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Quantum field theoretical study of an effective spin model in coupled optical cavity arrays
Physica B 407 (2012) 44–48
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Quantum field theoretical study of an effective spin model in coupled optical cavity arrays Sujit Sarkar n Poornaprajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 560080, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2011 Received in revised form 25 August 2011 Accepted 21 September 2011 Available online 24 September 2011
Atoms trapped in micro-cavities and interacting through the exchange of virtual photons can be modeled as an anisotropic Heisenberg spin-1/2 lattice. We do the quantum field theoretical study of such a system using the Abelian bosonization method followed by the renormalization group analysis. An infinite order Berezinskii–Kosterliz–Thouless transition is replaced by second order XY transition even when an infinitesimal anisotropy in exchange coupling is introduced. We predict a quantum phase transition between the photonic Coulomb blocked induce Mott insulating and photonic superfluid phases due to detuning between the cavity and laser frequency. A large detuning favors the photonic superfluid phase. We also perform the analysis of Jaynes and Cumming Hamiltonian to support the results of quantum field theoretical study. & 2011 Published by Elsevier B.V.
Keywords: Quantum many body models Polariton Cavity QED Spin chain model
1. Introduction The physics of a strongly correlated system is interesting in its own right and manifests in different branches of physics. The physics of strong correlation appears in natural oxide materials [1] and also in engineered materials, like the correlated physics in Josephson junction array [2], Bose–Einstein condensation and optical lattice [3,4]. Therefore, one can raise the question: what is the further source of correlated physics in the state of engineering? The recent experimental success in engineering strong interaction between the photons and atoms in high quality micro-cavities opens up the possibility to use light matter system as quantum simulators for many body physics [5–19]. In this paper we present the proper theoretical foundation through the quantum phase analysis based on Abelian bosonization and renormalization group (RG) study of the model Hamiltonian proposed in Refs. [6,7]. Before we proceed further, we would like to discuss the basic physics of micro-optical cavity very briefly. A micro-cavity can be created in a photonic band gap material by producing a localized defect in the structure of the crystal, in such a way that light of a particular frequency cannot propagate outside the defect area. Large arrays of such micro-cavities have been produced. Photon hopping between neighboring cavities has been observed in the microwave and optical domains. Many body Hamiltonians can be created and probed in coupled cavity arrays. Atoms in the cavity are used for detection and also for generation of interaction between photons in the same cavity. As the distance between
the adjacent cavities is considerably larger than the optical wave length of the resonant mode, individual cavities can be addressed. This artificial system can act as a quantum simulator. In this optical cavities system we study the different quantum phases of polariton (a combined excitations of atom–photon interactions) by using the spin model that conserves the total number of excitations. To the best of our knowledge, this is the first explicit quantum field theoretical calculations of this type of system. At first for the completeness, we discuss the generation of the spin model for such type of systems [6,7]. Micro-cavities of a photonic crystal are coupled through the exchange of photons. Each cavity consists of one atom with three levels in the energy spectrum, two of them are long lived and represent two spin states of the system and the other represents excited states (Figs. 1 and 2 of Refs. [6,7]). Externally applied laser and cavity modes couple to each atom of the cavity. It may induce the Raman transition between these two long lived energy levels. Under a suitable detuning between the laser and the cavity modes, virtual photons are created in the cavity which mediate interactions with another atom in a neighboring cavity. One can eliminate the excited states by choosing the appropriate detuning between the applied laser and cavity modes. Then one can achieve only two states per atom in the long lived state and the system can be described by a spin-1/2 Hamiltonian [6,7]. The Hamiltonian of the system consists of three parts H ¼ HA þ HC þ HAC :
ð1Þ
Hamiltonians are the following: n
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HA ¼
N X j¼1
oe 9ej S/ej 9þ oab 9bj S/bj 9,
ð2Þ
S. Sarkar / Physica B 407 (2012) 44–48
where j is the cavity index. oab and oe are the energies of the state 9bS and the excited state, respectively. The energy level of state 9aS is set as zero. 9aS and 9bS are the two stable states of an atom in the cavity and 9eS is the excited state of that atom in the same cavity. The following Hamiltonian describes photons in the cavity: H C ¼ oC
N X j¼1
ayj aj þ J C
N X
ðayj aj þ 1 þ h:cÞ,
ð3Þ
j¼1
where ayj ðaj Þ is the photon creation (annihilation) operator for the photon field in the jth cavity, oC is the energy of photons and J C is the tunneling rate of photons between neighboring cavities. Interaction between the atoms and photons and also by the driving lasers are described by N X Oa ioa t e HAC ¼ þ g a aj 9ej S/aj 9 þh:c þ ½a2b: ð4Þ 2 j¼1 Here g a and g b are the couplings of the cavity mode for the transition from energy states 9aS and 9bS to the excited state. Oa and Ob are the Rabi frequencies of the lasers with frequencies oa and ob , respectively. The authors of Refs. [6,7] have derived an effective spin model by considering the following physical processes: a virtual process regarding emission and absorption of photons between the two stable states of neighboring cavity yields the resulting effective Hamiltonian as N N X X J1 y J Hxy ¼ Bszj þ sj sj þ 1 þ 2 sj sj þ 1 þ h:c : ð5Þ 2 2 j¼1 j¼1 When J2 is real then this Hamiltonian reduces to the XY model. Where szj ¼ 9bj S/bj 99aj S/aj 9, sjþ ¼ 9bj S/aj 9, s j ¼ 9aj S/bj 9. The relation between the spin operator and atom–photon relation discussed explicitly in Ref. [20] Hxy ¼
N X
ðBszi þ J1 ðsxi sxiþ 1 þ syi syiþ 1 Þ þ J2 ðsxi sxiþ 1 syi syiþ 1 ÞÞ
i¼1
¼
N X
Bðszi þ J x sxi sxiþ 1 þJ y syi syiþ 1 Þ:
ð6Þ
i¼1
With J x ¼ ðJ 1 þJ 2 Þ and J y ¼ ðJ 1 J 2 Þ. Here we discuss very briefly about an effective z component of interactions (szi sziþ 1 ) in such a system. The authors of Refs. [6,7] have proposed the same atomic level configuration but having only one laser of frequency o that mediates the atom–atom coupling through virtual photons. Another laser field with frequency n is used to tune the effective magnetic field. In this case the Hamiltonian HAC changes but the Hamiltonians HA and HC are the same. N X Oa ioa t La ina t e e þ g a aj 9ej S/aj 9 þ h:c þ ½a2b: ð7Þ HAC ¼ 2 2 j¼1 Here Oa and Ob are the Rabi frequencies of the driving laser with frequency o on transition 9aS-9eS, 9bS-9eS, whereas La and Lb are the driving laser with frequency n on transition 9aS-9eS, 9bS-9eS. One can eliminate adiabatically the excited atomic levels and photons by considering the interaction picture with respect to H0 ¼ HA þHC [6,7]. They have considered the detuning parameter in such a way that the Raman transitions between two levels are suppressed and also chosen the parameter in such a way that the dominant two-photon processes are those that involve one laser photon and one cavity photon but the atom makes no transition between levels a and b. Whenever two atoms exchange a virtual photon both of them experience a Stark shift and play the role of an effective sz sz interaction [6,7]. Then the
45
effective Hamiltonian reduces to Hzz ¼
N X
ðB~ szj þ Jz szj szjþ 1 Þ:
ð8Þ
j¼1
~ J , J2 and J are given in Ref. [7]. Analytical expressions for B, 1 z These two parameters can be tuned independently by varying the laser frequencies. Finally, they have obtained an effective model by combining Hamiltonians Hxy and Hzz by using Suzuki– Trotter formalism. The effective Hamiltonian simulated by this procedure is ! N X X Hspin ¼ Btot szj þ J a saj saj þ 1 , ð9Þ j¼1
a ¼ x,y,z
~ It has been shown in Ref. [7] that J is less than where Btot ¼ B þ B. y Jx . From analytical expressions for Jx and Jy , it is clear that the magnitudes of J 1 and J2 are different. The result of numerical simulations trigger us also to define a model to study the quantum phases of this system. In the next section, we present the RG study of this model Hamiltonian to extract quantum phases and transition between themselves.
