PHYSICAL REVIEW A 93, 053603 (2016)

Sagnac with coherent vortex superposition states in - condensates

Frederick Ira Moxley III,1,* Jonathan P. Dowling,1 Weizhong Dai,2 and Tim Byrnes3,4,5 1Hearne Institute for Theoretical Physics, Department of Physics & Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA 2College of Engineering & Science, Louisiana Tech University, Ruston, Louisiana 71272, USA 3New York University, 1555 Century Avenue, Pudong, Shanghai 200122, China 4NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China 5National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan (Received 4 January 2016; published 4 May 2016)

We investigate prospects of using counter-rotating vortex superposition states in nonequilibrium exciton- polariton Bose-Einstein condensates for the purposes of Sagnac interferometry. We first investigate the stability of vortex-antivortex superposition states, and show that they survive at steady state in a variety of configurations. Counter-rotating vortex superpositions are of potential interest to gyroscope and seismometer applications for detecting rotations. Methods of improving the sensitivity are investigated by targeting high momentum states via metastable condensation, and the application of periodic lattices. The sensitivity of the polariton gyroscope is compared to its optical and atomic counterparts. Due to the large interferometer areas in optical systems and small de Broglie wavelengths for atomic BECs, the sensitivity per detected is found to be considerably less for the polariton gyroscope than with competing methods. However, polariton gyroscopes have an advantage over atomic BECs in a high signal-to-noise ratio, and have other practical advantages such as room-temperature operation, area independence, and robust design. We estimate that the final sensitivities including signal-to-noise aspects are competitive with existing methods.

DOI: 10.1103/PhysRevA.93.053603

I. INTRODUCTION [16]. On the other hand, exciton- (or polaritons for simplicity) typically have very high critical temperatures, due Atomic Bose-Einstein condensation (BEC) was first − to their exceedingly light effective mass (i.e., ∼10 5 times achieved in 1995 [1,2], which inspired a massive worldwide the bare electron mass). Consequently, this opens the door research effort into investigating its properties. Today, atomic for new quantum technologies based on room-temperature BECs are being considered for applications in a variety of superfluidity (i.e., “polaritonics”) [17], and avenues for novel fields, such as quantum simulation [3], quantum information light sources [9]. [4,5], and quantum metrology [6]. Recently, it has become One particular proposal based on BECs that has gathered apparent that BECs can be made in a variety of different recent interest is the matter analog of the Sagnac systems. In particular, a BEC of excitations in interferometer, also called a “quantum gyroscope.” An in- semiconductor systems consisting of exciton-polaritons have tuitive way to view these are as a neutral species version been observed [7,8]. Exciton-polariton BECs are formed in of a superconducting quantum interference device (SQUID) semiconductor microcavities [9], or planar Fabry- [18]. SQUIDs are well known to be the most sensitive Perot´ optical cavities [10], for example, and can be described magnetometers, with capabilities of detecting small magnetic as half-matter, half-light excitations that are typically created − fields in the region of 5 × 10 18 T[19]. Due to the close using laser injection methods [11,12]. Exciton-polariton BECs analogy of a magnetic field of a charged particle and rotation in are a nonequilibrium phenomenon that can be achieved at room the Schrodinger¨ equation, one may envision a device based on temperatures for suitable materials that support the polaritons BECs that is highly sensitive to rotations, instead of magnetic [13]. In materials where polaritons can be operated at room fields. For this reason, ultracold atomic gases have been temperature, interactions tend to be weaker, however, more considered to be an outstanding candidate physical system for recently the nonlinearities have been observed to be significant realizing such a Sagnac interferometer [20,21]. One advantage [14]. Despite the nonequilibrium nature of the polariton BEC, that atomic systems possess is the exquisite experimental it shares many of the properties of an atomic BEC, such control that is achievable, particularly for creating vortices as superfluidity, stable vortex formation, and high spatial in BECs [22]. Moreover, these ultracold atomic gases are coherence. highly coherent [23], meaning that they are particularly suited Although it is now commonplace to realize atomic BECs, for interferometry. However, from a practical perspective, typically they need to be cooled to very low temperatures in the one of the drawbacks of technologies that utilize ultracold region of ∼100 nK. Even for high-temperature superconduc- atomic gases is that in order to achieve BEC, experimental tors, which may be viewed as a type of charged condensation conditions need to be cooled to nanokelvin temperatures. [15], the highest critical temperature attained has been 138 K This can prove quite expensive, and difficult to achieve in a compact device design. In order to overcome this challenge, a natural extension of the idea is to use exciton-polariton *Corresponding author: [email protected] BECs [24,25], which may condense at room temperatures

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[26,27]. This offers a potential different configuration for achieving a novel Sagnac interferometer [28]. In this paper we investigate in detail the possibility of such a “polariton quantum gyroscope.” The type of measurement configuration investigated herein has the potential for providing a highly compact and stable Sagnac interferometer [29], which can also be used as a seismometer, for example. Future applications of quantum gyroscopes include the detection of earthquakes [30], gravitational [31,32], sidereal time [33], inertial sensing [34] capabilities, and even the rotational rate of our universe [35]. Such an exciton-polariton gyroscope would be a hybrid optical-matter wave Sagnac interferometer [36], as polaritons themselves consist of both and . In our scheme, one of the necessary ingredients for this interfer- FIG. 1. An experimental configuration for producing vortex- ometer will be the optical excitation of vortices [37]inthe antivortex superpositions in exciton-polariton Bose-Einstein conden- polariton BEC. The direct excitation of vortices by optical sates. A laser generates a TEM00 Gaussian beam profile. This light means is a convenient way of producing the required vortex is passed through a spiral phase plate (SPP), generating an orbital superposition state in the condensate, as techniques to generate angular momentum (OAM) state of light. Once the OAM state of the optical counterparts are well established. Here, it should light has traveled through the Mach-Zehnder interferometer with a be pointed out that the counter-rotating vortex superpositon Dove prism (DP) and mirrors (M), an OAM superposition state of state differs from the vortex-antivortex pair such as those light is made. The resulting OAM superposition state of light can be observed in Ref. [38]. The investigation of various vortex used to generate an interference pattern in a polariton BEC matter states in polariton BECs has continued to gain much interest, wave within the distributed Bragg reflector (DBR) microcavity. This interference pattern is imaged by a detector such as a charge-coupled with several works examining disorder-induced vortices [39], device (CCD), and can then be used to determine the Sagnac phase vortex-antivortex pairs [27,38,40], and vortex ring creation and therefore the rotational rate of the polariton BEC. via rotation of Gaussian laser beams [41], to name a few. One of the most efficient methods of vortex creation is using Laguerre-Gauss optical states which carry a well-defined external orbital angular momentum (OAM) [42–44]. However, II. VORTEX-ANTIVORTEX SUPERPOSITIONS IN for Sagnac interferometry, a superposition of a vortex and POLARITON BECS an antivortex is required, which has not been achieved in polariton BECs to date to our knowledge. While the vortex- A. Initialization by Laguerre-Gauss modes antivortex superposition has been achieved in atomic systems We first describe how the vortex-antivortex superposition is [45], this remains an outstanding problem for polaritons, created in the exciton-polariton BEC. The experimental setup and is interesting from both a fundamental and applications for the creation of a vortex-antivortex polariton BEC is shown perspective. in Fig. 1, which is similar to the traditional optical (i.e., Mach- The purpose of this article is to investigate the prospects of Zehnder) Sagnac interferometer. Unlike the traditional Mach- angular momentum superposition states in exciton-polariton Zehnder scheme, our polariton BEC interferometer is based BECs for potential applications in Sagnac interferometry. In on a principle that employs OAM superposition states of light. Sec. II we present and analyze a scheme for generating vortex- Our starting point is a source of OAM states of light, which antivortex superpositions via coherent excitation with OAM can readily be produced by established methods, for example, states of light. In the scheme, the polariton BEC is initially by using spiral phase plates (SPPs). This may be used to create seeded with the desired OAM superposition, then allowed OAM superpositions, which coherently excite the polaritons to evolve freely via the open-dissipative Gross-Pitaevskii in a DBR structure (e.g., semiconductor). This produces the equation. This allows for the polaritons to approach their own initial seed polariton condensate, from which the polariton dynamical equilibrium state, independent of the initial state of BEC is reinforced by the standard process of exciton-polariton light. We then study methods to improve the sensitivities of the cooling and scattering. polariton BEC gyroscope in Sec. III. This includes using ring Let us denote the bosonic annihilation operator for an geometries to target high angular momentum states, and ways OAM mode with angular momentum number l and radial of generating this using metastable condensation. A second node number p as al,p (see Appendix A for more details). enhancement scheme is discussed, where periodic lattices are Then the superposition state for the mode that we require 2 2 imposed in order to introduce high momentum components is αal,p + βa−l,p, where |α| +|β| = 1. This OAM super- to the wave function. In Sec. IV, we discuss the sensitivities position state of light has two counter-rotating components of the Sagnac phase for the vortex superposition state, and which will induce an interference pattern in the polariton BEC. compare with other existing approaches. In particular, we will Once the characteristic interference of the OAM superposition derive the Sagnac phase for the vortex superposition and ring (consisting of 2l lobes) has been transferred to the polariton geometry cases, and compare the performance to optical and BEC, the Sagnac phase can then be determined from the atomic Sagnac interferometers. Finally, we will summarize our angular velocity of the BEC cloud. We can obtain an ordinary results and conclude in Sec. V. vortex state here simply by setting either α or β = 0[46].

