A novel lemma of the the
Optical Equivalence Theorem
Global Webinar on Optics, Photonics, and Lasers (GWOPL-2023)
online event- India.
Orchidea Maria Lecian
Sapienza University of Rome, Rome, Italy.
25 MAR 2023
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Abstract
A novel lemma of the the optical equivalence theorem is presented.
Examples and applications are provided within mathematical models: quantum
correlation of multispatial modes for Gauss-Markoff models, the long-time limit for the
error estimation; chemistry methods: temperature-jump experiments for
protein-folding in molecular dynamics, jump processes between states; metrology:
quantum metrology, noisy metrology; Ramsey spectroscopy.
Furthermore, the new perspective of non-Hermitian systems is envisaged: the
calculations of the limits of non-diagonal elements of non-Hermitian-matrices systems
are newly refined and evaluated to be next orders.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Summary
New lemma of the Optical Equivalence Theorem:
• new definitions:
- operators,
- density matrix,
- states;
• applications:
- the long-time limit for the error estimation;
- quantum limit and states;
- quantum-mechanical noise in interferometry;
- specifications of time delays in optomechanically-induced lights;
- laser phase and frequency stabilization: optical resonator;
- Ramsey experiment;
- cold atomic ensembles;
- the Heisenberg limit;
- beyond the standard quantum limit of noisy metrology;
- quantum system coupled linearly to Gaussian white noise;
• outlook:
- non-Hermitian systems.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Introduction
A new lemma of the Optical Equivalence Theorem is
presented to establish a relation between the sequences of the
density matrix operators expansions and their expectation values.
The approximations necessary for the application to new systems
are therefore calculated.
In particular, the evaluation of the approximations of the weighting
function necessary for the expansion of the density matrix are
performed. The expansion of the density matrix is calculated at
the requested approximation order.
The new applications are provided with.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The optical equivalence Theorem
The optical equivalence theorem demonstrates the connection
between the classical probability distribution formalism and the
density matrix for a quantum-mechanical system.
J. Klauder, L. Streit, J. Math. Phys. 1974, 15, 760–763;
E. Sudarshan, Light Beams. Phys. Rev. 1963, 10, 277;
H. Leutwyler, J. Klauder, L. Streit,Nuovo Cim. A 1970, 66, 536–554;
K. Cahill, R. Glauber,Phys. Rev. 1969, 177, 1882–1902.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Demonstrations of the optical equivalence theorem
Among the several possible demonstrations, one of the proofs of the
theorem are not based on the number operator:
this allows one to construct a diagonal density operator ρ for the
coherent states | z >, i.e.
ρ ≡ π−1
Z
ϕ(z) | z >< z | d2
z
- possible to apply also to other degrees of freedom, where the
integration is extended on the entire suitable Fock occupation
space, where annihilation operators and creation ones are also
defined.
The interest in this demonstration relies on the particular hypotheses
assumed for the states | z > of the system.
For unbounded observables, the function ϕ(z) is replaced by a suitable
function, i.e.
ϕ(z) → ϕβ(z) ≡ Sβϕ(z),
where Sβ is a suitable support-controlling function for ϕ(z) in the
definition of ϕβ(z).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

To proceed further, the weighting function on compact support SM
can be further defined, which
- acts on ϕβ(z) as ϕβ,M (z), and
- defines the corresponding density matrix.
The change of variable
z′
e−β/2
→ z
leads to the definition of the function ϕM of compact support,
such that the corresponding density operator ρM, which becomes
finally diagonal.
The definition of the density operator ρM is a well-defined
approximating sequence for unbounded operators for the definition
of a density operator (which is, on the contrary, directly defined for
bounded operators).
Differently from other demonstrations, this procedure is not based
on any hypotheses for the Fock representation.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

