Quantum metrology in a driven-dissipation down-conversion system beyond the parametric approximation
Dong Xie Chunling Xu




xiedong@mail.ustc.edu.cn

xuchunling@guat.edu.cn
College of Science, Guilin University of Aerospace Technology, Guilin, Guangxi 541004, People's Republic of China




We investigate quantum metrology in a degenerate down-conversion system composed of a pump mode and two degenerate signal modes. In the conventional parametric approximation, the pump mode is assumed to be constant, not a quantum operator. We obtain the measurement precision of the coupling strength between the pump mode and two degenerate signal modes beyond the parametric approximation. Without a dissipation, the super-Heisenberg limit can be obtained when the initial state is the direct product of classical state and quantum state. This does not require the use of entanglement resources which are not easy to prepare. When the pump mode suffers from a single-photon dissipation, the measurement uncertainty of the coupling strength is close to 0 as
the coupling strength approaches 0 with a coherent driving. The direct photon detection is proved to be the optimal measurement. This result has not been changed when the signal modes suffer from the two-photon dissipation. When the signal modes also suffer from the single-mode dissipation, the information of the coupling strength can still be
obtained in the steady state. In addition, the measurement uncertainty of the coupling strength can also be close to 0 and become independent of noise temperature as the critical point between the normal and superradiance phase approaches. Finally, we show that a driven-dissipation down-conversion system can be used as a precise quantum sensor to measure the driving strength.

Quantum metrology in a driven-dissipation down-conversion system beyond the parametric approximation
    Chunling Xu
    April 29, 2024
====================================================================================================






§ INTRODUCTION

Quantum metrology mainly studies how to use quantum resources to improve the precision of parameter measurement over classical resources<cit.>.
The quantum resources, such as superposition and entanglement, offer a
possibility to make the measurement precision with linear generator scale as 1/N with N being the total average number of photons, which is the Heisenberg scaling<cit.>. However, classical resources only get the scaling 1/√(N) at most, which is the standard quantum limit<cit.>. By using nonlinear generator or time-dependent evolutions, the super-Heisenberg limit can be obtained<cit.>.

The parametric down-conversion is relatively easy to become a source of entangled photon pairs<cit.>, which plays an important role in quantum information processing.
Moreover, the parametric down-conversion can be used to build optical parametric amplifiers, which is the main difference between nonlinear SU(1,1)-interferometers and the linear SU(2)-interferometers.  SU(1,1)-interferometers offer a platform for multi-photon absorption<cit.>, phase measurement<cit.>, loss-tolerant quantum metrology<cit.>, Wigner function tomography<cit.>, and quantum state engineering<cit.>.

Most of the previous work has used the parametric approximation<cit.> to deal with the down-conversion process, where the number of pump mode is assumed to be constant. This parametric approximation can give an exact description of the system dynamics at short times, but the resulting state will become invalid as the number of pump photons decreases with time. Recently, K. Chinni and N. Quesada<cit.> used the cumulant
expansion method, perturbation theory, and the full numerical simulation of systems to deal with the closed down-conversion process beyond the parametric approximation. However, quantum metrology beyond parametric approximation in parametric down-conversion systems, especially in the case of driven-dissipation, has not been studied at present.

This inspires us to try to fill the gap. In this article, we investigate quantum metrology in a degenerate down-conversion system composed of a pump mode and two degenerate signal modes. In the closed system, we analytically obtain the quantum Fisher information of  the nonlinear coupling strength g between the pump mode and the signal modes. It shows that the super-Heisenberg limit can be obtained without using preparative entangled or squeezed states,  just the signal modes in the Fock state.
The reason for obtaining the super-Heisenberg  limit is the nonlinear generator of g.
When the pump mode suffers from a single-photon dissipation,
the measurement uncertainty of the coupling strength is close to 0 as the coupling strength approaches 0 due to a continuous coherent driving. By comparing with the results achieved by the quantum Fisher information, the direct photon detection is proved to
be the optimal measurement. This result has not been changed when the signal modes suffer from the two-photon dissipation. When the signal modes also suffer from the single-mode dissipation, the information of very weak coupling strength can still be
obtained with a finite precision under steady state. Finally,  the
measurement uncertainty of the coupling strength can also be close to 0 and become independent of the temperature of the noise as the critical point between the normal and superradiance phase approaches.

This article is organized as follows. In section II, quantum metrology in a closed degenerate down-conversion system is studied. In section III, quantum metrology in a driven-dissipation degenerate down-conversion system is studied in different cases. Quantum sensor of the driving strength is explored in section IV.  Finally, we make a simple conclusion and outlook in section V.


