Non-destructive cavity QED probe of Bloch oscillations in a gas of ultracold atoms
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Non-destructive cavity QED probe of Bloch oscillations in a gas of ultracold atoms
B. M. Peden, D. Meiser, M. L. Chiofalo, and M. J. Holland JILA, NIST, and Department of Physics, University of Colorado at Boulder, CO, USA Department of Mathematics and INFN, University of Pisa, Pisa, Italy (Dated: November 21, 2018)We describe a scheme for probing a gas of ultracold atoms trapped in an optical lattice andmoving in the presence of an external potential. The probe is non-destructive and uses the existinglattice fields as the measurement device. Two counter-propagating cavity fields simultaneously setup a conservative lattice potential and a weak quantum probe of the atomic motion. Balancedheterodyne detection of the probe field at the cavity output along with integration in time andacross the atomic cloud yield information about the atomic dynamics in a single run. The schemeis applied to a measurement of the Bloch oscillation frequency for atoms moving in the presence ofthe local gravitational potential. Signal-to-noise ratios are estimated to be as high as 10 . PACS numbers: 03.75.Lm, 37.10.Jk, 37.30.+i, 42.50.Ct
I. INTRODUCTION
The simulation of many-body models using gases ofultracold atoms trapped in optical lattices [1] has beensuccessful in investigating many systems in condensed-matter physics. Band physics in gases of non-interactingFermi gases in periodic potentials has been studied [2],quantum phase transitions such as the Mott insulatorto superfluid transition have been observed [3], andstrongly-correlated physics such as in one-dimensionalsystems [4, 5] has been investigated. In these exper-iments, techniques such as time-of-flight measurementsand Bragg spectroscopy are typically employed to probeatomic states and dynamics in optical lattices.In this paper, we present an alternative method for op-tically probing atomic gases in optical lattices subject toan external potential. The method is in situ and non-destructively measures properties of the atomic motionvia weak-coupling to the existing lattice fields. The tech-nique satisfies three main goals. The probe is weak sothat the atoms can be continuously monitored withoutaffecting their dynamics; the existing lattice fields areemployed as the probe, so that no external interrogationfields are necessary; and the signal-to-noise-ratio (SNR)is large enough for experimental detection. In a ring-cavity, two counter-propagating running-wave modes in-teract with a gas of ultracold atoms and simultaneouslyset up both a conservative, external lattice potential forthe atoms and a weak, quantum optical probe of theatomic center-of-mass dynamics. The probe field leaksout of the cavity and is detected with a balanced hetero-dyne scheme at the cavity output.This method is in a sense dual to strong measurementschemes such as time-of-flight absorption imaging andBragg spectroscopy. In these schemes, light from a strongsource is either absorbed by or scattered off of the atomiccloud. This allows for high resolution images and a strongsignal using only a single measurement, but the atomicsample is destroyed in the process. Here, the probe fieldis very weak so that a continuous measurement is madewithout affecting the atomic dynamics. Integration of the signal in time and across the atomic cloud yieldsmeasurements of dynamical properties of the atoms witha measurable SNR in a single experimental run at theprice of losing information about individual atoms andreal-time dynamics.The procedure is similar in nature to recent propos-als for optical detection of many-body atomic states.In one scheme, a weak probe beam is scattered offof atoms trapped in an optical lattice into a cavitymode, and signatures of many-body states such as Mottinsulators and superfluids appear in the out-coupledfields [6]. In another, atoms in a lattice interact withtwo-counter-propagating ring-cavity modes, and atomicnumber statistics can be inferred from the behavior ofthe cavity fields [7].Related techniques have been applied to nondestruc-tive