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Many lasers emit light beams which approximate a Gaus- relation as the injected light. In this configuration alone, there sian profile with l = p = 0. Using the TEM00 fundamental is no difference of using the OAM superposition states of transverse mode of the optical resonator and channeling it light to the polariton BEC, as the polaritons inherit their through a SPP, for example, can be used to generate OAM vortex nature from the light. However, once the initial coherent states of light (see Appendix A). Although these states excitation is induced, the OAM superposition is turned off, and are generally made with holographs, SPPs offer a simple the microcavity system is excited by conventional polariton and convenient way to obtain a pure OAM state al,p with pumping. This can be performed by an off-resonant pumping arbitrary integer l. Moreover, once the pure state al,p has where the excitons have a much higher energy than the been obtained, we can generate an OAM superposition state. polaritons, or by pumping at a high in-plane momenta [see One way to accomplish this is by using a Mach-Zehnder type Fig. 1(b) in Ref. [8]]. Either method creates a large population of configuration as seen in Fig. 1, where the Dove prism is of uncondensed polaritons from which the polariton BEC used to transform√ the al,p state into a a−l,p state to obtain is replenished. It is important to note that apart form the (al,p + a−l,p)/ 2 at the real output of the Mach-Zehnder initial seed polariton BEC which is induced coherently, the interferometer. If desired, one may choose the first beam remaining population thereafter is not pumped with the OAM splitter as an |α|2:|β|2 beam splitter, and the second one as superposition state. Hence in the second phase where the a 50:50 beam splitter to obtain an arbitrary superposition of conventional polariton BEC pumping is used, any OAM the type αal,p + βa−l,p. superposition state is a genuine steady state of the polariton Let us now consider the specific case as shown in Fig. 1, BEC, and possesses its own dynamics. where the two inputs of the first beam splitter are prepared in While both vortices and vortex-antivortex pairs have a coherent state in the LG mode al,p and the vacuum state, been shown theoretically and experimentally to be stable respectively. Hence, configurations of polariton BECs [38,47], whether or not a counter-rotating vortex superposition is a stable configuration = aIN al,p. (1) is a nontrivial problem. Such vortex-antivortex superpositions Assuming that the first beam splitter being used is a symmetric have a more complex phase relation than standard vortex con- 50:50 beam splitter, the OAM state of light has transformed figurations, and have the possibility to spontaneously separate like spatially, or relax to a zero momentum state. To investigate this, we have studied the vortex-antivortex superposition in a 1 aIN = al,p → √ (bl,p + cl,p), (2) variety of different circumstances to evaluate its stability. 2 The spatiotemporal evolution of the nonequilibrium where bl,p,cl,p denotes the LG modes in the two arms of exciton-polariton system is described by the open-dissipative the interferometer. The phase shift in the upper arm of Gross-Pitaevskii (dGP) equation, expressed as [15,48] the Mach-Zehnder interferometer will cause the coherent  amplitude to replaced by a → eiφa , and the Dove prism in ∂ψ(x,t)  l,p l,p i = − ∇2 + V (x) + g|ψ(x,t)|2 the lower arm of the Mach-Zehnder interferometer will cause ∂t 2m ext the coherent amplitude to be replaced by b → b− . Hence,  l,p l,p i the OAM state of light seen in Eq. (2) becomes + (P (x) − γ − η|ψ(x,t)|2) ψ(x,t), (5) 2 1 1 iφ √ (bl,p + cl,p) → √ (e bl,p + c−l,p). (3) 2 2 where ψ(x,t) is a complex valued condensate order parameter, Finally, neglecting the imaginary output of the interferometer, g is the polariton-polariton interaction constant, m is the the second beam splitter reverses the transformation and polariton mass, Vext(x) is a spatially dependent trapping produces the OAM superposition state of light, written as potential energy (see Fig. 5 in Ref. [8]), P (x) is a spatially dependent pumping rate, γ is the polariton loss rate, and η 1 iφ is the gain saturation [49,50]. Throughout this paper we will aOUT = √ (al,p + e a−l,p), (4) 2 assume that the time scale is in units of 2ma/, where a is which is our desired output. The superposition of a and the length scale set by the experiment (e.g., size of trapping l,p potential or pump spot diameter). a−l,p states in Eq. (4) can be detected using many different techniques, which typically includes imaging the interfer- We use the the generalized finite-difference time domain ence pattern of the light beams which constitute the OAM (G-FDTD) scheme to solve the dGP equation. G-FDTD is an superposition state. In particular once the polariton BEC is explicit method that permits an accurate solution with simple induced, one may use standard spatial and momentum space computation for solving Eq. (5) in multiple dimensions. To this imaging methods to analyze the coherent condensate state in end, the function ψ(x,t) is first split into real and imaginary the polariton microcavity [8]. components resulting in two coupled equations. The real and imaginary components are then approximated using higher- order Taylor series expansions in time, where the derivatives B. Exciton-polariton BEC dynamics in time are then substituted into the derivatives in space via The laser injection methods of the previous section can be the coupled equations. Finally, the derivatives in space are used to resonantly excite the polariton BEC. The phase of approximated using higher-order finite difference methods. the light is directly imprinted on the polariton BEC system, The G-FDTD has been successfully applied for solving both hence the injected polaritons simply follow the same phase linear and nonlinear Schrodinger¨ equations [51,52].

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FIG. 2. Time evolution of a polariton BEC initially seeded with FIG. 3. Time evolution of a polariton BEC initially seeded with a a vortex-antivortex superposition for a flat potential Vext(x) = 0. vortex-antivortex superposition for a flat potential Vext(x) = 0. Time Time frames for the (a) initial state t = 0; (b) intermediate state frames for the (a) initial state t = 0; (b) intermediate state t = 5; and t = 5; and (c) steady state t = 10 are shown. Parameters used were (c) steady state t = 10 are shown. Time scales are in units of /meV. 2 −(r/r0) P (x,y) = P0e ,whereP0 = 2, r0 = 5.35, and g = γ = η = 1. The following parameters were used in the simulations: g = 0.05, Time scales are in units of 2ma2/ where a is the length scale unit. γ = 1.0, and η = 0.1, where energies are in meV.

