PHYSICAL REVIEW A 90, 063824 (2014)

Antibunching and unconventional photon blockade with Gaussian squeezed states
Marc-Antoine Lemonde,1 Nicolas Didier,1,2 and Aashish A. Clerk1
1

2

Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8
Départment de Physique, Université de Sherbrooke, 2500 boulevard de l’Université, Sherbrooke, Québec, Canada J1K 2R1(Received 29 October 2014; published 16 December 2014)
Photon antibunching is a quantum phenomenon typically observed in strongly nonlinear systems where photon
blockade suppresses the probability of detecting two photons at the same time. Antibunching has also beenreported with Gaussian states, where optimized amplitude squeezing yields classically forbidden values of theintensity correlation, g (2) (0) < 1. As a consequence, observation of antibunching is not necessarily a signatureof photon-photon interactions. To clarify the significance of the intensity correlations, we derive a sufficientcondition for deducing whether a field is non-Gaussian based on a g (2) (0) measurement. We then show that theGaussian antibunching obtained with a degenerate parametric amplifier is close to the ideal case reached using
dissipative squeezing protocols. We finally shed light on the so-called unconventional photon blockade effectpredicted in a driven two-cavity setup with surprisingly weak Kerr nonlinearities, stressing that it is a particularrealization of optimized Gaussian amplitude squeezing.
DOI: 10.1103/PhysRevA.90.063824

PACS number(s): 42.50.Ct, 42.50.Dv, 42.50.Ex, 42.50.Ar

I. INTRODUCTION

Understanding, generating, and ultimately manipulating
nonclassical states of light are fundamental goals of the fieldof quantum optics [1]. They have been pursued in a wide
variety of physical systems ranging from atomic cavity QED
systems and nonlinear optical media, to recent experimentswith superconducting circuits; such states are also essential to
many approaches to quantum information processing [2]. A
common way to identify the quantumness of these states is toquantify their intensity fluctuations via the g (2) (0) correlation
function, defined as g (2) (0) = : Iˆ2 :/Iˆ2 , where Iˆ is the field
intensity and colons indicate normal ordering. While classicalintensity fluctuations always obey g (2) (0)  1, quantum states
can violate this bound; g (2) (0) < 1 is hence often used as acriterion to identify nonclassical states.
The standard mechanism for achieving g (2) (0) < 1 is known
as photon blockade [3–5]. A laser drives a nonlinear cavity inresonance with the 0 → 1 photon transition; however, becauseof the cavity’s nonlinear spectrum, the laser cannot add anotherphoton as the 1 → 2 photon transition is off resonant. One
thus obtains a strongly non-Gaussian state close to a singlephoton Fock state and strongly reduced intensity fluctuations.
The observation of photon blockade relies on the challenging
task of having systems with nonlinear interactions that exceed
the characteristic dissipation rate. Similarly, phonon blockade
arises in nonlinear mechanical resonators [6,7].
Spurred by both recent studies in optomechanics and circuit
QED [5,8–10], as well as by recent studies discussing a method
for achieving g (2) (0) < 1 with extremely weak nonlinearities[11–18], we reconsider in this work a somewhat underappreciated fact: the g (2) (0) < 1 condition for nonclassicality
can be achieved without photon blockade, by simply using
optimized amplitude-squeezed Gaussian states. The basic
mechanism is depicted in Fig. 1(a), and was discussed inseveral previous works [19–24]. The upshot is that stateswith nonclassical intensity fluctuations can be generated using
purely linear bosonic systems (i.e., systems described bya quadratic Hamiltonian), without the need of any spectral
1050-2947/2014/90(6)/063824(11)nonlinearity, and without any negativity in the Wigner functionof the state. It also leads to the conclusion that some Gaussianstates are more quantum than others [in that not all violate theclassical bound on g (2) (0)].
In this work, we start by characterizing the equal-time
intensity correlation function of the most general single-mode
Gaussian state (i.e., a displaced, squeezed thermal state),
identifying the full parameter regime where g (2) (0) < 1 (seealso Ref. [24]). We show that, for a given average field
amplitude, there is a minimum possible value of g (2) (0)consistent with a Gaussian state. Often, a finding of g (2) (0) < 1
is used implicitly (and incorrectly) as evidence of a nonGaussian state. The results presented here allow one to simply
identify when a measurement of g (2) (0) < 1 necessarily signals
the existence of a non-Gaussian state (see the dark shaded
region in Fig. 2).

FIG. 1. (Color online) (a) Ideally amplitude-squeezed Gaussianstates for a squeeze parameter r = ropt = 0.13 (blue), r = 0.3 (green),
and r = 0 (yellow). (b) Realizations of a degenerate parametricamplifier with a pumped Kerr nonlinearity (top) and cavity frequency
modulation (middle). Alternatively, highly pure intracavity squeezing
can be generated using quantum bath engineering (QBE), wherea cavity interacts with a structured reservoir (bottom). (c) Twocavity setup where unconventional photon blockade is predicted. As
pictured, the photon-photon interaction generates squeezing of theintracavity modes.

063824-1©2014 American Physical Society

LEMONDE, DIDIER, AND CLERKPHYSICAL REVIEW A 90, 063824 (2014)

FIG. 2. (Color online) (a) Equal-time intensity correlation function g (2) (0) for a Gaussian state as a function of the displacement ᾱ fordifferent values of the state purity (as quantified by n̄eff ). For all curves the squeeze parameter has been set to its optimal value r = ropt [ᾱ,n̄eff ];ropt is plotted in (c) as a function of ᾱ. The dotted black line corresponds to the degenerate parametric amplifier (DPA) (see Sec. III) wheren̄eff = sinh2 rDPA . The curve corresponding to n̄eff = 0 (solid blue) sets the minimum value of g (2) (0) possible for a Gaussian state with |â| = ᾱ;any values lying in the darkest shaded region necessarily correspond to non-Gaussian states. For finite n̄eff , g (2) (0) = 2 for α = 0 (r = 0 also),
as it should for a thermal state. (b) presents the minimal g (2) (0)|min (black full line) that can be achieved with a Gaussian state having a fixedvalue of n̄eff . For these curves, α = ᾱopt (red dot-dashed line) and r = ropt (black dashed line) [cf. Eq. (8)].

