From the Jaynes-Cummings-Hubbard to the Dicke model
Abstract
We discuss the Jaynes-Cummings-Hubbard model (JCHM) describing the superfluid-Mott insulator transition of polaritons (i.e., dressed photon-qubit states) in coupled qubit-cavity arrays in the crossover from strong to weak correlations. In the strongly correlated regime the phase diagram and the elementary excitations of lattice polaritons near the Mott lobes are calculated analytically using a slave boson theory (SBT). The opposite regime of weakly interacting polariton superfluids is described by a weak-coupling mean-field theory (MFT) for a generalised multi-mode Dicke model. We show that a remarkable relation between the two theories exists in the limit of large photon bandwidth and large negative detuning, i.e., when the nature of polariton quasiparticles becomes qubit-like. In this regime, the weak coupling theory predicts the existence of a single Mott lobe with a change of the universality class of the phase transition at the tip of the lobe, in perfect agreement with the slave-boson theory. Moreover, the spectra of low energy excitations, i.e., the sound velocity of the Goldstone mode and the gap of the amplitude mode match exactly as calculated from both theories.
pacs:
71.36.+c, 42.50.Ct, 64.70.?p, 73.43.NqI Introduction
The Jaynes-Cummings Hubbard model has been introduced by Greentree et al. Greentree et al. (2006) to describe a possible superfluid-Mott insulator transition of polaritons, i.e., quasiparticles of light and matter, in a coupled qubit-cavity array Greentree et al. (2006); Hartmann et al. (2006); Angelakis et al. (2007). In this model a single photonic mode is strongly coupled to a two-level system (2LS) in each cavity and photons can hop between cavities. Hereby, the qubits introduce a nonlinearity into the system, which gives rise to effective repulsive photon-photon interactions. The competition between repulsion (localization), and the photon hopping between cavities (delocalization) leads to an equilibrium quantum phase diagram featuring Mott lobes reminiscent of those of ultracold atoms in optical lattices as described by the Bose-Hubbard model Fisher et al. (1989).
In the strongly correlated regime of the JCHM, where the on-site repulsion dominates, photons become localised in a Mott-like state due to strong effective interactions. This extreme many-body state of light has been the subject of intense theoretical investigations (for recent reviews, see Houck et al. (2012); Schmidt and Koch (2013)). The quantum phase diagram and elementary excitations of the JCHM have been calculated using various methods, e.g., decoupling mean-field approximation Greentree et al. (2006), DMRG Rossini and Fazio (2007); Rossini et al. (2008), variational cluster approximation Aichhorn et al. (2008); Knap et al. (2010, 2011), strong coupling expansion Schmidt and Blatter (2009); Koch and Hur (2009); Schmidt and Blatter (2010); Nietner and Pelster (2012) and Quantum Monte Carlo simulations Pippan et al. (2009); Hohenadler et al. (2011, 2012). It was shown that the symmetry breaking phase transition of the JCHM is in the same universality class as the phase transition described by the BHM Schmidt and Blatter (2009); Koch and Hur (2009); Hohenadler et al. (2011). Major differences in the shape of the Mott lobes and the number of elementary excitations arise due to the -nonlinearity of the JCHM (as compared to a Kerr-like nonlinearity for the BHM) and the composite nature of polariton quasiparticles (in the JCHM denotes the number of polaritons per site). Possible experimental realizations of the JCHM include, e.g., cavity/circuit QED systems based on superconducting qubits in transmission line resonators Houck et al. (2012); Schmidt and Koch (2013) and phonon-polaritons in trapped ion systems Ivanov et al. (2009); Hohenadler et al. (2012).
