Quantum simulations with photons in one dimensional nonlinear waveguides 2

Chapter Pinning Quantum Phase Transition of Photons 3.1 Bose-Hubbard and Sine-Gordon Models The use of suitably controlled simple quantum systems to mimic the behavior of complex ones provides an exciting possibility to confirm existing physical theories and explore new physics. Two of the extremely successful models for describing a range of quantum many body effects and especially quantum phase transitions (QPT) [2, 44], are the Bose-Hubbard (BH) [6] and sine-Gordon (sG) models [2]. Cold atoms in optical lattices have so far been the most famous platform to implement these models, where it is possible to observe the MI to SF QPT for a weakly interacting gas in a deep lattice potential [6, 7]. More recently, it was made possible to tune up the interactions between the atoms in the gas leading to the realization of the sG model and the Pinning QPT [16, 45]. Alternative platforms in the field of quantum simulations of many body effects involve ions for quantum magnets [46], and photonic lattices for the understanding of in-and-out of equilibrium quantum many-body effects [47, 48, 49]. The photon-based ideas have initiated a stream of works in the many body properties of both closed and lossy cavity arrays [50, 51, 52, 53, 54, 55]. More recently, a new direction has appeared in 21 Chapter 3. Pinning Quantum Phase Transition of Photons the field of strongly correlated photons where hollowcore fibers filled with cold atomic gases or tapered fibers with cold atoms brought close to the surface of the fiber were considered [25, 26, 27, 28, 29, 30, 31, 32]. At low temperature, the speed of light can be substantially reduced, and the dephasing due to atomic collisions can be small enough to be ignored. The strong light confinement and the resulting large optical nonlinearities in the single photon level predicted for similar systems [22, 23, 24, 56] have motivated new proposals in the continuous polaritonic systems [57]. We will show here that it is possible to impose an effective lattice potential on the strongly interacting polaritonic gas in the fiber. This opens the possibilities for a large range of Hamiltonians to be simulated with photons. As examples we will study the simulation of the sG and BH models. We will show that the whole phase diagram of the MI to SF transitions for both models can be reproduced including a corresponding photonic “pinning transition”. We conclude with a discussion on the available tunability of the quantum optical parameters for the observation of the strongly correlated phases. The latter is possible by releasing the trapped polaritons and measuring the correlation on the photons emitted at the other end of the fiber. 3.2 Quantum Optical Simulator with OneSpecies Four-Level Atoms The considered atomic level structure is shown in Fig. 3.1. The 1D cold atomic ensemble is prepared outside the fiber using standard cold atom techniques and is transferred into or brought close to the surface of the fiber using techniques described in [25, 26, 27, 28, 29, 30, 31, 32]. The atoms are 22 3.2. Quantum Optical Simulator with One-Species Four-Level Atoms (a) (b) (c) (d) Figure 3.1: In (a) and (b) an ensemble of cold atoms with a 4-level structure is interacting with a pair of classical fields Ω± which create an effective stationary Bragg grating. The photons carried by the input pulse E+ coming in from the left are mapped to stationary excitations (polaritons) which are trapped in the grating. The strong photon nonlinearity which is induced by the 4th level leads to the creation of a strongly interacting gas (grey areas in (c) and (d). By modulating the density of the cold atomic ensemble, an effective lattice potential for polaritons/photons can be created-shown by red lines in (c) and (d). Tuning to the regime of weak interactions between polaritons and adjusting the lattice depth, the system undergoes a BH phase transition from SF (upper one) to MI (lower one) phase (c). In the opposite regime of strong interactions, the dynamics are described by the sG model where by adding an even shallow polaritonic potential, a “pinning transition” for polaritons (d) could be observed. As soon as the desired correlated state is engineered, Ω− is switched off, and the excitations propagate out of the fiber as correlated photons. The necessary correlations measurements to probe the phases of the system can be performed using standard optical technology on the photons exiting the fiber. 23 Chapter 3. Pinning Quantum Phase Transition of Photons initially in the ground state |a and the fiber is injected with a quantum coherent pulse Eˆ+ from the left side while a pair of classical fields Ω± are driving the atomic gas from both sides [Fig. 3.1(b)]. As shown in Fig. 3.1(a), the atomic configuration consisting of states |a, b, c, d comprises the typical stationary light set-up [20, 21, 22, 23, 24, 56]. We set the energy of atomic level a to be zero and the atomic levels b, c, and d to −ωQ + ∆0 , −ωcc + δ, and −ωcc − ωQ + ∆p . Here ωQ and ωC are central frequencies for Eˆ± and Ω± , ωcc is the frequency of level c, δ is the two-photon detuning, and ∆0 and ∆p are one-photon detunings. The Hamiltonian describing these four-level atoms, the quantum and classical fields with substantially different photon numbers, and atom-field interactions is then given by H = − na dz{δσcc + ∆0 σbb + ∆p σdd +[(gba σba + gdc σdc )(Eˆ+ eikQ z + Eˆ− e−ikQ z ) +σbc (Ω+ eikC z + Ω− e−ikC z ) + h.c.]