Synchronizing the Smallest Possible System

Alexandre Roulet and Christoph BruderDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

We investigate the minimal Hilbert-space dimension for a system to be synchronized. We firstshow that qubits cannot be synchronized due to the lack of a limit cycle. Moving to larger spinvalues, we demonstrate that a single spin 1 can be phase-locked to a weak external signal of similarfrequency and exhibits all the standard features of the theory of synchronization. Our findings relyon the Husimi Q representation based on spin coherent states which we propose as a tool to obtaina phase portrait.

Introduction.– The universal concept of synchroniza-tion applies to an impressively large variety of appar-ently unrelated systems. This ranges from the pendulumclocks of Huygens to heart beats, including light pulsesemitted by fireflies as well as an applauding crowd [1].The hallmark of this phenomena is the possibility to ad-just the phase of a self-sustained oscillator by perturbingit with an external periodic signal [2], or coupling it toone or more self-sustained oscillators of similar frequen-cies [3]. They key component for this is the absence of apreferred phase of the oscillator, which can thus be lockedwhile the limit cycle remains essentially intact.

The Van der Pol model [1, 4], one of the workhorsesfor theoretical studies of this effect has recently been for-mulated in the quantum framework, providing an excel-lent platform for studying the impact of quantum effectsonto synchronization [5, 6]. Building on the rapid exper-imental progress [7–10], an exciting direction is to studyhow synchronization emerges in a large network of suchquantum oscillators [11], similar in spirit to the classi-cal model proposed by Kuramoto [12]. However, whenmoving from a classical to a quantum framework, one isinevitably confronted with the exponential growth of theHilbert space as opposed to the linear classical scaling,which limits the tractable size of an arbitrary network ofquantum oscillators.

In this Letter, we approach this issue by addressingthe fundamental question of the minimal elementary unitwhich can be synchronized, using the Hilbert-space di-mension as a natural measure of size. While at first sightqubits and their associated Bloch representation appearto be the ideal candidate for the smallest possible sys-tem [13–16], we show that they lack a valid limit cycleand therefore do not fit into the standard paradigm ofsynchronization [1]. However, we find that the formalismof synchronization can be applied to the next smallestsystem: a single spin 1. To obtain a phase portrait appli-cable to spins in which we can identify the phase variablethat lies at the core of the synchronization formalism,we choose spin coherent states, which undergo an oscil-lation similar to the closed curve followed by the coher-ent states of a harmonic oscillator in position-momentumspace [17, 18]. In particular, we demonstrate how thespin formulation of the Husimi Q representation can beused to visualize the limit cycle and provide signaturesof synchronization.

Qubit.– A natural attempt to find the smallest possi-ble system to synchronize is to consider a single qubit.It comes equipped with a standard representation of the2-dimensional Hilbert space Hqubit known as the Blochsphere. Any unitary applied to a qubit can be visual-ized as a rotation of some angle about some direction.As a consequence, for a generic Hamiltonian pointing in

the direction Hqubit ∝ ~n · ~σ where the vector of Pauli

matrices is ~σ = (σx, σy, σz), any state different from theeigenstates | ± ~n〉 will rotate linearly in time about thedirection set by the unit vector ~n. This geometrical pic-ture provides the necessary phase variable for addressingthe question of synchronization, with the aim being tolock this oscillation in the Bloch sphere to an externalsignal.

To make contact with the standard paradigm of syn-chronization, we first need to establish a valid limit cyclefor the self-sustained oscillator. Specifically, adding lossand gain to the dynamics of the qubit, we look for afixed point of the dissipative map that does not possessany phase preference. That the phase of the limit cycleneeds to be free is a sine qua non condition which ensuresthat any perturbation neither grows nor decays, which isthe essence of synchronization [1].

In the case of a qubit, any state belonging to the space

Hqubit can be written as ρqubit = (1 + ~m · ~σ)/2 where|~m| ≤ 1 is maximized for pure states which lie on thesurface of the Bloch sphere. This implies that the set ofstates invariant under rotations with respect to the axis~n satisfy ~m = λ~n with −1 ≤ λ ≤ 1. In other words,they correspond to probabilistic mixtures of the eigen-states | ±~n〉, which are lying exactly on the rotation axiswhere the phase variable is not defined. Any attemptwith qubits is thus bound to fail due to the absence of aphase-symmetric state that is different from the extremaleigenstates. When put into the context of a Van der Poloscillator, this would mean being only able to stabilizethe vacuum state, which remains at the center of thephase space and does not provide a valid limit cycle forsynchronization. Indeed, the vacuum state is lacking self-sustained oscillations, which correspond to a stable closedcurve in phase space with an associated phase variable.