New lemma for the states with an
almost-infinite-expectation-valued operators
The partition function for the semiclassical description, for the
quantum-optics description and for the optical-systems descriptions, for n
the occupation number sequence, is calculated as
ρ(n; n′
) ≡ Πλ
R
d2
xλSβϕ(z)exp|xλ|2 z
nλ
λ (z∗
λ)n
′
λ
√
nλ!n
′
λ
.
In all the approximation here described, the weighting support
function S is therefore demonstrated to satisfy, by definition,
the properties
Sλρ(n; n′) ≡ ρ(z; z)[ϕβ(z)] + O(n; n′; β; λ)
for unbounded observables. The implications of such an
approximation are to be developped within the framework of
almost-infinite-valued-operator eigenstates for
extremely-high energetic processes.
OML, Estimations of the Optical Equivalence Theorem for Opto-Mechanical Systems for Investigation in General
Relativity and High-Energy Physics, Computation 8(3), 60 (2020).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Furthermore, the weighting-support function has the properties to
straightforward be extended to the convergence properties in
the trace class norm as well through the definition for the
orthonormal basis for the unbounded observables by
transitivity, where suitable higher-order corrections are
evaluated as
Sλ {ψk} ≡ {ψk} + O(n; n′
; β; λ)
and, for operators A, with weighted density matrix and the
spectral component ϕR(z) on compact support, for the
operator AR
AR ≡
1
π
Z
d2
zϕR(z) | z >< z | +O(n; n′
; β; λ),
respectively, in the projector operator | z >< z |.
The definition of the first-approximation correction orders
therefore very importantly depends on the definition of the
parameters β and λ in the definitions of the weighting
support-control function Sβ and Sλ,respectively.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The Optical Equivalents for Quantum-Mechanical
Operators
Possibility to apply the results found for the expressions of sequences
of observables of unbounded operators, such as the density matrix, to
the calculations of other operators defined in quantum-mechanical
systems to be compared to the corresponding optical system(s).
Be A operators with weighted density matrix and spectral component ϕR (z) on
compact support; then on finds that
AR ≡
1
π
Z
d2
zϕR (z) | z >< z | +O(n; n′
; β; λ),
respectively, in the projector operator | z >< z |.
The definition of the first-approximation correction orders have a specifying
functional dependence on the definition of the parameters β and λ.respectively.
The related results are obtained by considering the properties of quantization on
null-hypersurface quantization techniques. The investigation is consistent for systems
constituted of intense, non-monochromatic laser fields. The power spectrum of the
operators is therefore decomposed as a sequence obtained after the majorization of
the operators after those of the weighting function.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The power spectrum is therefore not needed to be expressed as a sequence (of
majorizations), where such majorization do ot apply to pure states. In the comparison
with the quasi-probability distributions for the density operator for an infinite
momentum which involves the Fock represetnation, infinite-momentum states can be
studied as suitable approximation for states in extremely-high intesity laser fields,
whose energy can be compared as its limit going to infinity. The observables for
interaction processes can be schemtazied within the approximation of the expectation
values of the density matrix corresponding to the infinite-energy (momentum) laser
fields, and the corresponding eigenstates.
The following remarks are in order.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Pure states ρ̃ in the momentum P representation define
observables by means of the density operator in the P
representation | ψ >< ψ |,
- the weight function P(ζ) in the momentum P representation allows one to classify
coherent states and incoherent states in a radiation background field, according to the
properties of the radiation background field.
- coherent states represented by a finite number of creation
operators, with, as coefficients, arbitrary complex numbers.
The weight functions P̃(ζ) are tempered distributions.
ρ̃ζ ≡
Z
P(ζ) | ζ >< ζ | d2
ζ
- d2ζ real element of area (even in the complex plane), - states are formed as a finite
number of creation operators in the Fock representation.
P(ζ) is a linear combination of 2-dimensional functions ˜
fβ and of a
finite number of its derivatives.
An infinitely-energetic P background laser filed for photons can be interpeted as a
superposition of incoherent states, within the due hypotheses.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