} (3.1) with σij ≡ σij (z, t) = |i j|, (3.2) Eˆ± ≡ Eˆ± (z, t) = ak e±i(k−kQ )z e−i(ωk −ωQ )t , (3.3) f e±i(k−kC )z e−i(ωk −ωC )t , (3.4) Ω± ≡ Ω± (z, t) = k k where na is the atomic density and σij are collective and continuous operators describing the average of |i j| over atoms in a small but macroscopic region around z. Here i, j go over a, b, c, and d. The fields Eˆ± couple the ground state |a to excited state |b with a strength given by gba , and |c to |d with a strength gdc . The metastable state |c and |b are coupled 24 3.2. Quantum Optical Simulator with One-Species Four-Level Atoms by classical, counter-propagating control fields Ω± . The quantum field and classical field envelopes are slowly varying operators and kQ and kC are the wavevectors corresponding to their central frequencies ωQ and ωC for Eˆ± and Ω± , respectively [22, 23, 24, 56]. The process to steer the system to the regime described by the previously mentioned strongly correlated models can be divided into four stages: preparation, turning on the interactions, creating an effective polaritonic lattice, and measurement/probing of the phase diagram. In the first stage, the atoms are initially in the ground state |a and the fiber is injected with a quantum coherent pulse Eˆ+ and a classical field Ω+ from the left side. Switching off the control field allows for the storage of the quantum pulse in the medium in the usual slow light manner. In the second stage, a pair of classical fields Ω± are subsequently switched on from both sides [Fig. 3.1(b)], making the stored excitation quasistationary [22, 23, 24, 56, 58]. During this part, the initially detuned 4th level is adiabatically brought closer to resonance, which allows for the required nonlinear interactions. The evolution of Eˆ± in the fiber is described by the Maxwell-Bloch equation √ (∂t ± v∂z )Eˆ± = −i∆ω Eˆ± + i 2πna (gba σab,± + gdc σcd,± ), (3.5) where σab,± and σcd,± are introduced by the following definition σij = σij,+ eikC z + σij,− e−ikC z (3.6) with i, j = a, c according to the basic EIT mechanism (details are shown in Appendix A). Here v = ωQ /kQ is the light speed in the empty waveguide and ∆ω is the difference between ωQ and ωC . 25 Chapter 3. Pinning Quantum Phase Transition of Photons We assume that the atoms are initialized to the ground state |a and the quantum field is a weak coherent state containing roughly ten photons. Following the standard methods for treating slow-light polaritons as analyzed in [22, 23, 24, 35, 56, 57], we introduce Ψ+ , Ψ− as the forward- and backward-going polaritons. These are the propagating excitations and are defined as √ Ψ± = cos θEˆ± − sin θ 2πna σca (3.7) with cos θ = Ω± + 2πg na (3.8) √ g 2πna . Ω2± + 2πg na (3.9) Ω2± and sin θ = For simplicity, we set the coupling constants gba = gdc = g. In the limit √ Ω± , i.e. sin θ 1, the excitations are mostly in the spin-wave g 2πna form, Ψ± = √ g ˆ 2πna E± , Ω± (3.10) vΩ2 /(πg na ), which corresponds to with a group velocity given by vg the product of vacuum light speed v and the photonic component in a polariton. Next, as shown in Appendix A, we adiabatically eliminate the fast rotating terms from Eq. (3.5) and introduce a stationary combination Ψ = (Ψ+ + Ψ− )/2. In the limit of a large optical depth1 , the equation of motion for Ψ reads (details shown in Appendix A) i∂t Ψ = − ∂z Ψ + V Ψ + 2χΨ† Ψ2 , 2m (3.11) The optical depth OD= na LΓ1D /Γ, with Γ1D = 4πg /v and Γ/Γ1D the ratio of the total spontaneous emission rate to the spontaneous emission rate into the waveguide. 26 3.3. Polaritons Trapped in an Effective Periodic Lattice with the effective mass ∆ω Γ1D na − , 2vvg 4∆0 vg (3.12) vg − ΛΓ1D δvg na v 4Ω2 (3.13) m=− the potential strength V = ∆ω and the interaction strength between polaritons χ = Λ2 ΞΓ1D vg /(2∆p ). (3.14) Here, we have introduced two dimesionless quantities as Λ = Ω2 /(Ω2 − δ∆0 /2) (3.15) Ξ = (∆p − δ/2)/(∆p − δ). (3.16) and 3.3 Polaritons Trapped in an Effective Periodic Lattice The nonlinear Schrödinger equation (3.11) determines the evolution of the trapped polaritonic field Ψ(z, t) as derived from the effective Hamiltonian H= dzΨ† (− ∇ + V )Ψ + χ 2m 27 dzΨ† Ψ† ΨΨ. (3.17) Chapter 3. Pinning Quantum Phase Transition of Photons 0) Ini'ally, the atoms are in the a-‐state. Their density is uniform. a 1) Switch on the microwave field (a standing wave) only, transfers some atoms from a-‐state to u-‐state. The u-‐state is not involved in the EIT process. ω M .W . = 150GHz, λM .W . = 2mm iπn z € ΩM .W .e ph + ΩM .W .e = 2ΩM .W . cos(πn ph z) −iπn ph z € 2) Switch off the microwave field. We get the modulated density. u (10%) a (90%) 3) Switch on lasers. Figure 3.2: A schematic diagram to illustrate how to modulate the atomic density in our scheme. 28 3.3. Polaritons Trapped in an Effective Periodic Lattice To add an effective polaritonic lattice, as illustrated in Fig. 3.2, we induce a periodic atomic density distribution by applying an external field such that the atoms in |a are now given by na = n0 + n1 cos2 (πnph z). Here nph is the photonic density. We keep n0 n1 which means that the modulation is only a perturbation in the atomic density and derive the new Hamiltonian which reads H = dzΨ† [− +χ ∇ + V0 + V1 cos2 (πnph z)]Ψ 2m dzΨ† Ψ† ΨΨ, (3.18) where V0 = ∆ω vg − ΛΓ1D δvg n0 v 4Ω2 (3.19) can be tuned to zero by tuning ∆ω and δ, and V1 = −ΛΓ1D δvg n1 /(4Ω2 ) is the resulting imposed polaritonic lattice depth. We note here the dependence of the effective polaritonic lattice on both the slow light parameters (group velocity, trapping laser detuning and strength), and the modulated atomic density. Finally, we note that the atomic lattice modulation should be chosen to be commensurate to the number of the photons in the initial pulse for the pinning transition to occur [16]. This means that the modulation length will approximately fall within the microwave regime as the numbers of trapped photons in the initial pulse is of the order of 10 and the fiber is a few cm in length. 