We also discuss how this Gaussian-state g (2) (0) suppression
can be realized using one of two simple and generic cavitybased setups: either via a degenerate parametric amplifier(DPA), as has been studied earlier [22–24], or via dissipative
squeezing interactions [25–31]. While dissipative squeezing
has the virtue of being able to produce pure squeezed
intracavity states, we find that in terms of g (2) (0) suppression,
it gives only a marginal improvement over a DPA. Further, weshow that the DPA exhibits a kind of optimality: for a given
level of state impurity, the amount of squeezing produced isexactly what is needed to allow a maximal g (2) (0) suppression.
We also show how these two generic approaches lead to photon
antibunching and nonmonotonic behavior of the two-time
intensity correlation function g (2) (τ ). These are additional nonclassical features characterized respectively by the conditions
g (2) (τ ) > g (2) (0) and |g (2) (τ ) − 1| > |g (2) (0) − 1| [32].
Finally, we use this Gaussian-state g (2) (0) suppression
mechanism to help clarify a series of recent studies [11–18] of
a novel driven nonlinear two-cavity setup where g (2) (0) < 1
is predicted despite having nonlinearities much weaker than
all dissipative rates. Nonlinearity was suggested to be the key
ingredient [13] behind this so-called unconventional photon
blockade (UPB) [17]. Here we show that the UPB is more
simply understood as yet another realization of the optimallysqueezed Gaussian state mechanism for g (2) (0) suppression.
II. INTENSITY FLUCTUATIONS OF GENERALGAUSSIAN STATES

earlier at t = 0; it can be measured using a Hanbury Brown
and Twiss type experiment [33,34]. For classical (commuting)fields, it directly follows from the Cauchy-Schwarz inequalitythat g (2) (0) ≥ 1, as well as g (2) (τ ) ≤ g (2) (0) [35]. In contrast,
a quantum field prepared in an appropriate state can violate
one or both of these classical bounds; such states are generally
termed “nonclassical.” Our first goal here will be to remind
the reader that a properly optimized Gaussian state can lead tosuch nonclassical signatures [19–24].
The most general single-mode Gaussian state, i.e., adisplaced squeezed thermal state, is described by the density
matrixρ̂α,ξ,n̄eff ≡ D̂(α)ρ̂n̄eff ,ξ D̂ † (α) = D̂(α)Ŝ(ξ )ρ̂n̄eff Ŝ † (ξ )D̂ † (α).
(2)
Here, D̂(α) = exp[α â † − α ∗ â] and Ŝ(ξ ) = exp[ 12 (ξ ∗ â 2 −ξ â †2 )] are respectively the displacement and squeezing operators [35], with α = ᾱeiϕ and ξ = reiθ (ᾱ > 0,r > 0). ρ̂n̄eff
is the density matrix of a thermal state with population n̄eff ;Tr[ρ̂n̄eff â † â] = n̄eff . The purity P of the density matrix in
Eq. (2) is set by n̄eff according to the relation
 2
P ≡ Tr ρ̂α,ξ,n̄eff =â † (0)â † (τ )â(τ )â(0)
g (τ ) ≡ †,â (0)â(0)â † (τ )â(τ )

g (2) (0) = 1 +2ᾱ 2 (n − s) + s 2 + n2,
(ᾱ 2 + n)2n̄tot ≡ â † â = ᾱ 2 + n,

(4a)
(4b)

with


n ≡ Tr[ρ̂n̄eff ,ξ â † â] = n̄eff + 12 cosh 2r − 12 ,s ≡ |Tr[ρ̂n̄eff ,ξ â â]| = n̄eff + 12 sinh 2r.

(1)

where â is the photon annihilation operator. g (2) (τ ) is proportional to the conditional probability for detecting a second
photon at time t = τ , given that a photon was also detected(3)

As one might expect, to minimize intensity fluctuations it
is always optimal to squeeze the amplitude quadrature, i.e., tochoose θ = 2ϕ. In this case, we findThe simplest measure of the intensity fluctuations of alight field (as measured by a photomultiplier) is the g (2) (τ )correlation function,(2)

1.
1 + 2n̄eff

(5a)
(5b)

Using Eq. (4a), it is now straightforward to find conditions
on the displacement, squeezing, and effective temperature of

063824-2

ANTIBUNCHING AND UNCONVENTIONAL PHOTON . . .

(a)

1 λ/κ = 0.25

0.6

10
1
0.1
0.01

=

0.
01

=

0.4
0.2

DPA
QBE

0

2

0

g (2) (τ )

κ
λ/

0.1

λ/
κ

g (2) (τ )

0.8

(b)

PHYSICAL REVIEW A 90, 063824 (2014)

4

0.3
0.2
0.1
0

0.4
2

0.2 ᾱ = 0.012
λ/κ = 0.02
0
0
2

4

ᾱ2opt0.5
0

0 0.1 0.2 0.3 0.4
λ/κ

κτ 6

1 ᾱ2 = 0.4
λ/κ = 0.25
0.8
0.6

1

P

8

ᾱ2opt0κτ 6ropt [ᾱ,n̄eff = 0] ≈ ᾱ 2 + O[ᾱ 4 ] ⇒ g (2) (0) ≈ 4ᾱ 2 + O[ᾱ 4 ].
(7)10

g (2) (0) < 1

(2)

ᾱ2+
ᾱ2ming (2) (0) > 1
0.1
0.2
0.3
λ/κ
8

10

FIG. 3. (Color online) Antibunching in the DPA versus QBE.
(a) Two-time intensity correlation g (2) (τ ) versus τ . Solid curves arefor the DPA system, with different curves corresponding to different
values of the strength of the squeezing interaction λ; for each curve,
the coherent displacement ᾱ = ᾱopt . The dashed lines are for the
QBE system. The intracavity squeeze parameter r is the same in
the two cases and set by the value of λ/κ. For QBE QBE = 0.9 κ.
The optimal displacement ᾱopt and the purity P of the intracavity
state [cf. Eq. (3)] is plotted in the inset as a function of λ/κ. Forκext , QBE generates a much better purity than the DPA.
QBE(b) Nonclassical nonmonotonic behavior of g (2) (τ ) for a DPA;
parameters for both curves are indicated in the plot. Inset: ᾱmin(green line) is the minimum value of ᾱ for the given λ/κ for whichg (2) (0) < 1, while ᾱ+ (red line) represents the minimum value of ᾱ
for which we have true antibunching [i.e., the slope of g (2) (τ ) at τ = 0
is positive]. Values of ᾱ between these curves (shaded region) will
yield g (2) (τ ) functions which are nonmonotonic.

