Measuring correlations of cold-atom systems using multiple quantum probes

Michael Streif,1, 2, ∗ Andreas Buchleitner,2 Dieter Jaksch,1, 3 and Jordi Mur-Petit1, †

1Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom2Physikalisches Institut, Albert-Ludwigs-Universitat Freiburg,

Hermann-Herder-Straße 3, 79104 Freiburg, Germany3Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore

(Dated: 10 October 2016)

We present a non-destructive method to probe a complex quantum system using multiple impurityatoms as quantum probes. Our protocol provides access to different equilibrium properties of thesystem by changing its coupling to the probes. In particular, we show that measurements withtwo probes reveal the system’s non-local two-point density correlations, for probe-system contactinteractions. We illustrate our findings with analytic and numerical calculations for the Bose-Hubbard model in the weakly and strongly-interacting regimes, under conditions relevant to ongoingexperiments in cold atom systems.

PACS numbers: 05.70.Ln, 67.85.-d 03.67.AcKeywords: quantum probing; strongly-correlated materials; non-equilibrium dynamics

I. INTRODUCTION

Different phases of matter are fundamentally associ-ated with different correlations among their constituents.These correlations can be encoded in various observables.For example, the ground state of a one-dimensional,single-component Fermi gas has the same density profileas a one-dimensional system of strongly repulsive bosons(Tonks-Girardeau gas), while their momentum distribu-tions are markedly different [1]. This stems from the factthat the momentum distribution contains further infor-mation on the two-particle correlations, which also affectother observables such as the excitation spectrum andthe structure factor of quantum systems [2, 3]. Whiletraditionally one could only access these properties viabulk measurements, e.g., neutron scattering off liquid he-lium, the advent of setups based on cold atoms in opti-cal lattices has opened up new possibilities. For exam-ple, the measurement of local two-particle correlations ina one-dimensional gas of bosonic atoms for various in-teratomic (repulsive) interaction strengths was found [4]to be in excellent agreement with theoretical calcula-tions [5–7]. Measurements of the momentum distribu-tion [8] and non-local density-density correlation func-tion [9] of one-dimensional bosons in a periodic potentialhave also been performed, and they agree with theoreticalfindings [8, 10]. More recently, Np-point non-local corre-lation functions up to Np = 10 between two quasi-one-dimensional Bose gases, were measured by matter-waveinterferometry [11]. These results underpin the necessityto account for conserved quantities in the description ofthe non-equilibrium evolution of quantum systems [12–16].

Common to all these experiments is that they use de-structive measurements to study the quantum systems,

∗ michael.streif@physics.ox.ac.uk† jordi.murpetit@physics.ox.ac.uk

most frequently the time-of-flight technique, where thetrapping potential is switched off and the system allowedto expand before light absorption images are recorded.Based on the development of new measurement and con-trol methods, such as the quantum gas microscope (whichenables access to quantum lattice systems with single-siteresolution) [17–21], an alternative approach is advancingwhich considers the use of other quantum objects, suchas photons, single atoms or ions as non-destructive quan-tum probes of many-body quantum systems [22–32].

The idea of using single quantum probes—which oftenare equipped with the simplest possible internal quan-tum structure of a qubit—has been implemented to inferdiverse properties of the host substrate, from Frohlich po-larons, to work statistics and quantum phase transitions,to the Efimov effect and more [33–44]. Yet it is clear thata single qubit probe in general cannot suffice to map outthe host’s characteristic properties exhaustively, since theprobe-system coupling and the thus defined local densityof states will generally limit the probe’s diagnostic hori-zon to a finite subset of the system’s Hilbert space. It istherefore natural to seek a systematic generalization ofthe quantum probe approach to larger numbers of probes,such as to complement the finite diagnostic power of asingle probe, e.g., by directly monitoring spatial correla-tions.

In the present contribution, we make a first step inthis direction by considering two impurities embeddedinto a host bosonic gas [45]. Specifically, we show thatthe coherence of a two-probe density matrix enables usto access the two-point correlation function of a strongly-correlated quantum system in a non-destructive way. Westart in Sec. II with a general presentation of our two-probe protocol. In Sec. III we study a specific model ofbosonic particles in a lattice, the Bose-Hubbard model(BHM), and show that our protocol enables us to deter-mine the average system density as well as the two-pointdensity-density correlation function, both in the super-fluid and in the insulating phases of the BHM. Finally,

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in Sec. IV, we conclude with a summary of our findingsand an outlook.

II. TWO-PROBE PROBING PROTOCOL

We consider a quantum system, S, coupled to twoprobes, which we label as L (for left) and R (right). TheHamiltonian of the composite system can be written as

Htot =HS ⊗ 1L ⊗ 1R + 1S ⊗ HL ⊗ 1R

+ 1S ⊗ 1L ⊗ HR + Hint , (1)

where HS is the Hamiltonian of the system and acts onthe Hilbert space HS , Hα (α = L,R) is the Hamilto-nian of the left (right) probe acting on its corresponding

Hilbert spaceHα, and Hint is the interaction Hamiltonianbetween the system and the two impurities and thereforeacts on Htot = HS ⊗HL ⊗HR.