29 Chapter 3. Pinning Quantum Phase Transition of Photons 30 V1/ER γ 20 10 p 64. 80 0.5 1.5 /Γ Ω (b) 0.1 75 0.0 /n n1 (a) 33. ∆ /Γ 02 .5 .0 .5 . .5 .0 /Γ Ω 0.5 Figure 3.3: Plots of the Lieb-Liniger interaction parameter γ as a function of the one-photon detuning ∆p /Γ and the Rabi frequency Ω/Γ of the classical laser field (a) and the lattice depth V1 /ER as a function of Ω/Γ and n1 /n (b). The parameters are taken as na = 107 m−1 , nph = 103 m−1 , Γ1D = 0.2Γ, ∆0 = 5Γ, and δ = 0.01Γ, with Γ 20MHz the typical atomic decay rate [57]. 3.4 Reaching Correlated Bose-Hubbard and Sine-Gordon Regimes The success of achieving a specific strongly correlated polaritonic/photonic state is characterized by the feasibility of tuning the Lieb-Liniger ratio of the interaction and kinetic energies γ, and the ratio of the depth of the polaritonic potential to the recoil energy V1 /ER to the relevant regimes [2, 6, 44]. In our system these two quantities read: Λ2 Ξ Γ21D n0 mχ =− , nph ∆0 ∆p nph Λ Γ21D δ n0 n1 = . 8π Ω2 ∆0 n2ph γ = V1 ER (3.20) (3.21) Assuming fixed atomic and photonic densities, both quantities can be controlled by tuning the one-photon detuning ∆p /Γ (shifting the fourth level |d to/from the resonance) and by changing the strength of the control Rabi frequency Ω/Γ. In Fig. 3.3, we plot the achievable regimes for γ and V1 /ER 30 Chapter Conclusions and Outlook We have shown that stationary light-matter excitations formed in nonlinear waveguides can be used to simulate many-body physics. One of the many diverse mechanisms to generate slow light is the EIT effect [21], where the photons of a probe field can be coherently transformed into the socalled dark-state polaritons, and stored in the medium for a length of time [22, 23, 24]. The photons can be coherently transformed back again by switching the control beam adiabatically. During the trapping, the photons can be tuned to exhibit different phases by controlling the detunings, the classical laser strengths, the atomic and photonic numbers with current or near future optical technologies. In our works based on slow-light-EIT quantum simulators, we have used photons to mimic phenomena predicted for material particles. We start from Chapter with a system consisting of one-type quantum field. By applying a microwave field, we have imposed an effective lattice potential on the strongly interacting polaritonic gas [33]. This opens the possibility of a large number of Hamiltonians to be simulated with photons. As examples, we have studied the simulations of sG and BH models. Contrast to the weakly interacting superfluids in a periodic potential studied in the BH phase transition, an arbitrarily weak optical lattice is capable of driving a pinning transition in a strongly interacting sG system from a photonic superfluid Luttinger liquid to an insulating state. The whole phase diagram 99 Chapter 7. Conclusions and Outlook of the MI to SF transitions for both models have been reproduced in our scheme. We have also analyzed the available tunability of the quantum optical parameters for the observation of the strongly correlated phases. The latter is possible by releasing the trapped polaritons and measuring the characteristic correlations on the emitted photons. Going beyond the single-species polaritonic systems, in Chapter 4, we drive the state of an effective two-component polaritonic system smoothly from a SF phase of BCS type (where the attraction is arbitrarily weak) to a BEC regime (where the attraction is arbitrarily strong) by tuning appropriately the optical parameters [34]. Therefore, we see that the BCS theory is intimately connected to a BEC regime, where the Cooper pairs are formed and statistically driven to a Bose condensate. Furthermore, a new strongly localized bosonic phase appears as the intra-species attraction is increased. This phase is termed by BB with almost all the bosonic molecules occupying the same site. Our work is motivated by the reason that the BCS-BEC-BB crossover is most directly seen in the behavior of their correlations, which are easy to detect in our photonic simulators. The resulting strongly correlated two-component polaritons can also be used to mimic the spin-charge separation. As shown in Chapter 5, we have proposed two schemes to simulate the two-component Lieb-Liniger model and the spin-charge separation. We utilize two types of four-level atoms to interact with two differently colored quantum beams, or we simplify the system by employing one types of multi-level atoms to interact with two oppositely circularly polarized quantum lights. In both schemes, the spincharge separation, which is an unusual behavior of fermions, is observable from two splitting peaks in the plot of the Fourier transformation of densitydensity correlation [35, 36]. The two peaks represent the spin and charge 100 waves, with the spin and charge densities being the difference and sum between two polaritonic densities. The spin-charge separation has not been directly observed in experiments, and our proposal provides a possible way to observe it directly. In Chapter 6, we show that by utilizing one-species multi-level atoms and two oppositely circularly polarized quantum lights [37], the bosonic Thirring model and effective fermionic Thirring model, as well as the massless and massive Thirring model, can be simulated in our system by controlling laser intensity strengths and one-photon detunings. The relevant scalings of the correlation functions, for any regime of interactions, have been analyzed and can be detected by standard quantum optical techniques. Significant advances in quantum optical simulators have opened up a new eva of studying many-body quantum systems. Recent theoretical proposals and experimental tests have demonstrated synthetic magnetic fields for neutral atoms by engineering the microscopic parameters of the system. To list some of these excellent works as examples: [104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114]. In a recent work, we have proposed a scheme to generate non-uniformly distributed Abelian and non-Abelian gauge fields for neutral atoms in ring and square lattices [115]. By exploiting Laguerre-Gauss-Laser assisted tunnelings, the dynamics of cold atoms can be made to resemble the Aharonov-Bohm effect of electrons, where the atoms tunnel quantum mechanically around an effective ideal solenoid. In future, by considering a combination of the ring-shaped trapping potential and the effective magnetic field generated in our scheme, we may create a space-time crystal for cold atoms with periodic structures in both space and time dimensions [116, 117, 118]. Synthetic gauge fields have also been demonstrated for dark-state po101 Chapter 7. Conclusions and Outlook laritons [119, 120, 121, 122, 123]. This provides the possibilities to emulate electronic topological states, design topologically non-trivial photonic states, study relativistic quantum random walks, and form quantum simulators to mimic fundamental complex Hamiltonians. The quantum magnetism in strongly correlated quantum systems still holds many secrets. Further progress may require more challenging controls over the system parameters, and we would like to explore this direction. In solid state physics, coupling of a spin system to bosonic and fermionic modes is one of the fundamental building blocks. When we insert a magnetic impurity in a polaritonic gas, where the polaritons are initially present on both sides of the impurity, we can expect to observe the Kondo effect for polaritons in this system [124]. Another way to break our discussed periodic polaritonic systems is to add a further potential with incommensurate lattice spacing. This will allow the realization of a pseudo random potential, and may lead to the study of disordered systems and appearance of a Bose glass phase or Anderson localization. For a BH model, negative temperature states have been proposed in cold atoms [125]. The negative temperature enable the investigation of Mott transition, the renormalization of parameters, and the study of color superfluidity and trion formation. In our polaritonic systems, wide tunabilities over parameters allow us to bridge a transition between positive and negative temperatures by driving the replusively interacting polaritons into a deep Mott insulating regime, switching the interaction strength and the lattice depth to negative values, and finally melting the Mott insulator by reducing the interaction strength [125]. Another possible application is that for two polaritonic ensembles, we can establish an effective Josephson junction by coupling the two polaritonic states by a Raman transition laser, 102 which may allow us to investigate a dynamical process of the fusing of two polaritonic ensembles and the cooling down of their relative phase [126]. Finally, a question arises as to what extent these simulations and the underlying physics can be adapted to provide a new perspective in the field of quantum computing and communications. The controlled manipulation of entanglement provides the basis of these applications, and we can study the possibilities of realizing fundamental quantum gates and creating large scale entanglement by methods like, e.g., bringing polaritons on different sites into contact. 103 Bibliography [1] H. Bethe. On the theory of metals. i. eigenvalues and eigenfunctions of the linear atom chain. Zeitschrift für Physik, 71:205–226, 1931. [2] T. Giamarchi. Quantum Physics in One Dimension. Oxford University Press and Oxford, 2004. [3] S. R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69:2863–2866, 1992. [4] R. P. Feynman. Simulating physics with computers. J. Theor. Phys., 21:467–488, 1982. [5] B. Sakita and K. Shizuya. Local chiral symmetry and charge-density waves in one-dimensional conductors. Phys. Rev. B, 42:5586–5593, 1990. [6] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonic atoms in optical lattices. Phys. Rev. Lett., 81:3108– 3111, 1998. [7] M. Greiner, O. Mandel, T. Esslinger, T. W. Hï¿ 12 nsch, and I. Bloch. Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature (London), 415:39–44, 2002. [8] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch. Tonks-girardeau gas of ultracold atoms in an optical lattice. Nature (London), 429:277– 281, 2004. [9] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser. Resonance superfluidity in a quantum degenerate fermi gas. Phys. Rev. Lett., 87:120406, 2001. [10] C. A. Regal, M. Greiner, and D. S. Jin. Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett., 92:040403, 2004. [11] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle. Condensation of pairs of fermionic atoms near a feshbach resonance. Phys. Rev. Lett., 92:120403, 2004. [12] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm. Collective excitations of a degenerate gas at the bec-bcs crossover. Phys. Rev. Lett., 92:203201, 2004. [13] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon. Experimental study of the bec-bcs crossover region in lithium 6. Phys. Rev. Lett., 93:050401, 2004. [14] T.-L. Ho. Arrival of the fermion superfluid. Science, 305:1114–1115, 2004. 105 Bibliography [15] B. Paredes and J. I. Cirac. From cooper pairs to luttinger liquids with bosonic atoms in optical lattices. Phys. Rev. Lett., 90:150402, 2003. [16] E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, M. Gustavsson, M. Dalmonte, G. Pupillo, and H.-C. Nägerl. Pinning quantum phase transition for a luttinger liquid of strongly interacting bosons. Nature (London), 466:597–600, 2010. [17] J. I. Cirac, P. Maraner, and J. K. Pachos. Cold atom simulation of interacting relativistic quantum field theories. Phys. Rev. Lett., 105:190403, 2010. [18] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner. A quantum gas microscope for detecting single atoms in a hubbardregime optical lattice. Nature, 462:74–77, 2009. [19] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr. Single-atom-resolved fluorescence imaging of an atomic mott insulator. Nature, 467:68–72, 2010. [20] H. Walther, B. T. H. Varcoe, B. G. Englert, and T. Becker. Cavity quantum electrodynamics. Rep. Prog. Phys., 69:1325, 2006. [21] M. Fleischhauer, A. Imamoglu, and J.P. Marangos. Electromagnetically induced transparency: Optics in coherent media. Rev. Mod. Phys., 77:633–673, 2005. [22] M. Fleischhauer and M. D. Lukin. Dark-state polaritons in electromagnetically induced transparency. Phys. Rev. Lett., 84:5094, 2000. [23] M. Bajcsy, A. S. Zibrov, and M. D. Lukin. Stationary pulses of light in an atomic medium. Nature, 426:638–641, 2003. [24] M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin. Efficient all-optical switching using slow light within a hollow fiber. Phys. Rev. Lett., 102:203902, 2009. [25] S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta. Resonant optical interactions with molecules confined in photonic band-gap fibers. Phys. Rev. Lett., 94:093902, 2005. [26] K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, and K. Hakuta. Optical nanofiber as an efficient tool for manipulating and probing atomic fluorescence. Opt. Express, 15:5431–5438, 2007. [27] T. Takekoshi and R.J. Knize. Optical guiding of atoms through a hollow-core photonic band-gap fiber. Phys. Rev. Lett., 98:210404, 2007. [28] C. A. Christensen, S. Will, M. Saba, G.-B. Jo, Y.-I. Shin, W. Ketterle, and D. Pritchard. Trapping of ultracold atoms in a hollow-core photonic crystal fiber. Phys. Rev. A, 78:033429, 2008. 106 Bibliography [29] S. Vorrath, S. A. Möller, P. Windpassinger, K. Bongs, and K. Sengstock. Efficient guiding of cold atoms through a photonic band gap fiber. New J. Phys., 12:123015, 2010. [30] M. Bajcsy, S. Hofferberth, T. Peyronel, V. Balic, Q. Liang, A. S . Zibrov, V. Vuletic, and M. D. Lukin. Laser-cooled atoms inside a hollow-core photonic-crystal fiber. Phys. Rev. A, 83:063830, 2011. [31] G. Sagué, E. Vetsch, W. Alt, D. Meschede, and A. Rauschenbeutel. Cold-atom physics using ultrathin optical fibers: Light-induced dipole forces and surface interactions. Phys. Rev. Lett., 99:163602, 2007. [32] E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel. Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber. Phys. Rev. Lett., 104:203603, 2010. [33] M.-X. Huo and D. G. Angelakis. Sine-gordon and bose-hubbard dynamics with photons in a hollow-core fiber. Phys. Rev. A, 85:023821, 2012. [34] M.-X. Huo, C. Noh, B. M. Rodriguez-Lara, and D. G. Angelakis. Probing the bcs-bec crossover with photons in a nonlinear optical fiber. Phys. Rev. A, 86:043840, 2012. [35] D. G. Angelakis, M.-X. Huo, E. Kyoseva, and L.C. Kwek. Luttinger liquid of photons and spin-charge separation in hollow-core fibers. Phys. Rev. Lett., 106:153601, 2011. [36] M.-X. Huo, D. G. Angelakis, and L. C. Kwek. Spinons and holons with polarized photons in a nonlinear waveguide. New J. Phys., 14:075027, 2012. [37] D. G. Angelakis, M.-X. Huo, D. Chang, L. C. Kwek, and V. Korepin. Mimicking interacting relativistic theories with stationary pulses of light. Phys. Rev. Lett., 110:100502, 2013. [38] M. Girardeau. Relationship between systems of impenetrable bosons and fermions in one dimension. J. Math. phys., 1:516, 1960. [39] M. Girardeau. Permutation symmetry of many-particle wave functions. Phys. Rev., 139:B500–B508, 1965. [40] E. H. Lieb and W. Liniger. Exact analysis of an interacting bose gas. i. the general solution and the ground state. Phys. Rev., 130:1605, 1963. [41] A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully. Slow, ultraslow, stored, and frozen light. Advances in Atomic, Molecular, and Optical Physics, 46:191–242, 2001. [42] H. Schmidt and A. Imamoglu. Giant kerr nonlinearities obtained by electromagnetically induced transparency. Opt. Lett., 21:1936–1938, 1996. 