Testing time reversal symmetry in artificial atoms

Testing Time Reversal Symmetry in Artificial Atoms

Frederico Brito,1, ∗ Francisco Rouxinol,2 M. D. LaHaye,2 and Amir O. Caldeira3

1Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, C.P. 369, Sao Carlos, SP,13560-970, Brazil2Department of Physics, Syracuse University, Syracuse New York 13244-1130, USA

3Instituto de Fısica Gleb Wataghin, Universidade Estadual de Campinas-UNICAMP, 13083-859 Campinas, SP, Brazil

Over the past several decades, a rich series of experiments has repeatedly verified the quantumnature of superconducting devices, leading some of these systems to be regarded as artificial atoms.In addition to their application in quantum information processing, these ‘atoms’ provide a testbed for studying quantum mechanics in macroscopic limits. Regarding the last point, we presenthere a feasible protocol for directly testing time reversal symmetry in a superconducting artificialatom. Time reversal symmetry is a fundamental property of quantum mechanics and is expected tohold if the dynamics of the artificial atom strictly follow the Schrodinger equation. However, thisproperty has yet to be tested in any macroscopic quantum system. The test we propose is based onthe verification of the microreversibility principle, providing a viable approach to verify quantumwork fluctuation theorems - an outstanding challenge in quantum statistical mechanics. For this, weoutline a procedure that utilizes the microreversibility test in conjunction with numerical emulationsof Gibbs ensembles to verify these theorems over a large temperature range.

Few concepts in nature are so simple and yet as pro-found as those related to symmetry. Indeed, the beautyof its manifestations has led to the modern view thatprinciples of symmetry dictate the forms of nature’s fun-damental laws[1], embodying striking implications thatrange from conservation principles to the classification ofelementary particles.

Time reversal symmetry (TRS) is a prominent ex-ample that underlies a large variety of phenomena. Inmany instances, the fundamental microscopic laws of na-ture are invariant under time reversal transformations.This invariance is at the heart of microscopic reversibil-ity (microreversibility)[2], which itself is crucial to pow-erful concepts such as the principle of detailed balance[3],the fluctuation-dissipation theorem[4], and the Jarzynskiequality[5], to name a few.

Yet TRS is not an exact symmetry of nature: in thevery least, it is observed to be broken in elementary pro-cesses that involve the weak force[6, 7], and moreover,there is evidence to suggest that it must also be violatedover a much broader range of conditions in order to ac-count for the prevalence of matter over anti-matter inthe universe[8, 9]. Manifestations of such violations po-tentially herald new phenomena and are thus the subjectof extensive experimental investigations in both atomicand particle physics[9].

While considerable effort has been invested in thesearch for violations of TRS in the interactions of funda-mental particles, experiments have not been conductedto investigate TRS in the physics of quantum systemsat the macroscopic scale. Specifically, the question thusarises: Once one properly takes into account dissipativeand decoherence effects, would TRS be observed in, say,a mesoscopic or even a macroscopic device? On the faceof it, there is no reason to expect that it breaks down: weknow the microscopic laws of quantum mechanics can beapplied to at least some (properly prepared) macroscopic

systems. Nonetheless, if it does break down, this must re-flect new physics, which could have potential connectionsto open questions like the nature of the quantum-classicaldivide[10, 11].

With these thoughts in mind, we delineate here a pro-tocol for directly testing TRS at the macroscopic scale inan artificial atom that is based upon a superconductingquantum device (SQD). While similar types of SQDsare known for their use as qubits in the development ofquantum computing architectures[12, 13], we propose toutilize an SQD as a multi-level artificial atom to test aspecific manifestation of TRS, namely the principle ofmicroreversibility.

Microreversibility and the artificial atomGenerally speaking, the principle of microreversibilitystates that for each state of motion that is accessible to agiven system, there is an equally probable time-reversedstate also accessible to it. In the context of quantummechanics, it manifests in a simple relationship for thetransition probabilities between any two states of a sys-tem whose Hamiltonian has undergone a time-dependenttransformation[14], namely that

Pn|m[λ] = Pm|n[λ] (1)

where Pn|m(Pm|n) is the probability for the system tomake a transition to state |m〉 (|n〉) when it starts in state|n〉 (|m〉). Here λ represents the forward-in-time transfor-mation of the system’s Hamiltonian and λ represents themotion-reversed process (Fig. 1a).

