Quantum Rabi model for N-state atoms

Abstract

A tractable N-state Rabi Hamiltonian is introduced by extending the parity
symmetry of the two-state model. The single-mode case provides a few-parameter
description of a novel class of periodic systems, predicting that the ground
state of certain four-state atom-cavity systems will undergo parity change at
strong coupling. A group-theoretical treatment provides physical insight into
dynamics and a modified rotating wave approximation obtains accurate analytical
energies. The dissipative case can be applied to study excitation energy
transfer in molecular rings or chains.

Full text

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arXiv:1112.0849v1 [cond-mat.mes-hall] 5 Dec 2011
Spin-boson models for periodic N-site systems
Victor V. Albert∗
Department of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520-8120, USA
(Dated: December 6, 2011)
This work presents a simple model for the interaction of a periodic system of N coupled sites with
one or more independent boson modes. The few-mode case results in a tractable multi-level Rabi
model, predicting that the ground state of certain four-level atom-cavity systems will undergo parity
change at large coupling. The dissipative case can be applied to study excitation energy transfer in
molecular rings or chains.
PACS numbers: 42.50.Pq, 42.50.-p, 03.65.Yz, 71.35.-y
Keywords: spin boson, quantum Rabi model, Jaynes Cummings model, molecular trimer, spin oscillator,
lambda and vee system
Interactions between spin systems and one or more
harmonic oscillators (boson modes) have been studied
for over 70 years [1–4].One ubiquitous model, the
single/multi-mode spin-boson [5], has formed a basis
of understanding of atom-cavity [6] and exciton-phonon
[7, 8] interactions and has numerous established appli-
cations in chemistry and physics (see [9, 10] and refs.
therein). Variations of this model continue to receive
considerable attention [11–17] and the single-mode case
was recently solved [18].
The single/multi-mode spin-boson is a two-site peri-
odic model with parity (Z2) symmetry. Many-site spin-
boson interaction, e.g. multi-level atom-light interaction
[19] or excitation energy transfer in multi-chromophoric
systems [20], continues to be a subject of significant inter-
est, dictating a need for extensions of the two-site model.
Existing symmetry-preserving extensions [8, 21] quickly
become intractable due to the inclusion of multiple nor-
mal modes. Other single-mode atom-cavity [22–24] and
infinite-mode (dissipative) [25–28] extensions ignore sym-
metries, thus containing arbitrary numbers of parame-
ters. The generalization presented here both preserves
symmetry and allows for coupling to any number of non-
interacting modes. The model determines the number
of independent parameters necessary to describe a peri-
odic N-site system coupled to a boson while maintain-
ing cyclic (ZN) symmetry, presenting a simple and rigor-
ous approach to both few-mode atom-cavity interactions
[19] and dissipative exciton-phonon systems [20, 29, 30].
Moreover, the theory of irreducible representations [21] is
used to partially diagonalize the Hamiltonian, providing
numerical advantages, physical insight, and a gateway
to accurate analytical approximations. All of the above
procedures are remarkably streamlined with the use of
generalized spin matrices [31], offering new insight into
the mathematical treatment of periodic systems.
The main idea of this work is as follows. One usually
assumes [25–28] a linear spin-boson coupling to be only
in position ˆ q ∝ b + b†, where the latter are the boson
lowering and raising operators. While true for the two-
site model, a many-site linear coupling has to include
momentum ˆ p ∝ i(b−b†) since ˆ q and ˆ p are basically inter-
changeable [32]. The general N-site symmetry-corrected
coupling for the nth site (with n = 0,1,...,N−1) involves
a discrete rotation in phase space:
bei2π
Nn+ b†e−i2π
Nn∝ ˆ q cos2πn
N+ ˆ psin2πn
N. (1)
The ability for an oscillator to be evenly smeared across
an N-site system introduces an extension that is simpler
and potentially more applicable than using N − 1 oscil-
lators to connect the sites, which has been the case of
previous classically motivated approaches [8, 21].
FIG. 1.
optical representations of the single-mode spin-boson are de-
picted in the left, center, and right panels, respectively. The
corresponding Hamiltonians for each representation, Eqs. (2-
4), are labeled in red along with the transformations that
connect them. (b),(c) analogously show the different repre-
sentations of the generalized case in Eq. (7) with N = 3,4,
respectively.
(color online) (a) Partially diagonalized, site, and
In this Letter, the N-site single-mode spin-boson
Hamiltonian H is first introduced in the exciton-phonon
(site) representation, then transformed into partially di-
agonalized form? H, and finally rotated into the atom-
the three representations are visualized in Fig. 1 for the
(a) two-, (b) three-, and (c) four-site cases. A set of con-
served quantum numbers is introduced, approximate an-
alytical energies are obtained, and applications to atom-
cavity interaction and dissipative systems are discussed.
For reference, the N = 2 case is reviewed first.
cavity (optical) representation? H. The Hamiltonians in