C. Stability of vortex-antivortex superpositions We now examine the stability of the vortex-antivortex leakage and gain, but stability is maintained such that there is superposition under the dynamical evolution of the dGP no net loss. equation. In Fig. 2, we show the spatiotemporal evolution of From the initial condition given by Eq. (6), the system is the nonequilibrium polariton BEC vortex superposition state time evolved under Eq. (5) until a steady state is reached. V = Figure 2 illustrates the condensate density distribution in a flat potential landscape ext 0. The initial condition was | |2 chosen as ψ(x,y,t) of the two-dimensional vortex nonequilibrium condensate in a flat potential at various times, and also −   P (x,y) γ LG LG illustrates the phase [i.e., the argument of ψ(x,y,t)] of the ψ(x,y,t = 0) = √ u = = + u =− = , (6) 2η l 1,p 0 l 1,p 0 two-dimensional vortex nonequilibrium condensate in a flat potential at various times. Upon examination of Fig. 2, we can where uLG are the spatial Laguerre-Gauss wave functions l,p see that our counter-rotating vortex-antivortex superposition P (x,y)−γ defined in Appendix A. The factor of η approximates state is coherent and stable as it is generated in the distributed the equilibrium density which the dGP equation converges to. Bragg reflector (DBR) microcavity outlined in Fig. 1. Here, it For densities near equilibrium, should be pointed out that the vortex-antivortex superposition P (x,y) − γ has the characteristic phase dependence of the lobes being |ψ(x,y,t)|2 = , (7) out of phase with each other by a factor of π. The system η quickly reaches steady state and little change is seen beyond the imaginary terms on the right-hand side of Eq. (5) vanish t = 10. The stability of the counter-rotating vortex-antivortex and we may obtain a zeroth order estimate for solutions superposition can be attributed to the same reason as for atomic [53]. Starting from this approximation, the numerics should BECs: Both the vortex and antivortex require a density defect converge towards the true solution for the dGP equation, which due to the phase velocity diverging at the vortex core. Thus will in general be different from the standard (conservative) it is energetically favorable for both to share the same vortex Gross-Pitaevskii solution. Throughout this study, we assumed core, effectively pinning them in the same location. that the wave function in the z direction has an insignificant In Fig. 3 we also simulate the vortex-antivortex superposi- effect on the vortex dynamics in the x − y plane. In the tion for experimental parameters as given in Refs. [14,54,55]. limit P = γ = η = 0, the condensate is conservative (i.e., the We see that generally the results are unchanged and the particle number is fixed and there is nothing entering or leaving interference lobes are stable under different parameters. As the condensate). With P − γ,η > 0, the condensate has both the vortex superposition state consists of lobes, the Sagnac

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FIG. 4. Time evolution of a polariton BEC initially seeded with FIG. 5. Time evolution of a polariton BEC initially seeded with a vortex-antivortex superposition for a disordered potential Vext = a vortex-antivortex superposition for a disordered potential Vext = rand(). Time frames for the (a) initial state t = 0; (b) intermediate rand(). Time frames for the (a) initial state t = 0; (b) intermediate state state t = 5; and (c) steady state t = 10 are shown. Parameters used t = 5; and (c) steady state t = 10 are shown. Time scales are in units 2 −(r/r0) were P (x,y) = P0e ,whereP0 = 2, r0 = 5.35, and g = γ = of /meV. The following parameters were used in the simulations: η = 1. Time scales are in units of 2ma2/ where a is the length scale g = 0.05, γ = 1.0, and η = 0.1, where energies are in meV. unit. phase can be determined from the rotation of the lobes about stable in polariton BECs. In the context of gyroscopy, rotations an axis of rotation. This will be discussed in more detail in are detected by changes in the lobe interference pattern as Sec. IV. found in Figs. 2–6. Thus for sensitive detections one would While Fig. 2 suggests that the vortex-antivortex superpo- like the lobes to be as sharply defined as possible, to be able sition is stable, a realistic semiconductor system does not to discriminate between the original and rotated interference patterns. For vortex-antivortex superpositions in polaritons, have a perfectly smooth potential Vext = 0. There are typically unavoidable sources of disorder, which may potentially affect this can be achieved using large angular momentum states l the stability of the superposition states. To investigate this, (see Fig. 2 of Ref. [56]). While this is possible in principle, we study the evolution of the vortex-antivortex superposition there are practical limits on the high angular momentum state in a disordered potential landscape. Upon examination of Laguerre-Gauss modes that can be excited. Hence alternative Fig. 4, we can see that in the presence of a disordered potential, strategies for sensitivity boosts are desirable for the polariton our vortex superposition state is coherent and stable as it is gyroscope. generated in the distributed Bragg reflector (DBR) microcavity outlined in Fig. 1. Figure 4 also illustrates the phase of A. Metastable condensation in ring geometries the two-dimensional vortex nonequilibrium condensate in a disordered potential, which shows in general the same In general, higher sensitivities are achieved by the use of behavior as the flat potential case. We have tried this for high momentum states due to their short wavelength, allowing several disorder types and strengths, and find that the vortex- for high resolutions. A variant of the vortex-antivortex super- antivortex superposition is stable in all cases. Again, upon position state is to use ring geometries, where the polariton examination of Fig. 5, we can see that with experimental condensate is placed in a circular narrow channel potential. parameters taken from Refs. [14,54,55] the results are largely In a BEC, high angular momentum states are energetically unchanged. disfavored, and typically multiquantized vortices break up into many l =±1 vortices [53]. This can be understood to result from the high momentum states that are near the vortex core III. SENSITIVITY ENHANCEMENTS FOR which have a rapid phase variation. By eliminating the vortex POLARITON GYROSCOPY core with the use of a ring potential, high angular momentum The results of Sec. II suggest that counter-rotating vortex- states become ultrastable, which is beneficial from a sensitivity antivortex superpositions can be coherently excited and are point of view.

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the superpositions states is via metastable condensation. In Ref. [57] it was experimentally found that in a one-dimensional periodic lattice, polariton condensates at high momentum states could be generated. This was attributed to the tradeoff between relaxation from high energy states and the finite decay time of the polaritons. Here we show that it is possible to target particular momentum state superpositions using the metastable condensation dynamics and spatially dependent pumping profile. For large rings we may model the ring geometry as discussed above by a toroidal geometry where the x direction is the direction along the channel, and y is in the radial direction. Then the ring geometry imposes that the wave function is periodic in x, ψ(x,y) = ψ(x + L,y), (11) where L is the circumference of the ring. We also take for convenience that ψ(x,y) = ψ(x,y + L) although this is not important for the purposes of our illustration, and other boundary conditions could be used just as efficiently. Our target configuration in this case is that we have two counterpropagating momentum modes ±k0. In a similar way as aforementioned, in the ideal case where only these two modes are initialized, this is a stable configuration of the FIG. 6. Counterpropagating currents in ring geometries. Cases polariton BEC. To examine what happens with an imperfect shown are (a) l =±1, t = 0; (b) l =±1, t = 10; (c) l =±5, t = initialization procedure, let us consider that we have in addition 0; (b) l =±5, t = 10. Parameters used are γ = η = V0 = g = 1, 2 a component with zero momentum, rmin = 5. A spatially dependent pump P (x,y) = P0|ψl (r,φ)| , with 2 P = 2 is used. Time scales are in units of 2ma / where a is the ik0x −ik0x 0 ψ(x,y,t = 0) = ξk=0 + ξk (e + e ). (12) length scale unit. 0 We compare two cases where the pump is spatially uniform,

