Towards the ultimate precision limits in parameter estimation: An - PowerPoint PPT Presentation

Towards the ultimate precision limits in parameter estimation: An introduction to quantum metrology Luiz Davidovich Instituto de Física - Universidade Federal do Rio de Janeiro

Outline of the lectures These three lectures will focus on recent developments in quantum metrology. The main questions to be answered are: (i) What are the ultimate precision limits in the estimation of parameters, according to classical mechanics and quantum mechanics? (ii) Are there fundamental limits? Is quantum mechanics helpful in reaching better precision? (iii) How to cope with the deleterious effects of noise? Our discussion is restricted to local quantum metrology: in this case, one is not interested in an optimal globally-valid estimation strategy, valid for any value of the parameter to be estimated, but one wants instead to estimate a parameter confined to some small range. The techniques to be developed are useful, for instance, for estimating parameters that undergo small changes around a known value, like sensing phase changes in gravitational-wave detectors or yet very small forces or magnetic fields — These are typical quantum sensing problems

Summary of the lectures The lectures will be organized as follows: LECTURE 1. Examples of metrological tasks. Quantum metrology and optical interferometers. Shot-noise and Heisenberg limits. Radiation pressure in gravitational-wave interferometers. Classical bounds on precision: The Cramér- Rao bound and introduction of the Fisher information. LECTURE 2. Extension of Cramér-Rao bound and Fisher information to quantum mechanics. Quantum Fisher information for noiseless systems. The role of entanglement. Application to atomic interferometry. Beyond the standard quantum limit: experimental results with optical interferometers and cavity QED. LECTURE 3. Noisy quantum-enhanced metrology: General framework for evaluating the ultimate precision limit in the estimation of parameters. Quantum channels. Application to optical interferometers. Quantum metrology and the energy-time uncertainty relation. Application to atomic decay and dephasing. For more details, see Lectures at College de France (2016): http://www.if.ufrj.br/~ldavid/eng/show_arquivos.php?Id=5

I.1 - General introduction: parameter estimation and classical limits on precision

Parameter estimation Depth of an oil well Time duration of a process Transition frequency A ∆ h = 33 cm g ∆ f = (4 . 1 ± 1 . 6) × 10 − 17 f Optical Clocks and Relativity C. W. Chou, * D. B. Hume, T. Rosenband, D. J. Wineland 24 SEPTEMBER 2010 VOL 329 SCIENCE www Laser Interferometer Weak forces or small Phase displacements in Gravitational Wave Observatory displacements interferometers LETTERS PUBLISHED ONLINE: 21 JULY 2013 | DOI: 10.1038/NPHOTON.2013.177 Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light The LIGO Scientific Collaboration*

Experiments: Parameter estimation beyond classical physics in the XXI century Phase resolution

Experiments: Parameter estimation beyond classical physics in the XXI century Atomic clocks

Experiments: Parameter estimation beyond classical physics in the XXI century Magnetometers

Experiments: Parameter estimation beyond classical physics in the XXI century Force, displacement, and tilt Published: 27 May 2019 Published: 27 May Published: 02 July 2019

Parameter estimation and uncertainty relations What is the meaning of ★ Time-energy uncertainty relation? Δ E Δ T ≥ ! / 2 ★ Number-phase uncertainty relation? Δ N Δφ ≥ ! / 2 We shall see that quantum parameter estimation allows to understand these relations in terms of uncertainties in the estimation of parameters: while Heisenberg uncertainty relations are associated with Hermitian operators, the theory of parameter estimation allows one to obtain uncertainty relations for parameters, like time or phase, with no need to associate them to suitable Hermitian operators.

Photons and beam splitters I ⎛ ⎞ ⎛ ⎞ a out a in ⎛ ⎞ r t ⎜ ⎟ ⎜ ⎟ a ⎟ = ⎜ ⎟ out b out t r b in ⎜ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a 2 = a in 2 ⇒ 2 + b out 2 + b in in a out b out 2 = 1, rt ∗ + r ∗ t = 0 2 + t r b in Balanced interferometer: ⎛ ⎞ ⎛ ⎞ a out a in ⎛ ⎞ r = 1 , t = i ⎟ = 1 1 i ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b out i 1 b in ⎜ ⎜ ⎟ 2 2 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Photons and beam splitters II Same for operators: a out ⎛ ⎞ ⎛ ⎞ ˆ ˆ a out a in ⎛ ⎞ ⎟ = 1 1 i ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ a ˆ ˆ ⎜ i 1 ⎜ ⎟ b out b in 2 in ⎝ ⎠ b ⎝ ⎠ ⎝ ⎠ out ⎦ = 1, ˆ b in , ˆ a in , ˆ a in , ˆ a in , ˆ ˆ ⎡ ⎤ ⎦ = 1, ˆ ⎡ ⎤ ⎦ = 0, ˆ ⎡ ⎤ ⎡ † ⎤ † † a in b in b in b in ⎦ = 0 ⎣ ⎣ ⎣ ⎣ b in EXERCISE 1: Show that this transformation preserves number of Heisenberg picture! photons and commutation relations Corresponding evolution operator: U = 1 U † ˆ a in + i ˆ ˆ a in ˆ ( ) = ˆ ˆ b in a out ⎡ ⎤ U = exp − i π a † ˆ a ˆ ˆ b † + ˆ ( ) ˆ 2 b ⎥⇒ EXERCISE 2: ⎢ 4 ⎣ ⎦ U = 1 Show this. U † ˆ a in + ˆ ) = ˆ ˆ b in ˆ ( i ˆ b in b out 2

