1. Introduction
Quantum sensing has emerged as a distinct and rapidly growing branch of research, which employs quantum mechanical systems as sensors to detect various physical quantities ranging from magnetic and electric fields, to time, frequency and rotations. Aiming at ultra-high sensitivity and precision, researchers have devised various of quantum sensors including atomic vapors sensors,[1–4] trapped ions sensors,[5–7] solid-state spins sensors,[8–11] and so on. By using quantum entanglement[12–14] and squeezing,[15–17] the sensitivity or precision of the quantum sensors have reached or even gone beyond the standard quantum limit, which is unreachable by classical measurement method. Undoubtedly, quantum sensing has opened a new door to the measurement areas.
The sensitivity of a quantum sensor scales as[18]
$$ \begin{eqnarray}{\rm{sensitivity}}\,\propto \displaystyle \frac{1}{\kappa \sqrt{{T}_{\chi }}},\end{eqnarray}$$ (1)
where κ is a transduction parameter related to the response of the quantum sensor to the signal to be measured, and Tχ is the coherence time of the quantum sensor. Better sensitivity requires longer coherence time besides a large transduction parameter. However, longer coherence time, which reflects greater immunity of the quantum sensor to its environment, indicates a narrower bandwidth of the sensor. This may be a intractable issue when applying the quantum sensors to practical applications.A well-known method in classical control theory to increase the bandwidth of a dynamical system is to the close-loop feedback control. In this paper, a simple model based on the analysis of feedback control of a quantum sensor is presented to show the relationship between bandwidth and sensitivity. The results indicate that bandwidth and sensitivity are two competing factors of the quantum sensor, when intrinsic noise of the sensor system due to its quantum characteristics is taken into consideration. This means that there must be a trade-off between bandwidth and sensitivity of the quantum sensor system.
2. Model of the quantum sensor
In this section, we give a simple model revealing the basic physics underlying the operational principles of the quantum sensor. At the same time, this model shows an unconquerable sensitivity limit imposed by the quantum feature of the sensor. The Hamiltonians of the sensor and the environment read
$$ \begin{eqnarray}\begin{array}{lll} & & {H}_{0}=\hslash {\omega }_{0}{a}^{\dagger }a+\hslash g\,\cos (\omega t)(a+{a}^{\dagger }),\end{array}\end{eqnarray}$$ (2)
$$ \begin{eqnarray}\begin{array}{lll} & & {H}_{{\rm{bath}}}=\hslash \displaystyle \sum _{j}{\omega }_{j}{b}_{j}^{\dagger }{b}_{j},\end{array}\end{eqnarray}$$ (3)
$$ \begin{eqnarray}\begin{array}{lll} & & {H}_{{\rm{I}}}=\hslash \displaystyle \sum _{j}{g}_{j}({a}^{\dagger }{b}_{j}+{\rm{h}}.{\rm{c}}.),\end{array}\end{eqnarray}$$ (4)
where H0 describes a harmonic oscillator driven by a classical driving field with frequency ω and coupling strength g. The environment is modeled by an ensemble of harmonic oscillators with j specifying different oscillator mode. The interaction of the oscillator with its environment, HI, will exert a fluctuating force on the oscillator, resulting in the dissipation of the harmonic oscillator.The quantum Langevin equation for the harmonic oscillator is
$$ \begin{eqnarray}\dot{a}(t)=-{\rm{i}}{\omega }_{0}a(t)-{\rm{i}}g\,\cos (\omega t)-\displaystyle \frac{\gamma }{2}a(t)+{f}_{{\rm{a}}}(t),\end{eqnarray}$$ (5)
where
$$ \begin{eqnarray}{f}_{{\rm{a}}}(t)=-{\rm{i}}\displaystyle \sum _{j}{g}_{j}{b}_{j}(0){{\rm{e}}}^{-{\rm{i}}{\omega }_{j}t}\end{eqnarray}$$ (6)