In Fig. 6 we show the stability of counterpropagating P (x,y) = P0, (13) polariton superfluid currents in ring geometries. The potential that is used is a “Mexican hat” form, and a spatially varying pump which has the same pumping   profile as the target wave function density, which in this case r4 r2 is cos2 k x. V (r) = V − 2 , (8) 0 ext 0 4 2 rmin rmin = 2 +  P (x,y) P0η cos (k0x) γ. (14) = 2 + 2 where r x y . Expanding around the minima of the In the case of Ref. [57], the spatially periodic pump modulation channel, one obtains an approximate wave function, occurs because of the metallic deposits on the surface of the  √  DBR which physically blocks the laser pump. These metal V0 2 −ilφ ψl(r,φ) = exp − (r − rmin) e , (9) deposits also have the effect of producing a periodic potential rmin of the form, where we have added a phase variation around the ring with V (x,y) = V cos(2k x), (15) angular momentum l. We then start in the initial state, ext 0 0 induced by the change in the photonic boundary conditions ψ (r,φ) + ψ− (r,φ) ψ(r,φ) = l √ l , (10) at the surface [58]. The factor of 2 arises because for π state 2 condensation as observed in Ref. [57], the phase in the wave and evolve the system using Eq. (5). Figure 6 shows our results function changes by π between adjacent lattices. for evolving l =±1 and l =±5 superpositions. We see that Figure 7 shows our results. We start in a state with = = = = =− = steady-state configurations are reached with the superpositions an√ equal distribution of momenta, ξk 0 ξk k0 ξk k0 being maintained for all times, in the same way as for the 1/ 3. Under the uniform pumping scheme Eq. (13), the vortex-antivortex superpositions. steady state evolves towards dominantly the zero momentum While the above shows that with a suitable initialization mode. There is a spatial dependence to the real space wave of the polariton BEC with counterpropagating currents is a function according to the potential Eq. (15), but the phase is stable configuration, a question is how in practice such a the same across all the sites, indicating that it is dominantly a state would be induced. The method of coherent excitations zero momentum mode. This is explicitly seen by the Fourier as discussed in Sec. II A is viable for low angular momenta space decomposition in Fig. 7(b). For the periodic pumping superpositions, but for higher angular momenta superpositions Eq. (14), the system evolves instead towards the momenta this can be problematic [56]. An alternative method of forming ±k0. The spatial distribution appears to be identical to the

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density phase momentum 20 20 (a) 2 10 10 1

0 y 0 y/a

y/a 0 k a

-10 -10 -1 -2 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -2 -1 0 1 2 x/a x/a k x a 20 20 (b) 2 10 10 1

y 0 k a 0 y/a 0 y/a

-10 -10 -1 -2 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -2 -1 0 1 2 x/a x/a k x a 20 20 (c) 2 10 10 1

0 y

y/a 0

y/a 0 k a

-10 -10 -1 2 -20 -20 - -20 -10 0 10 20 -20 -10 0 10 20 -2 -1 0 1 2 x/a x/a k x a 2 2 density 0 |ψ| /|ψmax | phase −π π

FIG. 7. Metastable condensation dynamics. (a) The initial state with equal populations of the three momenta k = 0,k0, −k0. Time-evolved = = = = steady-state distributions for (b) uniform pumping√ Eq. (13) and (c) periodic pumping Eq. (14). Parameters used are γ η V0 g 1, = = = = = 2  P0 2, k0 2π/10, ξk=0 ξk=k0 ξk=−k0 1/ 3. Time scales are in units of 2ma / where a is the length scale unit. uniform pumping case, but the phase changes by π for adjacent B. Periodic potentials maxima. This may be compared to the density and phase Another strategy that can be used to enhance the sensitivity profiles in the ring geometry in Fig. 6, which shows the same is via the use of periodic potentials on the underlying DBR variation. lattice. The origin of this effect is the increased effective This above shows that a spatially dependent pump may mass of the polaritons in the periodic lattice. In general, be used to target particular momentum modes with the same when a periodic potential is applied to a system, the energy- periodicity. We note that without an initial seed momentum momentum dispersion breaks up into bands. Within each state, the pumping technique cannot generate the desired band, the stronger the potential that is applied, the flatter momentum modes. The reason is that unlike the coherent the dispersion becomes. This originates from the fact that excitation technique of Sec. II A, the pumping profile P (x,y) the kinetic energy is suppressed due to the reduced tunneling does not contain any phase information, hence cannot impose between adjacent sites. Flatter bands result in a larger effective any particular phase on the condensate. Thus some population mass of the particles. As the de Broglie wavelength of the in the desired momentum mode must be there initially, but from particles is λ = h/mv, where v is the velocity of the particle, there the pumping can reinforce this population, and diminish the other contributions. In practice, below the condensation a large mass gives a shorter wavelength, and thus increased threshold many momentum modes are occupied and hence sensitivities. as it only crosses the critical pump density the system A more direct physical argument which shows the in- should naturally gravitate towards the target momentum creased sensitivity is shown in Fig. 8. Due to Bloch’s states. theorem, any eigenstate in a periodic potential may be

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(a) (b) ψ |ψ| 2 Re Vext =0

V ext Vext >0

x x

FIG. 8. Schematic for sensitivity enhancement effect for periodic potentials. (a) The real part of the wave function in a Bloch band (solid line) and the periodic potential (dashed line). (b) The resulting interference due to two waves of opposing momentum. Due to the spatial modulation of the Bloch wave function, the resolution of the interference parttern is enhanced. written, FIG. 9. Time evolution of a polariton BEC initially seeded with a vortex-antivortex superposition for a Kagome lattice potential given ikx = ψk(x) = e uk(x), (16) by Eq. (18). Time frames for the (a) initial state t 0; (b) intermediate state t = 5; and (c) steady state t = 10 are shown. Time scales where uk(x) is a periodic function with the same periodicity are in units of /meV. The following parameters were used in the as the crystal and k is the wave number. The envelope function simulations: g = 0.05, γ = 1.0, and η = 0.1, where energies are uk(x) can be decomposed into Wannier functions, which in meV. for strong potentials can be approximated by the localized potential around one of the potential minima in the periodic ik0px/(1+4p/3) where f1(x) = e cos[k0px/(1 + 4√p/3)], b1 = function. Thus in a periodic potential with counterpropagating (1/(1 + 4p/3),0),√ b2 = (−1/(2 + 8p/3), − 3/2), b3 = modes the interference pattern is (−1/(2 + 8p/3), 3/2), V > 0 is the lattice intensity, k 0 0 − 2 is a scalar quantity proportional to the lattice constant, and eik0x + e ik0x √ u (x) = cos2(k x)u2 (x). (17) p → 3/2[67,68]. We implement the above potential in the k0 0 k0 2 dGP equation Eq. (5), and evolve forward in time starting As illustrated in Fig. 8, the periodic potential thus gives a from a resonantly induced vortex-antivortex superposition state given by Eq. (6). finer periodicity than the ±k0 modes. By detecting shifts in the finer fringes this can give rise to higher sensitivities for the In Fig. 9 we show the spatiotemporal evolution of the gyroscope. The mechanism for the increased sensitivity origi- vortex superposition state in a Kagome lattice potential. Upon nates from the introduction of high momentum components in examination of Fig. 9, we can see that in the presence of a Kagome lattice, our vortex superposition state is coherent and the periodic potential uk(x). This is consistent with the more stable as it is generated in the distributed Bragg reflector (DBR) straightforward strategy of increasing k0, except that the high momentum modes are naturally present in a periodic lattice. microcavity outlined in Fig. 1. Figure 9 also illustrates the For the standard approach one must inject the system with a phase of the two-dimensional vortex superposition nonequi- librium condensate in a Kagome lattice potential, which shows momentum ±k0. To check the stability of vortex-antivortex superpositions in the characteristic phase difference between the lobes of π.It periodic lattice potentials, we again perform a time evolution should be pointed out that the flat band in the Kagome lattice under Eq. (5). Specifically, we choose a Kagome or a bas- corresponds to the third lowest energy band. In our simulations ketweave lattice [59]. A Kagome lattice consists of interlaced the polaritons predominantly occupy the first band, hence triangles and exhibits a higher degree of frustration when the only effect of the bands is to periodically modulate the compared with other 2D lattices (e.g., square and triangular interference. In order to target the flat band, techniques such as lattices) [60]. A Kagome lattice is particularly interesting as in metastable condensation are required to create the condensate the tight-binding limit with only nearest-neighbor tunneling in these high momentum states. the 2D Kagome lattice geometry possesses a completely flat band. In such flat bands, the polariton condensate order IV. SAGNAC PHASE parameters become tightly localized at the potential dips A. Comparison to optical Sagnac interferometers [61]. Experimentally, several schemes have been proposed to create flat bands in the 2D Kagome lattice, utilizing Optical Mach-Zehnder interferometers [69] are typically metallo-photonic [62], structures capable of achieving an excellent rotational measurement [63], or depositing a thin metal film [57,58,64,65]. Flat bands sensitivity, thanks to the large area that the interferometric can also be engineered in a frustrated lattice of micropillar loop circumscribes [70,71]. In this respect, atomic matter optical cavities [66]. The lowest order Fourier decomposition wave interferometers typically have small loop areas and short-time rotational sensitivities [72,73]. However, matter of a Kagome lattice potential Vext can be described using just three components, written wave interferometers have the advantage that their de Broglie wavelength is much shorter than the optical counterparts. ik0b1·x ik0b2·x ik0b3·x 2 Vext = V0|f1(x)e + e + e | , (18) For atomic BECs, this makes the rotational measurement

