Noise correlations of two-dimensional Bose gases
NNoise correlations of two-dimensional Bose gases
V. P. Singh
1, 2 and L. Mathey
1, 2 Zentrum f¨ur Optische Quantentechnologien and Institut f¨ur Laserphysik, Universit¨at Hamburg, 22761 Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany (Dated: November 9, 2018)We analyze density-density correlations of expanding clouds of weakly interacting two-dimensionalBose gases below and above the Berezinskii-Kosterlitz-Thouless transition, with particular focus onshort-time expansions. During time-of-flight expansion, phase fluctuations of the trapped systemtranslate into density fluctuations, in addition to the density fluctuations that exist in in-situ. Wecalculate the correlations of these fluctuations both in real space and in momentum space, andderive analytic expressions in momentum space. Below the transition, the correlation functionsshow an oscillatory behavior, controlled by the scaling exponent of the quasi-condensed phase, dueto constructive interference. We argue that this can be used to extract the scaling exponent of thequasi-condensate experimentally. Above the transition, the interference is rapidly suppressed whenthe atoms travel an average distance beyond the correlation length. This can be used to distinguishthe two phases qualitatively.
PACS numbers: 67.85.-d, 03.75.Hh, 03.75.Lm
I. INTRODUCTION
Phase coherence is a defining feature of degenerateBose gases. In three-dimensions (3D), a degenerate Bosegas shows long-range phase coherence, i.e., the single-particle correlation function approaches a constant atlarge distances. A weakly interacting Bose gas in two-dimensions (2D), however, shows quasi-long-range coher-ence [1, 2], i.e., the single-particle correlation function de-cays algebraically. As the temperature is increased, themagnitude of the algebraic scaling exponent increases,until it reaches a universal value, at which the systemundergoes a phase transition. Above the transition, thecorrelation function decays exponentially. This transitionwas predicted by Berezinskii [3] and by Kosterlitz andThouless [4], and is known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. It has been observed in sev-eral experiments, such as He films [5] and trapped Bosegases [6–9].Given the universal importance of correlations, detect-ing both single-particle and beyond single-particle cor-relations is of great interest. A seminal study was themeasurement of correlations in far-field intensity fluctua-tions of thermal light sources, observed first in Ref. [10],the Hanbury-Brown Twiss effect [11, 12]. In analogyto light sources, correlations in density fluctuations havebeen observed in expanding thermal Bose gases [13–18]and Fermi gases [19, 20]. The idea of probing many-body states of ultracold atoms using noise correlationswas first suggested in Ref. [21]. Since then, noise corre-lations have been used to study quantum phases, such asthe Mott-insulator transition in 2D Bose gas [22]. The-oretical investigations of noise correlations in 1D Fermigases were reported in Ref. [23, 24], of 1D Bose gases inRef. [24, 25], and of 2D Bose gases in Ref. [25, 26]. Thespectrum of these correlations in an expanding 2D Bosegas has been studied experimentally in Ref. [27, 28].In this paper we investigate if and how the BKT tran- sition in ultra-cold atom systems can be detected vianoise correlations, and what its signature is. For thispurpose, we study the density-density correlations in ex-panding clouds of 2D Bose gases, with particular focus onshort-time expansions, both above and below the tran-sition temperature. In experiments, the atoms are ini-tially trapped in a trap, and then released by turningoff the trap. During the subsequent free ballistic expan-sion, phase fluctuations present in the trapped systemtranslate into density fluctuations [29]. We calculate thecorrelations of these fluctuations for a homogeneous sys-tem of weakly-interacting bosons in the thermodynamiclimit, both in real space and in momentum space, forboth the temperature regime of algebraic scaling and ofexponential scaling. We find that in the quasi-condensedphase a long-lived interference pattern is visible in thenoise correlations, whereas in the thermal phase con-structive interference is suppressed after short times offlight. Furthermore, we argue that the shape of the in-terference pattern and its dependence on the algebraicscaling exponent can be used to determine this exponentexperimentally.This paper is organized as follows: In Sec. II we discusshow the in-situ correlations of the system are related tothe density-density correlations of two-dimensional Bosegases in a time-of-flight expansion. We first consideronly in-situ phase fluctuations, and calculate the density-density correlation function of an expanded cloud of Rbatoms below and above the BKT transition in Sec. III.In Sec. IV we derive an analytic expression for the spec-trum of the density-density correlations of the 2D quasi-condensate (below the BKT transition). We show a com-parison to numerical results, and discuss the properties ofthe analytic solution. In Sec. IV A we analyze the scalingbehavior of the peaks for the spectrum of density-densitycorrelations of the condensed phase, and derive an ana-lytic expression for the spectral peak locations. In Sec.IV B we show the spectrum of density-density correla- a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r FIG. 1. (Color online). Evolution of the two-point density correlation function g ( r , t ) is shown after successive expansiontimes t for the algebraic and exponential regime of a 2D Bose gas of Rb atoms. Panels (a)–(g) are for algebraic decay withscaling exponent τ = 0 . a = 1 µ m. Panels (h)–(n) are for exponential decay with the correlationlength r = 1 µ m and c = 1, see Eq. 12. tions for the exponential regime of 2D Bose gases (abovethe BKT transition). In Sec. V we expand the analysisto include in-situ density fluctuations of the 2D Bose gas.We analyze their impact on the density-density correla-tions and the spectrum of these correlations in time-of-flight expansion. We summarize our results and concludein Sec. VI. II. CORRELATIONS IN TIME-OF-FLIGHT