optical measurements of Rabi oscillations in gases ofCs atoms [8], of the Cs clock transition pseudo-spin [9],and of nonlinear dynamics in cold gases [10]. In addi-tion, state preparation such as atomic spin squeezing viameasurements on out-coupled cavity fields has been pro-posed [11, 12, 13]. Finally, it has been demonstrated thatthe motion of individual atoms in an optical cavity canbe tracked by the transmission of a probe field [14].We here provide a test of the technique for the con-ceptually simple motion of non-interacting atoms in anoptical lattice driven by a constant force, which leads toBloch oscillations [15]. Besides its simplicity, this choiceis motivated by the fact that Bloch oscillations can beviewed as a general probe for investigating quantum gasesin optical lattices. These oscillations may be used in themeasurement of fundamental constants [16], to providelevels of precision up to δg/g ≈ − in the measure-ment of the acceleration of gravity [17, 18, 19, 20], and tomeasure Casimir forces on small length scales [21]. Wheninteractions are significant, damping and destruction ofBloch oscillations provide information on correlation-induced relaxation processes [22, 23, 24, 25, 26]. Finally,this investigation is a jumping-off point for other opticalmeasurement schemes, such as periodically driven lat-tices that act as a spectroscopic probe of the atomic mo- FIG. 1: (Color online.) Schematic of the coupled atom-cavitysystem. In-coupled lasers set up two counter-propagatingfields within the cavity. The atoms interact with the cav-ity fields via the optical dipole potential. Photons from thecavity beams exit the cavity through the mirrors at a rate κ . tion [20].The paper is organized as follows. In Sec. II, wepresent the details of the system and detection scheme.In Sec. III, we apply this scheme to the detection ofBloch oscillations in an optical lattice. In Sec. IV, wesummarize the main results of the paper and concludewith prospects for measurements of many-body proper-ties of gases of ultracold atoms trapped in optical lattices. II. MODEL AND DETECTION SCHEME
Since the mathematical details of the system and de-tection scheme are somewhat complicated, we briefly out-line the physical basis of the model and approximationsused. This includes a discussion of obstacles in the way ofsatisfying the goals outlined above, avoiding these prob-lems, and the conditions required for the method to work.To set up a conservative lattice potential, many pho-tons must be present in the cavity field so that fluc-tuations can be neglected, and this necessitates strongpumping from the in-coupled lasers. On the other hand,the probe field amplitude must be small enough so thatit does not affect the atomic dynamics, requiring weakpumping. In addition, the probe and lattice fields coupleto each other through the scattering of photons off of theatoms. This acts as an extra source for probe dynamics.The probe field is then not a direct measure of atomicdynamics and can act back on the atoms, altering the properties we are attempting to measure.We can circumvent these problems by first choosingthe relative phase on the in-coupled lasers so that onlyone of two standing-wave modes in the cavity is pumped.Strong pumping and the properties of a bad cavity –where the fields are at all times in steady state – ensurethat the pumped mode acts as a lattice potential. Theother standing-wave mode is not pumped. Any field leak-ing out of the cavity from this mode arises solely becauseof events occurring in the cavity, and it can therefore actas a probe for system dynamics.Ensuring that the non-pumped mode acts as a probeof atomic dynamics requires that two conditions be met.The probe must have as its source only the motion ofthe atoms. This means that any probe dynamics due tothe effective coupling to the lattice field must be smallcompared to that induced by the motion of the atoms.The back-action of this mode on the atoms must also benegligible. Any atomic motion induced by coupling tothe probe field must be small compared to the motioninduced by both lattice and external potentials.Two conditions are also