our Gaussian state that reduce g (2) (0) below 1. Note that forarbitrary angles, (n − s) → n − s cos(θ − 2ϕ) in Eq. (4a).
We first investigate the ideal (and optimal) case where wehave a pure state, i.e., n̄eff = 0. Equation (4a) then simplifies tog (2) (0) = 1 +cosh 2rᾱ 2 (1 + sinh 2r).
−
ᾱ 2 + sinh2 r(ᾱ 2 + sinh2 r)2However, as ᾱ → ∞, the optimized g (2) (0) approaches theclassical value of 1 from below as g (2) (0) → 1 − 1/ᾱ 2 . The
full behavior of g (2) (0) versus ᾱ for this optimally squeezed
Gaussian state (for n̄eff = 0) is shown in Fig. 2.
For further insight, it is useful to consider the limit ofsmall displacements, ᾱ
1. The optimal squeezing and thecorresponding g (2) (0) are given byThus, one can make g (0) as small as one likes by simply
taking a small enough displacement ᾱ and always picking the
optimal (small) amount of squeezing.
The optimal parameter values in the small-ᾱ limit areeasily understood. To leading nonvanishing order in ᾱ andr, the probability for having two photons in our state is|2|D(α)S(ξ )|0|2 ≈ (ᾱ 2 − r)2 /2. The two terms here indicate
the two ways of getting two photons: either via the squeeze operator, or via the displacement operator. The optimal squeezing
condition thus simply corresponds to these two mechanisms interfering destructively [23]. Note that unlike photon blockade,we are not using the nonlinearity of the spectrum to suppress
the two-photon population, but rather the interference between
a coherent displacement and a squeeze operation.
Returning to the case of a general ᾱ, we see that for a given
average amplitude |â| = ᾱ, there is a minimum possibleg (2) (0) achievable with a Gaussian state; if one obtains alower g (2) (0), this then necessarily implies that the state is
non-Gaussian. For example, for ᾱ = 1 the smallest g (2) (0)achievable with a Gaussian state is g (2) (0) ≈ 0.71, which isachieved when the squeeze parameter r ≈ 0.28. This generalbound on the minimal g (2) (0) for a Gaussian state is shown in
Fig. 2: the dark shaded region indicates regimes where the intensity fluctuations are too small to be explained classically or
by a Gaussian state. We stress that for states with ᾱ → 0, an arbitrarily small value of g (2) (0) is possible with a Gaussian state,and hence in this case a g (2) (0) measurement cannot be used
to conclusively prove the existence of a non-Gaussian state.
We also show in Fig. 2 that g (2) (0) < 1 is possible with aGaussian state even if it fails to be pure, i.e., if n̄eff is nonzero
in Eq. (4a). Suppose we consider only Gaussian states whichhave a fixed effective thermal number n̄eff : if we optimize
both the state displacement ᾱ and the squeezing magnitude r
for such states, how small can we make g (2) (0)? The optimal
amount of squeezing ropt is, in this case, determined by(6)This simple expression already reveals some surprises. For
no squeezing (i.e., r = 0), we recover a coherent state andg (2) (0) = 1. One might have expected that g (2) (0) would
decrease monotonically if we now started to increase r (i.e., thegreater the squeezing, the smaller the intensity fluctuations).
This is clearly incorrect: Eq. (6) yields g (2) (0) → 3 as r → ∞,as was experimentally observed in Ref. [24]. Equation (6)instead reveals that for a fixed displacement ᾱ, g (2) (0) has
a minimum as a function of r; we denote this optimal value
ropt [ᾱ,n̄eff = 0]. If r is tuned to ropt [ᾱ,n̄eff = 0], we findthat the resulting g (2) (0) is always less than 1 (and hence
nonclassical), no matter how large the displacement ᾱ.

sinh2 ropt = n̄eff .

(8)

From Eq. (5a), one sees that sinh2 r is the mean numberof excitations in a vacuum squeezed state having squeeze
parameter r; for ropt , the contributions to the mean numberof excitations due to squeezing and due to thermal fluctuations
are equal.
The corresponding full expression for the optimal displacement ᾱ = ᾱopt is given in Appendix A [Eq. (A1)]; for the mostinteresting case of a small thermal population n̄eff
1, it is1/4
approximately given by√ᾱopt ∼ n̄eff . These choices lead to aminimal g (2) (0)|min ≈ 8 n̄eff [see Fig. 2(b) and Appendix A
for general n̄eff ]. Thus, for an impure state, one cannot
suppress g (2) (0) arbitrarily, even if one takes an arbitrarily

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LEMONDE, DIDIER, AND CLERKPHYSICAL REVIEW A 90, 063824 (2014)

small displacement ᾱ. It is also interesting to note that the
optimal relation between n̄eff and r given in Eq. (8) is exactly
satisfied by a DPA; we will see this explicitly below.

with τ  0 and the two-time correlations n(τ ) ≡ d̂ † (τ )d̂(0)
and s(τ ) ≡ |d̂(τ )d̂(0)| (with d̂ ≡ â − â) given by

III. ANTIBUNCHING USING COHERENT AND
DISSIPATIVE SQUEEZING INTERACTIONSHaving described the suppression of g (2) (0) below 1
for optimized amplitude-squeezed Gaussian states, we now
discuss a simple system which can realize this physics
via coherent squeezing interactions, namely, a cavity-based
DPA [22–24]. We also compare it to schemes that achievesqueezing via dissipative interactions [25–31]. Such schemes
have recently garnered interest in both the circuit QED [30]and optomechanics communities [31], and have the virtuethat they can in principle generate pure intracavity squeezing.
We focus here on the intracavity g (2) (τ ); this is of direct
relevance in experiments in optomechanics [10], and as weshow at the end of this section, it is simply related to thecorrelations of the output field. We then recover the factthat the DPA exhibits true photon antibunching [22–24]
and nonclassical nonmonotonic behaviors characterized by|g (2) (τ ) − 1| > |g (2) (0) − 1| [23,32]. We stress that in our case,the temporal evolution of the output correlations is due only
to the intracavity dynamics; in contrast, the time dependenceof g (2) (τ ) calculated in Ref. [24] reflected only the bandwidthof the chosen filter.
The general Hamiltonian of the DPA reads, in the interaction picture ( = 1),
Ĥ = − 2i λ[â †2 eiθ − â 2 e−iθ ] + [ â † +∗â] + Ĥκ ,(9)where the strength of the squeezing interaction λ and the angle
θ are controlled by the setup characteristics. The HamiltonianĤκ describes the coupling to a single input-output waveguide;
this coupling is characterized by the damping rate κ and istreated with a standard input-output approach [36,37] (seealso Appendix B). We assume that the phase of the parametriccoupling is always tuned to satisfy θ = 2 arg â, i.e., to get an
amplitude-squeezed intracavity state. The system is stable forλ  κ/2, and the steady state inside the cavity is a displacedsqueezed thermal state with the parameters tanh 2r = 2λ/κand n̄eff = sinh2 r. Remarkably, the latter relation is identicalto Eq. (8) (i.e., the optimal amount of squeezing given a fixed
effective thermal number).
For a fixed value of parametric coupling λ in the DPA,
the optimal displacement needed to minimize the intracavity
2g (2) (0) is given by ᾱopt= κλ/(κ − 2λ)2 ; this can be obtainedby tuning the drive strength . Alternatively, one could imagine
minimizing g (2) (0) for a fixed value of ᾱ (by tuning both λ andsimultaneously). The resulting minimal g (2) (0) versus ᾱ isplotted in Fig. 2 and is extremely close to the minimal value
allowed for an arbitrary Gaussian state with the same ᾱ. Whileg (2) (0) is minimal for infinitesimally small displacement andsqueezing, g (2) (0) < 1 can be obtained even for displacements
at the single- to the few-photon level.
To show furthermore that the field is antibunched, we
calculate the intracavity two-time intensity correlation g (2) (τ ),g (2) (τ ) = 1 +2ᾱ 2 [n(τ ) − s(τ )] + n2 (τ ) + s 2 (τ ),[ᾱ 2 + n(0)]2(10)