In this section, we relate the density-density correla-tions in time-of-flight to the in-situ correlations. Theexpansion of the atoms is assumed to be ballistic after re-lease from a tight transverse confinement. Due to the fastexpansion in the transverse direction, interaction effectsare suppressed quickly during time-of-flight. We considercorrelations of the column density, i.e., averaged over thetransverse direction ( z -axis), which reduces the analysisto two-dimensional expansion. The time evolution of thebosonic field operator is given by [30]ˆΨ( r , t ) = (cid:90) d r (cid:48) G ( r − r (cid:48) , t ) ˆΨ( r (cid:48) , . (1)Here, ˆΨ( r , t ) is the bosonic single-particle annihilationoperator at time t , and ˆΨ( r (cid:48) ,
0) is the initial single par-ticle operator. The Green’s function of free propagationfor a 2D system is defined as G ( r − r (cid:48) , t ) = G ( x − x (cid:48) , t ) G ( y − y (cid:48) , t ) , (2)where G ( ξ, t ) = (cid:114) m πi ¯ ht exp (cid:16) i mξ ht (cid:17) (3)with ¯ h being the reduced Planck constant, m the atomicmass, and t the expansion time. We now introduce the two-particle density matrix for the bosonic fields as ρ ( r , r , r , r ; t ) = (cid:104) ˆ ψ † ( r , t ) ˆ ψ † ( r , t ) ˆ ψ ( r , t ) ˆ ψ ( r , t ) (cid:105) . (4)The density-density correlations can be written in termsof the two-particle density matrix as (cid:104) ˆ n ( r , t )ˆ n ( r , t ) (cid:105) n ( r , t ) n ( r , t ) = ρ ( r , r , r , r ; t ) n ( r , t ) n ( r , t ) + δ ( r − r ) n ( r , t ) , (5)where n ( r , t ) = (cid:104) ˆ n ( r , t ) (cid:105) = (cid:104) ˆ ψ † ( r , t ) ˆ ψ ( r , t ) (cid:105) is the av-erage density at time t . The above expression hastwo terms on the right-hand side. The first termis the two-point density correlation function, and thesecond one is the shot-noise contribution that comesfrom the normal ordering of the bosonic field opera-tors. The two-point density correlation function is re-lated to the two-particle density matrix as g ( r , r ; t ) ≡ ρ ( r , r , r , r ; t ) / ( n ( r , t ) n ( r , t )). For homogeneoussystems, g ( r , r ; t ) depends only on the absolute valueof the relative distance | r − r | . Therefore, the density-density correlations for a homogeneous 2D system is (cid:104) ˆ n ( r , t )ˆ n (0 , t ) (cid:105) n = g ( r , t ) + δ ( r ) n . (6)Here, n is the average density for a homogeneous 2D sys-tem. Next, we calculate the two-point density correlationfunction g ( r , t ) of the expanded cloud. The observablemeasured experimentally is the density-density correla-tion function, which differs by the shot-noise term. Thefree evolution of the two-point density correlation func-tion for a 2D system is given by g ( r , r ; t ) = 1 n (cid:90) d r (cid:90) d r (cid:48) (cid:90) d r (cid:90) d r (cid:48) × G ( r , t ) G ( r , t ) G ∗ ( r (cid:48) , t ) × G ∗ ( r (cid:48) , t ) ρ ( r (cid:48) , r (cid:48) , r , r ; 0) , (7)where r ij ≡ r i − r j and r (cid:48) ij ≡ r i − r (cid:48) j . Using the trans-lational invariance of the two-point density correlation (a) r @ Μ m D g H r ,t L Τ= Τ= Τ= (b) r @ Μ m D g H r ,t L Τ= Τ= Τ= FIG. 2. (Color online). The density correlation function g ( r , t ) of an expanding cloud of Rb atoms below the BKTtransition (quasi-condensed phase) after two different expan-sion times t , (a) t = 0 . t = 3 ms. The blue,dot-dashed line corresponds to the scaling exponent τ = 0 . τ = 0 .
5; the red, solidline corresponds to τ = 0 . function, and Eqs. (2) and (3), we get the following ex-pression: g ( r , t ) = 1 n (cid:16) m π ¯ ht (cid:17) (cid:90) d r (cid:90) d r (cid:48) exp (cid:16) i m ht × (cid:2) ( r − r ) − ( r − r (cid:48) ) (cid:3)(cid:17) × ρ r ( r (cid:48) ; r ; 0) , (8)where ρ r ( r (cid:48) ; r ; t ) ≡ ρ (cid:16) r (cid:48) , − r (cid:48) , r , − r ; t (cid:17) . In the aboveexpression, we have used the center of mass transforma-tion, hence the reduced mass m/ g ( r , t ) = 1(2 πn ) (cid:90) d q (cid:90) d R cos q · r cos q · R × (cid:42) ˆ ψ † (cid:18) ¯ h q tm , (cid:19) ˆ ψ † ( R ,
0) ˆ ψ (cid:18) R + ¯ h q tm , (cid:19) ˆ ψ ( , (cid:43) , (9)which gives the two-point density correlation function attime t . (a) r @ Μ m D g H r ,t L r = Μ m r = Μ m r = Μ m (b) r @ Μ m D g H r ,t L r = Μ m r = Μ m r = Μ m FIG. 3. (Color online). The correlation function g ( r , t ) ofan expanded cloud of Rb atoms above the BKT transi-tion (thermal phase) after two different expansion times t ,(a) t = 0 . t = 3 ms. The blue, dot-dashed linecorresponds to the correlation length r = 1 µ m; the purple,dashed line corresponds to r = 2 µ m; the red, solid line cor-responds to r = 3 µ m. We set c to be 1.