§ DEGENERATE DOWN-CONVERSION SYSTEM

We consider that the Hamiltonian describing the
degenerate down-conversion process on resonance (ω_1=2ω_2)is given by


    H=ω_1a^† a +ω_2b^† b+g a b^2†+g a^† b^2,

where a (a^†) is the annihilation (creation) bosonic operator for
the pump mode with the frequency ω_1, and b (b^†) is the annihilation (creation) operator for
the signal mode with the frequency ω_2. The operators a and b satisfy the canonic commutation relation, i.e.,  [c,c^†]=1 with c={a,b}.  Without loss of generality, the coupling strength g is chosen to be the real quantity and ħ=1 through this article.

In the interaction picture, the Hamiltonian is rewritten as

    ℋ=g a b^2†+g a^† b^2.

For the pure state |ψ(g)⟩, the quantum Fisher information for the parameter g can be calculated by<cit.>

    ℱ(g)=4(⟨∂_θψ(g)|∂_θψ(g)⟩-|⟨ψ(g)|∂_gψ(g)⟩|^2),

where |ψ(g)⟩=e^-i g tℋ|ψ(t=0)⟩ and ∂_g=∂/∂ g.

 When the initial state is a semi-classical state (direct product of classical state and quantum state) |ψ(t=0)⟩=|α⟩|n⟩ with the coherent state |α⟩ and the Fock state |n⟩,
we can obtain that ℱ(g)=4[α^2(2n^2+2n+2)+n(n-1)]t^2. Given by the fixed total number of photons, α^2+n=N, the optimal estimation precision ℱ_o(g) is obtained with n=2/3N for n≫1,

    ℱ_o(g)=32/27 N^3 t^2.



When the initial state is a complete quantum state (direct product of quantum state and quantum state)  |ψ(t=0)⟩=|n_1⟩|n_2⟩ with n_1+n_2=N, ℱ(g)=4[n_1(2n_2^2+2n_2+2)+n_2(n_2-1)]t^2. The optimal estimation precision ℱ'_o(g) can be achieved when n_2=2/3N for n_2≫1,

    ℱ'_o(g)=32/27 N^3 t^2.

Obviously, the optimal measurement precision is the same for the complete quantum state and the semi-classical state, i.e., ℱ_o(g)=ℱ'_o(g).

For the complete classical initial state |ψ(0)⟩=|α_1⟩|α_2⟩, the according quantum Fisher information is ℱ(g)=α_2^4 t^2. Given by the constraint condition α_1^2+α_2^2=N, the maximal quantum Fisher information is achieved ℱ(g)=N^2 t^2 when α_1=0. It shows that the complete classical state can not perform better than the complete quantum state and the semi-classical state. In the down-conversion system, the super-Heisenberg limit can be obtained only if the initial state of the signal mode is quantum state, where entangled or squeezed states that are difficult to prepare are not required. The reason for obtaining the super-Heisenberg  limit is due to the fact that the generator a b^2†+ a^† b^2 is nonlinear.



§ DRIVEN-DISSIPATION SYSTEM


In this section, we consider that the degenerate down-conversion system suffers from an inevitably dissipative environment, as shown in Fig. <ref>. In order to resist the dissipation, we consider that there is an extra coherent driving H_d=iλ_a(a^†-a). The evolution of the density matrix ρ of the driven-dissipation system is described by the master equation

    ρ̇=-i[H+H_d,ρ]+γ_a𝒟[a]ρ+γ_b𝒟[b]ρ,

where the superoperator 𝒟[c]ρ=γ_c[2cρ c^†-c^† cρ-ρ c^† c] with c={a, b}.
The corresponding quantum Langevin-Heisenberg equation of a operator O is derived by<cit.>

    Ȯ=i[H+H_d,O]-∑_c=a,b{[O,c^†](γ_cc-√(2γ_c)c_in)
    
    -(γ_cc-√(2γ_c)c_in)[O,c]}.

Utilizing the above equation, let O=a, b, we can obtain the evolution dynamics

    ȧ   =-ig b^2-γ_a a+√(2γ_a )a_in+λ_a,
    ḃ   =-2ig ab^†-γ_b b+√(2γ_b )b_in,

where the expected values of the noise operators are ⟨ c_(t) ⟩=⟨ c_^†(t) ⟩=0, ⟨ c^†_(t)c_ (t')⟩=0 and ⟨ c_(t)c^†_ (t')⟩=δ(t-t') with c_={a_, b_}.



 §.§ In the case of γ_a≫γ_b

When the decay rate γ_a is much lager than γ_b (γ_a≫γ_b), the pump mode a can be adiabatically eliminated. Let ȧ=0, we obtain

    a=(-ig b^2+√(2γ_a )a_in+λ_a)/γ_a.

Substituting the above equation into Eq. (<ref>), we obtain that

    ḃ=   -2g^2 b^† b^2/γ_a-2igλ_ab^†/γ_a -2ig√(2/γ_b )b^† a_in
       -γ_b b+√(2γ_b )b_in.


