Abstract

Stimulated emission and absorption are two fundamental processes of light–matter interaction, and the coefficients of the two processes should be equal. However, we will describe a generic method to realize the significant difference between the stimulated emission and absorption coefficients of two nondegenerate energy levels, which we refer to as a nonreciprocal transition. 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. Moreover, a single-photon nonreciprocal transporter is proposed using two one-dimensional semi-infinite coupled-resonator waveguides connected by an atom with nonreciprocal transition effect. Our work opens up a route to design atom-mediated nonreciprocal devices in a wide range of physical systems.

© 2021 Chinese Laser Press

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 [58] and implemented experimentally [912]. Synthetic magnetism is an effective approach to achieve nonreciprocal transport of uncharged particles, such as photons [1316] or phonons [17,18], for potential applications in simulating quantum many-body phenomena [1925] and creating devices robust against disorder and backscattering [2630]. 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 [3739].

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

A general model of two nondegenerate energy levels |a and |b for nonreciprocal transition is shown in Fig. 1(a). The effective couplings between the two levels come from two different physical interactions. The first method is through a coherent interaction Hcoh, which is described by Hcoh=Ω|ab|+Ω*|ba| with complex coupling strength Ω. The simplest implementation of the coherent interaction Hcoh is driving the two levels with a coherent laser field.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram for generating nonreciprocal transition: two nondegenerate energy levels |a and |b are coupled to one another via a coherent interaction Hcoh, and they are also coupled to the same engineered reservoir. (b) Schematic diagram for implementation of a nonreciprocal transition in a cyclic three-level atom (characterized by |a, |b, and |c). A laser field (ΩabeiΦ) is applied to drive the direct transition between the two levels |a and |b, and they are also coupled indirectly by the auxiliary level |c through two laser fields (Ωca and Ωcb), where the decay of level |c is much faster than that of the other two levels, i.e., γcmax{γa,γb}, so the auxiliary level |c serves as a engineered reservoir.

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The second method is through coupling to a common engineered reservoir. A dissipative interaction Hdis between the two levels can be obtained by adiabatically eliminating the engineered reservoir. The effective Hamiltonian for the dissipative interaction Hdis can be written in a non-Hermitian form as Hdis=iγ(|ab|+|ba|) with positive real strength γ. 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

Hcoh+dis=(Ωiγ)|ab|+(Ω*iγ)|ba|.
When Ω=iγ and Ω*Ω, there is only transition |a|b but |b|a. Instead, when Ω*=iγ and Ω*Ω, there is only transition |b|a but |a|b.

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 (|a, |b, and |c) driven by three classical coherent fields (at rates Ωij, frequencies νij, phases ϕij, with i,j=a,b,c) that is described by a Hamiltonian (see Appendix A):

H=(Δabiγa)|aa|iγb|bb|+(Δcbiγc)|cc|+(ΩabeiΦ|ab|+Ωcb|cb|+Ωca|ca|+H.c.),
where Δij=ωijνij (i,j=a,b,c), ωij is the frequency difference between levels |i and |j; γi (i=a,b,c) are the decay rates. We assume that νab=νcbνca, so the detuning Δab=ΔcbΔca. The synthetic magnetic flux Φϕabϕcb+ϕca is the total phase of the three driving fields around the cyclic three-level atom and independent of the local redefinition of states |i. The time-reversal symmetry of the system is broken when we choose the phase Φnπ (n is an integer) even without spontaneous emissions (γa=γb=γc=0), and this is one of the key ingredients for nonreciprocal transition. In addition, we assume that the decays satisfy the conditions min{ωca,ωcb}γcmax{Ωca,Ωcb,γa,γb}, so that level |c serves as an engineered reservoir.

In order to show the nonreciprocal transition between levels |a and |b intuitively, we can derive an effective Hamiltonian by eliminating level |c (the engineered reservoir) adiabatically (see Appendix B) under the assumption that γcmax{γa,γb}. Then an effective Hamiltonian only including levels |a and |b is given by

Heff=(ΔaiΓa)|aa|+(ΔbiΓb)|bb|+Jab|ab|+Jba|ba|,
with the detunings ΔaΔabΩca2Δcb/(γc2+Δcb2) and ΔbΩcb2Δcb/(γc2+Δcb2), effective decay rates Γaγa+Ωca2γc/(γc2+Δcb2) and Γbγb+Ωcb2γc/(γc2+Δcb2), and effective coupling coefficients
JabΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2,
JbaΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2.
The effective coupling coefficients Jab and Jba include two terms: the first term comes from the coherent driving field and is dependent on the synthetic magnetic flux Φ, and the second term is induced by the auxiliary level |c. Under the resonant condition Δab=Δca=Δcb=0, the second term becomes purely imaginary, i.e., iΩcaΩcb/γc, and the effective Hamiltonian is the same as Eq. (2). Under the resonant conditions, the perfect nonreciprocal transition, i.e., Jab=0 and Jba0 (or Jba=0 and Jab0), is obtained when Φ=π/2 (or Φ=π/2) with Ωab=ΩcaΩcb/γc. More generally, we have |Jab|<|Jba| for 0<Φ<π and |Jab|>|Jba| for π<Φ<0.

To understand the nonreciprocal transition further, we take a view on the dynamical behavior of the transition probabilities between levels |a and |b. The time-evolution operator for the Hamiltonian H is given by U(t)=exp(iHt), and the probabilities for transitions |a|b and |b|a can be defined by Tba(t)|b|U(t)|a|2 and Tab(t)|a|U(t)|b|2, respectively. They are plotted as functions of time Ωabt in Figs. 2(a)–2(c). We can see that the transition probabilities are time dependent and the nonreciprocal behaviors emerge after a short time (1/γc). It is clear that Tab(t)Tba(t) for ϕ=π/2, Tab(t)Tba(t) for ϕ=π/2, and Tab(t)=Tba(t) for ϕ=0. The isolation for the nonreciprocal transition defined by I(t)Tab(t)/Tba(t) is plotted as a function of time Ωabt in Fig. 2(d). One can achieve I(t)>106 for Φ=π/2 and I(t)<106 for Φ=π/2 at time Ωabt=1.

 figure: Fig. 2.

Fig. 2. The transition probabilities Tab(t) and Tba(t) are plotted as functions of the time Ωabt for: (a) Φ=π/2, (b) Φ=0, and (c) Φ=π/2. (d) The isolation I(t) is plotted as a function of time Ωabt for Φ=π/2,0,π/2. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.

