Tomographic reconstruction of the Wigner function on the Bloch sphere
TTomographic reconstruction of the Wigner functionon the Bloch sphere
Roman Schmied and Philipp Treutlein
Departement Physik, Universit¨at Basel, Klingelbergstrasse 82, CH–4056 Basel,SwitzerlandE-mail: [email protected]
Abstract.
We present a filtered backprojection algorithm for reconstructing theWigner function of a system of large angular momentum j from Stern–Gerlach-type measurements. Our method is advantageous over the full determination ofthe density matrix in that it is insensitive to experimental fluctuations in j , andallows for a natural elimination of high-frequency noise in the Wigner functionby taking into account the experimental uncertainties in the determination of j ,its projection m , and the quantization axis orientation. No data binning andno arbitrary smoothing parameters are necessary in this reconstruction. Usingrecently published data [Riedel et al. , Nature :1170 (2010)] we reconstruct theWigner function of a spin-squeezed state of a Bose–Einstein condensate of about1250 atoms, demonstrating that measurements along quantization axes lying ina single plane are sufficient for performing this tomographic reconstruction. Ourmethod does not guarantee positivity of the reconstructed density matrix in thepresence of experimental noise, which is a general limitation of backprojectionalgorithms.
1. Introduction
The reconstruction of the quantum-mechanical state of a system from measurementsis an important topic of the emerging field of quantum technology [1]. Through partialor full state reconstruction we can estimate entanglement properties of multipartitequantum systems, and judge their usefulness for further experimental progress infields such as quantum metrology [2, 3, 4, 5, 6, 7, 8, 9], quantum simulation [10], andquantum computation [11, 12, 13, 14, 15].Particularly in quantum metrology, experiments often involve large numbersof particles, and single-particle resolution is unavailable in both control andmeasurement. Because of this limitation, standard methods for reconstructing thequantum-mechanical density matrix [16, 17, 13] cannot be applied. For instance, andcentrally to this work, in a Bose–Einstein condensate consisting of N atoms, with eachatom representing a pseudo-spin-1 / j = N/ N ↑ and N ↓ , in terms of which the total spin is j = ( N ↑ + N ↓ ) / m = ( N ↑ − N ↓ ) /
2. Since it isvery difficult to determine the populations N ↑ and N ↓ with atomic accuracy [18, 19],the density matrix, which requires knowledge of j , becomes impossible to reconstruct a r X i v : . [ qu a n t - ph ] M a r omographic reconstruction of the Wigner function on the Bloch sphere j + 1) degrees of freedom of the densitymatrix [16, 17, 20] requires at least as many uncorrelated measurements, and thereforethe experimental uncertainty in m will hinder this full determination. In the absenceof reliable data, there will be significant uncertainty and noise throughout the densitymatrix in its Dicke representation ρ mm (cid:48) = (cid:104) jm | ˆ ρ | jm (cid:48) (cid:105) , which severely limits itsusefulness. We need a method for calculating those components of ˆ ρ which aresignificant even in the presence of noise and for very large values of j , and a wayof determining which components must remain unknown.The Wigner function [21] is ideal for such a controlled reconstruction. It is a real-valued function on a sphere of radius (cid:126) (cid:112) j ( j + 1), represented in terms of orthonormalLaplace spherical harmonics as [22] W ( ϑ, ϕ ) = j (cid:88) k =0 k (cid:88) q = − k ρ kq Y kq ( ϑ, ϕ ) , (1)where ϑ is the polar angle measured from the + z axis, and ϕ is the azimuthalangle around the z axis. While this sphere is commonly called a generalized Blochsphere [4], its surface actually represents a two-dimensional phase space instead ofa Hilbert space as for the original Bloch sphere. This Wigner function contains thesame information as the density matrix for any spin- j system. While the marginalsof the better-known Wigner function in planar space [21, 23, 24, 25] are real-spaceor momentum-space probability distributions, the marginals of the spherical Wignerfunction are the projection quantum number distributions along all quantization axes[see (6) below]; further, the expectation value of the angular momentum vector isproportional to the “center of mass” of