By combining Eq. (<ref>) with Eq. (<ref>), we can get the reduced master equation of the signal mode, which is described as

    ρ̇=-i[H_b,ρ]+γ_b𝒟[b]ρ+κ𝒟[b^2]ρ,

where H_b=gλ_a/γ_a(b^2+b^2†) and κ=2g^2/γ_a.

The steady state can be analytically derived by a complex P-representation solution<cit.>  or  Keldysh-Heisenberg equation<cit.>. The general form can be expressed as

    ⟨ b^† lb^k⟩=1/𝒩√(2^l+k)∑_m=0^∞1/m! F^*_m+lF^*_m+k,

where the normalization factor 𝒩=∑_m=0^∞|μ|^2m/m! |_2F_1[-m,y;z;2] |^2, and the function F_m+k=(-μ)^m+k _2F_1[-(m+k),y;z;2], in which μ=i√(2iλ_a/g), y=γ_aγ_b/4g^2, and z=γ_aγ_b/2g^2.
The Gauss hypergeometric function _2F_1[-m,y;z;2] is described by

    _2F_1[-m,y;z;2]=∑_n=0^∞(-m)_n(y)_n 2^n/(z)_nn!,

where the function (r)_n=Γ(r+n)/Γ(r) and Γ(r) denotes the Gamma function.



  §.§.§ In the case of γ_b=0

When the decay rate γ_b=0, the expected value in Eq. (<ref>) can be simplified as

    ⟨ b^† lb^k⟩=(i√(-iλ_a/g))^l(-i√(iλ_a/g))^k.

With the specific measurement operator M, the measurement uncertainty of the parameter g can be calculated by the error propagation formula

    δ^2g=⟨ M^2⟩-⟨ M⟩^2/|∂⟨ M⟩/∂ g|^2.

For the direct photon detection with M_d=b^† b, the uncertainty of the parameter g is derived by the above equation

    δ^2g=g^3/λ_a.

For the homodyne detection with M_h=be^-iφ+b^† e^iφ, the uncertainty of the parameter g is given by

    δ^2g=2g^3/[λ_a(cosφ-sinφ)^2],

When φ=0, i.e., the Homodyne detection M_h=b+b^†, the above measurement uncertainty is optimized,

    δ^2g=2g^3/λ_a.

Comparing Eq. (<ref>) with Eq. (<ref>), we find that the direct photon detection performs better than the homodyne detection.
All results show that the uncertainty δ^2g is close to 0 as the parameter g→ 0. It is due to the fact that the subsystem of the signal mode can not arrive at the steady state when g=0. At this point, the total number of photons tends to infinity as g goes to 0, i.e., N_b=⟨ b^† b⟩=λ_a/g→∞. It is due to the fact that the infinite encoding time makes the measurement uncertainty become 0.
Using N_b, Eq. (<ref>) can be reexpressed as

    δ^2g=λ_a^2/N_b^3.

It shows that the super-Heisenberg scaling 1/N_b^3 of the single photon dissipation of the pump mode can be obtained by local measurement of the signal mode without considering the time consumption.


  §.§.§ Extra two-photon dissipation

Then, we consider that the signal mode suffers from an extra two-photon dissipation  with dissipation rate κ_e. The total evolution dynamics of the signal mode is described by
 
    ρ̇=-i[H_b,ρ]+γ_b𝒟[b]ρ+(κ+κ_e)𝒟[b^2]ρ,


When the decay rate γ_b=0, the expectated values can be simplified as

    ⟨ b^† lb^k⟩=(i√(-2igλ_a/γ_a(κ+κ_e)))^l(-i√(2igλ_a/γ_a(κ+κ_e)))^k.

Using the direct photon detection, the measurement uncertainty of the parameter g is given by

    δ^2g=g(κ_eγ_a+2g^2)^3/2λ_a(κ_eγ_a-2g^2)^2.


For the Gaussian state, the quantum Fisher information is derived by<cit.>

    ℱ(g)=   2d^2/4d^2+1[(𝒞^-1∂_g𝒞)^2]+
    8(∂_g d)^2/16d^4-1
       +⟨∂_g𝐗^⊤⟩𝒞^-1⟨∂_g𝐗⟩,

where 𝐗^⊤=(q,p) with quadrature operators defined as: p=1/√(2)(b+b^†), and q=1/i√(2)(b-b^†). And the entries of the covariance matrix are defined as 𝒞_ij=1/2⟨𝐗_i𝐗_j+𝐗_j𝐗_i⟩-⟨𝐗_i⟩⟨𝐗_j⟩. d is given by d=√(𝒞).

Based on the expected values in Eq. (<ref>), we obtain that

    𝒞=1/2(
    [ 1   0; 0   1 ] ).

Then, we derive that d=1. As a result, the quantum Fisher information can be obtained

    ℱ(g)=⟨∂_g𝐗^⊤⟩𝒞^-1⟨∂_g𝐗⟩
    
    =2λ_a(κ_eγ_a-2g^2)^2/g(κ_eγ_a+2g^2)^3.