2. Renormalization group study of model Hamiltonian To study the different quantum phases of the system described by the Hamiltonian (Eq. (9)), we express this Hamiltonian in more explicit way X z H2 ¼ ½ð1þ aÞSxn Sxn þ 1 þ ð1aÞSyn Syn þ 1 þ DSzn Szn þ 1 þhSn , ð10Þ n
where San are the spin-1/2 operators. We assume that the XY anisotropy a and the zz coupling D satisfy the condition 1 r D r1 and 0 o a r 1 and magnetic field strength is h Z0. Parameters correspondence between the micro-cavities and spin chain are the following, h Btot , D ¼ J z , J1 ¼ 1 and J 2 ¼ a. The XY anisotropy breaks the in plane rotational symmetry. The study of the quantum phases of this type of model Hamiltonian is not entirely a new one. There are few studies which have already been done for this model Hamiltonian in different context [21–23]. We follow one of our previous work on this model Hamiltonian, during the derivation of RG equations [23]. But we implement this RG equation in different context. Spin operators can be recast in terms of spinless fermions through Jordan–Wigner transformation and then finally one can express the spinless fermions in terms of bosonic fields [24]. We recast the spinless fermions operators in terms of field operators by this relation [24]. cðxÞ ¼ ½eikF x cR ðxÞ þ eikF x cL ðxÞ, where cR ðxÞ and cL ðxÞ describe the second-quantized fields of right- and the left-moving fermions, respectively, and kF is the Fermi wave vector. We express the fermionic pffiffiffiffiffiffiffiffiffi fields in terms of bosonic field by the relation cr ðxÞ ¼ ðU r = 2paÞeiðrfðxÞyðxÞÞ , where r denotes the chirality of the fermionic fields, right (1) or left movers ( 1). The operators Ur is the Klein factor to preserve the anti-commutivity of fermions. f field corresponds to the quantum fluctuations (bosonic) of spin and y is the dual field of f. They are related by the relations fR ¼ yf and fL ¼ y þ f. Hamiltonian R H0 ¼ ðv=2Þ dx½ð@x yÞ2 þð@x fÞ2 is non-interacting part of HXYZ, where v is the velocity of the low-energy excitations. It is one of the Luttinger liquid parameters and the other is K, which is related to D by [24,25] K¼
p , p þ2 sin1 ðDÞ
ð11Þ
where K takes the values 1 and 1/2 for D ¼ 0 (free field), and D ¼ 1 (isotropic anti-ferromagnet), respectively. The relation between K and D is not preserved under the renormalization, so this relation
46
S. Sarkar / Physica B 407 (2012) 44–48
is only correct for the initial Hamiltonian. The analytical form x of the spin of the bosonic fields are: pffiffiffiffiffiffiffioperatorsn in terms pffiffiffiffiffiffiffiSn ¼ pffiffiffiffiffiffiffiffiffi y ½c2 cosð2 pKp fffiffiffiffiffiffiffiffiffi Þ þ ð1Þ c3 cosð p=K Þ, Sn ¼ ½c2 cosð2 p fÞffi þ pKffiffiffiffiffiffi pyffiffiffiffiffiffiffiffiffi ð1Þn c3 sinð p=K yÞ, and Szn ¼ p=K @x f þð1Þn c1 cosð2 pK fÞ where cis are constants as given in Ref. [26]. The Hamiltonian H2 in terms of bosonic fields is the following: rffiffiffiffi Z p H2 ¼ H0 þa cos 2 yðxÞ dx K Z Z pffiffiffiffiffiffiffi þ D cosð4 pK fðxÞÞ dxh @x fðxÞ dx: ð12Þ One can get the HXY Hamiltonian by simply putting D ¼ 0 in the above Hamiltonian. In this derivation, different powers of coefficients ci have been absorbed in the definition of a,h and D. The integration of the oscillatory terms in the Hamiltonian yield negligible small contributions, the origin of the oscillatory terms occur due the spin operators. So it is a reasonably good approximation to keep only the non-oscillatory terms in the Hamiltonian. The Gaussian scaling dimension of these coupling terms, a, D are 1=K, 4K, respectively. The third term (D) of the Hamiltonian tends to order the system into density wave phase due to the photonic Coulomb blocked phase, whereas the second term (a) of the Hamiltonian favors the staggered order in the XY plane. Two sine-Gordon coupling terms are from two dual fields. Therefore, the model Hamiltonian consists of two competing interactions between the ordered phase and the XY order [28]. This Hamiltonian contains two strongly relevant and mutually nonlocal perturbation over the Gaussian (critical) theory. In such a situation the strong coupling fixed point is usually determined by the most relevant perturbation whose amplitude grows up according to its Gaussian scaling dimensions and it is not much affected by the less relevant coupling terms. However, this is not the general rule if the two operators exclude each other, i.e., if the field configurations which minimize one perturbation term do not minimize the other. In this case interplay between the two competing relevant operators can produce a novel quantum phase transition through a critical point or a critical line. Therefore, we would like to study the RG equation to interpret the quantum phases of the system. We now study how the parameters a, D and K flow under RG. The operators in Eq. (12) are related to each other through the operator product expansion; the RG equations for their coefficients therefore are coupled to each other. We use operator product expansion to derive these RG equations which is independent of boundary condition [27]. RG equations themselves are established in a perturbative expansion in coupling constant (g(l)), they cease to be valid beyond a certain length scale, where gðlÞ 1 [24]. The RG equations for the coefficients of Hamiltonian HXYZ are da 1 ¼ 2 a, dl K dD ¼ ð24KÞD, dl dK a2 ¼ K 2 D2 : dl 4
ð13Þ
We have followed Ref. [23] during the derivation of these RG equations. These RG equations have trivial (an ¼ 0 ¼ Dn ) fixed points for any arbitrary K. Apart from that these RG equations have also two non-trivial fixed lines, a ¼ D and a ¼ D for K ¼ 1=2. In our study, there are critical surfaces on which the system flows onto the non-trivial fixed lines (a ¼ 7 D). Now we present the quantum phase analysis based on these RG equations for the different limits of the Luttinger liquid
parameters: at first we present the quantum phases of anisotropy spin chain and then we present the corresponding photonic states of the optical cavity arrays. A commensurate density wave state can be characterized when K-0 or the staggered ordered when (K-1). Note that the transition occurring on them is second order. Infinitesimal amount of anisotropy changes the situation drastically, a gapless phase to a gapped phase in the presence of anisotropy. This transition correspondence to gapless photonic superfluid phase to gapped insulating phase of the micro-cavities arrays due to the presence of staggered order. This means that the presence of staggered order stops the photon transmission between the micro-cavities of the array that finally leads to the insulating state of the system. Since this gapped excitation is not directly related to magnetization, it will not favor creation of the plateau phase. Therefore, the analogy to the Mott insulator (MI) is not useful in the asymmetric XY ordered state. In the K-0 limit when the charge density wave phase is relevant then the system is in the plateau phase, the transition driven by the magnetic field always has z ¼ 2 (z is the dynamical critical exponent) [23] and thus the plateau shows a square-root behavior of magnetization. Therefore, the plateau phase