In the regime of strong hopping and weak correlations, polaritons in a coupled qubit-cavity array are expected to form a weakly interacting superfluid state. A BEC of weakly interacting polariton quasiparticles has already been observed experimentally with exciton-polaritons in a quantum well coupled to a semiconductor microcavity, formed by two Bragg mirrors Kasprzak et al. (2006); Balili et al. (2007). Spectacular experimental advances showed the existence of superfluidity Amo et al. (2009); Utsunomiya et al. (2008), quantized vortices Lagoudakis et al. (2008), and quantum solitons Amo et al. (2011) in these systems. A comprehensive review of these experiments, and their relation to other “quantum fluids of light” can be found in the recent review Carusotto and Ciuti (2013). One approach to modelling polariton BEC’s has been to use a generalised Dicke model in which many two level systems, representing the presence or absence of an exciton at a given site, coherently couple to the multimode spectrum of a large cavity with quadratic photon dispersion Eastham and Littlewood (2000, 2001); Keeling et al. (2004, 2005). The transition from a normal to a superfluid state then corresponds to the equilibrium superradiance transition as originally predicted for the single-mode Dicke model by Hepp and Lieb Dicke (1954); Hepp and Lieb (1973a); Wang and Hioe (1973); Hepp and Lieb (1973b).
Interestingly, the JCHM model is equivalent to the generalized Dicke model for small photon wave vectors. In this case the lattice dispersion can be expanded yielding a quadratic photon dispersion with an effective mass that is inversely proportional to the hopping strength in the JCHM. In this paper we investigate in detail the connection between these two models. In particular, we show that a weak coupling mean-field theory for the generalised Dicke model predicts the existence of Mott lobes and a change of the universality class of the phase transition at the tip of the lobe. This is in perfect agreement with results for the JCHM as obtained from a slave-boson theory Schmidt and Blatter (2010). In particular, the phase diagram and the elementary excitations at the phase boundary and inside the Mott lobes match exactly. This result is in strong contrast to similar weak-coupling approaches for the Bose-Hubbard model, which fail to predict the existence of a superfluid-Mott insulator transition van Oosten et al. (2001).
Our paper is organised as follows: In section II we introduce the Jaynes-Cummings Hubbard model and discuss its solution in the limit of vanishing hopping. In section III we present the details of the slave-boson theory and discuss the phase diagram and the elementary excitations of the JCHM in the strongly correlated regime. In section IV we map the JCHM to a generalised Dicke model and show that a weak coupling mean-field theory also predicts the existence of a superfluid-Mott insulator transition. In section V, we discuss the connection between the JCHM and the Dicke model at zero and finite temperatures. We conclude with a summary and outlook in section VI.
Ii The Jaynes-Cummings-Hubbard model (JCHM)
The Hamiltonian of the JCHM is given by
(1) |
where denotes the local Jaynes-Cummings Hamiltonian
(2) |
with site index , boson creation (annihilation) operators and qubit raising (lowering) operators . The bosonic mode frequency is , the two qubit levels are separated by the energy and the coupling is given by (we set ).
The on-site eigenstates of the Jaynes-Cummings Hamiltonian are labelled by the polariton number and upper/lower branch index . The mixed boson () - qubit () states define upper and lower polariton states
(3) |
where the weights are given by
(4) |
and
(5) |
with and the detuning parameter The mixing angle can be written more compactly as . The corresponding eigenvalues are
(6) |
The zero polariton state is a special case with .
Due to the mixing in (II), polaritons can be interpreted as dressed photons. This dressed photon inherits the anharmonicity of the matter component, i.e., the qubit, leading to a nonlinearity in the spectrum of the combined light-matter system, which is given by the energy difference
(7) |
The parameter describes the energy cost of adding a second photon to the cavity versus adding the first. It quantifies the effective on-site repulsive interaction among photons and is thus sometimes also called the particle-hole gap or effective Hubbard-U. The spectral shift in Eq. (7) is largest for zero qubit-cavity detuning (), where . It becomes vanishingly small for large detunings, i.e., . In this dispersive regime, photons and qubits barely interact with each other.
The second term in (1) describes the delocalization of bosons over the whole lattice due to hopping between nearest neighbour sites with amplitude . It competes with an effective on-site repulsion as given by , which is mediated by the coupling .
