about the signal after the outcome m in the measurement of M as [65,113–120] P (χ |m) ∝ P (m|χ )P (χ ), (43) where P(χ ) is the prior information about χ that may include the outcomes of previous measurements. The posterior distribution P(χ |m) can be used to infer the best value for χ and its uncertainty, for example via a maximum likelihood strategy. In any case, the minimum uncertainty of any unbiased and efficient estimator χ is given by ˜ the Cramer-Rao lower bound [65,113–118] 1 χ=√ , ˜ νF F = 1 P (m|χ ) ∂P (m|χ ) ∂χ (44) where F is the Fisher information. This is a measure of the information that m provides about χ . In particular, the Cramer-Rao lower bound coincides with the signal-to-noise ratio in Eq. (40) when P (m|χ ) is a Gaussian with a width independent of χ (assuming for simplicity that m is a continuous variable) P (m|χ ) = √ exp − 2π M while in general [35] M ∂ M ∂χ χ m− M 2 χ 2 2 ( M) (45) 1 ≥√ , F (46) where equality is reached for the minimum uncertainty states of Eq. (41) [see Eq. (52)]. The signal-to-noise ratio in Eq. (37) is a measure of how close M χ and M 0 are. Equivalently, the Fisher information is a measure of how close P (m|χ ) and P (m|χ = 0) are. For small enough χ 1 2 P (m|χ ) − m 2 P (m|0) =1− P (m|χ )P (m|0) 1 2 χ F. 8 (47) 8.3 Quantum Fisher Information and Intrinsic Resolution The two previous performance measures naturally depend on the actual measurement M performed. It can be also interesting to use assessments independent of M, representing the optimum results that can be obtained with any measurement. This can be addressed by considering the distance between the input |ψ and output |ψχ probe states. For example, in terms of the corresponding density matrices ρ and ρ χ , the Bures distance gives [119,120] D 2 = 1 − tr √ √ ρρχ ρ 1/2 1 2 χ FQ , 8 FQ = 2 k, (pk − p )2 | ψk |G|ψ |2 , pk + p (48) where pk , |ψk are the eigenvalues and eigenvectors, respectively, of ρ (for a simpler alternative via the Hilbert-Schmidt distance, see Ref. 85). It is worth noting that in general, FQ does not depend on χ [121]. SPIE Reviews 018006-15 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . FQ is the maximum F over all possible measurements M, so that F ≤ FQ [119], and the following quantum Cramer-Rao minimum uncertainty holds [79,117–119] χ= ˜ Since FQ ≤ 4 ( G)2 , where the equality holds for pure states, the minimum uncertainty scales as [119,120] 1 , χ= √ ˜ 2 ν G (51) (50) 1 . νFQ (49) ∂ M ∂χ χ 1 ≥√ ≥ F 1 1 , ≥ 2 G FQ (52) where all equalities are satisfied simultaneously for the minimum uncertainty states of Eq. (41). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png SPIE Reviews SPIE

Quantum-limited metrology with nonlinear detection schemes

SPIE Reviews , Volume 1 (1) – Jan 1, 2010

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1946-3251
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10.1117/6.0000007
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Complutense University of Madrid, Department of Optics, Faculty of Physical Sciences, 28040 Madrid, Spain alluis@fis.ucm.es Abstract. Quantum mechanics limit the resolution of detection schemes. Typical arrangements are based on linear processes, so that the corresponding quantum limits are usually understood as unsurpassable and ultimate. Recently it has been shown that nonlinear schemes allow signal detection and measurement with larger resolution than linear processes. In particular, this affects the quantum limits. We review the proposals introduced so far in this novel area of quantum metrology. C 2010 Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/6.0000007] Keywords: quantum metrology; uncertainty relations; nonclassical states; interferometry; quantum optics. Paper SR100110 received Apr. 30, 2010; accepted for publication Aug. 1, 2010; published online Sep. 23, 2010. 1 Introduction Quantum metrology investigates the ultimate limits that quantum physics places on the accuracy of the measurement of any magnitude, such as length, time, frequency, temperature, population, etc., referred to in general as the signal. Roughly speaking, the structure of detection schemes is quite universal. The signal modifies the state of a suitably prepared input probe state. This change is then detected by performing a measurement on the output probe state, whose outcome serves to estimate the value of the signal. By focusing on quantum limits, it is assumed that all sources of technical noise have been removed. In standard metrology the signal induces linear transformations, so that previously known quantum limits heavily depend on the assumption of linearity. Thus, a new frontier arises if we consider that the signal can be imprinted in the probe via nonlinear processes. The key point is that nonlinear schemes allow us to reach larger resolutions than the linear ones. This leads to new quantum limits, new experiments, and eventually new devices. The improvement holds even when using probes in classical states, which is very relevant concerning robustness against practical imperfections. Nonlinear transformations have played a distinguished role in quantum metrology, being extensively used for the preparation of nonclassical probe states (typically squeezed states), while signal transformation was always assumed to be linear. Nonlinear detection schemes change this perspective by showing that it is more advantageous to use nonlinear transformations to imprint the signal on the probe. The objective of this review is to examine the work exploring this possibility [1–25]. Also, we hope to provide a quick start on the subject for newly interested readers. Technical details are deferred to appendices in Secs. 6, 7, and 8. 