n(τ ) =λ e−κτ/2[2λ cosh(λτ ) + κ sinh(λτ )] ,
κ 2 − 4λ2

(11a)s(τ ) =λ e−κτ/2[2λ sinh(λτ ) + κ cosh(λτ )] .
κ 2 − 4λ2

(11b)

The time-dependent intensity correlations g (2) (τ ) are plottedin Fig. 3 for different values of λ/κ. The correlations start well
below unity and increase to reach 1 at long times. This behavior
is the signature of photon antibunching, which has beenmeasured experimentally in Refs. [22–24]. More precisely,
the condition to have a positive slope at τ = 0, ġ (2) (0) > 0,2
. This condition forcorresponds to ᾱ 2 > κλ/(κ 2 − 4λ2 ) ≡ ᾱ+antibunching is always satisfied for ᾱ = ᾱopt . Moreover, inFig. 3(b), we show an additional nonclassical behavior that canbe exhibited by g (2) (τ ), where |g (2) (τ ) − 1| > |g (2) (0) − 1|.
In order to observe this behavior, one has to tune ᾱ either
such that g (2) (0) < 1 and ġ (2) (0) < 0 or such that g (2) (0) > 1
and ġ (2) (0) > 0. These two conditions are represented by theshaded region in Fig. 3(b). The former case has been observedin [23].
The generation of intracavity vacuum squeezed states is also
possible using QBE approaches; unlike the DPA where one isusing a coherent (Hamiltonian) squeezing interaction, one is
now making use of dissipative squeezing interactions. There
are a variety of methods for achieving this kind of interaction,e.g., by modulating the cavity damping rate as a function oftime [30] or via two-tone driving [31]. The ideal versions ofsuch a scheme are equivalent to having effectively coupled thecavity to a squeezed reservoir (as could be realized directly by
driving the cavity with vacuum squeezed light [38]).
We wish to compare the ability of such dissipative squeezing approaches to generate states with g (2) (0) < 1 against thecoherent squeezing approach (i.e., using a DPA). We modelthe dissipative squeezing interaction by taking the cavity to
be coupled to a Markovian squeezed reservoir (in addition to
the main port used to drive the cavity and extract an output
field). This additional reservoir is characterized by a squeeze
parameter rQBE and an additional cavity damping rate QBE .
The damping rate QBE adds with the external damping rate
κext due to the coupling to the input-output waveguide to givethe total cavity damping κQBE . The quantum master equationρ̂˙ = L̂ρ̂ of the density matrix ρ̂ of the system induced by thetwo reservoirs is governed by the following Lindbladian:L̂ρ̂ = QBE b̂ρ̂ b̂† − 12 {b̂† b̂,ρ̂} + κext â ρ̂ â † − 12 {â † â,ρ̂} .
(12)
The Lindbladian has two contributions. The first term describes dissipative squeezing [30] and cools the Bogoliubov
mode b̂ = cosh rQBE â + eiθ/2 sinh rQBE â † to its vacuum; thiscorresponds to the vacuum squeezed state |rQBE eiθ  for the
intracavity field â. The second term cools the intracavity fieldto vacuum, thereby altering the squeezing purity. The purity
tends to unity for QBE
κext , as can be seen in Fig. 3. Further
details are given in Appendix C.
The comparison of the dissipative squeezing setup against
the DPA is shown in Fig. 3, where we plot the two-time

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ANTIBUNCHING AND UNCONVENTIONAL PHOTON . . .

PHYSICAL REVIEW A 90, 063824 (2014)

intensity correlation g (2) (τ ) for both approaches. To facilitatecomparison, we use parameters that ensure that the schemes
produce equivalent amounts of intracavity squeezing (seeAppendix C2). We also pick QBE = 9κext = 0.9κQBE toensure that the QBE scheme generates a highly pure squeezed
state, and that the total cavity damping rate is the same
in both approaches, κQBE = κ. Finally, in both schemes thecoherent drive is chosen so as to yield an optimal ᾱ, i.e., avalue which minimizes g (2) (0). Figure 3 shows that despite theadditional purity achieved using the dissipative scheme, it does
not perform significantly better except at the largest values ofλ/κ (which correspond to the largest values of r and of ᾱ).
The purity and the optimal g (2) (0) of the two setups are equal
when the two damping rates coincide, QBE = κext .
Finally, while we have discussed intracavity fields here, ourresults are easily extended to the g (2) (τ ) function of the output
light field leaving the coupled waveguide. For the single-sided
cavity case we focus on, the expressions for the output(2)(τ ) are identical to those for the intracavity√g (2) (τ ),field gout
except that one needs to replace α with α + αin / κ (seeAppendix B3).

IV. TWO-CAVITY UNCONVENTIONAL
PHOTON BLOCKADE

Having discussed in detail the fact that optimally squeezed
Gaussian states can lead to the nonclassical regime g (2) (0) < 1,
we reconsider the system introduced by Liew and Savona [11](subsequently studied in Refs. [12–17]). This work predicts
g (2) (0) < 1 in a two-cavity system having extremely weakKerr nonlinearities (i.e., on-site photon-photon interactions).
We show here that the main effect in the Liew and Savonasystem can be explained entirely using Gaussian states. The
only role of the Kerr interaction in the two-cavity system isthus to provide an effective (quadratic) squeezing term in the
Hamiltonian; the extremely weak nonlinearity of the system’s
spectrum plays no role. Moreover, the interpretation of the
UPB as a result of interference between different paths leading
to the two-photon state, as proposed in [13], is exactly the sameinterference condition that leads to maximal suppression ofg (2) (0) as discussed below Eq. (7) in Sec. II and in Ref. [23].
The Liew-Savona system is thus a particular realization of
the general Gaussian-state mechanism for the suppression ofg (2) (0) discussed above, albeit a more complicated one than
the single-cavity DPA.
The Liew-Savona system is composed of two bosonicmodes, e.g., optical modes in two separated cavities (aspictured in Fig. 1). In each cavity, Kerr-type photon-photon
interaction takes place with interaction strength Uk (k = 1,2).
The two cavities are linearly coupled together by a hopping
term (rate J ), and a weak drive F (at frequency ωd ) is applied
to the first cavity only. The corresponding Hamiltonian, in theframe rotating at the drive frequency, is given by

Ĥ =

2

[−

†
† †
k âk âk + Uk âk âk âk âk ]

k=1
†

†

†

+ J [â1 â2 + â2 â1 ] + F [â1 + â1 ] + Ĥκ .