Equation 1 is a fundamental and general result for non-dissipative quantum mechanical systems, deriving fromthe invariance of a system’s Hamiltonian under trans-formations by the anti-unitary time-reversal operator Θ[2, 14]. Thus it should hold true for all quantum systemsin which TRS is maintained. Naively, one would expectthis to include macroscopic systems for which the laws of

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quantum mechanics have been shown to apply, such asmechanical quantum systems[15, 16] and superconduct-ing cavities, circuits and devices[12, 13, 17–19]. However,a direct test of TRS in these systems has yet to be per-formed. As we show now, it should be technologicallyfeasible to perform a test of microreversibility, and henceTRS, via Eq. 1, in an artificial atom based upon an SQD.

The SQD here is a Cooper-Pair box (CPB), whichin our proposal consists of a nanofabricated supercon-ducting island (or box) that is formed by a pair ofJosephson junctions in a DC SQUID configuration (Fig.1b). The system is well-characterized by the followingHamiltonian[20]

H = 4EC

∑n

(n− ng)2 |n〉 〈n|

−∑n

[EJ(Φ)

2|n〉 〈n+ 1|+ E

∗J(Φ)

2|n+ 1〉 〈n|

]. (2)

The first term on the right-hand side of Eq. 2 repre-sents the electrostatic energy of the CPB for a givencharge state n (a discrete index labelling the number ofCooper-pairs on the island) and continuous polarizationcharge ng on a nearby electrode; the pre-factor EC isthe total charging energy of the CPB. The second termon the right in Eq. 2 represents the mixing of chargestates due to the Josephson coupling of each junction.

Here EJ(Φ) ≡ EJΣ

{cos(π Φ

Φ0

)+ iα sin

(π Φ

Φ0

)}is the

total Josephson energy of the two junctions; observe that|EJ(Φ)| is periodic in applied magnetic flux Φ with aperiod of one flux quantum Φ0. To account for asym-metry between the junctions, we define the parameterα ≡ (EJ1 − EJ2)/EJΣ, where EJΣ ≡ EJ1 + EJ2 is thesum of the individual junction Josephson energies EJ1

and EJ2.It is important to note that numerous experiments over

the past 15 years have shown that the two parameters Φand ng in Eq. 2 can be tuned in situ for experimen-tal implementation of unitary operations with the CPB[12, 13]. The proposal we put forth for testing Eq. 1exploits this coherent control. Specifically, it relies uponthe adjustment of Φ and ng to modify the characteristicsof the CPB’s energy eigenstates. To understand how thismight work, observe that, when Φ is adjusted so that |EJ |is relatively small (i.e β ≡ |EJ |/(4EC) � 1), and ng isadjusted near an integer, the eigenstates of the systemare essentially the charge states |n〉. On the other hand,if β & 1, or ng is near a half-integer, then the eigenstatesare no longer well-defined charge states, but instead areweighted superpositions of multiple values of |n〉. Thus,through the rapid tuning of Φ and ng, the CPB can beforced to undergo unitary evolution between various su-perpositions of charge states. Through repeated projec-tive measurements of the CPB’s charge state before (n)and after (m) identical forcing protocols, the transitionprobabilities Pn|m between any given pair of charge states

FIG. 1. Schematics of TRS for driven (nonau-tonomous) quantum systems and the Artificial Atom.a, While the unitary time evolution of the forward-in-timeprotocol takes the initial state |Ψi〉 to the evolved state |Ψf 〉 =UF [λ] |Ψi〉, the motion reversed state follows the dynamics∣∣ΨB

f

⟩≡ Θ |Ψi〉 = UB [λ]Θ |Ψf 〉, where Θ represents the time

reversal operator. In this generalization, Φ(t) and ng(t) rep-resent time-dependent parameters in the system’s Hamilto-nian which are tuned to change the state of the system. Ifa system parameter depends upon an applied magnetic field,then the field must be inverted to move from the Forwardto the Backward protocol as shown schematically with Φ(t).b, SQD based upon a CPB, used to implement the artificialatom in our protocol. The system dynamics can be controlledby adjusting the magnetic flux Φ(t) through the loop and thecharge ng(t) = CgVg(t)/2e on a nearby electrode, where Cg isthe capacitance of the CPB to the electrode and Vg(t) is an ex-ternally controlled voltage.The device features two Josephsonjunctions (red boxes), arranged in parallel, interrupting theloop. The physical dimensions assumed here are such that thegeometrical inductance is negligible compared to the Joseph-son inductances, leading to the Hamiltonian Eq. (2), where:EC/~ = 2π × 3GHz, EJΣ/~ = 2π × 10GHz, and α = 0.05.

|n〉 and |m〉 in the spectral decompositions of the initialand final states can be constructed.

