Mesoscopic Regulation of Atoms System Via Optical Microcavity Zeyang Li, Qi-Tao Cao, Pai Peng, Martina Hentschel, Heming Wang

Mesoscopic Regulation of Atoms System via Optical Microcavity Zeyang Li, Qi-Tao Cao, Pai Peng, Martina Hentschel, Heming Wang

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Zeyang Li, Qi-Tao Cao, Pai Peng, Martina Hentschel, Heming Wang. Mesoscopic Regulation of Atoms System via Optical Microcavity. 2017. hal-01545691

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Zeyang Li1, Qi-Tao Cao1, Pai Peng1, Martina Hentschel2, and Heming Wang1,3∗ 1State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University 2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China 3Technische Universitat¨ Ilmenau, Institut fur¨ Physik, D-98693 Ilmenau, Germany and 4T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA (Dated: June 23, 2017) The atom with fine structure is widely studied in the area of nuclear physics, atomic physics and condensed matter physics. The traditional way to manipulate them relates to static magnetic field; however, the scale area of a typical magnetic field is always macro. In this letter, we proposed a method to control atoms via vector the polarizability at mesoscopic scale by optical microcavity. The linear quantum control of optical microcavity, at the same time, is achieved, which could be a significant for on-chip device engineering. In addition, by applying magnetic field, a nonlinear phenomenon of transmission spectrum is observed, which could be used as a test pool for many fancy nonlinear dynamics.

PACS numbers: 00.00.00

I. INTRODUCTION introduced to WGM cavity due to its strong near-field intensity. We assume the atoms are set in the mesoscale Modern research on cavity quantum electrodynamics region, which is acme of both cavity mode intensity and (CQED) obtains great results about strong interactions the coupling strength. Due to the conversation of angular between light field and single atom. To obtain the in- momentum, different cavity mode which has its unique teraction a promising method is by increasing the energy orbital angular momentum could couple to each other via density, where using optical cavity to spatially confine each order of spin flipping process. photons is a promising approach. The strong interac- In Sec.II we establish a description of the system, tion has two remarkable consequences: on the one hand and discuss simplification which suits the ordinary exper- the atom is governed by the light field, and on the other imental condition. In Sec.III we discuss the influence hand the cavity modes are also modulated by the atoms. of this interaction to cavity mode where an interesting The former one is widely studied in numerous CQED nonlinear behavior is observed. In Sec.IV we then dis- systems such as graphene plasmonics [1], cavity polari- cuss how this universal interaction matters on atoms, and ton [2,3], quantum dot [4] and WGM cavity [5,6]. In two aspects are analyzed: first the population of spin as addition, perfect circular WGM cavities, though exhibit a function of input power and frequency is studied, and great quality factor, are hard to couple with fiber, while second we based on recent approach of deriving effective asymmetric WGM cavities are always good substitution large scattering length via time-dependent modulated ex- in reality. For single atom’s coupling with light field, the ternal field, where we extend it to light field rather than later phenomenon can hardly be observed at ordinary magnetic field to get better time-domain control. light intensity level for experiments; however, things are different for an atom ensemble interacts with light field. What makes atom ensemble unique in this interaction form rather than individual atom arises from the interac- II. MODEL CONSTRUCTION tion between atoms. Ordinarily, the interaction could be tuned by magnetic field via Feshbach resonance[7–9]. In Circular WGM cavity has a rotation symmetry, and general the Feshbach resonance occurs accompanied with its eigenmodes are all TM or TE polarized, i.e., have zero energy bound state. Magnetic field tune the Zeeman no circular polarizability. These properties would not be shifting to create a bound state, and it could also be done useful in a strong light-matter interaction for our study. via optical field by a so-called optical control of Feshbach By introducing asymmetry, such as deformation [18–20] resonance [10–17]. In the optical microcavity coupling to or defects to make annular cavity [21, 22], the near-field atom ensemble setup, this could also be important and intensity is increased at some particular positions, and have exotic phenomena both in cavity modes side and in the eigenmodes of the cavity have circular polarization. atom’s dynamics side. Set atom ensembles in the regions where field intensity In this work, we propose a system which for the first is large would therefore establish effective coupling be- time can directly couple WGM cavity with atom’s spin tween the cavity modes and atom. A fine-structured level. To achieve the necessary power, an asymmetry is atom could interact with small detuning monochromic light via a form as