We model the probes as two-level systems (qubits),and couple them separately to the system, so that theinteraction Hamiltonian reads

Hint = VSL ⊗(gL0 |0〉L 〈0|L + gL1 |1〉L 〈1|L

)⊗ 1R

+ VSR ⊗ 1L ⊗(gR0 |0〉R 〈0|R + gR1 |1〉R 〈1|R

).

(2)

Here, we have indicated the internal states of each probequbit by |0〉α , |1〉α, respectively, and the parameters gαqdescribe the interaction between the system and qubitα = L,R when in state q = |0〉 , |1〉.

Our probing protocol starts with the qubits uncoupledfrom the system, gαq(t = 0) = 0. The compound initialstate reads ρtot(t = 0) = ρS ⊗ |Φ+〉 〈Φ+|, i.e., with thetwo qubits not entangled with the system, and preparedin the Bell state |Φ+〉 = (|00〉+ |11〉) /

√2, with the usual

notation |00〉 = |0〉L ⊗ |0〉R and similarly for |11〉. Thisentangled state can be prepared from both qubits initiallyin the ground state |0〉 and then subjected to a Hadamardgate acting on the left qubit followed by a controlled-NOT gate (with the left qubit as control and the rightas target) [46].

At time t = 0, a unitary non-equilibrium evolutionis driven by changing the coupling of one of the in-ternal states of the qubits with the system, e.g., byusing a Feshbach resonance. For concreteness, we setgL0(t) = gR0(t) ≡ g(t) = 1 for t > 0, while keep-ing gL1(t) = gR1(t) = 0. The state of the compositesystem then evolves under the time evolution operator

U(t) = T e− i~∫ t0dt′Htot(t

′), where T is the time-orderingoperator, so that after a time t the composite system isin the state ρtot(t) = U(t)ρtot(0)U†(t). A trace over thesystem degrees of freedom yields the reduced density ma-trix operator of the two qubits, ρQ(t) = TrS(ρtot(t)). Wefocus our interest on the non-diagonal coherence element,whose time evolution can be expressed as 〈11|ρQ(t)|00〉 =

Δ c

a

a

FIG. 1. Schematic of a Bose gas (blue shading) in an opticallattice (black line). Two ancillary two-level quantum systems(circles) are coupled to the Bose gas at distinct sites on thelattice, separated by a distance ∆c in units of the lattice con-stant a.

14e−

i~ t∆ζ(t). Here, the exponential factor accounts for

the free evolution in terms of the energy splitting betweenthe internal states of the two probes, ∆ = E|11〉 − E|00〉;without loss of generality, we set this energy differenceto zero, i.e. ∆ = 0. The function ζ(t) characterizes thecoherence element’s time dependence due to the qubits’coupling to the system; we will refer to it as the coher-ence function. Note that ζ(t) will generally depend on thedistance between the probes, ζ(t) = ζ(t; ∆c) (cf. Fig. 1),which we indicate explicitly where necessary.

The moments of the interaction Hamiltonian deter-mine the derivatives of this coherence function [42]. Forexample,

dζ(t)

dt

∣∣∣∣t=0

=i

~〈Hint〉 , (3)

d2ζ(t)

dt2

∣∣∣∣t=0

= − 1

~2〈H2

int〉 , (4)

where the expectation values on the right hand sides arecalculated with ρtot(t). It follows that measurements ofζ(t) permit us to access several equilibrium expectationvalues of the system. These expectation values can berelated to observables of interest by a suitable choice ofthe interaction between probes and system. Below, weshow that, in particular, for contact probe-system inter-actions, measurements of the coherence function providea way to determine the density [Eq. (9)] and the two-point density correlation function [Eq. (10)] of the hostsubstrate.

Representing the internal state of each qubit as aspin operator, and using the Pauli spin matrices, σi(i = x, y, z), the real and imaginary parts of ζ(t) canbe written

Re(ζ(t)) =1

2〈σx ⊗ σx − σy ⊗ σy〉t (5)

Im(ζ(t)) =1

2〈σx ⊗ σy + σy ⊗ σx〉t, (6)

where the bracket 〈·〉t represents a trace overρQ(t). Thus, ζ(t) can be experimentally deter-mined by measuring the two-qubit correlation func-tions which enter Eqs. (5) and (6). Alternatively,

3

one can express ζ(t) in the Bell basis as Re(ζ(t)) =2 (ρQ,++(t)− ρQ,−−(t)), Im(ζ(t)) = 4 Im (ρQ,+−(t)),with ρQ,++(t) = 〈Φ+|ρQ(t)|Φ+〉 and analogously for

ρQ,−− and ρQ,+−, with |Φ−〉 = (|00〉 − |11〉)/√

2. Itfollows that ζ(t) can also be determined with Bell-statemeasurements.

III. APPLICATION TO THE BOSE-HUBBARDMODEL

We now apply the protocol described in Sec. II to thecase of N cold bosonic atoms loaded into the lowest en-ergy band of an optical lattice with M sites, describedby the Bose-Hubbard Hamiltonian [47, 48]

HS = −J∑〈i,j〉

a†i aj +U

2

M∑i=1

a†i a†i aiai + µ

M∑i=1

a†i ai . (7)

The operator a†i (ai) creates (annihilates) a boson at alattice site i = 1, . . . ,M , the index 〈i, j〉 indicates sum-mation over nearest neighbor pairs, and the parametersU , J , and µ are the on-site interaction energy, the hop-ping energy and the chemical potential, respectively. Weare interested in the translationally invariant system, i.e.,in the limit {N →∞, M →∞} with fixed average den-sity n = N/M .