According to the quantum Cramér-rao bound<cit.>, the
measurement uncertainty is given by

    δ^2g≥ 1/ℱ(g)=g(κ_eγ_a+2g^2)^3/2λ_a(κ_eγ_a-2g^2)^2.

Comparing with the result obtained by the direct detection in Eq. (<ref>), it obviously shows that the direct detection is the optimal measurement, which can obtain the optimal measurement precision.

The above result shows that the measurement uncertainty is also close  to 0 as g goes to 0. In order to explain this, we calculate the characteristic time for the system to reach steady state.
Let b=⟨ b⟩+δ b, keeping only the linear terms, the evolution of ⟨δ b⟩ is reduced to

    ⟨δ̇ ̇ḃ⟩= - 8g λ_a/γ_a(κ+κ_e)⟨δ b⟩.

According to the above equation, the characteristic time to steady state is given by

    τ=γ_a(κ+κ_e)/8g λ_a.

This result shows that the characteristic time τ is divergent at g=0.
The two-photon dissipation does not allow the signal mode to reach the steady state.
The intuitive explanation is that there are subspaces that are not subject to two-photon dissipation.




  §.§.§ Three-level approximation in the case of γ_a≫γ_b≠0

Expanding on the Fock basis {|n⟩, n = 0, 1, 2, . . . }, Eq. (<ref>) can be rewritten as

    ρ̇_n,n'=   -iλ_a g/γ_a[√((n+1)(n+2))ρ_n+2,n'
       +√((n-1)n)ρ_n-2,n'-√((n-1)n)ρ_n,n'-2
       -√((n'+1)(n'+2))ρ_n,n'+2]+
       +γ_b[√((n+1)(n'+1))ρ_n+1,n'+1-n+n'/2ρ_n,n']
       +κ [√((n+2)(n+1)(n'+2)(n'+1))ρ_n+2,n'+2
       -n(n-1)+n'(n'-1)/2ρ_n,n'].



We consider a weak coupling g, g≪γ_a/λ_a. The effective two-photon driving is weak. Therefore, the subsystem b is at a low energy level. We consider the three-level subsystem b with ρ_11=1-ρ_00-ρ_22. Keeping only the lowest three energy levels, Eq. (<ref>) is reduced to the following formulas

    ρ̇_00=   -i√(2)λ_a g/γ_a[ρ_20-ρ_02]+2γ_bρ_11+4κ'ρ_22,
    ρ̇_10=   γ_b[2√(2)ρ_21-ρ_10],
    ρ̇_22=   -4(κ'+γ_b)ρ_22-i√(2)λ_a g/γ_a(ρ_02-ρ_20),
    ρ̇_21=   -(2κ'+3γ_b)ρ_21-i√(2)λ_a g/γ_a(ρ_01),
    ρ̇_20=   -(2κ'+2γ_b)ρ_20-i√(2)λ_a g/γ_a(ρ_00ρ_22),

where κ'=κ+κ_e.
Let the left terms of the above equations be 0, the steady-state solutions are given by

    ρ_00=2(2g^2+γ_a(κ_e+γ_b))^2+g^2λ_a^2/2(2g^2+γ_a(κ_e+γ_b))^2+4g^2λ_a^2,
    ρ_11=g^2λ_a^2/(2g^2+γ_a(κ_e+γ_b))^2+2g^2λ_a^2,
    ρ_22=g^2λ_a^2/2(2g^2+γ_a(κ_e+γ_b))^2+4g^2λ_a^2,
    ρ_20=-ig(2g^2+γ_a(κ_e+γ_b))λ_a/2(2g^2+γ_a(κ_e+γ_b))^2+4g^2λ_a^2,
    ρ_10=0, ρ_21=0.

By calculating the expected value N_b=⟨ b^† b⟩ from the three-level approximation method  and the master equation in Eq. (<ref>), we can find that the three-level approximation is more and more accurate as the coupling strength g decreases, as shown in Fig. <ref>.

With the direct photon detection, the measurement uncertainty of the coupling strength g is given by

    δ^2 g|_g=0=3γ_a(κ_e+γ_b)^2/16λ^2.


With the homodyne detection,  the measurement uncertainty is given by

    δ^2 g|_g=0=γ_a(κ_e+γ_b)^2/λ^2.


When the density matrix is expressed in diagonal form with the eigenstate |k⟩, ρ=∑_k=0,1,2E_k|k⟩⟨ k|, the quantum Fisher information can be calculated by

    ℱ(g)=   ∑_k,E_k>0(∂_g E_k)^2/E_k
       +∑_k,k',E_k+E_k'>02(E_k-E_k')^2/E_k+E_k'⟨ k|∂_g |k'⟩.