2 Quantum Metrology Every quantum detection scheme contains five ingredients (see Fig. 1): the system, the input probe state, the transformation induced by the signal, the measurement performed on the output probe state, and the data analysis of the measurement outcomes. 1946-3251/2010/$25.00 SPIE Reviews 2010 SPIE 018006-1 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . estimation transformation probe signal Fig. 1 Detection schemes. measurement 2.1 System The first step is to identify elementary physical systems sensitive enough to changes induced by the signal. Two main possibilities have been considered in the bibliography: quantum light and atoms, usually in Bose-Einstein condensates (for more details, see the proposals reviewed in Sec. 4.1). Fortunately, the quantum theory allows us to describe them in a unified fashion. When the system is made of identical particles (light and Bose-Einstein condensates), we use bosonic annihilation operators aj representing the complex amplitude of the allowed modes (i.e., the degrees of freedom of the particles). Usually just two modes are considered, in which case the angular-momentum operators J are very useful (see Appendix A in Sec. 6). These are Stokes operators for light beams or collective spin operators for atoms. They allow us also to represent collections of distinguishable particles. For more details, see Appendix A in Sec. 6. 2.2 Input Probe State The system is prepared in a probe state |ψ (assumed pure for simplicity) that undergoes the transformation caused by the signal. The probe can be either in a classical or nonclassical state, usually entangled (see Appendix B in Sec. 7). The main advantage of nonclassical states is that they can provide better accuracy. The main advantage of classical states is that they are more robust against practical imperfections. Throughout this work probe realizations are assumed as statistically independent. 2.3 Transformation Information about the signal χ is imprinted in the probe state by letting it experience a signaldependent transformation |ψ → |ψχ . For simplicity we assume that the transformation is unitary |ψχ = exp (iχ G) |ψ , (1) where the generator G is Hermitian. A key point of this work is to distinguish between linear and nonlinear transformations. 2.3.1 Linear transformations By linear transformations, we mean that the output complex amplitudes exp(−iχ G)aj exp(iχ G) †are linear functions of the input ones aj , maybe including their Hermitian conjugates aj . In such a case the generators G are at most quadratic polynomials of aj , ak . Typical examples are phase ˆ shifts, generated by the number operator n = a †a, and rotations of the angular-momentum (k) operators J, generated by a J component, say Jz . For two-level atoms this is Jz ∝ k σz , where SPIE Reviews 018006-2 Vol. 1, 2010 †Luis: Quantum-limited metrology with nonlinear detection schemes. . . detection linear nonlinear Fig. 2 Linear versus nonlinear schemes. k indexes atoms and σ (k) are the Pauli matrices acting on the two-level space of each particle. Linear transformations correspond to free evolution, light propagation through optically linear media, beamsplitting, phase shifting, the most common interferometers, and atoms driven by external fields. Thus, most detection processes in interferometry and spectroscopy are linear. 2.3.2 Nonlinear transformations By nonlinear transformations, we mean that the output complex amplitudes are not linear †functions of the input ones. Their generators contain combinations of aj , ak with powers above ˆ the second. The most simple examples are G = n2 and G = Jz2 . For light, this corresponds to propagation through optically nonlinear media (such as the Kerr effect). In terms of single-atom (k) ( ) operators, Jz2 ∝ k, σz σz includes two-body interactions (see Sec. 4.1). Roughly speaking, in linear transformations each particle experiences the same transformation independent of the presence of other particles, irrespectively of whether the particles are prepared in independent or collective states (see Fig. 2). This can be exemplified by the famous saying that each photon interferes with itself [26]. In nonlinear schemes, particles interact among themselves during the imprinting of the signal, so that they experience a collective transformation. 2.4 Measurement The signal information encoded in the output probe state |ψχ must be disclosed by a measurement M with statistics P (m|χ ) = m|ψχ detection particles particles (2) where P(m|χ ) is the conditional probability of obtaining the outcome m when the true but unknown value of the signal is χ , and we have assumed for simplicity projection on pure states |m , usually the eigenstates of the measured observable M|m = m|m . In practical ˆ terms this is always a counting of atoms or photons per mode; this is M = nj , Jz . This is usually done after a linear coupling so that output populations depend via interference on the signal-induced phase shifts. The only exceptions seem to be the measurement of field quadratures X ∝ a + a †[4, 5], Jz2 [to avoid the vanishing of Jz for strong SU(2) squeezed probes] [4,9], parity [8], and phase difference [9]. Nevertheless, more or less directly these are also always particle-counting measurements. 