1. Introduction
Coherent manipulation of the quantum state is a crucial process for quantum information processing and quantum computing. How to implement high fidelity quantum state transfer is still an open question . The technique of the stimulated Raman adiabatic passage (STIRAP), which is robust against fluctuations of experimental parameters, has been widely applied in different physical systems [1-4]. This technique uses partially overlapping pulses to perfectly implement the population transfer of quantum states. However, this approach requires slow evolution of the quantum system. The decoherence may happen before the quantum information processing has been achieved. Therefore, the fast and high-fidelity quantum state transfer is an interesting topic.
On the other hand, several approaches [5] have been proposed to speed up adiabatic quantum state transfer, such as, transitionless quantum driving [6, 7], Lewis-Riesenfeld invariant-based inverse engineering [8-11], and dressed states [12, 13]. The transitionless quantum driving shows that by adding a supplementary Hamiltonian to the original Hamiltonian, so that the system evolves exactly following the adiabatic reference and diabatic transition can be suppressed. This shortcut approach has been demonstrated in recent experiments with optical lattice [14], nitrogen-vacancy (NV) center in diamonds [15, 16], and cold atoms [17]. Also, many theoretical schemes have been proposed to realize the transitionless process [10, 18-23]. For example, the scheme of a stimulated Raman shortcut-to-adiabatic passage (STIRSAP) in a quantum three-level system by the transitionless quantum driving has been proposed [18]. One considers a three-level system, which is described by a time-dependent Hamiltonian H0(t). The technique of transitionless quantum driving needs to construct an additional counterdiabatic field Hcd (t). So, the evolution of the system is guided by a joint Hamiltonian H0(t) + Hcd (t). This provides a fast and robust approach to quantum state transfer for a three-level system due to the additional counterdiabatic field Hcd (t). However, ‘there is no such thing as a free lunch.’ Hence, it is very interesting to study the energetic cost of a counterdiabatic driving field. Recently, a family of energetic cost functionals was introduced in [24]. The cost of the implementation of the fast rotation gates was analyzed [25]. The relationship between quantum speed limit and energetic cost in shortcuts to adiabaticity was studied [26]. It is necessary to investigate the energetic cost of implementing STIRSAP in a quantum three-level system by the transitionless quantum driving.
In this paper, we discuss how to inhibit the energetic cost of a STIRSAP in a quantum three-level system by the transitionless quantum driving. The expression of energetic cost for the three-level system is obtained. We find that the energetic cost can be controlled by adjusting detuning of the system. Under one-photon resonance condition (Δ=0), the energetic cost takes the minimum.
2. Stimulated Raman Shortcut-to-adiabatic passage
We briefly review the STIRSAP in the three-level systems. The configuration of the quantum system is shown in figure 1. The excited state is labelled as ∣2⟩ and two ground states are ∣1⟩ and ∣3⟩. For the two-photon resonance case, the Hamiltonian within the rotating wave approximation is [1]
$\begin{eqnarray}{H}_{0}(t)=\displaystyle \frac{1}{2}\left(\begin{array}{ccc}0 & {{\rm{\Omega }}}_{p}(t) & 0\\ {{\rm{\Omega }}}_{p}(t) & 2{\rm{\Delta }} & {{\rm{\Omega }}}_{s}(t)\\ 0 & {{\rm{\Omega }}}_{s}(t) & 0\end{array}\right).\end{eqnarray}$