Spin S > 1/2.– We now turn our attention to higher-spin systems. Since the large-spin limit can be mappedto a harmonic oscillator [19], it is interesting to ask whatis the minimal value of S required to observe the onset of

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synchronization. To investigate this case, we first need topropose an extension of the Bloch sphere representationto spins S ≥ 1/2. The well-known harmonic-oscillatorcoherent states can be extended to spin systems and areobtained by rotating the extremal state [17, 18],

|θ, φ〉 = exp(−iφSz) exp(−iθSy)|S, S〉, (1)

where the spin operators are generators of the rotationgroup SO(3) satisfying [Sj , Sk] = iεjklSl. Note thatthis parametrization in terms of a pair of Euler anglesθ and φ corresponds, in the case of a qubit, to the Blochsphere representation. Moreover, we can easily show thatthe time evolution of spin coherent states under the freeHamiltonian

H0 = ~ω0Sz (2)

is given by e−iH0t/~|θ, φ〉 = |θ, φ + ω0t〉, which corre-sponds to an oscillation with frequency ω0 as desired. Wethus have a family of states resembling a classical spin in a

unit sphere pointing in the direction 〈θ, φ| ~S|θ, φ〉 = ~nθ,φwhere ~nθ,φ = (cosφ sin θ, sinφ sin θ, cos θ), and precess-ing about the z axis. This analogy can be made morerigorous by noting that these states form an overcom-plete basis of the Hilbert space. Specifically, neighbor-ing directions have an overlap given by |〈θ′, φ′ |θ, φ〉|2 =[(1 + ~nθ,φ · ~nθ′,φ′)/2]2S , which vanishes in the limit ofa classical spin S → ∞, and the completeness relationreads ∫ π

0

dθ sin θ

∫ 2π

0

dφ |θ, φ〉〈θ, φ| = 4π

2S + 11. (3)

This motivates us to introduce the spin equivalent ofthe Husimi Q representation for discussing synchroniza-tion in spin systems, which is defined for any state ρas [20]

Q(θ, φ) =2S + 1

4π〈θ, φ|ρ|θ, φ〉. (4)

While it is normalized, this function should not be under-stood as a probability distribution, since the three com-

ponents of the spin operator ~S do not commute. Sim-ilar to other quasiprobability distributions such as theWigner function, it is employed to provide a phase por-trait, which in this case allows us to visualize any spinstate in terms of the most “classical” states [21]. More-over, it is easily computed with the use of the Wigner D-

matrix DSm′,m(φ, θ, 0) = 〈S,m′ |e−iφSze−iθSy |S,m〉 which

provides a representation of rotation operators in termsof the spin eigenbasis.

Limit cycle.– We will now apply this tool to studysynchronization in the context of a single spin 1. Thekey difference with the qubit case is the existence ofan additional phase-symmetric state |1, 0〉 that is notjust a combination of the extremal eigenstates |1,±1〉.Its Husimi Q representation Q(θ, φ) = 3 sin2(θ)/8π is

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(a)

|1, 1i

|1,�1i

|1, 0i

!0 S�Sz

!0 S+Sz

�d

�g

(b)

FIG. 1. The limit cycle. (a) The Q function of the targetstate |1, 0〉 represented using the Winkel tripel earth projec-tion. The north and south poles correspond respectively toθ = 0 and θ = π while φ specifies the longitude. The φ-symmetric distribution centered around the equator is rem-iniscent of the ring stabilized in the Wigner representationof a Van der Pol limit cycle [6]. (b) The spin-1 ladder, withthe eigenstates separated by the natural frequency ω0. Theaction of the dissipators is to transfer the populations fromthe extremal states towards the target state.

shown in Fig. 1(a) using the Winkel tripel projection of asphere [22]. It is independent of φ and centered aroundθ = π/2, corresponding to coherent states on the equatorprecessing about the z axis. The most straightforwardapproach to stabilize this target state is via the followingmaster equation (in a frame rotating with H0) illustratedin Fig. 1(b),

˙ρ =γg2D[S+Sz]ρ+

γd2D[S−Sz]ρ, (5)

where γg and γd are the respective gain and damping

rates, and D[O]ρ = OρO† − 12

{O†O, ρ

}is the Lindblad

superoperator [23]. The dynamics generated by this dis-sipative map is not only attracting any state towards theequator, thereby stabilizing the energy, but does so with-out any phase preference. This is a consequence of themaster equation being invariant under rotations aboutthe z axis. We have thus established a valid limit cycle,which we may now attempt to synchronize.Synchronization.– We model the external signal by a

semi-classical drive of strength ε at frequency ω, de-scribed by the Hamiltonian in the rotating-wave approx-imation