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Furthermore, the dependence of the transition probabilities Tab(t) and Tba(t) on the synthetic magnetic flux Φ is shown in Fig. 3(a). At time Ωabt=1, we have Tab(t)>Tba(t) for synthetic magnetic flux 0<Φ<π; in the contrast, we have Tab(t)<Tba(t) for synthetic magnetic flux π<Φ<0. As shown in Fig. 3(b), under the resonant condition Δcb=Δca=Δab=0, the optimal isolation I(t) is obtained with synthetic magnetic flux Φ=±π/2, which is consistent with the analytical predictions.

 figure: Fig. 3.

Fig. 3. (a) The transition probabilities Tab(t) and Tba(t) and (b) the isolation I(t) are plotted as functions of the synthetic magnetic flux Φ at time Ωabt=1. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.

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4. SINGLE-PHOTON NONRECIPROCAL TRANSPORT

As an important application, we will discuss how to realize a single-photon nonreciprocal transport between two 1D semi-infinite CRWs by the nonreciprocal transition effect. We assume that two 1D semi-infinite CRWs, with creation operators aj and bj and frequencies ωw,a and ωw,b for the jth cavity modes, are coupled by a -type three-level atom (|a, |b, and |g) with nonreciprocal transition |a|b as shown in Fig. 4. Here, ga (gb) is the coupling strength between CRW-a (CRW-b) and the transition |a|g (|b|g) with frequency ωag (ωbg). The system can be described by the total Hamiltonian under the rotating wave approximation Htot=l=a,bHl+H˜eff+Hint. Here, in the rotating reference frame with respect to Hrot=ωag(jajaj+|aa|)+ωbg(jbjbj+|bb|), the Hamiltonian Hl for the CRW-l is given by

Hl=Δlj=0+ljljξlj=0+(ljlj+1+H.c.),
with homogeneous intercavity coupling constants ξl and cavity-atom detunings Δl=ωw,lωlg (l=a,b); the effective Hamiltonian H˜eff for the -type three-level atom with nonreciprocal transition |a|b is obtained from Eq. (3) with Δa=Δb=0 as
H˜eff=Jab|ab|+Jba|ba|iΓa|aa|iΓb|bb|,
and the interaction Hamiltonian Hint between the zeroth cavity modes and the three-level atom is described by
Hint=gaa0|ag|+gbb0|bg|+gaa0|ga|+gbb0|gb|.
 figure: Fig. 4.

Fig. 4. Schematic of two 1D semi-infinite CRWs connected by a three-level atom characterized by |a, |b, and |g. CRW-a (CRW-b) couples to the three-level atom through the transition |a|g (|b|g) with strength ga (gb).

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The efficiency for nonreciprocity transport can be described by the scattering flow [4043] Ill for a single photon from CRW-l to CRW-l (l=a,b). The detailed calculations of the scattering flow Ill can be found in Appendix C. Nonreciprocal single-photon transport appears when IbaIab, which implies that the scattering flow from CRW-a to CRW-b is not equal to that along the opposite direction.

First of all, let us find the optimal conditions for perfect single-photon nonreciprocity, i.e., Iab=0 and Iba=1, analytically. For simplicity, we assume that the two semi-infinite CRWs have the same parameters, i.e., ξξa=ξb, kka=kb, gga=gb, and they are coupled to the atom resonantly with Δa=Δb=0 and ΓΓa=Γb. Then, Iab=0 can be obtained by setting Jab=0. Through a detailed derivation (see Appendix D), the condition for Iba=1 is |sink|=1, i.e., k=π/2 (0<k<π), in the case that |Jba|=2Γ and g2=Γξ. This fits the numerical simulation very well as shown in Figs. 5(a) and 5(b). Luckily, the parameters Jab and Jba as shown in Eqs. (4) and (5) depend on the parameters of the external driving fields, and the optimal conditions for perfect single-photon nonreciprocity can be achieved simultaneously by tuning the strengths and frequencies of the external driving fields.

 figure: Fig. 5.

Fig. 5. (a) Scattering flows Iab (black solid curve) and Iba (red dashed curve), (b) Iaa (black solid curve) and Ibb (red dashed curve), are plotted as functions of the wavenumber k/π for ξ/Γ=0.1. (c) Scattering flow Iab is plotted as a function of the wavenumber k/π for different ξ/Γ. (d) The width of the wavenumber Δk for single-photon nonreciprocity is plotted as a function of log10(ξ/Γ) given in Eq. (9). The other parameters are Jba=2Γ, Jab=0, ξ=Γ, Δa=Δb=0, g2=Γξ, ϕ=π/2.

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Now let us discuss the width of the wavenumber for single-photon nonreciprocity; see Appendix E. We define the width of the wavenumber Δk for single-photon nonreciprocity as the full width at half-maximum (FWHM) by setting Iba=1/2 for k=khalf[0,π/2):

Δkπ2khalf.
Under the conditions |Jba|=2Γ and g2=Γξ, there is a maximum FWHM for single-photon nonreciprocity at ξ=Γ/2, and the maximum FWHM Δkmax0.81π is obtained with khalf=arcsin(221222) in excellent agreement with Figs. 5(c) and 5(d).

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 [4451]. 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 Φe=fΦ0 (Φ0h/2e is the superconducting flux quantum, where h is Planck’s constant and fΦe/Φ0 is the reduced magnetic flux; e is the charge quantity of one electron). When the reduced magnetic flux f is a half-integer, the potential of the artificial atom is symmetric, and the interaction Hamiltonian has odd parity. However, when f 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 (|0 and |±1) of the spin operator along the NV’s symmetry axis z (i.e., Sz|m=m|m) are selected as a three-level atom [55,56]. The two degenerate levels |±1 can be split by applying an external magnetic field along z. We can use microwave magnetic fields to drive the transitions between |0 and |±1; the magnetic dipole-forbidden transition |+1|1 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 [6265], quantum nonreciprocal physics [6668], 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 (|a, |b, and |c) driven by three classical coherent fields (at rates Ωij, phases ϕij, frequencies νij, with i,j=a,b,c and νcb=νab+νca) that is described by a Hamiltonian given by

H˜=(ωabiγa)|aa|iγb|bb|+(ωcbiγc)|cc|+(Ωabeiϕabeiνabt|ab|+Ωcbeiϕcbeiνcbt|cb|+Ωcaeiϕcaeiνcat|ca|+H.c.),
where ωij is the frequency difference between levels |i and |j, and the three levels can decay to the other levels with the decay rates γi (i=a,b,c).

In the rotating frame respect to the operator W=ei(νab|aa|+νcb|cc|)t, we have

H=WH˜W+idWdtW=(Δabiγa)|aa|iγb|bb|+(Δcbiγc)|cc|+Ωabeiϕab|ab|+Ωcbeiϕcb|cb|+Ωcaeiϕca|ca|+H.c.,
with the detuning Δijωijνij (i,j=a,b,c). By local redefinition of the eigenstates, i.e., eiϕcbb|b| and eiϕca|a|a, the Hamiltonian can be rewritten as Eq. (2) in the main text with the synthetic magnetic flux Φϕabϕcb+ϕca.

