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Color-detuning-dynamics-based quantum sensing with dressed states driving

Hao Zhang, Guo-Qing Qin, Xue-Ke Song, and Gui-Lu Long

Open Access Open Access

Abstract

Exploring quantum technology to precisely measure physical quantities is a meaningful task for practical scientific researches. Here, we propose a novel quantum sensing model based on color detuning dynamics with dressed states driving (DSD) in stimulated Raman adiabatic passage. The model is valid for sensing different physical quantities, such as magnetic field, mass, rotation and so on. For different sensors, the used systems can range from macroscopic scale, e.g. optomechanical systems, to microscopic nanoscale, e.g. solid spin systems. The dynamics of color detuning of DSD passage indicates the sensitivity of sensors can be enhanced by tuning system with more adiabatic or accelerated processes in different color detuning regimes. To show application examples, we apply our approach to build optomechanical mass sensor and solid spin magnetometer with practical parameters.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum sensing [1], which uses the quantum property of a quantum system to build the sensor of a physical quantity, is a widely studied branch of quantum information and technology. According to the difference of physical quantities, such as magnetic field, temperature and mass, the required performance candidate has different physical systems known as solid spins [2–11], optomechanical systems [12–23], optical microcavity [24–31] and etc. With the advantages of large range scale and high sensitivity, quantum sensing plays a significant role in practical scientific researches and many interesting schemes have been proposed in recent years. For example, coupled with magnetic field, the nitrogen-vacancy (NV) color centre in diamond has been made for nanoscale magnetometer [4–8] and spins sensor [9,10]. Besides the solid spins, the sensors for detecting forces [32,33], acceleration [34,35] and masses [36,37] are proposed in optomechanical systems.

The perfect quantum operation is the goal pursued by quantum information tasks. For instance, quantum computing needs the fast quantum gate operations with high fidelity. To accurately control a quantum process, the adiabatic passage technique is proposed with good robustness [38–40]. The widely used techniques for two-level and three-level system are called rapid adiabatic passage [38] and stimulated Raman adiabatic passage (STIRAP) [39], respectively. However, the trade off between the robustness and speed limits the potential of adiabatic passage technique used for fast quantum operation. To solve the above problem, the shortcut to adiabaticity is developed [40–46] and applied successfully in experiment in different systems, including Bose-Einstein condensates in optical lattices [47], cold atoms [48], trapped ions [49,50] and NV centre in diamond [51–53]. The direct way, so called transitionless driving [41,42], to realize the shortcut to adiabaticity is to suppress or cancel the undesired transition terms of Hamiltonian in adiabatic frame and this approach is generalized and improved for realizable in situation without constructing the forbidden transition in three-level system by using dressed states driving (DSD) [43,52]. The three-level system with DSD has richer operations of quantum state for various applications, such as quantum computing and sensing.

In this paper, we propose a novel quantum sensing model based on color detuning dynamics with DSD passage. By investigating the detuning dynamic of three-level system governed by DSD, we find that the population of target state has different properties of sensitivity in degenerate or non-degenerate color detuning area. As the evolution speed is tuned, the slope of variation of the population can be larger and smaller for different color detuning areas. According to this property, different kind of quantum sensors can be built by constructing the relationship between detuning and corresponding physical quantities. We show the examples for realizing the improved sensitivity optomechanical mass sensor and solid spin magnetometer based on degenerate and non-degenerate color regime, respectively.

2. Universal model for quantum sensing in three-level system

To realize quantum sensing for unknown physical quantity, labelled with X, the first task is to find a controllable quantum system which can couple with corresponding X well. For a given initial state, the evolution paths of the system are different with different values of X. The key point is one should theoretically give a relationship between observable quantity Y of the system and X, the form can be written as

$$Y=f(X).$$
To detect the magnitude of Y, one can infer the quantity X via the relationship Eq. (1). The value of differential coefficient $\frac {\partial Y}{\partial X}$ is related to the sensitivity of the sensor.