053603-8 SAGNAC INTERFEROMETRY WITH COHERENT VORTEX . . . PHYSICAL REVIEW A 93, 053603 (2016) sensitivity of atomic matter wave interferometers per unit the area and wavelength dependence is reinstated, as we show area exceed that of optical ones by the ratio mc2/ω ∼ 1010 below. For the BEC case the Sagnac phase for polaritons can [36]. Typically polariton masses are extremely insignificant be written [74], −4 (i.e., meff ∼ 10 me), where me is the bare electron mass. 4mAloop This almost entirely negates the advantage of polaritons to φ (t) = N(t) , (22) 2  meffc /ω ∼ 1, which is the same level as photons. This is not surprising as polaritons themselves are part-matter, where N(t) is the number of times a polariton revolves around part-photon , hence much of their characteristics the loop in a time t. In a ring geometry of radius r as seen in are inherited from light. In terms of sensitivity, polaritons [46,75], we have for counter-propagating momenta ±k0, would appear to be ill-suited for such gyroscopic applications. tk0 We argue below that for counter-rotating vortex superposition- N(t) = . (23) based interferometers this argument does not directly apply, as 2πrm the Sagnac phase is independent of the de Broglie wavelength. This therefore gives Let us first review the state-of-the-art optical Sagnac φ (t) = 2k r t. (24) interferometers. Typically, in an optical interferometer, light of 0 wavelength λ propagates in a closed loop. For each completed We note here that each polariton in the BEC cloud has the propagation in the closed loop configuration, such as in a fiber same magnitude of the angular momentum, which contributes optic gyroscope, the Sagnac phase is given by to the signal-to-noise ratio as will be discussed below. The 8πA relation now has a boost in the sensitivity with the radius of φ = loop , (19) the ring, which arises from the fact that the velocity of a wave λc with momentum k0 has a value v = k0/m. This is in contrast where the λ is the wavelength of the light, is the angular to a vortex which decays with radius v = l/mr. (rotational) velocity, c is the speed of light, and Aloop is the area In the ring geometry we regain the enhancement due to of the Mach-Zender interferometer. In a ring laser gyroscope, the de Broglie wavelength in Eq. (24), as for heavy particles, the quantity that is measured is the beat frequency of two the typical wave number k0 = 2π/λ is much higher, which counterpropagating lasers. In terms of the phase difference, also results in enhanced sensitivities. In this sense, the use of this is now a time-evolving quantity of the form, periodic potentials as discussed in the previous section can πA t aid the sensitivity, as this produces effectively larger effective = 8 loop φ , (20) masses, and hence larger typical k0. We note that on setting λp 2 k0 = 2π/λ, Aloop = πr , and p = 2πr,Eq.(24) reduces to where p is the perimeter of the interferometric loop. The fiber Eq. (20), so that BEC-based gyroscopes can be considered to optic gyroscope and the ring laser gyroscope phases differ in be equivalent to the ring laser gyroscopes up to geometrical operation in that the latter is time dependent, while the former factors and the wavelength of the particles being used. is not. This means that as the system rotates, the fiber optic gyroscope produces a shift in the fringes only when there is B. Detection of Sagnac phase a rotation , while the ring laser gyroscope detects the total time-varying phase of the rotation. Therefore the longer one When the laboratory frame rotates in time, the two counter- waits between comparisons of the interference fringes, the rotating matter waves pick up different phases, and the = larger the angle difference, and therefore the sensitivity. transverse density profile at z 0 is given as In both types of optical gyroscopes, the phase is propor- {1 + cos[2l(φ + t)]}|ψ(x,t)|2, (25) tional to the area of the interferometric loop. This is quite different to the situation in a BEC where the phase around a or for the case of the ring geometry, vortex is fixed purely by the topological winding number. Thus (1 + cos[2k r(φ + t)])|ψ(x,t)|2, (26) independent of the path, a phase of 2πl is picked up around 0 a vortex. For a fixed l, the path can in principle enclose an signifying a rotation of the interference pattern as the Sagnac arbitrary area and yet will not pick up any additional phase. It phase accumulates over time. Following the rotation of the is also independent of the de Broglie wavelength λ for the same lobes in the interference pattern, one may determine the reasons. Therefore, for a counter-rotating vortex-antivortex angular velocity from the density profile of the polariton superposition, the Sagnac phase is independent of the area BEC. The condensate density will show an interference pattern of the BEC and the mass of the particles (i.e., polaritons) determined by the phase difference between the amplitudes involved. As the Sagnac phase directly originates from the and charges of the two vortex components, as seen in Figs. 2–6. interference of two counter-rotating angular momentum, it This allows the polariton BEC to be used for the measurement follows that the the Sagnac phase can then be written as of changes in the Sagnac phase caused by the rotation of the laboratory frame of reference (i.e., the interferometer itself). φ (t) = 2l t, (21) This can be performed using existing experimental techniques and can be determined from the of the vortex superposition where a CCD camera is used to image the photoluminescence state in a polariton BEC. from the polariton BEC. Is it possible to regain the geometry and wavelength Determining the Sagnac phase using our proposed polariton dependence of Eq. (19) in a BEC to further improve the Sagnac interferometer will depend on the vortex charge l of sensitivity? For the ring geometry considered in Sec. III A the polariton BEC superposition state. In practice, spiral phase