APPENDIX B: ADIABATIC ELIMINATION

We will derive the effective Hamiltonian Eq. (3) by eliminating the level |c (the engineered reservoir) adiabatically. The state vector for these three levels at time t can be written as

|ψ=A(t)|a+B(t)|b+C(t)|c.
The coefficients |A(t)|2, |B(t)|2, and |C(t)|2 denote occupying probabilities in states |a, |b, and |c, respectively. Then the dynamical behaviors for the coefficients can be obtained by the Schrödinger equation, i.e., i|ψ=H|ψ, given by
A˙(t)=(iΔabγa)A(t)iΩabeiΦB(t)iΩcaC(t),
B˙(t)=γbB(t)iΩabeiΦA(t)iΩcbC(t),
C˙(t)=(iΔcbγc)C(t)iΩcaA(t)iΩcbB(t).
Under the assumption that the decay of the state |c is much faster than decay of the states |a and |b with the conditions min{ωca,ωcb}γcmax{Ωca,Ωcb,γa,γb}, we can adiabatically eliminate level |c with C˙(t)=0 as
C(t)=iΩcaγc+iΔcbA(t)iΩcbγc+iΔcbB(t).
By substituting Eq. (B5) into Eqs. (B2) and (B3), then the dynamical equations of A(t) and B(t) become
A˙(t)=[i(ΔabΩca2Δcbγc2+Δcb2)+(γa+Ωca2γcγc2+Δcb2)]A(t)[iΩabeiΦ+ΩcaΩcb(γciΔcb)γc2+Δcb2]B(t),
B˙(t)=[iΩcb2Δcbγc2+Δcb2+(γb+Ωcb2γcγc2+Δcb2)]B(t)[iΩabeiΦ+ΩcaΩcb(γciΔcb)γc2+Δcb2]A(t).
Physically, the dynamic equations in Eqs. (B6) and (B7) correspond to the Schrödinger evolution of the effective Hamiltonian Eq. (3) in the main text.

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 Hrot as

Htot=l=a,bHl+H˜eff+Hint,
where the Hamiltonians Hl, H˜eff, and Hint are given in Eqs. (6)–(8) in the main text. As the total number of photons in the system is a conserved quantity (without dissipation), we consider the stationary eigenstate of a single photon in the system as
|E=j=0+[ua(j)aj+ub(j)bj]|g,0+A|a,0+B|b,0,
where |0 indicates the vacuum state of the 1D semi-infinite CRWs, ul(j) denotes the probability amplitude in the state with a single photon in the jth cavity of the CRW-l, and A (B) denotes the probability amplitude in the atom state |a (|b). Substituting the stationary eigenstate |E in Eq. (C2) and the total Hamiltonian Htot into the eigenequation Htot|E=E|E, we can obtain the coupled equations for the probability amplitudes as
Δaua(0)ξaua(1)+gaA=Eua(0),
Δbub(0)ξbub(1)+gbB=Eub(0),
iΓaA+gaua(0)+JabB=EA,
iΓbB+gbub(0)+JbaA=EB,
Δlul(j)ξl[ul(j+1)+ul(j1)]=Eul(j),
with j>0 and l=a,b.

If a single photon with energy E is incident from the infinity side of CRW-l, 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 (j0) are given by

ul(j)=eiklj+slleiklj,
ul(j)=slleiklj,
where sll denotes the single-photon scattering amplitude from CRW-l to CRW-l (l,l=a,b). Substituting Eq. (C8) or Eq. (C9) into Eq. (C7), the eigenvalue of the semi-infinite CRW-l in the rotating reference frame is given by [40]
E=Δl2ξlcoskl,0<kl<π,
where kl is the wavenumber of the single photon in the CRW-l. Without loss of generality, we assume that ξl>0 and 0<kl<π for semi-infinite CRW-l.

Now let us derive the scattering amplitudes for single-photon scattering by the atom with nonreciprocal transition. By solving Eqs. (C5) and (C6), the coefficients A and B can be expressed by

A=(E+iΓb)gaua(0)+Jabgbub(0)(E+iΓa)(E+iΓb)JbaJab,
B=(E+iΓa)gbub(0)+Jbagaua(0)(E+iΓa)(E+iΓb)JbaJab.
Substituting A and B into Eqs. (C3) and (C4), we have
(ΔaE+Δ¯a)ua(0)+Jabub(0)=ξaua(1),
(ΔbE+Δ¯b)ub(0)+Jbaua(0)=ξbub(1),
with the effective coupling strengths Jll and frequency shifts Δ¯l induced by the -type three-level atom defined by
Jab=Jabgagb(E+iΓa)(E+iΓb)JbaJab,
Jba=Jbagagb(E+iΓa)(E+iΓb)JbaJab,
Δ¯a=(E+iΓb)ga2(E+iΓa)(E+iΓb)JbaJab,
Δ¯b=(E+iΓa)gb2(E+iΓa)(E+iΓb)JbaJab.
When a single photon is input from CRW-a, we have ua(j)=eikaj+saaeikaj and ub(j)=sbaeikbj, and the scattering amplitudes saa and sba satisfy the following equations:
(ξaeika+Δ¯a)saa+Jabsba=ξaeikaΔ¯a,
Jbasaa+(ξbeikb+Δ¯b)sba=Jba.
Similarly, when a single photon is input from CRW-b, we have ub(j)=eikbj+sbbeikbj and ua(j)=sabeikaj, and the scattering amplitudes sab and sbb satisfy the following equations:
(ξaeika+Δ¯a)sab+Jabsbb=Jab,
Jbasab+(ξbeikb+Δ¯b)sbb=ξbeikbΔ¯b.
Equations (C19)–(C22) can be expressed concisely in matrix form as
LS=R,
with the scattering matrix
S=(saasabsbasbb)
and coefficient matrices
L=(ξaeika+Δ¯aJabJbaξbeikb+Δ¯b),
R=(ξaeika+Δ¯aJabJbaξbeikb+Δ¯b).
The solutions of Eq. (C23) are given by
saa=JabJba(ξaeika+Δ¯a)(ξbeikb+Δ¯b)(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sba=2iξaJbasinka(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sab=2iξbJabsinkb(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sbb=JabJba(ξaeika+Δ¯a)(ξbeikb+Δ¯b)(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba.
To quantify the efficiency for nonreciprocity transport, we define the scattering flow [4143] of a single photon from CRW-l to CRW-l as
Ill|sll|2ξlsinklξlsinkl,
where ξlsinkl (ξlsinkl) is the group velocity in the CRW-l (CRW-l).