Here we consider a generic three-level (mode) system and the detailed flowchart of our sensing proposal is given in Fig. 1. The first step is to find a three-level (mode) system which can couple with the physical quantity X. Before evolution, the system should be prepared on its initial state $|1\rangle$, e.g. the ground state of the system. This step is so-called initialization. If the operation is the first round, we apply two pulses designed without physical quantity X. After a time $\tau$, the system evolves to the final state and we measure the population of the separated level $|3\rangle$. The pulses used here are calculated for perfect population transfer between states $|1\rangle$ and $|3\rangle$. Because of existing X, the final population of level $|3\rangle$ is changed. If the sensor has low resolution, it should be operated in the second round with calibration step, i.e. the pulses should be calibrated to promise the sensor in higher sensitivity domain. In this case shown in Fig. 1, the sensing process has the second round.

 figure: Fig. 1.

Fig. 1. Flowchart for the quantum sensing in three level system. $X$ is an unknown physical quantity to be measured.

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3. Color detuning dynamics of DSD passage

We consider two different kind of structure of generic three-level systems, i.e. $\Lambda$ and $V$, shown in Fig. 2. With driving pulses $\Omega _{1}$ and $\Omega _{2}$, two structures have the Hamiltonian in the rotating frame given by

$$\begin{aligned} H_{1}(t)= \left[ \begin{array}{ccc} \delta_{1} & \Omega_{1}(t) & 0\\ \Omega_{1}(t) & 0 & \Omega_{2}(t)\\ 0 & \Omega_{2}(t) & \delta_{2}\\ \end{array} \right]. \end{aligned}$$
For multimode bosonic system, the matrix $H_{1}(t)$ satisfies equation $id\vec {v}(t)/dt=H_{1}(t)\vec {v}(t)$, where the vector operator is $\vec {v}(t)=[a_{1}(t),a_{2}(t),a_{3}(t)]^{T}$. The $a_{i}$ $(a^{\dagger }_{i})$ $(i=1,2,3)$ is defined as the annihilation (creation) operators for the corresponding $i$-th mode with the frequency $\omega _{i}$, respectively. In atomic system, the $\omega _{i}$ is energy of $i$-th bare level. $\delta _i$ is laser detuning with corresponding levels (modes). $\Omega _{i}(t)$ is the effective coupling strength between the corresponding levels (modes) shown in Fig. 2. When all the detuning is zero, i.e. $\delta _{i}=0$, the system has the eigenvalues $\lambda =0,\pm \Omega _{0}$ and the corresponding eigenstates (eigenmodes) are dark state (mode) $\psi _{d}=[-\Omega _{2}/\Omega _{0},0,\Omega _{1}/\Omega _{0}]^{T}$ and bright state (mode) $\psi _{\pm }=[\Omega _{1}/\Omega _{0},\pm 1,\Omega _{2}/\Omega _{0}]^{T}/\sqrt {2}$, where $\Omega _{0}=\sqrt {\Omega ^{2}_{1}+\Omega ^{2}_{2}}$.

 figure: Fig. 2.

Fig. 2. Schematic of generic three-level system. (a) $\Lambda$ type level structure with detuning. (b) $V$ type level structure with detuning. $\delta _1$ and $\delta _2$ represent the magnitude of detuning of coupling $\Omega _1$ and $\Omega _2$, respectively.

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To construct an STIRAP, we choose the adiabatic pulses with ‘Vitanov’ envelope given by [54]

