Ultracold atom interferometry with pulses of variable duration
UUltracold atom interferometry with pulses of variable duration
Valentin Ivannikov
Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne, Australia ∗ (Dated: 16 March 2017)We offer interferometry models for thermal ensembles with one-body losses and the phenomenolog-ical inclusion of perturbations covering most of the thermal atom experiments. A possible extensionto the many-body case is briefly discussed. The Ramsey pulses are assumed to have variable du-rations and the detuning during the pulses is distinguished from the detuning during evolution.Consequently, the pulses are not restricted to resonant operation and give more flexibility to op-timize the interferometer to particular experimental conditions. On this basis another model isdevised in which the contrast loss due to the unequal one-body population decays is cancelled bythe application of a non-standard splitting pulse. For the importance of its practical implications,an analogous spin-echo model is also provided. The developed models are suitable for the anal-ysis of atomic clocks and a broad range of sensing applications, they are particularly useful fortrapped-atom interferometers. PACS numbers: 03.75.Dg, 67.85.–d, 95.55.Sh, 06.30.Ft
I. INTRODUCTION
Most accurate experimental methods have been basedon interferometers, first invented for the measurement ofthe velocity of light and gradually extended to frequencystandards and metrology [1–9]. In recent years the inter-est in precision interferometry has been growing in thecontext of ultracold atomic systems [10–12], where atomchips that allow unparalleled control over atomic ensem-bles have become particularly promising [13, 14].In metrology it becomes a burden to interpret how in-advertently detuned pulses quantitatively influence re-sults. Our analytical models address most common cases.Imperfections are often compensated with sophisticatedtechniques [15–18] that may become an absolute neces-sity: The medium may distort the pulses in an uncon-trolled way or the pulse sources may suffer from imper-fections and lead to measurement inaccuracies. In thehigh-precision applications one employs the longest pos-sible times, limited by the saddle point of the Allan vari-ance graph [19] showing that longer integration times willgain no greater accuracy. As a consequence, the analysisshould include particle losses and cold atomic collisionssince they cause ensemble dephasings. In dense thermalclouds two-body processes may become a limiting factor;thus, they should also be included [20].In this contribution we offer a set of analytical modelsof various Ramsey-type interferometers [21, 22] with one-body losses and proper accounting for off-resonant cou-pling and ensemble dephasing. The models cover most ofthe experiments with thermal atoms and are extendablevia the inclusion of perturbations. The extension to thecase of two-body losses can be readily implemented [20]. ∗ [email protected]; Present address: Instituto de Física de SãoCarlos, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos, São Paulo, CES 13566-590, Brazil. T p1 T T p2 (a) T p1 Π T p2 T (cid:2) (cid:2) (b) FIG. 1. Ramsey interferometry with variable-duration pulses. (a)
Pulses are generally non- π/ with durations T p and T p . (b) Spin-echo sequence with variable-duration pulses.