(13)

FIG. 4. (Color online) g (2) (0) of the field inside cavity 1 of thetwo-cavity setup described in the main text as a function of the Kerrinteraction strength U . The full line is the solution of the linearized
Langevin equations and the dashed line is the solution of the quantum
master equation (see Appendix D2). In the inset, we compare thecorresponding squeeze parameter r1 (blue dashed line) to the optimalvalue ropt [ᾱ1 ,n̄eff,1 = 0] (green dot-dashed line) and we see that fullsuppression of g1(2) (0) corresponds to the case r1 = ropt [ᾱ1 ,n̄eff,1 =sinh2 r1 ]. For these curves, we use the same parameters as in[13], namely, κ1 = κ2 = κ, U1 = U2 = U , 1 = 2 = −0.275κ,F = 0.01κ, and J = 3κ. For such values n̄eff,1 ∼ 10−13 and the total
photon number inside the cavity is n̄tot,1 ∼ 10−7 [cf. Eq. (4b)].

Here, âk is the annihilation operator of mode k with detuning k and Ĥκ describes the coupling to the environment
characterized by the damping rates κk of each cavities. We
consider J and F to be real and positive without loss ofgenerality. From Eq. (13), we derive the Heisenberg-Langevinequations of motion for both cavities using the input-output
formalism [36,37]. One can solve these equations approximately by linearizing them about the classical steady-statesolution (see Appendix D). This approximation is equivalent totreating the interactions in Eq. (13) at a mean-field level: we arethus approximating the system with a quadratic Hamiltonian,and precluding any effects associated with non-Gaussian statesand spectral nonlinearity. In this approximation, the resulting
steady-state intracavity field of cavity 1 (where the interesting
physics is predicted) is Gaussian, and thus has the formρ̂α1 ,ξ1 ,n̄eff,1 given in Eq. (2), where we parametrize ξ1 = r1 eiθ1and α1 = ᾱ1 eiϕ1 .
In Fig. 4, we show how the cavity-1 squeeze parameter r1
and g1(2) (0) behave as functions of the interaction strength U =U1 = U2 for the same choice of parameters J , κk , and k as inRef. [13]. Results from the approximate linearized dynamics
are shown, compared against results of a numerical solutionof the full quantum master equation (see Appendix D2). The
two approaches are in excellent agreement for the range ofU yielding a minimal g (2) (0); in particular, the linearized
dynamics accurately describe the results obtained in Ref. [13].
This implies that the maximal g (2) (0) suppression seen here
is completely due to an optimally squeezed Gaussian state incavity 1.
The connection to optimized Gaussian squeezing can bemade even more precise. From the solutions of the linear
equations of motion, one can show that both the cavity-1

063824-5

LEMONDE, DIDIER, AND CLERKPHYSICAL REVIEW A 90, 063824 (2014)

squeeze angle θ1 and the cavity-1 average amplitude ᾱ1 eiϕ1are (to an excellent approximation) constant over the rangeof U considered in Fig. 4. For the set of parameters used,
they correspond almost precisely to amplitude squeezing(θ1 − 2ϕ1 ≈ 0.065). Also, for the whole range of U in Fig. 4,
the relation between r1 and n̄eff,1 is the same as a DPA (i.e.,n̄eff,1 = sinh2 r1 ). Thus, the only parameter determining thecavity-1 Gaussian state that varies with U in Fig. 4 is r1 , themagnitude of the cavity-1 squeezing. As shown in the inset of
Fig. 4, maximal suppression of g1(2) (0) exactly corresponds tothe case where r1 = ropt [ᾱ1 ,n̄eff,1 = sinh2 r1 ] [cf. Eq. (7)].
For further insight about the dynamics, one can eliminate
the cavity 2 from the linearized equation of motion of cavity 1(see Appendix D). Doing so, one can define a total squeezing
interaction λ1,tot [analoguous to the DPA; see Eq. (9)] whichis composed of two contributions: a direct interaction λ1
coming from the Kerr interaction inside cavity 1, and an
induced one λ1,ind [ω] coming from the interaction with cavity
2; this induced interaction is frequency dependent. In thelimit U1,2
κ (κ1 = κ2 = κ), these two contributions are( 1= 2= )λ1 = 2U1 α12 ,λ1,ind [ω = 0] ≈ 2U2 α22

Still in the limit U1,2
α12 ≈

(14)

κ, we also haveF ( + iκ/2),[( + iκ/2)2 − J 2 ]22J2.
2 + κ 2 /4

2

α22 ≈

[(

F 2J 2.
+ iκ/2)2 − J 2 ]2(15)

In the case where J is the largest scale in the system andwhere∼ κ (as in Fig. 4 and Refs [11,13]), the induced
squeezing interaction is enhanced compared to the directcontribution as λ1,ind /λ1 ∼ J 4 /κ 4 . This enhancement results
both from the explicit factor of J 2 in λ1,ind , and from thefact that α2 /α1 ∼ J /κ. Setting U1 = 0 would actually leadto almost the same results, as pointed out in [13]. In short,the second cavity in this system acts to produce an effective
squeezing interaction in cavity 1; by tuning U2 , the amount ofsqueezing can be tuned to give an optimal g (2) (0) suppression.
We stress that, in general, all that one needs is a sufficiently
tuned squeezing interaction; having two cavity modes is notnecessary.
The almost complete suppression of g1(2) (0) found in
Ref. [13] and shown in Fig. 4 thus corresponds to having
a displaced squeezed state which is infinitesimally close to
the vacuum, and where the squeezing is nearly optimally
matched to the displacement. While the large suppression of
g1(2) (0) is interesting, the fact that the state corresponds toalmost no photon in cavity 1 (i.e., n̄tot,1 [cf. Eq. (4b)] is of theorder of 10−7 ) makes the state inconvenient for applications.
While the general optimal-squeezing mechanism allows forg (2) (0) suppression at much larger average photon numbers,
the required tuning of parameters is hard to achieve in thetwo-cavity setup. If one simply increases the drive strength ing (2) (0)|opt = 1 −the two-cavity system to increase photon number, g (2) (0) is far
from optimal (e.g., for F = 10κ, n̄tot = 0.5, and g (2) (0) = 0.9
at U/κ = 0.006). To that extent, the DPA constitutes a simpler
and more efficient system for exploiting the nonclassicalfeatures of amplitude-squeezed Gaussian states [22–24].