is the noise operator and γ = 2π g2(ω0)D(ω0) is the decay rate of the oscillator induced by the environment.[19] Here is the density of the bath states at ω0, V is the quantization volume and c is the velocity of light. Then the average values of the two quadrature amplitudes of the oscillator X1 = (a + a†)/2 and X2 = (a – a†)/2 i follow the equations:
$$ \begin{eqnarray}\begin{array}{lll}\langle {\dot{X}}_{1}\rangle & = & {\omega }_{0}\langle {X}_{2}\rangle -\displaystyle \frac{\gamma }{2}\langle {X}_{1}\rangle,\\ \langle {\dot{X}}_{2}\rangle & = & -{\omega }_{0}\langle {X}_{1}\rangle -\displaystyle \frac{\gamma }{2}\langle {X}_{2}\rangle -g\cos (\omega t).\end{array}\end{eqnarray}$$ (7)
In deriving Eq. (7), the density operator of the composite system is assumed to be ρc = ρS ⊗ ρB at t = 0, where the harmonic oscillator and the bath are both in the thermal states
$$ \begin{eqnarray}\begin{array}{lll}{\rho }_{{\rm{S}}} & = & \displaystyle \frac{1}{{Z}_{{\rm{S}}}}{{\rm{e}}}^{-\displaystyle \frac{\hslash {\omega }_{0}{a}^{\dagger }a}{{k}_{{\rm{B}}}T}},\end{array}\end{eqnarray}$$ (8)
$$ \begin{eqnarray}\begin{array}{lll}{\rho }_{{\rm{B}}} & = & \displaystyle \frac{1}{{Z}_{{\rm{B}}}}{{\rm{e}}}^{-\displaystyle \frac{{H}_{{\rm{bath}}}}{{k}_{{\rm{B}}}T}}.\end{array}\end{eqnarray}$$ (9)
The normalization factors ZS and ZB are the partition functions of the quantum harmonic oscillator and the bath, respectively. T refers to the temperature and kB is the Boltzmann constant.We now turn to the rotating frame and define two dimensionless amplitudes: , and . The equations of motion for the average values of and can be derived from Eqs. (7):
$$ \begin{eqnarray}\begin{array}{lll} & & \langle {\dot{\bar{X}}}_{1}\rangle ={\rm{\Delta }}\omega \langle {\bar{X}}_{2}\rangle -\displaystyle \frac{\gamma }{2}\langle {\bar{X}}_{1}\rangle,\end{array}\end{eqnarray}$$ (10)
$$ \begin{eqnarray}\begin{array}{lll} & & \langle {\dot{\bar{X}}}_{2}\rangle =-{\rm{\Delta }}\omega \langle {\bar{X}}_{1}\rangle -\displaystyle \frac{\gamma }{2}\langle {\bar{X}}_{2}\rangle -\displaystyle \frac{g}{2},\end{array}\end{eqnarray}$$ (11)
where the terms that are oscillating at 2ω have been dropped, and Δω = ω0 – ω denotes the frequency detuning. In the regime of near resonance Δω ≪ γ, the steady average of is much smaller that of . This implies the average phase of the oscillator
$$ \begin{eqnarray}\langle \phi \rangle =\arctan \left(\displaystyle \frac{\langle {\bar{X}}_{1}\rangle }{\langle {\bar{X}}_{2}\rangle }\right)\end{eqnarray}$$ (12)
has a small steady-state phase shift from the resonant point 〈ϕ〉r = 0. Furthermore, the frequency shift is nearly linear in the frequency detuning
$$ \begin{eqnarray}{\langle \phi \rangle }_{{\rm{s}}}\approx \displaystyle \frac{{\rm{\Delta }}\omega }{\gamma /2}\equiv {T}_{2}{\rm{\Delta }}\omega,\end{eqnarray}$$ (13)
where T2 = 2/γ is the coherence time of the harmonic oscillator. Therefore, the oscillator can be used as a sensor to measure the input frequency ω by monitoring the phase shift 〈 ϕ〉s.The time evolution of the average phase 〈ϕ〉, when the oscillator is operating near the resonant point, satisfies the equation
$$ \begin{eqnarray}\langle \dot{\phi }\rangle \approx {\rm{\Delta }}\omega -\displaystyle \frac{1}{{T}_{2}}\langle \phi \rangle .\end{eqnarray}$$ (14)
Equations (13) and (14) show that longer coherence time T2 results in higher sensitivity at the expense of slower response (narrower bandwidth).Apart from the technique noise in the detection system, the phase measurement precision is ultimately limited by the fundamental uncertainty principle of quantum mechanics. In the oscillator model considered above, the phase shift 〈ϕ〉s cannot be measured with arbitrary precision. A key quantity characterizing how well we can determine the phase shift is the variance of the phase
