PHYSICAL REVIEW LETTERS 121, 203602 (2018)

Cavity-Free Optical Isolators and Circulators Using a Chiral Cross-Kerr Nonlinearity

Keyu Xia,1,2,3,* Franco Nori,3,4 and Min Xiao1,2,5
1
National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, and School of Physics,
Nanjing University, Nanjing 210093, China
2
Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
3
Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
4
Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
5
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
(Received 23 August 2018; published 14 November 2018)

Optical nonlinearity has been widely used to try to produce optical isolators. However, this is very
difficult to achieve due to dynamical reciprocity. Here, we show the use of the chiral cross-Kerr nonlinearity
of atoms at room temperature to realize optical isolation, circumventing dynamical reciprocity. In our
approach, the chiral cross-Kerr nonlinearity is induced by the thermal motion of N-type atoms. The
resulting cross phase shift and absorption of a weak probe field are dependent on its propagation direction.
This proposed optical isolator can achieve more than 30 dB of isolation ratio, with a low loss of less than
1 dB. By inserting this atomic medium in a Mach-Zehnder interferometer, we further propose a four-port
optical circulator with a fidelity larger than 0.9 and an average insertion loss less than 1.6 dB. Using atomic
vapor embedded in an on-chip waveguide, our method may provide chip-compatible optical isolation at the
single-photon level of a probe field.

DOI: 10.1103/PhysRevLett.121.203602

Introduction.—Optical isolation is highly desirable for nonreciprocity [30–32], and chirality [33–38]. Light propa-
lasers, optical information processing, and quantum net- gating in a “moving” Bragg lattice created in atoms is
works [1,2]. It requires optical nonreciprocity, i.e., breaking subject to a “macroscopic” Doppler effect and has dem-
of Lorentz reciprocity [3], but is very challenging to onstrated nonreciprocity [39–41]. By using a chiral quan-
achieve without applying magnetic fields. tum system, optical isolation has been achieved at the
Nonmagnetic optical isolation is chip compatible and single-photon level [42–45].
therefore is in great demand for integrated optical signal Optical chirality has been widely exploited to engineer
processing. It has been studied via dynamically modulating spin-orbital interaction of light [33–37]. In this Letter, we
material permittivity [4–7], inducing a photonic Berry propose how to achieve efficient optical isolation using
phase [8–11], twisting a resonator [12], a fast spinning chiral cross-Kerr (XKerr) nonlinearity induced in atoms.
resonator [13], or using optomechanical systems [10, Because of the chirality of atomic nonlinearity, the phases
14–16]. Over the past decades, optical nonlinearity (in and amplitudes of the forward- (right-) and backward-
particular, Kerr or Kerr-like nonlinearity) has attracted moving (left-moving) probe fields are very different after
intense research as a chip-compatible candidate for mag- passing through atoms along two opposite directions.
netic-free optical isolation [17–22]. Moreover, using a gain Therefore, both an optical isolator and a circulator can
medium has also been demonstrated for optical isolation be achieved with a high isolation ratio and low insertion
[19–21,23,24]. However, optical isolators with nonlinearity loss. Because the induced nonlinearity is chiral, our
or gain in the medium are subject to dynamic reciprocity proposals circumvent the problem of dynamic reciprocity
[25,26]. Therefore, this kind of device is nonreciprocal only and may provide a new cavity-free route for nonlinear
for strong signals with particular intensity but fails to optical isolators and circulators.
isolate weak signals. A chiral gain has been recently used to System and model.—Our setup is depicted in Fig. 1. We
overcome this fundamental barrier in nonlinear isolators first consider a waveguide (WG) embedded with N-type
[22,23]. However, a passive nonlinear isolator without atoms [46–50]; see the upper waveguide in Fig. 1(a). We
dynamic reciprocity would be of interest. Moreover, most apply the classical switching and coupling fields to induce
of the existing schemes for optical isolation require high- the phase shift ϕ and amplitude modulation ξ of the probe
quality resonators or cryogenic temperatures. field. To a good approximation, we treat the waveguide as a
Instead of classical optics, quantum optics provides a 1D space. If the forward and backward amplitude trans-
tool to control photon propagation, including electromag- missions ξf and ξb are sufficiently different after the probe
netically induced transparency (EIT) [27–29], optical field passes through the ensemble of atoms, then we can