the Wigner function, {(cid:104) S x (cid:105) , (cid:104) S y (cid:105) , (cid:104) S z (cid:105)} = (cid:113) j ( j +1)(2 j +1)4 π × (cid:82) π sin ϑ d ϑ (cid:82) π d ϕ { sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ } W ( ϑ, ϕ ).Most importantly, the Wigner function allows us to differentiate between moresignificant components ρ kq (with smaller values of k ) and more noise-prone components(with larger values of k ) in a natural way. Further, if only components with k (cid:28) j are reconstructed, then accurate knowledge of j is not necessary. As detailed insection 2, the transformation from j -space (the Dicke representation ρ mm (cid:48) of thedensity matrix) to k -space (the spherical harmonic decomposition ρ kq of the Wignerfunction) proceeds though coupling coefficients which, at low k , are smooth in both j and m ; this significantly reduces the impact of uncertainties in the experimentaldetermination of ( j, m ).Methods for reconstructing planar Wigner functions by inverse Radon transformare well established in the context of nonlinear optics [24, 25]. In the past theyhave also been applied to tomographic data on large-spin quantum systems, locallyapproximating the Bloch sphere by a tangental plane and neglecting its curvature [6].While this approximation is valid for spin states which are very localized on theBloch sphere and do not wrap around it, future experimental progress is expected toproduce ever more delocalized states ( e.g. , Schr¨odinger-cat states) whose propertiesare strongly influenced by the spherical shape of the Bloch sphere. Previous work onthe reconstruction of the Wigner function on the full Bloch sphere has used the Husimi- Q distribution as input [26], which is the convolution of the system’s Wigner functionwith that of a coherent state (see section 3). This convolution washes out featuresof the Wigner function that are smaller than a coherent state. Since the principalcharacteristic of spin-squeezed states is that their Wigner function possesses a peakwidth smaller than that of a coherent state, such a deconvolution-based reconstruction omographic reconstruction of the Wigner function on the Bloch sphere m of a quantum system along a certain quantization axis. In our case this correspondsto a single run of state preparation and population determination of a two-componentBose–Einstein condensate, yielding a single tuple ( j n , m n ). The equivalent for theoriginal experiment [27] is sending a single silver atom through the experimentalapparatus, and determining its deflection by the magnetic field gradient. On theother hand, a “Stern–Gerlach experiment” is a series of many single Stern–Gerlachmeasurements with fixed quantization axis, sufficient to determine the probabilitydistribution { p − j , p − j +1 , . . . , p j } while j is presumed fixed.
2. Wigner function reconstruction by filtered backprojection
The density matrix ˆ ρ of a system of total angular momentum j (assumed fixed here;this condition will be relaxed in section 2.1) is usually expressed in one of the twoforms ρ mm (cid:48) = (cid:104) jm | ˆ ρ | jm (cid:48) (cid:105) = j (cid:88) k =0 k (cid:88) q = − k ρ kq t jmm (cid:48) kq (2 a ) ρ kq = j (cid:88) m = − j j (cid:88) m (cid:48) = − j ρ mm (cid:48) t jmm (cid:48) kq , (2 b )with the transformation coefficients (in the following simply termed Clebsch–Gordancoefficients) [22] t jmm (cid:48) kq = ( − j − m − q (cid:104) j, m ; j, − m (cid:48) | k, q (cid:105) , (3)nonzero only if q = m − m (cid:48) . Both forms contain the same information and arecompletely interchangeable. While form (2 a ) is more common, form (2 b ) allowsexpressing the Wigner function on the Bloch sphere (1). Since our goal is thereconstruction of the Wigner function from experimental data, we focus on form (2 b ),in particular its low- k components.In order to determine the unknown quantum-mechanical state of a system oftotal spin j , it is necessary that many instances of this state can be generatedexperimentally [1], on which destructive measurements are performed. Further,projective Stern–Gerlach measurements must be performed along many differentquantization axis orientations ( ϑ n , ϕ n ). For the correctness of the followingreconstruction method it is crucial that these measured quantization axes aredistributed as evenly as possible over the hemisphere of orientations. Since thisrequirement may