A peculiar feature of the presented models is that the cou-pling field detuning from the atomic resonance is distin-guished from the energy level shifts during free evolution,and, since the effect of the off-resonant interrogation isoften significant, the detuning is assumed non-zero. Themodels are linked to the Bloch vector formalism to in-troduce the measurables and explain the underlying pro-cesses. Second, a more generic Ramsey-model is devel-oped where both pulses have variable durations. Then amethod to maximize the visibility by applying a non- π/ splitter pulse is devised. It is based on the fact that theinitialization of the two states having unequal populationdecays may, by the end of the evolution, result in thepopulation equalization highly desirable for result inter-pretation. Ramsey spectra with Rabi pedestals are givenanalytically for the one-body models. In Appendix Awe extend the Ramsey formalism to the off-resonant spinecho with one-body losses. II. INTERFEROMETRY MODELSA. Definitions
The ensemble of two-level atoms is initially prepared inthe state | ⟩ . A short π/ Rabi pulse (Fig. 1(a)) equates a r X i v : . [ phy s i c s . a t o m - ph ] M a r Initial state detectiontime Π T Π (a) (b) (c) (d) FIG. 2. (Color available online) Ramsey interferometry in the Bloch vector representation. The fans of vectors illustrate phasediffusion. The black circular arrows show the rotation by the effective torque acted on the vector. A fringe in P z is producedby varying T . In the phase-Ramsey method the fringe is produced at a fixed T by varying the phase of the second π/ -pulse. (a) Initially the atoms are in state | ⟩ , state | ⟩ is unpopulated. (b) After the first π/ -pulse the effective torque brings the vector to the phase-plane ( P z = 0 ). (c) Without interrogation the system relaxes, undergoing population loss and phase destruction, resulting in phase diffusion. (d)
The second π/ -pulse applies an effective torque projecting the vectors onto the axis of population difference. the populations placing the ensemble (pseudo-)spin tothe equatorial plane of the Bloch sphere where it canonly evolve in phase (Fig. 2). We assume the pulses areinstantaneous and neglect losses by satisfying Ω ≫ ∆ and /γ m ≫ d with m ∈ { , } for the resonant Rabifrequency Ω , detuning from the atomic resonance ∆ = ω atom − ω light , population loss rates γ m of the states | m ⟩ ,and the pulse duration d . The pulse areas equal π/ atany ∆ and have duration π /(2Ω R ) , with the generalizedRabi frequency Ω R = √ ∆ + Ω , to preserve the π/ -behavior away from resonance. It is distinct from theusual duration π /(2Ω) yielding a non- π/ pulse at ∆ ̸ =0 . The spectra of such systems differ as shown in theforthcoming discussion.After the first π/ -pulse, the system evolves for a time T . The phase difference between the two states startsgrowing. Before the second π/ -pulse arrives, the phaseis diffused due to the ensemble-related dephasing, trap-induced dephasing, and the driving frequency instabil-ity. The second π/ -pulse rotates the Bloch vector toaccomplish projective detection. It brings the imprintedphase to the axis of the normalized population difference P z . Locally dephased parts of the ensemble result in ablurred distribution of P z , the width of which expressesthe detection limit.The measurables are expressed in terms of the atomnumbers N n and density matrix elements ρ nn with thestate index n ∈ { , } , N = N + N ; P z = P − P : P = N N = ρ ρ + ρ , P = N N = ρ ρ + ρ . (1)The Bloch vector B is employed in Fig. 2 to articu- late the processes in the effective two-level system. TheLiouville–von Neumann equation for the resonant losslesscase reads ∂ B /∂t = Ω × B , where B = ( B x , B y , B z ) ⊤ =( ρ + ρ , Im { ρ − ρ } , ρ − ρ ) ⊤ is the pseudo-spinvector, and Ω = ( − Ω , , ∆) ⊤ acts on B as an effectivetorque. During free evolution, φ accumulates detuningand miscellaneous perturbations, e.g., collisional levelshifts, radiation shifts, etc, in general taking the formof a sum φ = ∆ + ∆ collisions + ∆ radiation + . . . . Separationof pulse ∆ and level shifts during evolution φ enablesthe model to sense perturbations. We shall refer to φ asthe Ramsey dephasing rate measured in rad / s. The pre-sented models are parametrized by the Ramsey evolutiontime T , the cumulative Ramsey dephasing rate φ duringfree evolution, and the phenomenological dephasing rate γ d . B. Master equation with one-body losses
Particle loss that causes dephasing, and the pure de-phasing that only occurs between the states and is notassociated with population loss, can be included in theLiouville–von Neumann equation. It is then written fora two-level system as ∂ρ∂t = 1 i ℏ [ H , ρ ] − { Γ , ρ } −
12 Ξ ◦ ρ, (2)where Γ is the loss operator that sets up γ , the popu-lation loss rate of state | ⟩ , and γ , the population lossrate of state | ⟩ . ρ is the density operator, [ • , • ] and {• , •} are commutator and anticommutator brackets, re-spectively, and H is the effective Hamiltonian of a spin- system; here we shall consider H in the rotating wave ap-proximation and interaction picture. The loss operatoris defined as a matrix: Γ = (cid:20) γ γ (cid:21) , Ξ = (cid:20) γ γ (cid:21) . (3)The Hadamard product allows us to conveniently intro-duce the off-diagonal phase relaxation rates γ d in thepure dephasing operator Ξ as a separate summand Ξ ◦ ρ of Eq. (2) where Ξ takes the form of a matrix with equalpure dephasing rates: γ = γ = γ d . Eq. (2) can bewritten explicitly as the following differential equations : ∂ρ ∂t = − γ ρ − i ρ − ρ ) ,∂ρ ∂t = − γ ρ + i ρ − ρ ) ,∂ρ ∂t = − γ ρ + i ρ − ρ ) + iξρ ,∂ρ ∂t = − γ ρ − i ρ − ρ ) − iξρ , (4)where ξ is included in H and the pure dephasing rate γ d in relaxation constant γ = ( γ + γ + γ d ) . We shalldistinguish two regimes: during coupling pulses ( ξ = ∆ )and during free evolution ( ξ = φ ).The Liouville–von Neumann equation [Eq. (2)] issolved with non-zero loss terms included with the as-sumption of lossless interrogation pulses. The solution ofEq. (2) for the off-resonant Ramsey sequence with non-negligible losses during evolution are ρ = 14Ω R (cid:16) Ω e − γ T + (cid:0) ∆ + Ω R (cid:1) e − γ T − k (cid:17) , (5a) ρ = 14Ω R (cid:0) Ω (cid:0) ∆ + Ω R (cid:1) (cid:0) e − γ T + e − γ T (cid:1) + k (cid:1) , (5b) P z = ∆ ( k − k ) − k e ( γ + γ ) T Ω R ( k + k ) , (5c)with the following auxiliary definitions: k = 2Ω e − γ T (Ω cos( φT ) − R sin( φT )) ,k = (cid:0) ∆ + Ω R (cid:1) e γ T ,k = Ω e γ T . (6)It is often the case that the interrogation is resonantwith the atomic transition. Then Eqs. (5) simplify as ρ = 14 (cid:0) e − γ T + e − γ T − e − γ T cos( φT ) (cid:1) , (7a) ρ = 14 (cid:0) e − γ T + e − γ T + 2 e − γ T cos( φT ) (cid:1) , (7b) P z = − e − γ d T sech (cid:18) γ − γ T (cid:19) cos( φT ) . (7c) ξ from these equations is erroneously typed as ∆ in Ref. [23]. C. Interferometry with variable-duration pulses