After a simple calculation according to the above equation, the quantum Fisher information is given by

    ℱ(g=0)=6λ^2/γ_a(κ_e+γ_b)^2


According to the quantum Cramér-rao bound, the measurement uncertainty is given by

    δ^2 g|_g=0≥1/ℱ(g=0)=γ_a(κ_e+γ_b)^2/6λ^2.

Comparing the above equation with Eq. (<ref>) and Eq. (<ref>), it shows that the direct homodyne detection is close to the optimal measurement.

When there are single photon dissipation, i.e., γ_b≠0, the characteristic time to steady state is given by τ=1/γ_b when the coupling strength g→ 0.
In the steady state, the total photon number is given by N_b=⟨ b^† b⟩=2g^2λ_a^2/(2g^2+γ_a(κ_e+γ_b))^2+2g^2λ_a^2. When g→ 0, the total photon number N_b=0. This result is interesting. Although the total photon number in the steady state is close to 0 and the interaction time is finite, the measurement uncertainty of the coupling strength g is not infinite. In other words, when the signal mode suffers from the single-photon and two-photon dissipation, the information of the weak coupling strength g can still be obtained in the steady state.



 §.§ The adiabatic elimination condition is not satisfied

In this subsection, we consider that the adiabatic elimination condition γ_a≫γ_b is not satisfied.

By taking the expected value of both sides of Eq. (<ref>) and Eq. (<ref>), we obtain that

    ⟨̇ ̇ȧ⟩̇   =-ig ⟨ b^2 ⟩-γ_a ⟨ a⟩+λ_a,
    ⟨̇ ̇ḃ⟩̇   =-2ig ⟨ a b^†⟩-γ_b ⟨ b⟩.


Let ⟨̇ ̇ȧ⟩̇=0 and ⟨̇ ̇ḃ⟩̇=0, the equations of motion for the steady-state mean values of the operators are derived as

    0   =-ig ⟨ b ⟩_s^2-γ_a ⟨ a⟩_s+λ_a,
    
    0   =-2ig ⟨ a⟩_s⟨ b⟩^*-γ_b ⟨ b⟩_s,

where we use  a mean-field like approximation ⟨ b^2 ⟩_s=⟨ b ⟩_s^2 and ⟨ a b^†⟩_s=⟨ a ⟩_s⟨ b ⟩_s^*.
We set that ⟨ a ⟩_s=x_a+i y_a and ⟨ b ⟩_s=x_b+i y_b, where x_a, x_b, y_a and y_b are real numbers. The above equations can be rewritten
as

    0   =-γ_ax_a+2gx_by_b+λ_a,
    
    0   =g(x_b^2-y_b^2)+γ_ay_a,
    
    0   =γ_bx_b+2gx_ay_b-2gx_by_a,
    
    0   =γ_by_b+2g(x_ax_b+y_ay_b).

By solving the above equations, we obtain

    :   x_a=λ_a/γ_a, x_b=y_a=y_b=0;
    :   x_a=γ_b/2g, x_b=±√(2gλ_a-γ_aγ_b)/2g,
       y_a=0, y_b=∓√(2gλ_a-γ_aγ_b)/2g.


When 2gλ_a-γ_aγ_b<0, the solution (2) does not exist due to the fact that x_b can not be a complex number.

By using that a=⟨ a⟩_s+δ a and b=⟨ b⟩_s+δ b, we can obtain the linearized quantum Langevin equations by keeping the terms up to the first
order of quantum fluctuation: 𝐡=(δ a, δ a^†, δ b, δ b^†)^⊤

    ḣ=𝐖𝐡+𝐡_

where the evolution matrix 𝐖 is described as

    𝐖=(
    [         -γ_a       0    -2ig⟨ b⟩_s       0;  0      -γ_a       0           2ig⟨ b⟩_s^*; -2ig⟨ b⟩_s^*   0    -γ_b        -2ig⟨ a⟩_s;     0       2ig⟨ b⟩_s   2ig⟨ a⟩_s^*   -γ_b ] ),

and the quantum noise operators are 𝐡_=(√(2γ_a)a_, √(2γ_a)a^†_, √(2γ_b)b_, √(2γ_b)b_)^⊤.

When 2gλ_a-γ_aγ_b<0, the steady-state solution (⟨ a⟩_s=λ_a/γ_a, ⟨ b⟩_s=0) ensures that the system is stable.
It is due to the fact that  all eigenvalues of the evolution matrix 𝐖 have negative real parts.

When 2gλ_a-γ_aγ_b>0, the solution (⟨ a⟩_s=λ_a/γ_a, ⟨ b⟩_s=0) makes the system unstable. In this case, (⟨ a⟩_s= γ_b/2g, ⟨ b⟩_s=∓√(2gλ_a-γ_aγ_b)/2g(1-i)) can make the system stable.
When γ_b=0, we achieve that ⟨ b⟩_s=∓√(λ_a/2g)(1-i). It recovers the previous result as shown in Eq. (<ref>).