In order to calculate the phase diagram of the JCHM it is convenient to work in the grand-canonical formalism. The global symmetry of the JCHM preserves the total number of polaritons . We thus introduce a chemical potential in the third term of (1), which fixes the number of polaritons. While this might be justified for some experimental setups with large coherence times, e.g., phonon-polaritons in trapped ion systems, it does not take into account the basic nature of most quantum optical applications: drive and dissipation. Some recent works have started to address this issue and suggest that remnants of the equilibrium SF-MI transition remain visible even in this strongly non-equilibrium situation Tomadin et al. (2010); Liu et al. (2011); Nissen et al. (2012); Kulaitis et al. (2013); Grujic et al. (2012, 2013). Note, that there have also been related studies of the driven, dissipative Bose-Hubbard model Hartmann (2010); Boité et al. . In this paper we assume that the system equilibrates and that the lifetime of the polariton quasiparticles is much larger than the measurement time.
Iii Strong correlations: Slave-Boson theory
An analytic strong-coupling theory for the phase diagram and the elementary excitations in the Mott phase of the JCHM has been derived based on a linked-cluster expansion (LCA). This strong-coupling theory has been generalised to the superfluid phase using a slave-boson approach Schmidt and Blatter (2010), which was previously applied to the BHM Huber et al. (2007). Below we present the details of this formalism with explicit algebraic expressions for the quantum phase diagram and the elementary excitations of the JCHM.
iii.1 Slave-boson formulation
A convenient starting point for our slave-boson approach is the polariton representation Koch and Hur (2009) of the boson operator
(8) |
in terms of standard algebra operators and matrix elements with for () at zero detuning (). It is straightforward to show that the polariton operators obey bosonic commutation relations if the constraint (completeness relation)
(9) |
is fullfilled at each site . In this new basis the JCHM becomes
In Schmidt and Blatter (2009) we have shown that the presence of the upper polariton branch with leads to additional high energy conversion modes in the Mott phase with small spectral weight and bandwidth. We thus neglect the upper branch as well as particle conversion hopping (i.e, processes where a polariton hops to another site and at the same time changes its nature from upper to lower or vice versa) from now on and drop the branch index . This leads to the simplified Hamiltonian
with and .
iii.2 Mean field theory
In order to calculate the phase boundary and static observables in the superfluid phase near a Mott lobe with filling , we restrict the Hilbert space to states with and bosons and make a Gutzwiller Ansatz for the ground-state wave function
where we also dropped the index and changed the notation to , , and . Note, that this variational wave function is normalized to unity and satisfies the completeness relation (9) in the restricted Hilbertspace of lower polaritons.
The expectation value yields the variational energy
(12) | |||||
which has to be minimized with respect to the variational parameters and . Here, denotes the dimension of a hypercubic lattice. We obtain the relations
(13) |
and
(14) |
The lobe boundaries are determined by the vanishing of the order parameter
(15) |
Setting (i.e, ) in (13) and (14) and eliminating yields the relation
(16) |
with
(17) |
Eq. (16) constitutes an expression for the mean-field boundaries of the Mott lobes in the JCHM shown in Fig. 1. Here, we point out that the size of all Mott lobes with filling factor decrease for any finite detuning , while the size of the lowest Mott lobe () increases with negative detuning (). Only in this latter case the nature of all polaritons in the lattice become qubit-like and are thus trivially localized. Fig. 1 thus already suggests that a weak-coupling mean-field approach might be suitable for a description of the lowest Mott lobe in the limit of large hopping and negative detuning. We will further elaborate on this in section V.
iii.3 Elementary excitations
In order to find the elementary excitations we define a new set of operators , which is obtained from the original polariton basis via a unitary transformation with
(18) |
The operator creates a new vacuum state, i.e., the mean-field ground state , and are orthogonal operators creating excitations above the ground-state. We express the Hamiltonian in terms of these new operators and eliminate by using the constraint (9) in the restricted Hilbert space
(19) |
Expanding the square root everywhere in the Hamiltonian to quadratic order in yields, after a Fourier transformation, an effective quadratic Hamiltonian
(20) |
where and is a matrix
(21) |
with denoting matrices defined in the appendix. The sum over runs over the first Brioullin zone. The effective Hamiltonian can be diagonalized by a bosonic Bogoliubov transformation yielding
(22) |
with a fluctuation-generated correction of the ground-state energy
(23) |
and creating excitations with energy
(24) |
with expressions for , , and given in the appendix.