$$\begin{aligned} &\Omega_{1}(t)=\Omega_{0}\sin[\theta(t)],\\ &\Omega_{2}(t)=\Omega_{0}\cos[\theta(t)],\\ &\theta(t)=\frac{\pi}{2}\frac{1}{1+e^{{-}t/\tau}}, \end{aligned}$$
where $\tau$ is pulse duration. We assume that the initial state is prepared in level 1 and the target state is level 3. The population can be transferred successfully via STIRAP. To speed up the STIRAP, the DSD method modulates the pulse envelope with [43]
$$\begin{aligned} &\widetilde{\theta}(t)=\theta(t)-\arctan(\dfrac{g_{x}(t)}{\Omega_{0}(t)+g_{z}(t)}),\\ &\widetilde{\Omega}_{0}(t)=\sqrt{(\Omega_{0}(t)+g_{z}(t))^{2}+g_{x}^{2}(t)}, \end{aligned}$$
where $g_{x}(t)$, $g_{z}(t)$ and $\mu$ are chosen with one of many choices as $g_{x}(t)=\dot {\mu }$, $g_{z}(t)=0$, $\mu =-\arctan (\dfrac {\dot {\theta }(t)}{\Omega (t)})$, respectively. Different detuning results in the different population of level 3 with the same initial conditions. As shown in Fig. 3, the population of level 3 is changed with respect to the detuning $\delta _1$ and $\delta _2$. From Fig. 3 (a) to (d), the pulse durations are chosen with $\tau =1.0\tau _m$, $2.0\tau _m$, $5.0\tau _m$ and $10.0\tau _m$, respectively. The $\tau _m$ is the minimum pulse duration time and has the relationship $\tau _m\simeq 1/(2.63\Omega _0)$ [43]. We define two different regimes as degenerate, i.e. two detuning $\delta _1$ and $\delta _2$ have same sign $+/-$, and non-degenerate color detuning, i.e. two detuning $\delta _1$ and $\delta _2$ have different sign $+/-$ areas. Degenerate and non-degenerate color detuning are I, III and II, IV areas in coordinate system shown in Fig. 3 (a), respectively. From Fig. 3 (a) to (d), degenerate color detuning has two different results. On the line satisfying $\delta _1=\delta _2$, the increasing domain of intermediate colors indicates the population of level 3 is less sensitive to the detuning as the evolution time becomes longer. When the duration time is in its limitation, i.e. $\tau =1.0\tau _m$, the variation of population is limited in the smallest length. But the situation out of this line is inverse, the variation of population is localized in a smaller area as the duration time becomes longer. The later result is same with the situation in non-degenerate color detuning regime. To show the results clearly, we plot the population with respect to the detuning and duration time with two special cases in Fig. 3 (e)-(h). The Fig. 3(e) and (f) are the situation of $\delta _1=\delta _2$ in degenerate color detuning. As the duration time increases, the area of intermediate colors increases in Fig. 3(e). The curves are plotted in Fig. 3(f), the curve with shorter duration time is steeper. But for $\tau =1.0\tau _m$, this result is valid in domain $-1/\tau _m\lesssim \delta \lesssim 1/\tau _m$. The situation of $\delta _1=-\delta _2$ is performed in Fig. 3 (g) and (h). The area of intermediate colors is decreases as the $\tau$ increases. This result indicates the population becomes more sensitive to the variation of detuning as the evolution time becomes longer. Some curves are given in Fig. 3 (h). For those two special cases, the results indicate that the accelerated passage enhances the sensitivity of population variation in $\delta _1=\delta _2$, but for $\delta _1=-\delta _2$, the sensitivity is enhanced by more adiabatic passage.

 figure: Fig. 3.

Fig. 3. Dynamics of three-level system with different detuning. (a)-(d) Population of level 3 vs detuning $\delta _1$ and $\delta _2$ with different pulse duration. (a) $\tau =1\tau _m$; (b) $\tau =2\tau _m$; (c) $\tau =5\tau _m$; (d) $\tau =10\tau _m$; (e) Population of level 3 vs same color detuning $\delta =\delta _1=\delta _2$ and pulse duration $\tau$. (f) Curves of some particular duration times for same color detuning $\delta =\delta _1=\delta _2$. (g) Population of level 3 vs different color detuning $\delta =\delta _1=-\delta _2$ and pulse duration $\tau$. (h) Curves of some particular duration times for different color detuning $\delta =\delta _1=-\delta _2$.

