Phase transitions and Heisenberg limited metrology in an Ising chain interacting with a single-mode cavity field

Abstract

We investigate the thermodynamics of a combined Dicke and Ising model that exhibits a rich phenomenology arising from the second-order and quantum phase transitions from the respective models. The partition function is calculated using mean-field theory, and the free energy is analyzed in detail to determine the complete phase diagram of the system. The analysis reveals both first- and second-order Dicke phase transitions into a super-radiant state, and the cavity mean field in this regime acts as an effective magnetic field, which restricts the Ising chain dynamics to parameter ranges away from the Ising phase transition. Physical systems with first-order phase transitions are natural candidates for metrology and calibration purposes, and we apply filter theory to show that the sensitivity of the physical system to temperature and external fields reaches the 1/N Heisenberg limit.

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Phase transitions in an Ising chain interacting with
a single mode cavity field
Søren Gammelmark, Klaus Mølmer
Lundbeck Foundation Theoretical Center for Quantum System Research,
Department of Physics and Astronomy, University of Aarhus, DK 8000 Aarhus
C, Denmark
E-mail: gammelmark@phys.au.dk
PACS numbers: 03.67.-a, 42.50.Dv, 64.60.De, 75.10.Jm
Abstract.
model which exhibits a rich phenomenology arising from the second order and
quantum phase transitions from the respective models. The partition function is
calculated using mean field theory, and the free energy is analyzed in detail to
determine the complete phase diagram for the system. The analysis reveals both
first- and second-order Dicke phase transitions into a super-radiant state, and the
cavity mean-field in this regime acts as an effective magnetic field, which restricts
the Ising chain dynamics to parameter ranges away from the Ising phase transition.
Physical systems with a first order phase transitions are natural candidates for
metrology and calibration purposes, and we apply filter theory to show that the
sensitivity of the physical system to temperature and external fields reaches the
1/N Heisenberg limit.
We investigate the thermodynamics of a combined Dicke- and Ising-
1. Introduction
The understanding of the remarkable and useful properties of matter in different phases
and of the critical behaviour near phase transitions presents ongoing challenges in
theoretical and experimental physics [1, 2, 3, 4, 5].
transition phenomena become relevant with the ability to control and engineer
microscopic interactions and systems, e.g., in cold atom experiments with spinor Bose-
Einstein condensates and with Fermi gases [6, 7]. Indeed, a whole branch of quantum
computing research attempts to use candidate systems for quantum computing as
quantum simulators in which quantum gates are operated so as to simulate suitable
inter-particle interactions [8] and in this way implement theoretical phase transition
models in a quantum analog computer.
Local interactions between nearest neighbours in a spin chain lead to Ising-,
Heisenberg-, and other interaction models with phase transitions at definite values of
the interaction strengths and external controllable parameters, such as a bias magnetic
field. While these interactions are reliable models of, e.g., magnetic interactions in
solids, they can also be engineered exactly among trapped atoms or ions, with the
added experimental possibility to control the sign and magnitude of the interactions
with laser beams and the spin temperature by optical pumping [9].
Atomic and optical systems also permit the engineering of interactions between a
large number of atoms and a quantized oscillator mode. Such systems are implemented
Entirely new types of phase
arXiv:1102.1905v1 [quant-ph] 9 Feb 2011

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Phase transitions in an Ising chain interacting with a single mode cavity field2
in various schemes for quantum computing, where the oscillator mode is used as
a data bus between the atomic quantum bits. Beyond a critical coupling strength
to the oscillator and below a critical temperature the system undergoes a phase
transition, and the thermodynamic ground state acquires a macroscopic excitation
of the oscillator mode. This phase-transition was first discussed by Dicke [10], and
the Dicke phase transition has since then been studied extensively [11, 12], and was
recently observed in experiments with cold atoms in an optical cavity [13] using
techniques similar to those described in [14].
It is conceivable that one can implement both the Ising and the Dicke interaction
Hamiltonians in many different ways using atoms and cavities, nano-mechanical
devices or collective vibrational motion of the atoms or ions. The partition function
of such a combined Dicke-Ising model was determined recently in the thermodynamic
limit [15], and in this article we study the phase diagram and further properties of the
system and possible applications.
First order phase transitions represent discontinuous changes and hence a high
sensitivity to variations in the values of the physical parameters of the model near
the critical point.Sensitivity of quantum systems in fundamental metrology is a
very active research field where much research has been devoted to identify how the
sensitivity depends on the number of particles N. The ”standard limit” 1/√N, is
replaced by the ”Heisenberg limit” 1/N within different realizations of non-interacting
particles [16, 17, 18, 19], while more rapid decrease with particle number has been
proposed in different models of interacting particles [20, 21, 22]. In this manuscript
we argue for a power law decrease similar to the Heisenberg limit for measurements
of temperature with our interacting system.
The manuscript is organized as follows: In section 2 we outline the model and
discuss the two limiting Ising and Dicke regimes and their phenomenology. In section
3 we review the phase diagram calculation [15] and we recast this calculation in terms
of mean-field theory. In section 4 we discuss the phase diagram in detail. In Section V
we analyze the application of the phase transition for metrology. Finally, we conclude
in section 6.
2. The Dicke-Ising model
2.1. The Ising Model
Consider a one-dimensional chain of N spins or two-level atoms realizing an Ising
model with a transverse magnetic field,
HIsing= −h
N
?
i=1
σz
i− J
?
i
σy
iσy
i+1
(1)
where h ≥ 0 is the transverse field and J ≥ 0 is the interaction strength between
neighbouring spins and σy
1.
The model is not trivially easy to diagonalize, as the transverse field operators
σz
transverse field dominates in which case we expect ?σy
strong coupling J ? h the interaction term favours parallel spin in the y-direction
so ?σz
?σy
N+1≡ σy
ido not commute with the interaction terms σy
iσy
i+1. In general, for h ? J the
i? ≈ 0 and ?σz
i? = 1. For a
i? ≈ 0 and the system possesses two states of equal energy with ?σy
i? = −1.
i? = +1, or