We now account for both probe impurities by a cou-pling mediated via a contact density-density interactionpotential,

VSα =

∫dxnα(x)Ψ†(x)Ψ(x) , α = L,R (8)

where Ψ(x) =∑j wj(x)aj is the bosonic field annihila-

tion operator of the system, with wj(x) the lowest energyWannier function at lattice site j = 1, . . . ,M , and nα(x)the density of qubit α at position x. Assuming that bothimpurities are strongly localized at distinct lattice sites(jL and jR), we find that they interact with the Wan-nier function of that very site only. Thus, the interactionterm can be written in terms of the boson number op-

erators at these sites, VSα = ηαa†jαajα , the parameter

ηα = J∫

dx|wα(x)|2nα(x) being a measure of the in-teraction strength between the bosons and the qubit atsite jα. For simplicity, we assume that the local interac-tion strengths at both probe locations are identical, i.e.,ηL = ηR ≡ η.

Substitution of Eq. (8) into Eq. (2), together withEq. (3), yields the expectation value of the interaction’scontribution to the total Hamiltonian which, due to thespecific form of VSα, is equal to the bosonic density

ρ(j) = a†j aj at site j:

2ρ = 〈ρ(j) + ρ(j + ∆c)〉 =~iη

dζ(t; ∆c)

dt

∣∣∣∣t=0

, (9)

For the first equality, we used that, for translationallyinvariant systems, 〈ρ(j)〉 = 〈ρ(j + ∆c)〉 ≡ ρ, with the

integer ∆c = jR−jL the distance between the two qubitsin units of the lattice constant a (see Fig. 1). (In anexperiment, this can be accomplished by trapping thetwo qubits in a separate optical lattice formed by crossingtwo laser beams; the inter-qubit distance ∆c can thenbe precisely tuned by changing the angle between thepropagation directions of the beams; see, e.g., [49].)

Similarly, using Eq. (4), we find the bosonic density-density correlation function Cor(∆c) = 〈ρ(j)ρ(j + ∆c)〉in terms of the qubits’ coherence function:

〈[ρ(j) + ρ(j + ∆c)]2〉 = −~2

η2

d2ζ(t; ∆c)

dt2

∣∣∣∣t=0

. (10)

Again, given the system’s translational invariance,〈ρ(j)2〉 = 〈ρ(j + ∆c)2〉, the last expression can be rewrit-ten as

Cor(∆c) =~2

2η2

d2

dt2

[1

2ζ(t; ∆c = 0)− ζ(t; ∆c)

]∣∣∣∣t=0

.

(11)

This result implies that measurements of the qubits’coherence function ζ(t) provide access to the system’sdensity-density correlation function. We remark that thisresult depends on the qubits-system coupling, Eq. (8),but not on the specific form of the system HamiltonianHS beyond its translational invariance. In the followingsections, we assess the experimental feasibility of our pro-tocol by simulating the outcome of the protocol in boththe superfluid (U/J � 1) and the insulating (U/J � 1)phases of the one-dimensional Bose-Hubbard model, andcomparing them with exact results for Cor(∆c) in bothlimits.

A. Weak interactions: Superfluid phase

In the regime of weak interactions (U/J � 1), wecan use Bogoliubov theory [50] to calculate both thecoherence function ζ(t) and the density-density cor-relation function Cor(∆c) analytically (see also [31]and [51]). We start from the Bose-Hubbard Hamilto-nian, Eq. (7), for a one-dimensional system of homoge-neous density n. We first transform the annihilation op-erators from the site basis, ai, to the momentum basis,

bk = (M)−1/2∑j aje

ikaj , and similarly for the creation

operator b†k. A Bogoliubov transformation to quasipar-

ticle operators, dk = uk bk + vk b†−k, brings the system

Hamiltonian into the diagonal form HS =∑k ~ωkd

†kdk,

with dk (d†k) the annihilation (creation) operator of Bo-goliubov quasiparticles of quasi-momentum k, and ωk =√εk(εk + 2Uρ) the quasiparticle dispersion relation in

terms of the single-particle energies εk = 2J(1−cos (ka)),with a the lattice constant and ρ the bosonic density [52].

