Quantum atom-light interfaces in the Gaussian description for spin-1 systems

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Phys. 15 103007 (http://iopscience.iop.org/1367-2630/15/10/103007) Home Search Collections Journals About Contact us My IOPscience Quantum atom–light interfaces in the Gaussian description for spin-1 systems Giorgio Colangelo1,4,5, Robert J Sewell1,4, Naeimeh Behbood1, Ferran Martin Ciurana1, Gil Triginer1,2and Morgan W Mitchell1,3 1ICFO—Institut de Ciencies Fotoniques, Mediterranean Technology Park, E-08860 Castelldefels, Barcelona, Spain 2Universitat Politecnica de Catalunya, E-08034 Barcelona, Spain 3ICREA—Instituci´ o Catalana de Recerca i Estudis Avan¸cats, E-08015 Barcelona, Spain E-mail: gior[email protected] New Journal of Physics 15 (2013) 103007 (25pp) Received 2 May 2013 Published 7 October 2013 Online at http://www.njp.org/ doi:10.1088/1367-2630/15/10/103007 Abstract. We extend the covariance matrix description of atom–light quantum interfaces, originally developed for real and effective spin-1/2 atoms, to include ‘spin alignment’ degrees of freedom. This allows accurate modelling of optically probed spin-1 ensembles in arbitrary magnetic fields. We also include technical noise terms that are very common in experimental situations. These include magnetic field noise, variable atom number and the effect of magnetic field inhomogeneities. We demonstrate the validity of our extended model by comparing numerical simulations to a free–induction decay measurement of polarized 87Rb atoms in the f=1 ground state. We qualitatively and quantitatively reproduce experimental results with no free parameters. The model can be easily extended to larger spin systems, and adapted to more complicated experimental situations. 4These authors contributed equally to this work. 5Author to whom any correspondence should be addressed. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 15 (2013) 103007 1367-2630/13/103007+25$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 Contents 1. Introduction 2 2. Formalism 3 2.1. Quantum polarization description ......................... 3 2.2. Description of spin-1 ensembles ......................... 4 2.3. Spin visualization ................................. 5 2.4. Commutation relationships ............................ 6 2.5. Collective spin operators ............................. 6 2.6. Covariance matrix ................................. 7 3. Dynamics 7 3.1. Light–atom interactions .............................. 7 3.2. Linearization ................................... 8 3.3. Optically induced decoherence .......................... 9 3.4. Atom–field interaction .............................. 10 3.5. Combined effects ................................. 12 4. Measurement 12 5. Initial state and technical noise contributions 13 6. An example: free-induction decay of collective atomic spin 13 7. Simulation of free-induction decay 16 8. Conclusions 20 Acknowledgments 20 Appendix A. Proof of equations (23) and (24) 21 Appendix B. Noise from uncertain atom number 22 Appendix C. Inhomogeneous magnetic fields 22 References 23 1. Introduction Atomic ensembles play an essential and growing role in quantum optics [1], with applications in quantum networking [2], generation of optical quantum resources [3,4] and quantumenhanced instruments [5–8]. The continuous-variable (CV) approach [9] efficiently describes experiments involving many quanta. The great majority of CV atomic ensemble experiments are performed with Gaussian states, although non-Gaussian atomic states [10] are required for some tasks [11]. Gaussian states can be described very economically in terms of mean values and variances, whereas general states require a description exponential in the size of the system. The Gaussian approximation can be justified via the Holstein–Primakoff approximation, including extensions to larger-spin systems [12]. We follow the approach of Kraus et al [13], Madsen and Mølmer [14], Hammerer et al [15] and Mølmer and Madsen [16] and use covariance matrix techniques to describe the collective atom and light variables and their interaction. A wide variety of effects, including spatial and temporal inhomogeneities, loss, decoherence, atomic transport and projective measurements have been incorporated into this framework [17]. An important omission until now has been the description of larger-spin New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 3 systems. Most atomic ensemble experiments are performed with alkali atoms and thus have ground-state spin of at least 1, which implies tensorial light shifts [14] and generalized Faraday rotation effects [18]. An extension of the covariance matrix techniques to include these effects will allow statistical description of many large-spin applications [19], including quantum state characterization [20] and preparation [21], quantum chaos [22], optical magnetometry [8,23,24] and quantum non-demolition measurement [25,26]. While several earlier works have applied the spin-1/2 framework to spin-1 or larger systems through the identification of a two-state ‘pseudo spin-1/2’ sub-system, there are scenarios in which the dynamics naturally involves more than two levels, and requires a more expanded description. A clear example is a spin-1 or larger atom in the presence of both magnetic and optical fields. The magnetic field couples Zeeman states differing by 1m=±1, whereas