APPENDIX D: PERFECT SINGLE-PHOTON NONRECIPROCITY

In this section, we will derive the conditions for perfect nonreciprocal single-photon transport, i.e., Iab=0 and Iba=1, analytically. For simplicity, we assume that the two semi-infinite CRWs have the same parameters, i.e., ξξa=ξb, kka=kb, gga=gb, and they are coupled to the atom resonantly with Δa=Δb=0 and ΓΓa=Γb. Iab=0 can be obtained by setting Jab=0 or Jab=0. In this case, we have

Iba=|sba|2=|2Jbag2ξsink[ξeik(2ξcosk+iΓ)+g2]2|2.
So the condition for Iba=1 is
|sink|=(|Jba|Γ)g2±Θ4(g2ξ2)ξ,
with
Θ=(|Jba|Γ)2g44(g2ξ2)[4(ξ2g2)ξ2+Γ2ξ2+g4].
As a simple example, the maximum scattering flow Iba=1 can be obtained at the maximum group velocity |sink|=1, with
|Jba|=(g2+Γξ)22g2ξ.
Furthermore, if g2=Γξ, then we have
|Jba|=2Γ.

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 Iba is given by

Iba=|sba|2=|2Jbag2ξsinkhalf[ξeik(2ξcoskhalf+iΓ)+g2]2|2=12.
Under the conditions that |Jba|=2Γ and g2=Γξ, we have
2(Γξ)ξ|sinkhalf|2+(122)Γ2|sinkhalf|+2(ξΓ)ξ+Γ2=0.
Defining ηξ/Γ and ζ4(1η)η, Eq. (E2) can be rewritten as
|sinkhalf|=(122)±(122)2ζ(2ζ)ζ.
The condition for maximum width Δkmax is
ddη|sinkhalf|=d|sinkhalf|dζdζdη=0,
which is satisfied with
η=12ξ=Γ2.
That is to say, the maximum width Δkmax is obtained at ξ=Γ/2 with
|sinkhalf|=221222,
and the maximum FWHM Δkmax is
Δkmaxπ2arcsin(221222)0.81π.

Funding

National Natural Science Foundation of China (12064010, 11904013, 11847165, 11775190); Natural Science Foundation of Jiangxi Province (20192ACB21002); National Basic Research Program of China (973 Program) (2014CB921401); Tsinghua University Initiative Scientific Research Program; Tsinghua National Laboratory for Information Science and Technology (TNList) Cross-discipline Foundation.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. A. Einstein, “On the quantum theory of radiation,” Phys. Z. 18, 121 (1917).

2. M. Scully and M. Zubairy, Quantum Optics (Cambridge University, 1997).

3. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015). [CrossRef]  

4. K. Fang, J. Luo, A. Metelmann, M. Matheny, F. Marquardt, A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017). [CrossRef]  

5. X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015). [CrossRef]  

6. X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016). [CrossRef]  

7. A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (2017). [CrossRef]  

8. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017). [CrossRef]  

9. F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016). [CrossRef]  

10. G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017). [CrossRef]  

11. N. Bernier, L. D. Toth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017). [CrossRef]  

12. S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. Dieterle, O. Painter, and J. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017). [CrossRef]  

13. J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82, 043811 (2010). [CrossRef]  

14. R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011). [CrossRef]  

15. Y.-P. Wang, W. Wei, Z.-Y. Xue, W. L. Yang, Y. Hu, and Y. Wu, “Realizing and characterizing chiral photon flow in a circuit quantum electrodynamics necklace,” Sci. Rep. 5, 8352 (2015). [CrossRef]  

16. F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19, 123039 (2017). [CrossRef]  

17. S. J. M. Habraken, K. Stannigel, M. D. Lukin, P. Zoller, and P. Rabl, “Continuous mode cooling and phonon routers for phononic quantum networks,” New J. Phys. 14, 115004 (2012). [CrossRef]  

18. A. Seif, W. DeGottardi, K. Esfarjani, and M. Hafezi, “Thermal management and non-reciprocal control of phonon flow via optomechanics,” Nat. Commun. 9, 1207 (2017). [CrossRef]  

19. M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013). [CrossRef]  

20. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015). [CrossRef]  

21. V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015). [CrossRef]  

22. V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016). [CrossRef]  

23. V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2015). [CrossRef]  

24. M. Minkov and V. Savona, “Haldane quantum hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016). [CrossRef]  

25. C. Brendel, V. Peano, O. Painter, and F. Marquardt, “Snowflake phononic topological insulator at the nanoscale,” Phys. Rev. B 97, 020102 (2018). [CrossRef]  

26. M. Hafezi, E. Demler, M. Lukin, and J. Taylor, “Robust optical delay lines via topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]  

27. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012). [CrossRef]  

28. L. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014). [CrossRef]  

29. K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015). [CrossRef]  

30. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015). [CrossRef]  

31. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum reservoir engineering with laser cooled trapped ions,” Phys. Rev. Lett. 77, 4728–4731 (1996). [CrossRef]  

32. X. Xu, T. Purdy, and J. M. Taylor, “Cooling a harmonic oscillator by optomechanical modification of its bath,” Phys. Rev. Lett. 118, 223602 (2017). [CrossRef]  

33. D. Kienzler, H.-Y. Lo, B. Keitch, L. Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. Home, “Quantum harmonic oscillator state synthesis by reservoir engineering,” Science 347, 53–56 (2014). [CrossRef]  

34. A. Miranowicz, J. C. V. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90, 033831 (2014). [CrossRef]  

35. C.-J. Yang, J.-H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92, 062311 (2015). [CrossRef]  

36. X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017). [CrossRef]  

37. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004). [CrossRef]  

38. M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014). [CrossRef]  

39. C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018). [CrossRef]  

40. L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013). [CrossRef]  

41. Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014). [CrossRef]  

42. X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017). [CrossRef]  

43. X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides,” Phys. Rev. A 96, 053853 (2017). [CrossRef]  

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48. D. Patterson and J. M. Doyle, “Sensitive chiral analysis via microwave three-wave mixing,” Phys. Rev. Lett. 111, 023008 (2013). [CrossRef]  

49. D. Patterson, M. Schnell, and J. Doyle, “Enantiomer-specific detection of chiral molecules via microwave spectroscopy,” Nature 497, 475–477 (2013). [CrossRef]  

50. S. Eibenberger, J. Doyle, and D. Patterson, “Enantiomer-specific state transfer of chiral molecules,” Phys. Rev. Lett. 118, 123002 (2017). [CrossRef]  