107 Bibliography [43] A. André, M. Bajcsy, A. S. Zibrov, and M.D. Lukin. Nonlinear optics with stationary pulses of light. Phys. Rev. Lett., 94:063902, 2005. [44] S. Sachdev. Quantum phase transitions. Cambridge University Press, Cambridge, England, 1999. [45] H. P. Büchler, G. Blatter, and W. Zwerger. Commensurateincommensurate transition of cold atoms in an optical lattice. Phys. Rev. Lett., 90:130401, 2003. [46] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, and T. Schaetz. Simulating a quantum magnet with trapped ions. Nature Phys., 4:757–761, 2008. [47] M. J. Hartmann, F. G. S. L. Brand ao, and M. B. Plenio. Strongly interacting polaritons in coupled arrays of cavities. Nature Phys., 2:849–855, 2006. [48] D. G. Angelakis, M. F. Santos, and S. Bose. Photon-blockade-induced mott transitions and xy spin models in coupled cavity arrays. Phys. Rev. A, 76:031805(R), 2007. [49] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg. Quantum phase transitions of light. Nature Phys., 2:856–861, 2006. [50] D. Rossini and R. Fazio. Mott-insulating and glassy phases of polaritons in 1d arrays of coupled cavities. Phys. Rev. Lett., 99:186401, 2007. [51] M. Aichhorn, M. Hohenadler, C. Tahan, and P. B. Littlewood. Quantum fluctuations, temperature, and detuning effects in solid-light systems. Phys. Rev. Lett., 100:216401, 2008. [52] D. Gerace, H. E. Türeci, A. Imamoglu, V. Giovannetti, and R. Fazio. The quantum-optical josephson interferometer. Nature Phys., 5:281– 284, 2009. [53] I. Carusotto, D. Gerace, H. E. Tureci, S. D. Liberato, C. Ciuti, and A. Imamoglu. Fermionized photons in an array of driven dissipative nonlinear cavities. Phys. Rev. Lett., 103:033601, 2009. [54] D. G. Angelakis, S. Bose, and S. Mancini. Steady-state entanglement between hybrid light-matter qubits. Europhys. Lett., 85:20007, 2009. [55] M. Kiffner and M. J. Hartmann. Master equation approach for interacting slow- and stationary-light polaritons. Phys. Rev. Lett., 82:033813, 2010. [56] E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan. Strongly interacting photons in hollow-core waveguides. Phys. Rev. A, 83:033806, 2011. [57] D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler. Crystallization of strongly interacting photons in a nonlinear optical fibre. Nature Phys., 4:884–889, 2008. 108 Bibliography [58] M. D. Lukin and A. Imamoglu. Nonlinear optics and quantum entanglement of ultraslow single photons. Phys. Rev. Lett., 84:1419, 2000. [59] R. Schützhold, M. Uhlmann, Y. Xu, and U. R. Fischer. Sweeping from the superfluid to the mott phase in the bose-hubbard model. Phys. Rev. Lett., 97:200601, 2006. [60] R. Zhang, S. R. Garner, and L. V. Hau. Creation of long-term coherent optical memory via controlled nonlinear interactions in boseeinstein condensates. Phys. Rev. Lett., 103:233602, 2009. [61] U. Schnorrberger, J. D. Thompson, S. Trotzky, R. Pugatch, N. Davidson, S. Kuhr, and I. Bloch. Electromagnetically induced transparency and light storage in an atomic mott insulator. Phys. Rev. Lett., 103:033003, 2009. [62] A. J. Leggett. What we know about high tc? 2:134–136, 2006. Nature Phys., [63] J. Zaanen and et al. Towards a complete theory of high tc. Nature Phys., 2:138, 2006. [64] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Phys. Rev., 108:1175–1204, 1957. [65] I. Bloch, J. Dalibard, and W. Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys., 80:885–964, 2008. [66] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys., 56:243–379, 2007. [67] P. Nozières and S. Schmitt-Rink. Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. J. Low Temp. Phys., 59:195–211, 1985. [68] K. Tanaka and F. Marsiglio. Even-odd and super-even effects in the attractive hubbard model. Phys. Rev. B, 60:3508–3526, 1999. [69] J. Kinnunen, M. Rodríguez, and P. Törmä. Pairing gap and in-gap excitations in trapped fermionic superfluids. Science, 305:1131–1133, 2004. [70] M. D. Lukin. Colloquium: Trapping and manipulating photon states in atomic ensembles. Rev. Mod. Phys., 75:457–472, 2003. [71] R. G. Unanyan, J. Otterbach, M. Fleischhauer, J. Ruseckas, V. Kudriašov, and G. Juzeli¯ unas. Spinor slow-light and dirac particles with variable mass. Phys. Rev. Lett., 105:173603, 2010. [72] J. Otterbach, J. Ruseckas, R. G. Unanyan, G. Juzeli¯ unas, and M. Fleischhauer. Effective magnetic fields for stationary light. Phys. Rev. Lett., 104:033903, 2010. [73] J. Otterbach, R. G. Unanyan, and M. Fleischhauer. Confining stationary light: Dirac dynamics and klein tunneling. Phys. Rev. Lett., 102:063602, 2009. 109 Bibliography [74] M. J. Hartmann, F. G. S. L. Brand ao, and M. B. Plenio. A polaritonic two-component bose-hubbard model. New J. Phys., 10:033011, 2008. [75] C. J. Foot. Atomic Physics. Oxford University Press, 2005. [76] S. Tomonaga. Remarks on bloch’s method of sound waves applied to many-fermion problems. Prog. Theor. Phys., 5:544–569, 1950. [77] C. Kim, A. Y. Matsuura, Z. X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa. Observation of spin-charge separation in one-dimensional srcuo2. Phys. Rev. Lett., 77:4054–4057, 1996. [78] P. Segovia, D. Purdie, M. Hengsberger, and Y. Baer. Observation of spin and charge collective modes in one-dimensional metallic chains. Nature (London), 402:504–507, 1999. [79] T. Lorenz, M. Hofmann, M. Grüninger, A. Freimuth, G. S. Uhrig, M. Dumm, and M. Dressel. Evidence for spin-charge separation in quasi-one-dimensional organic conductors. Nature, 418:614–617, 2002. [80] O. M. Auslaender, H. Steinberg, A. Yacoby, Y. Tserkovnyak, B. I. Halperin, K. W. Baldwin, L. N. Pfeiffer, and K. W. West. Spin-charge separation and localization in one dimension. Science, 308:88–92, 2005. [81] B. J. Kim, H. Koh, E. Rotenberg, S.-J. Oh, H. Eisaki, N. Motoyama, S. Uchida, T. Tohyama, S. Maekawa, Z.