Nonreciprocal transition between two indirectly coupled energy levels

Abstract

We propose a theoretical scheme to realize nonreciprocal transition between two energy levels that can not coupled directly. Suppose they are coupled indirectly by two auxiliary levels with a cyclic four-level configuration, and the four transitions in the cyclic configuration are controlled by external fields. The indirectly transition become nonreciprocal when the time reversal symmetry of the system is broken by the synthetic magnetic flux, i.e., the total phase of the external driving fields through the cyclic four-level configuration. The nonreciprocal transition can be identified by the elimination of a spectral line in the spontaneous emission spectrum. Our work introduces a feasible way to observe nonreciprocal transition in a wide range of multi-level systems, including natural atoms or ions with parity symmetry.

Graphical abstract

Keywords

nonreciprocal transition / time reversal symmetry / synthetic magnetic flux / spontaneous emission spectrum

1 Introduction

Time reversal symmetry is the hypothesis that certain physical quantities are unchanged under time reversal transformation, which is related to reversibility of the system, such as the principle of detailed balancing in kinetic systems [1]. With the principle of detailed balance as a background, A. Einstein proposed his quantum theory of radiation [2] in 1916, which is now considered as the theoretical foundation of the laser, and one of the important corollaries is the absorption coefficient should be equal to the stimulated emission coefficient between two nondegenerate energy levels. However, sometimes we need to break the time reversal symmetry of atomic systems to yield fantastic phenomena, e.g., cyclic population transfer [3], controllable electromagnetically induced transparency [4], gain without inversion [5].

One important approach to break the time reversal symmetry of atomic systems is based on the cyclic three-level transitions in the chiral molecules [3, 6, 7], and in the superconducting qubit circuit with three Josephson junctions [8, 9]. The cyclic three-level transitions in multilevel atomic systems have been used to generate many interesting phenomena in single-photon level, including single-photon quantum routing [10], single-photon second-order nonlinear processes [11–13]. When the three possible transitions in the cyclic three-level configuration are driven by three mutually phase-locked driving fields, the time reversal symmetry of atomic system can be broken by the magnetic flux synthesized from the driving-field phase. In a recent experiment, time-reversal symmetry breaking and cyclic population transfer were revealed in a single nitrogen-vacancy spin system by controlling the global phase of the driving fields [14].

On the basis above, a multi-level atomic system with cyclic three-level configuration was proposed to realize significant difference between the stimulated emission and absorption coefficients of two nondegenerate energy levels [15], which was referred to as nonreciprocal transition. Different from the closed-contour spin dynamics [14], besides synthetic magnetism, reservoir engineering was also employed to eliminate one of the transitions in opposite directions [15]. Nonreciprocal transition in the multilevel atomic system provides us a new physical mechanism for designing nonreciprocal photon devices [16, 17] at single-photon level with single atoms [15]. Moreover, it was shown that the nonreciprocal transition can lead to the elimination of a spectral line in the spontaneous emission spectrum [18], which has a potential application for the determination of enantiomeric excess of chiral molecules [18].

It is well known that due to the parity symmetry of the potential energy for usual natural atomic systems, described by SO(3) or SO(4), the parities of atomic eigenstates are well defined, and one-photon transitions between two energy levels require that the two corresponding eigenstates have opposite parities, which is referred to as the selection rule for the electric-dipole transitions. In a three-level atom system, at least two of the atomic eigenstates have the same parity and the one-photon dipole transition between them is forbidden. Thus, a cyclic three-level atom, i.e., the population cyclically transferred between three energy levels, cannot be realized with three one-photon transitions, except that the parity symmetry of the atomic system is broken. We note that the parity symmetry of a natural atom can be broken by applying a strong magnetic field [19, 20]. However, this technique is difficult to implement and is not commonly used.

In this paper, we propose a theoretical scheme to realize nonreciprocal transition between two energy levels that cannot coupled directly, which is significantly different from the models in Refs. [15, 18]. Most important, this provides a way to realize nonreciprocal transition without breaking the parity symmetry of the system, which open up a feasible way to observe nonreciprocal transition in wider systems, such as natural atoms or ions. Moreover, as two energy levels for nonreciprocal transition are not coupled directly, so there is not any strict restriction on the energy difference between them, and nonreciprocal transition can be realized between two degenerate energy levels, which goes beyond the limit of nondegenerate energy levels in Refs. [15, 18]. Our model can be used to design atom- or ion-mediated nonreciprocal devices with well-established technologies in the fields of cold atom [21] and ion trap [22, 23]. Such devices can find applications for quantum control of light in chiral quantum technologies [24] or topological photonics [25].

The remainder of this paper is organized as follows. In Section 2, a cyclic four-level configuration for two energy levels coupled indirectly by two auxiliary levels is introduced and the dynamical equations are given under the Weisskopf–Wigner approximation. The time evolution of the populations and the transition probabilities between the two energy levels are investigated, and nonreciprocal transitions are shown in Section 3. Moreover, the spontaneous emission spectra of the systems with nonreciprocal transitions are discussed in Section 4, which provide us a convenient way to measure nonreciprocal transitions in experiments. Finally, the conclusions is given in Section 5.