In the Mott phase, the spectra can be written explicitely as
(25) |
with
(26) |
and the single-particle spectrum
(27) |
We thus obtain two gapped modes corresponding to particle/hole like excitations (see, Fig. 2).
In the superfluid phase, we obtain a gapless, linear Goldstone mode with a finite sound velocity . At the phase boundary, the sound velocity of this Goldstone mode vanishes except at the tip of the lobe, where the sound velocity maintains a finite value different from zero. This leads to a change of the dynamical critical exponent of the SF-MI transition from its generic value to . The JCHM is thus in the same universality class as the BHM Schmidt and Blatter (2009); Koch and Hur (2009). This has been confirmed by large scale Quantum Monte-Carlo simulations Hohenadler et al. (2011).
A second mode, the so-called amplitude or Higgs mode, generally remains gapped with except for the tip of the lobe, where the gap vanishes. Thus, the amplitude mode at the tip of the lobe becomes linear consistent with a change of the dynamical critical exponent as discussed above. For a detailed discussion of the excitation spectra we refer to the caption in Fig. 2.
Iv Weak correlations: Bogoliubov-like theory
The quantum phase transitions in the JCHM separates a phase with a broken symmetry from “normal” insulating states. At finite temperature, as for the Bose-Hubbard model, the insulating Mott lobes join up to become a normal state (and the quantised occupation is destroyed at finite temperature). Viewed this way, there is a clear relation between the phase transitions in the JCHM and the Dicke-Hepp-Lieb Dicke (1954); Hepp and Lieb (1973a); Wang and Hioe (1973); Hepp and Lieb (1973b) superradiance phase transition of the Tavis-Cummings model Tavis and Cummings (1968), which is frequently referred to as the Dicke model (some authors make the distinction that the Dicke model contains also counter-rotating terms in the coupling between light and matter, however this naming convention is not followed by all authors). This section discusses how theories developed for the Dicke model can be used to understand the JCHM. Surprisingly, this reveals that the Dicke model shows Mott lobes, and that the generalised Dicke model defined below contains a point, where the universality class of the phase transition changes, just as for the JCHM.
iv.1 Mapping to the Dicke model
By Fourier transforming the photon operators to momentum space
(28) |
the JCHM can be written as
where with , and ( denotes the number of lattice sites). This Hamiltonian represents a many-mode Dicke model, as studied in Refs. Keeling et al. (2004, 2005). The case studied in those works, however, considered a quadratic photon spectrum, equivalent to expanding the lattice dispersion for small vectors yielding
(30) |
Here, we have defined a Dicke-model chemical potential , such that is required for thermodynamic stability. It is similarly useful to define a Dicke-model detuning , measuring the detuning between the 2LS energy and the bottom of the photon band so that . With this quadratic expansion the generalised Dicke model describes localised two-level systems coherently coupled to a continuum of photonic modes with an effective photonic mass .
The quadratic expansion of the dispersion removes behaviour arising when the bandwidth becomes small compared to other energy scales, i.e., the Dicke model with quadratic dispersion corresponds to the JCHM in the limit of large bandwidth . However, as discussed below, even with this restriction, the first two Mott lobes can still be reached. For the single mode Dicke model, in the limit, mean-field theory is exact, i.e., fluctuation corrections are suppressed as . However, for the generalised Dicke model this is not the case. Fluctuation corrections due to finite momentum photon modes can shift the phase boundary Keeling et al. (2004, 2005) and change the critical behaviour from mean-field to that of the XY model. This shift (and the size of the fluctuation dominated regime, as determined by the Ginzburg criterion) is, however, small if the density of states of finite momentum modes is small. As such, the limit in which the JCHM and Dicke models match is also the limit in which fluctuation corrections to mean field theory become negligible. In the following, we thus first discuss the mean field theory and the spectrum of fluctuations about this point. The following section relates these ideas to the effect of fluctuations on the phase boundary and further connections between the JCHM and Dicke model phase diagrams.