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Phase transitions in an Ising chain interacting with a single mode cavity field3
An (almost) exact diagonalization of the Ising Hamiltonian can be performed
[23, 24] using a Jordan-Wigner transformation which maps the spin-operators to
fermionic operators. The particular mapping we employ here is cn = i?
operators with phase factors, σz
j, depending on the spins at previous locations in
the spin-chain leads to the fermionic anti-commutator relations
The fermionic number operator is c†
all spins pointing in the z-direction is mapped to the fermionic vacuum while c†
flips a spin from up to down thereby creating a fermion. The inverse mapping reads
σ†
becomes −Jσy
conditions we cannot remove the intermediate (1−2c†
−Jσy
adding and subtracting J(c†
1+ c1) we can write the resulting Hamiltonian
as
HIsing= −h
i
j<nσz
jσ†
n,
where σ†
n= (σx
n− iσy
n)/2 is the n’th spin-z step-up operator. Multiplying the spin-
?
cn,c†
n?
?
= δn,n?.
ncn = σnσ†
n= (1 − σz
n)/2 and the state with
n
n= −i?
j<n(1−2c†
iσy
jcn)cnsuch that the y-y-interaction term in the Ising Hamiltonian
i+1= −J(c†
i−ci)(c†
i+1+ci+1) for i < N. Due to the periodic boundary
jcj)-factors in the final y-y-term
N− cN)(c†
N− cN)(c†
?
N
?
where the last term is of relative order 1/N since 1 +?
order 1.
By neglecting this less important final term the Hamiltonian is quadratic in
the fermionic operators, and can be diagonalized using a Bogoliubov-transformation.
The Bogoliubov-transformation can be decomposed into a single-particle basis change
to momentum space and quasi-particle operators connecting particles of opposite
momenta [23, 24, 25] and we get:
?
where the operators γkare the Bogoliubov transformed fermionic operators, ?(k) =
2(J2+h2−2Jhcos(k))1/2and k ∈ ZNhas the form 2πn/N and is a reciprocal lattice-
vector to the chain-lattice. The eigenmodes of HIsingcorrespond to free fermions with
a dispersion relation given by ?(k). Note that (2) has a single unique ground state in
contrast to (1) which has a two-fold degeneracy. This difference is due to the omitted
term and will not be relevant in the thermodynamic limit.
Since the spectrum of the Ising model is so simple we can calculate the partition
function for the system. The partition function is given by Z0
where β is the inverse temperature. We can easily calculate this trace,
?
=
k∈ZN
Using this result we can also calculate the free energy given by FIsing
−β−1logZ0
Nσy
1. This term can be written as J(c†
1+ c1)?N
j=1(1 − 2c†
jcj) and by
(1 − 2c†
ici) − J
?
(1 − 2c†
i
c†
ici+1+ c†
i+1ci+ ci+1ci+ c†
ic†
i+1
+J(c†
N− cN)(c†
1+ c1)(1 +
j=1
jcj))
j(1 − 2c†
jcj) is a projection
operator onto the subspace of even total spin in the z-direction and therefore is of
HIsing=
k∈ZN
?(k)
?
γ†
kγk−1
2
?
+ O(1/N), (2)
Ising= Tr(exp(−βHIsing))
?
Z0
Ising(β,h,J) =
k∈ZN
?
Tr
?
exp(−β?(k)(γ†
kγk− 1/2))
2cosh(β?(k)/2).
=
Ising. The diagonalization of HIsingis exact to order 1/N and to the same