As a useful extension to the Ramsey technique wepresent solutions for the interferometry with variable du-rations of the splitting and detecting pulses.The Liouville–von Neumann Eq. (2) is solved with thefollowing assumptions: the detuning of the interrogatingfield ∆ is arbitrary, no losses during the interrogatingpulses, no coupling during free evolution (i.e., Ω = 0 ).Then ρ and ρ at the interferometer output are ρ R e ( γ + γ ) T = 4Ω e γ T sin (cid:20) Ω R T p (cid:21) sin (cid:20) Ω R T p (cid:21) + k k e − γ T − e ( γ + γ − γ d ) T ( k − k ) ,ρ R Ω e ( γ + γ ) T = − k cos(Ω R T p ) − k cos(Ω R T p )+ k + k + 2 e ( γ + γ − γ d ) T ( k − k ) , (8)with auxiliary definitions for the sake of compactness: k = cos( φT ) (cid:20) Ω R sin(Ω R T p ) sin(Ω R T p ) − sin (cid:18) Ω R T p (cid:19) sin (cid:18) Ω R T p (cid:19) (cid:21) ,k = sin( φT ) Ω R ∆ [sin(Ω R T p ) + sin(Ω R T p ) − sin(Ω R T p + Ω R T p )] , (9) k = e γ T (cid:0) ∆ + Ω R + Ω cos(Ω R T p ) (cid:1) ,k = e γ T (cid:0) ∆ + Ω R + Ω cos(Ω R T p ) (cid:1) ,k = e γ T Ω (1 − cos(Ω R T p )) . The normalized population difference is then P z = Ω − R k + k (cid:104) ( k − k ) (cid:0) ∆ + Ω cos(Ω R T p ) (cid:1) − ( k − k ) e ( γ + γ − γ d ) T (cid:105) . (10)If an equal splitting is desired at an arbitrary detuning,the splitter pulse duration T p can be obtained from thelossless model [23] by solving the equation P z ( t ) = 0 : ρ = 12Ω R (cid:0) ∆ + Ω R + Ω cos (Ω R t ) (cid:1) ,ρ = Ω R (1 − cos (Ω R t )) . (11)The first pulse duration is then t π/ = T p = arccos (cid:18) − ∆ Ω (cid:19)(cid:30) Ω R , (12)where the sequence can be closed by a π/ -pulse defin-ing the duration T p = π/ (2Ω R ) . The T p is limited bythe detuning that is required to be not larger than theresonant Rabi frequency: Ω ≥ ∆ . If this condition is T (s) Ρ nn T (s) Ρ nn T (s) (cid:45) (cid:45) P z , Ρ nn T (s) (cid:45) (cid:45) P z , Ρ nn (cid:72) a (cid:76) (cid:72) b (cid:76)(cid:72) c (cid:76) (cid:72) d (cid:76) FIG. 3. (Color available online) Equalizing the state | ⟩ and | ⟩ instantaneous population decays by a variable splitter pulse toenhance P z . The model is evaluated for the following parameters: ∆ = 0 rad s − , T p = π/ (2Ω R ) , γ = 1 . s − , γ = 0 . s − , γ d = 0 s − , φ = 2 π × rad, and Ω = 2 π × rad s − . ρ (blue) and ρ (red) are from Eqs. (13), P z (dashed) from Eq. (10). (a) T p = π/ : equally split populations ρ nn at the end of free evolution, at T ; (b) T p = π/ : unequal splitting to produce a crossing of the population decays before the arrival of the second pulse; (c) T p = π/ : after the complete sequence, P z has a monotonic sech[( γ − γ ) T / envelope at γ ̸ = γ according to Eq. (7c); (d) T p = π/ : after the complete sequence a peak of visibility is observed at T = T optimal ≈ . s as expected from Eq. (14). not satisfied, the equation P z ( t ) = 0 gives an unphysi-cal imaginary result. The maximal possible off-resonant π/ -pulse duration that provides equal population split-ting is then found to be t π/ = π (cid:14)(cid:0) Ω √ (cid:1) .A direct application of the T p variation is to modelthe experimental imperfections, associated with the sec-ond pulse. The T p variation is of a more subtle charac-ter: The splitter allows us to initiate free evolution withunequal populations that may evolve into equal popula-tions. For this to happen, the state with the higher lossrate needs to be loaded more at the beginning of the evo-lution. The point where the unequally split populationsequalize gives the maximal normalized population value.By chasing the optimal value of T by accordingly correct-ing the splitter duration T p one can attain a perpetuallymaximal contrast of P z ( T ) . This is of benefit for datafitting since the envelope function becomes constant. Ofcourse, this method does not affect the signal-to-noiseratio defined by the fundamental limit, the Heisenberguncertainty. D. One-body loss asymmetry cancellation