Here, Ωs(t) and Ωp(t) are Rabi frequencies of Stokes and pump laser fields, and the detuning Δ=(E2 − E1)/ℏ − ωs = (E2 − E3)/ℏ − ωp, ωs and ωp are the laser frequencies of Stokes and pump laser, and Ej=1,2,3 are barebasis state energies. The instantaneous eigenstates are given by $\begin{eqnarray}\begin{array}{rcl}| {\lambda }_{+}(t)\rangle & = & \sin \theta (t)\sin \phi (t)| 1\rangle +\cos \phi (t)| 2\rangle \\ & & +\cos \theta (t)\sin \phi (t)| 3\rangle ,\\ | {\lambda }_{0}(t)\rangle & = & \cos \theta (t)| 1\rangle -\sin \theta (t)| 3\rangle ,\\ | {\lambda }_{-}(t)\rangle & = & \sin \theta (t)\cos \phi (t)| 1\rangle -\sin \phi (t)| 2\rangle \\ & & +\cos \theta (t)\cos \phi (t)| 3\rangle ,\end{array}\end{eqnarray}$
and their corresponding eigenvalues are given by $\begin{eqnarray}{E}_{+}=\displaystyle \frac{1}{2}{{\rm{\Omega }}}_{0}(t)\cot \phi (t),\,{E}_{0}=0,\,{E}_{-}=-\displaystyle \frac{1}{2}{{\rm{\Omega }}}_{0}(t)\tan \phi (t),\end{eqnarray}$
where parameters θ(t), φ(t), and Ω0(t) are defined by $\tan \theta (t)\,={{\rm{\Omega }}}_{p}(t)/{{\rm{\Omega }}}_{s}(t)$, $\tan \phi (t)={{\rm{\Omega }}}_{0}(t)/[{\rm{\Delta }}(t)$ + $\sqrt{{{\rm{\Delta }}}^{2}(t)+{{\rm{\Omega }}}_{0}^{2}(t)}],$ and ${{\rm{\Omega }}}_{0}(t)=\sqrt{{{\rm{\Omega }}}_{p}^{2}(t)+{{\rm{\Omega }}}_{s}^{2}(t)}$, respectively. When adiabatic conditions $\dot{\theta }\ll {{\rm{\Omega }}}_{0}$ and Ω0 ≫ 1/T (T is the operation time) are satisfied, the perfect population transfer from ∣1⟩ to ∣3⟩ is realized by the dark state ∣λ0(t)⟩. If Stokes and pump pulses are not adiabatic at all, the population is not completely transferred between quantum states ∣1⟩ and ∣3⟩ [27].
Figure 1. Λ-type three-level system driven by a pump and a Stokes field. Ωp (Ωs) presents the Rabi frequency of the pump (Stokes) field. Δ is the detuning. |
Although Stokes and pump pulses are not adiabatic at all, the transitionless quantum driving shows that the fast and perfect population transfer from quantum states ∣1⟩ to ∣3⟩ can be achieved by adding a supplementary Hamiltonian Hcd to the original Hamiltonian H0. The transitionless quantum driving equates to counter-diabatic driving. Also, a shortcut quantum state transfer is implemented by modifying the pump and Stokes pulses. The total Hamiltonian is [18]cd =i ∑∣∂tλn(t)⟩⟨λn(t)∣, respectively, and En are eigenvalues of the original Hamiltonian H0 and its corresponding instantaneous eigenstates ∣λn(t)⟩. The dynamics of the system will behave ideally adiabatically along the eigenvectors of the original Hamiltonian. For the STIRSAP case, the supplementary Hamiltonian is given by [18]
$\begin{eqnarray}H(t)={H}_{0}(t)+{H}_{{\mathtt{cd}}}(t),\end{eqnarray}$
where H0=∑∣λn(t)⟩En⟨λn(t)∣ and H $\begin{eqnarray}{H}_{{\mathtt{cd}}}(t)=\displaystyle \frac{1}{2}\left(\begin{array}{ccc}0 & 0 & {\mathtt{i}}\dot{\theta }\\ 0 & 0 & 0\\ -{\mathtt{i}}\dot{\theta } & 0 & 0\end{array}\right),\end{eqnarray}$
where the parameter $\dot{\theta }=2[{\dot{{\rm{\Omega }}}}_{p}(t){{\rm{\Omega }}}_{s}(t)-{{\rm{\Omega }}}_{p}(t){\dot{{\rm{\Omega }}}}_{s}(t)]/{{\rm{\Omega }}}_{0}^{2}(t)$. The fast and perfect population transfer from quantum state ∣1⟩ to ∣3⟩ has been shown within a shorter time [18].3. Energetic cost
3.1. Three-level system
In this section, we discuss the energetic cost of implementing STIRSAP in the three-level system by the transitionless quantum driving. The expense of energy in transitionless quantum driving can be written in general as [24] $C(T)={\nu }_{t,n}{\int }_{0}^{T}{\mathtt{d}}t| | H(t)| {| }^{n}$, where νt,n is a setup dependent constant, the index of the norm n depends on the nature of the applied fields, and the norm is defined by the Frobenius norm $| | A| | =\sqrt{{\mathtt{Tr}}[{A}^{\dagger }A]}$ with an operator A. The norm plays the most crucial role in defining the cost of driving. So, the expression of energetic cost can be reduced as [26]