$$ \begin{eqnarray}{\langle {\rm{\Delta }}\phi \rangle }_{{\rm{s}}}\approx \left|\displaystyle \frac{{({\rm{\Delta }}{\bar{X}}_{1})}_{{\rm{s}}}}{{\langle {\bar{X}}_{2}\rangle }_{{\rm{s}}}}\right|=\displaystyle \frac{\sqrt{2\bar{n}+1}}{{T}_{2}g},\end{eqnarray}$$ (15)
where is the average quanta number in the frequency ω0 and is the steady-state variance of . At absolute zero temperature,
$$ \begin{eqnarray}{\langle {\rm{\Delta }}\phi \rangle }_{{\rm{s}}}=\displaystyle \frac{1}{{T}_{2}g}\end{eqnarray}$$ (16)
gives the minimum phase uncertainty imposed by quantum fluctuations.3. Feedback control and bandwidth improvement
The bandwidth of a quantum sensor can be analyzed with the method of transfer functions.[20] The model is presented in Fig. 1 with a proportional-integral controller (PI-controller), which is widely used in classic control field. The quantum sensor G, as discussed in the above section, representing an oscillator operating near its resonance point, is described by the transfer function according to Eq. (14):
$$ \begin{eqnarray}G(s)=\displaystyle \frac{{T}_{2}}{1+{T}_{2}s},\end{eqnarray}$$ (17)
where s is the complex variable. The transfer function G(s) relates the frequency detuning Δω(s) to the phase shift signal ϕ(s), where Δω(s) and ϕ(s) are the Laplace transformation of the corresponding time-domain signals.
Figure 1.The block diagram of the close-loop control of a quantum sensor system. The open-loop transfer function of the oscillator is denoted by G. A second order filter F with filter time constant τ is applied to suppress high frequency noise. A PI-controller C is used to adjust the output frequency ωo so as to trace the variation of the input frequency ωi. The control parameter Kp represents the proportional term and Ki stands for the integral term. The quantum phase white noise Δϕ will be a disturbance to the output of the PI-controller. Finally, the phase shift ϕ is the output of the sensor, followed by the high-frequency-noise-filtered phase shift .
In the open-loop operation condition, the detection of the signal Δω can be described as follows. When the input frequency ωi is off-resonant from the resonant frequency of the oscillator G, the phase of the oscillator increases from zero value at the resonant point to a nonzero value ϕ. By monitoring the phase shift ϕ, the frequency detuning is deduced according to Eq. (13). Finally, equation (17) shows that the open-loop bandwidth is limited by the coherence time T2.
Although long coherence time brings about high sensitivity [see Eq. (1)], it significantly limits the measurement bandwidth. Small bandwidth will limit practical applications of the quantum sensors. Fortunately, the problem of small open-loop bandwidth can be resolved by feedback control method.
In the close-loop system depicted in Fig. 1, a PI-controller is utilized for frequency ωo feedback, whose performance is characterized by two parameters: the proportional term Kp and the integral term Ki. The central idea of the feedback is to keep the system in resonance by feedback controller. When the input frequency ωi deviates from the resonant point of the oscillator system (Δω ≠ 0), the quantum sensor will generate nonzero phase output ϕ. According to the value of the nonzero phase, the PI-controller takes action and updates the frequency ωo, which determines the new input detuning of G together with the input frequency ωi. In this way, the feedback control system continuously monitors the phase of the oscillator, and maintains its value as close as possible to zero by adjusting the frequency detuning. As Δω is close to zero with a feedback loop, ωi can be detected by the feedback frequency ωo.