behavior in the micro-cavity system for a proper choice of detuning between the cavity and laser fields. When a increases, a second order transition drives the system to the XY plane ordered phase, whose exponents depend on the initial couplings and hence are non-universal. Therefore, a finite J2 drives the system from the photonic superfluid phase to gapped photonic phase for micro-cavity arrays [29]. At around the transition point, the excitation is gapless, therefore z ¼ 1 [30–33]. A magnetic field larger than the relevant gap of the system drives the system to a gapless phase. This transition is the commensurate phase to incommensurate phase transition. This is the first RG study in the literature of this system. We also draw an analogy between the states of anisotropic quantum spin chain system and the optical cavity array system in the graphical representations and also in tabular form. The upper panel of Fig. 1 represents the isotropic situation (a ¼0) and the lower panel represents the anisotropic situation (a 40). The analogy between the quantum states of spin chain system and optical cavity arrays is explicit in this figure. We also present the analogy between the quantum states through the tabular form, the upper one represents the situation for K 41=2 and the lower one represents the situations for K o 1=2. aa0
K 4 1=2
a¼ 0
Spin chain system Optical cavity system
Luttinger liquid Staggered order Photonic super fluid
Staggered order photonic insulator
K o 1=2
a ¼0
aa0
Spin chain system Optical cavity system
Commensurate density wave Photonic Mott insulator
Commensurate density wave Photonic Mott insulator
Here we explain the physical meaning of different quantum states of the atom–cavity system that we have mentioned in the above paragraph. The photonic Mott insulator state of the atom–cavity system corresponds to the insulating state of the system where there is no transmission of photon between the micro-cavities in the array due to the interaction between photons. Photonic
S. Sarkar / Physica B 407 (2012) 44–48
a=0
K = 1/2 Luttinger Liquid
Spin Chain : Commensurate Density Wave Optical Cavity: Photonic Mott Insulator
Photonic Superfluid
a>0 Spin Chain : Commensurate Density Wave Optical Cavity: Photonic Mott Insulator
of the two level system, respectively. When we consider large values of photon and atomic frequencies compare to atom– photon coupling l, the number of excitations is conserved for this Hamiltonian. Suppose we consider a fixed number of excitations, n. The energy eigenvalues for n excitations are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi En7 ¼ noC þ
K = 1/2
Staggered Order Staggered Order Photonic Insulator
Fig. 1. The one dimensional schematic diagram for quantum phases as a function of Luttinger liquid parameters (K), in the presence and absence of anisotropy in XY coupling. In this figure, we draw an analogy between the spin chain system and the optical cavity array system.
superfluid state of the atom–cavity system corresponds to the gapless excitations of the system where the photon transmit from one cavity to the other without any blocking. When we consider the anisotropy in the exchange interaction of atom–cavity system it leads to the staggered order phase of the system. In this quantum phase, there is no transmission of photon between the microcavities of the array and the system is in the insulating phase. Here we discuss the effect of disorder in the optical cavity arrays. In our study we finally end up with the effective anisotropic Heisenberg Hamiltonian and extract all the quantum phases from the analysis of that effective Hamiltonian. Therefore, during this analysis we follow the prescription of the effect of disorder in quantum spin system. There are several kinds of weak randomness that have been studied in Ref. [34]. In the presence of random fields and random exchange directions the XY symmetry preserve, i.e., when a¼ 0 the photonic superfluid phase remains the same in the presence of random fields and random exchanges. They have found a transition from a quasi-long-range ordered ground state to one in which typical correlation function decays rapidly as the anisotropy parameter (D) is varied. In the case of random z-field, the disordered phase is expected to resemble a ‘‘Fermi glass’’ in which the elementary excitations are localized and spin correlation decay exponentially. The analogous phase in the optical cavity arrays is the photonic gapped Mott insulating phase. For the XY symmetric random exchange, on the other hand, they have found a ‘‘random singlet’’ phase in which spins are tightly coupled in singlet pairs. These singlet pairing can occur over the large distance. For these situations still there is a possibility for photonic superfluid phase. The authors of Ref. [7] have numerically studied the occupation probability to confirm the validity of their approximation during the foundation of the model. They have also numerically studied the von Neumann entropy of the reduced density matrix of the effective spin in multiples of ln 2. There is no analysis on quantum phase. The total magnetic field, Btot [7] increases for the larger values of detuning, therefore larger detuning drives the system from gapped (Mott-insulating) state to gapless superfluid state. This is the evidence of commensurate (gapped) phase to incommensurate (gapless) phase in micro-cavity arrays. We now discuss how the effective repulsion decreases as we increase the detuning between the atomic and laser frequency. It can be explained starting from the Jaynes–Cummings Hamiltonian [35,36]. Janes–Cummings Hamiltonian for a single atom is HJC ¼ oC ay a þ o0 9eS/e9 þ lðay 9gS/e9 þ a9eS/g9Þ,
ð14Þ
where oC and o0 are the frequencies of the resonant mode of the cavity and of the atomic transition, respectively. l is the Jaynes– Cumming coupling between the cavity mode and the two level system. ay (a) is the creation (annihilation) operator of a photon inside a cavity. 9gS and 9eS are the ground state and excited state
47
D 2
7
2
nl þ
D2 4
:
ð15Þ
Here D ¼ ðo0 oc Þ and n Z 1. Now we consider an array of cavities, the basic Hamiltonian for each cavity is the same as that of Eq. (14). Here, we consider the system with one excitation of energy E1 in each cavity and the lowest energy of two excitations in each cavity is E2 . Therefore, to create one additional excitation in each cavity requires the energy equal to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 2E1 ¼ 2
l2 þ
D2 4
2
2l þ
D2 D 4
2
:
One may consider this as an effective on-site repulsion because it measures the difference between the energy of two and single excitation (polariton in each cavity). This effective repulsion decreases as we increase the detune factor. Therefore, we conclude that for D ¼ 0, double occupation never occurs, indicating a Mott insulating behavior. When D is much larger than the coupling l, then the value of the occupation number as larger than one occurs as expected for a photonic superfluid regime. In our quantum field theoretical calculations, we have also predicted that the large detuning drives the system from gapped Mott insulating phase to the gapless superfluid phase.