053603-9 MOXLEY III, DOWLING, DAI, AND BYRNES PHYSICAL REVIEW A 93, 053603 (2016) plates (SPPs) generally work best for low integer values of l. TABLE I. Performance comparison for various Sagnac phase However, in principle, l can take on any integer value, although technologies. low integer values of l generally give a more stable vortex superposition. The proposed polariton Sagnac interferometer Sensitivity −1 −1/2 is based on imaging the standing wave of the polariton BEC, Sagnac Technology min [rad s Hz ] Comparison which is comparably much smaller than the area occupied by Polariton BECs (theory) 1 × 10−11 1 comparable optical technologies (e.g., ring-laser gyroscope). Ultracold Atoms 6 × 10−10 60 Thus the polariton ring geometry has the potential for higher Ring Laser Gyro (1m2)1.3 × 10−9 130 sensitivities. Superfluid Helium 2 × 10−7 20,000 We are now in a position to compare the rotational phases Fiber Optic Gyro (Commercial) 3 × 10−7 30,000 −6 per particle of the interference patterns φ . Due to the similar Nanofabricated Gratings 3.6 × 10 360,000 nature of the operation, it is simplest to compare the polariton Moire Classical Fringes 3.6 × 10−5 3,600,000 gyroscopes to the ring laser gyroscope, which both have an increasing Sagnac phase in the time domain. For the ring 2 laser gyroscope, taking Aloop ∼ 1m and p ∼ 1 m, and typical With regard to the stability of our proposed polariton optical wavelengths, one obtains the proportionality of t Sagnac interferometer, it bypasses the beam drift that is a 7 in Eq. (20)tobe8πAloop/λp ∼ 10 . For ultracold atom common cause of instability in the metrology that employs 2 implementations, Aloop ∼ 1mm , p ∼ 1 cm, and velocities atomic or optical beams, and no complicated atomic beam 6 v ∼ 1m/swehave8πAloop/λp ∼ 10 [76]. For the polariton configurations are required. The technical issues that optical −1 gyroscope with ring radii of r ∼ 100 μm and k0 ∼ 10 μm Sagnac interferometers face, such as the short-time rotational [57], this gives the coefficient of the rotational phase in sensitivity problem [78], and the change in phase velocity of 3 Eq. (24)ask0r ∼ 10 . Thus in terms of the best achievable light [79], should not be present for polaritons by construction. sensitivities per particle, polariton gyroscopes appear to have Moreover, thermal expansion over atomic beam trajectories is a large handicap compared to other methods due to their long not an issue for a polariton BEC, as atomic beams are not wavelengths and relatively small interferometric loop areas. needed. Interferometers which use atomic beams usually have However, we shall see in the next section that some of this beam drifts on the order of μ deg /h after hours of running deficit is made up by the superior signal-to-noise of polaritons. time [80]. The simplicity of our scheme (i.e., no large optical loops in a solid-state device), would suggest that our scheme should not suffer as seriously from stability as seen in Table I. C. Signal-to-noise ratio and stability It should be pointed out that although Sagnac interferometry We now discuss the signal-to-noise ratio (SNR) and the with coherent vortex superposition states in exciton polariton sensitivity achievable in such a polariton Sagnac interferome- condensates has not been verified experimentally to date, we ter. The SNR is given by the ratio of the rotational phase shift present in Table I a theoretical performance estimate against to the shot noise that depends on the number of photons N competing technologies which have. arriving at the CCD from the polariton condensate per second We note that for purposes of seismometry, since the vector [74,77], of angular velocity is proportional to the vector of ground  velocity v, SNR = φ (t)/ 1/N. (27) = 1 ∇ × 2 v. (31) The final SNR has a maximum at a certain rate of photon In this sense a quantum polariton seismometer can detect the detection, and the sensitivity min is calculated by setting SNR = 1. Thus, for the superposition in the geometries as ground velocity from the Sagnac phase by detecting rotations seen in Figs. 2–5, that originate from unaccounted rotations of the ground. √ min = 1/(2lt N), (28) V. SUMMARY AND CONCLUSION and for the vortex-antivortex superposition in the ring geome- In this study, we have introduced two schemes for inducing try of Fig. 6 we have counter-rotating vortex superposition states in a polariton BEC √ for gyroscopy. The first is via direct injection of orbital angular min = 1/(2k0rt N). (29) momentum (OAM) of light through a distributed Bragg reflec- tor (DBR) microcavity. Once the vortex superposition state is 9 −2 Typical polariton densities are in the region of 10 cm , and induced, it follows a free evolution under standard pump-loss 2 a polariton spot size of 100 μm gives a polariton number dynamics of the polariton BEC. The second is via metastable 3 in the region of 10 . The lifetime of the polaritons is in the condensation in periodic lattices, where particular momentum region of ∼10 ps, hence we can estimate that the number of modes are reinforced by using pumping profiles with the target 14 −1 signal photons is N ∼ 10 s for polaritons. This gives our periodicity. By directly numerically simulating the dynamics, estimate for the polariton vortex-antivortex superposition in we found that the vortex-antivortex superposition state is stable the ring geometry, under a variety of different conditions, in smooth, disordered, ∼ 10−11 rad s−1 Hz−1/2. (30) and periodic potentials. In all cases we found that the vortex min superposition state is stable and is long lived. This is comparable with state-of-the-art atomic beam gyro- We investigated the possibility that the superposition of scopes which are at the 10−10 rad s−1 Hz−1/2 level [72,76]. counter-rotating currents in the polariton BEC can be used as