Generally, it is defined as the normal phase when⟨ b⟩_s=0; and it is defined as the superradiance phase when⟨ b⟩_s≠0<cit.>. As the driving strength λ_a increases, the system changes from the normal phase to the superradiant phase.
The critical point of the phase transition occurs when the driving strength λ_a is equal to the critical driving strength λ_c=γ_aγ_b/2g.


In the normal phase, we can obtain the analytical solution.
Supposing that the system has reached the steady state after a long-time evolution, the solution of the quantum fluctuation operator δ b is

    δ b=∫_0^∞ dt   [e^-γ_b tcosh(gtλ_a/γ_a)b_(t)
        -ie^-γ_b tsinh(gtλ_a/γ_a)b^†_(t)].

Using the correlation functions of the noise operator b_(t), we can obtain the expected values

    ⟨δ b⟩   =⟨δ b^†⟩=0,
    ⟨δ b^†δ b ⟩   =2g^2 λ_a^2/γ_aγ_b-4g^2λ_a^2,
    ⟨(δ b)^2 ⟩   =-ig λ_aγ_aγ_b/2(γ_aγ_b-4g^2λ_a^2).

Using the decoupling relation
⟨ ABCD⟩=⟨ AB⟩⟨ CD⟩+⟨ AC⟩⟨ BD⟩+⟨ AD⟩⟨ BC⟩-2⟨ A⟩⟨ B⟩⟨ C⟩⟨ D⟩
as shown in ref.<cit.>, we obtain the expected value

    ⟨(δ b^†δ b)^2 ⟩   ≈2⟨(δ b^†δ b) ⟩^2 +⟨δ  b^†δ b ⟩+|⟨(δ b)^2⟩|^2-2|⟨δ  b⟩|^4
       =3g^2λ_a^2γ_a^2γ_b^2/(γ_a^2γ_b^2-4g^2λ_a^2)^2.


With the direct photon detection b^† b, the measurement uncertainty of g is given by

    δ^2 g   =⟨(δ b^†δ b)^2⟩ -⟨(δ b^†δ b)⟩^2 /|∂_g⟨(δ b^†δ b)⟩ |^2
       =(γ_a^2γ_b^2-4g^2λ_a^2)^2(3γ_a^2γ_b^2-4g^2λ_a^2)/16λ_a^2γ_a^4γ_b^4.



We can find that the measurement uncertainty δ^2 g is close to 0 as the phase transition point approaches, i.e., λ_a→λ_c.
As a cost, the time resources used will also tend to be infinite.




  §.§.§ Thermal noise with temperature T

When the system is subjected to a dissipative environment with a non-zero temperature T, the correlated values of the noises of the subsystem b are rewritten as

    ⟨ b^†_(t) ⟩   =⟨ b^†_(t) ⟩=0,
    ⟨ b^†_(t)b_(t') ⟩   =n(T)δ(t-t'),
    ⟨ b_(t') b^†_(t)⟩   =[n(T)+1]δ(t-t'),

where the average thermal photon number n(T)=1/exp (ω_2 T)-1.
With the direct photon detection b^† b, the measurement uncertainty of g is given by

    δ^2 g   =⟨(δ b^†δ b)^2⟩ -⟨(δ b^†δ b)⟩^2 /|∂_g⟨(δ b^†δ b)⟩ |^2
       =(γ_a^2γ_b^2-4g^2λ_a^2)^2[(3+2n)γ_a^2γ_b^2+4g^2λ_a^2(2n-1)]/16(1+2n)λ_a^2γ_a^4γ_b^4.

From the above equation, we can obviously see that the measurement uncertainty δ^2 g is also close to 0 as the phase transition point approaches. Near the critical point of the phase transition, (i.e. λ_a≃λ_c), we get a temperature-independent result

    δ^2 g=(γ_aγ_b-2gλ_a)^2/4λ_a^2γ a γ_b.

It shows that the influence of the temperature of the noise bath on the measurement precision of the coupling parameters tends to 0 near the phase transition point. That is, it is robust to the thermal fluctuations from the dissipative environment.



§ QUANTUM SENSOR OF THE DRIVING STRENGTH Λ_A

In addition to measuring the coupling strength g, it can also be used to measure the driving strength λ_a.
By the similar calculation, in the case of γ_b=0, the measurement precision of λ_a is

    δ^2λ_a=λ_a (2g^2+γ_a κ_e)/2g.

This result shows that the measurement uncertainty of the driving strength λ_a decreases with the decrease of λ_a. It means that the dissipative degenerate down-conversion system can be a sophisticated quantum sensor about the weak driving strength when the signal mode does not suffer from the single-photon dissipation.
With photon number of the signal mode N_b=2gλ_a/2g^2+γ_a κ_e,  the measurement precision of λ_a is rewritten as

    δ^2λ_a=λ_a^2 /N_b.