Forward and backward protocolsOur specific proposal to test Eq. 1 is outlined in Figs.

3

FIG. 2. Protocol scheme and system’s eigenenergiesversus time. a-b, Outline of the Forward and Backwardprotocols. The first step, the Preparation Protocol, is usedto construct an initial ensemble of several charge states. Af-ter the first measurement is performed, a driving protocolis used to implement a forward and backward-in-time evolu-tion, which is followed by another charge measurement. Aftermany runs, the transition probabilities between the allowedinitial and final states can be constructed. c-d, The time for-ward (solid line) and backward (dashed line) drive protocolsfor the gate charge ng and flux Φ. In order to maintain thetime reversal symmetry, the sign of the magnetic field must beinverted. e, The eigenenergies of the CPB as function of timeduring the driving protocol (the ground state energy is setzero). It is worth noticing the presence of several avoided levelcrossings, where Landau-Zener transitions are induced. Theeigenenergies are calculated for the same parameters statedin Fig. 1.

2a-d. It involves the application of two separate proto-cols to the CPB to measure Pn|m[λ] and Pm|n[λ], whichwe refer to as the forward (λ) protocol and the back-ward (λ) protocol respectively. For process λ, the CPBis taken through the following sequence of steps: (1)First with the CPB initialized in its ground state, apulse sequence consisting of the simultaneous applica-tion of time-varying signals ng(t) and Φ(t) (Figs. 2c-d)is applied to CPB, causing it to repeatedly pass throughavoided-level crossings in its energy spectrum (Fig. 2e).At each such crossing, the CPB can undergo a Landau-Zener transition[21, 22] between the adjacent states in-volved in the crossing, which leaves it in a superpositionof those two states. For the parameters considered here,after traversing the multiple crossings shown in Fig. 2e,the CPB state should be in a superposition of as many as5 charge states. (2) Immediately after the initial super-position state is prepared, a projective measurement ofthe CPB’s charge is made and recorded as state n. Werefer to those steps as the Preparation Protocol (Figs.2a-b), since they provide an effective way for preparing

an initial ensemble of charge states which can be usedfor measuring Pn|m -in our case, its composition is givenin (Fig. 3a). (3) Next, after the collapse to the chargestate |n〉, a second pulse sequence identical to the se-quence in step (1) is applied, again preparing the CPBin a superposition of charge states. (4) Finally, a sec-ond projective measurement of the CPB charge state ismade and recorded as m. After step (4), the CPB is al-lowed sufficient time to relax back to its ground state,after which time λ is repeated. In this manner, repeat-ing λ many times, the transitions probabilities Pn|m[λ]can be constructed. Figure 3b illustrates a histogram ofPn|m[λ] for this process calculated with numerical simu-lations using Eq. 2 and the pulse sequences in Figs. 2c-d(See Methods section).

To implement the time-reversed process λ and con-struct the corresponding transition probabilities Pm|n[λ],the same general procedure as outlined in the previousparagraph is followed. However, it is necessary tochange two physical quantities for the time-reversedprocess: First, the sign of the magnetic flux applied tothe CPB must be reversed to account for the reversalof momentum of the magnetic field’s source charges.Observe that such inversion leads to EJ(−Φ) = E∗J(Φ).Second, one should also invert the sign of the appropriatecanonical variable of the CPB during λ, which in thiscase turns out to be the effective phase difference ϕacross the CPB’s Josephson junctions. This step isnecessary to preserve the time-reversal invariance of thecanonical charge-phase commutation relations, sincecharge is considered an invariant under TRS. In ourparticular case, inverting the sign of ϕ has the effect ofconjugating EJ . Therefore, as one would expect, apply-ing those two changes leaves the system Hamiltonianinvariant. In addition, since we have a time-dependentHamiltonian (nonautonomous system)[14], we also haveto revert the forcing protocol applied to the system,i.e., |Φ(t)| → |Φ(−t)| and ng(t) → ng(−t). With thesechanges, numerical simulations of the backward protocolindeed predict that Eq. 1 should hold (Fig. 3c) (SeeMethods section).