With this transformation, we rewrite the density ma-trix of the lattice bosons by expressing the bosonic oper-

4

0 2 4 6 8 10 12 14 16 18 20∆c

-0.2

0

0.2

0.4

0.6

0.8

1g(2

) (∆c)

Protocol βJ=10Bogoliubov βJ=10Bogoliubov βJ=100

FIG. 2. Weak interactions. Normalized correlation functionobtained with Eq. (11) simulating Nexp = 104 experiments(crosses), compared with the analytic results Eq. (13) (lineswith symbols), for a system initially at equilibrium at inversetemperature βJ = 10 and βJ = 100. Other parameters usedare η = 0.4J and U/J = 0.1.

ators in terms of Bogoliubov quasiparticle operators

ρj = a†j aj =1

M

∑k,k′

b†k bk′eikaje−ik′aj

= ρ+

√N0

M

∑k

√εkωk

(d†ke

ikaj + dke−ikaj

). (12)

In the second line, we have applied Bogoliubov’s approx-imation, i.e., we assume that the occupation of k 6= 0modes is small [(N − N0)/N � 1], and neglect termsof quadratic (or higher) order in quasiparticle opera-tors [50, 52]. By inserting Eq. (12) into the definitionof the two-point density correlation function, we reachthe following analytic expression valid in the weakly-interacting limit

Cor(∆c) = ρ2 +ρ

M

∑k

εkωk

(2nthk + 1) cos (ka∆c) , (13)

where we have evaluated the occupations of the Bo-

goliubov modes in a thermal state, 〈d†kdk〉 = nthk =

1/(eβ~ωk − 1), with β the inverse temperature, and we

have dropped the anomalous averages 〈dkd−k〉, as theyare negligible at the low temperatures where the Bo-goliubov approximation applies [53]. At zero tempera-ture (β → ∞), Eq. (13) satisfies the sum rule estab-lished in Ref. [54] for density-density correlations in theground state, which re-expresses the sum rule relatingthe dynamic structure factor to the static structure fac-tor, which in turn is sensitive to two-body interactions inbosonic lattice systems [55].

Based on Eq. (13), we plot in Fig. 2 the normalizedsecond-order correlation function,

g(2)(∆c) =〈ρ(0)ρ(∆c)〉 − 〈ρ(0)〉 〈ρ(∆c)〉

〈ρ(0)〉2, (14)

tJ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

ζ(t)

0.8

0.85

0.9

0.95

1

Re(ζ(t))

Parabolic FittJ

0 1 2 3

ζ(t)

0

0.5

1

Re(ζ(t))

Im(ζ(t))

FIG. 3. Real part of the coherence function with added Gaus-sian noise (symbols) and a parabolic fit (solid line). Inset:Real (solid blue line) and imaginary (dashed red line) partsof ζ(t), Eq. (16), for the case ∆c = 5. Here, we used a sys-tem with M = 1000 lattice sites and N = 1000 bosons; and,therefore, an average density ρ = 1. The system is initially ina thermal state with βJ = 10; other parameters as in Fig. 2.

as a function of the inter-probe distance ∆c for dif-ferent temperatures. We see that, for all tempera-tures, the correlation vanishes for distances beyond afew lattice sites, which agrees with the picture that, inthe non-interacting limit, the system is effectively de-scribed by a product of on-site coherent states so that〈ρ(0)ρ(∆c)〉 = 〈ρ(0)〉 〈ρ(∆c)〉 [56]. Weakly-interactinghomogeneous one-dimensional Bose gases also convergeto this limit fairly quickly [57].

We proceed now to compare these analytic calculationswith the estimation by means of the coherence func-tion ζ(t). To evaluate the right-hand side of Eq. (11),we rewrite the system-qubit interaction Hamiltonian interms of Bogoliubov operators,

V = VSL + VSR = 2ρη +∑k

(η?kd†k + ηkdk

), (15)

where ηk = η√nεk/Mωk(e−ikajL + e−ikajR) is the cou-

pling strength of the qubits with the Bogoliubov mode ofquasi-momentum k. Substituting these expressions intoHint allows us to calculate analytically the time evolutionof the composite system and, therefore, to determine thecoherence function ζ(t). Full details of the derivation arereported in Appendix A (see also [45]); here we quoteonly the final result,

ζ(t) = e2iηρt exp

[−i∑k

|ηk|2

ω2k

[ωkt− sin(ωkt)]

]

× exp

[∑k

(−2|ηk|2

ω2k

sin2 ωkt

2coth

βωk2

)]. (16)

In an experiment, the coherence function ζ(t) canonly be measured at discrete times, tr. In addition, for

5

each time tr, the expectation value defining ζ(tr) is ob-tained upon accumulation of repeated measurements ofthe qubits’ state, with individual measurement outcomesexhibiting quantum (shot) noise. To simulate this un-avoidable spread of experimental measurement events,and to estimate how many measurements one would needfor their statistical average to converge to the expectationvalue, we follow the scheme in Ref. [41] and add Gaussiannoise to the calculated values of ζ(tr); see Appendix Bfor details on how to determine the corresponding vari-ance. As one would do in an experiment, to reduce theensuing uncertainty in ζ(tr), we repeat the simulated ex-periment a number Nexp of times and average over alloutcomes, for each inter-probe distance ∆c. The valuesof ζ(tr) estimated in this way are presented in Fig. 3 fora system with average density ρ = 1. Here, one can notethat the real part of ζ(t) has a parabolic dependence ontime, while the imaginary part is linear around t = 0.It follows that the second derivative will be real, in ac-cordance with our expectations for the density-densitycorrelation function [cf. Eq. (11)]. Thus, in practice itsuffices to measure only the real part of ζ(t), Eq. (5).