0031-9007=18=121(20)=203602(7) 203602-1 © 2018 American Physical Society

 PHYSICAL REVIEW LETTERS 121, 203602 (2018)

nonlinearity is strongly dependent on the effective detun-

ings and thus the Doppler shifts. Thus, these frequency
shifts change the optical nonlinearity in a way strongly
dependent on the propagation direction of the probe field
with respect to the switching and coupling fields, leading
to the chiral XKerr nonlinearity. We assume that
jks j ¼ jkc j ¼ jkp j ¼ k. Both the switching and coupling
laser beams are left moving such that ks vj ¼ kc vj . In the
above arrangement, the backward-moving (forward-
moving) probe field “sees” the same (opposite) Doppler
shift as the switching and coupling ones. Under the two-
photon resonance condition, i.e., Δc ¼ Δs ¼ δ, and jΩs j ≪
jΩc j leading to ρ11 ≈ 1, we can solve the master equation
with the perturbation approach [53–55] and obtain the total
XKerr nonlinearity averaged over the velocity distribution
FIG. 1. (a) Schematic diagram of our setup for an optical
as [56–60]
isolator and circulator. For an optical isolator, we consider only Z  
γ 23 1 1
the upper channel embedded with a cloud of N-type atoms. The χ f ¼ X0 þ  NðvÞdv ð1Þ
photon passing through the atoms suffers a phase shift ϕ and an ðiΔp þ γ 43 Þ ζ ζ
amplitude transmission of ξ, dependent on its propagation
direction. To perform an optical circulator, the lower channel for the forward-moving probe field and
is added to form a Mach-Zehnder interferometer, using beam
Z  
splitters BS1 and BS2, with the upper one. In the lower branch, a γ 23 1 1
phase shift ϑ is used to compensate the phase shift of the χ b ¼ X0 þ NðvÞdv ð2Þ
½iðΔp þ 2kvÞ þ γ 43  ζ ζ 
backward-moving photon in the upper branch. (b) Level diagram
of N-type atoms. The switching (Ωs ), coupling (Ωc ), and probe
(Ωp ) fields couple to the transitions j1i ↔ j2i, j3i ↔ j2i, and in the backward-moving case, where X0 ¼ 3πc2 γ 43 =
j3i ↔ j4i, with detunings Δs , Δc , and Δp , respectively. 8ω2p Γ3 ðγ 21 þ γ 23 Þ, ζ ¼ iðδ þ kvÞ þ ðγ 21 þ γ 23 þ Γ3 Þ þ
jΩc j2 =2Γ3 , and c is the vacuum light speed. The velocity
distribution is conventionally taken to be Maxwellian, i.e.,
realize a two-port optical isolator. By carefully choosing the 2 2 pffiffiffi

density and length of the atomic vapor and properly NðvÞ ¼ N a e−v =u = π u, where u is the room-mean-square
arranging the switching and coupling fields, we can obtain atomic velocity and ku ≈ 2π × 300 MHz for Rb atoms at
a phase shift difference, Δϕ ¼ ϕf − ϕb , approaching π room temperature [61]. In our arrangement, the linear
susceptibility of the probe light is vanishingly small and
with high transmissions ξf and ξb . This can provide a four-
can be neglected because ρ33 ≈ 0. Compared with the
port optical circulator by adding a lower waveguide to form
backward input case, where the Doppler broadening
a Mach-Zehnder interferometer (MZI).
significantly reduces the total XKerr nonlinearity [see
We consider an N-type configuration using rubidium
Eq. (2)], the Doppler shift “seen” by the forward-moving
(Rb) atoms to create the chiral XKerr nonlinearity. State j2i
probe field is partly compensated [see Eq. (1)], and
decays to states j1i and j3i with rates γ 21 and γ 23 ,
subsequently the nonlinearity remains large. This chirality
respectively. State j4i decays at a rate γ 43 . The dephasing
is a combination of thermal motion and the unidirection-
rates of both ground states j1i and j3i are Γ. For simplicity,
ality of the switching and coupling lasers. The Doppler
we assume γ 21 ¼ γ 23 ¼ γ 34 ¼ γ 0 and Γ ≪ γ 0 , and set γ 0 ¼
shift is due to the atomic thermal motion. The unidirec-
2π × 6 MHz [51]. The XKerr nonlinearity can be effi- tionally propagating switching and coupling lasers break
ciently induced between the probe and switching fields in the spatial symmetry, leading to a direction-dependent
the configuration shown in Fig. 1(b) and can be modified response to the probe laser. Without the switching and
by the coupling laser [52]. The switching, coupling, and coupling fields, the thermal motion sharply suppresses the
probe laser beams have carrier frequencies ωs , ωc , and ωp , atomic susceptibility in both directions. If the control fields
corresponding to wave vectors ks , kc , and kp , respectively. in EIT are applied to atoms from two opposite directions,
The switching (coupling, probe) field drives the transi- thermal motion will be detrimental [39,40]. In the two latter
tion j1i ↔ j2i (j3i ↔ j2i, j3i ↔ j4i) with a detuning cases, the chirality disappears. Note that the reduced
Δs (Δc , Δp ) in the absence of thermal motion. At room absorption in the “two-photon Doppler-free” configuration
temperature, the inevitable random thermal motion of the for EIT in a 3D atomic sample has been observed [62]. The
jth atom moving with velocity vj causes the “microscopic” two-port nonreciprocal transport has been experimentally
Doppler shifts ks vj , kc vj , and kp vj in the corresponding demonstrated as a result of atomic thermal motion and
atomic transitions, respectively. The strength of the the strong atom-cavity coupling [63]. However, cavity-free