51. C. Ye, Q. Zhang, and Y. Li, “Real single-loop cyclic three-level configuration of chiral molecules,” Phys. Rev. A 98, 063401 (2018). [CrossRef]  

52. Y.-X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005). [CrossRef]  

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56. V. Dobrovitski, G. Fuchs, A. Falk, C. Santori, and D. Awschalom, “Quantum control over single spins in diamond,” Annu. Rev. Condens. Matter Phys. 4, 23–50 (2013). [CrossRef]  

57. E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, “Mechanical spin control of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 111, 227602 (2013). [CrossRef]  

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60. A. Rosario Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121, 123601 (2018). [CrossRef]  

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References

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  1. A. Einstein, “On the quantum theory of radiation,” Phys. Z. 18, 121 (1917).
  2. M. Scully and M. Zubairy, Quantum Optics (Cambridge University, 1997).
  3. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
    [Crossref]
  4. K. Fang, J. Luo, A. Metelmann, M. Matheny, F. Marquardt, A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
    [Crossref]
  5. X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
    [Crossref]
  6. X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
    [Crossref]
  7. A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (2017).
    [Crossref]
  8. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
    [Crossref]
  9. F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
    [Crossref]
  10. G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).
    [Crossref]
  11. N. Bernier, L. D. Toth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
    [Crossref]
  12. S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. Dieterle, O. Painter, and J. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
    [Crossref]
  13. J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82, 043811 (2010).
    [Crossref]
  14. R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
    [Crossref]
  15. Y.-P. Wang, W. Wei, Z.-Y. Xue, W. L. Yang, Y. Hu, and Y. Wu, “Realizing and characterizing chiral photon flow in a circuit quantum electrodynamics necklace,” Sci. Rep. 5, 8352 (2015).
    [Crossref]
  16. F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19, 123039 (2017).
    [Crossref]
  17. S. J. M. Habraken, K. Stannigel, M. D. Lukin, P. Zoller, and P. Rabl, “Continuous mode cooling and phonon routers for phononic quantum networks,” New J. Phys. 14, 115004 (2012).
    [Crossref]
  18. A. Seif, W. DeGottardi, K. Esfarjani, and M. Hafezi, “Thermal management and non-reciprocal control of phonon flow via optomechanics,” Nat. Commun. 9, 1207 (2017).
    [Crossref]
  19. M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
    [Crossref]
  20. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
    [Crossref]
  21. V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).
    [Crossref]
  22. V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).
    [Crossref]
  23. V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2015).
    [Crossref]
  24. M. Minkov and V. Savona, “Haldane quantum hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016).
    [Crossref]
  25. C. Brendel, V. Peano, O. Painter, and F. Marquardt, “Snowflake phononic topological insulator at the nanoscale,” Phys. Rev. B 97, 020102 (2018).
    [Crossref]
  26. M. Hafezi, E. Demler, M. Lukin, and J. Taylor, “Robust optical delay lines via topological protection,” Nat. Phys. 7, 907–912 (2011).
    [Crossref]
  27. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
    [Crossref]
  28. L. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014).
    [Crossref]
  29. K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).
    [Crossref]
  30. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
    [Crossref]
  31. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum reservoir engineering with laser cooled trapped ions,” Phys. Rev. Lett. 77, 4728–4731 (1996).
    [Crossref]
  32. X. Xu, T. Purdy, and J. M. Taylor, “Cooling a harmonic oscillator by optomechanical modification of its bath,” Phys. Rev. Lett. 118, 223602 (2017).
    [Crossref]
  33. D. Kienzler, H.-Y. Lo, B. Keitch, L. Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. Home, “Quantum harmonic oscillator state synthesis by reservoir engineering,” Science 347, 53–56 (2014).
    [Crossref]
  34. A. Miranowicz, J. C. V. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90, 033831 (2014).
    [Crossref]
  35. C.-J. Yang, J.-H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92, 062311 (2015).
    [Crossref]
  36. X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
    [Crossref]
  37. P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
    [Crossref]
  38. M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
    [Crossref]
  39. C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018).
    [Crossref]
  40. L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
    [Crossref]
  41. Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
    [Crossref]
  42. X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
    [Crossref]
  43. X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides,” Phys. Rev. A 96, 053853 (2017).
    [Crossref]
  44. P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
    [Crossref]
  45. P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
    [Crossref]
  46. Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern-Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
    [Crossref]
  47. W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
    [Crossref]
  48. D. Patterson and J. M. Doyle, “Sensitive chiral analysis via microwave three-wave mixing,” Phys. Rev. Lett. 111, 023008 (2013).
    [Crossref]
  49. D. Patterson, M. Schnell, and J. Doyle, “Enantiomer-specific detection of chiral molecules via microwave spectroscopy,” Nature 497, 475–477 (2013).
    [Crossref]
  50. S. Eibenberger, J. Doyle, and D. Patterson, “Enantiomer-specific state transfer of chiral molecules,” Phys. Rev. Lett. 118, 123002 (2017).
    [Crossref]
  51. C. Ye, Q. Zhang, and Y. Li, “Real single-loop cyclic three-level configuration of chiral molecules,” Phys. Rev. A 98, 063401 (2018).
    [Crossref]
  52. Y.-X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
    [Crossref]
  53. J. Mooij, T. Orlando, L. Levitov, L. Tian, C. van der Wal, and S. Lloyd, “Josephson persistent-current qubit,” Science 285, 1036 (1999).
    [Crossref]
  54. A. Barfuss, J. Kölbl, L. Thiel, J. Teissier, M. Kasperczyk, and P. Maletinsky, “Phase-controlled coherent dynamics of a single spin under closed-contour interaction,” Nat. Phys. 14, 1087–1091 (2018).
    [Crossref]
  55. J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys. 13, 025025 (2011).
    [Crossref]
  56. V. Dobrovitski, G. Fuchs, A. Falk, C. Santori, and D. Awschalom, “Quantum control over single spins in diamond,” Annu. Rev. Condens. Matter Phys. 4, 23–50 (2013).
    [Crossref]
  57. E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, “Mechanical spin control of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 111, 227602 (2013).
    [Crossref]
  58. A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, “Strong mechanical driving of a single electron spin,” Nat. Phys. 11, 820–824 (2015).
    [Crossref]
  59. F. Ripka, H. Kübler, R. Löw, and T. Pfau, “A room-temperature single-photon source based on strongly interacting Rydberg atoms,” Science 362, 446–449 (2018).
    [Crossref]
  60. A. Rosario Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121, 123601 (2018).
    [Crossref]
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2020 (1)

2019 (1)

2018 (7)