3. Conclusions For the first time we have done the quantum field theoretical analysis of an effective spin model in coupled optical cavity arrays. We have predicted few quantum phases like, photonic Coulomb blocked induce Mott insulator, photonic superfluidity and staggered order insulator. An infinite order BKT transition has been replaced by the second order XY transition under the presence of exchange anisotropy. We have predicted the presence of plateau phase of this system and also commensurate phase to incommensurate phase transition. We have also discussed the effect of disorder in the optical cavity array system. The rigorous quantum field theoretical derivation of this work is absent in all previous studies and we also provide physical explanation of the transition process based on Jaynes–Cummings Hamiltonian.
Acknowledgments The author would like to thank The Center for Condensed Matter Theory of the Physics Department of IISc for facility extended. The author would like to thank Prof. N. Behera, Dr. B. Murthi and Dr. N. Sundaram for reading the manuscript very critically. References [1] C.N.R. Rao, T.V. Ramakrishnan, Superconductivity Today, Universities Press, Hyderabad, 1999. [2] K.K. Likharev, Dynamics of Josephson Junction and Circuits, Gordon and Breach, 1988. [3] J.H. Denschlog, et al., J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 3095. [4] D. Jaksch, P. Zoller, Ann. Phys. 315 (2005) 52. [5] A.D. Greentree, et al., Nat. Phys. 466 (2006) 856. [6] [a] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Nat. Phys. 462 (2006) 849; [b] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Laser Photon. Rev. 2 (2008) 527.
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S. Sarkar / Physica B 407 (2012) 44–48
[7] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Phys. Rev. Lett. 99 (2007) 160501. [8] A.C. Ji, X.C. Xie, W.M. Liu, Phys. Rev. Lett. 99 (2007) 183602. [9] T. Byrnes, N.Y. Kim, K. Kusudo, Y. Yamamoto, Phys. Rev. B 78 (2008) 075320. [10] I. Carusotto, et al., arXiv:0812.4195, 2008. [11] M.J. Bhaseen, M. Hohenadler, A.O. Silver, B.D. Simons, Phys. Rev. Lett. 102 (2009) 135301. [12] A. Tomadin, et al., arXiv:0904.4437, 2009. [13] J. Zhao, A.W. Sandvik, K. Ueda, arXiv:0806.3603, 2008. [14] P. Pippan, H.G. Evertz, M. Hohenadler, arXiv:0904.1350, 2009. [15] D. Rossini, R. Fazio, Phys. Rev. Lett. 99 (2007) 186401; N. Didier, S. Pugnetti, Y.M. Blanter, R. Fazio, arXiv:1007.471 (cond-mat), 2010. [16] M. Aichhorn, et al., Phys. Rev. Lett. 100 (2008) 216401. [17] S. Schmidt, G. Blatter, arXiv:0905.3344, 2009. [18] D.G. Angelakis, M.F. Santos, S. Bose, Phys. Rev. A 76 (2007) R031805. [19] J. Koch, K. Le Hur, Phys. Rev. A 80 (2009) 023811; K. Le Hur, 2009, arXiv/cond-mat-0909.4822. [20] Here we discuss the relation between the spin operators and the atom– photon system. Without loss of generality, we start with the well celebrated Janes–Cummings Hamiltonian. H ¼ gðsay þ h:cÞ. Cavity mode represents by the bosonic operators (a,ay ) and atomic mode represents by the fermionic operator (s, sy ). Where s is the Pauli matrix which transforms one excitation from the radiation field to the atomic field. Therefore, one can write 0 1 sy ¼ 0 0
[21] [22] [23] [24] [25]
similarly one can write for s with /19 ¼ ð1; 0Þ and /09 ¼ ð0; 1Þ. The excitation in this system only transfers between atom and photon in the cavity. For a fixed number (n) of total excitation, one can express the manifold: Hn ¼ f90,nS, 91,n1Sg provided nZ 1. Here 90,nS and 91,n1S represent atom in the ground state with n photon and excited states of the atom with (n1) photon, respectively. Now we explicitly establish the relation between the spin operator and the atom photon system: we consider the initial state 9e,n1S, we obtain the state 9g,nS by the following operation: 9g,nS ¼ sa þ 9e,n1S. Therefore, we may write the following relation based on the conservation of the number of excitation. sy a9e,n1S ¼ 0 ¼ say 9g,nS and sy a9g,nS ¼ 9e,n1S. These relations are nothing but the properties of spin operators acting on the spins in the z basis. B. Surtharland, J. Math. Phys. 11 (1970) 3183. Baxter, Phys. Rev. Lett. 26 (1971) 834. S. Sarkar, Phys. Rev. B 74 (2006) 052410. T. Giamarchi, Quantum Physics in One Dimension, Clarendon Press, Oxford, 2004. U. Schollwock, J. Richter, D.J.J. Farnell, R.F. Bishop (Eds.), Lecture Notes in Physics, vol. 645, Quantum Magnetism, Springer, Berlin, 2004.
[26] S. Lukyanov, A. Zamolodchikov, Nucl. Phys. B 493 (1997) 571. [27] J. Cardy, in: Scaling and Renormalization in Statistical Physics Cambridge University Press, Cambridge, 1996; I. Affleck, in: E. Brezin, J. Zinn-Justin (Ed.), Fields, Strings and Critical Phenomena, North-Holland, Amsterdam, 1989. [28] There is no spontaneous symmetry breaking in one dimension as we observe in higher dimensions. Therefore, we will not be able to define the order parameter as we define for higher dimensional systems. In one dimensional system only discrete symmetry is broken. Therefore, we have not defined any order parameter for the density wave and XY staggered phase. XY phase is related with the inplane broken rotational symmetry. In our work, we have studied the anamolous scaling dimensional of the sine-Gordon coupling terms of the Hamiltonian equation (12) to conclude the relevancy of these terms. [29] The quantum phases which we have discussed for one dimensional system is actually instabilities, i.e., the corresponding susceptibility will diverge for the one dimensional system. In one dimension, there is no spontaneous symmetry breaking, only discrete symmetry breaks in one dimension. These instabilities in one dimension are casually referred as a quantum phase. [30] In quantum phase transitions, in addition to the standard critical exponent, it is useful to define an additional exponent z, called the dynamical exponent, which tells us how a characteristic length in the time direction is related to a z length in the spatial direction xt xx . For the quantum problem the time plays a special role and this special direction has no reason to have the same exponent as the spatial one. Deep inside the Mott insulating phase, particle and hole excitations are gapped. The system is almost similar to the atomic limit in the deep Mott insulating phase. When we approach the phase boundary from the deep Mott insulating phase then the dispersion relation is 2 quadratic o k . (It is well known in the literature that the energy z dispersion at the transition point is o k , therefore z ¼ 2 during the transition from Mott insulating phase to the gapless superfluid phase.) But the situation is different at the end of CDW phase and the starting point of XY staggered order phase. At around the multical critical point z is 1 (around this point, energy dispersion is gapless, i.e., e k. Therefore, z ¼ 1). See Refs. [31–33] and for a detailed understanding of this subject. [31] S. Sachdev, Quantum Phase Transition, Cambridge University Press, Cambridge, 1998. [32] S.L. Sondhi, et al., Rev. Mod. Phys. 69 (1997) 315. [33] M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Phys. Rev. B 40 (1989) 546. [34] A. Doty Curtis, D.S. Fisher, Phys. Rev. B 45 (1992) 2167. [35] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. [36] S. Horoche, J.M. Raimond, Exploring the Quantum Atoms, Cavities, and Photons, Oxford University Press, 2006.