the optical fields couple also states with 1m=±2 through tensorial light shifts. This system has been much studied using density-matrix approaches [20], which describe fully the average single-atom properties but not the noise properties, which arise from correlations among the atoms. Consideration of the quantum noise in these systems motivates the current work, in which we extend the covariance matrix approach to spin-1 atoms. To the suite of techniques available for spin-1/2 ensembles [14,17], we add the ability to treat both vectorial and tensorial light shifts, technical noise due to uncertainty in the atom number and dephasing due to magnetic field inhomogeneities. The paper is organized as follows. In section 2we present the formalism, which employs eight orientation and alignment operators to describe the F=1 collective atomic spin. In section 3we analyse the spin dynamics in the presence of probing light and an external magnetic field, including coherent evolution, decoherence due to scattering of probe photons and dephasing due to inhomogeneous magnetic fields. In section 4we review the description of optical measurement within the covariance matrix formalism. In section 5we describe the initial state including technical noise from uncertain atom number. In sections 6and 7we compare numerical results of our model with experimental data and identify the practical limits of the Gaussian approximation in this system. 2. Formalism We work with collective operators describing macroscopic numbers of particles, for which a CV description is appropriate. Throughout, we use the covariance matrix techniques [14,17], which are sufficient to describe the Gaussian states encountered in the great majority of CV experiments. 2.1. Quantum polarization description Polarized light in CVs can be described with Stokes operators: ˆ Sx≡1 2ˆ a†σxˆ a,ˆ Sy≡1 2ˆ a†σyˆ a,ˆ Sz≡1 2ˆ a†σzˆ a,ˆ S0≡1 2ˆ a†1ˆ a,(1) where ˆ a≡(ˆa+,ˆa−)Tand ˆa+,ˆa−are the annihilation operators for the left and right circular polarization and σx,σy,σzthe Pauli matrices and 1is the identity matrix. The ˆ Sx,ˆ Sy,ˆ Szand ˆ S0Stokes operators represent, respectively, linearly polarized light horizontally or vertically, linearly polarized light on the ±45◦direction, left and right circularly polarized light and the pulse energy. They have the same commutation relations as angular momentum operators, [ˆ Sx,ˆ Sy]=iˆ Szand cyclic permutations and they all commute with ˆ S0. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 4 2.2. Description of spin-1 ensembles A single spin-1 atom is described by a density matrix with eight degrees of freedom, which we express in terms of eight single-particle operators ˆ λi. These generalize the Pauli matrices, in the sense that they are traceless, Hermitian, and obey the orthonormality relation Tr(λiλj)=2δi j . The first three operators are the components of the spin vector ˆ f, obeying [ ˆ fx,ˆ fy]=iˆ fz. For illustration, we give the spin-1 matrix representation: ˆ fx F=1 −→ 1 √2  0 1 0 1 0 1 0 1 0 , ˆ fy F=1 −→ 1 √2   0−i 0 i 0 −i 0 i 0   , ˆ fz F=1 −→   1 0 0 0 0 0 0 0 −1   . The others are rank-2 tensor operators, for which we use the symbol ˆ j, with components (again with the spin-1 representation for illustration): ˆx≡ˆ f2 x−ˆ f2 y F=1 −→   0 0 1 0 0 0 1 0 0   , ˆy≡ˆ fxˆ fy+ˆ fyˆ fx F=1 −→   0 0 −i 0 0 0 i 0 0   , ˆk≡ˆ fxˆ fz+ˆ fzˆ fx F=1 −→ 1 √2   0 1 0 1 0 −1 0−1 0   , ˆl≡ˆ fyˆ fz+ˆ fzˆ fy F=1 −→ 1 √2  0−i 0 i 0 i 0−i 0 , ˆm≡1 √3(2ˆ f2 z−ˆ f2 x−ˆ f2 y)F=1 −→ 1 √3   1 0 0 0−2 0 0 0 1   . New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 5 Figure 1. Contribution dW/d¯ λiof the fand joperators to the Wigner distribution Wρ(θ, φ) representing the state ρ=1 3I+1 2Piˆ λi¯ λi, as described in [28]. Radius indicates magnitude, warm (cold) colours indicate positive (negative) contributions. Axis markers indicate unity. With quantization axis along z,ˆmdescribes the population imbalance between mF=0 and other states, while ˆx,ydescribe mF=±1 coherences. ˆk,lrepresent mF=±1 coherences in other quantization axes. We note that the above operator definitions are spin-independent, and that the results in this paper follow from these operator definitions, not from the spin-1-specific matrix representations. As we shall see below, the most important coherent interactions: Larmor precession, Faraday rotation and tensorial light shifts, can be fully described using the above operators, even for larger spin. The formalism developed here is thus applicable to some scenarios involving spin-3/2 and higher. Not all processes can be explained using just ˆ fand ˆ joperators, however. For example, with spin-2 atoms modulated optical pumping in the presence of a B-field has been used to produce hexadecapole moment due to coherence between Zeeman states with 1mF=4 [27]. 2.3. Spin visualization From the orthogonality relation Tr[λiλj]=2δi j , an arbitrary single-atom density matrix ρcan be expressed as ρ=1 3I+1 2X iˆ λi¯ λj,(2) where ¯ λi≡Tr[ρˆ λi]. This suggests a visualization in terms of the spin Wigner distribution W(ρ), which is efficiently calculated as in Dowling et al [28]. In figure 1we show the differential contribution to the Wigner distribution dW/d¯ λi. This shows, for example, that any of ˆx,ˆy,ˆk and ˆlcan be obtained by rotation of ˆx,whereas ˆmcannot. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 6 Table 1. Commutation relationships for single-atom operators. [,]ˆ fxˆ fyˆ fzˆxˆyˆkˆlˆm ˆ fx0ˆ fz−ˆ fy−ˆlˆk−ˆy√3ˆm+ˆx−√3ˆl ˆ fy−ˆ fz0ˆ fx−ˆk−ˆl−√3ˆm+ˆxˆy√3ˆk ˆ fzˆ fy−ˆ fx0 2 ˆy−2ˆxˆl−ˆk0 ˆxˆlˆk−2ˆy0 2 ˆ fz−ˆ fyˆ fx0 ˆy−ˆkˆl2ˆx−2ˆ fz0ˆ fx−ˆ fy0 ˆkˆy√3ˆm− ˆx−ˆlˆ fy−ˆ fx0ˆ fz−√3ˆ fy ˆl−√3ˆm− ˆx−ˆyˆkˆ fxˆ fy−ˆ fz0√3ˆ fx ˆm√3ˆl−√3ˆk000 √3ˆ fy−√3ˆ fx0 2.4. Commutation relationships The operators ˆ f,ˆ jand ˆ Shave commutators given by [ˆ λa,ˆ λb]=icˆ λaˆ λb ˆ λkˆ λk,(3) where the ˆ λare ˆ for ˆ jcomponents and a sum is over kis implied. The structure constants cˆ λaˆ λb ˆ λk are completely antisymmetric in the three indexes, and cˆ fxˆ fy ˆ fz=1,cˆxˆy ˆ fz=2,cˆ fxˆl ˆm=cˆ fyˆm ˆk=√3,(4) cˆ fxˆy ˆk=cˆ fxˆl ˆx=cˆ fyˆk ˆx=cˆ fzˆk ˆl=1.(5) To this we can add cˆ Sxˆ Sy ˆ Sz=1.(6) All structure constants not given above are zero. The commutators are given explicitly in table 1. 2.5. Collective spin operators To describe the ensemble we define collective operators. If ˆ λ(i)describes atomic operators acting the ith of NAatoms, then ˆ 3≡PNA iˆ λ(i). Explicitly for the vector (ˆ f) and tensor (ˆ j) collective spin operators: ˆ F≡ NA X i=1ˆ f(i),ˆ J≡ NA X i=1ˆ j(i).(7) We note that these inherit their commutation relations from the microscopic operators: [ˆ 3a,ˆ 3b]=icˆ λaˆ λb ˆ λkˆ 3k. Finally, we define a phase-space vector to describe the state of the whole system ˆ V=B⊕ˆ F⊕ˆ J⊕ Npulses M i=1ˆ S(i),(8) New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 7 where ⊕indicates the direct sum and Bis the magnetic field vector at the location of the ensemble. It should be noted that Bis here a classical field, whereas the other components of ˆ Vare operators. The inclusion of Bin ˆ Vallows for classical uncertainty about the field to be incorporated in a natural way into the calculations [14], as we describe below. 2.6. Covariance matrix We work within the Gaussian approximation, i.e. we assume that ˆ Vis fully characterized by its average hˆ Viand by its covariance matrix 0V: 0V≡1 2hˆ V∧ˆ V+(ˆ V∧ˆ V)Ti−hˆ Vi∧hˆ Vi,(9) where ∧indicates the outer product. It will be convenient to define δˆ V≡ˆ V−hˆ Vi, the fluctuations of ˆ Vabout the mean, from which 0V=1 2hδˆ V∧δˆ V+(δ ˆ V∧δˆ V)Ti. 3. Dynamics We describe the most important dynamical effects for light–matter interfaces, namely the light–matter interaction that occurs when a pulse of probe light passes the ensemble, and the rotation due to a magnetic field. Both of these interactions produce coherent rotations, loss of coherence and addition of noise. 3.1. Light–atom interactions The light–atom interaction is described by an effective Hamiltonian which describes the dispersive effects of the electric dipole interaction in second order [29–31]. In simulations and in practice, it is very convenient to employ a train of optical pulses for probing the ensemble. Even for continuous probing it is useful to treat the probe as a train of continuous pulses, as this allows a course-grained description of the polarization evolution [14,16]. In F=1 atomic ensembles, during the time the mth pulse is passing through the ensemble, the effective Hamiltonian is H(m) eff =G1ˆ S(m) z τˆ Fz+G2 ˆ S(m) x τˆ Jx+ˆ S(m) y τˆ Jy+1 √3 ˆ S(m) 0 τˆ Jm!,(10) where G1and G2are coupling constants that depend on the geometry of the atomic ensemble and probe beam, the atomic structure and the detuning from resonance [30]. In situations where the approximation of homogeneous interaction breaks down, it is possible to model the system as in [17] by breaking it into smaller subsystems each of which is effectively homogeneous. In the situations simulated in sections 6and 7, the inhomogeneous coupling is due to the spatial distribution of atoms within a Gaussian probe beam, and is estimated to be a small effect for the examples discussed in this paper. We label the Stokes operators as ˆ S(m)(t), m=1,...,Npulses where tmindicates the time of arrival at the ensemble. Thus ˆ S(m)(t<tm) describes the polarization of the mth pulse before entering the ensemble, while ˆ S(m)(t>tm+τ), where τis the pulse duration, describes the polarization of the same pulse after exiting the ensemble. At time tm+τ, the mth pulse has left the ensemble (we assume the transit time is much less than τ) and the change in the system is described by ˆ 3(tm+τ) =ˆ 3(tm)−iτ[ˆ 3(tm), H(m) eff ] (11) New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 8 and ˆ S(m)(tm+τ) =ˆ S(m)(tm)−iτ[ˆ S(m)(tm), H(m) eff ].(12) All other polarizations ˆ S(n),n6=mare unchanged. These first-order difference equations will be accurate for sufficiently low-energy pulses, i.e. for small G1,2hˆ Si. For any given physical situation these conditions can be satisfied by subdividing longor high-energy pulses into subpulses with smaller τ, at the cost of additional computation time. Similarly, if the full ensemble produces large rotations of ˆ S, the ˆ 3can be subdivided as in Koschorreck and Mitchell [17]. In the simulations described below we subdivide ˆ Suntil the results converge. The evolution is compactly expressed in terms of a tensor H(m)containing the coupling constants G1,2and gFµ0and the structure factors cˆ Viˆ Vj ˆ Vk: ˆ Vi(tm+τ) =ˆ Vi(tm)+ˆ Vj(tm)Hi(m) jk ˆ Vk(tm)where (13) Hi(m) jk ≡G1δˆ S(m) z ˆ Vj cˆ Viˆ fz ˆ Vk+G2δˆ S(m) x ˆ Vj cˆ Viˆx ˆ Vk+δˆ S(m) y ˆ Vj cˆ Viˆy ˆ Vk+1 √3δˆ S(m) 0 ˆ Vj cˆ Vi J m ˆ Vk,(14) where δˆ Vj ˆ Viis 1 for ˆ Vi=ˆ Vjand 0 for ˆ Vi6= ˆ Vj, and we assume summation over repeated indices. 