V. CONCLUSION

In this work, we have discussed how one can achieveclassically forbidden values of the normalized intensity fluctuations of a light field [g (2) (0) < 1], antibunching [g (2) (0) <g (2) (τ )], and nonclassical nonmonotonic behaviors of g (2) (τ )[|g (2) (τ ) − 1| > |g (2) (0) − 1|] with Gaussian states. The key
ingredient is a well-tuned amount of amplitude squeezing.
We found that for a fixed average cavity amplitude, there
is a minimum possible value of g (2) (0) achievable witha Gaussian state; this minimum is attained using a purestate with an optimal amount of amplitude squeezing. This
result thus allows one to safely identify non-Gaussian statesfrom a measurement of g (2) (0). We then reviewed how ageneric DPA appears to be one of the simplest and mostefficient platforms to exploit these nonclassical signatures. We
also demonstrated that it compares favorably to the g (2) (0)suppression possible using dissipative squeezing interactions
generated via reservoir engineering. Finally, we have helped
clarify the origin of the so-called unconventional photon
blockade predicted in a driven two-cavity setup with weakKerr nonlinearity, showing that it is a particular realization ofthis general Gaussian-squeezed-state physics.
ACKNOWLEDGMENTS

We thank Christoph Marquardt, Florian Marquardt, Cristiano Ciuti, and Alexandre Blais for helpful conversations.
This work was supported by NSERC and FQRNT. N.D.
acknowledges support from CIFAR.
APPENDIX A: OPTIMIZATION OF g (2) (0) FORNONIDEAL SQUEEZING

In the case of a finite effective temperature n̄eff , g (2) (0)is minimized as follows. First, we choose the right angles
such that squeezing is along the displacement (amplitude
squeezing) and afterward, we minimize it as a function ofᾱ, keeping r and n̄eff constant. This then gives(2n̄eff + 1)(er n̄eff + sinh r) sinh 2r. (A1)
ᾱopt [r,n̄eff ] =e−3r [e2r − (2n̄eff + 1)]
Note√ that ᾱopt [r,n̄eff ] is real and positive only if r ln( 2n̄eff + 1). For squeezing below this minimal value, it
is possible to show that g (2) (0) > 1.
We then minimize g (2) (0) (with ᾱ = ᾱopt ) as a function of r,
keeping n̄eff constant, which leads to sinh2 ropt = n̄eff [Eq. (8)]
and

1
√

.
1 + 24n̄eff + 2n̄2eff [11 + 8n̄eff (2 + n̄eff )] + 8 1 + 6n̄eff + 12n̄2eff + 8n̄3eff n̄eff (n̄eff + 1)
063824-6(A2)

ANTIBUNCHING AND UNCONVENTIONAL PHOTON . . .

PHYSICAL REVIEW A 90, 063824 (2014)

APPENDIX B: DEGENERATE PARAMETRIC AMPLIFIERS(2)

1. Calculation of g (t)The Langevin equation induced by the general Hamiltonian
Eq. (9) of the DPA reads√∂t d̂(t) = −λeiθ d̂ † (t) − 12 κ d̂(t) − κ d̂in (t),(B1)
where we define d̂ = â − α, α = â, and we denote by κ thedamping rate of the resonator. The operator d̂in is the input
noise corresponding to vacuum, that is, d̂in (t) = 0 and the†only nonvanishing correlation function is d̂in (t)d̂in (0) = δ(t).
The Langevin equation is integrated to give t√
d̂(t) = − κdt  e−κ(t−t )/2 {cosh[λ(t − t  )]d̂in (t  )−∞− e sinh[λ(t − t  )]d̂in† (t  )}.
iθ

(B2)

In Eqs. (B9a) and (B9c), the second term is eliminated
due to causality: the input fluctuations at time t > 0 cannot
affect the intracavity field at time t = 0. The third and fourthterms also vanish since the incoming noise is vacuum noise.
Consequently, nout (t) = κn(t) and sout (t) = κs(t); note also†that [d̂out (t),d̂out (0)] = δ(t) and [d̂out (t),d̂out (0)] = 0, as theyshould.
Concerning the average value of the output field, αout =âout , we have
αout =√

κα + αin .

(B10)

From the general expression Eq. (B5) of g (2) (t), we see that(2)
gout(t) is equal√to g (2) (t) with a renormalized displacement:
α → α + αin / κ.

The two-time correlations are then equal toλ e−κt/2d̂ (t)d̂(0) = 2[2λ cosh λt + κ sinh λt] ,κ − 4λ2†d̂(t)d̂(0) = −APPENDIX C: SQUEEZED ENVIRONMENT(B3)

λ e−κt/2 eiθ
[2λ sinh λt + κ cosh λt] . (B4)κ 2 − 4λ2

Noting that n(t) = d̂ † (t)d̂(0), s(t) = |d̂(t)d̂(0)|, ᾱ = |α|,and ϕ = arg α, g (2) (t) is found fromg (2) (t) = 1 +2ᾱ 2 [n(t) − cos(θ − 2ϕ)s(t)] + n2 (t) + s 2 (t).
[ᾱ 2 + n(0)]2(B5)

1. Calculation of g (2) (τ )As discussed in the main text, we model QBE by considering a two-sided linear cavity coupled on one side to an
environment in a vacuum squeezed state and driven on theother side, both on resonance with the cavity frequency. The
damping rate of the former side is QBE and that of the latteris κext ; we denote κQBE = QBE + κext . The operator of thesqueezed vacuum is b̂in and the operator of the drive is ĉin . In
an interaction picture at the cavity resonance frequency, theLangevin equation of the intracavity field is∂t â(t) = − 12 κQBE â(t) −2. Thermal-squeezed-state parameters

QBE b̂in (t) −√κext ĉin (t), (C1)

At coinciding times, the two-time correlations give2λ2
d̂ d̂ = 2,
κ − 4λ2†

κλ eiθd̂  = − 2.
κ − 4λ22(B6)where the input noise corresponds to a displaced vacuum
squeezed state,†b̂in(t)b̂in (0) = sinh2 rQBE δ(t),Comparing these results to the case of a squeezed thermal state,Tr[d̂ † d̂ ρ̂n̄eff ,ξ ] = n̄eff cosh 2r + sinh2 r and Tr[d̂ 2 ρ̂n̄eff ,ξ ] =−(n̄eff + 12 ) sinh 2r eiθ , we gettanh 2r =2λ,κn̄eff = sinh2 r.

b̂in (t)b̂in (0) = − 12 sinh 2rQBE eiθ δ(t),κQBEĉin (t) = αin = − √α,2 κext(B7)

3. Output field

From input-output formalism [36,37], the output fieldâout = d̂out + αout is given by√d̂out (t) = κ d̂(t) + d̂in (t),
(B8)

(B9b)
√
d̂out (t)d̂out (0) = κd̂(t)d̂(0) + κd̂in (t)d̂(0) (B9c)√+ κd̂(t)d̂in (0) + d̂in (t)d̂in (0)= κd̂(t)d̂(0).