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Physica B journal homepage: www.elsevier.com/locate/physb
Quantum field theoretical study of an effective spin model in coupled optical cavity arrays Sujit Sarkar n Poornaprajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 560080, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 March 2011 Received in revised form 25 August 2011 Accepted 21 September 2011 Available online 24 September 2011
Atoms trapped in micro-cavities and interacting through the exchange of virtual photons can be modeled as an anisotropic Heisenberg spin-1/2 lattice. We do the quantum field theoretical study of such a system using the Abelian bosonization method followed by the renormalization group analysis. An infinite order Berezinskii–Kosterliz–Thouless transition is replaced by second order XY transition even when an infinitesimal anisotropy in exchange coupling is introduced. We predict a quantum phase transition between the photonic Coulomb blocked induce Mott insulating and photonic superfluid phases due to detuning between the cavity and laser frequency. A large detuning favors the photonic superfluid phase. We also perform the analysis of Jaynes and Cumming Hamiltonian to support the results of quantum field theoretical study. & 2011 Published by Elsevier B.V.
Keywords: Quantum many body models Polariton Cavity QED Spin chain model
1. Introduction The physics of a strongly correlated system is interesting in its own right and manifests in different branches of physics. The physics of strong correlation appears in natural oxide materials [1] and also in engineered materials, like the correlated physics in Josephson junction array [2], Bose–Einstein condensation and optical lattice [3,4]. Therefore, one can raise the question: what is the further source of correlated physics in the state of engineering? The recent experimental success in engineering strong interaction between the photons and atoms in high quality micro-cavities opens up the possibility to use light matter system as quantum simulators for many body physics [5–19]. In this paper we present the proper theoretical foundation through the quantum phase analysis based on Abelian bosonization and renormalization group (RG) study of the model Hamiltonian proposed in Refs. [6,7]. Before we proceed further, we would like to discuss the basic physics of micro-optical cavity very briefly. A micro-cavity can be created in a photonic band gap material by producing a localized defect in the structure of the crystal, in such a way that light of a particular frequency cannot propagate outside the defect area. Large arrays of such micro-cavities have been produced. Photon hopping between neighboring cavities has been observed in the microwave and optical domains. Many body Hamiltonians can be created and probed in coupled cavity arrays. Atoms in the cavity are used for detection and also for generation of interaction between photons in the same cavity. As the distance between
the adjacent cavities is considerably larger than the optical wave length of the resonant mode, individual cavities can be addressed. This artificial system can act as a quantum simulator. In this optical cavities system we study the different quantum phases of polariton (a combined excitations of atom–photon interactions) by using the spin model that conserves the total number of excitations. To the best of our knowledge, this is the first explicit quantum field theoretical calculations of this type of system. At first for the completeness, we discuss the generation of the spin model for such type of systems [6,7]. Micro-cavities of a photonic crystal are coupled through the exchange of photons. Each cavity consists of one atom with three levels in the energy spectrum, two of them are long lived and represent two spin states of the system and the other represents excited states (Figs. 1 and 2 of Refs. [6,7]). Externally applied laser and cavity modes couple to each atom of the cavity. It may induce the Raman transition between these two long lived energy levels. Under a suitable detuning between the laser and the cavity modes, virtual photons are created in the cavity which mediate interactions with another atom in a neighboring cavity. One can eliminate the excited states by choosing the appropriate detuning between the applied laser and cavity modes. Then one can achieve only two states per atom in the long lived state and the system can be described by a spin-1/2 Hamiltonian [6,7]. The Hamiltonian of the system consists of three parts H ¼ HA þ HC þ HAC :
ð1Þ
Hamiltonians are the following: n
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HA ¼
N X j¼1
oe 9ej S/ej 9þ oab 9bj S/bj 9,
ð2Þ
S. Sarkar / Physica B 407 (2012) 44–48
where j is the cavity index. oab and oe are the energies of the state 9bS and the excited state, respectively. The energy level of state 9aS is set as zero. 9aS and 9bS are the two stable states of an atom in the cavity and 9eS is the excited state of that atom in the same cavity. The following Hamiltonian describes photons in the cavity: H C ¼ oC
N X j¼1
ayj aj þ J C
N X
ðayj aj þ 1 þ h:cÞ,
ð3Þ
j¼1
where ayj ðaj Þ is the photon creation (annihilation) operator for the photon field in the jth cavity, oC is the energy of photons and J C is the tunneling rate of photons between neighboring cavities. Interaction between the atoms and photons and also by the driving lasers are described by N X Oa ioa t e HAC ¼ þ g a aj 9ej S/aj 9 þh:c þ ½a2b: ð4Þ 2 j¼1 Here g a and g b are the couplings of the cavity mode for the transition from energy states 9aS and 9bS to the excited state. Oa and Ob are the Rabi frequencies of the lasers with frequencies oa and ob , respectively. The authors of Refs. [6,7] have derived an effective spin model by considering the following physical processes: a virtual process regarding emission and absorption of photons between the two stable states of neighboring cavity yields the resulting effective Hamiltonian as N N X X J1 y J Hxy ¼ Bszj þ sj sj þ 1 þ 2 sj sj þ 1 þ h:c : ð5Þ 2 2 j¼1 j¼1 When J2 is real then this Hamiltonian reduces to the XY model. Where szj ¼ 9bj S/bj 99aj S/aj 9, sjþ ¼ 9bj S/aj 9, s j ¼ 9aj S/bj 9. The relation between the spin operator and atom–photon relation discussed explicitly in Ref. [20] Hxy ¼
N X
ðBszi þ J1 ðsxi sxiþ 1 þ syi syiþ 1 Þ þ J2 ðsxi sxiþ 1 syi syiþ 1 ÞÞ
i¼1
¼
N X
Bðszi þ J x sxi sxiþ 1 þJ y syi syiþ 1 Þ:
ð6Þ
i¼1
With J x ¼ ðJ 1 þJ 2 Þ and J y ¼ ðJ 1 J 2 Þ. Here we discuss very briefly about an effective z component of interactions (szi sziþ 1 ) in such a system. The authors of Refs. [6,7] have proposed the same atomic level configuration but having only one laser of frequency o that mediates the atom–atom coupling through virtual photons. Another laser field with frequency n is used to tune the effective magnetic field. In this case the Hamiltonian HAC changes but the Hamiltonians HA and HC are the same. N X Oa ioa t La ina t e e þ g a aj 9ej S/aj 9 þ h:c þ ½a2b: ð7Þ HAC ¼ 2 2 j¼1 Here Oa and Ob are the Rabi frequencies of the driving laser with frequency o on transition 9aS-9eS, 9bS-9eS, whereas La and Lb are the driving laser with frequency n on transition 9aS-9eS, 9bS-9eS. One can eliminate adiabatically the excited atomic levels and photons by considering the interaction picture with respect to H0 ¼ HA þHC [6,7]. They have considered the detuning parameter in such a way that the Raman transitions between two levels are suppressed and also chosen the parameter in such a way that the dominant two-photon processes are those that involve one laser photon and one cavity photon but the atom makes no transition between levels a and b. Whenever two atoms exchange a virtual photon both of them experience a Stark shift and play the role of an effective sz sz interaction [6,7]. Then the
45
effective Hamiltonian reduces to Hzz ¼
N X
ðB~ szj þ Jz szj szjþ 1 Þ:
ð8Þ
j¼1
~ J , J2 and J are given in Ref. [7]. Analytical expressions for B, 1 z These two parameters can be tuned independently by varying the laser frequencies. Finally, they have obtained an effective model by combining Hamiltonians Hxy and Hzz by using Suzuki– Trotter formalism. The effective Hamiltonian simulated by this procedure is ! N X X Hspin ¼ Btot szj þ J a saj saj þ 1 , ð9Þ j¼1
a ¼ x,y,z
~ It has been shown in Ref. [7] that J is less than where Btot ¼ B þ B. y Jx . From analytical expressions for Jx and Jy , it is clear that the magnitudes of J 1 and J2 are different. The result of numerical simulations trigger us also to define a model to study the quantum phases of this system. In the next section, we present the RG study of this model Hamiltonian to extract quantum phases and transition between themselves.