be difficult to fulfill experimentally, we assign weights c n to omographic reconstruction of the Wigner function on the Bloch sphere m n of the Stern–Gerlachmeasurements. In the ideal case of homogeneously distributed quantization axisorientations (for example through the vertices of a geodesic hemisphere), all theseweights are chosen equal and the data are used most efficiently.In this way, the results from M single Stern–Gerlach measurements along variousquantization axes orientations are assembled into a data set of tuples ( ϑ n , ϕ n , c n , m n )with n = 1 . . . M and (cid:80) Mn =1 c n = 1. Our filtered backprojection algorithm forreconstructing the Wigner function coefficients is then given by ρ (fbp) kq = (2 k + 1) M (cid:88) n =1 c n D kq ( ϕ n , ϑ n , t jm n m n k , (4)with D jm (cid:48) m ( α, β, γ ) = (cid:104) jm (cid:48) | e − i α ˆ J z e − i β ˆ J y e − i γ ˆ J z | jm (cid:105) a Wigner rotation matrix [28];in particular D kq ( ϕ, ϑ,
0) = (cid:113) π k +1 Y ∗ kq ( ϑ, ϕ ). This is formally equivalent to thefiltered backprojection algorithm used for planar inverse Radon transforms [29],with the factor 2 k + 1 representing the “filter”, and the summand representing thebackprojection. Our algorithm has all of the typical properties of planar inverse Radontransforms by filtered backprojection: no data binning is required, and there are no adhoc parameters to be chosen or optimized. Further, as the backprojection algorithmis a direct sum and does not include an inversion step (such as a straight inversionof the Radon transform would require), the impact of experimental noise is boundedin the result. It is this last property which makes backprojection algorithms fast andreliable in practical applications such as X-ray computed tomography [29].Our specific backprojection (4) can be interpreted in an intuitive way. Themeasured values of m n in the coordinate frame attached to the quantization axis( ϑ n , ϕ n ) are distributed according to the diagonal elements ρ m n m n and are convertedfrom j -space into k -space via the Clebsch–Gordan coefficients t jm n m n kq (cid:48) with q (cid:48) = 0 (seesection Appendix A for a numerical procedure). They are then rotated into the labframe through the rotation matrices D kqq (cid:48) ( ϕ n , ϑ n , χ n ) with the value of χ n irrelevant(set to zero) since q (cid:48) = 0.In the following, we demonstrate that this algorithm (4) works in the limit ofinfinite data. If all quantization axis orientations have been used with equal frequency,and infinitely many measurements have been performed along each quantizationaxis, the sum over measurements (cid:80) Mn =1 c n can be replaced by a normalized integral π (cid:82) π/ sin ϑ d ϑ (cid:82) π d ϕ over the hemisphere of axis orientations (by symmetry the otherhemisphere yields an identical result) and a sum over the measurement outcomes m , ρ (fbp) kq = 2 k + 12 π (cid:90) π/ sin ϑ d ϑ (cid:90) π d ϕ j (cid:88) m = − j p m ( ϑ, ϕ ) D kq ( ϕ, ϑ, t jmmk , (5)where the Stern–Gerlach probability distribution along a quantization axis ( ϑ, ϕ ) isgiven by the diagonal elements ρ mm of (2 a ) in the rotated frame, p m ( ϑ, ϕ ) = j (cid:88) k =0 k (cid:88) q = − k [ D kq ( ϕ, ϑ, ∗ ρ kq t jmmk . (6) omographic reconstruction of the Wigner function on the Bloch sphere j (cid:88) m = − j t jmmk t jmmk (cid:48) = δ kk (cid:48) , (7)and spherical harmonics, (cid:90) π/ sin ϑ d ϑ (cid:90) π d ϕ [ D kq (cid:48) ( ϕ, ϑ, ∗ D kq ( ϕ, ϑ,
0) = 2 π k + 1 δ qq (cid:48) , (8)it is easy to show that indeed ρ (fbp) kq = ρ kq , proving the validity of the reconstructionmethod in the limit of infinitely many homogeneously distributed Stern–Gerlachexperiments.In the more experimentally relevant case of a finite data set, the literature onthe two-dimensional inverse Radon transform by filtered backprojection [29] indicatesthat excellent results can still be recovered, albeit with aliasing artifacts present tosome degree. As a rough estimate, if Stern–Gerlach experiments are performed onlyalong certain quantization axes spaced by an average angle ∆ η , then the reconstructedpartial waves of the Wigner function become unreliable for k (cid:38) k max = π/ ∆ η . Further,if the number M of measurements is much less than the number of degrees of freedom( k max + 1) , then the reconstructed coefficients ρ kq will be dominated by noise, inparticular at large k . Both of these effects