∗ ∗ Heff = iuv(E × E ) · S (1) 2 where S is spin of an atom, and for the ideal two-level of a single resonance mode always has unitary distribu- 1 2 ∗ atom, uv = − α |hl = 1|d|l = 0i| is the so-called tion of E×E . However, when asymmetric is introduced 12∆e ~ vector polarization which is derived in textbooks and lit- into the microcavity, such as deformed microcavity or eratures [23–28] as the coupling strength of dipole in- annular microcavity, the TE mode electric field could be teraction, and the ∆e is the detuning of light frequency extremely different. In this letter, we take consideration from energy gap between the atom’s ground state and of annular cavity to make the cavity asymmetrical. Fig. first excited state: ~∆e = Ee − ~ω. This particular form 1 (a-d) shows an example for annular cavity and related of interaction is remarkable for artificial gauge field in ul- light intensity and |E×E∗| distribution. For a TM mode, tracold atom areas, considering it produces a vector shift the field component perpendicular to the plane is mag- of atom and explores new form of light-matter interac- netic field [22], therefore the TM mode’s electronic field tion, and could disappear only when the total angular can be derived as momentum F = 0. i E = − kzˆ × ∇ B , ∇ = (∂ , ∂ , 0) (2) ⊥ r2 ⊥ z ⊥ x y (a) (b) within cylinder coordinates, it’s i ∂B 1 ∂B E = − kzˆ × z e + z e (3) ⊥ r2 ∂r r r ∂θ θ and for a circular cavity this effect annihilates for θ symmetric and E is in same direction as E∗. For a TM mode, E × E∗ is pure imaginary and has no components on x, y direction, so we can suppose that ∗ E × E = iΩez/uv, an illustration see Fig.1. We then (c) (d) derive the effective Hamiltonian as

Heﬀ = −ΩSz (4) which causes the splitting of the alkali atom’s single out- most electron’s two spin degeneracy eigenstate. The physical process at resonance is important, for it obeys the conservation of energy and angular momentum at the same time, and hence the phenomenon is remarkable. The extra term we add into Hamiltonian is described as the following process: 1. the microcavity (e) mode |m, 1i is excited by taper, with annihilation op- erator am. Its emission, when treat as classical light, interacts with the atoms outside; 2. the spin-up atom absorbs photon with orbital momentum m~, then emit a Atom LevelMode photon with orbital momentum (m + 1)~ and falls into spin-down atom; 3. the interaction could be described as coupling between atom electron spin and photon, i.e., † † gb↓b↑am+1am + h.c; 4. the interaction happens because the photon spin is transferred into electron spin, how- Cavity ever, with even weak spin-orbital interaction of photon, this could happen. Hence, full Hamiltonian without input at resonance is given by

FIG. 1. (a) - (d): Comsol simulation of the distribution of an H = Hmodes + Hspin + Hint (5a) annular microcavity’s scar-like m = 16, 17 modes: (a, c) light X † Hmodes = (ωj − iΓj)a aj (5b) intensity |E|2 distribution, (b, d) imaginary part of E×E∗’s z ~ j j=m,m+1 component, i.e., Ω/uv in eq. (4), (e) energy level demonstra- † † † tion, where the left shaded region and right shaded region Hspin = ~(camam + dam+1am+1)b↑b↑ (5c) couple to each other based on the conservation of angular † † momentum. The coupling strength proportional to the over- Hint = (gb↓b↑am+1am + h.c) (5d) lapping of cavity mode within the area of atomic ensemble. where in the Hamiltonian, Hmodes, for the energy of photons; Hspin, for the splitting due to the contribution of For a symmetric optical microdisk, it could not exhibit the two optical modes (as in Fig.1, the two mode has any sort of property of this interaction, for the emission basically same strength of E×E∗, and hence the efficient 3 is chosen to be same). The interaction between atoms is III. REGULATION ON MICROCAVITY not taken into account for simplicity. The crossing of two electric field components is actually phenomenologically † In the first two sub-equations in (6a), (6b), the b↑b↑ described via the extra interaction term: Hint, for the in- term actually contributes a rather small effect, for firstly teraction between integrated state |m + 1, ↓i and |m, ↑i. the c is relatively a small number, and secondly the num- † † b↑b↑ + b↓b↓ = Ntot for the number of atoms. The Heisen- ber of atom is not large enough to influence the frequency berg equation gives the dynamics equation for modes and of light, compare to the high optical frequency we are in- spin population. The coupling coefficient is determined terested in. Based on this analyze, in principle we could by the overlapping of mode |mi and |m + 1i around a omit this extra term due to each spin number. given position, say around the place with high |E × E∗| We suppose that the input source set the frequency, as in Fig.1. Physically one would expect the g as a i.e., am, am+1, b↑, b↓ are both in a harmonic oscil- −iω1t −iω2t −iω3t function of the size and center displacement of annular lating way as am,0e , am+1,0e , b↓,0e and −iω4t cavity. The way to determine the coupling constant g b↑,0e . An obvious result for this is that ω1 = is by calculating the mode overlapping of |mi, |m + 1i ω2, ω3 = ω4, and due to the input as compulsory driven modes at one maximum place. At large N limits, which means the number of pho- 1.0 tons and atoms are large enough, we can apply a mean- field approach where all the operators could be treated 0.8 as numbers. By adding input term, we can write down the dynamics of cavity mode precisely as 0.6 Transmission Spectrum d † 0.4 Population of Spin up i~ am = ~(ωm − iΓm + cb b↑)am Population of Spin down dt ↑ (6a) ∗ ∗ √ −iωt + g b↑b↓am+1 + i κmP e , 0.2

