Generally speaking, a quantum sensor means the sensor utilizing quantum resources, such as quantum devices, quantum states, quantum effects, etc. [49, 50]. In [14], a paradigm in designing quantum sensors is proposed that several coupled cavities couple to a gain reservoir and a dissipative reservoir at a single point. Illuminated by this paradigm, the sensor we considered consists of a coupled double-cavity interacting with two reservoirs. The first cavity is coupled to a dissipative reservoir at x ₁ and x ₂, and the second cavity is coupled to it at x ₃ and x ₄, as shown in Figure 1. On the contrary, a gain reservoir couples to both cavities at the same point. In addition, a classical pump β with a noise input B _in enters the readout waveguide which only couples to the cavity 1, and its reflected field B _out is measured by homodyne detection. This model can be realized by superconducting quantum circuits, i.e., two LC resonators couple to three waveguides, where one of the waveguides is used for readout and the others are used as reservoirs. According to the model, the total Hamiltonian reads H tot = H 0 + H d + H I , (1) where H 0 = ∑ i , j = 1 2 H ij ε a i † a j + ∫ d k ω b , k b k † b k + ∫ d k ω c , k c k † c k + ∫ d k ω d , k d k † d k , (2a) H d = κ β e − i ω L t a 1 † + H . c . + κ ∫ d k 2 π a 1 † b k + H . c . , (2b) H I = ∑ i = 1 2 ∫ d k Y i a i † c k † + H . c . + ∫ d k Z 1 e i k x 1 + e i k x 2 a 1 † d k + Z 2 e i k x 3 + e i k x 4 a 2 † d k + H . c . . (2c)

Equation 2a describes the free Hamiltonian of the two cavities, the readout waveguide, the gain and dissipative reservoirs with bosonic annihilation operators a _i, b _k , c _k , and d _k , respectively. Here, we have assumed that the perturbation ɛ is small enough such that H _ij [ɛ] has a linear form ¹ [14, 51] H ij [ ε ] = H ij f + ε V ij , where H ij f is the unperturbed part of the coupled cavities and V _ij denotes the coupling of perturbation ɛ on the cavities. The first term in Eq. 2b represents a classical pump β with a driving frequency ω _L and a coupling strength κ that enters cavity 1 through the readout waveguide. The second term denotes the interaction between cavity 1 and the readout waveguide, which yields a noise input B _in to the cavity, as shown later. Eq. 2c describes couplings between the cavities and the reservoirs with strengths Y _i and Z _i, respectively. Notably, the position-dependent phase e i k x m , (m = 1, 2, 3, 4) with a wave vector k is introduced by the multi-point couplings.