are mitigated in section 2.2 for the presentreconstruction scheme. j We recall that for systems composed of many spin-1 / j = ( N ↑ + N ↓ ) / j , we notice that for k (cid:28) j the Clebsch–Gordan coefficients t jmmk depend smoothly on the total angular momentum j . Thisallows us to reconstruct the low-resolution part of the Wigner function even if j variesslightly between single Stern–Gerlach measurements. To this end we include themeasured values of j in the data tuples, extending them to ( ϑ n , ϕ n , c n , j n , m n ); thefiltered backprojection formula is modified to ρ (fbp) kq = (2 k + 1) M (cid:88) n =1 c n D kq ( ϕ n , ϑ n , t j n m n m n k . (9)Again we refer to Appendix A for a numerical method to evaluate this expression.The same smoothness of the Clebsch–Gordan coefficients at low k is usedin section 2.2 to treat measurement uncertainties in both j n and m n in aperturbative manner in (9). This is fundamentally different from a direct tomographicreconstruction of the Dicke matrix elements ρ mm (cid:48) , where such uncertainties introducelarge but correlated errors throughout the density matrix and make such a perturbativetreatment impossible. k damping It is natural to assume that M uncorrelated experimental measurements can only serveto reconstruct M coefficients ρ kq , suggesting an upper limit k max ≈ √ M (assuming omographic reconstruction of the Wigner function on the Bloch sphere k the angular power spectrum [30] C (fbp) k = 12 k + 1 k (cid:88) q = − k | ρ (fbp) kq | (10)tends to acquire large fluctuations because of insufficient experimental data (seefigure 3 for an example). However, simply cutting the reconstruction off at k max is unsatisfactory because it disregards that some useful information is still presentin these high- k partial waves. A more natural cutoff is introduced through the k -dependent sensitivity to experimental uncertainties. Assuming experimental variancesof (cid:104) N ↑ (cid:105) − (cid:104) N ↑ (cid:105) = (cid:104) N ↓ (cid:105) − (cid:104) N ↓ (cid:105) = σ N , we find that the uncertainties of (cid:104) j (cid:105) − (cid:104) j (cid:105) = (cid:104) m (cid:105) − (cid:104) m (cid:105) = σ N / (cid:104) jm (cid:105) = (cid:104) j (cid:105)(cid:104) m (cid:105) ) yield a leading orderdamping of the Clebsch–Gordan coefficients (cid:104) t jmmk (cid:105) ≈ t jmmk exp (cid:20) − σ N j (2 j − k ( k + 1) (cid:21) . (11)The rotation matrix elements are damped similarly: if the pointing direction of thequantization axis Ω = ( ϑ, ϕ ) has an uncertainty of σ Ω (cid:28) { in terms of the expectationvalue of the angle η ΩΩ (cid:48) between the ideal axis orientation Ω and its true experimentalvalue Ω (cid:48) we define σ = (cid:104) sin η ΩΩ (cid:48) (cid:105) = (cid:104) − [cos ϑ cos ϑ (cid:48) + sin ϑ sin ϑ (cid:48) cos( ϕ − ϕ (cid:48) )] (cid:105)} ,then for large k we find the rotation matrix elements to be damped as (cid:104) D kq ( ϕ, ϑ, (cid:105) ≈ D kq ( ϕ, ϑ,
0) exp (cid:20) − σ k ( k + 1) (cid:21) . (12)If σ N and σ Ω are equal for all measurements, the linearity of (9) yields a simplesmoothing ρ kq (cid:55)→ ρ kq e − αk ( k +1) with α = σ N j (2 j − + σ . In this way, these two dampingformulas (11,12) cut off the reconstruction at large k in a natural and smooth way. Inserting the resulting coefficients (9) into the form of the Wigner function (1) we findthe tomographically reconstructed Wigner function W (fbp) ( ϑ, ϕ ) = M (cid:88) n =1 c n j (cid:88) k =0 k (cid:88) q = − k (2 k + 1) D kq ( ϕ n , ϑ n , Y kq ( ϑ, ϕ ) t j n m n m n k = M (cid:88) n =1 c n Ξ j n ,m n [cos ϑ cos ϑ n + sin ϑ sin ϑ n cos( ϕ − ϕ n )] , (13)where the contributions can be simplified toΞ jm ( x ) = 1 √ π j (cid:88) k =0 (2 k + 1) / t jmmk P k ( x ) . (14)As is to be expected in spherical symmetry, the contribution of an individual Stern–Gerlach measurement (see figure 1) depends only on the relative angle cos η ΩΩ n =cos ϑ cos ϑ n + sin ϑ sin ϑ n cos( ϕ − ϕ n ) between the quantization axis orientation Ω n =( ϑ n , ϕ n ) of the measurement and the point Ω = ( ϑ, ϕ ) on the Bloch sphere (figure 2).Similarly to technical implementations of the planar inverse Radon transform [29], theWigner function is thus assembled from additive contributions due to the individual omographic reconstruction of the Wigner function on the Bloch sphere pp /2 p /4 3 p /4 h -20-15-10-505101520 m a nd X j m ( c o s h ) Figure 1.
