Ramsey interference in one-dimensional systems: the full distribution function of fringe contrast as a probe of many-body dynamics

Ramsey Interference in One-Dimensional Systems: The Full Distribution Function of Fringe Contrast as a Probe of Many-Body Dynamics

Takuya Kitagawa,¹Susanne Pielawa,¹Adilet Imambekov,²Jo¨rg Schmiedmayer,³Vladimir Gritsev,⁴and Eugene Demler¹

1Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA

2Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

3Atominstitut, TU-Wien, Stadionallee 2, 1020 Vienna, Austria

4Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland

We theoretically analyze Ramsey interference experiments in one-dimensional quasicondensates and obtain explicit expressions for the time evolution of full distribution functions of fringe contrast. We show that distribution functions contain unique signatures of the many-body mechanism of decoherence. We argue that Ramsey interference experiments provide a powerful tool for analyzing strongly correlated nature of 1D interacting systems.

PACS numbers: 67.85.d, 03.75.Dg

Introduction.—Recent progress in the field of ultracold atoms not only expanded our understanding of equilibrium properties of interacting 1D Bose gases [1,2] but also posed new theoretical challenges by studying far-from- equilbrium dynamics of such systems. Recent experiments addressed such questions as thermalization and integra- bility [3], decoherence after the splitting of two conden- sates [4] and spin dynamics of two-component Bose mixtures [5]. Motivation for such experiments comes not only from the basic interests in many-body dynamics [6]

but also from possible applications of ultracold atoms such as quantum information processing [7] and interferometric sensing [8]. In this paper, we theoretically analyze the decoherence dynamics of Ramsey interference fringes in one-dimensional quasicondensates. Such systems have been considered for possible applications in atomic clocks and quantum-enhanced metrology [9,10]. Here we show that a Ramsey interferometer is also a powerful tool for studying many-body dynamics of low dimensional quan- tum systems. We find that decoherence of Ramsey fringes is strongly affected by the multimode character of one- dimensional systems. Moreover we will demonstrate that the time evolution of the full distribution function of fringe contrast provides unique signatures of this many-body decoherence mechanism [11]. The idea of using noise and distribution functions to characterize equilibrium many- body states of ultracold atoms has been discussed in several theoretical papers [12] and applied in experiments [1,13].

This Letter constitutes the first proposal to study nonequi- librium dynamics of ultracold atoms with quantum noise.

The role of interactions in Ramsey interferometers with BEC was first addressed in the pioneering paper of Kitagawa and Ueda [10]. They used the single mode approximation to predict the interaction-induced decoher- ence of Ramsey fringes along with the appearance of spin- squeezed states. Their work stimulated ideas for quantum- enhanced metrology that take advantage of spin-squeezed states formed in interacting BECs [9,14]. For the analysis

of one-dimensional quasicondensates, however, the single mode approximation cannot be applied because these sys- tems do not have macroscopic occupations of a single state, even at zero temperature. The non mean-field character of the multimode spin dynamics in 1D quasicondensates was first reported in experiments by Wideraet al.[5]. However, this work did not provide a definitive demonstration of the many-body origin of Ramsey fringe decay. In the follow- ing, we argue that unambiguous signatures of the multi- mode decoherence are found in the full distribution function of the Ramsey fringe amplitudes. Such distribu- tion functions should be accessible in experiments with 1D quasicondensates realized on atom chips [15], because such systems do not average over multiple tubes and thus allow the measurements of shot-to-shot fluctuations [1].

Now we describe the Ramsey sequence considered in this Letter. Here we identify two hyperfine states as spin-up and spin-down states. The Ramsey sequence is carried out as follows: (i) all spins of the atoms are prepared in the spin up state; (ii) a=2pulse is applied to rotate each spin into thexdirection; (iii) spins freely evolve (precess) for timet;

(iv) another=2pulse is applied to map the transverse spin component to the z direction, which is then measured.

Measurements yield a net spin S~_lfor a segment of length land we assume thatlis smaller than the system size but large enough to contain a large number of particles N_l 1. In such a case, the simultaneous measurements of S^x_l and S^y_l are possible, because even though operators S^x_l and S^y_l generally do not commute, the noncommutativity gives only corrections of the order of 1= ffiffiffiffiffiffi

N_l

p relative to the average values [16]. Commutativity of S^x_l and S^y_l implies, in particular, that we can define the joint distribu- tion functionP^x;y_l for the two transverse spin components S^x_l andS^y_l. In experiments, measurements of P^x;y_l are pos- sible by mapping the spin orientations in x-y plane toz direction by =2 pulse, followed by the local measure- ments ofS_z[17].S^x_landS^y_las well as the magnitude of spin

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1

S^?_l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS^x_lÞ²þðS^y_lÞ² q

can be found by taking the integration overl. The analytic solution for the time evolution ofP^x;y_l constitutes the main result of this Letter. In addition, we assume thatlis larger than the spin healing length_s, so we can use Tomonaga-Luttinger liquid approach to describe the collective spin dynamics (see also below). For simplic- ity we work in the rotating frame of the Larmor precession, and consider the spins before the last=2pulse. Then the amplitude of Ramsey fringes, as it is conventionally de- fined, corresponds toS^x_l.