required for the pumped fieldto act as a conservative lattice potential. The back-actionof the atoms on the lattice field must be negligible, mean-ing that deviations from the mean field amplitude causedby coupling to the atoms is small compared to the meanfield amplitude itself. In addition, any atomic motion in-duced by fluctuations away from the mean lattice fieldmust be small compared to that induced by the externalpotential, since this is exactly the dynamics we want tomeasure.The scheme is realized in the setup illustrated in Fig.1. We consider N a ground-state atoms interacting withtwo counter-propagating running-wave cavity modes ina ring resonator setup. The two modes of the cavityhave frequency, ω ca , and wave vectors, ± k ca ˆ z . Thetwo cavity modes are coherently pumped at a detun-ing, ∆ p = ω L − ω ca , where ω L is the frequency of thepumping lasers. Photon decay through the cavity mirrorsis treated within the Born-Markov approximation. Wetreat the atom-cavity-field interaction in rotating waveand dipole approximations. The cavity-modes are far-detuned from atomic transitions. A. Model
The effective Hamiltonian for the coupled atom-cavitysystem is given byˆ H = X Z dz ˆΨ † ( z ) (cid:18) − ~ m d dz + V ext ( z ) (cid:19) ˆΨ( z ) + X k = ± k ca (cid:16) ~ η ˆ a k + ~ η ˆ a † k − ~ ∆ p ˆ a † k ˆ a k (cid:17) + Z dz ˆΨ † ( z ) ~ g (cid:16) ˆ a † k ca e − ik ca z + ˆ a †− k ca e ik ca z (cid:17) (cid:0) ˆ a k ca e ik ca z + ˆ a − k ca e − ik ca z (cid:1) ˆΨ( z ) . (1)Here, ˆΨ is the atomic field operator, and ˆ a k is the anni-hilation operator for the cavity mode, k . The parameter, m , is the the mass of the atom, g is the two-photonatom-cavity coupling, and η is the strength of the cavityfield pumping, taken to be real. Due to the far detuningof the cavity fields from the atomic transition, excitedstates of the atom have been adiabatically eliminated,and the atoms couple to the field intensity. The threeterms in Eq. (1) are respectively the atomic kinetic en-ergy and external potential, the bare cavity mode Hamil-tonian, and the atom-cavity interaction. Cavity lossesthrough the cavity mirrors are treated via a master equa-tion with Liouvillian,ˆ L ˆ ρ = − ~ κ X k = ± k ca (cid:16) ˆ a † k ˆ a k ˆ ρ + ˆ ρ ˆ a † k ˆ a k − a k ˆ ρ ˆ a † k (cid:17) , (2)where ˆ ρ is the reduced density matrix for the atom-cavitysystem, and κ is the cavity linewidth.We perform a canonical transformation of the cavitymode operators to symmetric and anti-symmetric modes,ˆ b ± = ˆ a k ca ± ˆ a − k ca √ . (3)The bare cavity Hamiltonian in terms of these operatorsis given byˆ H ca = √ ~ η (cid:16) ˆ b + + ˆ b † + (cid:17) − ~ ∆ p (cid:16) ˆ b † + ˆ b + + ˆ b †− ˆ b − (cid:17) . (4)The symmetric mode, ˆ b + , is pumped by the in-coupledlasers whereas the anti-symmetric mode, ˆ b − , is not. Theˆ b + mode has a mode function proportional to cos ( k ca z )and sets up the lattice potential, as follows.The equation of motion for the symmetric field ampli-tude is i d h ˆ b + i dt = (cid:16) − i κ − ∆ p (cid:17) h ˆ b + i + √ η + 2 g Z dz cos ( k ca z ) D ˆ b + ˆΨ † ( z ) ˆΨ ( z ) E + ig Z dz sin (2 k ca z ) D ˆ b − ˆΨ † ( z ) ˆΨ( z ) E . (5) We perform another transformation to a fluctuation op-erator, ˆ d + = ˆ b + − β , leaving the anti-symmetric modeunchanged, ˆ d − = ˆ b − . The mean steady-state amplitude, β , is given by β = h ˆ b + i = √ η ∆ p − g C (0) + iκ/ , (6)where C ( t ) = Z dz cos ( k ca z ) D ˆΨ † ( z, t ) ˆΨ( z, t ) E , (7)Assuming that the atom-field correlations factorizebetween atomic and field operators, e.g. h ˆ d + ˆΨ † ˆΨ i = h ˆ d + ih ˆΨ † ˆΨ i , the equations of motion for both ˆ d + and ˆ d − are given by i d h ˆ d − i dt = (cid:16) − ∆ p + 2 g S ( t ) − i κ (cid:17) h ˆ d − i− ig (cid:16) β + h ˆ d + i (cid:17) S ( t ) , (8)and i d h ˆ d + i dt = (cid:16) − ∆ p + 2 g C ( t ) − i κ (cid:17) h ˆ d + i + ig h ˆ d − iS ( t ) + 2 g β ( C ( t ) − C (0)) , (9)where S ( t ) = Z dz sin ( k ca z ) D ˆΨ † ( z, t ) ˆΨ( z, t ) E , (10) S ( t ) = Z dz sin (2 k ca z ) D ˆΨ † ( z, t ) ˆΨ( z, t ) E . (11)Finally, the equation of motion for the atomic field oper-ator is given by i ~ d ˆΨ( z ) dt = (cid:18) − ~ m d dz + V lat ( z ) + V ext ( z ) (cid:19) ˆΨ( z ) + i ~ g (cid:16) β ∗ ˆ d − − β ˆ d †− + ˆ d † + ˆ d − − ˆ d †− ˆ d + (cid:17) sin (2 k ca z ) ˆΨ( z )+ 2 ~ g ˆ d †− ˆ d − sin ( k ca z ) ˆΨ( z ) + 2 ~ g (cid:16) ˆ d † + ˆ d + + β ˆ d † + + β ∗ ˆ d + (cid:17) cos ( k ca z ) ˆΨ( z ) , (12)where V lat ( z ) = V cos ( k ca z ) is a conservative latticepotential of depth, V = 2 ~ g | β | .Aside from the conditions involving the external po-tential, the goals of simultaneously setting up both anoptical lattice potential and a weak probe require thatthe inequalities, |h ˆ d ± i| ≪ | β | , (13)are satisfied. This can be verified by examining equations(8), (9), and (12). These are necessary conditions, butfinding the exact criteria for neglecting the back-actionrequires a more careful analysis of the problem, includ-ing numerical simulations. This is left for future work.Ensuring that both the probe and the lattice fluctuationsinduce atomic motion that is negligible compared to thatinduced by the external potential requires explicit knowl-edge of the form of V ext and will therefore be left for thenext section.The two conditions in Eq. (13) can be be made moreexplicit. Equations (8) and (9) imply the scaling rela-tions, h ˆ d − i ∼ g β S ( t ) κ , (14) h ˆ d + i ∼ g β ( C ( t ) − C (0)) κ , (15)so that the two conditions are respectively equivalent to | g S ( t ) | ≪ κ and | g ( C ( t ) − C (0)) | ≪ κ . When theseare satisfied, we may neglect Eq. (9) altogether. In ad-dition, Eq. (8) can be solved approximately since in thislimit h ˆ d − i adiabatically follows the atomic motion. Fi-nally, we have ensured that both fluctuations in the lat-tice and the back-action of the probe field on the atomscan be neglected. We need then only keep the first termin Eq. (12).With these approximations in hand, the equation ofmotion for the atomic field operator is i ~ d ˆΨ( z ) dt = (cid:18) − ~ m d dz + V lat ( z ) + V ext ( z ) (cid:19) ˆΨ( z ) , (16)and the probe field amplitude is given by h ˆ d − ( t ) i = − ig β ∆ p − g S ( t ) + iκ/ S ( t ) . (17)These equations comprise a complete description of thecoupled atom-cavity dynamics. To the extent that the atoms are in the same center-of-mass state, | ψ ( t ) i , satisfying Eq. (16), we can in Eq.(17) make the replacement, S ( t ) → N a h ψ ( t ) | sin(2 k ca ˆ z ) | ψ ( t ) i . (18) B. Detection scheme
Through Eq. (17), h ˆ d − i provides a measure of theatomic dynamics within the cavity. In [12], two schemesfor detection of atomic motion using the out-coupled cav-ity fields were presented. We here briefly review the supe-rior case, where heterodyne detection of ˆ d − is performedby beating the field against a strong local oscillator, asillustrated in Fig. 2.According to input-output theory [27], the field at thecavity output is proportional toˆ d out = √ κ ˆ d − + ˆ d in . (19)By beating this field against a strong local oscillator,these photons can be detected with unit efficiency. Theinput field state is the vacuum, in which case h ˆ d out i = √ κ h ˆ d − i . The resulting signal is the difference signal atthe output of the photo-detectors, given by V ( t ) ∝ Im( √ κ α ∗ LO h ˆ d − i ( t )) , (20)which is a product of | α LO | withˆ q − = e − iφ LO ˆ d − − e iφ LO ˆ d †− , (21)a quadrature of the anti-symmetric mode field. The localoscillator amplitude is α LO = | α LO | e iφ LO . The SNR isthe ratio of the signal power to signal variance, given by SN R = Z dω |√ κ h ˆ q − i ( ω ) | . (22)The integrand is proportional to the power spectrum, S ( ω ), of the signal current in Eq. (20). III. RESULTS