Given the smooth character of ζ(tr), we fit a quadraticpolynomial through these values, which enables us to cal-culate the right-hand side of Eq. (11) and determine thetwo-point correlation function. We show the correspond-ing results for βJ = 10 in Fig. 2, which are in fair agree-ment with the analytic result (13). In particular, wesee that the value of g(2)(0) derived from the protocolshows the characteristic enhancement of the superfluidphase. Reducing the statistical uncertainty of g(2)(∆c)for ∆c � 1 requires a relatively large number of mea-surements Nexp, in line with previous experimental de-

terminations of g(2)(∆c) in cold atomic setups [58, 59]. Inthe framework of the present two-probe protocol, thesefluctuations, and correspondingly Nexp, can be reducedby running in parallel an arrangement with Npairs pairsof probes in a double-well superlattice [60–63]; a setupwith Npairs = 100 probe pairs would reach the precisionshown in Fig. 2 with only 100 measurement runs.

B. Strong interactions: Insulating phase

For stronger interactions U/J & 1, the correlationsbetween the bosons in the lattice invalidate an approachbased on the Bogoliubov treatment. An efficient methodto deal with this situation is Tensor Network Theory(TNT), which provides numerically exact ground stateproperties of strongly-correlated systems, in particular,of the one-dimensional BHM [64, 65]. Here, we applythis method to calculate g(2)(∆c) in the ground stateof this model using the implementation Oxford TNT li-brary [66]. As we are interested in investigating non-local correlation functions, we choose a large system withM = 101 lattice sites, and ρ = 1 as before, and calculateg(2)(∆c) around the central lattice site so that boundaryeffects are negligible and the system can still be consid-

0 5 10 15∆c

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

g(2) (∆

c)

U=3JU=15JU=100J

FIG. 4. Correlation function g(2)(∆c) for the ground statein the strongly-interacting regime for different interactionsstrengths, U/J = 3, 15, and 100, as indicated. These pointsrepresent the numerically exact expectation values of num-ber operator pairs, (〈ρiρj〉 − 〈ρi〉 〈ρj〉) / 〈ρi〉2, from the TNTcalculation.

ered (approximately) translationally invariant. For thecalculations presented below, we have checked that suffi-cient accuracy is reached bounding the site occupation toa maximum of four bosons per site and fixing a truncationparameter (maximum number of Schmidt coefficients) ofχ = 100.

The TNT method allows us to calculate directlythe expectation values of the number operator ateach lattice site, 〈ρi〉, and all pairs of number op-erators, 〈ρiρj〉. From these, we obtain directly the

normalized two-point correlation function g(2)(∆c) =

(〈ρiρi+∆c〉 − 〈ρi〉 〈ρi+∆c〉) / 〈ρi〉2; the results for increas-ing values of U/J are shown in Fig. 4. As expected, inthe limit U/J → ∞ we recover that g(2)(∆c) = 0 ∀∆cas the ground state is a product of on-site Fock stateswith no density fluctuations [56, 67]. These results con-stitute the test-bed corresponding to the left-hand site ofEq. (11), which we will compare to the outcome of theprotocol to obtain ζ(t) and its derivatives.

We calculate ζ(t) in the strongly-interacting regimein the following way: The coherence function can bewritten as a trace over system operators only, ζ(t) =

TrS(U1(t)U0(t)ρβ) [cf. Eq. (A1)]. Here, U0(t) is theevolution operator over a time t with the initial systemHamiltonian, while U1(t) is the evolution operator includ-ing the coupling to the qubits. For probe qubits localizedat lattice sites and coupled to the bosons by contact in-teractions of strength η, the effect of the probe-bosoncoupling amounts to a local shift of the bosons’ chemicalpotential, µ → µ − η, at the sites where the probes arelocated. Thus, we can obtain ζ(tr) at different time steps

tr by calculating the expectation value TrS(U1(t)U0(t)ρβ)with ρβ the ground state of the bosonic system in a latticewith modified local potential at the probe sites.