2. Renormalization group study of model Hamiltonian To study the different quantum phases of the system described by the Hamiltonian (Eq. (9)), we express this Hamiltonian in more explicit way X z H2 ¼ ½ð1þ aÞSxn Sxn þ 1 þ ð1aÞSyn Syn þ 1 þ DSzn Szn þ 1 þhSn , ð10Þ n
where San are the spin-1/2 operators. We assume that the XY anisotropy a and the zz coupling D satisfy the condition 1 r D r1 and 0 o a r 1 and magnetic field strength is h Z0. Parameters correspondence between the micro-cavities and spin chain are the following, h Btot , D ¼ J z , J1 ¼ 1 and J 2 ¼ a. The XY anisotropy breaks the in plane rotational symmetry. The study of the quantum phases of this type of model Hamiltonian is not entirely a new one. There are few studies which have already been done for this model Hamiltonian in different context [21–23]. We follow one of our previous work on this model Hamiltonian, during the derivation of RG equations [23]. But we implement this RG equation in different context. Spin operators can be recast in terms of spinless fermions through Jordan–Wigner transformation and then finally one can express the spinless fermions in terms of bosonic fields [24]. We recast the spinless fermions operators in terms of field operators by this relation [24]. cðxÞ ¼ ½eikF x cR ðxÞ þ eikF x cL ðxÞ, where cR ðxÞ and cL ðxÞ describe the second-quantized fields of right- and the left-moving fermions, respectively, and kF is the Fermi wave vector. We express the fermionic pffiffiffiffiffiffiffiffiffi fields in terms of bosonic field by the relation cr ðxÞ ¼ ðU r = 2paÞeiðrfðxÞyðxÞÞ , where r denotes the chirality of the fermionic fields, right (1) or left movers ( 1). The operators Ur is the Klein factor to preserve the anti-commutivity of fermions. f field corresponds to the quantum fluctuations (bosonic) of spin and y is the dual field of f. They are related by the relations fR ¼ yf and fL ¼ y þ f. Hamiltonian R H0 ¼ ðv=2Þ dx½ð@x yÞ2 þð@x fÞ2 is non-interacting part of HXYZ, where v is the velocity of the low-energy excitations. It is one of the Luttinger liquid parameters and the other is K, which is related to D by [24,25] K¼
p , p þ2 sin1 ðDÞ
ð11Þ
where K takes the values 1 and 1/2 for D ¼ 0 (free field), and D ¼ 1 (isotropic anti-ferromagnet), respectively. The relation between K and D is not preserved under the renormalization, so this relation
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S. Sarkar / Physica B 407 (2012) 44–48
is only correct for the initial Hamiltonian. The analytical form x of the spin of the bosonic fields are: pffiffiffiffiffiffiffioperatorsn in terms pffiffiffiffiffiffiffiSn ¼ pffiffiffiffiffiffiffiffiffi y ½c2 cosð2 pKp fffiffiffiffiffiffiffiffiffi Þ þ ð1Þ c3 cosð p=K Þ, Sn ¼ ½c2 cosð2 p fÞffi þ pKffiffiffiffiffiffi pyffiffiffiffiffiffiffiffiffi ð1Þn c3 sinð p=K yÞ, and Szn ¼ p=K @x f þð1Þn c1 cosð2 pK fÞ where cis are constants as given in Ref. [26]. The Hamiltonian H2 in terms of bosonic fields is the following: rffiffiffiffi Z p H2 ¼ H0 þa cos 2 yðxÞ dx K Z Z pffiffiffiffiffiffiffi þ D cosð4 pK fðxÞÞ dxh @x fðxÞ dx: ð12Þ One can get the HXY Hamiltonian by simply putting D ¼ 0 in the above Hamiltonian. In this derivation, different powers of coefficients ci have been absorbed in the definition of a,h and D. The integration of the oscillatory terms in the Hamiltonian yield negligible small contributions, the origin of the oscillatory terms occur due the spin operators. So it is a reasonably good approximation to keep only the non-oscillatory terms in the Hamiltonian. The Gaussian scaling dimension of these coupling terms, a, D are 1=K, 4K, respectively. The third term (D) of the Hamiltonian tends to order the system into density wave phase due to the photonic Coulomb blocked phase, whereas the second term (a) of the Hamiltonian favors the staggered order in the XY plane. Two sine-Gordon coupling terms are from two dual fields. Therefore, the model Hamiltonian consists of two competing interactions between the ordered phase and the XY order [28]. This Hamiltonian contains two strongly relevant and mutually nonlocal perturbation over the Gaussian (critical) theory. In such a situation the strong coupling fixed point is usually determined by the most relevant perturbation whose amplitude grows up according to its Gaussian scaling dimensions and it is not much affected by the less relevant coupling terms. However, this is not the general rule if the two operators exclude each other, i.e., if the field configurations which minimize one perturbation term do not minimize the other. In this case interplay between the two competing relevant operators can produce a novel quantum phase transition through a critical point or a critical line. Therefore, we would like to study the RG equation to interpret the quantum phases of the system. We now study how the parameters a, D and K flow under RG. The operators in Eq. (12) are related to each other through the operator product expansion; the RG equations for their coefficients therefore are coupled to each other. We use operator product expansion to derive these RG equations which is independent of boundary condition [27]. RG equations themselves are established in a perturbative expansion in coupling constant (g(l)), they cease to be valid beyond a certain length scale, where gðlÞ 1 [24]. The RG equations for the coefficients of Hamiltonian HXYZ are da 1 ¼ 2 a, dl K dD ¼ ð24KÞD, dl dK a2 ¼ K 2 D2 : dl 4
ð13Þ
We have followed Ref. [23] during the derivation of these RG equations. These RG equations have trivial (an ¼ 0 ¼ Dn ) fixed points for any arbitrary K. Apart from that these RG equations have also two non-trivial fixed lines, a ¼ D and a ¼ D for K ¼ 1=2. In our study, there are critical surfaces on which the system flows onto the non-trivial fixed lines (a ¼ 7 D). Now we present the quantum phase analysis based on these RG equations for the different limits of the Luttinger liquid
parameters: at first we present the quantum phases of anisotropy spin chain and then we present the corresponding photonic states of the optical cavity arrays. A commensurate density wave state can be characterized when K-0 or the staggered ordered when (K-1). Note that the transition occurring on them is second order. Infinitesimal amount of anisotropy changes the situation drastically, a gapless phase to a gapped phase in the presence of anisotropy. This transition correspondence to gapless photonic superfluid phase to gapped insulating phase of the micro-cavities arrays due to the presence of staggered order. This means that the presence of staggered order stops the photon transmission between the micro-cavities of the array that finally leads to the insulating state of the system. Since this gapped excitation is not directly related to magnetization, it will not favor creation of the plateau phase. Therefore, the analogy to the Mott insulator (MI) is not useful in the asymmetric XY ordered state. In the K-0 limit when the charge density wave phase is relevant then the system is in the plateau phase, the transition driven by the magnetic field always has z ¼ 2 (z is the dynamical critical exponent) [23] and thus the plateau shows a square-root behavior of magnetization. Therefore, the plateau phase behavior in the micro-cavity system for a proper choice of detuning between the cavity and laser fields. When a increases, a second order transition drives the system to the XY plane ordered phase, whose exponents depend on the initial couplings and hence are non-universal. Therefore, a finite J2 drives the system from the photonic superfluid phase to gapped photonic phase for micro-cavity arrays [29]. At around the transition point, the excitation is gapless, therefore z ¼ 1 [30–33]. A magnetic field larger than the relevant gap of the system drives the system to a gapless phase. This transition is the commensurate phase to incommensurate phase transition. This is the first RG study in the literature of this system. We also draw an analogy between the states of anisotropic quantum spin chain system and the optical cavity array system in the graphical representations and also in tabular form. The upper panel of Fig. 1 represents the isotropic situation (a ¼0) and the lower panel represents the anisotropic situation (a 40). The analogy between the quantum states of spin chain system and optical cavity arrays is explicit in this figure. We also present the analogy between the quantum states through the tabular form, the upper one represents the situation for K 41=2 and the lower one represents the situations for K o 1=2. aa0