An example: optical interferometry out a a ϕ 1 in ϕ est BS 1 BS 2 ϕ 2 b b out in Mach-Zender interferometer: a beam with complex amplitude a in is split on a balanced beam splitter BS 1 and the two resulting beams acquire phases ϕ 1 and , interfering on the second beam splitter BS 2 . The photon numbers ϕ 2 and are measured at the output ports. One could also have two n a out n b out incident beams, with complex amplitudes a in and b in . The outgoing fields are related to the incoming ones through the transformation (note that a out = a in , b out = b in when = =0, since ϕ 2 ϕ 1 [BS 1 ]X[BS 2 ]=1) : ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a out a in e i ϕ 1 ⎟ = 1 1 1 i 0 1 − i ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ BS 1 × BS 2 = 1 ⎜ ⎟ ⎜ ⎜ ⎟ b out i 1 e i ϕ 2 − i 1 b in 2 2 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ! # # " # # $ ! # # " ## $ BS 2 BS 1

̂ ̂ ̂ ̂ Optical interferometry (2) a out a in ϕ 1 ϕ est BS 1 BS 2 ϕ 2 b in b out Multiplying the matrices, and replacing the complex amplitudes by the corresponding photon annihilation operators, one gets: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ˆ ˆ ( ) ( ) a out a in cos ϕ / 2 − sin ϕ / 2 ϕ = ϕ 2 − ϕ 1 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ( ) /2 i ϕ 1 + ϕ 2 ⎟ = e ⎟ , ˆ ˆ ⎜ ⎟ ⎜ ⎜ ( ) ( ) sin ϕ / 2 cos ϕ / 2 b out b in ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ p where the operator â annihilates photons in mode a: ˆ a | N i = N | N � 1 i and is the Fock state with N photons, with , where a † ˆ | N i a | N i = N | N i ˆ is the number operator. The overall phase above can be neglected. a † ˆ ˆ a We use now the Jordan-Schwinger transformation, which allows to analyze the Mach-Zender interferometer in terms of the algebra of angular momentum operators.

̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ Optical interferometry and Jordan-Schwinger transformation This has the advantage of providing a unified formalism, which can also be applied to problems in atomic spectroscopy and magnetometry. J x = 1 J z = 1 J y = i a † ˆ b + ˆ 2(ˆ a † ˆ a − ˆ b † ˆ ˆ ˆ ˆ b † ˆ b † ˆ a † ˆ Let 2(ˆ a ) , a − ˆ b ) , 2(ˆ b ) ˆ ! ˆ N N J 2 = Then [ ˆ J i , ˆ J j ] = i ✏ ijk ˆ J k and ˆ ˆ a † ˆ a + ˆ b † ˆ 2 + 1 , N = ˆ b 2 EXERCISE 3: so these operators obey the angular momentum algebra. Show this. Transformations of operators and can be considered as rotations in angular a b U † ̂ U † ̂ a in ̂ b in ̂ U = exp( − i θ ̂ momentum space: , with , where J ⋅ ̂ a out = U , b out = U n ) unit vector is along the axis of rotation, with the correspondence: the n BS 1 → ˆ U = exp( − i π ˆ Phase delay → ˆ U = exp( − i φ ˆ J x / 2) J z ) φ = ϕ 2 − ϕ 1 → ˆ U = exp( i π ˆ BS 2 J x / 2)

Angular momentum operators for optical interferometry Corresponding transformation for the operators (Heisenberg picture!): ˆ J i ⎛ ⎞ ⎛ ⎞ ˆ ˆ out in ⎛ ⎞ J x J x ⎛ ⎞ ⎛ ⎞ cos ϕ − sin ϕ 0 ⎜ ⎟ ⎜ ⎟ 1 0 0 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ˆ ˆ out in J y 0 0 1 sin ϕ cos ϕ 0 0 0 − 1 J y = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 − 1 0 0 1 0 0 0 1 ⎜ ˆ ⎟ ⎜ ˆ ⎟ out in ⎝ ⎠ ⎝ ⎠ J z J z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ˆ in ⎛ ⎞ J x cos ϕ 0 sin ϕ Therefore, Mach-Zender ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ˆ transformation amounts to a in 0 1 0 J y = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ rotation around y axis of the − sin ϕ 0 cos ϕ ⎜ ⎟ ˆ in J z EXERCISE 4: ⎝ ⎠ ⎝ ⎠ angular momentum operators. Show this. J x π /2 ψ in − i ˆ The state transforms as ψ out = e i ˆ J z ϕ e − i ˆ J x π /2 e