2.5 Inference The last goal is to obtain the best hypothesis χ about χ along with its uncertainty χ . Both ˜ ˜ should be derived from the statistics P(m|χ ) of the measurement. In this work we are mainly interested in χ, since this allows us to compare the performance of different schemes. The ˜ estimation of the uncertainty can be done in many different ways. The most popular are recalled SPIE Reviews 018006-3 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . in Appendix C in Sec. 8. This subject is not trivial, since different data analysis can lead to different, or even contradictory, conclusions. Nevertheless, for the purpose of this work it is enough to mention that most approaches lead essentially to a minimal signal uncertainty of the form 1 , χ= √ ˜ 2 ν G ( G)2 = ψ|G2 |ψ − ψ|G|ψ 2 , (3) and ν is the number of independent repetitions of the measurement. Leaving aside the statistical factor ν, this recalls a Heisenberg-kind of uncertainty relation χ G ≥ 1/2, although χ is ˜ ˜ not necessarily the variance of an operator. 3 Quantum Limits The idea of precise detection is to have χ be as small as possible; this is G as large as ˜ possible. If no further restrictions are placed, nothing would prevent acquiring G as large as desired, so there would be no limit to accuracy [27]. Quantum limits can only emerge when we impose constraints. The most popular is to consider a fixed number of particles, although time limitations might also be considered [14,28] (see also Sec. 4.3). This is reasonable, since the number of available photons and atoms during the duration of the measurement seems to be always limited, for example, because of the maximum output power of lasers at hand. Nevertheless, this constraint is not trivial, since, for example, the quadrature coherent states have by definition an unbounded number of particles. Thus, there is the ambiguity of whether we refer to constraints in the total number or in the mean number. Because of this it can be advantageous to split the analysis for bounded and unbounded numbers of particles. 3.1 Bounded Number of Particles Let us say that we have just N particles. This is tantamount to say that the Hilbert space of the problem is of finite dimension, and thus G is bounded. In such a case the maximum G is reached by a 50% coherent superposition of the eigenstates of G, |Gmax,min , with maximum and minimum eigenvalues Gmax,min [28] G= 1 (Gmax − Gmin ) , 2 1 |ψ = √ (|Gmax + |Gmin ) . 2 (4) Let us particularize this to linear and nonlinear schemes. 3.1.1 Linear schemes To be more specific, let us consider G = Jz as the generator so that (see Appendix A in Sec. 6 for more details) Gmax = −Gmin = N , 2 G= N , 2 (5) where the optimum probe state (in the angular-momentum basis |j, m ) and the optimum resolution are 1 |ψ = √ (|j, j + |j, −j ) , 2 χ=√ ˜ 1 . νN (6) This can be compared with the maximum G and minimum χ achievable when the probe ˜ is in a classical state. This is an equatorial SU(2) coherent state with polar angle θ = π /2, SPIE Reviews 018006-4 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . leading to √ G= Let us draw some conclusions. 1. According to Eq. (6), the minimum uncertainty χ scales as the inverse of the number ˜ of particles N. This is usually referred to as the Heisenberg limit [28–40], considered the best that can be done with linear schemes. Nevertheless, other schemes where the probe state |ψ may be different for different repetitions can achieve a better use of resources, leading to uncertainties χ scaling as 1/(ν N) [40–43]. ˜ 2. The optimum resolution requires the probe to be prepared in a maximally entangled state (i.e., it cannot be expressed as a product of states in the corresponding modes) |ψ ∝ |j, j + |j, −j in the angular-momentum basis, equivalent to |ψ ∝ |N 1 |0 2 + |0 1 |N 2 in the particle-number basis. This is a coherent superposition of extreme distinguishable states, with all particles in one mode or the other (Schr¨ dinger o cat states or N00N states) [28,29,42–50]. 3. As an alternative to the N00N states, the Heisenberg limit can also be approached by probes prepared in SU(2) squeezed states [9,51–59]. 4. When using probes in coherent states (this is often regarded as equivalent to unentangled particles), the uncertainty in Eq. (7) scales as the inverse of the square root of the number of particles. This uncertainty is much larger than the Heisenberg limit in Eq. (6), referred to as the standard quantum limit [29,60]. 5. Some approaches to quantum linear metrology propose to reach the Heisenberg limit with classical states [14,28,42,43,61,62] by using multiround or sequential protocols, where the same probe experiences the same signal-induced transformation several times. As stated in Ref. 14, the number of interactions is a discrete version of the evolution time τ . Increasing evolution time generally improves sensitivity, since χ ∝ τ . As pointed out in Ref. 28, multiround protocols are an example of the frequent interplay between time and entanglement in quantum information. From a different perspective, this can be regarded as signal amplification with gain given by the number of rounds. Incidentally, nonlinear schemes might be regarded also as operating via signal amplification, as discussed in detail in Sec. 4.3. In any case, it seems that a proper comparison between different schemes (this is with