K 4 1=2
a¼ 0
Spin chain system Optical cavity system
Luttinger liquid Staggered order Photonic super fluid
Staggered order photonic insulator
K o 1=2
a ¼0
aa0
Spin chain system Optical cavity system
Commensurate density wave Photonic Mott insulator
Commensurate density wave Photonic Mott insulator
Here we explain the physical meaning of different quantum states of the atom–cavity system that we have mentioned in the above paragraph. The photonic Mott insulator state of the atom–cavity system corresponds to the insulating state of the system where there is no transmission of photon between the micro-cavities in the array due to the interaction between photons. Photonic
S. Sarkar / Physica B 407 (2012) 44–48
a=0
K = 1/2 Luttinger Liquid
Spin Chain : Commensurate Density Wave Optical Cavity: Photonic Mott Insulator
Photonic Superfluid
a>0 Spin Chain : Commensurate Density Wave Optical Cavity: Photonic Mott Insulator
of the two level system, respectively. When we consider large values of photon and atomic frequencies compare to atom– photon coupling l, the number of excitations is conserved for this Hamiltonian. Suppose we consider a fixed number of excitations, n. The energy eigenvalues for n excitations are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi En7 ¼ noC þ
K = 1/2
Staggered Order Staggered Order Photonic Insulator
Fig. 1. The one dimensional schematic diagram for quantum phases as a function of Luttinger liquid parameters (K), in the presence and absence of anisotropy in XY coupling. In this figure, we draw an analogy between the spin chain system and the optical cavity array system.
superfluid state of the atom–cavity system corresponds to the gapless excitations of the system where the photon transmit from one cavity to the other without any blocking. When we consider the anisotropy in the exchange interaction of atom–cavity system it leads to the staggered order phase of the system. In this quantum phase, there is no transmission of photon between the microcavities of the array and the system is in the insulating phase. Here we discuss the effect of disorder in the optical cavity arrays. In our study we finally end up with the effective anisotropic Heisenberg Hamiltonian and extract all the quantum phases from the analysis of that effective Hamiltonian. Therefore, during this analysis we follow the prescription of the effect of disorder in quantum spin system. There are several kinds of weak randomness that have been studied in Ref. [34]. In the presence of random fields and random exchange directions the XY symmetry preserve, i.e., when a¼ 0 the photonic superfluid phase remains the same in the presence of random fields and random exchanges. They have found a transition from a quasi-long-range ordered ground state to one in which typical correlation function decays rapidly as the anisotropy parameter (D) is varied. In the case of random z-field, the disordered phase is expected to resemble a ‘‘Fermi glass’’ in which the elementary excitations are localized and spin correlation decay exponentially. The analogous phase in the optical cavity arrays is the photonic gapped Mott insulating phase. For the XY symmetric random exchange, on the other hand, they have found a ‘‘random singlet’’ phase in which spins are tightly coupled in singlet pairs. These singlet pairing can occur over the large distance. For these situations still there is a possibility for photonic superfluid phase. The authors of Ref. [7] have numerically studied the occupation probability to confirm the validity of their approximation during the foundation of the model. They have also numerically studied the von Neumann entropy of the reduced density matrix of the effective spin in multiples of ln 2. There is no analysis on quantum phase. The total magnetic field, Btot [7] increases for the larger values of detuning, therefore larger detuning drives the system from gapped (Mott-insulating) state to gapless superfluid state. This is the evidence of commensurate (gapped) phase to incommensurate (gapless) phase in micro-cavity arrays. We now discuss how the effective repulsion decreases as we increase the detuning between the atomic and laser frequency. It can be explained starting from the Jaynes–Cummings Hamiltonian [35,36]. Janes–Cummings Hamiltonian for a single atom is HJC ¼ oC ay a þ o0 9eS/e9 þ lðay 9gS/e9 þ a9eS/g9Þ,
ð14Þ
where oC and o0 are the frequencies of the resonant mode of the cavity and of the atomic transition, respectively. l is the Jaynes– Cumming coupling between the cavity mode and the two level system. ay (a) is the creation (annihilation) operator of a photon inside a cavity. 9gS and 9eS are the ground state and excited state
47
D 2
7
2
nl þ
D2 4
:
ð15Þ
Here D ¼ ðo0 oc Þ and n Z 1. Now we consider an array of cavities, the basic Hamiltonian for each cavity is the same as that of Eq. (14). Here, we consider the system with one excitation of energy E1 in each cavity and the lowest energy of two excitations in each cavity is E2 . Therefore, to create one additional excitation in each cavity requires the energy equal to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 2E1 ¼ 2
l2 þ
D2 4
2
2l þ
D2 D 4
2
:
One may consider this as an effective on-site repulsion because it measures the difference between the energy of two and single excitation (polariton in each cavity). This effective repulsion decreases as we increase the detune factor. Therefore, we conclude that for D ¼ 0, double occupation never occurs, indicating a Mott insulating behavior. When D is much larger than the coupling l, then the value of the occupation number as larger than one occurs as expected for a photonic superfluid regime. In our quantum field theoretical calculations, we have also predicted that the large detuning drives the system from gapped Mott insulating phase to the gapless superfluid phase.
3. Conclusions For the first time we have done the quantum field theoretical analysis of an effective spin model in coupled optical cavity arrays. We have predicted few quantum phases like, photonic Coulomb blocked induce Mott insulator, photonic superfluidity and staggered order insulator. An infinite order BKT transition has been replaced by the second order XY transition under the presence of exchange anisotropy. We have predicted the presence of plateau phase of this system and also commensurate phase to incommensurate phase transition. We have also discussed the effect of disorder in the optical cavity array system. The rigorous quantum field theoretical derivation of this work is absent in all previous studies and we also provide physical explanation of the transition process based on Jaynes–Cummings Hamiltonian.