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The two-site single-mode spin-boson Hamiltonian is
H2= ωb†b + λ?b + b†?σz+ Jσx,
where σi are the usual Pauli matrices [10], ω is the os-
cillator frequency, J is the site coupling, and λ is the
spin-boson coupling. This represents a two-site periodic
system coupled to a phonon mode [2]. Of its many ap-
plications (e.g. see refs. in [16]), the dissipative case can
represent the single excitation manifold of a degenerate
molecular dimer coupled to a vibrational bath [15, 25].
The Hamiltonian’s Z2symmetry is manifested by the fact
that it is invariant under elements of the reflection group
{I,Rσx}, where the former is the identity and the latter
is a parity operator (with oscillator parity R = eiπb†b).
Using symmetry theory, Eq. (2) can be partially diago-
nalized via U2= (I − iRσy)/√2 [33]. The Hamiltonian
? H2= U†
?±|? H2|±? = ωb†b ± λ?b + b†?± JR.
Since the transformed commuting group of H2is {I,σz},
one can see that U2separates H2into two boson mani-
folds of different parity.
Using V2 = (σx+ σz)/√2 to transform Eq. (2) into
the optical representation, one obtains the well-known
quantum Rabi model [1]
? H2= ωb†b + λ?b + b†?σx+ Jσz.
Equation (4) represents a two-level atom of energy sep-
aration 2J coupled to a single mode of a quantized field
[34]. The two “counter-rotating” terms, bσ− and b†σ+,
can reasonably be ignored for some applications, re-
sulting in Eq. (4) in the rotating wave approximation
(RWA), i.e., the Jaynes-Cummings model [4].
Before introducing the N-site model, I give a brief in-
troduction to the generalized spin matrices [31]. Sup-
pressing dependence on N, generalized spin matrices for
0 ≤ j,k < N are defined (via modulo N) as
?
m=0
(2)
2H2U2is diagonal in the site subspace with en-
tries
(3)
(4)
Sj,k=
N−1
ei2π
Nmj|m??m + k| = (S1,0)j(S0,1)k,(5)
where |m? is a vector in the site space and S1,0and S0,1
are the generators of this unitary set. While S†
and S†
0,k= S0,−k, this is not true for arbitrary Sj,k. The
first generator elegantly expresses the spin-boson inter-
action from Eq. (1). The second generator describes the
site interactions for all possible neighbors. For
0 < k ≤ κ ≡?1
(with ⌊N⌋ the floor function), the kth neighbor interac-
tion term is simply S0,k+ S†
tional opposite-member interaction term is S0,N
j,0= S−j,0
2(N − 1)?
(6)
0,k. For even N, the addi-
2. The
N-site single-mode spin-boson Hamiltonian is thus
H = ωb†b + λ(bS1,0+ b†S†
1,0)
κ
?
k=1
+JS0,N
2+Jk(S0,k+ S†
0,k).(7)
The parameter Jkrepresents the kth neighbor interaction
defined earlier and the last two terms represent the site
coupling matrix. For the two-site case, S0,1= σx, S1,0=
σz, and one recovers Eq. (2).
Equation (7) is invariant under elements of the cyclic
group {RnS0,n}N−1
Rn= exp?i2π
It is clear that R†
One can utilize this ZN symmetry and partially diago-
nalize H in the site subspace using the prescription of
[21]. Remarkably, this method is significantly simplified
and the general transformation consists of compositions
of spin and boson operators:
n=0where the bosonic rotation
Nnb†b?= (R1)n.
n= R−nand RN/2= R (for even N).
(8)
U =
1
√N
N−1
?
k=0
ei2π
Nk2RkSk,k.(9)
This unitary U commutes with the spin-boson coupling
term and partially diagonalizes the site coupling matrix:
U†S0,kU = R†
kS†
k,0.(10)
The transformed H is diagonal in the site space with
? HN,n= ?n|? H|n? = ωb†b + λ(bei2π
+(−1)nJR +
k=1
Nn+ b†e−i2π
Nn)
κ
?
Jk(Rkei2π
Nnk+ R†
ke−i2π
Nnk). (11)
The above boson manifolds? HN,n(with n = 0,1,...,N−1)
well as physical insight.
Conserved quantum numbers.—The non-trivial mem-
bers of the commuting group {RnS0,n}N−1
cisely combined into the N-site commuting operator
are equivalent to H, providing numerical advantages as
n=0can be con-
N = JRS0,N
2+
κ
?
k=1
Jk(RkS0,k+ R†
kS†
0,k),(12)
which consists of the family of κ commuting Hermitian
operators multiplied by site couplings Jk, with the addi-
tional parity operator for even N. Transforming N by
U using Eq. (10) obtains the diagonalized site coupling
matrix
?N = JS N
2,0+
κ
?
k=1
Jk(Sk,0+ S†
k,0). (13)
For the nth diagonal of?N, the coefficients of the site cou-
plings Jkare conserved quantum numbers that describe
2