2 Hamiltonian and dynamical equations

We propose to realize nonreciprocal transition between two energy levels |a〉 and |b〉, under the assumption that they can not coupled directly. Suppose they are coupled indirectly through two auxiliary levels (|c〉 and |d〉) as a cyclic four-level configuration, and the four transitions are controlled by external fields with frequencies (ν_ca, ν_cb, ν_db, and ν_da), Rabi frequencies (Ω_ca, Ω_cb, Ω_db, and Ω_da) and phases (ϕ_ca, ϕ_cb, ϕ_db, and ϕ_da), as shown in Fig. 1. For different systems, the four (upper) levels may be coupled to the same lower level or to different lower levels respectively. Previous studies have shown that the spontaneous emission from multiple upper levels to the common lower level may result in spontaneous emission cancellation and spectral line elimination [26, 27], which is not the focus of this paper. In order to eliminate this effect, the spontaneous emission spectrum for the system will be derived for the case that the four upper levels are coupled to four different lower levels (|g_a〉, |g_b〉, |g_c〉, and |g_d〉) respectively with the same vacuum modes. The Hamiltonian is given by (\hslash =1)

\begin{array}{}\text{(1)}& \begin{array}{ll}H=\hfill & {\text{Ω}}_{ca}{\text{e}}^{\text{i}{\varphi }_{ca}}{\text{e}}^{-\text{i}{\text{Δ}}_{ca}t}|a\rangle \langle c|+{\text{Ω}}_{cb}{\text{e}}^{\text{i}{\varphi }_{cb}}{\text{e}}^{\text{i}{\text{Δ}}_{cb}t}|c\rangle \langle b|\hfill \\ \hfill & +{\text{Ω}}_{da}{\text{e}}^{\text{i}{\varphi }_{da}}{\text{e}}^{\text{i}{\text{Δ}}_{da}t}|d\rangle \langle a|+{\text{Ω}}_{db}{\text{e}}^{\text{i}{\varphi }_{db}}{\text{e}}^{-\text{i}{\text{Δ}}_{db}t}|b\rangle \langle d|\hfill \\ \hfill & +{\displaystyle \sum _k}[g_k^a{\text{e}}^{\text{i}({\omega }_{ag}-{\omega }_k)t}{v}_k|a\rangle \langle g_a|\hfill \\ \hfill & +g_k^b{\text{e}}^{\text{i}({\omega }_{bg}-{\omega }_k)t}{v}_k|b\rangle \langle g_b|]\hfill \\ \hfill & +{\displaystyle \sum _k}[g_k^c{\text{e}}^{\text{i}({\omega }_{cg}-{\omega }_k)t}{v}_k|c\rangle \langle g_c|\hfill \\ \hfill & +g_k^d{\text{e}}^{\text{i}({\omega }_{dg}-{\omega }_k)t}{v}_k|d\rangle \langle g_d|]\hfill \\ \hfill & +\text{H}\text{.c}\mathrm{.,}\hfill \end{array}\end{array}

where v_k and v_k^{†} are annihilation and creation operators fo r photons in the kth vacuum mode with frequency ω_k (k denotes both the momentum and polarization of the vacuum modes); ω_ij and ω_ig are the frequency differences between levels |i〉 and |j〉, and between |i〉 and |g_i〉 respectively, with (i, j = a, b, c, d); ∆_ij≡ω_ij−ν_ij is the detuning between the atomic transition |i〉 ↔ |j〉 and the driving field, and g_k^i is the coupling strength between the atomic transition |i〉 ↔ |g_i〉 and the kth vacuum mode. With the replacement |b\rangle \to {\text{e}}^{\text{i}{\varphi }_{cb}}|b\rangle , |d\rangle \to {\text{e}}^{\text{i}({\varphi }_{db}+{\varphi }_{cb})}|d\rangle , |a\rangle \to {\text{e}}^{\text{i}({\varphi }_{da}+{\varphi }_{db}+{\varphi }_{cb})}|a\rangle , g_k^b\to g_k^b{\text{e}}^{-\text{i}{\varphi }_{cb}} , g_k^d\to g_k^d{\text{e}}^{-\text{i}({\varphi }_{db}+{\varphi }_{cb})} , and g_k^a\to g_k^a{\text{e}}^{-\text{i}({\varphi }_{da}+{\varphi }_{db}+{\varphi }_{cb})} , the Hamiltonian is rewritten as

\begin{array}{}\text{(2)}& \begin{array}{ll}\text{H}=\hfill & {\text{Ω}}_{ca}{\text{e}}^{\text{i}\text{Φ}}{\text{e}}^{-\text{i}{\text{Δ}}_{ca}t}|a\rangle \langle c|+{\text{Ω}}_{cb}{\text{e}}^{\text{i}{\text{Δ}}_{cb}t}|c\rangle \langle b|\hfill \\ \hfill & +{\text{Ω}}_{da}{\text{e}}^{\text{i}{\text{Δ}}_{da}t}|d\rangle \langle a|+{\text{Ω}}_{db}{\text{e}}^{-\text{i}{\text{Δ}}_{db}t}|b\rangle \langle d|\hfill \\ \hfill & +{\displaystyle \sum _k}[g_k^a{\text{e}}^{\text{i}({\omega }_{ag}-{\omega }_k)t}{v}_k|a\rangle \langle g_a|\hfill \\ \hfill & +g_k^b{\text{e}}^{\text{i}({\omega }_{bg}-{\omega }_k)t}{v}_k|b\rangle \langle g_b|]\hfill \\ \hfill & \begin{array}{l}+{\displaystyle \sum _k[g_k^c{\text{e}}^{\text{i}\left({\omega }_{cg}-{\omega }_k\right)t}{v}_k|c\rangle \langle g_c|}\\ +g_k^d{\text{e}}^{\text{i}\left({\omega }_{dg}-{\omega }_k\right)t}{v}_k|d\rangle \langle g_d|]\\ +\text{H}\text{.c}\text{.},\end{array}\hfill \end{array}\end{array}