This means that for λ_a only the quantum limit is obtained, and nothing like g can obtain the super-Heisenberg limit in Eq. (<ref>). It is due to the fact that the generator of g is nonlinear and the generator of λ_a is linear.
Moreover, we find that the measurement precision of λ_a  can be enhanced by reducing the coupling strength g without the extra two-photon dissipation.
However, when g<√(γ_aκ_e), the measurement precision of λ_a  will decrease as g decreases. For g=√(γ_aκ_e), we can achieve the optimal precision of λ_a

    δ^2λ_a^o=λ_a√(2γ_a κ_e ).




§ CONCLUSION

We have investigated the quantum metrology in the degenerate down-conversion system.
Due to the nonlinear generator of the coupling strength, in the closed system the super-Heisenberg limit can be obtained with the semi-classical initial state. And the measurement precision increases with the evolution time, which gives an intuitive explanation of the results obtained in the dissipation case.

When γ_a≫γ_b, we obtain the analytical result by adiabatically eliminating the pump mode. In the case of γ_b=0, the direct photon detection is proved to the optimal measurement. The result shows that the measurement uncertainty δ^2g
is close to 0 as g→ 0. As a trade-off, the time to steady state also tends to infinity as g→ 0. Unexpectedly, the above result can not be influenced when the signal mode suffers from two-photon dissipation. The intuitive explanation is that there are subspaces that are not subject to two-photon dissipation. In the case of γ_b≠0, an interesting result shows that although the total photon number in the steady state is close to 0 and the interaction time is finite, the measurement uncertainty of the coupling g is not infinite. It means that the information of the coupling strength g can still be obtained in the steady state no matter how small g is.

When the adiabatic elimination condition is not satisfied, there is a critical phase transition point between the normal phase and superradiance phase. We can find that the measurement uncertainty δ^2 g is close to 0 as the phase transition point approaches. As before, it takes an infinite amount of encoding time to get that δ^2 g=0. And we show that the influence of the temperature of the
noise bath on the measurement precision of the coupling
parameters tends to 0 near the phase transition point.

Finally, the degenerate down-conversion system can be used as a quantum sensor to measure the coherent driving strength. We show that the measurement uncertainty of
the driving strength λ_a decreases with the decrease of the driving strength. Only the quantum limit is obtained for λ_a, which is due to the fact that the generator of g is nonlinear and the generator of λ_a is linear. This inspires us to discuss how to convert linear generators into nonlinear generators for improving the measurement precision.
Moreover, we find that the measurement precision of λ_a  can be enhanced by reducing the coupling strength g without the extra two-photon dissipation.
However, when g<√(γ_aκ_e), the measurement precision of λ_a  will decrease as g decreases.


It is worth further exploring the problems related to multi-photon absorption and multi-parameter simultaneous measurement in the down-conversion system. In experiment, the down-conversion system can be observed typically in uniaxial and noncentrosymmetric media<cit.>, a linear trapped three-ion crystal<cit.>, and a lithium niobate crystal<cit.>.




 Acknowledgements.-This research was supported by the National Natural Science Foundation of China (Grant No. 12365001 and No. 62001134), and Guangxi Natural Science Foundation ( Grant No. 2020GXNSFAA159047).
9