Our main results are summarized in Fig.1and can be understood from the following physical arguments. Strong fluctuations present in 1D systems forbid the existence of the long range coherence [18], and spatial fluctuations coming from different wavelengths strongly affect the dynamics in one dimension. Among those, fluctuations with wavelengths longer than the integration lengthlrotate S~_las a whole. So they decreaseS^x_l but not the magnitude of the spin S^?_l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðS^x_lÞ²þ ðS^y_lÞ² q

. Fluctuations with wave- lengths shorter than l decrease both S^x_l and S^?_l simulta- neously. Figures 1(a) and 1(b) show the situation where fluctuations with wavelength larger than l dominate the dynamics. In (a), we see that the magnitude of the spinS^?_l decays only slightly from the initial state but the direction of the spin is randomized during the time evolution. Note that in this case the distribution function ofS^x_l has a very peculiar shape with two peaks at large positive and nega- tive values. We call this regime, ‘‘spin diffusion’’ regime.

Figures1(c)and1(d)show the situation where fluctuations with wavelengths shorter thanldominate. In this caseS^?_l andS^x_l decay in the same time scale. We call this regime,

‘‘spin decay’’ regime. Below we argue that the crucial parameter of the system is a dimensionless ratio propor-

tional to the length of the integration region l₀ ¼_4K^2l

ss. Whenl₀1the system is in the spin diffusion regime and the other limitl₀1is the spin decay regime.

Model.—Following the first =2pulse we have a two- component Bose mixture with equal densities of both species. Tomonaga-Luttinger liquid (TLL) approach, which we use in this paper, focuses on the linearly dispers- ing modes in the low energy part of the spectrum. For simplicity, we consider the case when the interaction strength among particles with spin up is the same as that among particles with spin down. This condition can be reached for the hyperfine states jF¼1; m_F¼ 1i and jF¼2; m_F¼ þ1i of ⁸⁷Rb that are commonly used in experiments [1,5]. When this is the case, the charge and spin parts of the Tomonaga-Luttinger Hamiltonian de- couple and the spin part of the Hamiltonian is given by [18]

H_s¼c_s 2

ZL=2 L=2dr

K_s

ðr_sðrÞÞ²þ K_sn²_sðrÞ

: (1) Here L is the total system size,n_sðrÞ describes the local spin imbalance (i.e., z component of the spin) n_s¼ c^yð1₂^zÞc where c^y is the creation operator of a particle with spinand^aisacomponent of Pauli matrix.

_sðr; tÞdescribes the direction of the transverse spin com- ponent e^is ¼c^yð1₂^þÞc with being the average density for each species and^þ¼^xþi^y. Variablesn_s and _s obey canonical commutation relations ½n_sðrÞ;

_sðr⁰Þ ¼ i ðrr⁰Þ.K_sis spin Luttinger parameter rep- resenting interaction strength [5], and c_s is spin-wave velocity. In the weak interaction limit, these parameters are related to physical parameters asc_s¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g_s=m

p ,_s¼

c_s

gsandK_s¼_s=2, wheremis the mass of the particle

andg_s¼⁴_m^{2@2 a""þa##}₂^2a"#is the interaction strength in spin

channel where a_; refers to s-wave scattering length between hyperfine spin and. In this Letter, we focus on the regime g_s>0 when the system is miscible. Our approach can be extended tog_s<0but will be limited to times before z magnetization per atom becomes of the order of 1. Harmonic Hamiltonian (1) can be diagonalized by going to momentum space, and it takes the form

H_s¼X

k0

c_sjkjb^y_s;kb_s;kþc_s

2K_sn²_s;0: (2) The first and second term of (2) correspond tok0and k¼0part of the Hamiltonian, respectively, andn_s;k rep- resents the Fourier transform of n_sðrÞ. This Hamiltonian has a momentum cutoff set by the inverse of the spin healing length _s¹. The operatorsb^y_s;k create spin excita- tions with momentumk, and these spin excitations are the main focus of our study.