053603-10 SAGNAC INTERFEROMETRY WITH COHERENT VORTEX . . . PHYSICAL REVIEW A 93, 053603 (2016) a method for determining the Sagnac phase—and thus the an- Gaussian (i.e., LGl,p) beam mode of quantized OAM with gular velocity of the interferometer—effectively constituting quantum numbers l and p, and the Dove prism changes the a “quantum gyroscope.” The vortex-antivortex superposition winding of the quantized OAM from l to −l. In addition to has the same Sagnac phase dependence as atomic BECs, as the generation using spiral phase plates [82], cylindrical lens mode phase relation around a vortex is topologically fixed, regardless converters [83], and computer generated holograms [84]are of the particle mass or the area of the BEC. Optical Sagnac alternative methods to generate integer quantized OAM. The interferometers have large sensitivities by increasing the area LGl,p beam modes form a complete orthonormal basis set of of the interference loop, and atomic Sagnac interferometers solutions for paraxial light beams commonly found in lasers. have an advantage due to their small de Broglie wavelength. These modes can be expressed as [85] √ In this respect, polariton gyroscopes have relatively small loop  | |   2p! 2r l −r2 areas and long wavelengths. This can be enhanced using ring uLG(r,φ,z) = exp geometries or periodic potentials. In comparison with current l,p πw2(z)(|l|+p)! w(z) w2(z) technology it would appear that on a per polariton basis atomic   2r2 and optical interferometers have a larger sensitivity. However, × L|l| the polaritons do possess the great advantage that their p w2(z)    signal-to-noise ratio should be considerably larger than that kr2 of atomic BECs. In an atomic BEC the optical measurement × exp − i + lφ − ϕ(z) , (A1) R z has the limitation that it should not disturb the system [81], 2 ( ) hence there are restrictions on how bright the probe light where the radius of the beam squared is w2(z) = 2(z2 + should be. In contrast, for polaritons they are themselves b2)/kb, the radius of curvature is R(z) = (z2 + b2)/z, part light, hence illumination by laser light is intrinsic to the the Gouy phase ϕ(z) = (2p +|l|+1) tan−1(z/b),bis the |l| condensation process. Thus while the phase dependence of the Rayleigh range, k is the wave number, and Lp (r) describe polariton Sagnac interferometer starts with several orders of the Laguerre polynomials, magnitude deficit, the final estimated sensitivity is competitive p with existing methods. | | (l + p)! L l (r) = (−1)m rm. (A2) Polariton BEC-based gyroscopes have the additional practi- p − | |+ = (p m)!( l m)!m! cal advantage to atomic BECs that they should lead to compact m 0 and room temperature compatible devices, as it is based on In Eqs. (A1) and (A2), p is the number of nonaxial radial semiconductor technology. This could lead to a device with nodes, and the index l is known as the winding number. The the desirable characteristics of compact operation with very winding number describes the helical structure of the wave high sensitivity and stability. For the metastable condensation front and the number of times the phase jumps occur around technique, the ability of electrical injection of the condensate the beam path in the azimuthal direction. When l = p = 0, the would remove the need for optics entirely, simplifying the LGl,p beam is identical to the fundamental Gaussian beam, or device further. Due to the limitations in the size of the polariton TEM00. condensate, they are naturally suited towards small volume The higher order modes from the fundamental mode applications. The effective mass of the polaritons can be tuned may be calculated using the operator algebra formalism by engineering periodic structures, and conventional methods [86]. Analogous to the two-dimensional quantum harmonic such as changing the exciton fraction of the polaritons, which oscillator [87,88], at z = 0, give additional boosts to sensitivity. 1 Aˆx (0) = √ (kr cos(φ) + b cos(φ)∂r − br sin(φ)∂φ), 2bk ACKNOWLEDGMENTS (A3) F.I.M. would like to acknowledge support from NASA and EPSCoR & LaSPACE, Louisiana. J.P.D. would like to ac- 1 knowledge support from the AFOSR, the ARO, and the Aˆy (0) = √ (kr sin(φ) + b sin(φ)∂r + br cos(φ)∂φ). NSF. T.B. would like to acknowledge support from the 2bk Inamori foundation, NTT Basic Research Laboratories, and (A4) the Shanghai Research Challenge Fund. Shanghai Research These operators can be evolved in the z direction by a Challenge Fund, New York University Global Seed Grants for propagator Uˆ (z)for∂φ∂r = ∂r ∂φ according to Collaborative Research, National Natural Science Foundation   i of China Grant No. 61571301, and the Thousand Talents Uˆ (z) = exp − Pˆ2z Program for Distinguished Young Scholars. 2k   i = exp − (∂2 + r2∂2)z . (A5) APPENDIX: ORBITAL ANGULAR MOMENTUM 2k r φ STATES OF LIGHT Next, we can rewrite Eqs. (A3) and (A4)as OAM superposition states of light can be generated by † passing a TEM00 beam mode through a spiral phase plate Aˆx (z) = Uˆ (z)Aˆx (0)Uˆ (z), (A6) (SPP), followed by a Mach-Zehnder interferometer with a † Dove prism. The SPP converts the TEM00 to a Laguerre- Aˆy (z) = Uˆ (z)Aˆy (0)Uˆ (z), (A7)

053603-11 MOXLEY III, DOWLING, DAI, AND BYRNES PHYSICAL REVIEW A 93, 053603 (2016) where Higher-order LGl,p beam modes can be obtained from operat- 1 ingontheTEM00 according to [87] Aˆ±(z) = √ [Aˆx (z) ∓ iAˆy (z)] 2 1 † † 1 LG = ˆ n ˆ m = √ ∓iφ + ∓iφ + ±iφ + ul,p(z) [A−(z)] [A+(z)] TEM00(z), (A10) [kre (be 2ze )(∂r r∂φ)]. m!n! 2 bk (A8) where m = (l + p)/2 and n = (l − p)/2. We note that OAM LG LG The operators in Eq. (A8) obey the commutation rules, states ul,p(z) and u−l,p(z) differ only in the direction of the phase winding, that is, clockwise or counterclockwise, † † [Aˆ±(z),Aˆ±(z)] = 1,[Aˆ±(z),Aˆ∓(z)] = 0. (A9) respectively.

[1] W. Ketterle, M. R. Andrews, K. B. Davis, D. S. Durfee, D. M. [23] C. Yin, N. G. Berloff, V. M. Perez-Garc´ ´ıa,D.Novoa,A.V. Kurn,M.O.Mewes,andN.J.VanDruten,Phys. Scr. T66, 31 Carpentier, and H. Michinel, Phys.Rev.A83, 051605 (2011). (1996). [24] L. Gu, H. Huang, and Z. Gan, Phys.Rev.B84, 075402 (2011). [2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, [25] C. Sturm, D. Tanese, H. S. Nguyen, H. Flayac, E. Galopin, A. and E. A. Cornell, Science 269, 198 (1995). Lematre, I. Sagnes et al., Nature Commun. 5, 3278 (2014). [3] I. Bloch, J. Dalibard, and S. Nascimbene,` Nature Phys. 8, 267 [26] G. Lerario, A. Cannavale, D. Ballarini, L. Dominici, M. D. (2012). Giorgi, M. Liscidini, D. Gerace, D. Sanvitto, and G. Gigli, Opt. [4] T. Byrnes, K. Wen, and Y. Yamamoto, Phys. Rev. A 85, 040306 Lett. 39, 2068 (2014). (2012). [27] R. Dall, M. D. Fraser, A. S. Desyatnikov, G. Li, S. Brodbeck, M. [5] T. Byrnes, Phys. Rev. A 88, 023609 (2013). Kamp, C. Schneider, S. Hofling,¨ and E. A. Ostrovskaya, Phys. [6] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, Rev. Lett. 113, 200404 (2014). S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Kruger,¨ Nature [28] C. Gross, J.Phys.BAt.Mol.Opt.Phys.45, 103001 (2012). Phys. 1, 57 (2005). [29] Ch. J. Borde,´ Metrologia 39, 435 (2002). [7] H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. 82, 1489 [30] R. Michel, J. P. Ampuero, J. P. Avouac, N. Lapusta, S. Leprince, (2010). D. C. Redding, and S. N. Somala, IEEE TGRS 51, 695 (2013). [8] T. Byrnes, N. Y. Kim, and Y. Yamamoto, Nature Phys. 10, 803 [31] S. M. Dickerson, J. M. Hogan, A. Sugarbaker, D. M. S. Johnson, (2014). and M. A. Kasevich, Phys.Rev.Lett.111, 083001 (2013). [9] T. Byrnes, Y. Yamamoto, and P. V. Loock, Phys.Rev.B87, [32] K. Sun, M. M. Fejer, E. Gustafson, and R. L. Byer, Phys. Rev. 201301 (2013). Lett. 76, 3053 (1996). [10] J. D. Plumhof, T. Stoferle,¨ L. Mai, U. Scherf, and R. F. Mahrt, [33] M. A. Gasparini, Phys. Rev. D 72, 104012 (2005). Nature Mater. 13, 247 (2014). [34] M. Edwards, Nature Phys. 9, 68 (2013). [11] N. Masumoto, N. Y. Kim, T. Byrnes, K. Kusudo, A. Loffler,¨ S. [35] A. Delgado, W. P. Schleich, and G. Sussmann,¨ New J. Phys. 4, Hofling,¨ A. Forchel, and Y.Yamamoto, New J. Phys. 14, 065002 37 (2002). (2012). [36] J. P. Dowling, Phys.Rev.A57, 4736 (1998). [12] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. [37] G. Nandi, R. Walser, and W. P. Schleich, Phys.Rev.A69, Solnyshkov, G. Malpuech, E. Galopin, A. Lemaˆıtre, J. Bloch, 063606 (2004). and A. Amo, Phys. Rev. Lett. 112, 116402 (2014). [38] G. Roumpos, M. D. Fraser, A. Loffler,¨ S. Hofling,¨ A. Forchel, [13] R. Houdre,´ R. P. Stanley, U. Oesterle, M. Ilegems, and C. and Y. Yamamoto, Nature Phys. 7, 129 (2011). Weisbuch, Phys. Rev. B 49, 16761 (1994). [39] G. Grosso, G. Nardin, F. Morier-Genoud, Y. Leger,´ and B. [14] L. Ge, A. Nersisyan, B. Oztop, and H. E. Tureci,¨ Bull. Am. Deveaud-Pledran,´ Phys.Rev.Lett.107, 245301 (2011). Phys. Soc, 59 (2014). [40] G. Molina-Terriza, J. P. Torres, and L. Torner, Nature Phys. 3, [15] J. Keeling and N. G. Berloff, Contemp. Phys. 52, 131 (2011). 305 (2007). [16] A. Lesne and M. Lagues,¨ Scale Invariance: From Phase [41] A. I. Yakimenko, Y. M. Bidasyuk, O. O. Prikhodko, S. I. Transitions to Turbulence (Springer Science & Business Media, Vilchinskii, E. A. Ostrovskaya, and Y. S. Kivshar, Phys.Rev.A 2011). 88, 043637 (2013). [17] F. Medard,´ A. Trichet, Z. Chen, L. S. Dang, and M. Richard, [42] D. N. Krizhanovskii, D. M. Whittaker, R. A. Bradley, K. Guda, Phys. Quantum Fluids 177, 231 (2013). D. Sarkar, D. Sanvitto, L. Vina et al., Phys. Rev. Lett. 104, [18] G. Franchetti, N. G. Berloff, and J. J. Baumberg, 126402 (2010). arXiv:1210.1187 (2012). [43] D. Sanvitto, F. M. Marchetti, M. H. Szymanska,´ G. Tosi, M. [19] J. Clarke, Sci. Am. 271, 46 (1994). Baudisch, F. P. Laussy, D. N. Krizhanovskii et al., Nat. Phys. 6, [20] R. F. Max, P. Bohi,Y.Li,T.W.H¨ ansch,¨ A. Sinatra, and P. 527 (2010). Treutlein, Nature (London) 464, 1170 (2010). [44] K. P. Marzlin, W. Zhang, and E. M. Wright, Phys. Rev. Lett. 79, [21] A. Okulov, J. Low Temp. Phys. 171, 397 (2013). 4728 (1997). [22] F. Maucher, N. Henkel, M. Saffman, W. Krolikowski,´ S. Skupin, [45] K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, Phys. and T. Pohl, Phys.Rev.Lett.106, 170401 (2011). Rev. Lett. 102, 030405 (2009).