For the sake of sensing, we analyze how the output varies when the perturbation ɛ acts on the sensor, which can be done with the quantum Langevin equation. Before we proceed, we assume that the coupling points are equally spaced, i.e., d = x ₂ − x ₁ = x ₃ − x ₂ = x ₄ − x ₃. For simplicity, we let x ₁ = 0. Also, the linear dispersion relation holds in the dissipative reservoir, i.e., ω _d,k = v _g k with v _g being the group velocity [48, 52, 53]. With the abovementioned assumptions, the equations of motion for two cavities take the form a ̃ ̇ 1 t = F 11 ε a ̃ 1 t − 2 π | Z 1 | 2 a ̃ 1 t − τ e i ω L τ + F 12 ε a ̃ 2 t − M ̃ 1 in t , (3a) a ̃ ̇ 2 t = F 22 ε a ̃ 2 t − 2 π | Z 2 | 2 a ̃ 2 t − τ e i ω L τ + F 21 ε a ̃ 1 t − F 21 dir t − M ̃ 2 in t , (3b) where F 11 ε = i ω L − i H 11 ε + π | Y 1 | 2 − 2 π | Z 1 | 2 − κ 2 , (4a) F 22 ε = i ω L − i H 22 ε + π | Y 2 | 2 − 2 π | Z 2 | 2 , (4b) F 12 ε = − i H 12 ε + π Y 1 Y 2 ∗ , (4c) F 21 ε = − i H 21 ε + π Y 2 Y 1 ∗ , (4d) F 21 dir t = 2 π Z 2 Z 1 ∗ e i ω L τ a ̃ 1 t − τ + 2 a ̃ 1 t − 2 τ e i ω L τ + a ̃ 1 t − 3 τ e 2 i ω L τ , (4e) M ̃ 1 in t = i κ β + B ̃ in t − i 2 π Y 1 C ̃ in † t − i 2 π Z 1 D ̃ in t + D ̃ in t − τ e i ω L τ , (4f) M ̃ 2 in t = i 2 π Y 2 C ̃ in † t − i 2 π Z 2 e 2 i ω L τ D ̃ in t − 2 τ + D ̃ in t − 3 τ e i ω L τ , (4g) with τ = d/v _g being the time delay between the two neighboring points. Here, a ̃ i [ t ] = a i [ t ] e i ω L t denotes the slowly-varying operator. Also, B ̃ in t = B in t e i ω L t = 1 2 π ∫ d k b k 0 e − i ω b , k − ω L t , (5a) C ̃ in † t = C in † t e i ω L t = 1 2 π ∫ d k c k † 0 e i ω c , k + ω L t , (5b) D ̃ in t = D in t e i ω L t = 1 2 π ∫ d k d k 0 e − i ω d , k − ω L t (5c) are the inputs for the readout waveguide, gain, and dissipative reservoirs, respectively. In addition, the input-output relation for the field in the readout waveguide is given by B ̃ out t = B ̃ in t + β − i κ a ̃ 1 t , (6) where B ̃ out t = B out t e i ω L t = 1 2 π ∫ d k b k t 1 e − i ω b , k t − t 1 e i ω L t (7) is the output field in the waveguide at a final time t ₁.

Using Fourier transformation, the delayed differential Eqs. 3a, 3b can be solved as a ̄ 1 ω ; ε a ̄ 2 ω ; ε = ω L + ω I − H ε − π i G Y + 2 π i D Z + i κ ̃ 2 − 1 M ̄ in ω = χ ω ; ε i κ M ̄ in ω , (8) with κ ̃ = κ 1 0 0 0 , G Y = | Y 1 | 2 Y 1 Y 2 ∗ Y 1 ∗ Y 2 | Y 2 | 2 = Y 1 Y 2 Y 1 ∗ Y 2 ∗ = Y Y † , (9) D Z = | Z 1 | 2 1 + e i ω L − ω τ 0 Z 2 Z 1 ∗ e i ω L − ω τ + 2 e i 2 ω L − ω τ + e i 3 ω L − ω τ | Z 2 | 2 1 + e i ω L − ω τ , (10) and M ̄ in ω = ( κ 2 π β δ ω + B ̄ in ω 0 + 2 π Y 1 Y 2 C ̄ in † ω + 2 π Z 1 1 + e i ω L − ω τ Z 2 e i 2 ω L − ω τ + e i 3 ω L − ω τ D ̄ in ω ) . (11)