2D Bose gases
A two-dimensional Bose gas in equilibrium undergoesthe BKT phase transition, defined through the long-range behavior of the two-point correlation function. Thetwo-point correlation function of the field is defined as g ( r ) = (cid:104) ˆ ψ † ( r ) ˆ ψ ( ) (cid:105) n . (10)As discussed in Ref. [34] we write the single particle op-erator as Ψ( r ) ≈ exp( i ˆ θ ( r )) (cid:112) ˆ n ( r ). ˆ n ( r ) is the densityoperator and ˆ θ ( r ) is the phase. We write the densityoperator as ˆ n ( r ) = n + δ ˆ n ( r ), where δ ˆ n ( r ) is the opera-tor of density fluctuations and n is the average density.For now, we neglect density fluctuations and only con-sider the phase fluctuations of the gas. We will investi-gate the corrections due to density fluctuations in Sect.V. Thus, the single particle operator is approximatedby Ψ( r ) ≈ √ n exp( i ˆ θ ( r )). As described in Ref. [34]the single particle correlation function is approximately g ( r ) = n exp( −(cid:104) ∆ θ ( r ) (cid:105) / θ ( r ) ≡ θ ( r ) − θ ( ).In the quasi-condensed phase, the correlation function ofthe phase scales logarithmically. Therefore, the two-pointcorrelation function decays algebraically as g ( r ) ≈ (cid:18) a a + | r | (cid:19) τ/ ≡ F a ( r ) , (11)where a is a short distance cutoff [31]. τ is the scalingexponent [32, 33], which varies from 0 to 1 as the temper-ature is increased from 0 to the Kosterlitz-Thouless tem-perature T c . Above the critical temperature, the corre-lation function of the phase scales linearly, and thereforethe functional form of the two-point correlation changesto exponential decay as g ( r ) ≈ (cid:32) c c + 4 sinh ( | r | /r ) (cid:33) / ≡ F e ( r ) , (12)where r is the correlation length. To determine the two-particle density matrix we evaluate ρ ( r , r , r , r ) ∼ n (cid:104) e − i ˆ θ ( r ) − i ˆ θ ( r )+ i ˆ θ ( r )+ i ˆ θ ( r ) (cid:105) . (13)As described in Ref. [24] this gives ρ ( r , r , r , r ) ∼ n F i ( r ) F i ( r ) F i ( r ) F i ( r ) F i ( r ) F i ( r ) , (14)where i stands for algebraic or exponential regime as i = a, e . Therefore, the two-point density correlationfunction for the phase fluctuating 2D Bose gas is g ( r , t ) = 1(2 π ) (cid:90) d q (cid:90) d R cos q · r cos q · R × (cid:32) F i ( q t ) F i ( R ) F i ( R − q t ) F i ( R + q t ) (cid:33) , (15)where q t ≡ ¯ h q t/m . III. REAL SPACE DENSITY CORRELATIONS
In this section, we discuss g ( r , t ) of a cloud of Rbatoms below and above the BKT transition. We evaluate g ( r , t ) numerically by using Eq. (15). For a finite system L we replace the components x and y of r = ( x, y ) by x → L/π sin( πx/L ), and analogously for y . We calculate g ( r , t ) for the quasi-condensed phase (below the BKTtransition) with three different scaling exponents τ after0 . a to be 1 µ m. In Fig. 3 we show g ( r , t ) for the thermal phase (above the BKT transition)for three different correlation lengths r after 0 . (a) ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ Τ= Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D (b) ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ Τ= Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D FIG. 4. (Color online). The spectrum of the density-densitycorrelations (cid:104)| ρ ( q ) | (cid:105) /n of an expanded cloud of Rb atomsbelow the BKT transition (quasi-condensed phase) after twodifferent expansion times t . (a) t = 0 . t = 3 ms.The blue, solid line corresponds to τ = 0 .
1; the purple, solidline corresponds to τ = 0 .
2; the red, solid line corresponds to τ = 0 .
5. These lines are the analytic results obtained fromEq. (18). The filled circles are numerical results obtained fora finite size system from Eq. (17). overall magnitude of g ( r , t ) − τ below the BKT transition, and with decreasing correla-tion length r above the transition. Both cases, increas-ing τ and decreasing r , describe increasing temperature,and therefore an overall increase of the noise level. IV. CORRELATIONS IN MOMENTUM SPACE
In this section, we discuss the power spectrum density-density correlations, defined as [36] (cid:104)| ρ ( q , t ) | (cid:105) n = (cid:90) d r cos q · r [ g ( r , t ) − , (16)which is essentially the Fourier transform of g ( r , t ). Wethus Fourier transform Eq. (15) and get (cid:104)| ρ ( q ) | (cid:105) n = (cid:90) d r cos q · r (cid:32) F i ( q t ) F i ( r ) F i ( r − q t ) F i ( r + q t ) − (cid:33) . (17)This quantity can be obtained experimentally by deter-mining the density-density correlations for a single real- à à à à àæ æ æ æ æ n = =
15 10 15 2001234 t @ ms D q n @ H Μ m L - D FIG. 5. (Color online). Comparison of mean-field spectralpeak locations to the peak locations obtained numericallyfrom Eq. (17) for the quasi-condensed phase with exponent τ = 0 .