With the definitions obtained
- density operators in the momentum representations are weighted fucntionals P(ζ),
defining Wigner distribution W (ζ) continuous and uniformily bounded, as
< ζ | ˜
ρ | ζ >:
- quasi-probability distributions are expectation values for the corresponding density
operator(s),
- integral representations for the density operators can therefore be found:
P(ζ) are therefore the expectation values of Hermitian operator(s), whose eigenvalues
are infinite.
For incoherent states, a density matrix ˜
ρ̃′ can be rewritten in a limiting procedure
as a sum of neighbouring states
˜
ρ̃′
≡ (1 − ϵ̃)ρ̃ + ϵ̃ρ̃ | ζ >< ζ |,
with 0 < ϵ̃ < 1;
- in the limit ϵ̃ → 0, ˜
ρ̃′ is in the trace-class norm, and
- the corresponding weight functions P′(ζ), which correspond to the density operators
ρ̃′, are tempered distributions only for pure states;
- P(ζ) exhibits singularities not compatible with the form of the momentum P
representation.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Solved by a representation of the density operators, in the
case of (almost-) infinite laser background field independent, of the
Fock representation:
- discrepancies for quantum states, in the power spectrum, are
expressed for optical systems by the terms
O(n; n′; β; λ),
which depend both
- on the found weighting function ϕ, and
- on the (non-equal) non-trivial weights characterizing the Fock
states expressed by the parameters β and λ.
Corresponding Fourier decompositions:
- corrections at the proper order,
- ensures one to avoid the mixing and the superpositions of
the corrections at different orders, and
- ensures one to avoid the mixing and the superpositions of
the Fourier decomposition of the spectral modes.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Applications 1: optomechanical systems
The application of the new lemma allows one to obtain results
more precise than those obtained numerically and than those
obtained after the analytical approximations induced after
the Wigner quasi-distribution.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The long-time limit for the error estimation
Control paradigms for fundamental tests of quantum
mechanics,
long-time limit for the error estimation
can be formulated theoretically, (and by numerical methods) as with a
power spectrum S(ω)
S(ω) =
C2
(ω − Ω)2 + γ2
with
C a real parameter,
Ω the mechanical resonance frequency, and
γ the dumping rate.
S. Ang, G. Harris, W Bowen, M Tsang,New J. Phys. 2013, 15, 103028;
J. Du, L. Cao, K. Zhang, J Jing, Appl. Phys. Lett. 2017, 110, 241103.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The long-time limit of the error estimation is here newly
calculated exactly as
S(ω) ≃
2C2Ω
2Ω4 + 2γ2Ω2 + γ4
ω +
C2
Ω2 + γ2
+ O(ω2
, Ω2
; C2
; γ−4
)
where the correction term
C2
Ω2 + γ2
+ O(ω2
, Ω2
; C2
; γ−4
) (1)
consists of
- a non-trivial summand, plus
- the corrections due to the other parameters,
i.e., C a real parameter, Ω the mechanical resonance frequency, γ the
dumping rate.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

- Fourier decomposition of the modes corresponding to a
systems of an intense laser beam:
⇒ approximated by one containing particles also with infinite
momentum on Minkowski spacetime by means of standard
quantization techniques. For the standard quantization techniques,
the polarization tensor is evaluated at all orders.
• The peculiarities of intense laser fields containing particle with very large values of
the momentum, but not an infinite momentum, can be ascribed to the properties of
semiclassical optical systems, for which the dimensions of the system are larger than
the Planck length, but whose constituents are of a quantum nature.
• For particles with very high value of the momentum, such as in an intense laser field,
the Expansion Equation can be calculated for the intense laser beam modes.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Quantum limit and states
- N classical states:
- standard quantum limit in estimation precision scaling as 1/
√
N;
- N entangled particle: precision scaling as 1/N
quantum frequency estimation for N qubits subject to independent
Markovian noise
Francesco Albarelli, Matteo A. C. Rossi, Dario Tamascelli1, Marco G. Genoni, Restoring Heisenberg scaling in noisy
quantum metrology by monitoring the environment, Quantum 2, 110 (2018).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Quantum-mechanical noise in interferometry
Photon-counting error, and
radiation-pressure error
- probable error in the determination of distances with respect to
the beam-splitter can be no smaller than the standard quantum
limit
-squeezed-state technique (quasi-probability densities).
D. F. Walls, Squeezed states of light, Nature 306(5939), 141–146 (1983).
Carlton M. Caves, Quantum-mechanical noise in an interferometer, Phys. Rev. D 23,
1693.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Specifications of time delays in optomechanically-induced
lights
- ideal optomechanically-induced transparency,
- inverted transparency window,
k1 decay rate of the passive cavity
k2 gain rate of the active cavity
- the aim of the procedure is to study the mean values in the
’steady’ states
- the presence of a probe field allows for the expression of the
states as the steady states and the fluctuations, for which the
phase space is described after a linearized Langevin system at a
mechanical damping rate γm;
- the response to the probe filed is investigated:
- strong coupling of the cavities and the resonator at the
near-resonant frequency energy spectrum δ after the frequencies
ωm
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