Transverse part of the spin operatorS~_lis given by S^x_l¼Zl=2

l=2drcosðsðrÞÞ; S^y_l¼Zl=2

l=2drsinðsðrÞÞ:

(3)

0 0.05

0 0.05 0 0.02 0.04 0.06

a

c d

1 0 -1 1 0 -1 1 0 -1 1 0

-1-1 0 1-1 0 1-1 0 1 0 0.02 0.04 0.06

b

time=0 time=2.4 time=4.8

time=7.2 time=9.6 time=12

time=0 time=2.4 time=4.8

time=7.2 time=9.6 time=12

time=0 time=2.4 time=4.8

time=7.2 time=9.6 time=12

time=0 time=2.4 time=4.8

time=7.2 time=9.6 time=12

1

0 1

1

- -1 0 -1 0 1

FIG. 1 (color). (a),(c) Evolution of joint FDFP^x;y_l with short integration length l=s¼10 [top, (a)] and long integration lengthl=_s¼30 [bottom, (c)]. (b),(d) Corresponding FDF for spinx,P^x_l with short integration lengthl=s¼10[top, (b)] and long integration length l=_s¼10 [bottom, (d)]. Here L=_s¼ 200,K_s¼20. Time is measured in units of_s=c_s.

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When describing spin dynamics, one typically considers the time evolution of expectation valueshS^a_lðtÞi. However important information is also contained in the shot-to-shot fluctuations of S^a_lðtÞ. Such quantum noise is captured by the full distribution functions (FDF) of spin operators, P^a_lð; tÞ[11]. In particular, high moments ofS^a_lðtÞcan be obtained from these FDFsP^a_lð; tÞ. Physically,P^a_lð; tÞd is the probability that a single measurement of the spin operatorS^a_l at timetgives a value betweenandþd.

In experiments,P^a_lð; tÞcan be obtained by making histo- grams of the measurement results ofS^a_lðtÞ.

To describe the time evolution of spin operators (3), we need to characterize the initial state of the system after the first =2rotation [19]. Classically one expects the initial state to be the eigenstate of_sðrÞwith eigenvalue zero for all r. However, such a state is unphysical in quantum mechanics since it leads to infinite uncertainty of the conjugate variablen_sðrÞ, and thus, to infinite energy. It is more sensible to consider an initial state that has a small uncertainty in_sat the expense of enhanced fluctuations in n_s, which is a squeezed state in the_s,n_svariables [5,21].

We observe that the spin of each individual atom is inde- pendently rotated by the first=2pulse into thexdirection, so the initial state satisfies hS^zðrÞS^zðr⁰Þi ¼₂ ðrr⁰Þ.

Thus we find a Gaussian state for the spin operatorS^z in momentum space

jc0i ¼ 1

N expX

k0

W_kb^y_s;kb^y_s;k

j0ijcs;k¼0i; (4)

where2W_k¼ ð1_kÞ=ð1þ_kÞ,_k ¼ jkjK_s=andN is the overall normalization of the state. For the uniform part of the spin operator we also have a squeezed state hn_s;0jcs;k¼0i ¼expð1=ð2Þn²_s;0Þ. We note that model (1) has a short distance cutoff, so the function in hS^zðrÞS^zðr⁰Þi should be understood as rounded off on the scale of_s, which is implicit in the momentum cutoff in Eq. (4).

The time evolution of the state (4) leads to W_k ! W_ke^2icsjkjt. From the resulting expression for the state at time t, one can readily calculate the decay of Ramsey fringes given byhS^x_lðtÞi, which is independent of integra- tion lengthl(See also [5,21]). To calculate the time evo- lution of FDF, we define instantaneous annihilation operators _ksðtÞ such that applying _ksðtÞ on the state expðW_ke^2icsjkjtb^y_s;kb^y_s;kÞj0i gives zero. Using operators _ksðtÞ, one can apply the approach described in Refs. [1,11] for calculating distribution functions of equi- librium systems. After direct calculation we find [20]

P^x;y_l ð; tÞ ¼Y

k

Z1

1e^2rsk=2d_rskZ d_sk

Zl=2

l=2dre^{iðr;t;fjskgÞ}

; (5)

ðr; t;f_jskgÞ ¼X

k

_rsk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hj_s;kðtÞj²i

L s

sinðkrþ_skÞ;

hj_s;k0j²i ¼ jkjK_s

₂sin²ðc_sjkjtÞ

2 þcos²ðc_sjkjtÞ 2 ; hj_s;k¼0j²i ¼ 1

2þc_st K_s

₂

2;

(6)

where the real and imaginary part of corresponds tox andycomponent ofS~_l, respectively. Equation (5) and (6) allow a simple physical interpretation. Functionðr; t;fgÞ defines the local direction of transverse magnetization, which results from the summation over spin-wave like modes sinðkrþ_skÞ. Amplitudes of individual modes are given by the time dependent expectation values hj_s;kðtÞj²iand by the set of random variables_rsk drawn from a Gaussian ensemble. Equation (5) and (6) reflect the key feature of dynamics of the quadratic Luttinger model (1): an initial Gaussian state for_s;k remains Gaussian at all times.