where Φ ≡ ϕ_ca+ϕ_db+ϕ_cb+ϕ_da is the total phase of the four strong driving fields through the cycle-transition |a〉→|d〉→|b〉→|c〉→|a〉, i.e., the synthetic magnetic flux. Moreover, real coupling strength g_k^i is assumed for the phase of g_k^i does not matter in the following discussions. For simplicity, we also make the assumption of resonance ∆_c ≡ ∆_ac = ∆_bc and ∆_d ≡ ∆_da = ∆_db.

In the following, we will discuss the indirect transition between levels |a〉 and |b〉 based on the Schrödinger equation under the Weisskopf–Wigner approximation [28, 29]. We assume that the system is prepared in level |a〉 or |b〉 initially, i.e., |ψ(0)〉 = |a〉|0〉 or |ψ(0)〉 = |b〉|0〉 with |0〉 denoting the vacuum state. According to the Schrödinger equation, d |ψ(t)〉 /dt = −iH |ψ(t)〉, the state vector at time t can be written as

\begin{array}{}\text{(3)}& \begin{array}{ll}|\psi (t)\rangle =\hfill & \left[A(t)|a\rangle +B(t)|b\rangle +C(t)|c\rangle +D(t)|d\rangle \right]|0\rangle \hfill \\ \hfill & +{\displaystyle \sum _{i=a\mathrm{,}b\mathrm{,}c\mathrm{,}d}}{\displaystyle \sum _k}{G}_k^i(t)v_k^{†}|g_i\rangle |0\rangle \mathrm{,}\hfill \end{array}\end{array}

with occupation populations |A(t)|², |B(t)|², |C(t)|², |D(t)|², and |G_k^i(t){|}^2 in the corresponding levels. Under the Weisskopf–Wigner approximation [28, 29], the dynamical behaviors for the coefficients are obtained as

\begin{array}{}\text{(4)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}A(t)=\hfill & -\frac{{\gamma }_a}{2}A(t)-\text{i}{\text{Ω}}_{ca}{\text{e}}^{\text{i}\text{Φ}}{\text{e}}^{-\text{i}{\text{Δ}}_{c}t}C(t)\hfill \\ \hfill & -\text{i}{\text{Ω}}_{da}{\text{e}}^{-\text{i}{\text{Δ}}_{d}t}D(t)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(5)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}B(t)=\hfill & -\frac{{\gamma }_b}{2}B(t)-\text{i}{\text{Ω}}_{cb}{\text{e}}^{-\text{i}{\text{Δ}}_{c}t}C(t)\hfill \\ \hfill & -\text{ i}{\text{Ω}}_{db}{\text{e}}^{-\text{i}{\text{Δ}}_{d}t}D(t)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(6)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}C(t)=\hfill & -\frac{{\gamma }_c}{2}C(t)-\text{i}{\text{Ω}}_{ca}{\text{e}}^{-\text{i}\text{Φ}}{\text{e}}^{\text{i}{\text{Δ}}_{c}t}A(t)\hfill \\ \hfill & -\text{i}{\text{Ω}}_{cb}{\text{e}}^{\text{i}{\text{Δ}}_{c}t}B(t)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(7)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}D(t)=\hfill & -\frac{{\gamma }_d}{2}D(t)-\text{i}{\text{Ω}}_{da}{\text{e}}^{\text{i}{\text{Δ}}_{d}t}A(t)\hfill \\ \hfill & -\text{i}{\text{Ω}}_{db}{\text{e}}^{\text{i}{\text{Δ}}_{d}t}B(t)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(8)}& \frac{\text{d}}{\text{d}t}{G}_k^a(t)=-\text{i}{g}_k^a{\text{e}}^{-\text{i}\left({\omega }_{ag}-{\omega }_k\right)t}A(t)\mathrm{,}\end{array}

\begin{array}{}\text{(9)}& \frac{\text{d}}{\text{d}t}{G}_k^b(t)=-\text{i}{g}_k^b{\text{e}}^{-\text{i}\left({\omega }_{bg}-{\omega }_k\right)t}B(t)\mathrm{,}\end{array}

\begin{array}{}\text{(10)}& \frac{\text{d}}{\text{d}t}{G}_k^c(t)=-\text{i}{g}_k^c{\text{e}}^{-\text{i}\left({\omega }_{cg}-{\omega }_k\right)t}C(t)\mathrm{,}\end{array}