(C3)
(C4)

where ĉin = αin + d̂in . The operator d̂in describes the same
vacuum noise as in Appendix B. The intracavity quantum
fluctuations d̂ = â − α are then given byd̂(t) = −giving rise, for t  0, to the output correlations√†
d̂out (t)d̂out (0) = κd̂ † (t)d̂(0) + κd̂in† (t)d̂(0) (B9a)√+ κd̂ † (t)d̂in (0) + d̂in† (t)d̂in (0)= κd̂ † (t)d̂(0),

(C2)

 t
QBE

√
− κext

 t



−∞

−∞

dt  e−(1/2)κQBE (t−t ) b̂in (t  )dt  e−(1/2)κQBE (t−t ) d̂in (t  ).

(C5)

The correlations of the fluctuations are(B9d)063824-7d̂ † (t)d̂(0) = η sinh2 rQBE e−(1/2)κQBE t ,(C6)d̂(t)d̂(0) = − 12 η sinh 2rQBE eiθ e−(1/2)κQBE t ,(C7)

LEMONDE, DIDIER, AND CLERK

where we denote η =PHYSICAL REVIEW A 90, 063824 (2014)

QBE /κQBE . The two-time second-order correlations read

g

(2)

1
|α|2 (1 − e−2rQBE ) e−(1/2)κQBE t − sinh2 rQBE cosh 2rQBE e−κQBE tη(t) = 1 −.
1
2
|α|2 + sinh2 rQBE
η

Note that the output field on the driven side is obtained from√√
αout = κext α + αin ,(C9)d̂out = κext d̂ + d̂in ,with the output correlations†

d̂out (t)d̂out (0) = κext d̂ † (t)d̂(0),(C10)d̂out (t)d̂out (0) = κext d̂(t)d̂(0)

(C11)

for t  0. The output intensity correlations are then given by√
Eq. (C8) by replacing α → αout / κext .
2. Comparison with the DPA

The intensity correlations obtained with QBE are comparedto the case of the DPA by setting the same total damping rateκ and the same intracavity squeeze parameter r (amplitude
squeezing), which is achieved by setting
e2rQBE =2λ(1 − η) +4λ2 (1 − 2η) + η2 κ 2.
η(κ − 2λ)PDPA =

1 − 4λ2 /κ 2

(C13)

and
PQBE = 1 +

2(1 − η)(4λ2 − η2 κ 2
κ 2 − 4λ2

+ 4λ2 (1 − 2η) + η2 κ 2 )cavities, we treat the environment as a Markovian dissipative
bath with zero effective temperature. Using input-output
formalism [36,37], we write the full nonlinear equations of
motion asκk√∂t âk = −i[âk ,Ĥ ] − â − κk ξ̂k ,2κk†âk − 2iUk âk âk âk − iJ âl= i k−
2√− κk ξ̂k − iF δk,1 .
(D1)Here, κk is the damping rate of cavity k, ξ̂k is the annihilation
operator of the corresponding incoming vacuum noise, l = 1(l = 2) if k = 2 (k = 1), and δk,1 is the Kronecker function.
The only nonvanishing correlation function of the vacuum
noise is†ξ̂k (t)ξ̂k (t  ) = δ(t − t  ).

(C12)

The purity P = 1/(1 + 2n̄eff ) of the two systems is then

(C8)

(D2)

As presented in the main text, we want to show thatthe corresponding linearized theory is enough to recover the
results predicted in [13]. To do this, we first displace the field
operators by their steady-state mean values, i.e., âk → αk + d̂kwith d̂k  = 0. Doing so, one gets∂t d̂k = i(

−1/2

κk
2
k − 4Uk |αk | ) −2††d̂k − 2iUk αk2 d̂k†− 2iUk (d̂k d̂k + αk∗ d̂k + 2αk d̂k )d̂k − iJ d̂l −

.

(D3a)

(C14)
The purity from QBE is higher than the state of the DPA forη > 1/2 and always tends to unity as η → 1 for any value ofλ/κ.
For a given intracavity squeezing, the optimal equal-time
intensity correlation is equal to
=1−
g (2)
opt
QBE

2
e4rQBE + 2e2rQBE − 1

,

(C15)

obtained at the optimal displacementᾱ 2opt = 14 η(e4rQBE − 1).

∂t αk = ik −κkαk − 2iUk |αk |2 αk − iJ αl − iF δk,1 .
2(D3b)

From there, we drop the nonlinear terms in the quantum
equations of motion Eq. (D3a). This then leads toκkd̂k∂t d̂k = i( k − 4Uk |αk |2 ) −2√†− 2iUk αk2 d̂k − iJ d̂l − κk ξ̂k ,
(D4a)

(C16)

0= i

QBE

k −

κk
αk − 2iUk |αk |2 αk − iJ αl − iF δk,1 .
2

The optimal g (2) (0) is lower with QBE for η > 1/2. The purity
and optimal g (2) (0) coincide at η = 1/2, i.e., for QBE = κext .
APPENDIX D: SOLUTION OF THE QUANTUM DYNAMICS
OF THE TWO COUPLED CAVITIES
1. Linearized equations of motion

In this section, we give details about the linearization of theequations of motion of the two coupled cavities [see Eq. (13)]and the technique used to solve them. As is standard for optical√κk ξ̂k ,(D4b)
The coupled nonlinear classical equations are straightforward
to solve, but the solutions are too cumbersome to be shown
here. In order to solve the quantum counterpart of the
Heisenberg-Langevin equations, one can go into the frequency
domain, defining
 ∞
 ∞
†
†
iωt
d̂k [ω] ≡d̂k (t)e ,d̂k (t)eiωt (D5)d̂k [ω] ≡063824-8−∞−∞

ANTIBUNCHING AND UNCONVENTIONAL PHOTON . . .