1, after several expansion times t . The light blue, solidline and the orange, dashed line are the square of wavevec-tors corresponding to the first ( n = 1) and the second ( n = 2)spectral peak, respectively, obtained by the mean-field results,Eq. (21), for 3 ∼
20 ms expansion times. The filled squaresand the filled circles are the numerical results correspondingto n = 1 and n = 2 spectral peak, respectively, obtainednumerically from Eq. (16) for a finite system. æ æ æ æ æà à à à à ò ò ò ò ò n = = = t @ ms D l n @ È D q n È H Μ m L - D FIG. 6. (Color online). The shift ∆ q n of spectral peaklocations of the density-density correlations for the quasi-condensed phase with exponent τ = 0 . t . The filled circles, the filled squares, and thefilled triangles are results for the shift corresponding to thefirst ( n = 1), the second ( n = 2), and the third ( n = 3) spec-tral peak, respectively, obtained numerically from Eq. (17)for a finite system. The solid lines are shifts of the corre-sponding spectral peaks, obtained from Eq. (20) and plottedas function of expansion time. The dashed lines are shifts ofthe corresponding spectral peaks, obtained from Eq. (22). ization, and then repeating the measurement and averageover realizations. We first derive an analytic expressionfor the spectrum of density-density correlations for thequasi-condensed phase by expanding the two-point corre-lation function, Eq. (11), to first order in τ (see AppendixA for details). We get the following analytic result: (cid:104)| ρ ( q ) | (cid:105) n ≈ πaτ K ( aq ) q (cid:32) a a + q ¯ h t m (cid:33) τ/ × (cid:18) − cos (cid:18) q ¯ htm (cid:19)(cid:19) , (18)where K is the Bessel function of second kind and q ≡| q | . The above expression is a product of three terms onthe right-hand side. The first one is an exponential decaywith the short distance cutoff a for the quasi-condensedphase of the Bose gas. This term is time-independent,and purely a result of the system having a short rangecutoff. The third one is the mean-field term, see [25]. Ifthe system had perfect coherence, this term generates anundamped interference pattern. The second term is dueto quasi-long-range order: As the atomic cloud expands,the atoms interfere with other atoms from the systemfurther and further apart. As a result the constructiveinterference diminishes following a power-law.We evaluate the spectrum of the density-density cor-relations by numerical integration of Eq. (17) for a finitesystem and compare these results to the analytic resultsobtained from Eq. (18) for a cloud of Rb atoms be-low the BKT transition. We consider a two-dimensionalfinite system of length L for the numerical calculations.Figure 4 shows a direct comparison of the analytic re-sults to the numeric ones for three different temperaturedependent exponents τ after 0 . τ .We observe that the magnitude of the spectral peaksincreases with both the expansion time and the scal-ing exponent τ . In particular for small momenta, cor-responding to large distances, we can expand Eq. (18)and obtain: (cid:104)| ρ ( q ) | (cid:105) n ≈ πτ ¯ h q t m . (19)We note that the dependence on a drops out, and thatthe overall magnitude of the noise just scales linearly in τ . We propose to detect τ by fitting the power spectrumwith either Eq. (18) or Eq. (19). A. Spectral peak locations
We now discuss the scaling behavior of the spectralpeaks, as shown in Fig. 4, and derive analytic expressionsfor the spectral peak locations from Eq. (18). The shiftof the spectral peak locations owing to the temperature (a) q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D r = Μ m r = Μ m r = Μ m (b) q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D r = Μ m r = Μ m r = Μ m FIG. 7. (Color online). The spectrum of the density-densitycorrelations (cid:104)| ρ ( q ) | (cid:105) /n for a cloud of Rb atoms above theBKT transition (thermal phase) after two different expansiontimes t . (a) t = 0 . t = 3 ms. The blue, dashedline corresponds to the correlation length r = 1 µ m; the pur-ple, dot-dashed line corresponds to r = 2 µ m; the red, solidline corresponds to r = 3 µ m. We set c to be 1. and the expansion time is given by (see Appendix B)∆ q n ≈ q n L n (cid:16) q n t τ ¯ h K ( aq n ) + 2 aL n m K ( aq n ) (cid:17) × (cid:104) q n (cid:16) a m ( a − q nt ) + t (cid:0) − a τ + L n ( τ − q n q nt ) (cid:1) ¯ h (cid:17) K ( aq n ) + 2 am L n × (cid:0) L n + q nt τ (cid:1) K ( aq n ) (cid:105) − , (20)where K and K are the Bessel functions of the secondkind, and L n ( t ) = a + q nt , where q nt ≡ q n ¯ htm . Here, q n is defined as q n = (cid:114) (2 n − πm ¯ ht , (21)which is the location for the spectral peaks obtained fromthe mean-field term in Eq. (18). n is the peak ordernumber.To analyze Eq. (20), we consider two different cases forthe shift of the spectral peak locations as either aq n (cid:28) aq n (cid:29)
1. For aq n (cid:28)
1, i.e., t (cid:29) ma ¯ h (2 n − π , asimplified expression for the spectral peak shift is given (a) æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à àá á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á r @ Μ m D g H r ,t L á Τ= H density & phase fluc. L à Τ= H phase fluc. L ç Τ= H density & phase fluc. L æ Τ= H phase fluc. L (b) æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à àá á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á á r @ Μ m D g H r ,t L á Τ= H density & phase fluc. L à Τ= H phase fluc. L ç Τ= H density & phase fluc. L æ Τ= H phase fluc. L FIG. 8. (Color online) We show g ( r , t ) for the quasi-condensed phase of Rb atoms with a density of n =40 µ m − after two different expansion times t . In panel (a)we use t = 1 ms, and in panel (b) t = 3 ms. The dashedand the dot-dashed lines show the result based on Eq. (27),which takes into account density fluctuations. For compari-son, we show the result for the phase fluctuations only, basedon Eq. (15), as solid lines. In particular, the blue line withfilled circles and the red line with filled squares correspondto τ = 0 . τ = 0 .