Contributions Ξ jm (cos η ) to the Wigner function (14) for j = 20 and m = − . . . + 20. All curves have been divided by 100 and offset vertically by m . The bold curve for m = +16 is used in figure 2. Notice that the m = ± j contributions have lower spatial resolution (∆ η ∼ / √ j ) than those with m ≈ η ∼ /j ); see section 3. Figure 2.
Contribution to the Wigner function (14) for j n = 20 and m n = 16(see figure 1); colors as in figure 4 but scaled to the maximum value of +163. Thecontribution Ξ , (cos η ) depends only on the angle η between the quantizationaxis Ω n and the direction Ω in which the Wigner function is measured. omographic reconstruction of the Wigner function on the Bloch sphere jm (cos η ) ultimately determinethe spatial resolution of the reconstructed Wigner function: if the Wigner functionis composed predominantly of contributions with m n ≈ ± j n its angular resolution islimited by that of a coherent state, ∆ η (cid:38) / (cid:112) (cid:104) j (cid:105) ; if on the other hand the majorityof contributions has m n ≈ η (cid:38) / (cid:104) j (cid:105) .We make use of this observation in sections 3 and 4, where a spin-squeezed state isreconstructed and the increased spatial resolution is critically important. It is well known that only positive semi-definite density matrices represent validquantum-mechanical states of a system [1]. Unfortunately, the filtered backprojectionmethod (9) does not assure that the reconstructed ˆ ρ is positive semi-definite when usedwith a finite and noisy data set. For the purpose of displaying the Wigner functiongraphically, this is of no concern (see figure 4); however, when the tomographicallyreconstructed coefficients ρ (fbp) kq are used in quantitative calculations (see section 4)positivity can be crucial. This is a similar problem as the requirement for a positiveabsorption density in medical computed tomography (CT) imaging [29]. It is alsopresent in many quantum-state reconstruction schemes, and has been discussedextensively in the quantum tomography literature [1].We do not offer a solution for assuring the positivity of the reconstructed densitymatrix. Here we merely point out that in other reconstruction schemes, such asmaximum-likelihood estimates [31], the ansatz ˆ ρ = ˆ T † ˆ T forces the density matrix ˆ ρ tobe positive semi-definite; but a direct tomographic reconstruction of ˆ T similar to (9)is currently lacking.
3. Quantization axes lying in a single plane
When the spin- j system’s quantum-mechanical state is fairly localized on the Blochsphere, not every choice of quantization axis orientation has the same potential forextracting information about the state. When the axis is close to parallel to thestate, most Stern–Gerlach measurements will yield | m | ≈ j , with a limited angularresolution ∼ / √ j given by the size of a coherent state on the Bloch sphere [26]. Ifthe axis is close to perpendicular to the state, on the other hand, the distribution ofmeasured values m represents the structure of the state’s Wigner function much moreaccurately, with an angular resolution ∼ /j . This difference in scaling of the angularresolution, visible in figure 1, suggests that for large j it may be advantageous tofocus on performing Stern–Gerlach measurements with quantization axes in a planeperpendicular to the quantum state, instead of covering the entire hemisphere of axisorientations. As a consequence much fewer measurements are needed, and we can getmuch more rapid convergence of the reconstruction in practice. But it is not a priori clear that this restriction of the quantization axes to a single plane has the potentialfor reconstructing the full quantum-mechanical state of the system.As it turns out, a modification to the “filter” function in (9) results in a fullreconstruction of the mirror-symmetric part of the Wigner function. Defining thecoordinate system such that the state is localized near the + z axis and all quantization omographic reconstruction of the Wigner function on the Bloch sphere xy plane, the in-plane filtered backprojection formula is ρ (fbp , P) kq = ( k − q +12 ) ( k + q +12 ) π M (cid:88) n =1 c n D kq ( ϕ n , π , t j n m n m n k , (15)where ( a ) n = Γ( a + n ) / Γ( a ) is a Pochhammer symbol, and ( a ) ≈ √ a − / (8 √ a ).We again prove this reconstruction in the infinite-data