d † i~ am+1 = ~(ωm+1 − iΓm+1 + db b↑)am+1 0.0 dt ↑ (6b) ∗ √ −iωt 190 200 210 220 + gb↓b↑am + i κm+1P e , and also the dynamics of spin population is given by FIG. 2. Numerically calculated transmission spectrum (rescaled by T → (T − 0.8)/0.2) and spin population for the d † † eﬀective two mode system. The frequency of the two optical i~ b↑ = ~(camam + dam+1am+1)b↑ dt (6c) mode is set at ωm = 205 and ωm+1 = 210. The coupling on ∗ ∗ + g b↓amam+1, the one hand causes expel of peak position, on the other hand d give rises of bistable modes. i b = gb a a∗ . (6d) ~dt ↓ ↑ m m+1 ∗ ∗ with all these four equations, one can exactly solve the force, ω1 = ω2 = ω. Deﬁne G1 = g b↑,0b↓,0/~, G2 = ∗ ∗ dynamical evolution of cavity mode and spin population. g am,0am+1,0/~; y ≡ am+1,0/am,0, z ≡ b↓,0/b↑,0 as some However, on the other hand, knowing the exact dynam- parameters. Based on the condition that the number of 2 2 ics is relatively useless for the physics behind the whole total spin is conserved, i.e., |b↓,0| + |b↑,0| = Ntot, the framework. Therefore we might need other way to treat equations (6a) could be transformed that the amplitude these equations, as in Sec.III. obeys the following coupled algebraic equations:

q 1 2 2 2 2 2 2 z = − (c|am,0| + d|am+1,0| ) ± (c|am,0| + d|am+1,0| ) + 4|G2| (7a) 2G2 1 q ω = ω = (c|a |2 + d|a |2) ± (c|a |2 + d|a |2)2 + 4|G |2 (7b) 3 4 2 m,0 m+1,0 m,0 m+1,0 2 √ √ κ1(ω − ωm+1 + iΓm+1) + κ2G1 iP am,0 = 2 (7c) (ω − ωm+1 + iΓm+1)(ω − ωm + iΓm) − |G1| √ √ κ2(ω − ωm + iΓm) + κ1G1 iP am+1,0 = 2 (7d) (ω − ωm+1 + iΓm+1)(ω − ωm + iΓm) − |G1|

At this level, the phenomenon is purely linear: for scaling input power, the am, am+1 scales linearly which 4

relation of spin population to the frequency around the first major absorb, i.e., the ω ∼ 150 − 160 could be read out. This is actually due to the mode beating, where the 4 two optical mode are mixed together and the dominant one would rule which spin direction has more population. It’s the same reason that the two spin orientations dominant at different input power, as in Fig.3. The transmission spectrum has a shape much different from 3 the traditional Lorentzian result. The spin plays a role of regulating the transmission. Due to it’s nonlinear ef-