8, respectively, with phase fluctuationsonly. The blue, dashed line with empty circles and the red,dot-dashed line with empty squares correspond to τ = 0 . τ = 0 .
8, respectively, with both density and phase fluctu-ations. For the case of τ = 0 . λ T ≈ . µ m, and for τ = 0 . λ T ≈ . µ m. by (see Appendix B for details)∆ q n ∼ − q n (4 + τ )4(2 n − π + ma ¯ ht (2 n − π − τ − , (22)which is the first-order correction to the spectral peaklocations.In Fig. 5, we compare locations of the spectral peaksobtained from the mean-field results, Eq. (21), to thespectral peak locations obtained numerically from Eq.(17) for a finite system. We observe that the spectralpeak locations are shifted towards lower wavevectors q than the mean-field locations. However, the shift of thespectral peak locations becomes negligible for large peakorder numbers n .Figure 6 shows the shift of the spectral peak loca-tions on semi-logarithmic scale for different peak ordernumbers n after various expansion times for the quasi- q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D t = = = = FIG. 9. (Color online). The spectrum of density-density cor-relations (cid:104)| ρ ( q ) | (cid:105) /n , including the contribution due to in-situ density fluctuations, for a quasi-condensate of Rb atomswith a density of n = 40 µ m − after several expansion times t , for τ = 0 .
8. The gray, solid line corresponds to the originalquasi-condensate ( t = 0); the purple, dot-dashed line cor-responds to t = 0 . t = 1 ms; the light blue, dotted line corresponds to t = 1 . λ T ≈ . µ m. condensed phase. Here, we show a comparison betweenthe spectral peak shifts obtained numerically from Eq.(17) for a finite system and the shifts obtained from theanalytic expression, Eq. (20), for a cloud of Rb atoms.Both the analytic results and the numerical ones are ingood agreement. The dashed lines in Fig. 6 are the re-sults obtained from the analytic expression, Eq. (22),that approximates the scaling behavior for the spectralpeak shift.Overall we find a nearly linear behavior due to themean-field contribution, with a small, negative ∼ /n correction, as can be seen from Eq. (22), which is due tothe fluctuations and the short-range cut-off. B. Power spectrum above the BKT transition
Now we discuss the spectrum of the density-densitycorrelations of 2D Bose gases above the BKT transi-tion (thermal phase). We calculate the spectrum of thedensity-density correlations numerically using Eq. (17)for a finite size system. Figure 7 shows the spectrum ofthe density-density correlations of an expanded cloud of Rb atoms for three different correlation lengths r after0 . V. INFLUENCE OF DENSITY FLUCTUATIONSON DENSITY CORRELATIONS
So far, we considered only the phase fluctuations ofthe condensed phase, as expressed by Eq. (11). In thissection, we include the density fluctuations and analyzetheir effect on the density-density correlations and the spectrum of these correlations in time-of-flight expansion.We consider small fluctuations of density, and thereforeexpand √ ˆ n up to second order in δ ˆ n as (cid:112) n ( r ) + δ ˆ n ( r ) ≈ √ n (cid:104) δ ˆ nn − δ ˆ n n (cid:105) . (23)The contribution of small fluctuations of density to thetwo-point correlation function is [34] g ( r ) ∼ = F a ( r ) (cid:104) − (cid:104) (∆ δ ˜ n ) (cid:105) (cid:105) , (24)where ∆ δ ˜ n ≡ δ ˜ n ( ) − δ ˜ n ( r ) and δ ˜ n ( r ) = δ ˆ n ( r ) /n ( r ).We calculate the expectation value of (∆ δ ˜ n ) using Bo-goliubov theory. We find that for temperatures smallcompared to the chemical potential, k B T < µ , the den-sity fluctuations are suppressed, and the two-point cor-relation function is principally governed by phase fluc-tuations. This corresponds to the regime λ T > ξ ,where λ T is the thermal de Broglie wavelength, λ T ≡ (cid:113) π ¯ h / ( mk B T ), and ξ is the healing length, related tothe chemical potential ξ = ¯ h/ √ mµ . For temperatures k B T > µ , i.e., λ T < ξ , density fluctuations are of sizeablemagnitude. They are predominantly due to the thermalsingle particle excitations of the Bogoliubov spectrum,and are approximately (cid:104) δ ˜ n ( r i ) δ ˜ n ( r j ) (cid:105) ≈ n λ T e − π | r i − r j | λ T . (25)With this, Eq. (24) becomes g ( r ) ∼ = F a ( r ) (cid:104) − A (cid:0) − e − π | r | λ T (cid:1)(cid:105) , (26)where A = 1 / (4 n λ T ).In order to