different |ψ and G) should be done on an equal-time basis, considering the same number of applications of the transformation. When this is done it arises that for linear schemes, resolution beyond coherent states requires nonclassical probes [21]. N , 2 χ=√ ˜ 1 νN . (7) 3.1.2 Nonlinear schemes To be more specific, we consider G = Jz2 so that, for N Gmax = and G= N2 , 8 χ=√ ˜ 4 . νN 2 (9) N2 4 Gmin , 1, (8) Thus, with the same number of particles N, the uncertainty in nonlinear schemes can be much smaller than in linear schemes, so we can say that nonlinear schemes provide resolution SPIE Reviews 018006-5 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . beyond the Heisenberg limit. It might be argued that for different generators the Heisenberg limit should be different [24]. Leaving aside terminology, we stress that the key point is the different dependence on the number of particles N in Eqs. (6) and (9). Also, for nonlinear schemes optimum resolution requires entangled probes, perhaps with the only exception of the scheme proposed in Ref. 12. The optimum uncertainty can be approached also using SU(2) squeezed states [9]. When considering probes prepared in SU(2) coherent states, the maximum variance for Jz2 is obtained for θ = π/4, leading to [10] G= N 3/2 , 4 2 χ = √ 3/2 . ˜ νN (10) √ This uncertainty is larger than Eq. (9), although this is still N times smaller than the Heisenberg limit in Eq. (6). We emphasize that this is perhaps the key point of nonlinear detection schemes: improvement of resolution by robust classical states. 3.2 Unbounded Number of Particles When there is no restriction on the total number of particles, we have that G is usually unbounded, so there is no way to establish an upperbound on G, even for a fixed mean number of particles [27,38–40]. As a very simple example let us consider the variance of the number ˆ operator G = n in the probe state (in the photon-number basis) |ψ = 1 − p|0 + √ p|n/p , (11) where 1 ≥ p > 0. It should be understood that n/p is an integer. The mean number of photons is ˆ n = n for all p, while its uncertainty is ˆ n=n 1−p . p (12) ˆ Therefore, when p → 0, we have n → ∞. It seems that we would have arbitrarily high ˆ resolution χ → 0 with any fixed mean number of photons n = n, in contradiction with the ˜ usual idea of quantum limit. In a similar approach, Ref. 63 claims to beat the Heisenberg limit in a linear scheme with G = Jx if the probe is prepared in the mixed state (in the photon-number basis) ρ = (1 − p)|0, 0 0, 0| + p|n/p, n/p n/p, n/p|, where |n1 , n2 = |n1 1 |n2 (13) and n/p is an integer, leading to 1 2 n2 +n , p ˆ N = 2n, (14) ( Jx )2 = SPIE Reviews 018006-6 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . rather than to the Heisenberg limit to emphasize the dependence with the number of particles, rather than claim the existence of an absolute lower bound. Leaving aside these details, most analyses assume that the same limits of the bounded case ˆ ˆ hold simply by replacing the total number N by the mean number of particles N or n . ˆ Concerning nonlinear schemes, the most simple example in the single-mode case is G = nk . ˆ Variance scaling as G ∝ n k can be reached by quadrature squeezed states, while quadrature ˆ coherent states lead to G ∝ n k−1/2 [7,10,16]. Within this same framework, it has been shown that in nonlinear schemes, mixed classical states can lead to larger resolution than quadrature coherent states [21], which are pure states, contrary to what happens in linear schemes. This is revealed by the following probe state with density matrix √ √ ρ = (1 − p)|0 0| + p|α/ p α/ p|, (15) √ where |α/ p is a coherent state, |0 is the vacuum, and 1 ≥ p > 0, which includes the coherent ˆ ˆ state |α for p = 1. The mean number of photons is the same n = |α|2 for every p. For G = n2 and |α| 1, it holds that G=2 ˆ n 3/2 , p p χ= √ ˜ ˆ 4 ν n , (16) 3/2 4 Nonlinear Detection Schemes Next we provide a more detailed account of the material published on quantum limits of nonlinear detection schemes, commenting on some specific items. 4.1 Practical Implementations Essentially three basic schemes have been considered: atom-light interaction, light-light coupling via nonlinear optics, and atom-atom interactions, typically in Bose-Einstein condensates. Only the proposal in Ref. 14 dealing with nanomechanical resonators is somewhat out of this scope, although it can be clearly pictured as a nonlinear interferometer. 4.1.1 Atom-light interaction Most proposals based on atom-light interactions are nonresonant, with a typical generator ˆ G = nJz , where Jz refers to a component of the collective atomic spin [1–3]. This is essentially the same transformations arising in Faraday rotation [80–82] and the basis of signal amplification further discussed in Sec. 4.3. In Ref. 3 there is a slight variation in the form G = X Jz , where ˆ X ∝ a + a †is a field quadrature, that can be obtained from G = nJz by a large displacement of the field complex amplitude a. Reference 12 proposes a generalized resonant Jaynes-Cummings model, where a light mode of frequency Nω interacts with a molecule made of N two-level atoms of internal energies ω, SPIE Reviews 018006-7 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . via a coupling of the form (k) (k) G ∝ a ⊗ σx + iσy + HC, k=1 (k) N (17) where σx,y are the Pauli matrices for the k’th atom, and HC means Hermitian conjugated. This coupling produces the following evolution