limit. In the case ofa homogeneous distribution of all azimuthal axis orientation angles ϕ we use therelationship1 π (cid:90) π d ϕ [ D kq (cid:48) ( ϕ, π , ∗ D kq ( ϕ, π , δ qq (cid:48) ( k − q +12 ) ( k + q +12 ) π if k + q even0 if k + q odd, (16)which remains true in the experimentally more relevant case of a finite number A ofequally-spaced axis orientations (replacing π (cid:82) π d ϕ (cid:55)→ A (cid:80) A − a =0 with ϕ = aπ/A ) aslong as k < A . Together with (6) and (7) we thus find that ρ (fbp , P) kq = ( k − q +12 ) ( k + q +12 ) (cid:90) π d ϕ p m ( π , ϕ ) D kq ( ϕ, π , t j n m n m n k = (cid:40) ρ kq if k + q even0 if k + q odd. (17)Thus in the infinite-data limit such an in-plane reconstruction exactly determines thecoefficients ρ kq for which k + q is even, while giving no information on the coefficientsfor which k + q is odd. Since the parity of k + q is the z ↔ − z reflection parity ofthe spherical harmonics Y kq ( ϑ, ϕ ), the in-plane formula (15) reconstructs the positive-parity component W + ( ϑ, ϕ ) of the Wigner function W ( ϑ, ϕ ) = W + ( ϑ, ϕ ) + W − ( ϑ, ϕ ),with W ± ( π − ϑ, ϕ ) = ± W ± ( ϑ, ϕ ). If we know from other measurements that the stateis fully localized on the “northern” Bloch hemisphere ( z > W ( ϑ, ϕ ) = (cid:40) W + ( ϑ, ϕ ) if 0 ≤ ϑ < π π < ϑ ≤ π , (18)which has the decomposition ρ (fbp , P , N) kq = (cid:90) π sin ϑ d ϑ (cid:90) π d ϕ Y ∗ kq ( ϑ, ϕ ) W ( ϑ, ϕ ) = j (cid:88) k (cid:48) =0 Υ qkk (cid:48) ρ (fbp , P) k (cid:48) q (19)in terms of the overlap integrals Υ qkk (cid:48) given in Appendix B. We conclude that the dataacquired by Stern–Gerlach measurements with quantization axes lying solely within aplane are sufficient for an exact reconstruction of the Wigner function. k damping Measurement uncertainties can be introduced in (15) in the same way as in section 2.2.However, in an in-plane measurement series we can additionally separate out theazimuthal axis orientation uncertainty: since the rotation matrix elements D kq ( ϕ, ϑ, e − i qϕ , a variance (cid:104) ϕ (cid:105) − (cid:104) ϕ (cid:105) = σ ϕ leads to a damping (cid:104) D kq ( ϕ, π , (cid:105) = D kq ( ϕ, π ,
0) exp( − q σ ϕ ) . (20) omographic reconstruction of the Wigner function on the Bloch sphere -11 -10 -9 -8 -7 -6 -5 -4 -3 C k (f bp , P ) undampeddamped Figure 3.
Angular power spectrum (10) of the reconstructed Wigner functionof figure 4. Without damping ( (cid:31) ) the power in modes k (cid:38)
70 is too large anddominated by noise and aliasing effects; experimental uncertainties damp theangular power at large k in a natural way ( • , see sections 2.2 and 3.1). Odd- k modes contain less power than even- k modes because of the approximate pointsymmetry of the Wigner function (see figure 4).
4. Demonstration with experimental data
In this section we reconstruct the Wigner function from a data set describing ensemblesof N = 1250(45) atoms acquired in our group [6]. In contrast to [6] we rotatethe coordinate system such that all quantization axes lie in the xy plane and thestate is localized around the + z axis; in this way the procedure of section 3 can beemployed directly. The data set consists of three experimental runs spanning differentranges of ϕ with different angular resolutions, owing to the fact that the need forhomogeneity in ϕ for the filtered backprojection algorithm (15) was not known at thetime of data acquisition. We use weights c n adjusted such that the weighted densityof Stern–Gerlach measurements is as close to homogeneous as possible over the range ϕ = 0 . . . π of azimuthal quantization axis orientations. As discussed in section 3the planar arrangement of quantization axis orientations leads to a Wigner functionwhich is peaked along both the + z and − z directions, featuring two identical copies ofthe quantum state. An additional Ramsey experiment [6] was used to experimentallydetermine the correct location of the state on the northern ( z >
0) Bloch hemisphere.High- k damping (section 3.1) is achieved with an experimental uncertainty of σ N ≈
11 atoms (11) and with an experimental error model dominated by phase noise: σ ϕ ≈ σ sin( | ϕ | ) / √ σ ph = 8 . ◦ [6]. In figure 3the effect of this damping is shown to be crucial for partial waves k (cid:38) ϑ . omographic reconstruction of the Wigner function on the Bloch sphere Figure 4.