PHYSICAL REVIEW A 90, 063824 (2014)(and the same for ξ̂k [ω]). In the frequency domain, Eq. (D4a) becomes
√

d̂[ω]
= − κχ [ω]ξ̂ [ω],

(D6)

†
†
†
†

with d̂[ω]= [d̂1 [ω],d̂2 [ω],d̂1 [ω],d̂2 [ω]] , ξ̂ [ω] = [ξ̂1 [ω],ξ̂2 [ω],ξ̂1 [ω],ξ̂2 [ω]] , and the susceptibility⎡⎤−1iJ
2iU1 α120κ1 /2 − i( 1 + ω)⎢iJ⎥
02iU2 α22κ2 /2 − i( 2 + ω)⎢⎥
χ [ω] ≡ ⎢⎥ .
⎣−2iU1 (α1∗ )2⎦0
κ1 /2 − i(− 1 + ω)
−iJ
∗ 2
0
−2iU2 (α2 )−iJ
κ2 /2 − i(− 2 + ω)

In terms of the χ [ω] matrix, one has [see Eq. (5)]

2. Quantum master equation†nk ≡ d̂k (t = 0)d̂k (t = 0)
 ∞dω=(|χk,3 [ω]|2 + |χk,4 [ω]|2 )
−∞ 2π

(D7)

(D8)

and
sk ≡ |d̂k (t = 0)d̂k (t = 0)|

 ∞


dω
∗
∗

(χk+2,3 [ω] χk,3 [ω] + χk+2,4 [ω] χk,4 [ω]).
=
2π

The quantum dynamics of the two coupled cavities can besolved numerically by finding the steady state of the density
matrix ρ̂ of the system. The time evolution of the density
matrix is governed by the Hamiltonian Ĥ of Eq. (13) andthe Lindbladian L̂ of the Markovian environments at zerotemperature,∂t ρ̂ = −i[Ĥ ,ρ̂] + L̂ρ̂.

(D13)

L̂ = κ1 D̂[â1 ] + κ2 D̂[â2 ](D14)

The Lindbladian

−∞

(D9)
†

Finally, using Eq. (D4a), one can express d̂2 [ω] in terms†of only d̂1 [ω], d̂1 [ω], and noise operators and then formallyeliminate cavity 2 from the equation of motion for cavity 1. It
leads to (the frequency dependence of the operators is implicit
for clarity)d̂1 = i(ω + 1 ) − κ1 /2 − iJ 2 GR2 [ω] d̂1
2J 2 U2 α22†
2R− i 2U1 α1 −G [ω] d̂1
ω − 2 + iκ2 /2 2√√
2U2 α22†R− κ2 J G2 [ω] ξ̂2 +ξ̂ − κ1 ξ̂1 ,ω − 2 + iκ2 /2 2

8×10−7 (a)4×10−70
4.12 (b)
4.08
4.04
4
0.0008 (c)0.0004with1
ω+

4U22 |α2 |4
2 + iκ2 /2 + ω− 2 +iκ2 /2

.

(D11)

From Eq. (D10), one sees that one of the net effects of cavity
2 is to generate an additional parametric interaction in cavity
1. If we compare with the DPA [see Eq. (B1)], one can define
the total parametric interaction strength λ1tot as
λ1tot ≡ 2U1 α12−(ω −2J 2 U2 α22.
24
2 + iκ2 /2)(ω + 2 + iκ2 /2) − 4U2 |α2 |(D12)As discussed in the main text, for the parameters used in Fig. 4,
the main source of squeezing comes from cavity 2.

ᾱ
ᾱopt
g (2) (0)01.2 (d)0.8
0.4
0−13 (e)(D10)

GR2 [ω] ≡

r
ropt
θ
2ϕ

6×10
4×10−132×10−13
0

0

0.02

0.04

U/κ

sinh2 r
n̄eﬀ
0.06
0.08

0.1

FIG. 5. (Color online) Steady-state solution of the quantum master equation Eq. (D13) for cavity 1: (a) Squeeze parameter r1 andoptimal value ropt1 ; (b) squeezing angle θ1 compared to twice thedisplacement argument ϕ1 ; (c) displacement ᾱ1 and optimal valueᾱopt1 ; (d) intensity correlation at equal times g1(2) (0); (e) effective
photon number n̄eff1 and sinh2 r1 . The parameters are plotted against
U ≡ U1 = U2 for κ1 = κ2 ≡ κ and the same parameters as in Fig. 4.
The vertical line is placed at the minimal value of the intensity
correlation, and coincides with r = ropt , θ = 2ϕ, and ᾱ = ᾱopt .

063824-9

LEMONDE, DIDIER, AND CLERKPHYSICAL REVIEW A 90, 063824 (2014)is expressed in terms of the damping rates κ1,2 and thedissipator superoperator [37],D̂[â]· = â · â † − 12 {â † â,·}.

withtan 2φ =

(D15)
†

2J
1−

(D17)

,
2





1
2 + 2 iκ2 F


,
1
1
2
1 + 2 iκ1
2 + 2 iκ2 − J

The steady-state field â1 , photon number â1 â1 , squeezing â12 , equal-time intensity correlation g1(2) (0), and the parameters of the corresponding displaced squeezed thermal state
for the cavity 1 are plotted in Fig. 5 as functions of the interaction strength U . The results are in very good agreement withthe linearized theory, especially around the minimum of the intensity correlation. As predicted from the linearized theory, thisminimum coincides with the optimal condition for the squeeze
parameterr = ropt ≈ ᾱ 2 /(1 + 2ᾱ 2 ) (for r1), and the displacement ᾱ = ᾱopt . Moreover, considering the full nonlinear
dynamics, we see that θ depends on U and that the minimumof g (2) (0) coincides with amplitude squeezing, i.e., θ = 2ϕ.
For large tunnel coupling and drive, it is more convenient
to work with the normal modes b̂1 and b̂2 that diagonalize the
Hamiltonian without nonlinearity. These normal modes aredefined so as to eliminate the linear and tunneling terms,     
â1
cos φ
sin φ
b̂1
α1
=
,
(D16)
+
â2
α2
− sin φ cos φ b̂2

The Hamiltonian then reads1†Ĥ = − 1 cos2 φ + 2 sin2 φ + J sin 2φ b̂1 b̂1
2


1
†
2
2
−
1 sin φ + 2 cos φ − J sin 2φ b̂2 b̂2
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α1 = 
α2 = 

+2


1
1 + 2 iκ1

† †Uk âk âk âk âk .



JF
1
2 + 2 iκ2



− J2

.

(D18)
(D19)

(D20)

k=1

One has finally to express the Kerr nonlinearities as well as
the Lindbladian in terms of the b̂k .

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ANTIBUNCHING AND UNCONVENTIONAL PHOTON . . .

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