lab1S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nat. Photonics 12,724-733 (2018).
lab2C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
lab3V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
lab4V. Giovanetti, S. Lloyd, L. Maccone Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330 (2004).
lab5D. Xie, and A. Wang, Quantum metrology in correlated environments, Phys. Lett. A 378, 2079 (2014).
lab6C. W. Helstrom, Quantum Detection and Estimation Theory, Academic, New
York, 1976.
lab7J. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Lukin, High-sensitivity diamond magnetometer with nanoscale resolution, Nat. Phys. 4: 810 (2008).
lab7a1C. Napoli, S. Piano, R. Leach, G. Adesso, and T. Tufarelli, Towards superresolution surface metrology: Quantum estimation of angular and axial separations. Phys. Rev. Lett. 122, 140505 (2019).
lab7a2Z. Hou, Y. Jin, H. Chen, J. Tang, C. Huang,
H. Yuan, G. Xiang, C. Li, and G. Guo, Super-Heisenberg and Heisenberg scalings achieved
simultaneously in the estimation of a rotating field. Phys Rev Lett
126, 070503 (2021).
lab7a3X. He, R. Yousefjani, and A. Bayat, Stark localization as a resource for weak-field sensing with super-Heisenberg precision, Phys. Rev. Lett. 131, 010801(2023)
lab7a4J. Yang, S. Pang, A. d. Campo, A. N. Jordan, Super-Heisenberg scaling in Hamiltonian parameter estimation in the long-range Kitaev chain, Phys. Rev. Research 4, 013133(2022).
lab8M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Theory of two-photon entanglement in type-II optical parametric down-conversion, Phys. Rev. A 50, 5122 (1994).
lab9C. Sanchez Muñoz, G. Frascella, and F. Schlawin, Quantum metrology of two-photon absorption, Phys. Rev. Research 3, 033250 (2021).
lab10S. Panahiyan, C. S. Muñoz, M. V. Chekhova, and F. Schlawin, Two-photon absorption measurements in the presence of single-photon losses, Phys. Rev. A 106, 043706 (2022).
lab11C. Drago and J. E. Sipe, Aspects of two-photon absorption of squeezed light: The continuous-wave limit, Phys. Rev. A 106, 023115 (2022).
lab12S. Panahiyan, C. S. Muñoz, M. V. Chekhova, and F. Schlawin, Nonlinear Interferometry for Quantum-Enhanced Measurements of Multiphoton Absorption, Phys. Rev. Lett. 130, 203604 (2023).
lab12aM. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer, Phys. Rev. Lett. 119, 223604 (2017).
lab13Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, Loss-tolerant quantum dense metrology with su(1,1) interferometer, Opt. Express 26, 27705 (2018).
lab14W. Du, J. Kong, G. Bao, P. Yang, J. Jia, S. Ming, C.-H. Yuan,
J. F. Chen, Z. Y. Ou, M. W. Mitchell, and W. Zhang, Su(2)-in su(1,1) nested interferometer for high sensitivity, loss-tolerant quantum metrology, Phys. Rev. Lett. 128, 033601 (2022).
lab15M.Kalash, and M. V. Chekhova, Wigner function tomography via optical parametric amplification, Opica 10, 1142 (2023).
lab16S. Lemieux, M. Manceau, P. R. Sharapova, O. V. Tikhonova, R. W. Boyd, G. Leuchs, and M. V. Chekhova, Engineering the frequency spectrum of bright squeezed vacuum via group velocity dispersion in an su(1,1) interferometer, Phys. Rev. Lett.
    117, 183601 (2016).
lab17J. Su, L. Cui, J. Li, Y. Liu, X. Li, and Z. Y. Ou, Versatile and precise quantum state engineering by using nonlinear interferometers, Opt. Express 27, 20479 (2019).
lab18P. D. Drummond and M. Hillery, The quantum theory of
nonlinear optics (Cambridge University Press, 2014).
lab19K. Chinni, and N. Quesad, Beyond the parametric approximation:
pump depletion, entanglement and squeezing in macroscopic down-conversion, arXiv:2312.09239 (2023).
lab20C. W. Helstrom, Quantum Detection and Estimation Theory (New
York: Academic) 1976.
lab21A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Amsterdam: North Holland) 1982.
lab22C. Gardiner, and P. Zoller, Qauntum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, vol. 56 (Springer, Berlin, 2004).
lab23M. Reitz, C. Sommer, and C. Genes, Langevin approach to quantum optics with molecules, Phys. Rev. Lett. 122, 203602 (2019).
lab24D. Xie, C. Xu, and A. M. Wang, Quantum thermometry with a dissipative quantum Rabi system, Eur. Phys. J. Plus 137:1323 (2022).
lab25N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable wigner function multimodality and dissipative phase transitions. Phys. Rev. A 94, 033841 (2016).
lab26Y. Zhang, and G. Chen, Method for driven-dissipative problems: Keldysh- Heisenberg equations, Phys. Rev. A 102, 062205 (2020).
lab27A. Monras, Phase space formalism for quantum estimation of Gaussian states, arXiv:1303.3682 (2013).
lab28L. Garbe, M. Bina, A. Keller, M. G. A. Paris, S. Felicetti, Critical quantum metrology with a finite-component quantum phase transition. Phys. Rev. Lett.
124, 120504 (2020).
lab29H. Cramér, Mathematical Methods of Statistics (Princeton University, Princeton, 1946).
lab30C.R. Rao, Linear Statistical Inference and Its Applications (Wiley, NewYork, 1973).
lab31S.L. Braunstein, C.M. Caves, Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439 (1994).
lab32A. Baksic and C. Ciuti, Controlling Discrete and Continuous Symmetries in Superradiant Phase Transitions with Circuit QED Systems, Phys. Rev. Lett. 112, 173601 (2014).
lab33J. Naikoo, K. Thapliyal, A. Pathak, S. Banerjee, Probing nonclassicality in an optically driven cavity with two atomic ensembles. Phys. Rev. A 97,063840 (2018).
lab34R. W. Boyd, Nonlinear Optics (Academic Press, 2020).
lab35J. Flórez, J. S. Lundeen, and M. V. Chekhova, Pump depletion in parametric down-conversion with low pump energies, Optics Letters 45, 4264 (2020).
lab36S. Ding, G. Maslennikov, R. Hablützel, and D. Matsukevich, Quantum simulation with a trilinear hamiltonian,
Physical Review Letters 121, 130502 (2018).



