calculate the corrections to g ( r , t ) due todensity fluctuations, we consider Eq. 9, and expand theproduct of four single particle operators to second or-der in δ ˜ n , using Eq. (23). Here we focus on the high-temperature contributions, and ignore further shot noisecontributions due to the commutators between δ ˜ n and ˜ θ .With this, g ( r , t ) is given by g ( r , t ) = 1(2 π ) (cid:90) d q (cid:90) d R cos q · r cos q · R × (cid:32) F a ( q t ) F a ( R ) F a ( R − q t ) F a ( R + q t ) (cid:33)(cid:32) (cid:104) (cid:104) δ ˜ n ( R ) δ ˜ n ( ) (cid:105) + (cid:104) δ ˜ n ( q t ) δ ˜ n ( ) (cid:105) (cid:105) + 14 (cid:104) (cid:104) δ ˜ n ( R − q t ) δ ˜ n ( ) (cid:105) + (cid:104) δ ˜ n ( R + q t ) δ ˜ n ( ) (cid:105) (cid:105) − (cid:104) δ ˜ n ( ) (cid:105) (cid:33) . (27)We use this expression to calculate g ( r , t ) numericallyfor a finite system, as described above. In Fig. 8, wecompare the two-point density correlation function for (a) Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D (b) Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D FIG. 10. (Color online) Power spectrum (cid:104)| ρ ( q ) | (cid:105) /n of aquasi-condensate of Rb atoms with density n = 40 µ m − after two different expansion times t , (a) t = 1 ms, and (b) t = 3 ms. The blue, solid line and the red, solid line cor-respond to τ = 0 . τ = 0 .
8, respectively. These showthe spectrum for phase fluctuations only, see Eq. (17). Theblue, dashed-line and the red, dot-dashed line correspond to τ = 0 . τ = 0 .
8, respectively, with both density and phasefluctuations, based on the Fourier transform of Eq. (27). Forthe examples with τ = 0 . λ T ≈ . µ m, for τ = 0 . λ T ≈ . µ m. the density and phase fluctuating quasi-condensate to thephase fluctuating quasi-condensate. We observe that thecorrelation function is enhanced by density fluctuationsfor distances r < ∼ λ T and it is mainly given by phasefluctuations for distances r > ∼ λ T , with a small reduc-tion of contrast. We also note that this effect of thedensity fluctuations decreases with increasing expansiontime, because of the small length scale associated withthe density fluctuations.Next, we calculate the spectrum of the density-densitycorrelations, including the effect of density fluctuations,i.e., we Fourier transform g ( r , t ), given in Eq. (27). InFig. 9, we show the spectrum of the density-density cor-relations as it evolves for short expansion times. Initially,at t = 0, the power spectrum is just the spectrum of in-situ density fluctuations, given by (cid:104)| ρ ( q ) | (cid:105) n = 1 n e − q λ T π . (28)As the atomic cloud expands, the phase fluctuationstranslate into density ripples, and an oscillatory patternemerges. Two further examples are shown in Fig. 10. (a) Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D (b) Τ= Τ= q @H Μ m L - D X È Ρ H q ,t L È \ (cid:144) n @ H Μ m L D FIG. 11. (Color online) The solid lines are the same data asin Fig. 10, the power spectrum of a quasi-condensate of Rbatoms with density n = 40 µ m − , with only phase fluctua-tions taken into account. The blue, dashed-line and the red,dot-dashed line also correspond to the same parameters as inFig. 10, however, now the contribution due to in-situ densityfluctuations has been subtracted, Eq. (28). Since we want to extract the scaling exponent τ whichonly controls the phase fluctuations, we have to distin-guish between the features which are due to the densityfluctuations and which are due to the phase fluctuations.We note that these two contributions occur on very dif-ferent length scales. This can be easily seen from thesingle-particle correlation function shown in Eq. (24).While the phase fluctuations lead to a slowly decayingfunction F a ( r ), the density correlations fall off quicklyon a length scale λ T , over which the function F a ( r ) isnearly constant. As a result the phase and the densitycontributions nearly separate into a sum. A similar ob-servation holds for g ( r , t ). As a result, the spectrumnearly separates into a sum of two terms, one due to thein-situ density fluctuations and the other due to phasefluctuations.We therefore propose to subtract the contribution ofin-situ density fluctuations from the spectrum, as shownin Fig. 11. We observe that the resulting spectrum isessentially just that of a purely phase fluctuating conden-sate, up to wavevectors q < ∼ /λ T . Especially for q → τ . VI. CONCLUSIONS