In this paper, we consider the motion of atoms confinedin the optical lattice in the presence of gravity, V ext ( z ) = mgz, (23) FIG. 2: (Color online.) Schematic of the balanced heterodynedetection scheme. The out-coupled cavity beams, ˆ a ± k ca , arecombined to form symmetric (ˆ b + ) and anti-symmetric (ˆ b − )modes. The antisymmetric mode beats against a strong localoscillator (LO), ˆ a LO and photodetectors count the numberof photons in the quadratures of ˆ b − . The difference of thesecounts is the signal. and use the scheme outlined in the previous section toprobe the motion of the atoms. Gravity measurementsare important for instance for optical lattice clocks [28].For this reason, we treat the specific system of a gasof Sr atoms, though the method certainly applies tomany species of atoms. The parameters for the coupledatom-cavity system are chosen to reflect current exper-imental conditions. They are λ ca = 2 π/k ca = 813 nm, ~ κ = 100 E R , ∆ p = 0, ~ g = 10 − E R , and N a = 10 ,implying derived parameters of E R ≈ π ~ and mgd ≈ . E R ; E R = ~ k ca / m is the recoil energy ofthe lattice, and d = π/k ca is the lattice spacing.We have to ensure that the back-action of both ˆ d − andˆ d + on the atoms is still negligible. Specifically, the cou-pling strengths in Eq. (12) must be small compared tothe characteristic coupling strength of V ext , ~ ω B = mgd .These conditions are met if | g β | S ( t ) /κ ≪ ω B and | g β | |C ( t ) − C (0) | /κ ≪ ω B . These inequalities are well-satisfied for the parameters above. Again, while theseconditions are necessary, the exact criteria for being ableto neglect the back-action of the fields on the atoms re-quires more careful numerical study, which will be leftfor future work.Within this setup, we envision an experiment in whichthe atoms are initially loaded into a harmonic trap. Avertical one-dimensional optical lattice is slowly rampedon so that the atoms are in the ground state of the com-bined potential of trap and lattice for a non-interactinggas. The trap is then switched off, and the gas is allowedto evolve under gravity. In the presence of such a con-stant force, the atoms undergo Bloch oscillations. Thisdynamics is briefly reviewed in the following discussion. A. System Dynamics
The central result of the theory describing Bloch os-cillations is based on a semi-classical equation of mo-
FIG. 3: Example of system dynamics for V = − E R andinitial state a Gaussian of width σ = 2 d projected into thefirst band. (a) Atomic density in the first band plotted ver-sus quasi-momentum. White corresponds to zero population,black to maximal population. Population in the second bandis at most 0 . N a . (b) Expectation value of atomic momen-tum reflecting Bloch oscillations. (c) Number of photons inthe probe field. tion [15], which states that the average quasi-momentumof a wave-packet restricted to the first band increaseslinearly in time until it reaches the Brillouin zone (BZ)boundary, at which point it is Bragg-reflected. Explicitly,this is ~ h q i ( t ) = ~ h q i (0) + mgt, (24)where the quasi-momentum, q , is restricted to the range, − k ca / ≤ q ≤ k ca /
2. Since the group velocity of theatomic wave-packet is given by the derivative of the dis-persion relation [15], the periodic nature of the quasi-momentum implies that the atomic momentum oscillatesat a frequency, ω B = mgd/ ~ . These Bloch oscillationswill persist as long as there is negligible Landau-Zenertunneling to higher bands. Each time the wave-packetreaches the BZ boundary, a fraction of population istransferred to the second band, given by [29] P LZ = exp (cid:18) − π mgdE R (cid:19) , (25)where ∆ is the band-gap at the boundary. When ∆ < mgdE R , the population transfer is appreciable, and vi-brational dynamics significantly alter the behavior of theatoms. For this reason, we restrict our attention to lat-tice depths greater than 3 E R , where P LZ is at most 10 − for our choice of parameters.In order to understand how Bloch oscillations are re-flected in the time-dependence of the probe field, we care-fully consider Eq. (17). The operator, sin(2 k ca ˆ z ), is peri-odic in space with period d and has odd parity, implyingthat it connects two Bloch states, | ψ ( n ) q i and | ψ ( n ′ ) q ′ i , onlyif the quasi-momenta are equal, q = q ′ , and the bandssatisfy n − n ′ = odd. Taking | ψ i = X n,q c ( n ) q | ψ ( n ) q i , (26)we can write the matrix element in Eq. (17) approxi-mately as h ψ | sin(2 k ca ˆ z ) | ψ i ≈ X q ρ (1 , q,q + h.c., (27)where ρ (1 , q,q = c (1) ∗ q c (2) q is the coherence between bandsone and two. This assumes an initial state confined tothe first band in the case of negligible coupling to bandsthree