In our numerical calculations, we take tr = r∆t with∆t = ~ × 0.01/J and r = 0, . . . , 20. As for the weakly-

6

(a)

0 0.1 0.2 0.3 0.4 0.5tJ

0

0.2

0.4

0.6

0.8

1

ζ(t)

Re(ζ(t))Im(ζ(t))

(b)

0 1 2 3 4 5 6 7 8 9 10∆c

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

g(2) (∆

c)

Simulated pointsTNT calculation

FIG. 5. Strong interactions. (a) Real (asterisks) and imagi-nary (crosses) parts of the coherence function ζ(t) for U/J =3. Other parameters are ∆c = 5, M = 101, ρ = 1, andη = J . (b) Normalized correlation function g(2)(∆c) for thesame parameters. The results of the measurement protocolwith Nexp = 104 (crosses) agree with the numerically exactvalues calculated with the TNT method (circles).

interacting regime, we simulate the uncertainty in an ex-periment by adding noise to each simulated data point,ζ(tr), and calculate the numerical second derivative att = 0. We repeat this procedure for all integer dis-tances between the two qubits 0 ≤ ∆c ≤ 15 (� M toavoid boundary effects). The coherence function, ζ(t)obtained in this way is shown in Fig. 5(a). We observethat both real and imaginary parts exhibit broadly a be-havior similar to that of the weakly interacting system.However, the correlation function that one obtains fromthis according to Eq. (11) is notably different, as shownin Fig. 5(b), where we compare the value of g(2)(∆c)obtained from the coherence function by using Eq. (11)with the numerically exact values derived from the TNTground state (the latter values are the same as those inFig. 4 for U/J = 3). We see that there is a good agree-ment between the two calculations, as happened in theweakly interacting regime. In particular, the estimationof the correlation function using our protocol is able todetect the reduction in g(2)(0) as the system gets deeperinto the Mott insulating phase, U/J � 1. To illustratethis point, we show in Fig. 6 the normalized correlation

10-1 100 101

U/J

-0.2

0

0.2

0.4

0.6

0.8

1

g(2) (∆

c)

Exact - ∆c=0Exact - ∆c=1Exact - ∆c=2

Protocol - ∆c=0Protocol - ∆c=1Protocol - ∆c=2

FIG. 6. Correlation function g(2)(∆c) for ∆c ∈ {0, 1, 2}and different values of U/J . Bogoliubov theory was used forU/J ≤ 0.4 with a βJ = 1000 thermal state (shaded region),and TNT for larger values of U/J . For clarity, we do not in-clude error bars for the ∆c = 2 calculation; they are similarto those for ∆c = 1.

function g(2)(∆c) at selected distances ∆c for differentvalues of U/J across the Mott insulator–to–superfluidtransition. First, we observe that the outcome of ourprotocol in each case is very close to the exact result(calculated with Bogoliubov theory for weak interactionsand with TNT for stronger interactions). Physically, thelocal correlation, g(2)(0), decreases steadily as the repul-sion between bosons increases, and it vanishes in the limitU/J � 1. Correlations at larger distances are negative(meaning, it is less probable to find a particle at distance∆c in the actual ground state than what one would pre-dict by relying only on the average density) and generallyof smaller magnitude than the local correlation; they alsovanish in the strongly repulsive limit, as expected for aMott insulator.

IV. DISCUSSION AND OUTLOOK

In this paper, we have developed a framework to studycorrelation functions in cold atom systems by using mul-tiple atomic impurities as quantum probes, a setup real-ized in recent experiments where potassium [27, 68, 69]or cesium [29, 34] atomic impurities were immersed inlarger rubidium Bose gases.

We have presented a protocol which is able to measurethe density-density correlations of the system relying onmeasuring the internal states of two probes and studyingan off-diagonal element of their reduced density matrix.We have shown that the results of this protocol agree withthose of analytic and numerically exact calculations for aone-dimensional Bose-Hubbard model in both the weaklyand the strongly interacting regimes. In particular, wehave shown that the protocol is able to witness the changein correlations across the superfluid–to–Mott insulatortransition.

7

Non-local density correlations in quantum gases havepreviously been measured by various methods, includ-ing noise interferometry, Bragg spectroscopy, and matter-wave interferometry. Let us briefly contrast our proposalwith these techniques. In Bragg spectroscopy, some ofthe atoms in the system are excited by two-photon Braggscattering into a state of given momentum and energy.This provides access to the dynamic structure factor ofthe gas, which is the Fourier transform of the densitycorrelation function [55, 70–72]. This method inevitablydestroys the initial quantum state of the system, in con-trast to our proposal, which is inherently non-destructiveand, thus, could permit a time-dependent monitoring ofthe evolution of correlations. In addition, our protocolcan be extended to using N > 2 quantum probes to de-termine N -point correlation functions.

Matter-wave interferometry [73] is a destructive mea-surement method especially suited to probing the phasestructure of bosonic quantum gases. As mentioned ear-lier, it has been used recently to measure density corre-lation functions up to 10th order between two quasi-one-dimensional bosonic gases [11]. However, the applica-tion of this method to higher-dimensional systems wouldrequire a rather involved analysis of the correspondingmulti-dimensional phase interference pattern. In con-trast, it is straightforward to see that our protocol appliesto systems of any dimensionality.