when the initial atomic state is |j, m = j (i.e., all particles in the same mode), eiχG |j, j → cos 2N−1 χ |j, j − i sin 2N−1 χ |j, −j . (18) As pointed out in Ref. 6, in this case the generator depends on N, so that when the number of atom changes, the interaction changes, which is somewhat peculiar. This might be the reason why this scheme reaches a resolution scaling exponentially χ = 1/2N instead of the most ˜ typical polynomial form. Within this same framework we can place the proposals of the magnetometer in Refs. 17 and 18, where a far-detuned laser beam passes twice through an atomic sample as it undergoes the Larmor precession, continuously observing the polarization of the light beam using techniques of quantum filtering. As stated by the authors, this double-pass scheme mimics a nonlinear magnetic interaction, although it is not trivial to explain the emergence of nonlinearity from effective linear couplings and formulate it resembling more typical nonlinear or multibody interactions. Finally, the approach in Refs. 22 and 25 considers a near-resonant atom-light interaction 2 leading to coupling terms of the form G ∝ S Jk + Sm Jn , where S are field Stokes operators and Jk are components of the collective atomic spin, with the possibility of tuning the linear and nonlinear parts. Within this scheme, Ref. 25 presents an experimental demonstration of the sensitivity scaling in Eq. (10) obtained by letting pulses of polarized light interact with an ensemble of 106 cold 87 Rb atoms in a dipole trap, and probing one component of the collective angular momentum via measurements of the polarization changes in the light. 4.1.2 Nonlinear optics Typical effective generators for light propagation in nonresonant nonlinear media are given by ˆ powers and products of photon-number operators of the form G = nk in a single-mode approach [4,5,7,8,16,24], G = Jz2 in a two-mode scheme [4], G = ⊗k nj in a multimode configuration [7], jˆ ˆj or even G = j nk [23]. 4.1.3 Bose-Einstein condensates Most of the proposals based on atomic systems focus on two-mode Bose-Einstein condensates [6,9–11,13,15,19], where each mode corresponds to a different internal atomic state, typically two hyperfine levels. The most common generator is of the form G = Jz2 , arising from two-body collisions. Nevertheless, under some specific circumstances nonlinear terms of the form Jz2 can ˆ be negligible, and the relevant leading term can be of the form G = N Jz [13,15,19], which is extremely interesting concerning the role of entanglement and amplification, as shown later. The signal χ is the strength of the two-body interaction being proportional to the scattering lengths. They can be tuned by varying an external magnetic field, and depend on the electron-to-proton mass ratio [9]. In Refs. 13,15, and 19, there are complete analyses of potential problems and challenges of such implementation, including the dependence of the nonlinear coupling with the number of atoms. In Refs. 6 and 10, there are very general analyses in terms of generators involving symmetric k-body terms. In Ref. 9, the implementation via fermionic atoms in an optical lattice loaded with one atom per site is discussed, with the advantage that losses are suppressed. SPIE Reviews 018006-8 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 4.2 Role of Entanglement Entanglement enters at two stages: entanglement carried by the probe and entanglement caused by the nonlinear transformation during detection. 4.2.1 Entangled probes As we have shown in Sec. 3.1, optimum probes are usually entangled [28,29,83,84]. The only exception is in the proposal in Ref. 12, where the optimum resolution is seemingly reached by a separable probe. For linear schemes, improved resolution beyond the one provided by coherent states requires first and foremost nonclassicality of the probe [21,85] (the situation is different for nonlinear schemes, as shown in Sec. 3.2 [21]). In turn, for distinguishable two-level particles, nonclassicality is equivalent to entanglement, because all states in 2-D spaces are classical [59]. For identical particles such as photons or Bose-Einstein condensates, the role of entanglement is much more obscure. On the one hand, entanglement plays no role in single-mode schemes. On the other hand, for two-mode schemes there are mode-factorized nonclassical states |ψ1 |ψ2 , where |ψj is in the mode with complex amplitude aj (such as the product of quadrature squeezed states) that reach Heisenberg scaling [86]. Thus, it seems that the key point for improved resolution is nonclassicality rather than entanglement. From a more practical perspective, most analyses focus on the resolution achievable with classical or separable probes. Actually, there are few works considering in some detail nonclassical or entangled probes. In Ref. 9, the probe is prepared in an SU(2) squeezed state reaching maximum resolution for G = Jz2 , while in Ref. 16, quadrature squeezed probes are considered ˆ in a single-mode configuration with G = nk , showing that they can reach maximum resolution with the same amount of squeezing for all k. 4.2.2 Signal-induced entanglement It has been debated whether the improved resolution provided by nonlinear schemes is due to entanglement caused by the signal-induced nonlinear transformation during the detection process, as more or less directly suggested in Refs. 6,9, and 12. In fact, the same nonlinear generators considered