Acknowledgments The author would like to thank The Center for Condensed Matter Theory of the Physics Department of IISc for facility extended. The author would like to thank Prof. N. Behera, Dr. B. Murthi and Dr. N. Sundaram for reading the manuscript very critically. References [1] C.N.R. Rao, T.V. Ramakrishnan, Superconductivity Today, Universities Press, Hyderabad, 1999. [2] K.K. Likharev, Dynamics of Josephson Junction and Circuits, Gordon and Breach, 1988. [3] J.H. Denschlog, et al., J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 3095. [4] D. Jaksch, P. Zoller, Ann. Phys. 315 (2005) 52. [5] A.D. Greentree, et al., Nat. Phys. 466 (2006) 856. [6] [a] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Nat. Phys. 462 (2006) 849; [b] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Laser Photon. Rev. 2 (2008) 527.
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[7] J. Hartmann Michael, G.S. Fernando, L. Brando, B. Plenio Martin, Phys. Rev. Lett. 99 (2007) 160501. [8] A.C. Ji, X.C. Xie, W.M. Liu, Phys. Rev. Lett. 99 (2007) 183602. [9] T. Byrnes, N.Y. Kim, K. Kusudo, Y. Yamamoto, Phys. Rev. B 78 (2008) 075320. [10] I. Carusotto, et al., arXiv:0812.4195, 2008. [11] M.J. Bhaseen, M. Hohenadler, A.O. Silver, B.D. Simons, Phys. Rev. Lett. 102 (2009) 135301. [12] A. Tomadin, et al., arXiv:0904.4437, 2009. [13] J. Zhao, A.W. Sandvik, K. Ueda, arXiv:0806.3603, 2008. [14] P. Pippan, H.G. Evertz, M. Hohenadler, arXiv:0904.1350, 2009. [15] D. Rossini, R. Fazio, Phys. Rev. Lett. 99 (2007) 186401; N. Didier, S. Pugnetti, Y.M. Blanter, R. Fazio, arXiv:1007.471 (cond-mat), 2010. [16] M. Aichhorn, et al., Phys. Rev. Lett. 100 (2008) 216401. [17] S. Schmidt, G. Blatter, arXiv:0905.3344, 2009. [18] D.G. Angelakis, M.F. Santos, S. Bose, Phys. Rev. A 76 (2007) R031805. [19] J. Koch, K. Le Hur, Phys. Rev. A 80 (2009) 023811; K. Le Hur, 2009, arXiv/cond-mat-0909.4822. [20] Here we discuss the relation between the spin operators and the atom– photon system. Without loss of generality, we start with the well celebrated Janes–Cummings Hamiltonian. H ¼ gðsay þ h:cÞ. Cavity mode represents by the bosonic operators (a,ay ) and atomic mode represents by the fermionic operator (s, sy ). Where s is the Pauli matrix which transforms one excitation from the radiation field to the atomic field. Therefore, one can write 0 1 sy ¼ 0 0
[21] [22] [23] [24] [25]
similarly one can write for s with /19 ¼ ð1; 0Þ and /09 ¼ ð0; 1Þ. The excitation in this system only transfers between atom and photon in the cavity. For a fixed number (n) of total excitation, one can express the manifold: Hn ¼ f90,nS, 91,n1Sg provided nZ 1. Here 90,nS and 91,n1S represent atom in the ground state with n photon and excited states of the atom with (n1) photon, respectively. Now we explicitly establish the relation between the spin operator and the atom photon system: we consider the initial state 9e,n1S, we obtain the state 9g,nS by the following operation: 9g,nS ¼ sa þ 9e,n1S. Therefore, we may write the following relation based on the conservation of the number of excitation. sy a9e,n1S ¼ 0 ¼ say 9g,nS and sy a9g,nS ¼ 9e,n1S. These relations are nothing but the properties of spin operators acting on the spins in the z basis. B. Surtharland, J. Math. Phys. 11 (1970) 3183. Baxter, Phys. Rev. Lett. 26 (1971) 834. S. Sarkar, Phys. Rev. B 74 (2006) 052410. T. Giamarchi, Quantum Physics in One Dimension, Clarendon Press, Oxford, 2004. U. Schollwock, J. Richter, D.J.J. Farnell, R.F. Bishop (Eds.), Lecture Notes in Physics, vol. 645, Quantum Magnetism, Springer, Berlin, 2004.
[26] S. Lukyanov, A. Zamolodchikov, Nucl. Phys. B 493 (1997) 571. [27] J. Cardy, in: Scaling and Renormalization in Statistical Physics Cambridge University Press, Cambridge, 1996; I. Affleck, in: E. Brezin, J. Zinn-Justin (Ed.), Fields, Strings and Critical Phenomena, North-Holland, Amsterdam, 1989. [28] There is no spontaneous symmetry breaking in one dimension as we observe in higher dimensions. Therefore, we will not be able to define the order parameter as we define for higher dimensional systems. In one dimensional system only discrete symmetry is broken. Therefore, we have not defined any order parameter for the density wave and XY staggered phase. XY phase is related with the inplane broken rotational symmetry. In our work, we have studied the anamolous scaling dimensional of the sine-Gordon coupling terms of the Hamiltonian equation (12) to conclude the relevancy of these terms. [29] The quantum phases which we have discussed for one dimensional system is actually instabilities, i.e., the corresponding susceptibility will diverge for the one dimensional system. In one dimension, there is no spontaneous symmetry breaking, only discrete symmetry breaks in one dimension. These instabilities in one dimension are casually referred as a quantum phase. [30] In quantum phase transitions, in addition to the standard critical exponent, it is useful to define an additional exponent z, called the dynamical exponent, which tells us how a characteristic length in the time direction is related to a z length in the spatial direction xt xx . For the quantum problem the time plays a special role and this special direction has no reason to have the same exponent as the spatial one. Deep inside the Mott insulating phase, particle and hole excitations are gapped. The system is almost similar to the atomic limit in the deep Mott insulating phase. When we approach the phase boundary from the deep Mott insulating phase then the dispersion relation is 2 quadratic o k . (It is well known in the literature that the energy z dispersion at the transition point is o k , therefore z ¼ 2 during the transition from Mott insulating phase to the gapless superfluid phase.) But the situation is different at the end of CDW phase and the starting point of XY staggered order phase. At around the multical critical point z is 1 (around this point, energy dispersion is gapless, i.e., e k. Therefore, z ¼ 1). See Refs. [31–33] and for a detailed understanding of this subject. [31] S. Sachdev, Quantum Phase Transition, Cambridge University Press, Cambridge, 1998. [32] S.L. Sondhi, et al., Rev. Mod. Phys. 69 (1997) 315. [33] M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Phys. Rev. B 40 (1989) 546. [34] A. Doty Curtis, D.S. Fisher, Phys. Rev. B 45 (1992) 2167. [35] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. [36] S. Horoche, J.M. Raimond, Exploring the Quantum Atoms, Cavities, and Photons, Oxford University Press, 2006.