3.2. Linearization The difference equation (13) are bilinear in the components of ˆ V. Although nonlinearity can in some cases lead to non-Gaussian phase-space distributions [22,32], in practice Gaussian or near-Gaussian distributions are far more common, and indeed producing measurably nonGaussian distributions is non-trivial [10]. This motivates a linearization of the above equations. Symbolically we write ˆ V(tm+τ) =ˆ V(tm)+ˆ V(tm)·H(m)·ˆ V(tm)(15) and separate ˆ Vinto the average ¯ Vand the fluctuations δˆ V: ˆ V(tm+τ) =¯ V(tm+τ) +δˆ V(tm+τ) =¯ V(tm)+¯ V(tm)·H(m)·¯ V(tm)+δˆ V(tm)+¯ V(tm)·H(m)·δˆ V(tm) +δˆ V(tm)·H(m)·¯ V(tm)+δˆ V(tm)·H(m)·δˆ V(tm). (16) When the last term can be neglected, the dynamics reduce to ¯ V(tm+τ) =¯ V(tm)+¯ V(tm)·H(m)·¯ V(tm)(17) =¯ V(tm)+U(m)·¯ V(tm), (18) where U(m)≡¯ V(tm)·H(m), which describes a nonlinear evolution of the average ¯ V, and δˆ V(tm+τ) =δˆ V(tm)+¯ V(tm)·H(m)·δˆ V(tm)+δˆ V(tm)·H(m)·¯ V(tm) =T(m)·δˆ V(tm), (19) which describes a linear evolution of the fluctuations δˆ Vin terms of the matrix T(m) ik ≡δk i+ VjHi(m)+Hi(m) kj jk Vj.(20) The covariance matrix 0evolves as 0(tm+τ) =T(m)·0(tm)·[T(m)]T.(21) New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 15 Figure 2. (a) Experimental setup. PD: photodiode; L: lens; WP: wave plate; BS: beam splitter; PBS: polarizing beam splitter. Atoms are loaded into a singlebeam optical dipole trap and probed with pulses of light propagating along the trap axis. An external magnetic field is applied to coherently rotate the atomic spins, and gradient field components ∂Bi/∂zare actively cancelled. The initial atomic state is prepared via optical pumping with circularly polarized light either propagating perpendicular to or along the trap axis, to prepare an ˆ Fy-polarized or ˆ Fz-polarized CSS, respectively. Also shown are experimental data of (b) the average signal ¯ φfor an input ˆ Fy—and ˆ Fz—polarized CSS (blue and orange, respectively), and (c) the evolution of the variance var(φ) for the same input states. Error bars represent ±1σstatistical errors. The lines connect experimental points to aid visualization. ˆ Fzand into the alignment variables ˆ Jx,yat a rate ωG2=(G2Sx/2)(1/P1)=2π×0.43 kHz. The revival is accompanied by a πphase shift in the oscillations (see figure 3below for an illustration of this effect). The observed variance var(φ) oscillates at a frequency 2ω0and undergoes a similar but much more pronounced collapse and revival driven by the tensorial light shifts. Note that the observed variance in this experiment is dominated by technical noise; for our experimental parameters, the quantum noise contributions from both the atomic and light variables are 1 mrad2. The dominant technical noise contributions are due to uncertainty in the atom number δN2 A, and magnetic field noise, described by the covariance matrix 0B, coupled into the observed variable via tensorial light shifts. We illustrate the effect of these terms separately in figure 3, by running numerical simulations, as described in detail in section 7, using experimental parameters from the data shown in figure 2and setting variously the 0B,δN2and G2terms equal to zero. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 16 Figure 3. Theoretical curves illustrating the effect of technical noise terms and tensorial light shifts on the mean ¯ φand variance var(φ) of the observed signal. (a) Tensorial light shifts rotate population out of ˆ Fzand into the alignment variables ˆ Jx,yleading to a collapse and revival of the mean ¯ φ. The revival is accompanied by a πphase shift in the oscillations, as is evident in the simulation with G2=0, which removes the effect of tensorial light shifts. (b)–(d) The observed variance var(φ) undergoes a more pronounced collapse and revival driven by the tensorial light shifts, which couple technical noise from the atomic variables ˆ 3and the magnetic field covariance matrix 0Binto the observed variable. The contribution to the observed signal in (b) is due to magnetic field noise. Technical noise in the atomic variables due to uncertainty δN2 Ain the atom number only adds significant noise only during the early stages of the evolution. This is illustrated more clearly in the magnified plots (c) and (d). The dot-dashed magenta line in plot (d) illustrates the quantum noise contributions due to the light (shot-noise) and atoms (projection-noise). 7. Simulation of free-induction decay In order to simulate the FID experiment, we need to estimate a number of experimental parameters, including the input state vector ¯ Vand covariance matrix 0V, the coupling constants of equation (10) and the decoherence terms in equations (26)–(29) and (39). The atom–light coupling constant G1is calibrated in an auxiliary experiment as described in Kubasik et al [39]. From this calibration we calculate the effective atom–light interaction area A, the coupling New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 17 constant G2of equation (10) and the single-photon scattering probability ηγused to describe the optically induced decoherence terms in equations (26)–(29) (see [41] for details). We estimate the average magnetic field ¯ Bfor a single experimental data set by fitting the observed signal hφ(t)iwith equation (50), as described detail in Behbood et al [36]. Since equation (50) neglects the effect of tensorial light shifts, we fit only the first 250 µs of each measured signal to minimize the systematic error that this