-X. Shen, and C. Kim. Distinct spinon and holon dispersions in photoemission spectral functions from one-dimensional srcuo2. Nature Phys., 2:397–401, 2006. [82] Y. Jompol, C. J. B. Ford, J. P. Griffiths, I. Farrer, G. A. C. Jones, D. Anderson, D. A. Ritchie, T. W. Silk, and A. J. Schofield. Probing spin-charge separation in a tomonaga-luttinger liquid. Science, 325:597–601, 2009. [83] D. Porras and J. I. Cirac. Effective quantum spin systems with trapped ions. Phys. Rev. Lett., 92:207901, 2004. [84] A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller. Spin-charge separation in ultracold quantum gases. Phys. Rev. Lett., 90:020401, 2003. [85] L. Kecke, H. Grabert, and W. Häusler. Charge and spin dynamics of interacting fermions in a one-dimensional harmonic trap. Phys. Rev. Lett., 94:176802, 2005. [86] C. Kollath, U. Schollwöck, and W. Zwerger. Spin-charge separation in cold fermi gases: A real time analysis. Phys. Rev. Lett., 95:176401, 2005. [87] A. Kleine, C. Kollath, I. P. McCulloch, T. Giamarchi, and U. Schollwöck. Spin-charge separation in two-component bose gases. Phys. Rev. A, 77:013607, 2008. 110 Bibliography [88] A. Iucci, G. A. Fiete, and T. Giamarchi. Fourier transform of the 2kf luttinger liquid density correlation function with different spin and charge velocities. Phys. Rev. B, 75:205116, 2007. [89] M. R. Sprague and et al. Efficient optial pumping and high optical depth in a hollow-core photonic-crystal fibre for broadband quantum memory. New J. Phys., 15:055013, 2013. [90] W. E. Thirring. A soluble relativistic field theory. Annals Phys., 3:91–112, 1958. [91] V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin. Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge, England, 1993. [92] B. Klaiber. Lectures in theoretical physics and Vol. X and Part A. Gordon and Breach, New York, 1968. [93] K. Furuya, R. E. G. Saraví, and F.A. Schaposnik. Path-integral formulation of chiral invariant fermion models in two dimensions. Nucl. Phys. B, 208:159–181, 1982. [94] C. M. Naon. Abelian and non-abelian bosonization in the pathintegral framework. Phys. Rev. D, 31:2035–2044, 1985. [95] V. Fateev, D. Fradkin, S. Lukyanov, A. Zamolodchikov, and A. Zamolodchikov. Expectation values of descendent fields in the sine-gordon models. Nucl. Phys. B, 540:587, 1999. [96] F.A. Smirnov. Form-factors in completely integrable models of quantum field theory. Singapore, World Scientific, 1992. [97] J. Cardy and G. Mussardo. Universal properties of self-avoiding walks from two-dimensional field theory. Nucl. Phys. B, 410:451, 1993. [98] A. Fring, G. Mussardo, and P. Simonetti. Form factors for integrable lagrangian field theories, the sinh-gordon models. Nucl. Phys. B, 393:413, 1993. [99] L. Tonks. The complete equation of state of one, two and threedimensional gases of hard elastic spheres. Phys. Rev., 50:955, 1936. [100] T. Bhattacharyya. Quantum integrability of bosonic massive thirring model in continuum. J. Math. Phys., 46:012301, 2005. [101] F. A. Smirnov. Connection between the sine-gordon model and the massive bose thirring model. Teoreticheskaya i Matematicheskaya Fizika, 53:323, 1982. [102] S. Coleman. Quantum sine-gordon equation as the massive thirring model. Phys. Rev. D, 11:2088–2097, 1975. [103] V. E. Korepin. Direct calculation of the s matrix in the massive thirring model. Theor. Math. Phys., 41:169, 1979. [104] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. S. De, and U. Sen. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys., 56:243, 2007. 111 Bibliography [105] T.-L. Ho. Bose-einstein condensates with large number of vortices. Phys. Rev. Lett., 87:060403, 2001. [106] A. L. Fetter. Rotating trapped bose-einstein condensates. Rev. Mod. Phys., 81:647, 2009. [107] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman. Bose-einstein condensate in a uniform lightinduced vector potential. Phys. Rev. Lett., 102:130401, 2009. [108] Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman. Synthetic magnetic fields for ultracold neutral atoms. Nature, 462:628, 2009. [109] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein. Cold atoms in non-abelian gauge potentials: From the hofstadter “moth” to lattice gauge theory. Phys. Rev. Lett., 95:010403, 2005. [110] D. Jaksch and P. Zoller. Creation of effective magnetic fields in optical lattices: the hofstadter butterfly for cold neutral atoms. New J. Phys., 5:56, 2003. [111] F. Gerbier and J. Dalibard. Gauge fields for ultracold atoms in optical superlattices. New J. Phys., 12:033007, 2010. [112] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. Öhberg. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys., 83:1523–1543, 2011. [113] N. Goldman, J. Beugnon, and F. Gerbier. Detecting chiral edge states in the hofstadter optical lattice. Phys. Rev. Lett., 108:255303, 2012. [114] N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman. Direct imaging of topological edge states in cold-atom systems. arXiv:1212.5093. [115] M.-X. Huo, N. Wei, D. A.W. Hutchinson, and L. C. Kwek. Interference signatures of abelian and non-abelian aharonov-bohm effect on neutral atoms in optical lattices. arXiv:1210.8008. [116] A. Shapere and F. Wilczek. Classical time crystals. Phys. Rev. Lett., 109:160402, 2012. [117] F. Wilczek. Quantum time crystals. Phys. Rev. Lett., 109:160401, 2012. [118] T. Li, Z.-X. Gong, Z.-Q. Yin, H. T. Quan, X. Yin, P. Zhang, L.-M. Duan, and X. Zhang. Space-time crystals of trapped ions. Phys. Rev. Lett., 109:163001, 2012. [119] G. Juzel¯ unas, P. Öhberg, J. Ruseckas, and A. Klein. Effective magnetic fields in degenerate atomic gases induced by light beams with orbital angular momenta. Phys. Rev. A, 71:053614, 2005. [120] J. Ruseckas, G. Juzel¯ unas, P. Öhberg, and M. Fleischhauer. Nonabelian gauge potentials for ultracold atoms with degenerate dark states. Phys. Rev. Lett., 95:010404, 2005. 112 Bibliography [121] G. Juzel¯ unas and P. Öhberg. Slow light in degenerate fermi gases. Phys. Rev. Lett., 93:033602, 2004. [122] R. O. Umucalï¿ 12 lar and I. Carusotto. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A, 84:043804, 2011. [123] K. Fang, Z. Yu, and S. Fan. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photonics, 2012. [124] J. Kondo. Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys., 32:37, 1964. [125] S. Braun, J. P. Ronzheimer, M. Schreiber, S. S. Hodgman, T. Rom, I. Bloch, and U. Schneider. Negative absolute temperature for motional degrees of freedom. Science, 339:52–55, 2013. [126] D. Jaksch, S. A. Gardiner, K. Schulze, J. I. Cirac, and P. Zoller. Uniting bose-einstein condensates in optical resonators. Phys. Rev. Lett., 86:4733–4736, 2001. [127] F.E. Zimmer, A. André, M.D. Lukin, and M. Fleischhauer. Coherent control of stationary light pulses. Optics Communications, 264:441– 453, 2006. 113 [...]... nΓ1D ∂z 2 (4 .20 ) In addition, the losses come from the nonlinear intra- and inter-species interactions are given by ∂Ψj ∂t ∂Ψj ∂t † 1 Γ1D Γv g Ψj 2 j ∼ , 2 2 ∆4 ∼ 2 † b Γ1D Γ[ 2 + (δ1 )2 + 2 ]v g Ψj Ψj Ψj 4 b b [(∆4 )2 − (δ1 )2 ][ 2 − (δ1 )2 + 2 ] 4 (4 .21 ) (4 .22 ) In the strong coupling regime, the largest spatial component is given by ∆z ∼ (nph )−1 , where nph is the photonic density Since the... as (nph )2 v g Γ , nΓ1D 1 Γ1D Γv g nph = , 2 2 4 b Γ1D Γ[ 2 + (δ1 )2 + 2 ]v g nph 4 = 2 b b [(∆4 )2 − (δ1 )2 ][ 2 − (δ1 )2 + 2 ] 4 κl = κns κnd (4 .23 ) (4 .24 ) (4 .25 ) Here κl is the linear loss rate, and κns , κnd are the nonlinear loss rates coming from same- and different-species interactions For the parameters given in Fig 4 .2, i.e., n/nph = 104 , nph = 300m−1 , Γ 20 MHz, η = Γ1D /Γ = 0 .2, v g ∼... repulsive intra-species interactions (U > 0), and attractive interspecies interactions (V < 0), which can be achieved by setting χ > 0 and χ 12 < 0 The ratios between the inter- and intra-species interactions V 2 2 = 2 4 b 2, U ∆4 − (δ1 ) (4.30) and the ratio of the hopping to the intra-species repulsion 1 /2 4µ1 /2 ER exp( 2 µ/ER ) t √ = U 2 χnph (4.31) determine the physics of the Hamiltonian (4 .26 ) completely... the jth quantum field Here, we have as- 43 Chapter 4 Simulating Cooper Pairs with Photons ˆ sumed that the quantum fields E1 ,2 drive the transitions |2 x 1| and |4 x 3| x with the same strength g1 ,2 The one- photon detunings are denoted as ∆x 2 and ∆x ; the two-photon detunings as ∆x ; the quantum pulse detunings are 3 4 (1) (2) a a b a a b written as δj with 2 = ωQ − ωQ , δ1 = − 2 , and δ1 = 2 = 0 ˆ... Photons transition lines corresponding to the sG and BH model occurring at V1 /ER = 2 / γ − γ 3 /2 / (2 ) − 4 (3 .29 ) and (U/J)c = √ 2 exp (2 V1 /ER )γ/[4π(V1 /ER )1 /2 ] 3.85 (3.30) respectively [2, 6, 7, 44] In our case, these are probed by adjusting the detuning and the laser coupling accordingly The pinning transition is expected to occur for any value ∆p /Γ less than 20 when Ω is increased to be larger... the interspecies repulsions is χ 12 g g na (Λa Ωb )2 Ξa Γ1D v1 nb (Λb Ωa )2 Ξb Γ1D v2 2 1 + , = a b nb 2 (∆a − 2 ) na 2 (∆b − δ1 ) 4 a 4 b (4.9) where two dimensionless quantities x x Λxj = 2 j /( 2 j − ∆3 j 2 j /2) x x (4.10) and x x x x Ξxj = (∆4 j − ∆3 j /2) /(∆4 j − ∆3 j ) have been introduced When the EIT conditions |∆3 | Λx , Ξx (4.11) 2 Ω | 2 |, ∆4 hold, 1 As one would expect, the above nonlinear. .. system under study In a fiber setup (a hollow-core version is shown here [25 , 26 , 27 , 28 , 29 ] but a tapered fiber approach [ 32] could also be used), cold atoms are interacting with a ˆ pair of quantum fields E1 ,2 , and a pair of classical fields Ωa,b The resulting stationary light-matter excitations in the waveguide can be steered to a strongly interacting regime mimicking an effective FH model with highly tunable... (ER) MI 1.0 1.5 Ω/Γ 2. 0 2. 5 3.0 (b) U J SF 0.10 MI 0.05 0.00 0.03 0.06 0.09 n1/n0 0. 12 0.15 Figure 3.5: The interaction and tunnelling strength as functions of Ω/Γ with n1 /n = 0.1 in (a) and n1 /n0 with Ω = Γ in (b) with ∆p = 50Γ for the weakly interacting gas in the BH regime The red dot line at Ω/Γ 1.03388 in (a) and n1 /n = 0.093 in (b) corresponds to the Mott phase transition point (U/J)c 3.85 The... of a σ-type polariton at ith site and i, j stands for nearest neighbors The coupling strength 4 1/4 tσ = √ µ3/4 ER exp( 2 µ/ER ), π (4 .27 ) the intra-species interaction strength Uσ = √ 2 ph χn (µ/ER )1/4 , 2 (4 .28 ) and the inter-species interaction strength V = √ 2 χ 12 nph (µ/ER )1/4 , 2 (4 .29 ) where ER = π 2 (nph )2 /(2m) is the recoil energy As t↑ = t↓ and U↑ = U↓ , we drop their subscripts from... a two-component Lieb-Liniger Hamiltonian: 2 H = Ψ† j dz j=1 +χ 12 1 2mj 2 + Vj Ψj + χj Ψ† Ψ† Ψj Ψj j j dzΨ† Ψ1 Ψ† 2 1 2 (4.5) Here, the effective masses of polaritons are mj = − Γ1D nxj j x g, 4 2 j vj (4.6) g where vj = vj 2 j /(πg 2 nxj ) is the group velocity of j-type polaritons in the x nonlinear medium, and Γ1D = 4πg 2 /vj is the spontaneous emission rate of j a single xj -type atom into the . described in [25 , 26 , 27 , 28 , 29 , 30, 31, 32] . The atoms are 22 3 .2. Quantum Optical Simulator with One- Species Four-Level Atoms (a) (b) (c) (d) Figure 3.1: In (a) and (b) an ensemble of cold atoms with. using standard optical technology on the photons exiting the fiber. 23 Chapter 3. Pinning Quantum Phase Transition of Photons initially in the ground state |a and the fiber is injected with a quantum coherent. and the quantum field is a weak coherent state containing roughly ten photons. Following the standard methods for treating slow-light polaritons as analyzed in [22 , 23 , 24 , 35, 56, 57], we introduce

Định dạng
Số trang	94
Dung lượng	11,43 MB