- ultra-slow light δ = ωm
obtained at the transparency window after adjusting the dissipation rates
of the cavities (with the usual mechanical linewidth)
- τmax maximum time delay at the transparency window δ = ωm
τmax =
4k1
(k1 − k2)k2
2 + 4(γm + mk1 − k2)ω2
m
γm(k1 − k2)2(k2
2 + 4ω2
m)
from which the analysis of the dispersion allows one to infer that the
steepest the slope of the dispersion, the slowest the light
- if the value of γm is small, the ’extreme’ value of τmax independent of
ωm is found as
τmax |γm1=
4k1
γm(k1 − k2)
- if k1 − k2 γm is small, the following value of τmax is found
τmax |k1−k2γm =
16k1ω2
m
(k1 − k2)2(k2
2 + 4ω2
m)
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

ultra-fast light δ = −ωm
obtained after tuning the coupling strength and the decay rates
- τmax maximum time delay at the transparency window δ = −ωm
τmax = −
4k1
k2(k2
2 + 4ω2
m − k1k2) + 2γmω2
m
γm(k2
2 + 42
m − k1k2)
(2)
- if the value of γm is small, the ’extreme’ value of τmax is found as
τmax |γm1= −
4k1k2
γm(k2
2 + ω2
m − k1k2)
(3)
from which one comments that all systems with different mechanical
decay rates γm tend to the same type of system, and τmax in the
opportune limit
- if k1 − k2 γm is small, the following value of τmax is found
τmax |k1−k2γm = −
8k1ω2
m
k2
2 + 4ω2
m − k1k2
(4)
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

ultra-slow light: the ’steady’ states
expansion on the ’steady’ states leads to
- maximum time delay at the transparency window newly found
as
τmax ∼
4K − 1
(k1 − k2)γm
+
16k1
(k1 − k2)2k2
2
ω2
+ Rτ (ω4
; k1, k2) (5)
- the 0 − th-order addend only depends on γm;
- when k1 − k2 γm
τmax |k1−k2γm ∼
16k1
(k1 − k2)2k2
2
ω2
+ Rτ (ω4
; k1, k2) (6)
the two results are found to agree
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

ultra-fast light: the ’steady’ states
expansion on the ’steady’ states leads to
maximum time delay at the detuning is newly found as
δ = −ωm
τmax ∼ −
k1k2
γm
+
8k1
k2(k1 − k2)
ω2
+ O(ω4
)
depends on γm only in the 0 − th-order addend
at k1k2 γm
the time delay is newly found as
τmax |k1−k2γm ∼
8k1
k2(k1 − k2)
ω2
−
32k1
k2
2 (k1 − k2)2
ω3
+ O(ω4
)
new addend if found
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Laser Phase and Frequency Stabilization: Optical
Resonator
- signals reflected form a stable Fabry-Perot reference interferometer
- frequency variation associated with the shot-noise limit
- measurements with average time τ
- frequency fluctuations due to the measurement shot noise
- high sensitivity of detection of resonances
δνc reference cavity full width (at half the maximum)
P0 mode-matched laser-input power
T on-resonance transmission efficiency of the optical cavity
η photodetector quantum efficiency
average of time τ
τ ≥
ℏω
P0Tη
(δνc )2
(δνtau)
the states are discrimined after the photodetector quantum efficiency
R. W. P. Drever, J. L. Hall et al., Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B
31, 97 (1983)
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Applications 2
- experimental apparati techniques can be obtained in a wide range of
applications, such as cold atomic ensembles and cold atomic trapped ions
(at different optical scales), temperature-jump experiment for
protein-folding in molecular dynamics,
Jan-Hendrik Prinz, Hao Wu, Marco Sarich, Bettina Keller, Martin Senne, Martin Held,
John D. Chodera, Christof Schuette, and Frank Noé, Markov models of molecular
kinetics: Generation and validation, The Journal of Chemical Physics 134, 174105
(2011);
N. Djurdjevac, M. Sarich, and C. Schuette, Estimating the eigenvalue error of Markov
state models, Multiscale Model. Simul. 10(1), 61 (2012);
Marcus Jaéger, Houbi Nguyen, Jason C. Crane, Jeffery W. Kelly, Martin Gruebele,
The Folding Mechanism of a b-Sheet: The WW Domain, J. Mol. Biol. (2001) 311,
373-393.
-jump processes between states,
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