iv.2 Mean-field theory of the Dicke model
We now first consider a mean-field approximation for the single photon mode with zero wave vector, i.e., . As first discussed by Hepp and Lieb Hepp and Lieb (1973a, b), for a single mode, there is a transition to a superradiant state (i.e., a superfluid state of the JCHM with broken symmetry) if
(31) |
Here, denotes the inverse temperature given in units of the Boltzman constant . The original idea of the superradiant phase transition for the ground state of the Dicke model was later questioned by Rzazewski et al. Rzazewski et al. (1975) who pointed out that diamagnetic terms prevent the phase transition of two-level systems coupled to a photon mode, leading to a “no-go theorem” for the superradiance transition. The presence of a chemical potential avoids this no-go theorem: increasing the density of excitations by increasing reduces the critical to a regime where diamagnetic terms can be neglected Eastham and Littlewood (2000).
This is particularly clear at . In this case the mean-field self-consistency equation (gap equation) for becomes
(32) |
where . Thus, the photonic condensate is given by
(33) |
The transition from superfluid to normal phase is signalled by the vanishing of the order parameter, i.e., , corresponding to (the modulus sign appearing here can be understood from the limit of equation (31)).
The zero temperature phase diagram is shown in Fig. 3. As , the critical coupling strength goes to zero and so a superradiant region is always seen near . For , more complicated behaviour vs occurs — two superradiant (superfluid) regions exist, separated by a normal state. Near there is a high susceptibility of the two-level systems towards polarisation and hence a superradiant region exists near . The normal state for has all two-level systems inverted and thus corresponds to a Mott lobe with one excitation per cavity.
With this identification of the Mott lobe, the tip of the lobe can be found as the value of for which two branches of the phase boundary merge. Since the lobe occurs within the region the modulus sign in the expression for the critical can be removed and the phase boundary of this lobe becomes
(34) |
hence the lobe tip is at as is clear from Fig. 3. As we are going to show in the next section, this result exactly agrees with those obtained from the slave boson theory derived in the previous section.
iv.3 Excitation spectra
We now look at the excitation spectra. These can be found by writing an effective action for the Dicke model and expanding around the saddle point corresponding to the mean-field solution Eastham and Littlewood (2001); Keeling et al. (2005). Identifying the poles of the Green’s function then gives the excitation spectrum. In general, they can be written as
(35) |
with
(36) | |||||
(37) |
In the normal phase with , these expressions simplify to
(38) |
with
(39) |
where .
It follows that the gap of the amplitude mode is given by . Thus, this gap is non-zero everywhere on the to superradiant boundary (where ). At the tip of the to superradiant boundary, i.e., at , there is a vanishing gap of the amplitude mode. Similarly there is vanishing sound velocity of the gapless mode everywhere on the phase boundary, except at the tip of the lobe where a vanishing gap leads to a linear dispersion with sound velocity .
V Connection between the two limits
v.1 Lobe tips in quantum theory
In order to compare the two approaches valid in the weak and strong correlation limits, we evaluate the slave boson theory at infinite bandwidth while keeping and fixed, corresponding to infinite negative detuning. With the expressions in the appendix we obtain for the phase boundary matching exactly Eq. (34), with the tip of the lobe at and . Here, one mode remains gapless, the other maintains a gap away from the tip. The sound velocity vanishes everywhere except at the tip of the lobe where . These results agree exactly with those obtained from a weak-coupling mean-field theory in the previous section.
Thus, the weak-coupling mean-field theory describes correctly the superfluid-Mott insulator transition of the lowest Mott lobe with in the limit of large hopping and large negative detuning. In Fig. 1 one can see the reason for this success: the size of the lowest Mott lobe increases for large and large negative , while the size of all other lobes decreases. Thus, only one lobe survives in Fig. 3. All other modes are pushed towards and vanish. The success of the weak-coupling theory for the JCHM is in strong contrast to the Bose-Hubbard model, where a Bogoliubov-like theory describing weakly interacting atomic BEC‘s, fails to predict the existence of Mott lobes and gapped Higgs modes at weak interactions van Oosten et al. (2001). In the polariton picture, one source of this difference is clear: the mean field theory of the Dicke model has normal modes that result from hybridisation of the photon modes and spin-waves of the two-level systems, leading to two polariton branches Pekar (1958); Hopfield (1958). The nature of these branches depends on the detuning parameter , which does not exist for the BHM. At the phase boundary, one of these branches turns into the Higgs mode, the other corresponds to the Goldstone mode in Fig. 2. It is interesting to note that the agreement of these results occurs even despite the upper polariton state having been eliminated in the slave-boson theory. To understand how this can be so, one should note that this elimination is in terms of the modes defined on a single cavity, whereas the weak coupling theory describes polaritons arising from the delocalised photon mode, thus corresponding to different mixtures of qubit and photon as well as phase and amplitude degrees of freedom.