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4. Applications for sensing

4.1 Degenerate color optomechanical mass sensor

To build a mass sensor, we consider a multimode optomechanical model composed of two optical cavities and a mechanical membrane. The detailed setup is shown in Fig. 4(a). By pumping two laser fields with frequency $\omega _{d1}$ and $\omega _{d2}$, two optical cavities couple to middle mechanical oscillator simultaneously. The frequency of left, right cavity and mechanical oscillator are $\omega _{1}$, $\omega _{2}$ and $\omega _{m}$, respectively. The black small dot in mechanical membrane is deposited mass with magnitude $\delta m$ to be measured. After the standard linearization procedure, the Hamiltonian of this multimode interactions structure is given by [17–19,21–23]

$$H_{op1}\!=\omega_{m}b^{\dagger}b\!+\!\!\!\sum_{i=1,2} \!\!\left[-\Delta_{i}a^{\dagger}_{i}a_{i}\!+\!G_{i}(a^{\dagger}_{i}\!+\!a_{i})(b\!+\!b^{\dagger})\right]\!\!,\;\;$$
where $a_{i}$ $(a^{\dagger }_{i})$ $(i=1,2)$ and $b$ $(b^{\dagger })$ are the annihilation (creation) operators for the $i$-th cavity mode and the mechanical mode, respectively. Here, $\Delta _{i}=\omega _{di}-\omega _{i}$ and $G_{i}=G_{0i}\sqrt {n_{i}}$ are the pump laser detuning with cavity mode and the effective optomechanical coupling strength, respectively. $G_{0i}$ and $n_{i}$ are the single-photon optomechanical coupling and the intra-cavity photon number produced by the driving field, respectively. When we consider the case that all the cavities are driven with red detuning laser shown in Fig. 4 (b), under the rotating-wave approximation, the Hamiltonian can be rewritten as
$$H_{op2}=\sum_{i=1,2}\delta_{i}a_{i}^{\dagger}a_{i}+G_{i}(a_{i}^{\dagger}b+b^{\dagger}a_{i}).$$
Here, the detuning is $\delta _{i}=-\Delta _{i}-\omega _{m}$. Calculating the Heisenberg equations of above Hamiltonian, the effective coefficient matrix is equivalent to the Eq. (2) by defining the vector operator with $\vec {v}_{op}(t)=[a_{1}(t),b(t),a_{2}(t)]^{T}$. The coupling strength $G_i$ is chosen with the form in Eq. (4) to realize the population transfer from cavity 1 to 2. To realize the mass sensor, we first assume that the mechanical membrane is deposited a mass represented with a black dot shown in Fig. 4(a). Therefore, the frequency of mechanical mode has a little drift and the resonance coupling is broken by inducing the two small detuning with degenerate color. This additional mass impacts the result of the population in cavity 2. The relationship between frequency shift $\delta \omega _m$ and the deposited mass $\delta m$ is given by [37]
$$\delta\omega_m=R\delta m.$$
The parameter $R=\omega _m/2m$ is mass responsivity, where $m$ is the mass of the mechanical resonator for supporting the deposition. By choosing the detuning with $\delta _1=\delta _2=0$ without deposited mass, when we assume that the mechanical resonator is added with a deposited mass $\delta m$, the system evolves under the degenerate color detuning $\delta _1=\delta _2=-R\delta m$. Therefore, the variation of population of cavity 2 infers the magnitude of $\delta m$. According to the Fig. 3 (e) and (f), one can choose the fastest evolution path, i.e. duration $\tau =1.0\tau _m$, to design the mass sensor with high sensitivity and consider the domain in the rectangle in Fig. 4(c). To show the practical example, the curve of sensing deposited mass of mechanical oscillator is performed in Fig. 4(d) with given parameters as $\omega _m=2\pi \times 6$ GHz and $m=10^{-15}$ g. The $G_0$ which is equal to the $\Omega _0$ of Eq. (3) is assumed with $G_0\sim 1.0$ MHz. The results show if the resolution of population is about $0.1$, the optomechanical mass sensor has the resolution with scale $10^{-20}$ g for a mechanical oscillator with mass $m=10^{-15}$ g.

 figure: Fig. 4.

Fig. 4. Schematic of degenerate color optomechanical mass sensor. (a) Generic setup for optomechanical system. Two optical cavities couple to each other by driving a mechanical oscillator. The small dot represents a deposited mass on the mechanical oscillator. (b) Red detuning driving for two optomechanical interactions. (c) The curve for population vs detuning with duration $\tau =1.0\tau _{m}$. (d) The mass sensor in the domain labelled with dashed rectangle in (c). (e) The population transfer of modes in optomechanical system with thermal noises $n_{th}=200$ in mechanical mode. (f) The curves for population of optical mode 2 vs detuning with different mechanical thermal noises. The parameters are $\tau =1.0\tau _{m}$ and $\gamma _{m}=0.5$kHz in (e) and (f).