In conclusion, we have analyzed the density-densitycorrelations of an ultra-cold atomic Bose gas in two-dimensions, below and above the BKT transition, duringtime-of-flight. As a typical example, we considered con-densates of Rb atoms. We first discussed the density-density correlations and the spectrum of these correla-tions in time-of-flight taking into account only the phasefluctuations, and later included the contribution due tothe density fluctuations. We have shown that the quasi-condensed and the thermal phase of the 2D Bose gasescan be distinguished from the interference pattern ob-served for the density-density correlations. Below theBKT transition, this interference pattern is controlledby the scaling exponent of the quasi-condensed phase.Above the transition, the oscillating pattern is quicklysuppressed for expansion distances beyond the correla- tion length. We propose to use the power spectrum ofthe density ripples to detect the algebraic scaling expo-nent τ . We have demonstrated that the spectrum of thedensity-density correlations is a superposition of the insitu density fluctuations term and the phase fluctuationsterm. The analytic expression for the power spectrumwithout the in situ density fluctuations term can be usedas a fitting function to experimentally detect the alge-braic scaling exponent of the phase fluctuating 2D quasi-condensates, after removing the contribution due to in-situ density fluctuations. ACKNOWLEDGEMENTS
We gratefully acknowledge valuable discussions withJean Dalibard, R´emi Desbuquois, Selim Jochim, EiteTiesinga, and Yong-il Shin. We acknowledge supportfrom the Deutsche Forschungsgemeinschaft through theHamburg Centre for Ultrafast Imaging and the SFB 925,and from the Landesexzellenzinitiative Hamburg, whichis supported by the Joachim Herz Stiftung. [1] N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966).[2] P. Hohenberg, Phys. Rev. , 383 (1967).[3] V. L. Berezinskii, Sov. Phys. JETP , 610 (1972).[4] J. M. Kosterlitz and D. J. Thouless, J. Phys. C , 1181(1973).[5] D. J. Bishop and J. D. Reppy, Phys. Rev. Lett. , 1727(1978).[6] Z. Hadzibabic, P. Kr¨uger, M. Cheneau, B. Battelier, andJ. Dalibard, Nature (London) , 1118 (2006).[7] P. Kr¨uger, Z. Hadzibabic, and J. Dalibard, Phys. Rev.Lett. , 040402 (2007).[8] P. Clad´e, C. Ryu, A. Ramanathan, K. Helmerson, andW. D. Phillips, Phys. Rev. Lett. , 170401 (2009).[9] J. Choi, S. W. Seo, and Y. Shin, Phys. Rev. Lett. ,175302 (2013).[10] R. Hanbury Brown, and R. Q. Twiss, Nature , 27(1956).[11] U. Fano, Am. J. Phys. , 539 (1961).[12] R. J. Glauber, Phys. Rev. , 2529 (1963).[13] M. Yasuda, and F. Shimizu, Phys. Rev. Lett. , 3090(1996).[14] S. F¨olling, F. Gerbier, A. Widera, O. Mandel, T. Gericke,and I. Bloch, Nature (London) , 481 (2005).[15] M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes,D. Boiron, A. Aspect, and C. I. Westbrook, Science ,648 (2005).[16] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen,V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang,D. Boiron, A. Aspect, and C. I. Westbrook, Nature (Lon-don) , 402 (2007).[17] S. S. Hodgman, R. G. Dall, A. G. Manning, K. G. H.Baldwin, and A. G. Truscott, Science , 1046 (2011).[18] A. Perrin, R. B¨ucker, S. Manz, T. Betz, C. Koller, T.Plisson, T. Schumm, and J. Schmiedmayer Nature Phys. , 195 (2012).[19] M. Greiner, C. A. Regal, J. T. Stewart, and D. S. Jin,Phys. Rev. Lett. , 110401 (2005).[20] T. Rom, Th. Best, D. van Oosten, U. Schneider, S.F¨olling, B. Paredes, and I. Bloch, Nature (London) ,733 (2006).[21] E. Altman, E. Demler, and M. D. Lukin, Phys. Rev. A , 013603 (2004).[22] I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys.Rev. Lett. , 080404 (2007).[23] L. Mathey, A. Vishwanath, and E. Altman, Phys. Rev.Lett. , 240401 (2008).[24] L. Mathey, A. Vishwanath, and E. Altman, Phys. Rev.A , 013609 (2009).[25] A. Imambekov, I. E. Mazets, D. S. Petrov, V. Gritsev,S. Manz, S. Hofferberth, T. Schumm, E. Demler, and J.Schmiedmayer, Phys. Rev. A , 033604 (2009).[26] I. E. Mazets, Phys. Rev. A , 055603 (2012).[27] J. Choi, S. W. Seo, W. J. Kwon, and Y. Shin, Phys. Rev.Lett. , 125301 (2012).[28] S. W. Seo, J. Choi, and Y. Shin, arXiv:1402.3862.[29] S. Dettmer, D. Hellweg, P. Ryytty, J. J. Arlt, W. Ert-mer, K. Sengstock, D. S. Petrov, G. V. Shlyapnikov, H.Kreutzmann, L. Santos, and M. Lewenstein, Phys. Rev.Lett. , 160406 (2001).