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optical isolation exploiting thermal motion is conceptually nonlinear medium. The two BSs are chosen to be identical
different and admirable, because its realization can be with reflection and transmission amplitudes of sin θ and
simpler and it can implement multiport optical circulators. cos θ, respectively. The relative phase in these amplitudes
Unlike the configurations for quantum gates [51] and is φ. Their operation on photons is determined by HBS ¼
nondestructive detection of photons [64], the applied θeiφ â†in b̂in þ θe−iφ âin b̂†in [68]. A fixed phase shift ϑ in the
switching and coupling modes are chosen here to be much lower path compensates the phase shift ϕb of the backward-
stronger than the probe laser beam. The backaction on the moving probe laser beam caused by the nonlinear medium.
switching field due to the probe photon is negligible. Thus, Therefore, the backward-moving probe photons entering
they can be considered as constant in atoms. We apply the BS1 have the same phase in the upper and lower wave-
slowly varying envelope approximation to the probe field. guides. Applying H BS and the transmission relation Eq. (5),
The backscattering is negligible during the propagation, we obtain the forward transmission matrix elements
and the probe photon propagates unidirectionally [51,65]. between the input and output ports as
When jΩc j ≫ jΩs j ≫ jΩp j, the propagation of the probe
 2
pulse in atoms is described by Maxwell equations by taking a 
into account the XKerr nonlinearity [66,67]: T 12 ¼  o  ¼ jξf eiðϕf −ϑÞ cos2 θ − sin2 θj2 ; ð6aÞ
ain
 2
∂Ωfp ðz; tÞ 1 ∂Ωfp ðz; tÞ a 
þ ¼ −χ f jΩs j2 Ωfp ðz; tÞ; ð3Þ T 32 ¼  o  ¼ jð1 þ ξf eiðϕf −ϑÞ Þ cos θ sin θj2 ; ð6bÞ
∂z c ∂t bin
∂Ωbp ðz0 ; tÞ 1 ∂Ωbp ðz0 ; tÞ  2
b 
þ ¼ −χ b jΩs j2 Ωbp ðz0 ; tÞ ð4Þ T 14 ¼  o  ¼ jð1 þ ξf eiðϕf −ϑÞ Þ cos θ sin θj2 ; ð6cÞ
∂z0 c ∂t ain
for the forward- and backward-moving probe pulses,  2
b 
respectively, and z0 ¼ L − z. When χ f ¼ χ b as in the usual T 34 ¼  o  ¼ jcos2 θ − ξf eiðϕf −ϑÞ sin2 θj2 ; ð6dÞ
Kerr nonlinear isolators, the medium is reciprocal for the bin
probe beam. However, the medium can be nonreciprocal
where T mn is the transmission coefficient from port m to
even for two weak counterpropagating probe beams coex-
port n, with m, n ¼ 1, 2, 3, 4. Exchanging the inputs and
isting in the medium simultaneously when χ f and χ b are
the outputs and replacing ξf and ϕf with ξb and ϕb in T mn ,
very different. Therefore, optical isolators or circulators
respectively, we obtain the transmission matrix element
using this chiral medium can overcome the dynamical
T nm for the backward-moving case. Optical nonreciprocity
reciprocity in conventional nonlinear isolators [25]. We
requires T mn ≠ T nm for m ≠ n. We have T mm ¼ 0 in
focus on the steady-state solution, where a long probe
the circulator. Also, the backscattering to ports at the same
pulse is constant in time at position z [51], such that
side as the input is negligible so that T 31 ¼ T 13 ¼ T 41 ¼
ð1=cÞð∂Ωfp =∂tÞ ≈ 0 and ð1=cÞð∂Ωbp =∂tÞ ≈ 0. After passing T 14 (see details in Supplemental Material [56]). An ideal
through the atomic medium with length L, the probe fields circulator, in which the photons flow along 1 → 2 → 3 →
become 4 → 1, has a transmission matrix T id with elements
12 ¼ T 23 ¼ T 34 ¼ T 41 ¼ 1 and others zero. Note that
T id id id id
Ωjp ðLÞ ¼ ξj eiϕj Ωjp ð0Þ; ð5Þ Tr½T T  ¼ 4.
id id;T