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The optomechanical system is sensitive to the thermal noises in mechanical mode. We investigate the system with adding the noise term $\gamma _{m}D[b]\rho /2$, where the term is $D[b]\rho =(n_{th}+1)(2b\rho b^{\dagger }-b^{\dagger }b\rho -\rho b^{\dagger }b)+n_{th}(2b^{\dagger }\rho b-bb^{\dagger }\rho -\rho bb^{\dagger })$, in master equation. The population transfer between modes is given in Fig. 4 (e) with thermal occupation $n_{th}=200$. The results show that the thermal noise has significant influence on the final population of optical mode 1, but little influence on optical mode 2. So the target population chosen with optical mode 2 is stable. In Fig. 4 (f), by changing the thermal noise from 0 to 300, we show the larger thermal noises will increase the populations of target mode. In low thermal noise, the slope of curves is changed slightly and the mass sensor keeps good robustness.

4.2 Non-degenerate color solid spin magnetometer

As a solid spin system, NV centre in diamond is a promising platform for designing magnetometer due to its good coupling between spin and external magnetic field. As shown in Fig. 5, the NV centre has a rich level structure. In the environment with a weak static magnetic field along the NV principle axis, the Hamiltonian of electron-spin ground triplet is given by [2]

$$H_{NV}=DS_{z}^{2}+\gamma_{e}B_{z}S_{z},$$
where $D=2.87\ \textrm {GHz}$ is zero-field splitting between spin triplet $m_{s}=0$ and $m_{s}=\pm 1$. $S_{z}$ is three dimension electron-spin operator. $\gamma _{e}=2.8025\ \textrm {MHz}\cdot \textrm {Gauss}^{-1}$ is the electronic gyromagnetic ratio. The first term describes the zero-field splitting and the second one is electronic-spin Zeeman splitting term. The NV centre in diamond with the Hamiltonian as Eq. (8) has the level structure shown in the right rectangle of Fig. 5 (b). The splitting of levels in $m_{s}=\pm 1$ induced by magnetic field produces non-degenerate color detuning, as the detuning in Fig. 5 (b) has relationship with $\delta =\delta _{1}=-\delta _{2}=\gamma _{e}B_{z}$. Therefore, by applying two microwave pulses in the non-degenerate color regime, the population of level 3 of NV centre infers the magnitude of the magnetic field via the electronic gyromagnetic ratio.

 figure: Fig. 5.

Fig. 5. Schematic of solid spin for detecting magnetic field. (a) The structure of an NV color centre in diamond. (b) Level structure of NV color centre in diamond. The levels in the right rectangle are triplet structure in ground state. (c) The curve for population vs detuning with duration $\tau =10\tau _{m}$. (d) The magnetometer in the domain labelled with dashed rectangle in (c).

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To build the magnetometer, we design the driving pulses with the parameters $\tau =10\tau _m$. The interaction domain of detuning is chosen in the rectangle shown in Fig. 5 (c). Therefore, if the magnetic field is so weak, the frequency of pulse should be calibrated in effective detuning domain, i.e. in the dashed rectangle in Fig. 5 (c), before formal measurement, as shown in the first round of flowchart Fig. 1. When the parameter is given with $\Omega _0\sim 1.0$ MHz, the sensing curve of magnetic field is shown in Fig. 5 (d). The population of each level can be performed via optically detected magnetic resonance technique [55]. When the resolution of population of level 3 is $1\%$, the scale of the magnetometer can be about $10^{-3}\ \textrm {Gauss}$.

5. Conclusion

To conclude, we use the three-level (mode) system to design a novel quantum sensing model by using DSD. Driving with different pulses, three-level system has two different interaction regimes, i.e. degenerate and non-degenerate color detuning. In degenerate color detuning, the variation of population in target state is more sensitive to the detuning value as the process is accelerated on the line with $\delta _1=\delta _2$. On the contrary in degenerate with $\delta _1\neq \delta _2$ and non-degenerate color detuning, the process is more adiabatic, the sensitivity of population of target state is higher. Those properties indicate the system can be used to design different kinds of quantum sensors with higher sensitivity by chosen with different interaction regime. To show the examples of practical sensor, we perform how to apply our model for building degenerate color mass sensor and non-degenerate color magnetometer.