up/down fect, the Fig.2 only presents a low-power case, and the strong power situation would causes more results, as in 2 Fig.3. When the power is too strong where the photon number is remarkably high, the coupling between spin T&N up and down is remarkable. This leads to a far devia- tion from half-half population at resonance, and on the other hand results in the power hard to permit inside 1 the cavity, which in the spectrum means a transmission transparency. This kind of transparency has a dramat- ically sharp shape and small width, mainly because it happens when the spin population experience a nonlinear bifurcation. It’s a sudden phenomenon, results in a discontinuity in physical quantity, and hence the trans- 100 150 200 250 300 mission spectrum would only exhibit a sharp peak. ω There are two things should be discussed carefully. One is about T > 1 for some particular frequency region. This could be understood as atoms gain energy FIG. 3. Transmission spectrum and corresponding spin pop- from magnetic field, and the transfer this energy into ulation at different power. In Fig.2 the power is P0, and photons and flip back into low energy state, and not vio- here is 50/100/200/300P0. As power increases, the frequency where the spin population first present near saturated occu- late energy conservation law, as expect. The other is the pation (50/100P0: spin down; others: spin up) tends to be regularization of peak position by this approach is actu- lower, and the saturation’s frequency range tends to broaden; ally limited by the total number of atoms, and hence it besides, the similar saturation is arose at second dip. saturates at high input energy. make the z, ω = ω unchanged. The effective coupling IV. OPTICAL FESHBACH RESONANCE, 3 4 ATOMIC INTERACTION AND FLOQUET strength is therefore a function of total atom number N, COLLECTIVE EXCITATION as illustrated. We can show that at the experimental practical region, where cN ω, the effective coupling Each atom could interact with each other where the strength is linear increase of Ntot. This would result in a linear quantum control of cavity mode, as a useful tool interaction could be described well by contact potential for quantum engineering based on on-chip device. How- U = gδ(r) ever, the control is determined exclusively by the atom I number and cavity shape (g), both hard to vary dur- where ing experiments, and hence is not efficient for practical 2 use. The following approach provides a solution to this 4π as g = ~ (8) dilemma. m In addition, by adding a constant magnetic field and as is s-wave scattering length. The scattering length around, i.e., adding additional term µBB|↑ih↑|, the sys- could be tuned by magnetic field due to the emergence tem shows a very complicated behavior of transmission of zero energy bound state spectrum, and begin regularized by the input power. In other words, the system enters nonlinear region with ∆ a(B) = abg 1 − (9) nonzero magnetic field. In this region, the quantum B − B0 control over microcavity is derived: by controlling the input power P , one would slightly control the peak It could also be tuned by |E × E∗| term in same man- position in transmission spectrum, which is defined as ner, and the formula just replaces B by αI, I for light √ √ T = |P − κmam,0 − κm+1am+1,0|/|P |. Numerically intensity and α is the coefficient, as in Fig.4. It could calculated spectrum is shown in Fig.2, where a sensitive be seen that the contact potential diverges when B = B0 5

simple. The light intensity has a nonlinear dependency of input power, due to the interacting between cavity mode and atom. We here can show a direct mapping from [29] the magnetic field approach to our optical approach that if the input power is a function of time I(t) = I + ∆I cos(Ωt), the scattering length exhibits 0 0 following behavior 1 1 Ω − Ω = 0 + iγ a(Ω) a¯ Ω − Ω0 − δ and therefore being able to tune the scattering length and at the same time avoid heating and atom loss due to resonance, i.e., it can enhance the real part of scattering length while still having small imaginary part. The advantage for this setup is that the controlling of light FIG. 4. Illustration of scattering length as a function of light intensity is much more flexible than the controlling of intensity. Shaded region for scattering length exceeds the area magnetic field, for the latter one always involves com- of near field light spot and therefore the description of system plicated current control due to large inductor. The light is no longer valid. intensity is quite simple to control even at high frequency. V. SUMMARY

In this work we briefly introduce circular polarization or αI = B0, indicating a strong interacting. However, around these resonance place, the atoms got heated very to the annular microcavity, which absences in circular mi- fast, results in a remarkable atom loss. Recently a floquet crocavity. This on the other hand could perform a vector approach [29] showed a modulated magnetic Feshbach polarization on atom and therefore could regularize both resonance which means that a time periodic driven mag- the cavity’s and the atom’s behavior. We then analyze netic field could effectively reach higher scattering length both sides, and find some nontrivial behavior and poten- so that might solve the atom loss issue when reaching tial application. In particular, this could be useful for strong interaction. However, even though the required controlling the interaction between atom. The previous frequency for driving the magnetic field is not high for difficulties in controlling interaction between atoms are 87Rb, this could limit a potential application to other either the loss due to resonance or the technique in mag- species of atom. Considering the driven frequency of light netic field precise control. In our setup both difficulties intensity could be really high, this approach could also be are solved. used in an optical control of Feshbach resonance, where This project was supported by Ministry of Science and all procedures are exactly the same. Technology of the People’s Republic of China (MOST) (2016YFA0301302, 2013CB921904, and 2013CB328704); However, to obtain a same form of time-dependent National Natural Science Foundation of China (NSFC) scattering length a(t), the control of input power is not (61435001 and 11474011).

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