Reconstructed Wigner function using the data set described insection 4. The Wigner function takes values from − .
93 to +3 .
54. The coordinatesystem is rotated from [6] (see text).
We demonstrate the quantitative use of the reconstructed Wigner function byestimating the amount of spin squeezing in the system. Given a set of reconstructedWigner function coefficients ρ kq we can calculate the probability distribution for theangular momentum projection quantum number onto any quantization axis orientation( ϑ, ϕ ) from (6). In principle the variance V ( ϑ, ϕ ) = (cid:104) m (cid:105) ( ϑ, ϕ ) − [ (cid:104) m (cid:105) ( ϑ, ϕ )] measuresthe amount of spin noise obtained experimentally. The expectation values of smallinteger powers of the projection quantum number m depend only on the low- k components of the Wigner function, which are particularly insensitive to experimentalnoise (11,12,20); in particular, (cid:104) m (cid:105) ( ϑ, ϕ ) = j (cid:88) m = − j m p m ( ϑ, ϕ ) = (cid:114) (2 j ) (cid:88) q = − [ D q ( ϕ, ϑ, ∗ ρ q (cid:104) m (cid:105) ( ϑ, ϕ ) = j (cid:88) m = − j m p m ( ϑ, ϕ )= j ( j + 1) √ j + 13 ρ + (cid:114) (2 j − (cid:88) q = − [ D q ( ϕ, ϑ, ∗ ρ q . (21)In figure 5 we plot the resulting variances for quantization axes in the xy plane,and compare them to a coherent state centered on the + z axis. In the presence ofimaging noise the variance of such a coherent state is given by (21) with ρ (coh) kq = omographic reconstruction of the Wigner function on the Bloch sphere -100 -50 0 50 100 j -505101520 V ( p / , j ) / V c oh [ d B ] from data [6]from r kq from p m fits Figure 5.
Normalized variance V = (cid:104) m (cid:105) − (cid:104) m (cid:105) as a function of azimuthalquantization axis orientation ϕ . Open circles show variances calculated directlyfrom Stern–Gerlach experiments along a given quantization axis [6]. The dashedline was calculated directly from the coefficients ρ (fbp , P) kq through (21). The solidline shows the results of Gaussian fits (figure 6) to the probability distributions p m ( π , ϕ ) given in (6). As in [6] we first subtract the experimental noise ( σ N / σ N = 11) from the calculated variances, and then divide by the variance ofa coherent state, V coh = (cid:104) j (cid:105) / (cid:104) j (cid:105) = 630 [see (22)]. A spin-squeezed state ischaracterized by negative values (in dB). -600 -400 -200 0 200 400 600m00.010.020.03 p m j =6.7 o j =-88.0 o Figure 6.
Probability distributions p m of the projection quantum number m along the minimum-variance axis ( ϕ s = 6 . ◦ ) and the maximum-variance axis( ϕ = − . ◦ ) of figure 5, assuming j = 629 [found from ρ = (cid:104) (2 j + 1) − / (cid:105) ≈ . p m indicate that the reconstructed density matrix(as shown in figure 4) is not positive semi-definite and therefore does not strictlyrepresent a physical state (see section 2.4). Gaussian fits used for figure 5 areshown as continuous lines. omographic reconstruction of the Wigner function on the Bloch sphere t jjjkq e − σ N j (2 j − k ( k +1) , (cid:104) m (cid:105) (coh) = j ( j + 1)3 − j (2 j − e − σ N j (2 j − = j + σ N O ( σ N /j ) . (22)In figure 5 the experimental variance and the coherent-state variance are compared without imaging noise, i.e. , the leading-order imaging noise contribution σ N / p m ( ϑ, ϕ ) along a given quantization axis through (6) and fit it with a Gaussian curve(see figure 6); the variance of this fit then serves as an estimate of V ( ϑ, ϕ ). In sucha fit the positivity of the p m is no longer a crucial ingredient. In figure 5 we showthat this produces results that are very close to the variances calculated directly fromStern–Gerlach experiments along the various quantization axes. The deviations closeto the squeezing maximum ( ϕ s ≈ . ◦ ) result from the fact that the reconstructedWigner function contains contributions from all measurements, and therefore theextracted variance along a given quantization axis may be contaminated. Nonethelessthe reconstructed Wigner function delivers a very concise picture of the structureof the multiparticle state, even for a data set with a non-uniform distribution overquantization axis orientations and with fluctuating values of j