v.2 Finite temperature phase transition and mean-field theory
As noted above, the mean field theory of the Dicke model is only correct in the limit of large bandwidth (small photon mass), where the density of states for fluctuation corrections vanishes. In the context of the Dicke model with a quadratic dispersion, a heavier mass does not mean a finite bandwidth, but does mean fluctuation corrections are important. In the absence of a finite bandwidth, tight-binding (strong correlations) approaches cannot so easily be applied. An alternate approach to account for the fluctuation corrections is to consider the fluctuation correction to the effective action, following Nozières & Schmitt-Rink Nozieres and Schmitt-Rink (1985); Randeria (1995). When such an approach is carried out for the Dicke model Keeling et al. (2005) one notable effect is that even for , there can be multiple disconnected normal regions, i.e., fluctuations can suppress the superradiant phase, leading to the appearance of the Mott lobe. This suppression of superradiance is most clearly seen by considering the critical temperature for the superradiance transition. Including fluctuations, this critical temperature can be suppressed to , leading to two disconnected superradiant “bubbles” in Fig.4.
The suppression of due to fluctuations has a natural interpretation in the context of the JCHM. Including fluctuations corresponds to increasing the photon mass, or decreasing the effective hopping. For the JCHM, it is self-evident that reducing hopping will lead to the suppression of superfluidity. This is shown in Fig. 4 which compares the critical temperature coming from the standard decoupling mean field theory (as opposed to weak-coupling mean-field theory) of the JCHM to the critical temperature of the corresponding Dicke model. These clearly show how the Dicke limit is recovered (both panels). Note, that the quantum phase diagram as obtained from decoupling mean-field theory agrees exactly with the results of the slave-boson theory.
Vi Conclusions and Outlook
In this paper, we have shown that a remarkable relation exists between a slave-boson theory for the JCHM and a weak-coupling mean-field theory for the multi-mode Dicke model. We found that both theories match exactly in the limit of infinite bandwidth and negative detuning. In this special limit a single Mott lobe survives and thus a weak-coupling mean-field theory is capable of describing the superfluid-Mott insulator transition with correct critical exponents. So far, our work was based on the equilibrium assumption, i.e., a chemical potential was introduced in order to fix the number of polaritons inside the cavity-array. It would be very interesting to see whether predictions of the weak-coupling mean-field theory for the driven dissipative generalizations of the Dicke model Szymanska et al. (2006, 2013); Dimer et al. (2007); Baumann et al. (2010); Nagy et al. (2010); Keeling et al. (2010); Nagy et al. (2011); Kónya et al. (2012); Bhaseen et al. (2012); Öztop et al. (2012); Torre et al. (2013) also allow predictions for the fate of the SF-MI transition under non equilibrium conditions.
Acknowledgements.
This work was supported by a SNF Ambizione award (Swiss National Science Foundation).Appendix: Abbreviation in the slave-boson formalism
The matrix elements of the two-by-two matrices in Eq. (21) are given by
(40) | |||||
(41) | |||||
(42) | |||||
(43) |
and
(44) | |||||
(45) | |||||
(46) | |||||
(47) |
with the definitions
(48) | |||||
(49) | |||||
(50) | |||||
(51) |
and
(52) | |||||
(53) | |||||
(54) | |||||
(55) |
The fluctuation correction to the ground-state energy in Eq. (23) reads
(56) | |||||
Finally, the expressions describing the excitation spectra in Eq. (24) are given by
(57) | |||||
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