\begin{array}{}\text{(11)}& \frac{\text{d}}{\text{d}t}{G}_k^d(t)=-\text{i}{g}_k^d{\text{e}}^{-\text{i}\left({\omega }_{dg}-{\omega }_k\right)t}D(t)\mathrm{,}\end{array}

with the decay rates {\gamma }_a=2\pi {(g_k^a)}^2\rho ({\omega }_{ag}) , γ_b = 2\pi {(g_k^b)}^2\rho ({\omega }_{bg}) , {\gamma }_c=2\pi {(g_k^c)}^2\rho ({\omega }_{cg}) , and γ_d = 2\pi {(g_k^d)}^2\rho ({\omega }_{dg}) , and mode density ρ(ω_k). The dynamical equations (4)–(7) can be rewritten with constant coefficients as

\begin{array}{}\text{(12)}& \frac{\text{d}}{\text{d}t}A(t)=-\frac{{\gamma }_a}{2}A(t)-\text{i}{\text{Ω}}_{ca}{\text{e}}^{\text{i}\text{Φ}}\tilde{C}(t)-\text{i}{\text{Ω}}_{da}\tilde{D}(t)\mathrm{,}\end{array}

\begin{array}{}\text{(13)}& \frac{\text{d}}{\text{d}t}B(t)=-\frac{{\gamma }_b}{2}B(t)-\text{i}{\text{Ω}}_{cb}\tilde{C}(t)-\text{i}{\text{Ω}}_{db}\tilde{D}(t)\mathrm{,}\end{array}

\begin{array}{}\text{(14)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}\tilde{C}(t)=\hfill & \left(-\text{i}{\text{Δ}}_c-\frac{{\gamma }_c}{2}\right)\tilde{C}(t)-\text{i}{\text{Ω}}_{ca}{\text{e}}^{-\text{i}\text{Φ}}A(t)\hfill \\ \hfill & -\text{i}{\text{Ω}}_{cb}B(t)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(15)}& \begin{array}{ll}\frac{\text{d}}{\text{d}t}\tilde{D}(t)=\hfill & \left(-\text{i}{\text{Δ}}_d-\frac{{\gamma }_d}{2}\right)\tilde{D}(t)-\text{i}{\text{Ω}}_{da}A(t)\hfill \\ \hfill & -\text{i}{\text{Ω}}_{db}B(t)\mathrm{,}\hfill \end{array}\end{array}

with the definitions \tilde{C}(t)\equiv {\text{e}}^{-\text{i}{\text{Δ}}_{c}t}C(t) , \tilde{D}(t)\equiv {\text{e}}^{-\text{i}{\text{Δ}}_{d}t}D(t) . It is worth mentioning that, here we have ignored the effects of the spontaneous decay between the four upper excited states, based on the assumption that the spontaneous relaxation between the four upper excited states is much slower than the spontaneous decay from the excited states to the ground states [30], and also much slower than the coherent transition between the excited states induced by the strong driving fields.

3 Transition probabilities

To gain insight into the transition probabilities between energy levels |a〉 and |b〉, we will solve the dynamic equations (12)–(15) both analytically and numerically. The dynamic equations (12)–(15) can be rewritten as an effective Schrödinger equation

\begin{array}{}\text{(16)}& \text{i}\frac{\text{d}\text{Ψ}(t)}{\text{d}t}=H_{\text{eff}}\text{Ψ}(t)\mathrm{,}\end{array}

with state vector \text{Ψ}(t)={\left[\begin{array}{c}A(t)\mathrm{,}B(t)\mathrm{,}\tilde{C}(t)\mathrm{,}\tilde{D}(t)\end{array}\right]}^T , and effective Hamiltonian as

\begin{array}{}\text{(17)}& H_{\text{eff}}=(\begin{array}{cccc}-\text{i}\frac{{\gamma }_a}{2}& 0& {\text{Ω}}_{ca}{\text{e}}^{\text{i}\text{Φ}}& {\text{Ω}}_{da}\\ 0& -\text{i}\frac{{\gamma }_b}{2}& {\text{Ω}}_{cb}& {\text{Ω}}_{db}\\ {\text{Ω}}_{ca}{\text{e}}^{-\text{i}\text{Φ}}& {\text{Ω}}_{cb}& {\text{Δ}}_c-\text{i}\frac{{\gamma }_c}{2}& 0\\ {\text{Ω}}_{da}& {\text{Ω}}_{db}& 0& {\text{Δ}}_d-\text{i}\frac{{\gamma }_d}{2}\end{array})\mathrm{.}\end{array}

The formal solutions for Eq. (16) can be written as

\begin{array}{}\text{(18)}& \text{Ψ}(t)=U(t)\text{Ψ}\left(0\right)\mathrm{,}\end{array}

with initial vector Ψ (0) = [A (0), B (0), C (0), D (0)]^⊺, and time-evolution matrix

\begin{array}{}\text{(19)}& U(t)\equiv {\text{e}}^{-\text{i}{H}_{\text{eff}}t}\mathrm{.}\end{array}