The visibility in long Ramsey experiments decreasesdue to loss asymmetry as one of the dominant factors[23]. A look at the population decays in Figs. 3(a,b)suggests that if the populations start from unequal val-ues N ( T = 0) > N ( T = 0) , then N ( T ) and N ( T ) cross. P z has a maximum at this point; i.e., the lossasymmetry is cancelled. One can tailor a sequence withvariable splitter π/ -pulse duration to obtain a Ramseyfringe that gives unitary visibility at a desired location.To derive the expression for the optimal first-pulse du-ration, the density operator ρ ( t ) is propagated until theend of free evolution, before the second π/ -pulse, wherethe populations are ρ ( t ) = 12Ω R e − γ t (cid:0) ∆ + Ω R + Ω cos (Ω R T p ) (cid:1) ,ρ ( t ) = 12Ω R e − γ t Ω (cid:0) − cos (Ω R T p ) (cid:1) . (13)Then the crossing of the populations is found by solv-ing equation ρ ( t ) = ρ ( t ) with respect to time t anddiscarding irrelevant solutions. The solution gives timewhere the maximum visibility of P z occurs as a functionof T p ; we label this time T optimal further on: T optimal = 1 γ − γ ln (cid:32) Ω R Ω csc (cid:18) Ω R T p (cid:19) − (cid:33) . (14)In Fig. 3 the effect of loss compensation is shown witha set of test parameters. Figure 3(a) shows the dynamicsof the freely evolving populations following the applica-tion of the standard π/ splitting pulse. The populationsare plotted before the arrival of the detecting π/ -pulse.In contrast, Fig. 3(b) shows how the splitter can affect (cid:2) (cid:2) (cid:2)
10 10 20 30 (cid:3) Π (cid:2) kHz (cid:3) (cid:2) (cid:2) P z (cid:2) (cid:2) (cid:2)
10 10 20 30 (cid:3) Π (cid:2) kHz (cid:3) (cid:2) (cid:2) P z (cid:2) (cid:2) z (cid:2) (cid:2) (cid:2)
10 10 20 30 (cid:3) Π (cid:2) kHz (cid:3) (cid:2) (cid:2) P z (cid:2) (cid:2) (cid:2)
10 10 20 30 (cid:3) Π (cid:2) kHz (cid:3) (cid:2) (cid:2) P z (cid:2) (cid:2) z (cid:2) a (cid:3) (cid:2) b (cid:3)(cid:2) c (cid:3) (cid:2) d (cid:3) FIG. 4. (Color available online) Ramsey spectra. The following values are used:
Ω = 2 π × (510 Hz ) , T = 5 ms. The measurable P z (∆ , T ) with the Ramsey condition φ = ∆ (the one-body model of Eqs. (5)) is plotted in (a) . Losses are neglected. The Rabipedestal in (a) has a narrow line shape at T = 0 and has a single-peak coming from Eq. (10) where the pulses are assumed tobe π/ for all ∆ . Whereas in (c) , corresponding to the model of Eqs. (5), the pulses are π/ only at resonance; at ∆ ̸ = 0 theybecome non- π/ -pulses distorting the spectrum. (b) and (d) are differences P z − I z of (a) and (c) , correspondingly. the populations and lead to their balance at an arbitrarytime T . Figures. 3(c,d) show the populations and mea-surable P z after the full interferometric sequence with anon-zero φ producing a fringe. In accordance with theexpectations, Fig. 3(d) indicates an extremum in the vis-ibility at T = T optimal defined by Eq. (14). III. RABI PEDESTALS & RAMSEY SPECTRA
As in the case of the Rabi model of Eqs. (11), theRamsey spectrum (Fig. 4, lossless, φ = ∆ ) has a combof resonances at around ∆ = 0 that narrow down withincreasing evolution time T . At resonance the visibilityis highest and the slope is steepest, which is ultimatelyconverted to the best interferometer accuracy.For the measurables P , P , and P z the correspond-ing Rabi pedestal [22] functions I , I , and I z with al-ways resonant π/ -pulses given by T p = T p = π/ / Ω R ,forming the baseline for the Ramsey oscillations, are { + g , − g , g } , where g = ∆ / Ω R . It is remark-able that the pedestals, and, consequently, the oscilla-tion envelopes, are more flat at around ∆ = 0 , thanthe Lorentzians of the Rabi spectra [23]. More generalpedestal functions are obtained by averaging the measur-ables from the variable-pulse model: I Ω R = (2 + a + b )∆ Ω + (1 + ab )Ω + 2∆ , I Ω R = (cid:0) Ω + ∆ (2 − b ) − a (cid:0) ∆ + b Ω (cid:1)(cid:1) Ω , I z Ω R = (cid:0) a Ω + ∆ (cid:1) (cid:0) b Ω + ∆ (cid:1) , (15)with a = cos (Ω R T p ) and b = cos (Ω R T p ) . The av-erages of the standard Ramsey pulses can be modelled by setting T p = T p = π/ / Ω to