The principle of bandwidth improvement by feedback method is demonstrated in the following. In the absence of noise, the close-loop transfer function of the system is given by
$$ \begin{eqnarray}{\varPhi }_{{\rm{c}}}(s)=\displaystyle \frac{G(s)F(s)C(s)}{1+G(s)F(s)C(s)},\end{eqnarray}$$ (18)
related to input ωi(s) and output ωo(s). We first focus on the limit where the filter time constant τ → 0. In this limit, the transfer function F(s) ≈ 1, and the close-loop transfer function
$$ \begin{eqnarray}{\varPhi }_{{\rm{c}}}(s)\approx \displaystyle \frac{C(s)}{1+G(s)C(s)}G(s)=A(s)G(s),\end{eqnarray}$$ (19)
where
$$ \begin{eqnarray}\begin{array}{ll}A(s) =C(s)/[1+C(s)G(s)].\end{array}\end{eqnarray}$$ (20)
The function A(s) is a key quantity that is responsible for increasing the bandwidth ωB from the open-loop one, 1/T2, to a much larger one.The close-loop frequency response can be separated into two parts:
$$ \begin{eqnarray}|{\varPhi }_{{\rm{c}}}(j\omega ){|}^{2}=\mathop{A}\limits^{\sim }(\bar{\omega }){A}_{\bar{G}}(\bar{\omega }),\end{eqnarray}$$ (21)
where
$$ \begin{eqnarray}\begin{array}{lll} & & \mathop{A}\limits^{\sim }(\bar{\omega })=\displaystyle \frac{{k}_{{\rm{p}}}^{2}{\bar{\omega }}^{4}+({k}_{{\rm{p}}}^{2}+{k}_{{\rm{i}}}^{2}){\bar{\omega }}^{2}+{k}_{{\rm{i}}}^{2}}{{\bar{\omega }}^{4}+\left[-2{k}_{{\rm{i}}}+{(1+{k}_{{\rm{p}}})}^{2}\right]{\bar{\omega }}^{2}+{k}_{{\rm{i}}}^{2}},\end{array}\end{eqnarray}$$ (22)
$$ \begin{eqnarray}\begin{array}{lll} & & {A}_{\bar{G}}(\bar{\omega })=\displaystyle \frac{1}{1+{\bar{\omega }}^{2}}.\end{array}\end{eqnarray}$$ (23)
Here, we have introduced three dimensionless parameters: , kp = KpT2 and .In the region ki < (1 + kp)2/2, the function can be analyzed in the log-log coordinate, and the numerator and denominator of can be well approximated by several broken lines representing terms proportional to , and the constant terms as shown in Fig. 2. There are four boundary points characterizing the dominant contributions of constant, , terms in different regions:
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{2}^{({\rm{n}})}=\displaystyle \frac{{k}_{{\rm{i}}}}{\sqrt{{k}_{{\rm{p}}}^{2}+{k}_{{\rm{i}}}^{2}}},\end{array}\end{eqnarray}$$ (24)
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{24}^{({\rm{n}})}=\displaystyle \frac{\sqrt{{k}_{{\rm{p}}}^{2}+{k}_{{\rm{i}}}^{2}}}{{k}_{{\rm{p}}}},\end{array}\end{eqnarray}$$ (25)
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{2}^{({\rm{d}})}=\displaystyle \frac{{k}_{{\rm{i}}}}{\sqrt{-2{k}_{{\rm{i}}}+{(1+{k}_{{\rm{p}}})}^{2}}},\end{array}\end{eqnarray}$$ (26)
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{24}^{({\rm{d}})}=\sqrt{-2{k}_{{\rm{i}}}+{(1+{k}_{{\rm{p}}})}^{2}},\end{array}\end{eqnarray}$$ (27)
where is the cross point between and the terms in the numerator, and is the cross point between the terms and the terms in the numerator. The and are corresponding cross points in the denominator of . In the regimes kp ≫ ki and kp ≫ 1,
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{2}^{({\rm{n}})}\approx {\bar{\omega }}_{2}^{({\rm{d}})}\approx \displaystyle \frac{{k}_{{\rm{i}}}}{{k}_{{\rm{p}}}},\end{array}\end{eqnarray}$$ (28)
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{24}^{({\rm{n}})}\approx 1,\end{array}\end{eqnarray}$$ (29)
$$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\omega }}_{24}^{({\rm{d}})}\approx {k}_{{\rm{p}}}.\end{array}\end{eqnarray}$$ (30)
Under this condition, the numerator terms , the denominator term and are well approximated by Fig. 2. The close-loop frequency response is the product of and . As demonstrated by Fig. 2, the bandwidth has been increased from the open-loop value to the close-loop value .