and higher. Using Eq. (16), we can derive anapproximate equation of motion for the coherence; it is i dρ (1 , q,q dt = ∆ (1 , q ρ (1 , q,q + ω B ρ (1) q , (28)where ρ (1) q is the population of the q -quasi-momentumstate in the first band, and ~ ∆ (1 , q = E (2) q − E (1) q is theenergy difference between the q -quasi-momentum Blochstates in the first two bands. Since ∆ (1 , q ≫ ω B , thecoherence follows the first-band population adiabatically.In this approximation, ρ (1 , q,q = − ρ (1) q ω B / ∆ (1 , q .Combining this expression for a wave-packet that isnarrow in quasi-momentum with Eq. (24), Eq. (17) ap-proximately becomes h ˆ d − ( t ) i ≈ ig βN a ∆ p − g S ( t ) + iκ/ ω B ∆ (1 , mgt . (29)This expression implies that the probe field amplitude islargest when the atomic wave-packet is centered at theBZ boundary since ∆ (1 , mgt is smallest at this point.Equations (16) and (17) are numerically integrated foran initial state that is a Gaussian of spatial width, σ , pro-jected into the first band. This approximates the ground FIG. 4: Examples of the (a) signal power and (b) signal powerspectrum, computed with parameters, V = − E R , ~ g =10 − E R , N a = 10 , and ~ κ = 100 E R . The signal displays aclear oscillation at the Bloch frequency, ω B = 0 . E R / ~ . state of the combined potential of lattice and harmonictrap for a non-interacting gas. An example of the sys-tem dynamics is illustrated in Fig. 3, where V = − E R ,and σ = 2 d . A vertical slice through Fig. 3(a) is thewave function density in the first band plotted versusquasi-momentum at an instant in time. The center ofthis wave-packet moves linearly in time and is reflectedat the BZ boundary ( q = k ca / B. Signal and SNR
As described in Sec. II, the probe field is combinedat the cavity output with a strong local oscillator, andthe resulting signal is proportional to a quadrature ofthe probe field, Eq. (21). An example of such a signalis plotted in Fig. 4. There is a clear peak at the Blochoscillation frequency in the signal power spectrum, butthere are also several harmonics present. In calculatingthe SNR, Eq. (22), we place a notch-filter about ω B andcount only the total number of photons out-coupled fromthe quadrature at this frequency.There are three properties of the system that can affectthe SNR. First, the width of the initial wave-packet hasan effect. It is easiest to see why this is so by taking asthe initial state a Wannier function, which is a coherentsuperposition of Bloch states in a single band, populatedequally. According to Eq. (24), the wave-packet is con-tinuously reaching the BZ boundary, and the oscillationin the signal is washed out. Second, when the lattice istoo deep, the first two bands are essentially flat, in whichcase ∆ (1 , q does not change with quasi-momentum, elimi-nating the oscillations in the signal according to Eq. (29).The temperature of the atomic gas can also signifi-cantly influence the SNR. In a thermal cloud the replace-ment, Eq. (18), cannot be made, since the atoms donot all occupy the same state. In this case, atoms indifferent lattice sites may contribute to the signal withrandom phases, in which case the SNR scales with N a rather than N a . The temperature and chemical poten-tial of the gas also determine the relative populationsof the various Bloch states, and appreciable populationin higher bands can destroy Bloch oscillations. A propertreatment of thermal effects is necessary for exact results,but here we assume the replacement, Eq. (18), is a goodapproximation.Equation (12) is numerically integrated for a time t = 400 ~ /E R . The resulting wave function is used tocompute the probe field amplitude, Eq. (17), whichis Fourier-transformed and squared, yielding the powerspectrum. The SNR is computed and scaled up linearlyto an interrogation time of 1s, which assumes that coher-ence time of the Bloch oscillations is longer than 1s.The results are plotted in Figs. 5 and 6. The SNRclimbs from zero for small wave-packet widths and satu-rates near σ = 2 d . The decrease in SNR for σ < d is aresult of the fact that the wave-packet is wide in quasi-momentum, which means that a significant portion of thewave-packet is at the the BZ boundary for all times. Weget a maximum when the lattice depth is relatively small, | V | ≈ E R , and the SNR decreases with increasing lat-tice depth. IV. CONCLUSION