here for the signal transformation were proposed previously to produce the nonclassical probes reaching optimum resolution in linear schemes. However, there are some conclusive arguments showing that entanglement plays no role, and even might be counterproductive [7,10,11,13,15,23]. A very first proof is provided by the singlemode schemes in quantum optics [1–5,8,16], since for single-mode photons there is no room for entanglement. Moreover, in Ref. 7 there is a simple comparison between single-mode and multimode configurations showing that, ceteris paribus, single-mode transformations provide better performance than multimode entangling ones. A similar conclusion has been reached in Ref. 23, showing that for nonlinear generators the best strategy is to concentrate all resources in a single mode instead of splitting them over many modes. Further conclusive proofs are provided in Refs. 10 and 13. In Ref. 10 there is a very illustrative model showing that signal-generated entanglement becomes negligible (see the linearization in Sec. 4.3). An even more convincing argument is presented in Ref. 13, where a ˆ ˆ generator G = N Jz is suitably engineered, where N is the total number operator, so that the signal-induced transformation does not produce entanglement at all. Moreover, it is argued in Refs. 11,13, and 15 that entanglement may lead to phase dispersion that, far from being an aid, prevents reaching optimum resolution. Next we discuss that built-in signal amplification, rather than entanglement, serves to account for the improved performance of nonlinear schemes. SPIE Reviews 018006-9 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 4.3 Phase Amplification Here we present an intuitive picture of the enhancement of resolution caused by nonlinear ˆ processes. Let us consider the simplest single-mode situation, comparing the linear G = n and ˆ nonlinear schemes G = n2 , with a probe in a coherent state |α in both cases. In the linear arrangement, the mean value of the complex amplitude in the transformed state is ψχ |a|ψχ = αeiχ , |ψχ = exp(iχ G)|α , (19) ˆ where |ψχ = exp(iχ n)|α = |αeiχ is a coherent state with complex amplitude α exp(i χ ). On ˆ the other hand, in the nonlinear case with G = n2 , we have ψχ |a|ψχ = αeiχ α|αei2χ = αeiχ e−2|α| , 1 (20) ˆ ˆ where we have used af (n) = f (n + 1)a. This can be simplified if χ ψχ |a|ψχ ˆ αei2 n χ , 1 and |α|χ (21) ˆ with n = |α|2 . Comparing Eqs. (19) and (21), we see that in the nonlinear case there is an ˆ effective and noiseless amplification of the signal χ → 2 n χ [1,2,13], which is referred to as increased or enhanced rotation in Refs. 10 and 15. The signal amplification is also very clear in Ref. 12 [see Eq. (18)]. Moreover, in Refs. 17 and 18, it is explicitly stated that the improvement arises from amplification of the Larmor precession. A different approximation leading to an equivalent conclusion for our purposes considers, ˆ prior to performing any transformation, the linearization of n2 around its mean value (roughly ˆ ˆ speaking valid provided that n n ), ˆ n2 ˆ ˆ ˆ ˆ ˆ ˆ n 2 + 2 n (n − n ) → 2 n n, (22) ˆ where the constant terms n 2 can be ignored, since they lead to global phases that play no essential role. This is the uniform-fringe approximation Jz2 → 2 Jz Jz considered in Refs. 10 and 15. Within this limit both phase dispersion and signal-induced entanglement disappear, and ˆ |α exp(i2 n χ ) . an initial coherent probe state remains always coherent |ψχ 4.3.1 Phase ambiguity Phase-shift amplification unavoidably implies signal ambiguity, since in the previous example ˆ in Eq. (21) we have that χ and χ + π/ n produce the same effects. This parallels the free spectral range in spectroscopic measurements using Fabry-Perot interferometers or diffraction gratings. It can be avoided by suitably distributing the measurement into several steps of different precision [10,40–43,61,62]. Otherwise, the signal must be known with a prior uncertainty of the ˆ order of 1/ n . The posterior resolution after ν repetitions of the measurement with the probe in √ ˆ a coherent state is of the order of 1/( ν n 3/2 ), so even in such a case there is an improvement ˆ of resolution by a factor ν n from the prior to posterior situations. 4.4 Robustness Against Practical Imperfections A relevant feature of nonlinear detection is the possibility of reducing uncertainty with classical probe states. This is an interesting property, because nonclassical states tend to be fragile against practical imperfections [87–90]. Thus, unlike other applications of quantum theory, quantum metrology might not face formidable problems fighting decoherence [12]. The robustness of nonlinear schemes against photon and atom losses, inefficient detectors, damping, and atomic decoherence has been addressed in most of the proposals [2,3,5,9,10,14–16]. The conclusion is that for classical probes, practical imperfections lead to a reduction of sensitivity without affecting scaling with the number of particles [10]. In this context, there are SPIE Reviews 018006-10 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 5 Conclusions We hope to have included all proposals introduced so far in the new area of quantum nonlinear metrology. These works confirm that nonlinear schemes allow us to put quantum limits farther than linear schemes. Much remains to be done, especially concerning experimental confirmations and applications, and extending its usefulness to a wider range of signals. To this end, a key point is the possibility of using classical probes because of their robustness against practical imperfections. In any case, we think that this issue provides a deeper understanding of quantum metrology. 