Eqs. (15). Thesebaseline functions only depend on the pulse parame-ters Ω , ∆ , T p , and T p . Hence, they isolate Ramsey-interference and Rabi-pulse-related contributions. Dif-ferences { P − I , P − I , P z − I z } contain only Ramsey-related interference patterns (Figs. 4(b,d)).In the model of Eqs. (5) the π/ -pulses split the pop-ulations of the two states : , even off resonance with Ω ̸ = Ω R . This is different from the standard π/ -pulsewhose duration is adjusted while at resonance and keptconstant when the detuning ∆ is changed. Such pulseswith the ∆ -dependent duration produce a Rabi pedestalwith a single broad peak as shown in Fig. 4(a). In theRamsey approach the π/ -pulse durations are kept con-stant, i.e., T p = T p = π/ in Eq. (10). In this casethe pulses split the populations into halves at resonance,but provide an unequal splitting away from resonance.The corresponding Ramsey spectrum has an infinite se-quence of maxima in the envelope function in Fig. 4(c).Losses in Fig. 4 are neglected for they are system spe-cific; however, in a more realistic model the effects of theMaxwell-Boltzmann velocity distribution, atomic motionand miscellaneous inhomogenieties [24] may suppress ordistort the Ramsey features away from ∆ = 0 , resultingin a spectrally more localized envelope [25, 26].These two models (Fig. 4) have different assumptionsabout how the π/ -pulse duration is defined in experi-ment. Typically in applications, near resonant operationis desirable to benefit from high visibility; hence in thepresent discussion we neglect the contrast loss away from ∆ = 0 , and the envelope shift away from ∆ = 0 andwe limit ourselves to the dominant effect of Ω R solelyforming the broad spectral envelope in Fig. 4. Individualfeatures of the spectral comb are also affected by the colli-sional shift [27] and the effects of the trap [24]; e.g., phasedifference acquired during the evolution, if any, shifts theinterference pattern. In clocks this effect is undesirable,but it is routinely used in sensing applications. IV. CONCLUSIONS
In this work we presented a set of Ramsey-type mod-els in which Ramsey interference and Rabi-pulse-relatedpedestal were separated, and one-body models were gen-eralized into a variable-duration pulse model with thedetunings separately defined for the periods of pulse cou-pling and free evolution.The presented models are of general interest; theycan be employed when many-body physics is negligible.A many-body model would be described by a systemof nonlinear differential equations difficult to solve inan exact analytical form [20]. The one-body model isparametrized with the detuning and with variable pulsedurations to model realistic systems where pulses maybe generated with imperfections or be distorted by themedium. Such a flexible model allows to find an optimalsplitter-pulse duration that cancels the effect of unequalone-body losses on the P z visibility [23]. It turns out thatthe expression for the optimal splitter pulse duration hasa simple analytical form. The cancellation strategy canbe extended to the many-body case; however, a greaternumber of decay channels would ensue a more complexanalysis.Equation (14) is valid for one-body-decay limited sys-tems. The many-body counterpart of Eqs. (4) can also beobtained [20]; in practice this implies numerical integra-tion to search the corresponding T optimal . The attainedeffect of constant P z ( ϕ, T ) visibility can facilitate, e.g.,atom-clock stability analysis. It should be noted thatthis technique does not improve the signal-to-noise ratio.The approach is equally valid for time- and phase-domainRamsey experiments [23]. ACKNOWLEDGMENTS
The author thanks Andrei Sidorov and Peter Han-naford for stimulating discussions.