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the corresponding manifold? HN,n. Note that some man-
case,?N → Jσz, resulting in manifolds of parity p = ±1,
ifolds have the same set of coefficients. For the two-site
as reviewed earlier and shown in Fig. 1(a).
Analytical insight.—To obtain an understanding of the
spectrum of H, Eq. (11) allows for the application of the
symmetrized generalized RWA (S-GRWA) [10], a related
form of the RWA more suited for strong coupling. The
1-by-1 S-GRWA is derived by applying the generalized
displacement operator
D(S1,0λ/ω) = exp[(bS1,0− b†S†
1,0)λ/ω] (14)
to
D†? HN,nD in the boson Fock space. This removes the lin-
elements) obtains energies (for m = 0,1,2,...)
? H and writing out each transformed manifold
ear bosonic coupling terms and (ignoring all off-diagonal
ES-GRWA
N,n,m
= ωm−λ2
ω+Je−2λ2/ω2Lm(4λ2/ω2)(−1)n+m+
κ
?
k=1
2Jke−1
2|αkλ/ω|2Lm
?
|αkλ/ω|2?
cos
?2π
Nk (n + m) + θk
?
where θk = (λ/ω)2sin(2πk/N), αk = 1 − ei2πk/N, and
the third term is only for even N. In the site interpreta-
tion, the energies consist of contributions from standing
waves along the ring/chain of sites weighted by site cou-
pling Jk, displaced by θk, and decaying as O(λ2). The
corresponding eigenfunctions consist of the original ba-
sis of (7) rotated along the ring by U†and displaced
vibrationally by D†. The two limits of the system are
λ ≫ {Jk}, which simplifies H into a collection of sym-
metrically displaced harmonic oscillators, and λ ≪ {Jk},
which results in an uncoupled N-site system and an os-
cillator.
Application to atom-cavity interaction.—An important
result of this work is the generalization of the rotation V2
used to take H2from the site to the optical representa-
tion:
V =
1
√N
N−1
?
k=0
ei2π
Nk2Sk,kS1,0. (15)
Remarkably, the transformed H results in a type of multi-
level Rabi model,
? H = ωb†b + λ(bS2,1ei4π
that can be described in the same framework as the orig-
inal Rabi Hamiltonian [34].
N-level atom with energies determined by?N coupled to
option for more modes [35]). Rotating/counter-rotating
interactions are divided among any degenerate energy
levels and?N determines whether excitation or relaxation
Hamiltonian (4). The three- and four-level cases are re-
viewed below.
N = 3: Setting J1 = K in Eq. (7), the three-site
single-mode spin-boson Hamiltonian H3− ωb†b is


N + b†S†
2,1e−i4π
N) +?N,(16)
Equation (16) models an
a field mode via complex dipole interactions (with the
has occurred. For N = 2, Eq. (16) reduces to the Rabi
λb + λb†
K
K
KK
K
λbei2π
3 + λb†e−i2π
K
3
λbe−i2π
3 + λb†ei2π
3

.
Applying U3 transforms H3 into three boson manifolds
? H3,n with n = 0,1,2. The commuting operator N3 is
quantum number δ = 2,−1 [Fig. 1(b)]. Rotating into
the optical representation by V3obtains