The transition probabilities T_ba(t) for |a〉→|b〉 and T_ab(t) for |b〉→|a〉 can be defined from the time-evolution matrix U (t) as

\begin{array}{}\text{(20)}& T_{ba}(t)\equiv {|U_{21}(t)|}^2\mathrm{,}\end{array}

\begin{array}{}\text{(21)}& T_{ab}(t)\equiv |U_{12}(t){|}^2\mathrm{,}\end{array}

where the subscripts 1 and 2 are the row and column numbers of the matrix U (t). Moreover, the isolation for the nonreciprocal transition between levels |a〉 and |b〉 is defined by

\begin{array}{}\text{(22)}& I(t)\equiv \frac{T_{ab}(t)}{T_{ba}(t)}\mathrm{.}\end{array}

Under the assumption that the decay rates of the upper levels |c〉 and |d〉 are much larger than the other parameters, i.e., min {γ_c, γ_d} ≫ max {γ_a, γ_b, Ω_ca, Ω_cb, Ω_da, Ω_db}, we can eliminate the auxiliary levels |c〉 and |d〉 adiabatically. By setting \begin{array}{l}d\tilde{C}(t)/dt=d\tilde{D}(t)/dt=0\hfill \end{array} in Eqs. (14) and (15), we obtain

\begin{array}{}\text{(23)}& \tilde{C}(t)=-\frac{\text{i}{\text{Ω}}_{ca}}{\text{i}{\text{Δ}}_c+\frac{{\gamma }_c}{2}}{\text{e}}^{-\text{i}\text{Φ}}A(t)-\frac{\text{i}{\text{Ω}}_{cb}}{\text{i}{\text{Δ}}_c+\frac{{\gamma }_c}{2}}B(t)\mathrm{,}\end{array}

\begin{array}{}\text{(24)}& \tilde{D}(t)=\frac{-\text{i}{\text{Ω}}_{da}}{\text{i}{\text{Δ}}_d+\frac{{\gamma }_d}{2}}A(t)+\frac{-\text{i}{\text{Ω}}_{db}}{\text{i}{\text{Δ}}_d+\frac{{\gamma }_d}{2}}B(t)\mathrm{.}\end{array}

Substitute these into Eqs. (12) and (13), the effective dynamic equations for levels |a〉 and |b〉 can be written as

\begin{array}{}\text{(25)}& \frac{\text{d}}{\text{d}t}A(t)=-\left(\frac{{\gamma }_{a\mathrm{,}\text{eff}}}{2}+\text{i}{\text{Δ}}_{a\mathrm{,}\text{eff}}\right)A(t)-J_{ab}B(t)\mathrm{,}\end{array}

\begin{array}{}\text{(26)}& \frac{\text{d}}{\text{d}t}B(t)=-\left(\frac{{\gamma }_{b\mathrm{,}\text{eff}}}{2}+\text{i}{\text{Δ}}_{b\mathrm{,}\text{eff}}\right)B(t)-J_{ba}A(t)\end{array}

with effective decay rates

\begin{array}{}\text{(27)}& {\gamma }_{a\mathrm{,}\text{eff}}={\gamma }_a+\frac{4{\gamma }_c{\text{Ω}}_{ca}^2}{4{\text{Δ}}_c^2+{\gamma }_c^2}+\frac{4{\gamma }_d{\text{Ω}}_{da}^2}{4{\text{Δ}}_d^2+{\gamma }_d^2}\mathrm{,}\end{array}

\begin{array}{}\text{(28)}& {\gamma }_{b\mathrm{,}\text{eff}}={\gamma }_b+\frac{4{\gamma }_c{\text{Ω}}_{cb}^2}{4{\text{Δ}}_c^2+{\gamma }_c^2}+\frac{4{\gamma }_d{\text{Ω}}_{db}^2}{4{\text{Δ}}_d^2+{\gamma }_d^2}\mathrm{,}\end{array}

effective detunings

\begin{array}{}\text{(29)}& {\text{Δ}}_{a\mathrm{,}\text{eff}}=-\frac{4{\text{Δ}}_c{\text{Ω}}_{ca}^2}{4{\text{Δ}}_c^2+{\gamma }_c^2}-\frac{4{\text{Δ}}_d{\text{Ω}}_{da}^2}{4{\text{Δ}}_d^2+{\gamma }_d^2}\mathrm{,}\end{array}

\begin{array}{}\text{(30)}& {\text{Δ}}_{b\mathrm{,}\text{eff}}=-\frac{4{\text{Δ}}_c{\text{Ω}}_{cb}^2}{4{\text{Δ}}_c^2+{\gamma }_c^2}-\frac{4{\text{Δ}}_d{\text{Ω}}_{db}^2}{4{\text{Δ}}_d^2+{\gamma }_d^2}\mathrm{,}\end{array}

and effective coupling coefficients

\begin{array}{}\text{(31)}& J_{ab}=\frac{{\text{Ω}}_{ca}{\text{Ω}}_{cb}{\text{e}}^{\text{i}\text{Φ}}}{\text{i}{\text{Δ}}_c+\frac{{\gamma }_c}{2}}+\frac{{\text{Ω}}_{da}{\text{Ω}}_{db}}{\text{i}{\text{Δ}}_d+\frac{{\gamma }_d}{2}}\mathrm{,}\end{array}

\begin{array}{}\text{(32)}& J_{ba}=\frac{{\text{Ω}}_{ca}{\text{Ω}}_{cb}{\text{e}}^{-\text{i}\text{Φ}}}{\text{i}{\text{Δ}}_c+\frac{{\gamma }_c}{2}}+\frac{{\text{Ω}}_{da}{\text{Ω}}_{db}}{\text{i}{\text{Δ}}_d+\frac{{\gamma }_d}{2}}\mathrm{.}\end{array}

The condition for nonreciprocal transition is J_ab ≠ J_ba, i.e., Ф ≠ nπ (n is an integer). This can be understood intuitively that Ф ≠ nπ breaks the time-reversal symmetry of the Hamiltonian given in Eq. (2).