Funding

China Postdoctoral Science Foundation(2019M650620); National Natural Science Foundation of China (61727801); National Key Research and Development Program of China (2017YFA0303700); The Key Research and Development Program of Guangdong province (2018B030325002).

Disclosures

The authors declare no conflicts of interest.

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Figures (5)
Fig. 1.
Fig. 1. Flowchart for the quantum sensing in three level system. $X$ is an unknown physical quantity to be measured.

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Fig. 2.
Fig. 2. Schematic of generic three-level system. (a) $\Lambda$ type level structure with detuning. (b) $V$ type level structure with detuning. $\delta _1$ and $\delta _2$ represent the magnitude of detuning of coupling $\Omega _1$ and $\Omega _2$, respectively.

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Fig. 3.
Fig. 3. Dynamics of three-level system with different detuning. (a)-(d) Population of level 3 vs detuning $\delta _1$ and $\delta _2$ with different pulse duration. (a) $\tau =1\tau _m$; (b) $\tau =2\tau _m$; (c) $\tau =5\tau _m$; (d) $\tau =10\tau _m$; (e) Population of level 3 vs same color detuning $\delta =\delta _1=\delta _2$ and pulse duration $\tau$. (f) Curves of some particular duration times for same color detuning $\delta =\delta _1=\delta _2$. (g) Population of level 3 vs different color detuning $\delta =\delta _1=-\delta _2$ and pulse duration $\tau$. (h) Curves of some particular duration times for different color detuning $\delta =\delta _1=-\delta _2$.

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Fig. 4.
Fig. 4. Schematic of degenerate color optomechanical mass sensor. (a) Generic setup for optomechanical system. Two optical cavities couple to each other by driving a mechanical oscillator. The small dot represents a deposited mass on the mechanical oscillator. (b) Red detuning driving for two optomechanical interactions. (c) The curve for population vs detuning with duration $\tau =1.0\tau _{m}$. (d) The mass sensor in the domain labelled with dashed rectangle in (c). (e) The population transfer of modes in optomechanical system with thermal noises $n_{th}=200$ in mechanical mode. (f) The curves for population of optical mode 2 vs detuning with different mechanical thermal noises. The parameters are $\tau =1.0\tau _{m}$ and $\gamma _{m}=0.5$kHz in (e) and (f).

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Fig. 5.
Fig. 5. Schematic of solid spin for detecting magnetic field. (a) The structure of an NV color centre in diamond. (b) Level structure of NV color centre in diamond. The levels in the right rectangle are triplet structure in ground state. (c) The curve for population vs detuning with duration $\tau =10\tau _{m}$. (d) The magnetometer in the domain labelled with dashed rectangle in (c).

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Equations (8)

Equations on this page are rendered with MathJax. Learn more.

Y = f ( X ) .
H 1 ( t ) = [ δ 1 Ω 1 ( t ) 0 Ω 1 ( t ) 0 Ω 2 ( t ) 0 Ω 2 ( t ) δ 2 ] .
Ω 1 ( t ) = Ω 0 sin ⁡ [ θ ( t ) ] , Ω 2 ( t ) = Ω 0 cos ⁡ [ θ ( t ) ] , θ ( t ) = π 2 1 1 + e − t / τ ,
θ ~ ( t ) = θ ( t ) − arctan ⁡ ( g x ( t ) Ω 0 ( t ) + g z ( t ) ) , Ω ~ 0 ( t ) = ( Ω 0 ( t ) + g z ( t ) ) 2 + g x 2 ( t ) ,
H o p 1 = ω m b † b + ∑ i = 1 , 2 [ − Δ i a i † a i + G i ( a i † + a i ) ( b + b † ) ] ,
H o p 2 = ∑ i = 1 , 2 δ i a i † a i + G i ( a i † b + b † a i ) .
δ ω m = R δ m .
H N V = D S z 2 + γ e B z S z ,
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