approximation introduces. From the fits we also determine the coherence time Tthat is used in equation (39) From the same fits we estimate the number of atoms hˆ Fi(0)i= NA/2. Assuming an input state of the form R≡ρ⊗NA ˆ fi for the atoms, this specifies the initial atomic state via ¯ 3=NATr[ˆ λρˆ fi]. The number of photons NLin each pulse is independently measured via a reference detector, as shown in figure 2. Together, these estimates specify the initial state of the vector ¯ V. Statistics from the fits to equation (50) across the data set allow us to estimate the covariance matrix 0B. Similarly, we estimate the uncertainty δN2 Ain the atom number from var(ˆ Fi)=var(NA/2)(which includes contributions from variable trap loading and state preparation efficiency as well as measurement uncertainty). This allows us to estimate the initial covariance matrix 0Λusing equation (48). Since the measurement is shot-noise limited, the input light covariance matrix is 0S=(NL/4,NL/4,NL/4). Together, via equation (49), these specify the initial covariance matrix 0V(0). As an example, we give the experimental parameters in detail of the first example described in section 6, shown in figures 2(b) and (c). For this experiment, we set the detuning of the probe to 1=−700 MHz and probe with a sequence of 1 µs long h-polarized pulses of light with on average NL=7.19 ×106photons per pulse at 10 µs intervals. The measured average magnetic field was B=(11.92,−4.08,−3.23)mG, with a covariance matrix 0B=   0.091 0.005 −0.023 0.005 0.116 0.010 −0.023 0.011 0.007   mG2.(51) We estimate NA=6.05 ×105and 1NA=1.6×103, giving an initial atomic vector for the ˆ Fy–polarized input Λ(t=0)=(0,1,0,−0.5,0,0,0,−0.29)×NAand covariance matrix 03(t=0)=                 0.5 0 0 0 0.5 0 0 0 0 4.98 0 −2.49000−1.43 0 0 0.5 0 0 0 0.5 0 0−2.49 0 1.49 0 0 0 0.28 0.5 0 0 0 0.5 0 0 0 0 0 0 0 0 1 0 0 0 0 0.5 0 0 0 0.5 0 0−1.43 0 0.28 0 0 0 1.16                 ×NA.(52) We further estimate a coherence time T=450 ±10 µs. The calibrated coupling constants were G1=1.8±0.2×10−7rad atom−1and G2=−9.3±0.8×10−9rad atom−1, and the atom–light scattering parameter ηγ=1.1×10−9. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 18 Together these parameters determine the initial state vector ¯ V(t=0)and covariance matrix 0V(t=0)as described in section 5, the atom–light coupling constants of equation (10), the magnetic field components of equation (33), and the decoherence terms in equations (26)–(29) and equation (39). With these input parameters, we then run numerical simulations of the evolution of the state vector ¯ Vand covariance matrix 0Vfollowing the procedure described in section 3, keeping track of V(tm)and 0V(tm)at each time step. As in the experiment, we alternate between a time interval of 1 µs in which the light is present and an interval of 9 µs with no light present. For the numerical calculations, we divide these intervals into 50 and 100 sub-steps, respectively, which is sufficient to ensure numerical convergence of the results. The results of the simulations (dark blue curves) are plotted along with the experimental data (light blue circles) in figures 4(a) and (b) for both the mean ¯ φand variance var(φ) with an input ˆ Fy-polarized atomic state. We observe excellent qualitative and quantitative agreement between the simulations and the observed mean ¯ φand variance var(φ) in the rotation angle with no further free parameters adjusted in the calculations. For the mean ¯ φ, we reproduce the observed behaviour in our simulations over the entire 1 ms of observation, as shown in figures 4(b) and (d). For the variance var(φ) the quantitative agreement is initially good, but breaks down at longer times. This can be explained by the effect of the uncertainty in the magnetic field, described by the covariance matrix 0B, which eventually drives the atomic variables out of the Gaussian approximation. Because magnetic precession is cyclic, not linear, an initially Gaussian spin distribution will become non-Gaussian due to uncertainty in the precession frequency. As some parts of the distribution precess faster than others, the distribution begins to ‘wrap around’ the Bloch sphere, forcing a non-Gaussian shape on the distribution. It is convenient to define a time τgauss ≡π/(γF1Bk), where 1Bkis the uncertainty in Balong the average field direction, found using 0Bof equation (51). τgauss indicates the moment at which the precession angle uncertainty becomes π. Perhaps surprisingly, the observed and predicted variances (see figures 2(c) and (e)) agree very well up to nearly t=τgauss ≈0.45 ms, showing that the theory gives accurate results even for significant departures from Gaussianity. The agreement in the average values persists even for t> τgauss. This changes accumulates with time and becomes much more significant than the technical noise in the number of atoms so that while the atomic noise is important early on, the dephasing effect of magnetic field inhomogeneities becomes the dominant contribution for larger times. The collapse and revival of the oscillations in figures 4(a) and (b) is due to rotations driven by the tensorial light shift. The effect of the tensorial light shifts can be reduced either by probing further off resonance, or by probing the atoms alternately with h– and v–polarized pulses, as described in detail in [38]. We illustrate this in figures 4(c)–(d), where we compare these data to FID