- quantum correlation of multispatial modes for Gauss-Markoff models,
Du, J.; Cao, L.; Zhang, K.; Jing, J. Experimental observation of multi-spatial-mode
quantum correlations in four-wave mixing with a conical pump and a conical probe.
Appl. Phys. Lett. 2017, 110, 241103.
- uncertainties estimations in quantum metrology,
in decoherence and dissipation and for mixed system-collections for new
highly-intense laser fields after the application for the of the pertinent
expansions after the calculations. i.e. to be appplied to noisy metrology
beyond the standard quantum limit.
R. Chaves, J. B. Brask, M. Markiewicz, J. Kolodynski, A. Acin, Noisy metrology
beyond the standard quantum limits, Phys. Rev. Lett. 111, 120401 (2013)
[arxiv:1212.3286];
M. Napolitano, M.W. Mitchell, Nonlinear metrology with a quantum interface, New
Journ. of Phys. 12, (2010).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Applications 3
Ramsey experiment:
collective phase noise on the signal sensitivity obtained after
experimenting on an ensemble of N atoms with a laser exhibiting
phase noise
signal sensitivity
δΩ =
p
Nsinh(γτ τ) + cosh(γτ τ)
τ
√
N
saturates as
[δΩ]sat ∼
p
sinh(γττ)
τ
D. Plankensteiner et al., Laser noise imposed limitations of ensemble quantum
metrology, J. Phys. B: At. Mol. Opt. Phys. 49 (2016) 245501.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

the signal sensibility is newly found to scale as
δΩ ≃
1
τ
√
N
+
1
2
γτ
√
N +
1
4
1 − 1
2N2
√
N
γ2
τ τ + O(
√
Nτ2
) + O(N5/2
τ2
)
contains one term in τ, while the remainder in τ2 is limited after N
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

more precise than the newly found saturated signal sensitivity
[δΩ]sat ≃
√
γtau
√
τ
+
1
12
γ5/2
τ τ3/2
which contains only a τ3/2 term.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Cold atomic ensembles
Many-body interactions complex dynamics produces higher optical
densities understanding the dynamics of cooling processes and trapping
ones deep artificial neural network to for optimization efficiency of cold
atomic ensembles improved increasing the number of atoms reducing the
temperature signal-to-noise ratio is measured in magneto-optical trap:
signal-to noise ratio
I(t)
I0
= e
OD
γ2/4
∆2+γ2/4
with
OD resonant optical depth of the ensemble
γ excited state decay rate
∆ probe detuning
A. D. Tranter et al., Multiparameter optimization of a magneto-optical trap using
deep learning, Nat. Commun. 9, 4360 (2018).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