One necessary condition for optimal nonreciprocal transition is one of the effective coupling coefficients (J_ab or J_ba) equal zero, i.e.,

\begin{array}{}\text{(33)}& {\text{e}}^{\pm \text{i}\text{Φ}}=-\frac{(\text{i}2{\text{Δ}}_c+{\gamma }_c)}{(\text{i}2{\text{Δ}}_d+{\gamma }_d)}\frac{{\text{Ω}}_{da}{\text{Ω}}_{db}}{{\text{Ω}}_{ca}{\text{Ω}}_{cb}}\mathrm{.}\end{array}

The transition probabilities from |a〉 to |b〉 [T_ba(t)] and from |b〉 to |a〉 [T_ab(t)] can be obtained approximately under this condition. For simplicity, we choose the symmetric parameters: Ω_ca = Ω_cb = Ω_da = Ω_db, γ_a = γ_b, γ_c = γ_d, ∆_c = −∆_d. The condition J_ab = 0 (J_ba = 0) is satisfied for Φ = −π/2 (Φ = π/2) with detuning ∆_c = −∆_d = γ_c/2 = γ_d/2, and we have transition probabilities

\begin{array}{}\text{(34)}& T_{ba}(t)\approx {\left|J_{ba}t\exp \left[-\left(\frac{{\gamma }_{b\mathrm{,}\text{eff}}}{2}+\text{i}{\text{Δ}}_{b\mathrm{,}\text{eff}}\right)t\right]\right|}^2\mathrm{,}\end{array}

for initial conditions A (0) = 1 and B (0) = C (0) = D (0) = 0, and

\begin{array}{}\text{(35)}& T_{ab}(t)\approx {\left|J_{ab}t\exp \left[-\left(\frac{{\gamma }_{a\mathrm{,}\text{eff}}}{2}+\text{i}{\text{Δ}}_{a\mathrm{,}\text{eff}}\right)t\right]\right|}^2\end{array}

for initial conditions B (0) = 1 and A (0) = C (0) = D (0) = 0. The transition probabilities are time dependent, and the time for maximal transition probability is

\begin{array}{}\text{(36)}& t_M\approx \frac{2}{{\gamma }_{b\mathrm{,}\text{eff}}}=\frac{2}{{\gamma }_{a\mathrm{,}\text{eff}}}\mathrm{,}\end{array}

with transition probabilities

\begin{array}{}\text{(37)}& T_{ba}(t_M)\approx {\left|\frac{2}{e}\frac{J_{ba}}{{\gamma }_{b\mathrm{,}\text{eff}}}\right|}^2\mathrm{,}\end{array}

\begin{array}{}\text{(38)}& T_{ab}(t_M)\approx {\left|\frac{2}{e}\frac{J_{ab}}{{\gamma }_{a\mathrm{,}\text{eff}}}\right|}^2\mathrm{,}\end{array}

where e is the mathematical constant approximately equal to 2.71828.

The populations |A (t)|² (black solid curve), |B (t)|² (red dashed curve), and |C (t) |² = |D (t) |² (blue dashed-dot curve) obtained from Eqs. (17)–(19) are plotted as functions of the time t in Figs. 2(a)–(d). It is clear that the population can transfer from the level |b〉 to level |a〉 for Φ = π/2, but almost no population will transfer from the level |a〉 to level |b〉. In contrast, the population can transfer from the level |a〉 to level |b〉, but almost no population will transfer from the level |b〉 to level |a〉 when Φ = −π/2.

The transition probabilities from |b〉 to |a〉 [T_ab(t)] and from |a〉 to |b〉 [T_ba(t)] can also be obtained from Eqs. (19)–(21). They are plotted as functions of time t in Figs. 2(e) and (f). It is clear that T_ab(t) ≫ T_ba(t) for ϕ = π/2, and T_ab(t) ≪ T_ba(t) for ϕ = −π/2, i.e., the transitions between levels |b〉 and |a〉 are nonreciprocal. The approximate analytical results given in Eqs. (34) and (35) are shown by open squares in Figs. 2(e) and (f), which agree well with the solid and dashed curves obtained from Eqs. (19)–(21).

Furthermore, the dependence of the transition probabilities T_ba(t) and T_ab(t) on the synthetic magnetic flux Φ is show in Fig. 3(a). At time t = t_M, we have T_ba(t) < T_ab(t) for synthetic magnetic flux 0 < Φ < π; in the contrast, we have T_ba(t) > T_ab(t) for synthetic magnetic flux −π < Φ < 0. As shown in Fig. 3(b), under the conditions Ω_ca = Ω_cb = Ω_da = Ω_db and ∆_c = −∆_d = γ_c/2 = γ_d/2, the optimal isolation I(t) is obtained with synthetic magnetic flux Φ = ±π/2.