This three-level Rabi model describes the Λ-system [22],
a three-level atom with a degenerate ground state and en-
ergy separation 3K coupled to one cavity mode [35] (with
inversion of K obtaining the V-system). This setup al-
lows for dipole transitions to occur between all three lev-
els while extending the symmetry and maintaining the
relative simplicity of the original Rabi model. The bot-
tom left entry in Eq. (17) describes the process in which
the atom makes a transition from the upper to the lower
level and a photon is annihilated [34]. The RWA (with
respect to ωb†b +?N3) removes this transition, relating
model [22]. However, the interaction between degenerate
levels remains, demonstrating that it may play an impor-
tant role even when the degeneracy is not broken. The
numerical energies for each manifold (color) and the S-
GRWA energies (dashed) for the resonant case (ω = 3K)
are depicted in Fig. 2(a), where one can see the famil-
iar braid-like behavior of the two-level Rabi Hamiltonian
with the addition of a third level.
N = 4: This richer case has two symmetries, one for
each site coupling J,K. Transforming N4by U4obtains
transformed into a diagonal matrix?N3 with conserved
? H3=
ωb†b + 2K
λb†ei2π
λb
λbe−i2π
ωb†b − K
λb†e−i2π
3
λb†
3
λbei2π
ωb†b − K
3
3

.(17)
(17) to established extensions of the Jaynes-Cummings
?N4= diag{J + 2K,−J,J − 2K,−J}.
Eigenstates of H4thus have two quantum numbers: par-
ity p = ±1 and cascade number c = 0,±2. Depending
on the relation between J > 0 and K [Fig. 1(c)], one
can rotate by V4to obtain either a symmetric double-Λ,
tripod, or ♦ four-level system [24]. Inversion of J ob-
tains inverted tripod and double-Λ systems; inversion of
(18)
3

Page 4

(a)
0.20.4 0.6 0.8 1.01.21.4
?5
0
5
10
Λ?Ω
E?K
(b)
0.2 0.40.60.8 1.01.21.4
?10
0
10
20
Λ?Ω
E?J
FIG. 2.
coupling λ for (a) H3, a resonant Λ-system with K = ω/3,
and (b) H4, a ♦-system with K = ω/4 and J = ω/6 (ω = 1).
The numerical energies belong to manifolds? HN,n (for n <
N = 3,4) represented as red, blue, green, and cyan, in that
order, while approximate S-GRWA energies are dashed. The
model predicts that the ♦-system may have a different ground
state at λ/ω ≈ 1 than at weak coupling.
(color online) Correlation diagrams of energy vs.
K leaves the system invariant just like inversion of J for
N = 2. Only two parameters are needed to describe all
four levels and the two degenerate levels do not interact,
contrary to the three-level case. Another surprising fea-
ture is that the ground state can change for increasing
values of λ, a property not seen at N < 4. Shown in
Fig. 2(b) for a particular ♦-system, the ground state at
deep-strong coupling (λ/ω ≈ 1) [12, 14] is from a dif-
ferent manifold (? H4,1, blue) than that at λ ≪ ω (? H4,2,
Application to dissipative systems.—The many-mode
extension, where {b,ω,λ} → {bi,ωi,λi} in Eq.
is an effective model for the single excitation manifold
of a molecular ring or periodic chain interacting with
a collective uncorrelated vibrational bath. This model
specifically includes the geometrical structure of the sys-
tem, a property shown to be important in excitation en-
ergy transfer [36]. Couplings Jkbetween all sites in the
chain are included, allowing one to model systems with
interactions other than nearest-neighbor. This version
can model photoexcitation dynamics of molecular trimers
[28, 29] such as Ru(bpy)+2
3
green).
(7),
light-harvesting compounds
[37] while higher-N versions can model larger protein
rings [26, 27, 38]. Recently developed methods [20] can
readily be applied to this symmetric case to reveal similar
insight into many-site systems as previous methodologies
[5, 15–17, 39] have revealed in the simplest two-site case.
Summary.—This work introduces an extension of the
two-site spin-boson model to describe dynamics of a more
general N-site periodic system using a minimal number
of parameters. The symmetry of the system is utilized
in a group-theoretical approach, revealing insight into its
energies and conserved quantities while also simplifying
numerical analysis. A recently developed class of ma-
trices [31] provides an elegant method for obtaining the
above results. The equivalence between the single-mode
spin-boson Hamiltonian and the (two-level) Rabi model
[1] is extended to the many-level case, revealing applica-
tions to few-mode atom-cavity interaction. Finally, the
proposed infinite-mode extension generalizes the two-site
spin-boson model [5] to dissipative periodic N-site sys-
tems.
Discussions with S. M. Girvin, F. Iachello, G. D. Sc-
holes, and A. Nazir are gratefully acknowledged. I wish
to thank J. I. V¨ ayrynen, K. A. Velizhanin, and D. Bokhan
for help with preparation of this manuscript. This work
is supported by an NSF Graduate Research Fellowship.
∗victor.albert@yale.edu
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