Noise interferometry retrieves information on particlecorrelations in atomic gases by analyzing the shot-to-shot fluctuations in absorption images of the system aftertime-of-flight evolution [9, 10, 74]. In strongly correlatedphases, where the time-of-flight technique is not suitable,one could implement noise interferometry by imaging theatoms with a quantum gas microscope [17–21, 75–82] toanalyze correlations in optical lattice setups. Our pro-posal constitutes a complementary approach of similarexperimental complexity, particularly suited to multi-component setups with impurities [27, 29, 34, 68, 69],with the distinctive feature of allowing non-destructivemeasurements.

The main challenge of our proposal may lie in thedynamical control of the probe-system coupling. Ma-nipulation via Feshbach resonances is an option if theseare available between the atomic species involved. Moregenerally, one could envisage probing a one- or two-dimensional gas by allowing the impurities to “fallthrough” it, pulled by gravity or driven by an externalfield. This would turn on and off the interactions withoutchanging the state of the system appreciably (given thatthere are many more atoms in the background than im-purities). This approach could be implemented exploit-ing existing experimental schemes in which the impuritiesare trapped near but outside the system and then driveninto it for a fixed amount of time [29, 34], or made topenetrate it periodically [83].

In summary, the framework presented here opens upnew possibilities for the experimental investigation ofquantum many-body systems and, especially, systems of

cold atoms in optical lattices. The protocol can be ex-tended in various ways, e.g., to estimate N -point cor-relation functions. Another possibility stems from thefreedom of choosing the kind of interaction Hamiltonianbetween qubits and system, different choices allowing oneto gain access to different observables. For example, byusing Raman transitions [84], the evolution of the probesbecomes sensitive to the phase of the matter wave andone could measure cross-correlation functions [85].

ACKNOWLEDGMENTS

The authors would like to thank J. J. Mendoza-Arenas,T. H. Johnson, and M. Mitchison for useful discussions.This work was supported by the EU H2020 FET Collabo-rative project QuProCS (Grant Agreement No. 641277),EU Seventh Framework Programme (FP7/2007-2013)Grant Agreement No. 319286 Q-MAC, and ErasmusPlacements (M.S.). D. J. thanks the Graduate Schoolof Excellence Material Science in Mainz for hospitalityduring part of this work.

Appendix A: Bogoliubov treatment of the weaklyinteracting system

We briefly expand on the explicit calculation of thecoherence function ζ(t) for weak interactions, with thehelp of Bogoliubov theory, and following the procedureoutlined in [41]. The first step is to introduce a set of pro-jection operators on the Hilbert space of the two qubits,

P11 = |11〉 〈11| , P10 = |10〉 〈10| ,P01 = |01〉 〈01| , P00 = |00〉 〈00| .

This enables us to rewrite the full Hamiltonian in a moreconvenient form.

Htot = P11 ⊗(E1 + HS + gL1VSL + gR1VSR

)+

+ P10 ⊗(E2 + HS + gL1VSL + gR0VSR

)+

+ P01 ⊗(E3 + HS + gL0VSL + gR1VSR

)+

+ P00 ⊗(E4 + HS + gL0VSL + gR0VSR

)As stated in the text, we are interested in the time evo-lution of the qubits only. Therefore, after calculating thetime evolution of the composite system, we trace out thedegrees of freedom of the bosons. After that, we concen-trate on the coherence element of the two-qubit densitymatrix, 〈11|ρQ|00〉. We find that the coherence functioncan be determined by calculating the expectation value

ζ(t) = TrS(U1(t)U0(t)ρS) (A1)

with the initial state of the system ρS . In this expression

U0(t) = T exp

(− i

~

∫ t

0

dt′HS

)

8

is the time evolution operator with the unperturbed sys-tem Hamiltonian, and

U1(t) = T exp

(− i

~

∫ t

0

dt′(HS + g(VSL + VSR))

)is the time evolution operator with the Hamiltonian in-cluding the coupling to the probes, where we have usedthat gL0 = gR0 = 1 and gL1 = gR1 = 0. It is worth notingthe similarity of ζ(t) to the Loschmidt echo [86, 87], whichis a function that enables us to characterize memory ef-fects in the dynamics of quantum systems (see, e.g., [88]).

For simplicity, we change into the interaction picture,where U0 = 1. The remaining time evolution operatorsimplifies to a more convenient expression:

U1(t) = T exp

(− i

~

∫ t

0

dt′Vint(t′)

).

Here, Vint(t) is the interaction part of the Hamiltonian inthe interaction picture,

Vint(t) = 2ρη +∑k

(η?ke

iωktb†k + ηk bke−iωkt

).

We can simplify the expression for U1 by applying theMagnus expansion [89]. To this end, we introduce an

operator A by

T exp

(− i

~

∫ t

0

dt′Vint(t′)

)= eA .

This operator can be expressed as a sum of operatorsA =

∑i Ai which are related to commutators of the in-

teraction Hamiltonian:

A1 = −i

∫ t

0

dt′Vint(t′)

A2 =1

2

∫ t

0

dt′∫ t′

0

dt′′[Vint(t′), Vint(t

′′)]

....