Figure 2.Schematic diagram of bandwidth improvement by PID feedback. These curves are depicted in the log–log coordinates, where and represent the numerator and denominator of , respectively. The numbers +2 and +4 indicate that the terms proportional to and are dominant in those frequency intervals. The number –2 indicates that the functions decay as in those frequency intervals.
Figure 2 shows that the close-loop frequency response is maximum in the zero frequency and decreases with when ki ≪ kp. As ki grows and exceeds a critical point
$$ \begin{eqnarray}\begin{array}{ll}{k}_{{\rm{i}}}^{{\rm{c}}}= {k}_{{\rm{p}}}+1/2,\end{array}\end{eqnarray}$$ (31)
the close-loop frequency response will exhibit resonant behavior. In this case it acquires a maximum value at some positive frequency as demonstrated in Fig. 3. This resonant behavior corresponds to the condition that
$$ \begin{eqnarray}\displaystyle \frac{\partial }{\partial {k}_{{\rm{i}}}}|{\varPhi }_{{\rm{c}}}(j\omega ){|}^{2}=0\end{eqnarray}$$ (32)
has a positive frequency solution.
Figure 3.Demonstrations of the resonant and non-resonant behaviors of the close-loop frequency responses Aclose(ω). The black curve represent the regime ki ≪ kp, while the red curve present resonant behavior with . In this graph, kp = 100 and ki = (1 + kp)2.
The above discussion shows that the bandwidth of the close-loop system can be improved without a limit by increasing kp, keeping ki ≪ kp at the same time. When finite filter time constant τ is taken into consideration, the close-loop bandwidth will be ultimately limited by τ. However, in practical applications the filter time constant τ can be adjusted to a value such that 1/τ ≫ ωB ≈ kp/T2. In this case, the bandwidth can still be improved by the method we discussed above. Figure 4 shows that when the bandwidth ωB is much smaller than 1/τ, ωB is still determined by kp. The filter F begins to take effect at frequency 1/τ and suppresses high-frequency noise.
Figure 4.Comparisons of frequency responses with filter F present or not. The black curve represents the frequency response of the open-loop system G. The pink and green curves describe the close-loop frequency responses of Φ1(s) = CGF/(1 + CGF) and Φ2(s) = CG/(1 + CG) when the filter F is present or not, respectively. The blue and red curves correspond to the frequency responses of A1(s) = CF/(1 + CGF) and A2(s) = A(s), respectively. The parameters used in this graph are T2 = 10 s, τ = 10−4 s, kp = 1000 and ki = 100.
4. Competition between bandwidth and sensitivity
As discussed above, the feedback method indeed improves the bandwidth of the sensor, when PI parameters are properly chosen. In this section, we consider the effect of the noise on the performance of the quantum sensor. The noise under consideration is assumed to be due to purely quantum mechanical origin, which roots in the quantum characteristic of the sensor, as given by Eq. (16).