signal-to-noise ratio newly calculated as
I(t)
I0
∼ 1 +
1
4
OD
∆2
γ2
+ O(γ4
)
where the role of the decay rate has been outlined.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The same methods apply to
Cygan, A. , Lisak, D. , Wojtewicz, S. , Domyslawska, J. , Hodges, J. , Trawinski, R.
and Ciurylo, R. (2012), High signal-to-noise ratio laser technique for accurate
measurements of spectral line parameters, Phys. Rev. A 85, 022508 (2012).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Classical system and quantum ones
The Heisenberg limit
probes of non-interacting particles
best possible scaling 1
√
ν
N
shot-noise: standard quantum limit
uncertainty 1
√
nν
ν = T/t number of ’evolve-and-measure’ rounds
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Beyond the standard quantum limit of noisy metrology
N independent qubits
Laser spectroscopy of atoms and molecules:
Markoff model of quantum fluctuations
⇒ fundamental limits, and
non-reciprocal approaches in
non-Hermitian quantum sensing
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

phase ωt decoherence γt
uncertainty δω
ω unknown frequency to be estimated
fundamental limits of non-reciprocal approaches in non-Hermitian
quantum sensing
- classical system of N independent qubits
δω
√
T ≥
r
2γe
N
(7)
- quantum approach:
optimal time parallel noise
δω
√
T ≥
q
1 + (e2γt − 1)N/tn2 (8)
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

in the quantum approach, the new limit of the optimal-time
parallel noise in the quantum approach is exactly calculated
δω
√
T ≃
1
n
√
t
+ γ
√
t +
1 −
1
2
N
γ2t3/2
+ O(t5/2
)
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

A quantum system coupled linearly to Gaussian white noise
Functional methods:
- integration over the noisy field in the evolution operator;
- equivalent non-Hermitian Hamiltonian: the quantum state evolves with
a dissipative dynamics;
-integration over the noisy field : for the time evolution of the density
matrix:
⇒ a gain contribution from the fluctuations can be accessed in addition
to the loss contribution from the non-hermitian Hamiltonian dynamics.
Space and time localisation given by new effective Hamiltonian
proportional to the variance of a Gaussian distribution, the new
non-hermitian interaction being proportional to the square of the
position of the center of mass of the identical particle system. The
non-hermitian operator is responsible for the damping in time of the
many-body system and also by its localisation in space.
O. Oliveira, W. de Paula, T. Frederico, M. S. Hussein Quantum Dynamics in Noisy
Backgrounds: from sampling to dissipation and fluctuations, arXiv: 1512.06456
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Non-Hermitian systems
Non-Hermitian systems can have ultra-high precision of quantum
sensing,
- measurement precision depends on the position of the coherent
drive in the regime of linear response
Markovian features of the full dynamics,
quantum Gaussian white noise
hopping amplitude
amplification factor
non-linear perturbative regime
complete Hamiltonian containing a perturbation term, where the latter is
modulated after a parameter ϵ
L. Bao, Non-Hermitian quantum sensing in the regime beyond linear response,2022
41st Chinese Control Conference (CCC), Hefei, China, 2022, 5670.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Hamiltonian parameter ϵ:
- estimation improved after integrating the output field over
a long time duration of τ;
- the non-diagonal elements can be expanded and calculated
to be higher-order by means of the weighting function.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

The same methods apply to
B. J. Falaye et al., Investigating quantum metrology in noisy channels, Sci. Rep. 7,
16622 (2017);
B. Escher et al., General framework for estimating the ultimate precision limit in noisy
quantum-enhanced metrology, Nature Phys 7, 406 (2011);
S. Zhou et al., Achieving the Heisenberg limit in quantum metrology using quantum
error correction, Nat. Commun. 9, 78 (2018);
M. M. Mueller et al., Noise-robust quantum sensing via optimal multi-probe
spectroscopy, Sci. Rep. 8, 14278 (2018);
M. A. Rodrı́guez-Garcı́a et al., Determination of the asymptotic limits of adaptive
photon counting measurements for coherent-state optical phase estimation, Quantum
Inf. 8, 94 (2022);
F. Roccati et al., Non-Hermitian Physics and master equations, Open Systems and
Inf. Dyn. 29, 2250004 (2022).
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob

Thanks
Thank You for Your attention.
Orchidea Maria Lecian Sapienza University of Rome, Rome, Italy.
A novel lemma of the the Optical Equivalence Theorem Glob