Given the form of Vint above, the commutators atdifferent times are c-numbers, [Vint(t

′), Vint(t′′)] =

−2i∑k |ηk|2 sin (ωk(t′ − t′′)). Therefore, all terms of the

expansion beyond the second term vanish. Thus, we canwrite the coherence function as

ζ(t) = e2iρηt exp

[−i∑k

|ηk|2

ω2k

[ωkt− sin (ωkt)]

]

× Tr

[exp

{−i∑k

(γkd†k + γ?k dk

)}ρS

](A2)

where we have defined γk =η?kωk

(eiωkt−1

i

). We are left

with the task of calculating the trace over the initial stateρS . A close investigation of this expression reveals that

the operator acting on ρS is a displacement operator,

D(α) = eαd†−α?d, for each Bogoliubov mode with corre-

sponding displacement iγk. Due to this and the commu-

tation relations of Bogoliubov operators, [d†k, dk′ ] = δk,k′ ,we can write the trace in the last line of Eq. (A2) as theexpectation value of a product of displacement operators

trace = Tr

[∏k

Dk(iγk)ρβ

]

=∑{nk}

∏k

〈ρk〉nk

(1 + 〈ρk〉)nk+1〈{nk}|

∏k′

Dk′(iγk′)|{nk}〉 .

Here, we have considered that initially the system is ina thermal equilibrium state at inverse temperature β, sothat ρS = exp(−βHS)/Z, with the partition function

Z = Tr[exp(−βHS)], and then used the diagonal repre-sentation of the thermal state in the Fock basis.

The action of a displacement operator on a Fock state|n〉 is to generate a displaced Fock state |n, γ〉. The re-maining overlap of two of these states can be expressedby [90]

〈n, γ|m,α〉

= 〈γ|α〉√n!

m!(γ? − α?)m−nLm−nn [(γ − α)(γ? − α?)] ,

where Lan(x) are the generalized Laguerre polynomialsand 〈γ|α〉 = exp[− 1

2

(|γ|2 + |α|2 − 2γ?α

)] is the overlap

of two coherent states. This enables us to calculate thetrace as

trace =∑{nk}

∏k

〈nk〉nk

(1 + 〈nk〉)nk+1〈{nk}|{nk}, {iγk}〉

=∏k

∑{nk}

〈nk〉nk

(1 + 〈nk〉)nk+1e−

12 |γk|

2

L0nk

(|iγk|2) .

This expression can be simplified with the generat-ing function of Laguerre polynomials,

∑∞n=0 t

nLn(x) =1

1−te− tx

1−t [91], which leads to

trace = exp

{∑k

[−1

2|γk|2 coth

(β~ωk

2

)]},

where we have used 〈nk〉 = 1/(exp(β~ωk)−1) for a ther-mal state. Substituting this result into Eq. (A2) providesEq. (16).

Appendix B: Calculation of the variance

We show how to estimate the uncertainty in the mea-surement of Re(ζ(t)) due to the projection noise on themeasurement of the state of the qubits. In this way, wedetermine the noise which has to be added to the cal-culated values of the coherence function to simulate theoutcome of experiments.

9

In accordance with Eq. (5),

Re(ζ(t)) =1

2〈σx ⊗ σx − σy ⊗ σy〉 , (B1)

the real part of the coherence function can be determinedby measuring the expectation value of a combination ofPauli matrices on the state of the qubits. Hence, westart by calculating the variance associated with thisexpectation value. Introducing the shorthand notationσxx = σx ⊗ σx, and similarly for σyy and σzz, we have

Var(σxx − σyy) = 〈(σxx − σyy)2〉 − 〈σxx − σyy〉2 .

The last term is directly related to the coherence func-tion 〈σxx − σyy〉2 = 4Re(ζ(t))2, whereas the first can becalculated as

〈(σxx − σyy)2〉 = 〈σ2xx + σ2

yy − σxxσyy − σyyσxx〉= 2 〈14 + σzz〉 (B2)

where 14 is the 4 × 4 identity matrix. In the last line,we have used that the Pauli matrices fulfill the algebraicrelation σa σb = δab12 + i

∑c=x,y,z εabc σc. We observe

that the right-hand side of Eq. (B2) is a diagonal matrix.Since the time evolution does not affect the diagonal el-ements, we can evaluate this expectation value over theinitial Bell state, resulting in 〈(σxx − σyy)2〉 = 4. Thus,

Var(σxx − σyy) = 4[1− Re(ζ(t))2

].

Substituting this into Eq. (B1), it follows that the vari-ance of the real part of the coherence function is con-nected to the function itself via

Var(Re(ζ(t)) =1

4Var(σxx − σyy) = 1− Re(ζ(t))2 .

For the error on the imaginary part of the coherence func-tion, the calculation is analogous.

Having determined the variances of the real and imag-inary parts of ζ(t), we simulate the uncertainty in ex-periments by adding Gaussian noise of zero mean andstandard deviations σRe =

√1− Re(ζ(t))2 and σIm =√

1− Im(ζ(t))2 to the real and imaginary parts, respec-tively.

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