Hereafter, we take δ ¼ 0 and ωp =2π ∼ 384 THz for the

where ξj ¼ expð−Re½χ j jΩs j2 LÞ and ϕj ¼ −Im½χ j jΩs j2 L,
D1 line of Rb atoms and choose the parameters N a ¼
with j ¼ f, b, are the corresponding transmission ampli-
5 × 1012 cm−3 , Γ3 ¼ 0.1γ 0 , Ωc ¼ 20γ 0 , and Ωs ¼ 4γ 0 ,
tude and phase shift, respectively. When jΔp j ≫ γ 43
yielding ρ11 ≈ 0.96. Such large switching light can enhance
and jΩc j2 =2Γ3 ≫ jδ þ kvj, to a good approximation, we the cross phase modulation of the probe field.
have ϕf ≈ N a Lð3πc2 =4ω2p Þðγ 0 =Δp ÞðjΩs j2 =jΩc j2 Þ and ξf ≈ Isolator.—For a centimeter-scale medium, e.g., L ¼
exp ½−N a Lð3πc2 =4ω2p Þðγ 20 =Δ2p ÞðjΩs j2 =jΩc j2 Þ. The trans- 2 cm, the medium is absorptive. The forward and backward
mission is calculated as jξj j2 . In contrast, the transmission transmissions are very different; see Fig. 2. As the probe
and phase modulation of the backward-moving probe laser detuning increases, the forward transmission T 12 rapidly
are much smaller. increases to a high value of 0.80 at Δp ¼ 35.6γ 0 , corre-
Obviously, an optical isolator can be realized when sponding to an insertion loss of 1 dB. Because of Doppler
ξf ≫ ξb . For ξf ≈ ξb and ϕf − ϕb ≈ π, an optical circulator broadening, the backward-moving probe field suffers a
could be made by inserting the atomic vapor in an MZI, as larger absorption. Therefore, the backward transmission
shown in Fig. 1(a). To achieve that, two beam splitters T 21 is much smaller than T 12 , when 35.6γ 0 < Δp < 60.6γ 0 .
(BSs) are needed to first divide the input probe pulse into In this region, the insertion loss is smaller than 1 dB, while
two paths and then mix them after passing through the the isolation ratio is larger than 15 dB. The isolation ratio

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 PHYSICAL REVIEW LETTERS 121, 203602 (2018)

FIG. 3. Circulator performance versus the detuning Δp .