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121, 153601 (2018).
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F. Ripka, H. Kübler, R. Löw, and T. Pfau, “A room-temperature single-photon source based on strongly interacting Rydberg atoms,” Science 362, 446–449 (2018).
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A. Rosario Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121, 123601 (2018).
[Crossref]

C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018).
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C. Ye, Q. Zhang, and Y. Li, “Real single-loop cyclic three-level configuration of chiral molecules,” Phys. Rev. A 98, 063401 (2018).
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A. Barfuss, J. Kölbl, L. Thiel, J. Teissier, M. Kasperczyk, and P. Maletinsky, “Phase-controlled coherent dynamics of a single spin under closed-contour interaction,” Nat. Phys. 14, 1087–1091 (2018).
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C. Brendel, V. Peano, O. Painter, and F. Marquardt, “Snowflake phononic topological insulator at the nanoscale,” Phys. Rev. B 97, 020102 (2018).
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2017 (13)

X. Xu, T. Purdy, and J. M. Taylor, “Cooling a harmonic oscillator by optomechanical modification of its bath,” Phys. Rev. Lett. 118, 223602 (2017).
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X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
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K. Fang, J. Luo, A. Metelmann, M. Matheny, F. Marquardt, A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
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G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).
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N. Bernier, L. D. Toth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
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S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. Dieterle, O. Painter, and J. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
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A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (2017).
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L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
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F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19, 123039 (2017).
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A. Seif, W. DeGottardi, K. Esfarjani, and M. Hafezi, “Thermal management and non-reciprocal control of phonon flow via optomechanics,” Nat. Commun. 9, 1207 (2017).
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X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
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X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides,” Phys. Rev. A 96, 053853 (2017).
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S. Eibenberger, J. Doyle, and D. Patterson, “Enantiomer-specific state transfer of chiral molecules,” Phys. Rev. Lett. 118, 123002 (2017).
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2016 (4)

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
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X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
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M. Minkov and V. Savona, “Haldane quantum hall effect for light in a dynamically modulated array of resonators,” Optica 3, 200–206 (2016).
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V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).
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2015 (10)

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2015).
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M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
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V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).
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Y.-P. Wang, W. Wei, Z.-Y. Xue, W. L. Yang, Y. Hu, and Y. Wu, “Realizing and characterizing chiral photon flow in a circuit quantum electrodynamics necklace,” Sci. Rep. 5, 8352 (2015).
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K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).
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R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
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X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
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A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
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A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, “Strong mechanical driving of a single electron spin,” Nat. Phys. 11, 820–824 (2015).
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C.-J. Yang, J.-H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92, 062311 (2015).
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2014 (5)

M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
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Z. H. Wang, L. Zhou, Y. Li, and C. P. Sun, “Controllable single-photon frequency converter via a one-dimensional waveguide,” Phys. Rev. A 89, 053813 (2014).
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L. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014).
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D. Kienzler, H.-Y. Lo, B. Keitch, L. Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. Home, “Quantum harmonic oscillator state synthesis by reservoir engineering,” Science 347, 53–56 (2014).
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A. Miranowicz, J. C. V. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90, 033831 (2014).
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2013 (6)

M. Rechtsman, J. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
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L. Zhou, L.-P. Yang, Y. Li, and C. P. Sun, “Quantum routing of single photons with a cyclic three-level system,” Phys. Rev. Lett. 111, 103604 (2013).
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D. Patterson and J. M. Doyle, “Sensitive chiral analysis via microwave three-wave mixing,” Phys. Rev. Lett. 111, 023008 (2013).
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D. Patterson, M. Schnell, and J. Doyle, “Enantiomer-specific detection of chiral molecules via microwave spectroscopy,” Nature 497, 475–477 (2013).
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V. Dobrovitski, G. Fuchs, A. Falk, C. Santori, and D. Awschalom, “Quantum control over single spins in diamond,” Annu. Rev. Condens. Matter Phys. 4, 23–50 (2013).
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E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, “Mechanical spin control of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 111, 227602 (2013).
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2012 (2)

S. J. M. Habraken, K. Stannigel, M. D. Lukin, P. Zoller, and P. Rabl, “Continuous mode cooling and phonon routers for phononic quantum networks,” New J. Phys. 14, 115004 (2012).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
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2011 (5)

M. Hafezi, E. Demler, M. Lukin, and J. Taylor, “Robust optical delay lines via topological protection,” Nat. Phys. 7, 907–912 (2011).
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R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
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M. Johnson, M. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. Berkley, J. Johansson, P. Bunyk, E. Chapple, C. Enderud, J. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
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W. Z. Jia and L. F. Wei, “Probing molecular chirality by coherent optical absorption spectra,” Phys. Rev. A 84, 053849 (2011).
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J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys. 13, 025025 (2011).
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2010 (3)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
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W. Z. Jia and L. F. Wei, “Gains without inversion in quantum systems with broken parities,” Phys. Rev. A 82, 013808 (2010).
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J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A 82, 043811 (2010).
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2007 (1)

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern-Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
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2005 (1)

Y.-X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit,” Phys. Rev. Lett. 95, 087001 (2005).
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2004 (1)

P. Rabl, A. Shnirman, and P. Zoller, “Generation of squeezed states of nanomechanical resonators by reservoir engineering,” Phys. Rev. B 70, 205304 (2004).
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2003 (1)

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
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2001 (1)

P. Král and M. Shapiro, “Cyclic population transfer in quantum systems with broken symmetry,” Phys. Rev. Lett. 87, 183002 (2001).
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1999 (1)

J. Mooij, T. Orlando, L. Levitov, L. Tian, C. van der Wal, and S. Lloyd, “Josephson persistent-current qubit,” Science 285, 1036 (1999).
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1996 (1)

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum reservoir engineering with laser cooled trapped ions,” Phys. Rev. Lett. 77, 4728–4731 (1996).
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1992 (1)

O. Kocharovskaya, P. Mandel, and Y. V. Radeonychev, “Inversionless amplification in a three-level medium,” Phys. Rev. A 45, 1997–2005 (1992).
[Crossref]

1989 (2)

S. E. Harris, “Lasers without inversion: interference of lifetime-broadened resonances,” Phys. Rev. Lett. 62, 1033–1036 (1989).
[Crossref]

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
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1917 (1)

A. Einstein, “On the quantum theory of radiation,” Phys. Z. 18, 121 (1917).

Alù, A.

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref]

Amin, M.

M. Johnson, M. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. Berkley, J. Johansson, P. Bunyk, E. Chapple, C. Enderud, J. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
[Crossref]

An, J.-H.

C.-J. Yang, J.-H. An, W. Yang, and Y. Li, “Generation of stable entanglement between two cavity mirrors by squeezed-reservoir engineering,” Phys. Rev. A 92, 062311 (2015).
[Crossref]

Asjad, M.