The noise transfer in the closed-loop sensor system is shown in Fig. 5(a). The noise will cause an output frequency fluctuation ωn(s) in the PI controller, thus limiting the sensitivity of the sensor. The transfer function relating ωn(s) to Δϕ(s) is
$$ \begin{eqnarray}{\varPhi }_{{\rm{n}}}(s)=\displaystyle \frac{{\omega }_{{\rm{n}}}(s)}{{\rm{\Delta }}\phi (s)}=\displaystyle \frac{CF}{1+CGF}.\end{eqnarray}$$ (33)
The output frequency fluctuation ωn(t) thus has a power spectral density (PSD)[21]$$ \begin{eqnarray}\begin{array}{ll} & {S}_{{\rm{n}}}(\omega )=|{\varPhi }_{{\rm{n}}}(j\omega ){|}^{2}{S}_{0},\end{array}\end{eqnarray}$$ (34)
where S0 is the power spectral density of the input phase noise. The magnitude of S0 can be estimated as follows. The Wiener–Khinchin theorem shows that the variance of the phase white noise equals the integral of the corresponding PSD over the frequency domain. Thus we can estimate S0 by
$$ \begin{eqnarray}{S}_{0}=\displaystyle \frac{1}{{({T}_{2}g)}^{2}}\displaystyle \frac{1}{B},\end{eqnarray}$$ (35)
where B refers to the largest bandwidth in the measurement system, and can be taken as 1/τ, since τ is the minimum time constant under consideration. Then Sn(ω) has the form
$$ \begin{eqnarray}{S}_{{\rm{n}}}(\omega )=|{\varPhi }_{{\rm{n}}}(j\omega ){|}^{2}\displaystyle \frac{\tau }{{({T}_{2}g)}^{2}}.\end{eqnarray}$$ (36)
Figure 5.(a) The block diagram of the noise transfer. The noise Δϕ(s), which is here the Laplace transformation of Δϕ(t), causes a output frequency fluctuation in the PI controller, as represented by ωn(s) in the above diagram. (b) The power spectral density of the noise output ωn(t). The red curve is depicted with kp = 2000, which is twice over that of the black one. Other parameters used are ki = 100, T2 = 10 s, τ/T2 = 10−5, and T2g = 103.
The plot of versus the dimensionless angular frequency is depicted in Fig. 5(b) with different PI parameters. As the proportional constant Kp increases, the noise is further amplified by the feedback loop. The increased noise power will degrade the performance of the sensor, because the signal to be detected is determined by the frequency increment Δω and ωn will contaminates the output of ωo. This reveals the competitive relationship between bandwidth and sensitivity, since increasing Kp will improve the bandwidth and degrade the sensitivity inevitably at the same time.
The competitive relationship between bandwidth and sensitivity can also be perceived qualitatively and visually. Equation (33) shows that the transfer function Φn(s) is identical to Φc(s)/G(s), and Fig. 4 indicates that the function of Φn(s) is to increase the bandwidth of the sensor from the open-loop value 1/T2 to the closed-loop value kp/T2. For the sake of achieving large bandwidth, the frequency response of Φn(s) must compensate for the drop of G(s) when ω > 1/T2, which indicates the amplification of the noise with frequency in this frequency range. Therefore, the sensitivity of the sensor is further reduced when larger kp is chosen.
Another point to be emphasized is that the original quantum phase noise Δϕ can influence the sensor’s sensitivity in a non-white form by means of frequency output fluctuation ωn(t), since the sensor’s sensitivity is determined by . As shown in Fig. 5, has a positive slope 1, which indicates a f2 noise contribution in the noise spectrum Sn(ω). In higher frequency intervals, the noise spectrum becomes more complex, and is not white noise as well.
5. Conclusions
In summary, we have investigated the relationship between bandwidth and sensitivity of a quantum sensor based on the transfer function method. The results indicate that the bandwidth of sensor are greatly increased by feedback when choose proper PI parameters. The proportional term Kp is the key parameter in improving bandwidth. Unfortunately, the quantum noise inherent to the quantum sensor system will be further amplified when larger bandwidth is achieved. The bandwidth and sensitivity are two competing performance parameters. Therefore, there must be a trade-off between bandwidth and sensitivity in applying the quantum sensor into practical application. In addition, an original quantum white noise may present itself in form of f2-noise due to the feedback control.