6 Appendix A: Atom-Photon Equivalence Bose-Einstein condensates and photons can be described alike in terms of annihilation operators †or complex amplitude operators ak with commutation relation [ak , a ] = δk, . The index k typically represents internal energies, minima of trapping potentials, spin or polarization states, frequency, or spatial distributions, such as input, internal, or output arms of interferometers. For two-mode situations, the following operators are especially useful [91,92] ††ˆ N = a1 a1 + a2 a2 , Jx = 1 2 1 2 a2 a1 + a1 a2 , a1 a1 − a2 a2 , ††††Jy = i 2 a2 a1 − a1 a2 , Jz = ††(23) n=x,y,z k, ,n Jn , ˆ [N, J] = 0, (24) with J2 = ˆ N 2 ˆ N +1 , 2 (25) ˆ where k, ,n is the fully antisymmetric tensor with x,y,z = 1. The operator N is the total number of particles, while Jz is proportional to the difference of the number of particles in the two modes. There is the following useful correspondence |j, m = |n1 = j + m |n2 = j − m , (26) between the standard angular-momentum basis |j, m of simultaneous eigenvectors of Jz and J 2 , i. e., Jz | j, m = m| j, m , J 2 | j, m = j (j + 1)| j, m , and the product of number states in †the two modes |n1 1 |n2 2 with aj aj |nj j = nj |nj j . For N two-level atoms, we can consider also the collective spin picture ˆ N= k=1 σ0 , (k) J= σ (k) , k=1 (27) where σ (k) are the Pauli matrices acting on the two-level space of the k’th particle spanned by | ↑, ↓ k σ0 = | ↑ (k) σy (k) k ↑|+|↓ ↑|−|↑ ↓ |, σx = ↓ (k) |) , σz (k) i 2 (| ↓ (| ↓ (| ↑ k k ↑|+|↑ ↑|−|↓ k k ↓ |) , ↓ |) . (28) SPIE Reviews 018006-11 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 7 Appendix B: Classical and Nonclassical States Nonclassicality is a relevant issue in quantum metrology. The common wisdom is that the ultimate quantum limits can be approached exclusively by nonclassical or entangled states, this being a paradigm of practical applications of quantum physics. 7.1 Classical States The most famous classical states, and the only ones that are pure states, are quadrature coherent states |α , which are eigenstates of the complex amplitude operator a|α = α|α [93–96]. In the number basis read |α = e −|α|2 /2 ∞ n=0 αn √ |n . n! (29) The most general classical state is given by a random distribution of coherent states |α , where the random amplitude α follows any real, positive, and normalized distribution P(α). Nevertheless, the classical/nonclassical frontier is somewhat misty [97–103]. Two-mode coherent states |α1 |α2 define the SU(2) coherent states |N, as [104,105] |α1 |α2 = e−r with (up to a global phase) /2 r N eiNδ | j = N/2, √ N! N=0 ∞ (30) ⎛ ⎝ 2j j +m ⎞1/2 ⎠sin θ 2 | j, m=−j j +m cos θ 2 j −m e−imφ |j, m , (31) where represents the pair of state parameters θ, φ. The quadrature and SU(2) state parameters are connected by the relations θ α1 = r sin eiδ e−iφ , 2 θ α2 = r cos eiδ . 2 (32) SU(2) coherent states are considered classical concerning just angular-momentum variables [106], but they are nonclassical from a wider two-mode perspective. For example, the polar SU(2) coherent states θ = 0, π, are the product of number states |n 1 |0 2 , |0 1 |n 2 , and therefore nonclassical. For distinguishable particles, the SU(2) coherent states factorize as the product of identical single-particle states | j = N/2, where | j = N/2, = ⊗ |j = 1/2, k=1 k, (33) are SU(2) coherent states for j = 1/2 θ θ = sin e−iφ/2 | ↑ k + cos eiφ/2 | ↓ k , 2 2 (34) | j = 1/2, and it can be appreciated that for j = 1/2, every pure state is an SU(2) coherent state. This relation supports the general understanding of coherent states as unentangled, and accordingly, nonclassical states as entangled. In this regard, note that SU(2) coherent states are in general SPIE Reviews 018006-12 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . mode-entangled in the sense that | j, corresponding mode aj . = |ψ1 1 |ψ2 2 , where |ψj are arbitrary states in the 7.2 Nonclassical States Most famous nonclassical states related to precision metrology are as follows. 1. Two-mode number states |n1 1 |n2 2 , with n1 n2 = 0, this is to say |j, m = ±j , especially cases with the same number of photons in each mode |n 1 |n 2 = | j = n, m = 0 [107– 109], which can be regarded as the limit of SU(2) squeezed states [36,37,51–58]. 2. Quadrature squeezed states: these are states with fluctuations of a rotated quadrature Xϕ = eiϕ a + e−iϕ a †smaller than for quadrature coherent states [93–95]. 3. SU(2) squeezed states: there are several definitions of SU(2) or spin squeezing [36,37, 51–58]. The most meaningful criteria are based on the angular-momentum components J⊥ orthogonal to the average vector J , so that by definition J⊥ = 0. The general idea is that J⊥ , or a suitable function of J⊥ and | J |, should be smaller than its value for SU(2) coherent states with J⊥ = 0. From the single-particle perspective discussed before, SU(2) squeezing would imply entanglement. 4. Schr¨ dinger cat states or N00N states [44–50]: these are coherent superpositions of o distinguishable states of the form (in the number and spin bases) 1 1 √ (|n 1 |0 2 + |0 1 |n 2 ) = √ (|j, j + |j, −j ) . 