(a) Phase shifts (blue curves) and transmission amplitudes (red
curves) for right- (solid curves) and left-moving (dashed lines)
probe fields as a function of the probe detuning Δp . (b) Fidelity
(green curves) and average insertion loss (blue dashed curves)
of an optical circulator as a function of Δp . The vertical black
FIG. 2. Transmission of an isolator for the right-moving (blue dashed lines in the two figures show the optimal detuning
curves) and left-moving (red curves) probe fields and the isolation Δopt opt opt opt opt
p =γ 0 ¼ 7.77 when ϕf − ϕb ¼ π and ξf ≈ ξb ≈ 0.66.
ratio (green curves associating with the right vertical axis) as a The length of the atomic medium is 3.33 mm. Other parameters
function of the probe detuning Δp . Solid (dashed) curves are are as in Fig. 2.
for L ¼ 2ð4Þ cm. Other parameters are N a ¼ 5 × 1012 cm−3 ,
Γ3 ¼ 0.1γ 0 , Ωc ¼ 20γ 0 , Ωs ¼ 4γ 0 , and δ ¼ 0.
another important figure characterizing the four-port cir-
can be considerably improved with a large forward trans- culator. We scan the probe frequency to find the working
mission by using a longer medium or, equivalently, window of the circulator in Fig. 3. As the detuning Δp
increasing the atomic density. The nonreciprocal window varies from 6γ 0 to 20γ 0 , the fidelity first rises up rapidly,
of frequency slightly moves to larger detuning. For reaches the maximum 0.944 at Δopt p ¼ 7.77γ 0 , and then
L ¼ 4 cm, the isolation ratio can reach more than 30 dB decreases to a small value of 0.63. During this sweep, η
in the range of 50γ 0 < Δp < 60γ 0 , yielding an isolation increases from 0.68 to about 0.83. Although the photons
bandwidth of 2π × 60 MHz. At the same time, the insertion have a larger probability to survive at a large detuning, the
loss remains low, less than 1 dB. Thus, we can simply use fidelity is low. As a trade-off, the circulator operating
this medium as an isolator. within the frequency range 6.6γ 0 < Δp < 9.7γ 0 can achieve
Circulator.—For a short medium, the transmissions of a fidelity larger than 0.9 at the expense of η > 0.69. The
the forward- and backward-moving probes can be compa- corresponding working window is about 2π × 20 MHz,
rably high. However, at a particular probe frequency, the and the average insertion loss is about 1.6 dB. If ϑ ¼ 0, the
phase shift difference between these two opposite propa- fidelity and insertion loss reduce only very slightly.
gating probes can approach π. As shown in Fig. 3(a), for At Δopt
p , we obtain F ¼ 0.944 and η ¼ 0.72, yielding
L ¼ 3.33 mm, the phase shift ϕb is always small, specifi- an insertion loss of 1.42 dB. The corresponding trans-
cally about 0.011π at Δp ¼ 7.77γ 0 . In contrast, ϕf expo- mission matrix is shown in Fig. 4. The obtained matrix
nentially decays from about 2π at Δp ¼ 3.5γ 0 to 0.5π at is close to that of the ideal circulator, implying that a
Δp ¼ 15.5γ 0 . At the optimal point Δopt p ¼ 7.77γ 0 , the good optical nonreciprocity is obtained. We can also
difference of phase shifts, ϕf − ϕb , reaches the optimal quantify the circulator performance by the isolations I i ¼
value of π. At this point, ξopt opt −10 logðT iþ1;i =T i;iþ1 Þ of the four optical isolators formed
f ≈ ξb ≈ 0.66. Thus, a high-
performance circulator can be made by inserting this between adjacent ports [43] and have fI i g ¼ f41.7;
nonlinear medium into an MZI composed of unbalanced 13.8; 13.8; 8.2g dB with i ¼ f1; 2; 3; 4g, implying nonre-
BSs. Here, we set ϑ ¼ 0.01π and sin2 θ ¼ 0.4 ≈ ξ̄=ð1 þ ξ̄Þ ciprocal photon circulation along 1 → 2 → 3 → 4 → 1.
The achieved performance is already useful for practical
with ξ̄ ¼ ðξopt opt
f þ ξb Þ=2. optical isolation [43].
The performance of a circulator can be quantified with Implementation.—The required 1D nonlinear waveguide
the fidelity F and the average photon survival probability embedded with alkali atoms can be made with various
η [43]. The fidelity is evaluated as the overlap of the methods and platforms [42,46–50,69–74]. A feasible plat-
renormalized transmission matrix T̃ ¼ ðT ij =ηi Þ with the
P form can be a centimeter-scale hollow-core photonic crystal
ideal one, T id . Here, ηi ¼ k T i;k is the survival probability fiber filled with Rb atoms at room temperature [49,74]. A
of the probe photons entering port i. The average operation few-photon-level memory and a strong XKerr nonlinearity
fidelity of the circulator is then F ¼ Tr½T̃T id;T = have been demonstrated with a weak control field in such a
Tr½T id T id;T , giving the probability of a correct circulator platform [49,74]. For an on-chip realization, we consider a
operation averaged over various inputs. The minimum zigzag waveguide cladded with high-density Rb atoms,
fidelity is F ¼ 0, while an ideal operation yields P F ¼ 1. allowing a coherent light-atom interaction [71–73]. To
The average photon survival probability η ¼ i ηi =4 is conduct an N-type configuration, we couple the lasers

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 PHYSICAL REVIEW LETTERS 121, 203602 (2018)

Research and Development (AOARD) (Grant

No. FA2386-18-1-4045), Japan Science and Technology
Agency (JST) (the ImPACT program and CREST Grant
No. JPMJCR1676), Japan Society for the Promotion of
Science (JSPS) (JSPS-RFBR Grant No. 17-52-50023 and
JSPS-FWO Grant No. VS.059.18N), RIKEN-AIST
Challenge Research Fund, and the John Templeton
Foundation.

*
keyu.xia@nju.edu.cn
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