Appendix A: Off-resonant spin-echo interferometrywith one-body losses
In spin-echo interferometry (Fig. 1(b)) B also under-goes rotations by Ω (Fig. 5). In the middle of the se-quence a phase-reversal π -pulse flips the dephasing direc-tion, and the pseudo-spins start rephasing and refocusingby the second π/ -pulse. The solution of Eq. (2) for anoff-resonant spin-echo sequence with losses during freeevolution is parametrized by the total sequence duration T , and the cumulative Ramsey dephasing rates in thetwo arms of the interferometer φ and φ that allow oneto include miscellaneous physical mechanisms displacingthe energy levels.It is convenient to isolate interference from decay terms a = e − γ T , a = e − γ T , a = e − γ γ T , a = e − γ T ,a = e − γ γ T , a = e − γ γ T , a = cos (cid:16) φ T (cid:17) ,a = cos (cid:16) φ T (cid:17) , a = sin (cid:16) φ T (cid:17) , a = sin (cid:16) φ T (cid:17) ,b = e ( γ + γ ) T , b = e ( γ + γ + γ ) T , b = e ( γ + γ ) T ,b = e ( γ + γ ) T , b = e ( γ + γ ) T , b = e ( γ + γ − γ ) T ,k = ∆ Ω b Ω R + ∆ b Ω R − Ω b a Ω R + 2Ω b Ω R +2∆ Ω b a Ω R − b a Ω R + 2∆Ω b a Ω R +∆ b Ω R , (A1)from ρ , ρ , and P z : ρ · R = ∆ a Ω R − Ω a a − a a − Ω a a a − Ω a a a + ∆ Ω a Ω R + 2∆ Ω a Ω R + 2∆ Ω a a Ω R − Ω a a Ω R + 2∆ Ω a a Ω R − Ω a a Ω R + ∆ a Ω R + 2Ω a a a Ω R + 2∆ Ω a a Ω R + 2∆ Ω a a Ω R − a a Ω R − Ω a a a Ω R + 2∆ a Ω R + 2Ω a Ω R + 2∆ Ω a a Ω R + 2∆ Ω a a Ω R + 2Ω a a a Ω R + 2∆Ω a a Ω R + 2∆Ω a a Ω R + 6∆ Ω a a a Ω R , (A2a) ρ · R = ∆ Ω a Ω R + ∆ Ω a Ω R − Ω a a − Ω a a − a a + 4∆ Ω a a a + 4∆ Ω a a a + 4∆ Ω a Ω R + 4∆ Ω a Ω R − a a Ω R + 2Ω a Ω R + 2∆ Ω a a Ω R − Ω a a Ω R − Ω a a Ω R − Ω a a Ω R + 2∆ Ω a a Ω R − Ω a a a Ω R − a a a Ω R + 2∆Ω a a Ω R − Ω a a Ω R + 2∆Ω a a Ω R + 4∆ Ω a a a Ω R + ∆ Ω a Ω R + ∆ Ω a Ω R − Ω a a Ω R − a a a Ω R , (A2b) P z · k = 2∆ Ω b a − Ω b a a Ω R + 2∆Ω b a Ω R + 2∆ Ω b a − Ω b a a − Ω b a a − ∆ Ω b Ω R − Ω b Ω R + 2∆ b Ω R + ∆ Ω b Ω R + 2∆ Ω b a Ω R + 2∆ Ω b a Ω R − Ω b a Ω R + 4∆ Ω b a Ω R + 2∆ Ω b a Ω R + 6∆ Ω b a a Ω R + 2Ω b a a Ω R − b a Ω R + 2∆ Ω b a Ω R + 2∆ Ω b a + 2Ω b a a Ω R + 2∆ Ω b a . (A2c)Provided the detuning is zero, which is physically justi- Initial state detectiontime Π Π Π (cid:72) Φ (cid:76) T (cid:144) (cid:144) (a) (b) (c) (d) (e) (f) FIG. 5. (Color available online) Spin-echo interferometry in the Bloch vector representation. ϕ is the phase of the second π/ -pulse. As in Ramsey interferometry (Fig. 2), ϕ modulation can be used at a fixed T to record a P z fringe. Insets (a) , (b) , (c) show dynamics identical to that of Figs. 2(a,b,c). In (d) the spin-echo pulse implements phase reversal, and the ensemblestarts rephasing. (e) After the second π/ -pulse the Bloch vectors are refocused. (f ) The ensemble spin spread is nullified. fied in the case of ∆ ≪ Ω , Eqs. (A2) become ρ = 12 e − γ T (cid:20) e γd T + cos (cid:18) φ − φ T (cid:19)(cid:21) , (A3a) ρ = 12 e − γ T (cid:20) e γd T − cos (cid:18) φ − φ T (cid:19)(cid:21) , (A3b) P z = e − γ d T cos (cid:18) φ − φ T (cid:19) . (A3c)It follows from Eqs. 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