2 2 (35) 5. Phase states [92]: deep down, interferometry is related with phase-difference statistics. The states that represent this variable in the quantum domain are the phase-difference states 1 e−imφ |j, m . |j, φ = √ 2j + 1 m=−j (36) There are no simple procedures of generating or measuring these states, so that sometimes they are replaced by equatorial SU(2) coherent states | j, with θ = π/2 [110–112]. 8 Appendix C: Data Analysis In our context, our main interest is placed in the estimation of the signal uncertainty χ , since ˜ this allows us to compare schemes with different generators and determine optimum probes and measurements. Throughout we assume that the signal χ is small enough to simplify some expressions by series expansions on powers of χ retaining just the lower one. This is justified, since quantum limits are usually concerned with the detection of very weak signals. 8.1 Signal-to-Noise Ratio A convenient performance measure is the signal-to-noise ratio (see Fig. 3) √ M S = ν N SPIE Reviews χ − M M √ χ ν ∂ M ∂χ χ χ=0 (37) Vol. 1, 2010 018006-13 Luis: Quantum-limited metrology with nonlinear detection schemes. . . P(m 0) P(m ) Fig. 3 Signal-induced shift χ of the statistics of the observable M. where M is the variance of M in the probe state |ψ , ν is the number of repetitions of the measurement (assumed independent), and we considered the weak-signal approximation M being ∂ M ∂χ χ χ=0 χ = ψχ |M|ψχ +χ ∂ M ∂χ χ χ=0 (38) = | ψ|[M, G]|ψ |. (39) The minimum signal that can be detected (referred to as leading to unit signal-to-noise ratio S =1→ N χ=√ ˜ ν M ∂ M ∂χ χ χ ) can be estimated as the signal ˜ 1 ≥ √ , 2 ν G χ=0 (40) where the Heisenberg uncertainty relation has been used, G M≥ 1 ∂ M 1 | [M, G] | = 2 2 ∂χ χ (41) Equation (40) is essentially an error propagation in the inversion of Eq. (38), χ= ˜ M χ − M χ ∂ M ∂χ (42) χ=0 In principle, from Eqs. (40) and (41) it seems that optimum resolution requires both equality in the uncertainty relation in Eq. (41) and maximum G. However, these two conditions may be not compatible [16]. The main advantage of the signal-to-noise ratio is simplicity, allowing meaningful conclusions with very simple calculus. Nevertheless, relevant situations are excluded. For example, it may happen that M χ = 0 for all χ . This holds in linear schemes for relevant probes such as |j, 0 and |j, j + | j, −j . This might be avoided by considering that M k χ = 0 for a suitable integer k (otherwise, the measurement would be useless). Moreover, M χ may be a periodic function of χ , so that no single χ can be inferred from Eqs. (38) or (42), for example (see Sec. ˜ 4.3). SPIE Reviews 018006-14 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . 8.2 Fisher Information A more powerful analysis considers a Bayesian strategy to get a probability distribution representing our knowledge about the signal after the outcome m in the measurement of M as [65,113–120] P (χ |m) ∝ P (m|χ )P (χ ), (43) where P(χ ) is the prior information about χ that may include the outcomes of previous measurements. The posterior distribution P(χ |m) can be used to infer the best value for χ and its uncertainty, for example via a maximum likelihood strategy. In any case, the minimum uncertainty of any unbiased and efficient estimator χ is given by ˜ the Cramer-Rao lower bound [65,113–118] 1 χ=√ , ˜ νF F = 1 P (m|χ ) ∂P (m|χ ) ∂χ (44) where F is the Fisher information. This is a measure of the information that m provides about χ . In particular, the Cramer-Rao lower bound coincides with the signal-to-noise ratio in Eq. (40) when P (m|χ ) is a Gaussian with a width independent of χ (assuming for simplicity that m is a continuous variable) P (m|χ ) = √ exp − 2π M while in general [35] M ∂ M ∂χ χ m− M 2 χ 2 2 ( M) (45) 1 ≥√ , F (46) where equality is reached for the minimum uncertainty states of Eq. (41) [see Eq. (52)]. The signal-to-noise ratio in Eq. (37) is a measure of how close M χ and M 0 are. Equivalently, the Fisher information is a measure of how close P (m|χ ) and P (m|χ = 0) are. For small enough χ 1 2 P (m|χ ) − m 2 P (m|0) =1− P (m|χ )P (m|0) 1 2 χ F. 8 (47) 8.3 Quantum Fisher Information and Intrinsic Resolution The two previous performance measures naturally depend on the actual measurement M performed. It can be also interesting to use assessments independent of M, representing the optimum results that can be obtained with any measurement. This can be addressed by considering the distance between the input |ψ and output |ψχ probe states. For example, in terms of the corresponding density matrices ρ and ρ χ , the Bures distance gives [119,120] D 2 = 1 − tr √ √ ρρχ ρ 1/2 1 2 χ FQ , 8 FQ = 2 k, (pk − p )2 | ψk |G|ψ |2 , pk + p (48) where pk , |ψk are the eigenvalues and eigenvectors, respectively, of ρ (for a simpler alternative via the Hilbert-Schmidt distance, see Ref. 85). It is worth noting that in general, FQ does not depend on χ [121]. SPIE Reviews 018006-15 Vol. 1, 2010 Luis: Quantum-limited metrology with nonlinear detection schemes. . . FQ is the maximum F over all possible measurements M, so that F ≤ FQ [119], and the following quantum Cramer-Rao minimum uncertainty holds [79,117–119] χ= ˜ Since FQ ≤ 4 ( G)2 , where the equality holds for pure states, the minimum uncertainty scales as [119,120] 1 , χ= √ ˜ 2 ν G (51) (50) 1 . νFQ (49) ∂ M ∂χ χ 1 ≥√ ≥ F 1 1 , ≥ 2 G FQ (52) where all equalities are satisfied simultaneously for the minimum uncertainty states of Eq. (41).

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