4 Spontaneous emission spectrum

In this section, we will show that the nonreciprocal transitions can be identified by measuring the spontaneous emission spectra of the system. The spontaneous emission spectra of the system can be derived analytically by the Laplace transform method [26, 27]. Taking the Laplace transform, i.e., \overline{O}(s)={\displaystyle {\int }_0^{+\infty }}O(t){\text{e}}^{-st}\text{d}t , the solution of the effective Schrödinger equation (16), \overline{\text{Ψ}}(s)= {\left[\begin{array}{c}\overline{A}(s)\mathrm{,}\overline{B}(s)\mathrm{,}\overline{\tilde{C}}(s)\mathrm{,}\overline{\tilde{D}}(s)\end{array}\right]}^{\text{T}} , is given by

\begin{array}{}\text{(39)}& \overline{\text{Ψ}}(s)=M^{-1}\text{Ψ}\left(0\right)\end{array}

with coefficient matrix

\begin{array}{}\text{(40)}& M=\left(\begin{array}{cccc}s+\frac{{\gamma }_a}{2}& 0& \text{i}{\text{Ω}}_{ca}{\text{e}}^{\text{i}\text{Φ}}& \text{i}{\text{Ω}}_{da}\\ 0& s+\frac{{\gamma }_b}{2}& \text{i}{\text{Ω}}_{cb}& \text{i}{\text{Ω}}_{db}\\ \text{i}{\text{Ω}}_{ca}{\text{e}}^{-\text{i}\text{Φ}}& \text{i}{\text{Ω}}_{cb}& s+\frac{{\gamma }_c}{2}+\text{i}{\text{Δ}}_c& 0\\ \text{i}{\text{Ω}}_{da}& \text{i}{\text{Ω}}_{db}& 0& s+\frac{{\gamma }_d}{2}+\text{i}{\text{Δ}}_d\end{array}\right)\mathrm{.}\end{array}

The spontaneous emission spectrum of the system S({\omega }_k)={\displaystyle {\sum }_{i=a\mathrm{,}b\mathrm{,}c\mathrm{,}d}}{|G_k^i(+\infty )|}^2\rho ({\omega }_k) is obtained from the Fourier transform of 〈E⁻(t + τ) E⁺ (t)〉_t→+∞ [18, 26, 27], with G_k^i(+\infty )\equiv G_k^i(t){|}_{t\to +\infty } as the long time behavior (t→+∞) of G_k^i(t) . By integrating time t in Eqs. (8)–(11), G_k^i(+\infty ) is given by

\begin{array}{}\text{(41)}& G_k^a(+\infty )=-\text{i}{g}_k^a\overline{A}\left[i\left({\omega }_{ag}-{\omega }_k\right)\right]\mathrm{,}\end{array}

\begin{array}{}\text{(42)}& G_k^b(+\infty )=-\text{i}{g}_k^b\overline{B}\left[i\left({\omega }_{bg}-{\omega }_k\right)\right]\mathrm{,}\end{array}

\begin{array}{}\text{(43)}& G_k^c(+\infty )=-\text{i}{g}_k^c\overline{\tilde{C}}\left[\text{i}\left({\omega }_{cg}-{\text{Δ}}_c-{\omega }_k\right)\right]\mathrm{,}\end{array}

\begin{array}{}\text{(44)}& G_k^d(+\infty )=-\text{i}{g}_k^d\overline{\tilde{D}}\left[\text{i}\left({\omega }_{dg}-{\text{Δ}}_d-{\omega }_k\right)\right]\mathrm{.}\end{array}

Thus, we find the spontaneous emission spectrum as

\begin{array}{}\text{(45)}& \begin{array}{ll}S({\omega }_k)=\hfill & \frac{{\gamma }_a}{2\text{π}}{\left|\overline{A}\left[\text{i}\left({\omega }_{ag}-{\omega }_k\right)\right]\right|}^2+\frac{{\gamma }_b}{2\text{π}}{\left|\overline{B}\left[\text{i}\left({\omega }_{bg}-{\omega }_k\right)\right]\right|}^2\hfill \\ \hfill & +\frac{{\gamma }_c}{2\text{π}}{\left|\overline{C}\left[\text{i}\left({\omega }_{cg}-{\text{Δ}}_c-{\omega }_k\right)\right]\right|}^2\hfill \\ \hfill & +\frac{{\gamma }_d}{2\text{π}}{\left|\overline{\tilde{D}}\left[\text{i}\left({\omega }_{dg}-{\text{Δ}}_d-{\omega }_k\right)\right]\right|}^2\hfill \end{array}\end{array}

with \overline{A}(s) , \overline{B}(s) , \overline{\tilde{C}}(s) , and \overline{\tilde{D}}(s) given by Eqs. (39) and (40).