Here, I denotes a 2 × 2 identity matrix and χ[ω; ɛ] is the dimensionless state transfer matrix. Operators with a bar ⋅ ̄ denote the Fourier transformation of the corresponding operators in the frequency domain. The diagonal terms in the gain matrix (9) and dissipative matrix (10) describe decays to the reservoirs, while the off-diagonal terms represent indirect couplings between the two cavities induced by the shared reservoir. Different from Eq. 9, the non-Hermitianity of Eq. 10 shows that the arrangement of the giant cavities can induce a non-reciprocal coupling a ₁ → a ₂ which results from the delayed coupling term F 21 dir [ t ] (4e). This non-reciprocal coupling means that the cavity 2 can affect the excitation of the cavity 1, but not vice-versa. The reasons lie in that: for the cavity 1, the interaction provided by the cavity 2 occurs at a later time, such that the dynamics of the cavity 1 does not include this interaction at the current time. However, for the cavity 2, the interaction provided by the cavity 1 comes from a previous moment, such that the dynamics of the cavity 2 preserves this interaction at the current moment. Or equivalently from a steady-state viewpoint, the time-delay property makes the exchange of photons between two cavities via the shared reservoir unidirectional, i.e., the a ₁ → a ₂ exchange is allowed but the a ₂ → a ₁ exchange is forbidden. It should be noticed that this directional coupling is an inherent delay effect and thus it does not involve interferences between cavities. Therefore, our proposal requires no other non-linear elements in acquiring non-reciprocity, e.g., Faraday rotators [20, 21] and Josephson parametric converters [26, 28]. In addition, the delayed differential Eqs. 3a, 3b indicate a non-Markovian effect, i.e., the dynamics of the system depends on a moment in the past. In the frequency domain, the non-Markovian effect behaves as the dependence of the matrix D _Z on the driving frequency ω _L . These two forms of non-Markovian effect are connected with the spatial non-locality resulting from the multiple-point couplings [48]. Another change induced by the arrangement lies in the last term of the input matrix (11), where exponents describe delayed inputs. Similarly, the input-output relation (6) in the frequency domain reads B ̄ out ω = 1 − χ 11 ω ; ε B ̄ in ω + 2 π β δ ω − 2 π κ C ̄ in † ω χ 11 ω ; ε Y 1 + χ 12 ω ; ε Y 2 − 2 π κ D ̄ in ω χ 11 ω ; ε Z 1 1 + e i ω L − ω τ + χ 12 ω ; ε Z 2 e i 2 ω L − ω τ + e i 3 ω L − ω τ . (12)

We have provided a description of our sensor in the Heisenberg picture. From the above derivation, we can investigate how the unknown parameter affects the output of the detecting field. Different from the existing sensors, the dynamics of our sensor involve non-reciprocity induced by time-delayed terms which would improve the performance of the sensor.

As we have introduced, our sensor employs homodyne detection to extract the perturbation, where the photon current of the output field I t = κ 2 e i φ B out t + H . c . (13) is measured. All the information of ɛ is contained in the real part of e ^iφ B _out [t]. Note that the current is measured in a steady-state of the system such that we can evaluate the response of the system to the perturbation at the zero frequency; i.e., ω = 0. Also, for small ɛ, the expectation value of the output is assumed to be in a linear response to ɛ [14], i.e., 〈 B ̄ out 0 〉 ε = 〈 B ̄ out 0 〉 0 + λ ε , (14) where ⟨⋅⟩ _z denotes taking expectations at ɛ = z. Using this relation, the response coefficient λ reads λ = lim ε → 0 〈 B ̄ out 0 〉 ε − 〈 B ̄ out 0 〉 0 ε = − 2 π β δ 0 d χ 11 0 ; ε d ε | ε = 0 = 2 π i β δ 0 κ χ ̃ V χ ̃ 11 , (15) whose phase φ = − arg  λ determines the angle in Eq. 13.

To estimate the performance of the sensor, we further define a measurement operator m[ω] as the windowed Fourier transformation of current I [t], i.e., m ω = 1 T ∫ − T / 2 T / 2 d t I t e − i ω t , (16) where the segment T should be much greater than 1/κ such that the sensor can reach the steady states during the measurement window. Under this condition, the integral limits can be extended to ±∞. Notably, this definition of m[ω] makes it have a unit of A / H z [54].

The power associated with the signal can be defined as the square of the difference of measurement operator m [0] between the perturbed and unperturbed cases, i.e., S = 〈 m 0 〉 ε − 〈 m 0 〉 0 2 = 2 κ ε 2 T | λ | 2 . (17)

In addition, the total average photon number induced by the classical input can be calculated as n tot = ∑ i = 1 2 〈 a ̄ i † 0 ; ε 〉 0 〈 a ̄ i 0 ; ε 〉 0 = | 2 π δ 0 β | 2 κ χ ̃ † χ ̃ 11 , (18) where the mean-field approximation [55, 56] has been used. With this definitaion, the signal per photon can be expressed as S n tot = 2 ε 2 T | χ ̃ V χ ̃ 11 | 2 χ ̃ † χ ̃ 11 , (19) where we let χ ̃ = χ [ 0 ; 0 ] for brevity.