measurements (and simulations) made with two alternate probing strategies. In figures 4(c) and (d) we set the detuning to 1=−1.5 GHz and repeat the single–polarization probing sequence. In figures 4(e) and (f) we set the detuning to 1=−700 MHz and probe with pairs of alternately hand v-polarized pulses separated by 3 µs are sent through the atoms 20 µs intervals (for clarity we plot only the h–polarized pulses). This results in the same total number of photons used per unit time as in the single polarization probing strategy. With these data we observe similar behaviour in the mean ¯ φand variance var(φ) at both detunings, indicating the effective cancellation of tensorial light shifts in these experiments. We also observe non-zero minima in var(φ) at all phases in the experimental data for t>0.5 ms, which are not reproduced in our calculations. Looking at the data globally, we also note that lower envelope rises more New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 19 Figure 4. Comparison of experimental data (light blue circles) with numerical simulations (dark blue curves) of the mean ¯ φ(a), (c) and (e) and variance var(φ) (b), (d) and (f) of the optical rotation of the probing light. For (a) and (b) we set the detuning of the probe to 1=−700 MHz and probed the atoms with a sequence of 1 µs long h–polarized pulses of light with on average NL=7.19 ×106photons per pulse at 10 µs intervals. For (c) and (d) we set the detuning of the probe to 1=−1.5 GHz and probed with the same measurement sequence with on average NL=1.35 ×107photons per pulse. For (e) and (f) we set the detuning of the probe to 1=−700 MHz and probed the atoms with a sequence pairs of 1 µs long pulses with alternating h– and v–polarization, separated by 3 µs and sent through the atoms 20 µs intervals with on average NL=2.10 ×106photons per pulse. Error bars represent ±1σstatistical errors. See text for details. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 20 in the data than in the simulation. This may due to the curvature of the Bloch sphere: for small uncertainty in precession angle 1θ =tγF1Bk, we will have 1Fz∝1θ∂θFz. For flat parts of the curve Fz(θ) there will be zero ∂θFzand thus zero 1Fzbut for larger 1θ, we need to take into account higher derivatives, i.e. the curvature ∂2 θFzand higher. This implies a first-order insensitivity of the measured component to rotations, and thus (to first order) zero contribution to the measured 1θ. When the distribution describing the state ‘wraps around’ the Bloch sphere, the second-order effects, which are not taken into account in the Gaussian approximation, become important and give a contribution to the measured variance proportional to δB2. This contribution is necessarily positive, and raises the lower envelope above the level predicted by the simulations. 8. Conclusions We have presented a method for describing the quantum dynamics of spin-1 atomic ensembles, extending the method introduced for quantum light interfaces [13,15,16], developed for spin-1/2 atomic ensembles by Madsen and Mølmer [14] and generalized by Koschorreck et al [17] and Toth et al [42]. Our approach, which explicitly includes the so-called ‘spin alignment’ degrees of freedom, fits naturally with the light–matter interaction, which couples both spin alignment and spin orientation to the optical Stokes parameters. For spin-1 our description is complete within the Gaussian approximation, while for larger spins it is still useful when octopole and higher spin moments can be neglected. We also include the important technical noise associated with magnetic fields and noise due to uncertain atom number, as typically arises due to stochastic trap-loading processes. Finally, we give explicit formulae for the noise introduced by spontaneous scattering during the optical probing process and due to dephasing in an inhomogeneous magnetic field. We have tested the model against experiment in a scenario involving all of these effects. We compute the evolution of the spin orientation average and variance for atomic ensembles with uncertain atom number, undergoing a combination of free-induction decay and alignment-toorientation conversion in the presence of a noisy magnetic field. The simulation is compared to experimental observations made with a cold 87Rb ensemble held in an optical dipole trap, probed by shot-noise-limited Faraday rotation with near-resonant light. We find good agreement within the Gaussian regime. In addition to validating the model, the experiments provide a heuristic guide to the limits of the Gaussian approximation in these systems. Given that most atomic ensemble experiments are performed with spin-1 or larger atoms, the technique described here will allow more accurate modelling of established quantum optical protocols, e.g. quantum memory [15], quantum non-demolition measurement [26,38], dynamical decoupling [25], spin squeezing [8] and vector magnetometry [36], as well as proposed applications such as generation of macroscopic singlet states [42,43] and planar squeezed states [37]. Acknowledgments We thank G´ eza T´ oth, Mario Napolitano and Graciana Puentes for helpful discussions. This work was supported by the Spanish MINECO under the project MAGO (ref. no. FIS201123520), by the European Research Council under the project AQUMET and by Fundaci´ o Privada Cellex Barcelona and Fundaci´ o Catalunya-La Pedrea. New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 21 Appendix A. Proof of equations (23) and (24) For brevity, we write the covariance as C(A,B)≡1 2hA∧B+(B∧A)i−hAi∧hBi,(A.1) where the expectation is taken on the state of the ensemble. In this notation, 03≡C(ˆ 3,ˆ 3) where as above ˆ 3=Piˆ λ(i). We assume that all atoms are statistically equivalent, so that hˆ λ(i)∧ˆ λ(j)i=hˆ λ(1)∧ˆ λ(2)ifor all i6= jand hˆ λ(i)∧ˆ λ(i)i=hˆ λ(1)∧ˆ λ(1)ifor all i. We then have 03=X i,j C(ˆ λ(i),ˆ λ(j))=NAC(ˆ λ(1),ˆ λ(1))+NA(NA−1)C(ˆ λ(1),ˆ λ(2)), (A.2) which we solve for C(ˆ λ(1),ˆ λ(2))to get C(ˆ λ(1),ˆ λ(2))=03−NAC(ˆ λ(1),ˆ λ(1)) NA(NA−1).