Time evolution ofhj_s;kðtÞj²ifollowing the first=2ro- tation can be understood as the free dynamics of a har- monic oscillator. From the conjugate nature of _s;k and n_s;k, we findhj_s;kð0Þj²i ¼¹_{4 hjn} ¹

s;kð0Þj²i¼₂¹ att¼0. Subse- quentlyhj_s;kðtÞj²ioscillates between the minimal value in the initial state and some maximum value hj_s;kj²i_max at the frequency of a harmonic oscillator c_sjkj. hj_s;kj²i_max can be estimated from energy conservation. Since the initial state was squeezed with respect to_sk, most of the energy of the mode is stored in the interaction termjn_s;kj². Therefore the total energy of the harmonic oscillator for momentumkcan be approximated by^c_K^s

s , which in turn giveshj_s;kj²i_max²_K2²

sk² ¼₂¹ ð_jkjK

sÞ². These considerations lead to the dynamics of phase fluctuation amplitude of the form in (6). We note that the spin fluctuations are domi- nated by small momentums since the maximum fluctuation amplitude is suppressed as1=k²for large momentum. This justifies our analysis based on the Tomonaga-Luttinger theory.

Results of numerical plots based on Eq. (5) and (6) are shown in Fig.1. In Fig.2(a)we also present the distribution functionP^?_l of the magnitude square of the integrated spin ðS^?_l Þ², which clearly demonstrates the difference between the ‘‘spin diffusion’’ and spin decay regimes. The character of these distribution functions can be understood from the following arguments. We first discuss the ‘‘spin diffusion’’

regime, where the characteristic wavelength of spin fluc- tuations is longer than the integration length l[Figs. 1(a) and1(b)]. In this regime, all spins withinlessentially point in the same direction and S^?_l remains large even after a long time evolution. Thus we find a peak at ðS^?_l Þ² 1in the distribution functionP^?_l (red line). In the other regime of spin decay [Figs.1(c)and1(d)], the typical length scale of spin fluctuations is shorter than the integration lengthl.

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In this case, integration of spins overlis akin to taking a random walk in 2D plane and accordingly, the distribution for ðS^?_l Þ² approaches exponential form with a peak at ðS^?_l Þ² ¼0 in P^?_l (blue line). In the intermediate regime, we observe two peak structure forP^?_l , where the distribu- tion exhibits both characteristic peaks (green line). We can understand the condition that separates these spin decay and spin diffusion type dynamics in the following way.

Deviation of spin angles atr¼lrelative tor¼0can be estimated from ¹ffiffiffi

L p P

k_rsk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hj_s;kj²i_max

q sinðklÞ. A

typical magnitude of is given by hðÞ²i where the average is taken over fluctuations of_rsk. The factorsinðklÞ in effectively limits momentum integration range to k >2=l, sohðÞ²i _2K^2l

ss. WhenhðÞ²i¹⁼² is smaller than2the system is in the spin diffusion regime. When hðÞ²i¹⁼² becomes of the order of 2 and larger, the system enters the spin decay regime. The crossover takes place around_4K^2l

ss1.

In the spin diffusion regime, the dynamics ofS^?_l andS^x_l display different time scales as can be seen in Fig.2(b). In order to understand this separation of time scale, we note that the magnitude of the integrated spin S^?_l is only affected by fluctuations with short wavelengths, < l, for which dynamics takes place at short time scale.

Hence hS^?_l i decays until the time t?2l=c_s [see Eq. (6)] and then it reaches a saturated value. On the other hand, S^x_l is affected by excitations of all wavelengths, so hS^x_lidecays until it reaches0. These behaviors are shown in Fig.2(b), where we compared the decay ofhS^?_l ifor various segment lengthlandhS^x_li. Nontrivial time evolution of the distribution functions P^x_l, P^x;y_l and especially the striking contrast of the spin diffusion and spin decay regimes should provide unique signatures of the non-mean-field character and multimode dynamics of 1D systems.

Summary.—We provided a theoretical analysis of Ramsey interference experiments with 1D quasiconden- sates, and showed that the time evolution of the full distri-

bution function contains unique signatures of the many- body dynamics in one dimension.

This work was supported by NSF grant DMR-0705472, Harvard MIT CUA, DARPA OLE program, AFOSR MURI, and Swiss NSF.

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0 4 8 12

0 0.2 0.4 0.6 0.8 1

Time 0

0.1 time =0 time =2.4 time =4.8

0 0.5 1

0

0.1time =7.2

0 0.5 1

time =9.6

0 0.5 1

time =12

l/ξs =10 l/ξs =30 l/ξs =50

a

b

FIG. 2 (color). (a) Time evolution of the distribution P^?_l for the magnitude of spinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS^?_l Þ². (b) time evolution of hS^x_li and

hðS^?_l Þ²i q

with various integration lengthl=_s¼10, 30, 50. Here we setK_s¼20,L=_s¼200.

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