Similarly, the power of the output noise is defined as the fluctuation of the measurement operator m [0] in the unperturbed case; i.e., N = 〈 m 2 0 〉 0 − 〈 m 0 〉 0 2 = κ 2 T ( 1 + | χ ̃ 11 | 2 − χ ̃ 11 + χ ̃ 11 ∗ + 2 π κ χ ̃ G Y χ ̃ † 11 + 2 π κ 1 + e i ω L τ 2 χ ̃ Z ̃ Z ̃ † χ ̃ † 11 ) = κ 2 T ( 1 + 2 Ξ ⋅ θ Ξ + 4 π κ 1 + cos ω L τ 1 + e i ω L τ | Z 1 χ ̃ 11 + Z 2 χ ̃ 12 e 2 i ω L τ | 2 ) , (20) where Z ̃ = Z 1 Z 2 e 2 i ω L τ T , Ξ = | χ ̃ 11 − 1 | 2 − 1 , and θ[⋅] is the Heaviside step function introduced by the semi-defined positivity of the matrix χ ̃ G Y χ ̃ † . In the derivation, we have assumed that both reservoirs and the waveguide are initially prepared in the vacuum states. Note that the output noise (20) is complex due to the exponent e i ω L τ , which is in contrast to Refs. [14, 51]. However, one can define its real part Re(N) as the measured noise. The constant part is the so-called shot noise [14], which describes the minimum noise of the sensor. The second term denotes the reflective gain resulting from the gain reservoir. When the sensor has a reflective gain, i.e., | χ ̃ 11 − 1 | > 1 , the output noise must be greater than the simple shot noise. Or equivalently speaking, a linear amplification for signal also amplifies the noise. And the third term results from the dissipative noise of the dissipative reservoir.

Combining Eqs. 19, 20, one can obtain the signal-to-noise ratio (SNR) per photon S N n tot = 4 ε 2 κ | χ ̃ V χ ̃ 11 | 2 1 + 2 Ξ ⋅ θ Ξ + 4 π κ 1 + cos ω L τ 1 + e i ω L τ | Z 1 χ ̃ 11 + Z 2 χ ̃ 12 e 2 i ω L τ | 2 χ ̃ † χ ̃ 11 , (21) which is the sensitivity of the sensor. Notably, the state transfer matrix χ ̃ is now independent in the perturbation ɛ, which means that the SNR has a purely parabolic response to the changes of ɛ for a determined χ ̃ .

For comparison, we also consider the sensor made up of two small cavities that couple to the dissipative reservoir in a single-point way. This is a standard model of two-mode quantum sensors [14, 51], which is used as a benchmark. In this case, the second line in interaction Hamiltonian Eq. 2c is rewritten as H I − D S = ∑ i = 1 2 ∫ d k Z i a i † d k + H . c . . (22)

This induces a modification on Eq. 10 D Z S = 1 2 | Z 1 | 2 Z 1 Z 2 ∗ Z 2 Z 1 ∗ | Z 2 | 2 = 1 2 Z 1 Z 2 Z 1 ∗ Z 2 ∗ = 1 2 Z Z † (23) and Eq. 11 M ̄ in S 0 = ( κ 2 π β δ 0 + B ̄ in 0 0 + 2 π Y 1 Y 2 C ̄ in † 0 + 2 π Z 1 Z 2 D ̄ in 0 ) , (24) and the gain matrix G _Y (9) remains the same. Hereafter, we use superscript S to label the corresponding quantities of the sensor composed of small cavities.

An interesting fact is that, the third term in Eq. 20 then reduces to 4 π κ | Z 1 χ ̃ 11 S + Z 2 χ ̃ 12 S | 2 in this case, which is an unavoidable and untunable noise. However, in our proposal, one can adjust ω _L or τ to eliminate the dissipative noise such that the output noise N can remain at a lower level.