(A.3) Because of the symmetry, removing atoms does not change C(ˆ λ(1),ˆ λ(1))or C(ˆ λ(1),ˆ λ(2)). If a fraction 1 −Xof the atoms is removed, the covariance matrix 0(XNA) 3of the remaining atoms can be calculated as in equation (A.2), but summing iand jfrom 1 to XNA. We find 0(XNA) 3=XNAC(ˆ λ(1),ˆ λ(1))+XNA(XNA−1)C(ˆ λ(1),ˆ λ(2)) =0(NA) 3 X(XNA−1) NA−1+0λX(1−X)N2 A NA−1, where 0λ≡C(ˆ λ(1),ˆ λ(1))(A.4) is the single-atom covariance matrix. Dropping terms of order 1/NAand smaller, 0(XNA) 3=0(NA) 3X2+0λX(1−X)NA.(A.5) This accounts for the change in 0Λdue to removing (1−X)NAatoms. We must also account for the noise of returning a fraction pof these atoms to the ensemble in a decohered state. We assume they are completely random, and thus add the noise of a thermal state (i.e. variance f(f+ 1)/3 per atom). 0(XNA) 3=0(NA) 3X2+0λX(1−X)NA+ 2p(1−X)NA1/3.(A.6) Here 0λ=2 318×8−¯ λ∧¯ λ+Pk¯ λkM(k), M(k) i j ≡1 4Tr[ˆ λk{ˆ λi,ˆ λj}],(A.7) where {·,·}indicates the anti-commutator. This can be shown using the expectation Tr[ρˆ λi]=¯ λi and the orthonormality condition Tr[λiλj]=2δi j from which Tr[{λi, λj}]=4δi j . We find the single-particle state ρ=1 31+1 2Pi¯ λiˆ λiand the covariances cov(λi, λj)≡1 2h{ˆ λi,ˆ λj}i−hˆ λiihˆ λji =2 3δi,j+1 4Pk¯ λkTr[ˆ λk{ˆ λi,ˆ λj}]−¯ λi¯ λj.(A.8) New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 22 Appendix B. Noise from uncertain atom number We consider the statistics of ˆ 3for ensembles with NAatoms in a permutationally invariant product state R(NA)=ρ⊗NA, and taking a statistical average over NA. We indicate averages with subscripts, e.g. hˆ 3iR=Tr[ ˆ 3R] indicates an expectation with respect to the state R, while hˆ 3iR,NA=hTr[ ˆ 3R(NA)]iNAindicates a statistical average of hˆ 3iRover the distribution of NA. Due to the structure of R(NA),hˆ λ(k)iR(NA)=hˆ λ(1)iρ≡hˆ λiρis independent of both kand of NA, so that hˆ 3iˆ 3jiR= NA X k,l=1hˆ λ(k) iˆ λ(l) jiρ =NAhˆ λiˆ λjiρ+NA(NA−1)hˆ λiiρhˆ λjiρ,(B.1) and thus hˆ 3iˆ 3jiR,NA=¯ NAhˆ λiˆ λjiρ+(δN2 A+¯ N2 A−¯ NA)hˆ λiiρhˆ λjiρ,(B.2) where δN2 A≡hN2 Ai−hNAi2indicates the variance. At the same time hˆ 3iiR,NA=¯ NAhˆ λiiρ,(B.3) so that cov(ˆ 3i,ˆ 3j)R,NA≡1 2hˆ 3iˆ 3j+ˆ 3jˆ 3iiR,NA−hˆ 3iiR,NAhˆ 3jiR,NA(B.4) =¯ NAcov(ˆ λi,ˆ λj)ρ+δN2 Ahˆ λiiρhˆ λjiρ.(B.5) In terms of the single-atom covariance matrix 0λof equation (A.4), 0=¯ NA0λ+δN2 A(¯ λ∧¯ λ). (B.6) Appendix C. Inhomogeneous magnetic fields The microscopic spin operators evolve as fi(t)=R(zi,t)fi(0), (C.1) where fiis the spin of the ith atom with position ziand R(z,t)=exp[γFt|B(z)|AB],(C.2) where AB≡    0−ˆ Bzˆ By ˆ Bz0−ˆ Bx −ˆ Byˆ Bx0     (C.3) is the generator of rotations about Band ˆ B≡B/|B|. Expanding the field as B(z)≈B0+(B0 k+ B0 ⊥)z+O(z2), where B0 kis parallel to B0and B0 ⊥is perpendicular. We note that a change in the magnitude of Bhas an accumulating effect on the spin precession, i.e. the change in fgrows with t. In contrast, a change in the direction of Bhas a fixed effect: from the perspective of the measurement, a rotation of Bis equivalent to a rotation of both the initial state and the measured New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 23 component Fz. For small gradients ∂zBB/latoms, where latoms is the length of the cloud, we can ignore B0 ⊥. This approximation, along with the fact that An+2 B=−An B, allows us to write R(z,t)≈I+AB0sin ω(z)t+A2 B0[1 −cos ω(z)t],(C.4) where ω(z)=γF|B0+zB0 k|. In our trap, we observe an atomic density ρ(z)well approximated by a Lorentzian ρ(z)= w/π(z2+w2)where wis the FWHM extent of the ensemble. The collective spin F≡Pifi then evolves as F(t)=Zdzρ(z)R(z,t)f(0)(C.5) =[I+A2 B0]F(0)+ e−wγF|B0 k|t(AB0sin ω0t−A2 B0cos ω0t)F(0). (C.6) In the first term I+A2 B0describes a projector onto the direction of B0. This is the steady-state polarization. The second line describes a decaying oscillation about B0of the perpendicular components with a coherence time T=1/(wγF|B0 k|). References [1] Hammerer K, Sørensen A S and Polzik E S 2010 Quantum interface between light and atomic ensembles Rev. Mod. Phys. 82 1041–93 [2] Julsgaard B, Sherson J, Ignacio Cirac J, Fiurasek J and Polzik E S 2004 Experimental demonstration of quantum memory for light Nature 432 482–6 [3] Matsko A B, Novikova I, Welch G R, Budker D, Kimball D F and Rochester S M 2002 Vacuum squeezing in atomic media via self-rotation Phys. Rev. A66 043815 [4] Mikhailov E E and Novikova I f 2008 Low-frequency vacuum squeezing via polarization self-rotation in Rb vapor Opt. Lett. 33 1213–5 [5] Giovannetti V, Lloyd S and Maccone L Quantum metrology Phys. Rev. Lett. 96 010401 [6] Louchet-Chauvet A, Appel J, Renema J J, Oblak D, Kjaergaard N and Polzik E S 2010 Entanglement-assisted atomic clock beyond the projection noise limit New J. Phys. 12 065032 [7] Napolitano M, Koschorreck M, Dubost B, Behbood N, Sewell R J and Mitchell M W 2011 