Conditions for unitarity and the measurementprotocolTo claim a true test of TRS through verification of mi-croreversibility (Eq. 1), it is essential that the CPB’stime evolution be predominantly unitary during the λand λ protocols. This requires that the protocols be im-plemented on a time scale τp that is much faster thanany environmental effects. By applying the methodologyintroduced by Burkard-Koch-DiVincenzo[23], one findsthat the figure of merit for quantifying such effects in ourprotocol is the relaxation time T1. It can be shown thatthe decoherence time T2 is determined by T1 (T2 ∼ 2T1),except for the regime β � 1, which corresponds to atiny window of ∼ 0.2ns in the protocol, during which

4

!!

61.3%

7.6%4.1% 5.5%

21.5%

|n = 2i|n = 1i|n = 0i|n = �1i

|n = �2i

Ensemble charge spectral

decompositionForward Protocol Backward Protocol

(a) (b) (c)

60.2%

9.1%

4.8%

5.3%

20.5%

7.3% 39.9%

12.8%

1.2%38.8%

4.2%13.5%

32.1% 45.9%

4.4%

5.6%0.7%

45.5%

36.4%

11.8%

22.6% 36.8%

4.7%11.2%

24.5%

|n = 2i|n = 1i

|n = 0i|n = �1i

|n = �2i

|m = �2i

|m = �1i

|m = 0i|m = 1i

|m = 2i

20.5%

5.3%

4.8%9.1%

60.2%

38.8%

1.2%

12.8%39.9%

7.3%

4.4%45.9%

32.1%

13.5%

4.2%

11.8%36.4%

45.5%

0.7%

5.6%

24.5%

11.2%

4.7%36.8%

22.6%

|n = 2i|n = 1i

|n = 0i

|n = �1i

|n = �2i

|m = �2i|m = �1i

|m = 0i|m = 1i

|m = 2i

Pm|n[�]Pn|m[�]

61.3

7.64.1 5.5

21.5

FIG. 3. Compiled probability distributions for preparation, forward and backward protocols. a, The chargeensemble distribution (%) prepared after starting the system in the ground state and performing the Preparation Protocol. Forthe parameters considered here, the spectral decomposition obtained is predominantly (> 99.9%) comprised of charge states{−2,−1, 0, 1, 2}. b(c), The probability transitions Pn|m (%) between the initial |n〉 and final |m〉 charge states determined for

the Forward λ (Backward λ) protocol. The leakage probability of leaving the charge subspace {−2,−1, 0, 1, 2} is determined tobe . 0.1%. Observe that microreversibility demands comparing columns of (b) with rows of (c). The spectral decompositionand transition probabilities are calculated for the same parameters stated in Fig. 1.

T2 ∼ 0.02T1 (See Methods section). Thus, even for amodest T1 of 50ns, which is readily achievable with cur-rent technology[24], the designed protocol with τp ∼ 1ns(Figs. 2a-b) should provide a satisfactory unitary evolu-tion.

It is also important that the projective charge mea-surements are made within a time-scale τmeas � T1.That T1 sets the relevant time-scale can be understoodby realizing that decoherence effects becomes innocuousif one chooses projective measurements in the eigenen-ergy basis, since such effects would not lead to changesin the system state eigenenergy spectral decomposition.Notice from Fig. 2d that our protocol complies withthis case: at the end of a protocol, when a projectivemeasurement of charge is made, the CPB is biased sothat the charge states are quasi-eigenenergy states of thesystem (i.e. β � 1). Hence these measurements shouldalso each be performed on a time scale τmeas . 10ns.A natural and viable possibility for performing suchhigh-speed, high-sensitivity charge measurements wouldbe to use a superconducting single electron transistor(SSET)[25–27]. When operated in RF mode, SSETs canhave bandwidth in excess of 100 MHz [25] and chargedetection sensitivity approaching the limit allowed byquantum mechanics [26, 27]. Indeed, assuming thedetection sensitivity achieved in Ref. [26], it should bepossible to resolve the CPB’s charge state with an errorof ∼ 0.5% in a time scale of tmeas ∼ 20ns (See Methodssection).