C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018).
[Crossref]

Aumentado, J.

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).
[Crossref]

Awschalom, D.

V. Dobrovitski, G. Fuchs, A. Falk, C. Santori, and D. Awschalom, “Quantum control over single spins in diamond,” Annu. Rev. Condens. Matter Phys. 4, 23–50 (2013).
[Crossref]

Bajer, J. C. V.

A. Miranowicz, J. C. V. Bajer, M. Paprzycka, Y.-X. Liu, A. M. Zagoskin, and F. Nori, “State-dependent photon blockade via quantum-reservoir engineering,” Phys. Rev. A 90, 033831 (2014).
[Crossref]

Barfuss, A.

A. Barfuss, J. Kölbl, L. Thiel, J. Teissier, M. Kasperczyk, and P. Maletinsky, “Phase-controlled coherent dynamics of a single spin under closed-contour interaction,” Nat. Phys. 14, 1087–1091 (2018).
[Crossref]

A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, “Strong mechanical driving of a single electron spin,” Nat. Phys. 11, 820–824 (2015).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. Dieterle, O. Painter, and J. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref]

Berkley, A.

M. Johnson, M. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. Berkley, J. Johansson, P. Bunyk, E. Chapple, C. Enderud, J. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
[Crossref]

Bernier, N.

N. Bernier, L. D. Toth, A. Koottandavida, M. Ioannou, D. Malz, A. Nunnenkamp, A. Feofanov, and T. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
[Crossref]

Bhave, S. A.

E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, “Mechanical spin control of nitrogen-vacancy centers in diamond,” Phys. Rev. Lett. 111, 227602 (2013).
[Crossref]

Brendel, C.

C. Brendel, V. Peano, O. Painter, and F. Marquardt, “Snowflake phononic topological insulator at the nanoscale,” Phys. Rev. B 97, 020102 (2018).
[Crossref]

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2015).
[Crossref]

V. Peano, C. Brendel, M. Schmidt, and F. Marquardt, “Topological phases of sound and light,” Phys. Rev. X 5, 031011 (2015).
[Crossref]

Bruder, C.

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern-Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[Crossref]

Bunyk, P.

M. Johnson, M. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. Berkley, J. Johansson, P. Bunyk, E. Chapple, C. Enderud, J. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
[Crossref]

Cao, H.

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

Carusotto, I.

R. O. Umucalılar and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011).
[Crossref]

Chapple, E.

M. Johnson, M. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. Berkley, J. Johansson, P. Bunyk, E. Chapple, C. Enderud, J. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
[Crossref]

Chen, A.

Chen, A.-X.

X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Single-photon nonreciprocal transport in one-dimensional coupled-resonator waveguides,” Phys. Rev. A 95, 063808 (2017).
[Crossref]

X.-W. Xu, A.-X. Chen, Y. Li, and Y.-X. Liu, “Nonreciprocal single-photon frequency converter via multiple semi-infinite coupled-resonator waveguides,” Phys. Rev. A 96, 053853 (2017).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Chu, Y.

J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys. 13, 025025 (2011).
[Crossref]

Cicak, K.

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).
[Crossref]

Cirac, J. I.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum reservoir engineering with laser cooled trapped ions,” Phys. Rev. Lett. 77, 4728–4731 (1996).
[Crossref]

Clercq, L.

D. Kienzler, H.-Y. Lo, B. Keitch, L. Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. Home, “Quantum harmonic oscillator state synthesis by reservoir engineering,” Science 347, 53–56 (2014).
[Crossref]

Clerk, A.

C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. Matheny, F. Marquardt, A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
[Crossref]

V. Peano, M. Houde, C. Brendel, F. Marquardt, and A. Clerk, “Topological phase transitions and chiral inelastic transport induced by the squeezing of light,” Nat. Commun. 7, 10779 (2015).
[Crossref]

Clerk, A. A.

A. Metelmann and A. A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95, 013837 (2017).
[Crossref]

V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, “Topological quantum fluctuations and traveling wave amplifiers,” Phys. Rev. X 6, 041026 (2016).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
[Crossref]

M. J. Woolley and A. A. Clerk, “Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir,” Phys. Rev. A 89, 063805 (2014).
[Crossref]

Cohen, D.

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref]

Combes, J.

A. Rosario Hamann, C. Müller, M. Jerger, M. Zanner, J. Combes, M. Pletyukhov, M. Weides, T. M. Stace, and A. Fedorov, “Nonreciprocity realized with quantum nonlinearity,” Phys. Rev. Lett. 121, 123601 (2018).
[Crossref]

Dai, Y. T.

F. X. Sun, D. Mao, Y. T. Dai, Z. Ficek, Q. Y. He, and Q. H. Gong, “Phase control of entanglement and quantum steering in a three-mode optomechanical system,” New J. Phys. 19, 123039 (2017).
[Crossref]

Damskägg, E.

C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. Clerk, F. Massel, M. Woolley, and M. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556, 478–482 (2018).
[Crossref]

DeGottardi, W.

A. Seif, W. DeGottardi, K. Esfarjani, and M. Hafezi, “Thermal management and non-reciprocal control of phonon flow via optomechanics,” Nat. Commun. 9, 1207 (2017).
[Crossref]

Demler, E.

M. Hafezi, E. Demler, M. Lukin, and J. Taylor, “Robust optical delay lines via topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Devoret, M. H.

K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).
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Figures (5)

Fig. 1.
Fig. 1. (a) Schematic diagram for generating nonreciprocal transition: two nondegenerate energy levels |a and |b are coupled to one another via a coherent interaction Hcoh, and they are also coupled to the same engineered reservoir. (b) Schematic diagram for implementation of a nonreciprocal transition in a cyclic three-level atom (characterized by |a, |b, and |c). A laser field (ΩabeiΦ) is applied to drive the direct transition between the two levels |a and |b, and they are also coupled indirectly by the auxiliary level |c through two laser fields (Ωca and Ωcb), where the decay of level |c is much faster than that of the other two levels, i.e., γcmax{γa,γb}, so the auxiliary level |c serves as a engineered reservoir.
Fig. 2.
Fig. 2. The transition probabilities Tab(t) and Tba(t) are plotted as functions of the time Ωabt for: (a) Φ=π/2, (b) Φ=0, and (c) Φ=π/2. (d) The isolation I(t) is plotted as a function of time Ωabt for Φ=π/2,0,π/2. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.
Fig. 3.
Fig. 3. (a) The transition probabilities Tab(t) and Tba(t) and (b) the isolation I(t) are plotted as functions of the synthetic magnetic flux Φ at time Ωabt=1. The other parameters are γa=γb=Ωab/10, γc=100Ωab, Ωca=Ωbc=10Ωab, and Δcb=Δca=Δab=0.
Fig. 4.
Fig. 4. Schematic of two 1D semi-infinite CRWs connected by a three-level atom characterized by |a, |b, and |g. CRW-a (CRW-b) couples to the three-level atom through the transition |a|g (|b|g) with strength ga (gb).
Fig. 5.
Fig. 5. (a) Scattering flows Iab (black solid curve) and Iba (red dashed curve), (b) Iaa (black solid curve) and Ibb (red dashed curve), are plotted as functions of the wavenumber k/π for ξ/Γ=0.1. (c) Scattering flow Iab is plotted as a function of the wavenumber k/π for different ξ/Γ. (d) The width of the wavenumber Δk for single-photon nonreciprocity is plotted as a function of log10(ξ/Γ) given in Eq. (9). The other parameters are Jba=2Γ, Jab=0, ξ=Γ, Δa=Δb=0, g2=Γξ, ϕ=π/2.