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[46] A. Dreismann, P. Cristofolini, R. Balili, G. Christmann, F. [66] F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Pinsker, N. G. Berloff, Z. Hatzopoulos, P. G. Savvidis, and J. J. Lemaˆıtre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. Tureci,¨ Baumberg, Proc. Natl. Acad. Sci. USA 111, 8770 (2014). A. Amo, and J. Bloch, Phys. Rev. Lett. 116, 066402 (2016). [47] G. Tosi, F. M. Marchetti, D. Sanvitto, C. Anton,´ M. H. [67] M. J. Ablowitz, S. D. Nixon, and Y. Zhu, Phys. Rev. A 79, Szymanska, A. Berceanu, C. Tejedor et al., Phys. Rev. Lett. 053830 (2009). 107, 036401 (2011). [68] K. J. H. Law, A. Saxena, P. G. Kevrekidis, and A. R. Bishop, [48] K. G. Lagoudakis, M. Wouters, M. Richard, A. Baas, I. Phys. Rev. A 79, 053818 (2009). Carusotto, R. Andre,´ L. S. Dang, and B. Deveaud-Pledran,´ [69] W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, Nature Phys. 4, 706 (2008). W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985). [49] M. Wouters and I. Carusotto, Phys.Rev.Lett.99, 140402 (2007). [70] R. Colella, A. W. Overhauser, and S. A. Werner, Phys.Rev.Lett. [50] M. Wouters and I. Carusotto, Phys.Rev.Lett.105, 020602 34, 1472 (1975). (2010). [71] F. Riehle, Th. Kisters, A. Witte, J. Helmcke, and Ch. J. Borde,´ [51] F. I. Moxley, T. Byrnes, B. Ma, Y. Yan, and W. Dai, J. Comput. Phys. Rev. Lett. 67, 177 (1991). Phys. 282, 303 (2015). [72] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys. Rev. [52] F. I. Moxley, D. T. Chuss, and W. Dai, Comput. Phys. Commun. Lett. 78, 2046 (1997). 184, 1834 (2013). [73] J. M. McGuirk, M. J. Snadden, and M. A. Kasevich, Phys. Rev. [53] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation Lett. 85, 4498 (2000). (Clarendon Press, Oxford, 2003). [74] S. Thanvanthri, K. T. Kapale, and J. P. Dowling, J. Mod. Opt. [54] F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. Andre,´ and B. 59, 1180 (2012). Deveaud-Pledran,´ Phys. Rev. Lett. 107, 106401 (2011). [75] V. K. Kalevich, M. M. Afanasiev, V. A. Lukoshkin, D. D. [55] M. Wouters, T. Liew, and V. Savona, Phys. Rev. B 82, 245315 Solnyshkov, G. Malpuech, K. V. Kavokin, S. I. Tsintzos, Z. (2010). Hatzopoulos, P. G. Savvidis, and A. V. Kavokin, Phys.Rev.B [56] R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, 91, 045305 (2015). S. Ramelow, and A. Zeilinger, Science 338, 640 (2012). [76] B. Barrett et al. C. R. Physique 15, 875 (2014). [57] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, [77] F. E. Zimmer and M. Fleischhauer, Phys.Rev.A74, 063609 M. D. Fraser, T. Byrnes et al., Nature (London) 450, 529 (2007). (2006). [58] N. Y.Kim, C. W. Lai, S. Utsunomiya, G. Roumpos, M. Fraser, H. [78] K. C. Wright, L. S. Leslie, and N. P. Bigelow, Phys.Rev.A77, Deng, T. Byrnes et al., Phys. Status Solidi B 245, 1076 (2008). 041601 (2008). [59] M. Mekata, Phys. Today 56(2), 12 (2003). [79] G. E. Stedman, Rep. Prog. Phys. 60, 615 (1997). [60] B. G. Levi, Phys. Today 60(2), 16 (2007). [80] D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, Phys. Rev. [61] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, Lett. 97, 240801 (2006). 1003 (1998). [81] E. Ilo-Okeke and T. Byrnes, Phys.Rev.Lett.112, 233602 (2014). [62] S. Endo, T. Oka, and H. Aoki, Phys.Rev.B81, 113104 [82] G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and (2010). M. J. Padgett, Opt. Commun. 127, 183 (1996). [63] H. Takeda, T. Takashima, and K. Yoshino, J. Phys.: Condens. [83] Chr. Tamm and C. O. Weiss, JOSA B 7, 1034 (1990). Matter 16, 6317 (2004). [84] N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, [64] N. Y. Kim, K. Kusudo, C. Wu, N. Masumoto, A. Loffler,¨ Opt. Lett. 17, 221 (1992). S. Hofling,¨ N. Kumada, L. Worschech, A. Forchel, and Y. [85] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Yamamoto, Nature Phys. 7, 681 (2011). Woerdman, Phys. Rev. A 45, 8185 (1992). [65] T. C. H. Liew, A. V. Kavokin, T. Ostatnicky,´ M. Kaliteevski, [86] D. Stoler, JOSA 71, 334 (1981). I. A. Shelykh, and R. A. Abram, Phys.Rev.B82, 033302 [87] G. Nienhuis and L. Allen, Phys.Rev.A48, 656 (1993). (2010). [88] A. Messiah, Mecanique´ Quantique 2, 444 (1995).

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