In Fig. 4, the spontaneous emission spectra of the system are plotted with different initial conditions: (a) A (0) = 1 and B (0) = C (0) = D (0) = 0; (b) B (0) = 1 and A (0) = C (0) = D (0) = 0. In Fig. 4 (a), we assume that the system is prepared in level |a〉 initially, and the black solid curves are for phase Φ = π/2 and the red dashed curves for Φ = −π/2. When Φ = −π/2, as the population can transfer from the level |a〉 to level |b〉, so there is a peak around the resonant frequency ω_bg in the spontaneous emission spectra. In the meantime, the population can transfer from both the levels |a〉 and |b〉 to levels |c〉 and |d〉, and the interferences between different population transferring paths suppress the population decay from levels |c〉 and |d〉 [see |C (t) |² = |D (t)|² (blue dashed-dot curve) in Fig. 2(c)] and induce the dips around frequencies ω_cg−∆_c and ω_bg−∆_d. Instead, there is almost no population transferring from the level |a〉 to level |b〉wh en Φ = π/2, so that the peak around the frequency ω_bg is eliminated. Meanwhile, the population transfer only from the level |a〉 to levels |c〉 and |d〉, so that the population in levels |c〉 and |d〉 decay nearly exponentially [see |C (t) |² = |D (t) |² (blue dashed-dot curve) in Fig. 2(a)] and there are single peaks around the frequencies ω_cg−∆_c and ω_bg−∆_d. In Fig. 4 (b), we assume that the system is prepared in level |b〉 initially. When Φ = π/2, as the population can transfer from the level |b〉 to level |a〉, so there is a peak around the resonant frequency ω_ag and dips around frequencies ω_cg−∆_c and ω_bg−∆_d in the spontaneous emission spectra. Instead, there is almost no population transferring from the level |b〉 to level |a〉 when Φ = −π/2, so that the peak around the frequency ω_ag is eliminated and there are single peaks around the frequencies ω_cg−∆_c and ω_bg−∆_d.

In addition, the dependence of the spontaneous emission spectra S(ω_k = ω_ag) and S(ω_k = ω_bg) on the synthetic magnetic flux Φ are shown in Fig. 5, with the system initially prepared in level |b〉 and |a〉 respectively. We have S(ω_k = ω_ag) > S(ω_k = ω_bg) for synthetic magnetic flux 0 < Φ < π, which corresponds with T_ba(t) < T_ab(t) in Fig. 3(a); in the contrast, we have S(ω_k = ω_ag) < S(ω_k = ω_bg) for synthetic magnetic flux −π < Φ < 0, conforming to T_ba(t) > T_ab(t). Thus we can also observe the nonreciprocal transitions by contrasting the difference of the spontaneous emission spectra S(ω_k = ω_ag) and S(ω_k = ω_bg) with the system prepared in level |b〉 or |a〉 initially. Moreover, almost all the populations have decayed to the ground states at the time t = 2/γ, as shown in Figs. 2(a)–(d), so the elimination of spectral lines can be observed within the time scale of 2/γ.

5 Conclusions

Let us now discuss the experimental feasibility of our proposal. Take ⁴⁰Ca⁺ for example, let us show the corresponding relationship between the energy levels in Fig. 1 and the energy levels of ⁴⁰Ca⁺: |a〉 = |3D_5/2〉, |b〉 = |3D_3/2〉, |c〉 = |5P_3/2〉, |d〉 = |4P_3/2〉. First of all, the one-photon dipole transitions |4P_3/2〉 ↔ |3D_5/2〉, |4P_3/2〉 ↔ |3D_3/2〉, |5P_3/2〉 ↔ |3D_5/2〉, and |5P_3/2〉 ↔ |3D_3/2〉 are allowed, but the one-photon dipole transitions |5P_3/2〉 ↔ |4P_3/2〉 and |3D_5/2〉 ↔ |3D_3/2〉 are forbidden, i.e., the cyclic transitions for the four levels can be realized. To avoid the effects of Doppler broadening, the calcium ions should be laser cooled primarily. A single calcium ion can be trapped and laser cooled to 1 mK by Doppler cooling in experiments [31]. According to the experiments with single cold calcium ions [31–34] or few cold calcium ions [35], |3D_5/2〉 and |3D_3/2〉 are metastable levels with lifetimes about 1.2 seconds [31–33, 35], the lifetimes of |4P_3/2〉 is about 6.6 nanoseconds [34, 36], and the lifetimes of |5P_3/2〉 must also be much shorter than 1.2 seconds, i.e., the conditions for adiabatic approximation (min {γ_c, γ_d} ≫ max {γ_a, γ_b}) are satisfied. Hence if four laser fields with appropriate frequencies, strengths, and phases are applied to the ⁴⁰Ca⁺ system, the proposal is realizable. Moreover, similar energy structures can be found in other singly charged alkaline-earth metals ions, such as ⁸⁸Sr⁺ [37, 38] and ¹³⁸Ba⁺ [39–42].

In summary, a theoretically scheme was proposed to realize nonreciprocal transition between two indirectly coupled energy levels in a multi-level atomic system with cyclic four-level configuration. The spontaneous emission spectra of the multi-level system with nonreciprocal transition was also investigated. The nonreciprocal transition results in the elimination of a spectral line in the spontaneous emission spectrum, which can be used to identify the nonreciprocal transition experimentally. As the multi-level atomic system with cyclic four-level configuration can be realized under the rules of electric dipole transitions, the scheme in this paper can be applied for observing nonreciprocal transition in the natural atoms or ions with parity symmetry, which will widely broaden the application sphere of nonreciprocal transition.

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