Gibbs ensemble emulation and quantum workfluctuation relationIn addition to testing TRS in new macroscopic quan-

tum limits, the investigations that we have outlined herewould have implications for at least one contemporaryavenue of investigation: quantum work fluctuation theo-rems. Indeed, to derive these theorems, it is sufficientto make two hypotheses: the microreversibility prin-ciple and the assumption that the system is initiallyin thermal equilibrium at temperature T (a Gibbsiandistribution)[14].

One paradigm of such work fluctuation relations is thequantum Bochkov-Kuzovlev equality[28]

〈e−W/kBT 〉 = 1 (3)

which states that the work distribution of any drivingprotocol applied to a system initially in thermal equi-librium at temperature T is a random quantity, whoseexpected value does not depend on the details of eitherthe system or the driving protocol. To test Eq. 3 usingan SQD would require running an experiment at very lowtemperature (T ∼ 30mK), in which case the SQD’s ini-tial thermal state is predominantly the population of thesystem ground state. Unfortunately this leads to verypoor statistics for Eq. 3. In principle, this problemcould be solved by just increasing the system tempera-ture, but in order to obtain a Gibbsian distribution com-prised of a reasonable number of states, e.g., 5 states,one should perform the experiment at T ∼ 1K, whichcertainly would destroy the unitary evolution requiredfor microreversibility to hold.

Here, we envision a solution to this problem by con-structing a thermal state out of the initial charge en-semble obtained in the preparation protocol, which wename an emulation of an initial Gibbs ensemble. Theprocedure consists of randomly selecting the outcomes of

5

the first measurement following the probability rule im-posed by the Boltzmann weight exp(−H/kBT ). If thenumber of experimental events is sufficiently large, suchdistribution can be obtained for a given temperature T .Indeed, as Table I shows, for 106 events, one can emulatea Gibbs ensemble comprised of those 5 states for temper-atures above 1K, and verify with a small statistical errorthe quantum Bochkov-Kuzovlev equality (See Methodssection).

TABLE I. Emulation for 106 Events

Temperature (K) 1− 〈e−W/kBT 〉1 (−0.4± 5.8)× 10−2

10 (−2.7± 7.9)× 10−4

20 (−2.1± 4.2)× 10−4

30 (−0.6± 3.0)× 10−4

40 (−0.1± 2.2)× 10−4

50 (−0.3± 1.7)× 10−4

In the present paper we have created a protocol forthe preparation, time evolution and measurement of thequantum state of an SQD in order to test TRS in a newregime using state-of-the-art technology and techniques.Our numerical simulations show that the repeated appli-cation of this protocol to the SQD would enable verifi-cation of the microreversibility principle. An immediateapplication of this result is the confirmation of the fluctu-ation relations for which microreversibility and an initialGibbsian distribution are necessary conditions. Using theresults of our simulations, we show how to fulfil this latterrequirement by emulating a possible initial equilibriumstate of the system, which together with microreversibil-ity allowed us to numerically verify one of the possibleforms of the fluctuation relations, namely, the Bochkov-Kuzovlev equality.

Although one could argue that our results are for-mally expected, their experimental observation wouldbe of utmost importance and the reason is threefold.Firstly, it would provide the first direct test of themicroreversibility of transitions between the states ofa macroscopic quantum system. Secondly, operatingon a single system, one would be able to emulatethe results one would have obtained starting from anensemble of systems in thermodynamic equilibrium, andconsequently confirm the quantum fluctuation relationswithout actually raising the temperature of the system.Finally, if microreversibility is indeed observed in thiskind of system it may constitute additional possibleevidence to rule out alternative theories for quantummechanics such as the dynamical reduction models[29].Since the latter assumes a non-linear stochastic dynamicsto macroscopic observables of a given system, it cannotbe time reversal invariant, contrary to our predictions.

MethodsNumerical simulationsThe system’s state time evolution was determined throughnumerical simulations of the unitary time-ordered evolutionoperator due to the Hamiltonian Eq. 2. The calculation wasperformed taking into account an N = 51 charge dimensionalHilbert space. Considering the time discretization procedure andthe Hilbert space truncation, we estimated a maximum relativeerror of ∼ 0.05% for the probabilities quoted in the main text.The specific flux and charge pulses used in our protocol read:Φ(t) = (Φ0/2) cos(2π× 3

2× t) and ng(t) = 0.05− 2 cos(2π× 3

2× t),

with time in unit of nanoseconds.

