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Manipulating Quantum Materials with Quantum Light PHYSICAL REVIEW B 99, 085116 (2019) Manipulating quantum materials with quantum light Martin Kiffner,1,2 Jonathan R. Coulthard,2 Frank Schlawin,2 Arzhang Ardavan,2 and Dieter Jaksch2,1 1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 2Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom (Received 18 June 2018; revised manuscript received 25 January 2019; published 11 February 2019) We show that the macroscopic magnetic and electronic properties of strongly correlated electron systems can be manipulated by coupling them to a cavity mode. As a paradigmatic example we consider the Fermi-Hubbard model and find that the electron-cavity coupling enhances the magnetic interaction between the electron spins in the ground-state manifold. At half filling this effect can be observed by a change in the magnetic susceptibility. At less than half filling, the cavity introduces a next-nearest-neighbor hopping and mediates a long-range electron- electron interaction between distant sites. We study the ground-state properties with tensor network methods and find that the cavity coupling can induce a phase characterized by a momentum-space pairing effect for electrons. DOI: 10.1103/PhysRevB.99.085116 I. INTRODUCTION Here we show that shaping the vacuum via a cavity allows one to manipulate macroscopic properties like the magnetic The ability to control and manipulate complex quantum susceptibility of a quantum material. As a paradigmatic model systems is of paramount importance for future quantum tech- of quantum materials we consider the Fermi-Hubbard model nologies. Of particular interest are quantum hybrid systems [40] which captures the interplay between kinetic fluctua- [1,2] where different quantum objects hybridize to exhibit tions and strong, local, electron-electron interactions [32–34]. properties not shared by the individual components. Examples While the interaction of these systems with strong, classical of this hybridization effect in cold atom systems coupled light fields has been investigated, for example, in [41–44], the to optical cavities are given by self-organization phenomena intriguing possibility of coupling them to quantum light fields [3–7] as well as the occurrence of quantum phase transitions has not been explored yet. and exotic quantum phases [8–11]. More specifically, we consider a one-dimensional Hubbard Recently, the class of available hybrid systems has been model coupled to a single mode of an empty cavity. For an extended by solid-state systems that couple strongly to mi- electronic system at half filling we find that the electron-cavity crowave, terahertz, or optical radiation [12–30]. For example, interaction enhances the magnetic interactions between spins the coupling of microwave cavities to magnon and spinon in the ground-state manifold. This effect can be experimen- excitations in magnetic materials has been investigated in tally observed by measuring the magnetic susceptibility. [12–16] and [17], respectively. Two-dimensional electron At less than half filling, the cavity coupling introduces gases in magnetic fields can couple very strongly to tera- (i) a next-nearest-neighbor hopping, (ii) an on-site energy hertz cavities [18–20] such that Bloch-Siegert shifts become shift, and (iii) a long-range electron-electron interaction be- observable [21], and tomography of an ultrastrongly cou- tween distant sites. We investigate the ground state of the pled polariton state was presented in [22] using magneto- electronic system at less than half filling and in the pres- transport measurements [23]. A recent experiment [24] has ence of the cavity with density-matrix renormalization-group demonstrated that coupling of an organic semiconductor to (DMRG) techniques [45,46]. We find that the cavity induces an optical cavity enhances the electric conductivity, which momentum-space pairing for mesoscopic electron systems. can be understood in terms of delocalized exciton polaritons The transition to this phase is a collectively enhanced effect [25,26]. and does not require ultrastrong coupling on the single- A special class of solid-state systems are quantum mate- √ electron level and scales with 1/ v , where v is the volume rials [31–34] where small microscopic changes can result in uc uc of the unit cell of the crystal. large macroscopic responses due to strong electron-electron This paper is organized as follows. In Sec. II we introduce interactions. Coupling these systems to cavities opens up the our model for the system shown in Fig. 1. Our results are fascinating possibility of investigating the ultimate quantum presented