Equations (61)

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

Hcoh+dis=(Ωiγ)|ab|+(Ω*iγ)|ba|.
H=(Δabiγa)|aa|iγb|bb|+(Δcbiγc)|cc|+(ΩabeiΦ|ab|+Ωcb|cb|+Ωca|ca|+H.c.),
Heff=(ΔaiΓa)|aa|+(ΔbiΓb)|bb|+Jab|ab|+Jba|ba|,
JabΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2,
JbaΩabeiΦiΩcaΩcb(γciΔcb)γc2+Δcb2.
Hl=Δlj=0+ljljξlj=0+(ljlj+1+H.c.),
H˜eff=Jab|ab|+Jba|ba|iΓa|aa|iΓb|bb|,
Hint=gaa0|ag|+gbb0|bg|+gaa0|ga|+gbb0|gb|.
Δkπ2khalf.
H˜=(ωabiγa)|aa|iγb|bb|+(ωcbiγc)|cc|+(Ωabeiϕabeiνabt|ab|+Ωcbeiϕcbeiνcbt|cb|+Ωcaeiϕcaeiνcat|ca|+H.c.),
H=WH˜W+idWdtW=(Δabiγa)|aa|iγb|bb|+(Δcbiγc)|cc|+Ωabeiϕab|ab|+Ωcbeiϕcb|cb|+Ωcaeiϕca|ca|+H.c.,
|ψ=A(t)|a+B(t)|b+C(t)|c.
A˙(t)=(iΔabγa)A(t)iΩabeiΦB(t)iΩcaC(t),
B˙(t)=γbB(t)iΩabeiΦA(t)iΩcbC(t),
C˙(t)=(iΔcbγc)C(t)iΩcaA(t)iΩcbB(t).
C(t)=iΩcaγc+iΔcbA(t)iΩcbγc+iΔcbB(t).
A˙(t)=[i(ΔabΩca2Δcbγc2+Δcb2)+(γa+Ωca2γcγc2+Δcb2)]A(t)[iΩabeiΦ+ΩcaΩcb(γciΔcb)γc2+Δcb2]B(t),
B˙(t)=[iΩcb2Δcbγc2+Δcb2+(γb+Ωcb2γcγc2+Δcb2)]B(t)[iΩabeiΦ+ΩcaΩcb(γciΔcb)γc2+Δcb2]A(t).
Htot=l=a,bHl+H˜eff+Hint,
|E=j=0+[ua(j)aj+ub(j)bj]|g,0+A|a,0+B|b,0,
Δaua(0)ξaua(1)+gaA=Eua(0),
Δbub(0)ξbub(1)+gbB=Eub(0),
iΓaA+gaua(0)+JabB=EA,
iΓbB+gbub(0)+JbaA=EB,
Δlul(j)ξl[ul(j+1)+ul(j1)]=Eul(j),
ul(j)=eiklj+slleiklj,
ul(j)=slleiklj,
E=Δl2ξlcoskl,0<kl<π,
A=(E+iΓb)gaua(0)+Jabgbub(0)(E+iΓa)(E+iΓb)JbaJab,
B=(E+iΓa)gbub(0)+Jbagaua(0)(E+iΓa)(E+iΓb)JbaJab.
(ΔaE+Δ¯a)ua(0)+Jabub(0)=ξaua(1),
(ΔbE+Δ¯b)ub(0)+Jbaua(0)=ξbub(1),
Jab=Jabgagb(E+iΓa)(E+iΓb)JbaJab,
Jba=Jbagagb(E+iΓa)(E+iΓb)JbaJab,
Δ¯a=(E+iΓb)ga2(E+iΓa)(E+iΓb)JbaJab,
Δ¯b=(E+iΓa)gb2(E+iΓa)(E+iΓb)JbaJab.
(ξaeika+Δ¯a)saa+Jabsba=ξaeikaΔ¯a,
Jbasaa+(ξbeikb+Δ¯b)sba=Jba.
(ξaeika+Δ¯a)sab+Jabsbb=Jab,
Jbasab+(ξbeikb+Δ¯b)sbb=ξbeikbΔ¯b.
LS=R,
S=(saasabsbasbb)
L=(ξaeika+Δ¯aJabJbaξbeikb+Δ¯b),
R=(ξaeika+Δ¯aJabJbaξbeikb+Δ¯b).
saa=JabJba(ξaeika+Δ¯a)(ξbeikb+Δ¯b)(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sba=2iξaJbasinka(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sab=2iξbJabsinkb(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba,
sbb=JabJba(ξaeika+Δ¯a)(ξbeikb+Δ¯b)(ξaeika+Δ¯a)(ξbeikb+Δ¯b)JabJba.
Ill|sll|2ξlsinklξlsinkl,
Iba=|sba|2=|2Jbag2ξsink[ξeik(2ξcosk+iΓ)+g2]2|2.
|sink|=(|Jba|Γ)g2±Θ4(g2ξ2)ξ,
Θ=(|Jba|Γ)2g44(g2ξ2)[4(ξ2g2)ξ2+Γ2ξ2+g4].
|Jba|=(g2+Γξ)22g2ξ.
|Jba|=2Γ.
Iba=|sba|2=|2Jbag2ξsinkhalf[ξeik(2ξcoskhalf+iΓ)+g2]2|2=12.
2(Γξ)ξ|sinkhalf|2+(122)Γ2|sinkhalf|+2(ξΓ)ξ+Γ2=0.
|sinkhalf|=(122)±(122)2ζ(2ζ)ζ.
ddη|sinkhalf|=d|sinkhalf|dζdζdη=0,
η=12ξ=Γ2.
|sinkhalf|=221222,
Δkmaxπ2arcsin(221222)0.81π.