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Phase transitions in an Ising chain interacting with a single mode cavity field4
precision we can replace the sum over k ∈ ZN by an integral,
FIsing(β,h,J) = −1
β
?
?π
k∈ZN
log(2cosh(β?(k)/2))
≈ −N
−π
dk
2πβlog(2cosh(β?(k,h,J))).(3)
Note that β?(k,h,J)/2 = βJ(1+(h/J)2−2(h/J)cos(k))1/2, and hence it is convenient
to parametrize FIsingwith the dimensionless quantities˜β = βJ and˜h = h/J in the
following.
The Ising model exhibits an infinite order quantum phase-transition between a
paramagnetic h ? J and ferromagnetic J ? h phase with critical point at J = h where
a non-analyticity arises in ?(k) at k = 0. This system has been studied extensively [25]
and is a prime example of a quantum phase transition. The quantum phase-transition
does not survive to finite temperatures, but it has important consequences for the
finite-temperature behaviour of the system near the critical point.
2.2. The Dicke model
Another phase transition model consists of spins or two-level atoms coupled to a
harmonic oscillator mode with frequency ω,
Hosc= ωa†a,(4)
through the interaction
V =
g
√N
?
i
σx
i(a + a†), (5)
where σx
step down operator. We denote the coupling-strength of the oscillator to a single spin
by g/√N. For a physical implementation with atoms inside a cavity with a mode
volume V , the quantum field strength per photon is proportional to 1/√V , and our
scaling thus corresponds to atoms with a constant spatial density which is well defined,
also in the thermodynamic limit N → ∞.
The explicit form of the interaction is well established in quantum optical systems,
and occurs both for two-level systems and for, e.g., Raman processes between two
states via an intermediate excited state through absorption and stimulated emission
of the quantum field and a classical control field.
Cummings model of a single two-level atom and a single field mode, the rotating wave
approximation retains only the terms σ†a + σa†in V and provides a considerable
simplification of the problem. The partition function for the Dicke-model has been
calculated analytically in the thermodynamic limit and a second order phase transition
has been identified [11].While that calculation pertained to the rotating wave
approximation it has been shown [12] that this does not change the Dicke phase-
transition qualitatively. Within our application of a mean field approximation to the
combined Dicke-Ising model we shall retain the full interaction (5) as, the rotating
wave approximation is more difficult to deal with.
The Dicke model Hosc+ V can be realized experimentally as described in [14]
where a dynamical version of the standard Dicke model is investigated in a cavity
using four-level atoms coupled by Raman channels. The model parameters of this
system are functions of the atomic and field parameters applied and can be tuned
over large ranges.
iis the Pauli x-matrix acting on the i’th spin and a is the harmonic oscillator
We note that in the Jaynes-

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Phase transitions in an Ising chain interacting with a single mode cavity field5
3. The partition function for the Dicke-Ising model
The combined Dicke-Ising model has total Hamiltonian H = HIsing+ Hosc+ V with
three parameters g, h, and J which can be varied independently in atomic simulators
of the model.With J = 0 the Hamiltonian realizes the Dicke model of N two-
level atoms interacting with a cavity-field via the dipole interaction. In this regime
2h correspond to the energy-splitting of the individual two-level atoms and g is the
coupling strength to the cavity-field.
3.1. Coherent state integral
We will proceed by writing first the expression of the partition function for the full
problem Z = Tr(exp(−βH)) with H = HIsing+ Hosc+ V . Following [11] we write H
as
?
We will evaluate the trace over the oscillator-mode in the partition function using
the coherent state representation of the field, and for this purpose we should bring
exp(−βH) on normal ordered form with respect to the scaled operators b = a/√N
and b†= a†/√N. Due to the scaling, their commutator is [b,b†] = 1/N and vanishes
for N → ∞.
The approximation of neglecting the commutator in the thermodynamic limit
can be illuminated by Wick’s theorem [26]: Any string of creation- and annihilation
operators can be written as their normally ordered form plus additional terms. These
terms are given by means of a contraction defined by C(AB) = AB−:AB:, where
:AB: is the normal ordering of AB. Using this notation, Wick’s theorem states that
H =
N
i=1
?
−hσz
i− Jσy
iσy
i+1+ gσx
i
?
a
√N
+
a†
√N
?
+ ωa†
√N
a
√N
?
.
ABC ...Q =:ABC ...Q:+
?
?
:one contraction:
+:two contractions:+...
Now, in our case C(bb†) = bb†− b†b = [b,b†] = 1/N and C(b†b) = 0.
applying Wick’s theorem to the expansion of the exponential in Z = Tr(exp(−βH)),
and assuming that the limits in Z = limN→∞limR→∞
interchanged, we see that all terms involving contractions will be of order 1/N or
higher and can therefore be neglected in the thermodynamic limit. More generally any
expression of the form Tr(f(b,b†)e−βH), where f is a polynomial, can be calculated
to an accuracy of 1/N using the normal order :f(b,b†)e−βH:. The validity of this
truncation for the Dicke-model was discussed in [12].
When performing the trace of a normally ordered operator it is convenient to use
the coherent states, |α?, eigenstates of the annihilation operator: a|α? = α|α?, which
yields
?d2α
The remaining trace over the spin degrees of freedom is exactly equivalent to the the
original Ising model calculation, where the term involving the real part of the complex
field argument acts as an additional magnetic field in the x direction, and hence the
Ising model is biased by an effective magnetic field in the xz-plane with magnitude
Hence,
?R
r=0(−βH)r/r! can be
Z =
π
e−βω|α|2Trspin
?
exp
?
−β
?
i
?
−hσz
i− Jσy
iσy
i+1+2g?(α)
√N
σx
i
???
+ O(1/N).