Hsignal = i~ε

2(eiωtS− − e−iωtS+). (6)

Moving to a frame rotating with the drive frequency, thesystem is thus described by the master equation

˙ρ = −i[∆Sz + εSy, ρ] +γg2D[S+Sz]ρ+

γd2D[S−Sz]ρ, (7)

where ∆ = ω0−ω is the detuning between the frequencyof the autonomous oscillator and the drive. The steadystate can be obtained analytically, yet its expression israther uninformative as such. Instead, Fig. 2 illustratesthe power of the Q function to provide signatures of syn-chronization. In particular, focusing first on the reso-nant scenario we find that phase locking is achieved with

3

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0

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(a)

0 0.05 0.1

⇡

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0

Q(✓,�)

(b)

⇡

⇡

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0

Q(✓,�)0 0.05 0.1

(c)

FIG. 2. Phase locking to a resonant (∆ = 0) external signal represented by the steady-state Q function. The signal strengthis chosen as ε = 0.1γmin where γmin = min(γg, γd), ensuring that we are in the weakly-perturbed regime. (a) γg/γd = 0.1,γg = 10ε, the distribution is localized around φ = 0 while remaining approximately on the equator. (b) γg/γd = 1, γg = 10ε,the limit cycle is slightly distorted with no visible signature of phase locking. (c) γg/γd = 10, γd = 10ε, same as (a) but thedistribution is now localized around φ = π. Thus, phase locking is possible when one of the rates dominates.

equal or opposite phases when one of the rates domi-nates. This is one of the main results of our Letter. Theπ phase difference between these two situations can bephysically understood by noting that they are equivalentunder time reversal with the relabelling |1, 1〉 ↔ |1,−1〉.Intriguingly, the transition regime where the rates arecomparable shows no visible locking.

To better understand these different synchronizationregimes, we define the following synchronization measure,

S(φ) =

∫ π

0

dθ sin θ Q(θ, φ)− 1

2π, (8)

which vanishes everywhere in the absence of synchro-nization. On the other hand, any non-uniform distri-bution of the phase will lead to a positive maximum ofthis measure, indicating a locking to the correspondingphase [24, 25]. We now take advantage of the small di-mension of the Hilbert space and derive analytically thetime evolution of this measure using the master equa-tion (7). Synchronization effects being of first order inthe perturbation [1], we perform a first-order expansionin the signal strength, yielding

S(φ) =3ε

16(e−γgt/2−e−γdt/2) cos(φ−∆t)+O

([ε

γmin]2),

(9)where γmin = min(γg, γd). This expression allows to com-pute the phase distribution at any time given the ini-tial condition S(φ) = 0 ∀φ, corresponding to the phase-symmetric limit cycle (see Fig. 1).

The first observation is that indeed, no synchroniza-tion is expected in the balanced case γg = γd where

S(φ)→O(

[ εγmin

]2)

. The weak signal is not able to adjust

the phase, and increasing the power further will start todeform the limit cycle, thereby leaving the paradigm ofsynchronization. When one of the rates dominates, sayγd (γg), the exponential prefactor contributes positively

(negatively) over a timescale of order γ−1g (γ−1d ). Thedynamics is then that of synchronization to a phase ref-erence that is linearly varying in time for non-zero detun-ing. When |∆| � γmin, this phase reference is effectively

constant and leads to significant in-phase (anti-phase)locking. This is the situation illustrated in Fig. 2, forwhich the steady-state distribution is given from Eq. (9)by S(φ) ≈ 3ε

8 (1/γg − 1/γd) cos(φ). As the detuning is in-creased, the spin stays phase-locked to the shifted phasereference up to a regime |∆| � γmin where it cannotfollow its fast rotation. The effect of synchronization isthen averaged out, cos(φ−∆t)→ 0, as the signal is toooff-resonant to adjust the phase.

This behavior is illustrated in Fig. 3(a), where we showhow the distribution S(φ) is dragged along as the refer-ence phase is rotated by the detuning ∆. Note that theamplitude of the peak is also decreasing, as expected fora fixed signal strength ε. To further characterize this de-pendence, we now fix the detuning in Fig. 3(b), and findthat S(φ) tends to be more and more localized as we in-crease the signal strength. This is characteristic of syn-chronization in the presence of noise, where a strongersignal is able to reduce the tendency of phase slips in-duced by the fluctuations of the phase. This can be seenin Eq. (9), which predicts that the synchronization dy-namics is characterized by a rate of order ε. As the sig-nal strength approaches γmin, we thus expect the phaselocking to be more and more pronounced. These featurescan be nicely summarized by the classic Arnold tongue,shown in Fig. 3(c).

