Abstract
1. INTRODUCTION
According to Einstein’s phenomenological radiation theory [1], the absorption coefficient should be equal to the stimulated emission coefficient between two nondegenerate energy levels. When the spontaneous emission can be neglected, a two-level system undergoes optical Rabi oscillations under the action of a coherent driving electromagnetic field [2]. However, can we make the absorption coefficient different from the stimulated emission coefficient for the transition between two energy levels with different eigenvalues, i.e., nonreciprocal transition between two nondegenerate energy levels? The answer is yes. In this paper, we describe a generic method to realize a nonreciprocal transition between two nondegenerate energy levels, and we show that the absorption and stimulated emission coefficients can be controlled via a combination of synthetic magnetism and reservoir engineering.
Theoretical research has shown that [3] a combination of synthetic magnetism and reservoir engineering can be used to implement nonreciprocal photon transmission and amplification in coupled photonic systems, and this has been confirmed by a recent experiment [4]. Based on a similar mechanism, many different schemes for nonreciprocal photon transport are proposed theoretically [5–8] and implemented experimentally [9–12]. Synthetic magnetism is an effective approach to achieve nonreciprocal transport of uncharged particles, such as photons [13–16] or phonons [17,18], for potential applications in simulating quantum many-body phenomena [19–25] and creating devices robust against disorder and backscattering [26–30]. Reservoir engineering [31] has been a significant subject for generating useful quantum behavior by specially designing the couplings between a system of interest and a structured dissipative environment, such as cooling mechanical harmonic oscillators [32]; synthesizing quantum harmonic oscillator states [33]; and generating state-dependent photon blockades [34], stable entanglement between two nanomechanical resonators [35,36], and squeezed states of nanomechanical resonators [37–39].
In this paper, we introduce the concept of nonreciprocity to investigate the transitions between different energy levels and generalize the general strategy for nonreciprocal photon transmission [3] to atomic systems to achieve a nonreciprocal transition between two nondegenerate energy levels. As a simple implementation, a cyclic three-level atom system, comprising two nondegenerate energy levels and one auxiliary energy level, is employed to show a nonreciprocal transition via a combination of synthetic magnetism and reservoir engineering.
In application, the atomic systems with nonreciprocal transitions allow one to generate nonreciprocal devices. In this paper, a single-photon nonreciprocal transporter is proposed in a system of two one-dimensional (1D) semi-infinite coupled-resonator waveguides (CRWs) connected by an atom based on the nonreciprocal transition effect. The nonreciprocal transition effect provides a new routine to design atom-mediated nonreciprocal devices in a variety of physical systems.
2. GENERAL METHOD FOR NONRECIPROCAL TRANSITION
Figure 1.(a) Schematic diagram for generating nonreciprocal transition: two nondegenerate energy levels
The second method is through coupling to a common engineered reservoir. A dissipative interaction between the two levels can be obtained by adiabatically eliminating the engineered reservoir. The effective Hamiltonian for the dissipative interaction can be written in a non-Hermitian form as with positive real strength . This dissipative version of interaction can be implemented by an auxiliary energy level, which is damping much faster than the two levels. The details of the realization will be shown in the next section.
Based on the two distinct methods, the total Hamiltonian for the interaction between the two levels is
3. NONRECIPROCAL TRANSITION WITH CYCLIC THREE-LEVEL TRANSITION
To make the method more concrete, we show how to implement nonreciprocal transition in a cyclic three-level atom as depicted in Fig. 1(b). We consider a cyclic three-level atom (, , and ) driven by three classical coherent fields (at rates , frequencies , phases , with ) that is described by a Hamiltonian (see Appendix A):
In order to show the nonreciprocal transition between levels and intuitively, we can derive an effective Hamiltonian by eliminating level (the engineered reservoir) adiabatically (see Appendix B) under the assumption that . Then an effective Hamiltonian only including levels and is given by
Figure 2.The transition probabilities
Figure 3.(a) The transition probabilities
4. SINGLE-PHOTON NONRECIPROCAL TRANSPORT
Figure 4.Schematic of two 1D semi-infinite CRWs connected by a three-level atom characterized by
The efficiency for nonreciprocity transport can be described by the scattering flow [40–43] for a single photon from CRW- to CRW- (). The detailed calculations of the scattering flow can be found in Appendix C. Nonreciprocal single-photon transport appears when , which implies that the scattering flow from CRW- to CRW- is not equal to that along the opposite direction.