[30] R. P. Feynman and A. R. Hibbs, Quantum Mechanicsand Path Integrals (McGraw-Hill, New York, 1965).[31] For a Bose gas in finite two-dimensional systems the shortdistance cutoff a is defined as a = λ T / (2 πξ h ), where λ T isthe thermal de Broglie wavelength, and ξ h is the healinglength.[32] N. Prokof’ev et al. , Phys. Rev. Lett. , 270402 (2001).[33] N. Prokof’ev and B. Svistunov, Phys. Rev. A , 043608(2002).[34] C. Mora and Y. Castin, Phys. Rev. A , 053615 (2003). [35] L. Cacciapuoti et al. , Phys. Rev. A , 053612 (2003).[36] In the right-hand side of Eq. (16) we subtract the asymp-totic value of g ( r , t ), which does not affect the behaviorof the spectrum at non-zero wavevectors. APPENDIX A
In this appendix, we derive the analytic result, Eq.(18), for the quasi-condensed phase. The expressionfor the spectrum of the density-density correlations forthe phase fluctuating quasi-condensate in two-dimensions[Eq. (17)] is given by (cid:104)| ρ ( q ) | (cid:105) n = (cid:90) d r cos q · r (cid:32) F a ( q t ) F a ( r ) F a ( r − q t ) F a ( r + q t ) − (cid:33) , (A1)where q t ≡ ¯ h q t/m and F a ( r ) is the algebraic decay forthe quasi-condensate, which is given by F a ( r ) ≡ (cid:18) a a + | r | (cid:19) τ/ . (A2)We consider only the x -component ( q x ) of the wavevector q and expand the above integral, Eq. (A1), to first orderin the exponent τ . We get the following first order term: τ F a ( q tx ) (cid:90) dx cos( q x x ) (cid:90) dy ln (cid:32)(cid:16) a + ( x − q tx ) + s ( y ) (cid:17)(cid:16) a + ( x + q tx ) + s ( y ) (cid:17)(cid:16) a + x + s ( y ) (cid:17) − (cid:33) (A3)with s ( y ) defined as s ( y ) = L/π sin( πy/L ). Here, weexplicitly kept the expression for a finite size systemand used q tx ≡ ¯ hq x t/m . After integrating over the y -dimension of the system and Fourier transforming along x -axis, we get an analytic expression for the spectrum ofthe density-density correlations, which is given by (cid:104)| ρ ( q ) | (cid:105) n ≈ πaτ K ( aq ) q (cid:32) a a + q ¯ h t m (cid:33) τ/ × (cid:18) − cos (cid:18) q ¯ htm (cid:19)(cid:19) , (A4)which is Eq. (18). APPENDIX B
In this appendix, we derive the analytic expressions,Eqs. (20) and (22), from the analytic result for the spec-trum of density-density correlations, Eq. (A4). The first-order correction to the location of the spectral peaks is given by∆ q n = 2 m ( a m q n + q n t ¯ h ) (cid:104) a m K ( aq n )+ q n t ¯ h (cid:0) τ K ( aq n ) + 2 aq n K ( aq n ) (cid:1)(cid:105) × (cid:104) am K ( aq n )( a m + q n t ¯ h ) × (cid:0) a m + q n t ( τ + 3)¯ h (cid:1) + q n K ( aq n ) (cid:16) a m − a m t τ ¯ h + m q n t ¯ h (cid:0) τ ( τ + 2) − a q n (cid:1) − q n t ¯ h (cid:17)(cid:105) − , (B1)where K and K are the Bessel functions of second kind.After rearranging the above equation and neglectingthe second order term in exponent τ , we arrive at∆ q n ≈ q n L n (cid:16) q n t τ ¯ h K ( aq n ) + 2 aL n m K ( aq n ) (cid:17) × (cid:104) q n (cid:16) a m ( a − q nt ) + t (cid:0) − a τ + L n ( τ − q n q nt ) (cid:1) ¯ h (cid:17) K ( aq n ) + 2 am L n × (cid:0) L n + q nt τ (cid:1) K ( aq n ) (cid:105) − , (B2)which is Eq. (20). Here L n ( t ) = a + q nt , and q nt ≡ q n ¯ htm .We now consider two cases either aq n (cid:28) aq n (cid:29) aq n (cid:28)
1, the Bessel functions in Eq. (B2) canbe replaced as K ( X ) ∼ /X , and K ( X ) ∼ /X . Thelimit aq n (cid:28)
1, i.e., t (cid:29) ma ¯ h (2 n − π translates L n ( t ) ∼ q nt . And we get∆ q n ∼ − q n (4 + τ )2 q n ( a + 2 q nt ) − τ − − q n (4 + τ )4(2 n − π + ma ¯ ht (2 n − π − τ − , (B3)which is Eq. (22). When aq n (cid:29)
1, the Bessel func-tions K ( X ) and K ( X ) can be replaced by their asymp-totic values as K ( X ) = K ( X ) ∼ (cid:112) π X exp( − X ).Since, aq n (cid:29)
1, i.e., t (cid:28) ma ¯ h (2 n − π and L n ( t ) = a + ¯ htm (2 n − π , these lead to two limiting cases: either t (cid:28) ma ¯ h n − π or t (cid:29) ma ¯ h n − π . If t (cid:28) ma ¯ h n − π that means L n ( t ) ∼ a . We obtain∆ q n ∼ − q n (2 a m + q n t τ ¯ h ) × (cid:104) q n t (4 q n q nt + τ )¯ h − m (3 a + a q n − q n q nt + aq nt τ ) (cid:105) − . (B4)The condition t (cid:29) ma ¯ h n − π translates into L n ( t ) ∼ q nt , And we get∆ q n ∼ − (2 aq n + τ )2 a ( aq n − τ −
3) + 4 q n q nt ..