We have described a general cavity QED system inwhich properties of atomic dynamics can be probed insitu and non-destructively. One cavity field is strongenough to act as a conservative lattice potential for theatoms, and the other cavity field is weak so that it actsas a non-destructive probe of atomic motion. This tech-nique is applied to the detection of Bloch oscillations.Balanced heterodyne detection of the probe field at thecavity output combined with integration in time andacross the atomic cloud allows for SNRs as high as 10 .Examining Eqs. (17) and (22), we can see that theSNR can be increased by either decreasing the cavitylinewidth, κ , at fixed lattice depth and atom-cavity cou-pling or increasing the coupling constant, g , at fixed V and κ . The linewidth can be increased as long asthe system remains in the bad cavity limit. However,a linewidth of κ = 100 E R is already very small froman experimental standpoint, so increasing it beyond this FIG. 5: (Color online.) Signal-to-noise ratio as a functionof the initial wave-packet width for an interrogation time of1s. The different plots correspond to lattice depths of (fromlargest to smallest SNR) V = − , − , − , − , − E R . For σ < d , the SNR is reduced due to parts of the wave-packetconstantly moving past the Brillouin zone boundary, wherethe signal peaks. The SNR saturates near σ = 2 d .FIG. 6: Signal-to-noise ratio as a function of lattice depth for σ = 2 d and an interrogation time of 1 s . For | V | < E R , sig-nificant Landau-Zener tunneling to the second band destroysBloch oscillations. For | V | > E R , the SNR decreases dueto the increasing flatness of the lowest band, which in turndecreases the amplitude in momentum space of the Bloch os-cillations. level is a technological challenge. On the other hand, g can be varied merely by varying the detuning betweenthe cavity fields and atomic transitions. In addition, theSNR scales with the square of the number of atoms, soincreasing N a beyond the 10 level assumed in this paperis also desirable. This can all be done to the extent thatthe conditions outlined in Sec. II and Sec. III are stillmet.This scheme can be extended for use in detection ofvarious atomic properties, and the measurement of Blochoscillations itself can be viewed as a general DC probefor atomic dynamics and states. For instance, Bloch os-cillations may be used for measurement of fundamentalconstants [16] and for Casimir forces [21]. Varying thedetuning between two lattice beams gives rise to an ef-fective acceleration of the lattice [29], and band physicsmay be probed by varying the Bloch oscillation frequencyin such a setup. Breakdown of Bloch oscillations are asignature of many-body effects in an atomic gas [22], andthis is signalled by a reduction in SNR compared to thenon-interacting case.Two generalizations of this measurement technique arereadily realizable. We may implement a periodic forcingwhose varying driving frequency can be a spectroscopicprobe of atomic dynamics. The simplest examples of thisinclude shaking the lattice [20] and modulating the am-plitude of the lattice [30]. Another important extension of the method involves measuring higher-order correla-tion functions of the out-coupled probe field. Since onecavity field operator couples to two atomic field opera-tors (see for instance Eq. (8)), higher-order properties ofthe atoms such as density-density correlations can easilybe measured with standard quantum optical techniques.The use of higher-order correlation functions of the probefield is a starting point for generalizing this technique toprobe many-body physics in optical lattices.We acknowledge useful conversations with Jun Ye, AnaMaria Rey, and Victor Gurarie. This work was sup-ported by DARPA, NIST, DOE, NSF, DFG (DM), andASI grant WP4200 (MC). [1] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, andP. Zoller, Phys. 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