Decoherence and relaxation rates for the CPBThe methodology introduced by Burkard-Koch-DiVincenzo[23] al-lows one to use circuit theory for describing the dissipative elementsof the circuit with a bath of oscillators model, from which it is pos-sible to calculate the dissipative effects for multilevel superconduct-ing devices. From this modelling, the system-bath coupling derivedis a functional of the charge number operator n. Therefore, it willonly connect the system energy eigenstates that have at least onecharge state in common in their spectral decomposition. For thephysical CPB parameters and the flux and charge protocols consid-ered in our proposal, we found that only neighbouring eigenstatesshare one charge state in their spectral decomposition. Thus, witha good approximation, the dissipative process can be viewed as a se-quence of multiple processes involving only two eigenstates. Underthis perspective, one can obtain the relaxation (T1) and decoher-ence (T2) times concerning those two levels. In the Born-Markovapproximation, the relation between T1 and the pure dephasing Tφis found to be

Tφ

T1∼

4| 〈ek|n |ek+1〉 |2

| 〈ek|n |ek〉 − 〈ek+1|n |ek+1〉 |2ek+1 − ek

2kBTcoth

ek+1 − ek2kBT

,

where ek is the instantaneous value of the eigenenergy state k,and T−1

2 = T−11 /2 + T−1

φ . Performing the calculation of the

matrix elements above for each time instant of our protocol,we found that the decoherence time T2 is determined by T1,i.e., Tφ � T1, except for the regime β � 1, during whichT2 ∼ 0.02T1. Observe that for β � 1 (the SQD charge regime),the charge operator n almost commutes with the system Hamil-tonian, which explains why T1 becomes the longest time scale here.

Estimate of the CPB measurement uncertaintyTo estimate the CPB charge-state measurement uncertainty,we first assume that the CPB is probed using an SSET that iscoupled to the CPB through a capacitance CC . It is furtherassumed that the SSET charge sensitivity SQ is dominated bythe noise of the pre-amplifier used to read-out the SSET. In thiscase, for each Cooper-pair number state N , the inferred charge(QC) on CC will have a Gaussian distribution pN (QC) withR.M.S. of σQ =

√(SQ/τmeas), where τmeas is the measurement

time. We then define the measurement uncertainty through theuse of the Kolmogorov (trace) distance [30], which is given byD(pN (QC), p(N+1)(QC)) = (1/2)

∫|pN (QC)− p(N+1)(QC)|dQC .

The probability to correctly identify from which of two adjacentprobability distributions pN (QC) and p(N+1)(QC) an outcome ofa measurement QC comes is thus given by PD = (1 + D)/2. Forexample, using SQ

1/2 = 1.7µe/√

Hz, which was achieved in [26],a measurement time τmeas = 20ns, and realistic parameters forthe total capacitance of the CPB island CΣ = 6.5fF(correspondingto EC/~ = 2π × 3GHz) and CC = 0.20fF, we find PD = 99.5%,corresponding to a measurement uncertainty of 0.5%.

Quantum work and the Gibbs ensemble generatorThe quantum Bochkov-Kuzovlev equality Eq. 3 is derived[28] con-sidering the exclusive viewpoint for the definition of quantum work.Such definition considers that the quantum work performed in aspecific process λ is determined as the difference of the outcomes ofthe eigenenergy measurements of the unperturbed system Hamil-tonian done at the initial and final process times. In our case,

6

the Hamiltonian Eq. 2 can be viewed as H(t) = H0 + Hp[λ(t)],where the unperturbed Hamiltonian H0 is set as H(t = 0), andthe force-dependent Hamiltonian perturbation Hp[λ(t)] is given byH(t)−H0.

The Gibbs ensemble emulation is constructed using a standardpseudorandom routine to select the states out of the initialensemble obtained from the preparation protocol. The pseudo-random choice is weighted by the Boltzmann weight for a giventemperature T .

AcknowledgementsFB and AOC are supported by Instituto Nacional deCiencia e Tecnologia - Informacao Quantica (INCT-IQ)and by Fundacao de Amparo a Pesquisa do Estado deSao Paulo (FAPESP) under grant number 2012/51589-1.MDL and FR acknowledge support for this work pro-vided by the National Science Foundation under GrantNo. DMR-1056423 and Grant No. DMR-1312421.

Author contributionsAll authors contributed equally to this work.

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