$\begin{eqnarray}\begin{array}{rcl}C(T) & = & \displaystyle \frac{1}{T}{\displaystyle \int }_{0}^{T}{\mathtt{d}}t| | H(t)| | ,\\ & = & \displaystyle \frac{1}{T}{\displaystyle \int }_{0}^{T}{\mathtt{d}}t\sqrt{{\mathtt{Tr}}[{H}_{0}^{2}(t)+{H}_{{\mathtt{cd}}}^{2}(t)]}.\end{array}\end{eqnarray}$
In order to evaluate C(T), we adopt the basis of eigenstates of the original Hamiltonian. So, the energetic cost becomes $\begin{eqnarray}C(T)=\displaystyle \frac{1}{T}{\int }_{0}^{T}{\mathtt{d}}t\sqrt{\displaystyle \sum _{n}[{E}_{n}^{2}+{\mu }_{n}(t)]},\end{eqnarray}$
where μn(t)=⟨∂tλn(t)∣∂tλn(t)⟩ − ∣⟨λn(t)∣∂tλn(t)⟩∣2, En and ∣λn(t)⟩ are the set of energies and eigenstates of the adiabatic Hamiltonian H0.For the STIRSAP in the three-level system, according to equations (2 ), (3 ) and (7 ) we obtain the expression of the energetic cost as follows8 ), we can see that the energetic cost C(T) can be controlled by changing parameters φ and θ. Therefore, the energetic cost is effectively inhibited. If the parameter φ is time-independent, i.e., $\dot{\phi }=0$, the energetic cost C(T) can be reduced ascd . This explicitly shows that a transitionless evolution has an energetic cost larger than its corresponding adiabatic evolution. C″(T) is independent with parameter φ. Comparing equations (9 ) with (10 ), we obtain the inequality $C^{\prime} (T)\geqslant C^{\prime\prime} (T)$, which means that the energetic cost is inhibited further. In order to avoid the expense of energy in transitionless quantum driving, we need to control system work in a resonance interaction regime.
$\begin{eqnarray}C(T)=\displaystyle \frac{1}{T}{\int }_{0}^{T}{\mathtt{d}}t\sqrt{{{\rm{\Omega }}}_{0}^{2}({\cot }^{2}\phi +{\tan }^{2}\phi )/4+2{\dot{\theta }}^{2}+2{\dot{\phi }}^{2}},\end{eqnarray}$
where $\dot{\phi }=({\dot{{\rm{\Omega }}}}_{0}{\rm{\Delta }}-{{\rm{\Omega }}}_{0}\dot{{\rm{\Delta }}})/2({{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}_{0}^{2})$. From the equation ( $\begin{eqnarray}C^{\prime} (T)=\displaystyle \frac{1}{T}{\int }_{0}^{T}{\mathtt{d}}t\sqrt{{{\rm{\Omega }}}_{0}^{2}({\cot }^{2}\phi +{\tan }^{2}\phi )/4+2{\dot{\theta }}^{2}}.\end{eqnarray}$
From the equation $\dot{\phi }=({\dot{{\rm{\Omega }}}}_{0}{\rm{\Delta }}-{{\rm{\Omega }}}_{0}\dot{{\rm{\Delta }}})/2({{\rm{\Delta }}}^{2}+{{\rm{\Omega }}}_{0}^{2})$, this condition $\dot{\phi }=0$ is satisfied by adjusting Ω0=KΔ, where K is a time-independent constant. It is obvious, we have the inequality $C(T)\geqslant C^{\prime} (T)$. Thus, we can reduce the energetic cost by adjusting parameter φ. And then, ${\cot }^{2}\phi +{\tan }^{2}\phi $ takes the minimum, i.e., $\tan \phi =\cot \phi =\pm 1$ (or φ=± π/4). This condition can be satisfied under the resonance interaction (Δ=0), which is easily realized by controlling the Rabi frequencies Ωs(t) and Ωp(t). So, the energetic cost C(T) becomes $\begin{eqnarray}C^{\prime\prime} (T)=\displaystyle \frac{1}{T}{\int }_{0}^{T}{\mathtt{d}}t\sqrt{{{\rm{\Omega }}}_{0}^{2}/2+2{\dot{\theta }}^{2}},\end{eqnarray}$
where the first and second terms are the energetic cost of the adiabatic Hamiltonian H0 and the supplementary Hamiltonian H3.2. Effective two-level system
On one-photon resonance (Δ=0), the Hamiltonian (1) of a three-level system is reduced to an equivalent two-level model with an effective Rabi frequency Ωp(t)/2 and an effective detuning Ωs(t)/2 [28]. The effective Hamiltonian can be written as [27, 28]7 ), (12 ), and (13 ), the energetic cost of the effective two-level system is10 ) and (15 ) we obtain the Σ(T)=C″(T)/2. Therefore, the quantum three-level system is reduced as an effective two-level model, the energetic cost of shortcut population transfer is half only than former.