in Sec. III, and their experimental realization is limit where macroscopic properties of quantum materials are discussed in Sec. IV. A brief discussion and conclusion are determined by quantum light fields and vice versa. First steps provided in Sec. V. in this direction have been undertaken recently [27–30,35]. For example, quantum counterparts of light-induced super- II. MODEL conductivity [36–39] have been investigated in [27–29,35]us- ing terahertz and microwave cavities, and a superradiant phase The system shown in Fig. 1 is comprised of an elec- of a cavity-coupled quantum material has been predicted tronic system coupled to a single-mode cavity. We introduce in [30]. the Hamiltonian describing this quantum hybrid system in 2469-9950/2019/99(8)/085116(9) 085116-1 ©2019 American Physical Society KIFFNER, COULTHARD, SCHLAWIN, ARDAVAN, AND JAKSCH PHYSICAL REVIEW B 99, 085116 (2019) The operator Dˆ is diagonal in the basis of Wannier states [40], |x, s=cˆ† ...cˆ† |0 , (7) xN ,sN x1,s1 E where |0E is the vacuum state of the electronic system and x = (x1,...,xN ), (8a) s = (s1,...,sN ), (8b) are row vectors with x j ∈{1,...,L}, s j ∈{↑, ↓}, and j ∈ {1,...,N}. The vectors in Eq. (8) describe the spatial distri- FIG. 1. The system of interest is given by an electronic system bution of N electrons and their spin state in a one-dimensional coupled to a single-mode cavity. The electronic system is described lattice with L sites. by the Fermi-Hubbard model with on-site interaction U and hopping The Wannier states are eigenstates of Dˆ and form degen- amplitude t between neighboring sites. erate manifolds with energies kU, where k is an integer that counts the total number of doubly occupied sites in |x, s.In Sec. II A, and discuss its gross energy structure for the pa- the following we refer to doubly occupied sites as doublons. rameters of interest in Sec. II B. The projector onto the manifold with k doublons is given by [40] A. System Hamiltonian − k Pˆ D = ( 1) ∂k , k x G(x) (9) We begin with the description of the system Hamiltonian k! x=1 with the single-mode cavity with resonance frequency ω .The c where Hamiltonian for the cavity photons with energy = h¯ωc is L † Pˆ = aˆ aˆ, (1) G(x) = (1 − x nˆ j,↑nˆ j,↓ ) (10) j=1 wherea ˆ † (ˆa) is the bosonic photon creation (annihilation) operator. The eigenstates of Pˆ are the photon number states is the generating function. The spectral decomposition of Dˆ is | jP with Pˆ | jP= j| jP. The spectral decomposition of Pˆ thus given by can thus be written as L ∞ Dˆ =U k Pˆ D. (11) ˆ = Pˆ P, k P j j (2) k=0 j=0 Note that in general, each manifold with a given number where of doublons contains a large number of electronic states. For example, the ground-state manifold with no doublons contains Pˆ P =| | j jP jP (3) L # Pˆ D = 2N (12) is the projector onto the state with j photons. 0 N The electronic system is described by the one-dimensional Fermi-Hubbard model [40] with Hamiltonian states for a system with N L electrons. The preceding defi- nitions for Hˆ and Pˆ allow us to write the total Hamiltonian ˆ = ˆ + ˆ , FH HFH T D (4) of the hybrid system in Fig. 1 as where Hˆ = Hˆ + Pˆ + Vˆ , (13) FH ˆ =− † + T t (ˆc j,σ cˆk,σ H.c.) (5) where Vˆ accounts for the electron-photon interaction. In jkσ Appendix A, we outline the derivation of this interaction term from first principles and find accounts for hopping between neighboring sites jk with < † † j k, t is the hopping amplitude, andc ˆ j,σ (ˆc j,σ ) creates Vˆ = g(ˆa + aˆ )Jˆ, (14) (annihilates) an electron at site j in spin state σ ∈{↑, ↓}. where The second term in Eq. (4) describes the on-site Coulomb † † interaction between electrons, Jˆ =−i (ˆc cˆ ,σ − cˆ cˆ ,σ ) (15) j,σ k k,σ j jkσ Dˆ =U nˆ j,↑nˆ j,↓, (6) j is the dimensionless current operator. The parameter g = tη in Vˆ determines the coupling strength between the electrons and = † where U is the interaction energy andn ˆ j,σ cˆ j,σ cˆ j,σ counts photons, and the dimensionless parameter the number of electrons at site j in spin state σ. Each site can de accommodate at most two electrons with opposite spins, and η = √ (16) ε ω in the following we refer to doubly occupied sites as doublons. 2¯h 0 cv 085116-2 MANIPULATING QUANTUM MATERIALS WITH QUANTUM LIGHT PHYSICAL REVIEW B 99, 085116 (2019) Energy where = − U is the energy difference between the cavity and the doublon transition, j ∈{0,...,n} denotes the number ⎫ of photons, and n − j is the number of doublons. The corre- 2 ˆ(0) ⎬ ( j) U P2 sponding projector onto the submanifold with energy En is Δ ˆ(1) ˆ P2 P2 Δ (2) ⎭ Pˆ ( j) = Pˆ D ⊗ Pˆ P, ˆ n n− j j (20) P2 Pˆ D Pˆ P where k and j are defined in Eqs.
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