n . Further, with ourmethod the variance V ( ϑ, ϕ ) can be calculated along any quantization axis orientation.In practice any proof of spin squeezing will not proceed through the reconstructionof the Wigner function followed by either a fit to the projection (6) or a direct studyof the projection noise (21). Instead, once the direction of squeezing ϕ s has beendetermined, a full Stern–Gerlach experiment will be performed along this axis inorder to directly estimate the probability distribution p m ( ϕ s ), as in [6] and in figure 5(circles). In this way, problems associated with the positivity of the reconstruction(section 2.4) and with the influence of experimental data and noise from directions ϕ (cid:54) = ϕ s are strictly eliminated.In future experiments providing data for the present tomographic reconstructionmethod, we plan to perform Stern–Gerlach measurements along many morequantization axes, but with as little as a single measurement per axis. Further, wewill pay attention to cover the entire range of quantization axes uniformly [either theentire equator for (15) or the entire sphere for (9)] in each experimental run. In thisway, we expect to need only minimal data preprocessing before reconstructing theWigner function, and will be able to use the acquired data in the most efficient wayby using equal weights c n = 1 /M for all data points. We also expect that for such animproved data set the variance of the simple estimate given by (21) will be closer tothat of the quantum-mechanical state. omographic reconstruction of the Wigner function on the Bloch sphere
5. Conclusions
We have presented a simple method for a tomographic reconstruction of the Wignerfunction of a spin- j system, applicable even to experimental settings where j is largeand fluctuates between measurements. While the general procedure (9) requiresStern–Gerlach type measurements spread uniformly over all possible quantizationaxis orientations, a more specialized and faster procedure (15) determines the Wignerfunction using only a single plane of quantization axis orientations. We have shownthat this latter procedure is capable of reconstructing the Wigner function of a spin-squeezed state from a recently published experimental data set [6]. Acknowledgments
We thank Jonathan Dowling and Wolfgang Schleich for helpful discussions, and MaxRiedel for help with the interpretation of the experimental data. This research wassupported by the Swiss National Science Foundation and by the European Communitythrough the project AQUTE.
Appendix A. Numerically evaluating Clebsch–Gordan coefficients
We have used a recursion relation [32] to evaluate the Clebsch–Gordan coefficients τ j,mk = t jmmk = ( − j − m (cid:104) j, m ; j, − m | k, (cid:105) from (3): τ j,jk = π / √ k + 12 j +1 / (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:18) j + 12 j − k (cid:19) (2 j + 1) τ j,j − k = (cid:18) − k ( k + 1)2 j (cid:19) τ j,jk τ j,mk = 2 j ( j + 1) − m + 1) − k ( k + 1) j ( j + 1) − m ( m + 1) τ j,m +1 k − j ( j + 1) − ( m + 1)( m + 2) j ( j + 1) − m ( m + 1) τ j,m +2 k τ j, − mk = ( − k τ j,mk . (A.1)This procedure is numerically stable even at very large values of j and k . Appendix B. Hemispherical overlap integrals of spherical harmonics
The hemispherical overlap integrals of the spherical harmonics are [33]Υ qkk (cid:48) = 2 (cid:90) π/ sin ϑ d ϑ (cid:90) π d ϕ Y ∗ kq ( ϑ, ϕ ) Y k (cid:48) q ( ϑ, ϕ ) omographic reconstruction of the Wigner function on the Bloch sphere k = k (cid:48) ( − k − k (cid:48)− q − k + k (cid:48)− (cid:112) (2 k + 1)(2 k (cid:48) + 1)( k − k (cid:48) )( k + k (cid:48) + 1) × (cid:115) ( k − q )!( k (cid:48) − q )!( k + q )!( k (cid:48) + q )! ( k (cid:48) + q )!!( k + q − k (cid:48) − q − )!( k − q )!if k − q even and k (cid:48) − q odd( − k − k (cid:48)− q − k + k (cid:48)− (cid:112) (2 k + 1)(2 k (cid:48) + 1)( k − k (cid:48) )( k + k (cid:48) + 1) × (cid:115) ( k − q )!( k (cid:48) − q )!( k + q )!( k (cid:48) + q )! ( k + q )!!( k (cid:48) + q − k − q − )!( k (cid:48) − q )!if k (cid:48) − q even and k − q odd0 otherwise . (B.1) References [1] M. Paris and J. ˇReh´aˇcek, editors.
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