Interaction-based quantum metrology showing scaling beyond the Heisenberg limit Nature 471 486–9 [8] Sewell R J, Koschorreck M, Napolitano M, Dubost B, Behbood N and Mitchell M W 2012 Magnetic sensitivity beyond the projection noise limit by spin squeezing Phys. Rev. Lett. 109 253605 [9] Braunstein S L and van Loock P 2005 Quantum information with continuous variables Rev. Mod. Phys. 77 513–77 [10] Dubost B, Koschorreck M, Napolitano M, Behbood N, Sewell R J and Mitchell M W 2012 Efficient quantification of non-Gaussian spin distributions Phys. Rev. Lett. 108 183602 [11] Giedke G and Ignacio Cirac J 2002 Characterization of Gaussian operations and distillation of Gaussian states Phys. Rev. A66 032316 [12] Kurucz Z and Mølmer K 2010 Multilevel Holstein-Primakoff approximation and its application to atomic spin squeezing and ensemble quantum memories Phys. Rev. A81 032314 [13] Kraus B, Hammerer K, Giedke G and Ignacio Cirac J 2003 Entanglement generation and Hamiltonian simulation in continuous-variable systems Phys. Rev. A67 042314 [14] Madsen L B and Mølmer K 2004 Spin squeezing and precision probing with light and samples of atoms in the Gaussian description Phys. Rev. A70 052324 [15] Hammerer C, Mølmer K, Polzik E S and Cirac J I 2004 Light–matter quantum interface Phys. Rev. A 70 044304 New Journal of Physics 15 (2013) 103007 (http://www.njp.org/) 24 [16] Mølmer K and Lars Bojer Madsen 2004 Estimation of a classical parameter with Gaussian probes: magnetometry with collective atomic spins Phys. Rev. A70 052102 [17] Koschorreck M and Mitchell M W 2009 Unified description of inhomogeneities, dissipation and transport in quantum light–atom interfaces J. Phys. B: Atom. Mol. Opt. Phys. 42 195502 [18] de Echaniz S R, Koschorreck M, Napolitano M, Kubasik M and Mitchell M W 2008 Hamiltonian design in atom–light interactions with rubidium ensembles: a quantum-information toolbox Phys. Rev. A77 032316 [19] Norris L M, Trail C M, Jessen P S and Deutsch I H 2012 Enhanced squeezing of a collective spin via control of its qudit subsystems Phys. Rev. Lett. 109 173603 [20] Smith G A, Chaudhury S, Silberfarb A, Deutsch I H and Jessen P S 2004 Continuous weak measurement and nonlinear dynamics in a cold spin ensemble Phys. Rev. Lett. 93 163602 [21] Pustelny S, Koczwara M, Cincio Ł and Gawlik W 2011 Tailoring quantum superpositions with linearly polarized amplitude-modulated light Phys. Rev. A83 043832 [22] Chaudhury S, Smith A, Anderson B E, Ghose S and Jessen P S 2009 Quantum signatures of chaos in a kicked top Nature 461 768–71 [23] Budker D, Kimball D F, Rochester S M and Yashchuk V V 2000 Nonlinear magneto-optical rotation via alignment-to-orientation conversion Phys. Rev. Lett. 85 2088–91 [24] Pustelny S, Wojciechowski A, Gring M, Kotyrba M, Zachorowski J and Gawlik W 2008 Magnetometry based on nonlinear magneto-optical rotation with amplitude-modulated light J. Appl. Phys. 103 063108 [25] Koschorreck M, Napolitano M, Dubost B and Mitchell M W 2010 Quantum nondemolition measurement of large-spin ensembles by dynamical decoupling Phys. Rev. Lett. 105 093602 [26] Sewell R J, Napolitano M, Behbood N, Colangelo G and Mitchell M W 2013 Certified quantum nondemolition measurement of a macroscopic material system Nature Photon. 7517–20 [27] Acosta V M et al 2008 Production and detection of atomic hexadecapole at earth’s magnetic field Opt. Express 16 11423–30 [28] Dowling J P, Agarwal G S and Schleich W P 1994 Wigner distribution of a general angular-momentum state: applications to a collection of two-level atoms Phys. Rev. A49 4101–9 [29] Geremia J M, Stockton J K and Mabuchi H 2006 Tensor polarizability and dispersive quantum measurement of multilevel atoms Phys. Rev. A73 042112 [30] de Echaniz S R, Mitchell M W, Kubasik M, Koschorreck M, Crepaz H, Eschner J and Polzik E S 2005 Conditions for spin squeezing in a cold 87RB ensemble J. Opt. B: Quantum Semiclass. Opt. 7s548 [31] Deutsch I H and Jessen P S 2010 Quantum control and measurement of atomic spins in polarization spectroscopy Opt. Commun. 283 681–94 [32] Juli´ a-D´ ıaz B, Zibold T, Oberthaler M K, Mel´ e-Messeguer M, Martorell J and Polls A 2012 Dynamic generation of spin-squeezed states in bosonic josephson junctions Phys. Rev. A86 023615 [33] Vasilyev D V, Hammerer K, Korolev N and Sørensen A S 2012 Quantum noise for faraday light–matter interfaces J. Phys. B: Atom. Mol. Opt. Phys. 45 124007 [34] Petersen V and Mølmer K 2006 Estimation of fluctuating magnetic fields by an atomic magnetometer Phys. Rev. A74 043802 [35] Demoen B, Vanheuverzwijn P and Verbeure A 1977 Completely positive maps on the C.C.R-algebra Lett. Math. Phys. 2161–6 [36] Behbood N, Martin Ciurana F, Colangelo G, Napolitano M, Mitchell M W and Sewell R J 2013 Real-time vector field tracking with a cold-atom magnetometer Appl. Phys. Lett. 102 173504 [37] Puentes G, Colangelo G, Sewell R J and Mitchell M W 2012 Planar quantum squeezing via quantum nondemolition measurements in cold atomic ensembles arXiv:1211.4862 [quant-ph] [38] Koschorreck M, Napolitano M, Dubost B and Mitchell M W 2010 Sub-projection-noise sensitivity in broadband atomic magnetometry Phys. Rev. Lett. 104 093602 [39] Kubasik M, Koschorreck M, Napolitano M, de Echaniz S R, Crepaz H, Eschner J, Polzik E S and Mitchell M W 2009 Polarization-based light–atom quantum interface with an all-optical trap Phys. Rev. A 79 043815 New Journal of Physics 15 (2013) 103007 (http://www.njp.org/)