Figure 5.(a) Scattering flows
Now let us discuss the width of the wavenumber for single-photon nonreciprocity; see Appendix E. We define the width of the wavenumber for single-photon nonreciprocity as the full width at half-maximum (FWHM) by setting for :
5. CONCLUSIONS AND DISCUSSION
In summary, we have shown theoretically that nonreciprocal transition can be observed between two nondegenerate energy levels. A general method has been presented to realize nonreciprocal transition between two nondegenerate energy levels based on a combination of synthetic magnetism and reservoir engineering. As a simple example, we explicitly show an implementation involving an auxiliary energy level, i.e., a cyclic three-level atom system. The generic method for realizing a nonreciprocal transition can be applied to design nonreciprocal phonon devices. A single-photon nonreciprocal transporter has been proposed by the nonreciprocal transition effect. The atom-mediated nonreciprocal devices based on the nonreciprocal transition are suitable for applications in building hybrid quantum networks.
To realize a nonreciprocal transition with a cyclic three-level atom, one ingredient is breaking the symmetry of the potential of the atom. The cyclic three-level transition has been proposed and observed in chiral molecules [44–51]. In addition, the potential of the atom can also be broken by applying an external magnetic field. We can consider a qubit circuit composed of a superconducting loop with three Josephson junctions [52,53] that encloses an applied magnetic flux ( is the superconducting flux quantum, where is Planck’s constant and is the reduced magnetic flux; is the charge quantity of one electron). When the reduced magnetic flux is a half-integer, the potential of the artificial atom is symmetric, and the interaction Hamiltonian has odd parity. However, when is not a half-integer, the symmetry of the potential is broken, and the interaction Hamiltonian does not have well-defined parity. In this case, transitions can occur between any two levels.
Alternatively, cyclic transitions in a three-level atom can be realized by a single nitrogen-vacancy (NV) center embedded in a mechanical resonator [54]. Three eigenstates ( and ) of the spin operator along the NV’s symmetry axis (i.e., ) are selected as a three-level atom [55,56]. The two degenerate levels can be split by applying an external magnetic field along . We can use microwave magnetic fields to drive the transitions between and ; the magnetic dipole-forbidden transition can be driven by a time-varying strain field through the mechanical resonator [57,58].
Besides the implementations in a cyclic three-level atom, the nonreciprocal transition can also be implemented in the other physical systems, such as a four-level atom system [59], two qubits in a one-dimensional waveguide [60], and even qubit arrays [61]. The nonreciprocal transition can be extended to explore lasing without inversion [62–65], quantum nonreciprocal physics [66–68], and topological phases [69] in a single multilevel atom or qubit array.
APPENDIX A: HAMILTONIAN FOR CYCLIC THREE-LEVEL ATOM
We consider a cyclic three-level atom (, , and ) driven by three classical coherent fields (at rates , phases , frequencies , with and ) that is described by a Hamiltonian given by
In the rotating frame respect to the operator , we have
APPENDIX B: ADIABATIC ELIMINATION
We will derive the effective Hamiltonian Eq.?(
APPENDIX C: SCATTERING FLOW
To study the nonreciprocal single-photon transport, we discuss the scattering of a single photon in the system with the total Hamiltonian in the rotating reference frame with respect to as
If a single photon with energy is incident from the infinity side of CRW-, the -type three-level atom will result in photon scattering between different CRWs or photon absorption by the dissipative of the atom. The general expressions of the probability amplitudes in the CRWs () are given by
Now let us derive the scattering amplitudes for single-photon scattering by the atom with nonreciprocal transition. By solving Eqs.?(
APPENDIX D: PERFECT SINGLE-PHOTON NONRECIPROCITY
In this section, we will derive the conditions for perfect nonreciprocal single-photon transport, i.e.,? and , analytically. For simplicity, we assume that the two semi-infinite CRWs have the same parameters, i.e.,?, , , and they are coupled to the atom resonantly with and . can be obtained by setting or . In this case, we have
APPENDIX E: MAXIMUM FULL WIDTH AT HALF-MAXIMUM
We will derive the maximum full width at half-maximum (FWHM) for perfect nonreciprocal single-photon transport. The half-maximum of the scattering flow is given by
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