Figures 3(b) and (c) appear to indicate that increasingthe signal strength ε can only improve the resulting phaselocking. Yet, we expect a breakdown of this scaling fromthe standard theory of synchronization [1]: on increas-ing the signal beyond the weakly-perturbed regime, theenergy of the self-oscillator starts to be affected and thestability of the limit cycle is compromised. This is shownfor ∆ = 0 in Fig. 4, where we find that as the strengthenters the gray region ε > 0.1γmin, the energy of the spinis not stabilized anymore and the Q function starts tomove away from the equator, i.e. from its natural limitcycle.

Discussion.– The framework developed here providesa natural platform to explore how synchronization scaleswith the dimension of the Hilbert space. In particular,

4

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FIG. 3. The distribution of the phase S(φ) as a measure of phase locking for γg = 0.1γd. (a) The phase locks to differentvalues when the detuning ∆ of the signal is varied. This is indicated by the dashed line which follows the peak maxS(φ) of thedistribution. (b) Increasing the signal strength ε leads to a more pronounced phase locking. (c) The Arnold tongue. Significantphase-locking is achieved over a broader range of detuning ∆ by increasing the signal strength ε. The tongue does not touchthe bottom axis due to the presence of noise.

��-� ��-� ��-� �-���

-���

���

���

���

�g/�d = 0.1

�g/�d = 10

FIG. 4. Influence of the signal strength ε onto the limitcycle for zero detuning, ∆ = 0. In the grey-shaded area, theeffect of the signal is not limited to a perturbation of the freephase φ but instead substantially deforms the limit cycle byincreasing or decreasing the energy of the spin, depending onthe ratio γg/γd. The corresponding Q functions are shown forε = γmin.

one may conjecture an improvement of the phase lockingas the spin approaches the classical limit and becomesless susceptible to quantum noise. To investigate this di-rection, we apply the master equation (7) to the case ofa spin 2. We find that the spin synchronizes but witha weaker phase locking: maxS(φ) = S(0) ≈ 0.001 atresonance for ε = 0.1γg and γg = 0.1γd. This is to becompared with the value obtained for a spin 1 with thesame parameters: maxS(φ) = S(0) ≈ 0.032. This re-sult highlights a key open question in the field of quan-tum synchronization: given a target limit cycle, what isthe optimal dissipative map that will yield the strongestphase locking once a signal is applied? For a generic spinnumber, the dynamics described by (5) is just one outof the many possibilities for stabilizing the limit cycle tothe equator. For instance, an alternative choice in thehigh-spin limit would be to reproduce the Van der Polstructure of a linear gain and non-linear damping.

An advantage of the spin unit introduced here is that itcan be implemented in a wide variety of physical systems.Specifically, any effective spin-1 ladder provides a suit-able basis, such as that obtained in trapped ions [26, 27]or NV-centres [28]. The main requirement is the possibil-ity to stabilize the target state |1, 0〉, thereby providingthe limit cycle to be synchronized. Incoherent gain anddamping, for instance of the form of (5), could be exper-imentally engineered by using the multi-level structureof these systems [29, 30]. The signal (6) can then beimplemented by a laser drive.Conclusion.– We have shown that the smallest possible

system that can be synchronized is a single spin 1. Wehave demonstrated how the general theory of synchro-nization applies to this quantum system with no classicalanalogue. This was achieved by relying on the Husimi Qrepresentation, which we propose as a powerful tool tostudy synchronization in spin systems.

The system presented here leads to many interestingresearch directions. In particular, it can be used as anelementary building block for studying synchronizationof limit cycles in a network of quantum spins, minimiz-ing the dimension of the resulting Hilbert space. Thefirst step towards this direction would be a pair of spins,for which the interaction could be modeled by a directeffective Hamiltonian or a mediating bosonic field. Ona more fundamental level, studying how synchronizationscales with the spin number leads to the open questionof half-integer spins greater than 1/2 which do not haveaccess to an eigenstate centered around the equator.

ACKNOWLEDGMENTS

We would like to thank R. Fazio and N. Lorch for dis-cussions. This work was financially supported by theSwiss SNF and the NCCR Quantum Science and Tech-nology.

5

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