$\begin{eqnarray}{H}_{{\mathtt{e}}}=\displaystyle \frac{1}{2}\left(\begin{array}{cc}{{\rm{\Omega }}}_{s}(t)/2 & {{\rm{\Omega }}}_{p}(t)/2\\ {{\rm{\Omega }}}_{p}(t)/2 & -{{\rm{\Omega }}}_{s}(t)/2\end{array}\right),\end{eqnarray}$
whose instantaneous eigenstates are $\begin{eqnarray}\begin{array}{rcl}| {\lambda }_{+}(t)\rangle & = & \cos \displaystyle \frac{\theta }{2}| 1\rangle +\sin \displaystyle \frac{\theta }{2}| 3\rangle ,\\ | {\lambda }_{-}(t)\rangle & = & \cos \displaystyle \frac{\theta }{2}| 3\rangle -\sin \displaystyle \frac{\theta }{2}| 1\rangle ,\end{array}\end{eqnarray}$
with eigenvalues $\begin{eqnarray}\begin{array}{rcl}{E}_{+}^{{\prime} } & = & \displaystyle \frac{1}{4}\sqrt{{{\rm{\Omega }}}_{s}^{2}+{{\rm{\Omega }}}_{p}^{2}}=\displaystyle \frac{1}{4}{{\rm{\Omega }}}_{0},\\ {E}_{-}^{{\prime} } & = & -\displaystyle \frac{1}{4}\sqrt{{{\rm{\Omega }}}_{s}^{2}+{{\rm{\Omega }}}_{p}^{2}}=-\displaystyle \frac{1}{4}{{\rm{\Omega }}}_{0}.\end{array}\end{eqnarray}$
The supplementary Hamiltonian of transitionless quantum driving is [18] $\begin{eqnarray}{H}_{{\mathtt{cd}}}=\displaystyle \frac{1}{2}\left(\begin{array}{cc}0 & -{\mathtt{i}}\dot{\theta ^{\prime} }\\ {\mathtt{i}}\dot{\theta ^{\prime} } & 0\end{array}\right),\end{eqnarray}$
where the parameter $\dot{\theta ^{\prime} }=[{\dot{{\rm{\Omega }}}}_{p}(t){{\rm{\Omega }}}_{s}(t)-{{\rm{\Omega }}}_{p}(t){\dot{{\rm{\Omega }}}}_{s}(t)]/{{\rm{\Omega }}}_{0}^{2}(t)$. The total Hamiltonian is H=He + Hcd. Also, the perfect and fast population transfer can be implemented by this Hamiltonian [27]. According to the equations ( $\begin{eqnarray}{\rm{\Sigma }}(T)=\displaystyle \frac{1}{T}{\int }_{0}^{T}{\mathtt{d}}t\sqrt{{{\rm{\Omega }}}_{0}^{2}/8+{\dot{\theta }}^{2}/2}.\end{eqnarray}$
From equations (4. Discussion and conclusion
In recent experiments, the STIRSAP has been realized with a NV center in diamond [16] and cold atoms [17], respectively. An NV center has a relatively long coherence time and the possibility of coherent manipulation at room temperature [29]. And it has a rich energy level structure, which includes ground state, metastable, and excited state. The ground state is a spin triplet with a zero-field splitting, Dgs =2.87 GHz, between ∣ms=0⟩ and the nearly degenerate sublevels ∣ms=± 1⟩ in the absence of external magnetic field [30]. The degeneracy of spin sublevels ∣ms=± 1⟩ can be removed by an external magnetic field. The three-level system is composed of ∣ms=0⟩, ∣ms=− 1⟩, and ∣ms=1⟩ (or ∣ms=− 1⟩, ∣ms=1⟩, and excited state). Also, this three-level system exists in the cold 87Rb atom ensemble, two ground states ∣F=1, mF=0⟩, ∣F=2, mF=0⟩ and one excited state 52P3/2.
In summary, the energetic cost of a STIRSAP in the three-level system was studied. Through calculating energetic cost function, we found that the energetic cost can be inhibited by adjusting detuning Δ. When the detuning Δ=0, the three-level system can be reduced as an effective two-level system. The energetic cost of shortcut population transfer is inhibited further.