Noise Correlations in low-dimensional systems of ultracold atoms
We derive relations between standard order parameter correlations and the noise correlations in time of flight images, which are valid for systems with long range order as well as low dimensional systems with algebraic decay of correlations. Both Bosonic and Fermionic systems are considered. For one dimensional Fermi systems we show that the noise correlations are equally sensitive to spin, charge and pairing correlations and may be used to distinguish between fluctuations in the different channels. This is in contrast to linear response experiments, such as Bragg spectroscopy, which are only sensitive to fluctuations in the particle-hole channel (spin or charge). For Bosonic systems we find a sharp peak in the noise correlation at opposite momenta that signals pairing correlations in the depletion cloud. In a condensate with true long range order, this peak is a delta function and we can use Bogoliubov theory to study its temperature dependence. Interestingly we find that it is enhanced with temperature in the low temperature limit. In one dimensional condensates with only quasi-long range (i.e. power-law) order the peak in the noise correlations also broadens to a power-law singularity.
The ability to trap ultracold atoms in tightly confined tubes formed by an optical lattice has opened the door for the controlled study of one dimensional physics. Such systems have been used to investigate ground state correlations (1); (2) as well as dynamics (3) and transport (4) of strongly interacting bosons in one dimension. One dimensional traps of ultracold fermions have also been realized. The ability to control the interactions has been demonstrated using s-wave Feshbach resonances for fermions with spin (5), and with p-wave resonances for spin-polarized fermions (6).
From the theoretical viewpoint, one dimensional systems provide good starting points to study strong correlation physics. Because of the enhanced quantum fluctuations, continuous symmetries cannot be broken in generic one dimensional systems, and mean field theories fail. Nevertheless the Luttinger liquid framework provides a well developed formalism to treat these systems theoretically (see for example (7)). In place of a mean field order parameter the tendencies to ordering manifest themselves by slowly decaying algebraic correlations and correspondingly divergent susceptibilities. A diverging susceptibility in a particular channel implies that coupling an array of tubes in the transverse direction would lead to true order in that channel. Thus weakly coupled one dimensional systems provide a possible theoretical route to investigate open questions regarding competing orders in higher dimensions.
But the absence of long range order, which makes one dimensional systems interesting also complicates the ways by which these systems can be probed. Various methods have been proposed and used to probe specific correlations. For example time of flight (1) as well as interference experiments (8) can probe single particle correlations. Bragg (9); (10) and lattice modulation (3) spectroscopies can measure dynamic density correlations . In (11) we proposed that noise correlations can be used as a highly flexible probe of 1D Fermi systems. In particular this method is sensitive to a wide range of correlations including spin, charge and pairing, and it treats the various channels on an equal footing. These results, obtained using the effective Luttinger liquid theory were confirmed by Luscher et. al. using a numerical simulation of a microscopic model.
In this paper we provide the detailed theory of quantum noise interferometry in one dimensional Fermi systems and extend it to interacting Bose systems.
Atomic shot noise in time of flight imaging was proposed in Ref. (13) as a probe of many body correlations in systems of ultra cold atoms. Specifically what is measured using this approach is the momentum space correlation function of the atoms in the trap
where are spin indices, are momenta, and and are the occupation operators of the corresponding momentum and internal state. This relies on the assumption that the atoms are approximately non-interacting from the time they are released from the trap (even if they were rather strongly interacting in the trap). Here we envision a system of one dimensional tubes, that allows expansion only in the axial direction. To ensure weak interactions during time of flight, the radial confinement in the tubes can be reduced simultaneously with release of the atoms from the global trap.
Recent experiments demonstrated the ability to detect many-body correlations by analysis of the noise correlations (1). For example long ranged density-wave correlations (induced by external lattice) were seen in a system of ultracold bosons(14); (15) and fermions(16) in deep optical lattices. The sign of the correlation, peak or dip, depended on the statistics as expected, demonstrating that the effect is essentially a generalization of the Hanbury-Brown Twiss effect. Pairing correlations in fermions originating from a dissociated molecular condensate were also observed in experiment(17). All of these demonstrations involved states with true long range order that are easily related to simple two particle effects such as Hanbury-Brown Twiss or bound state formation. However, one of the main points of Ref. (13) was that the noise correlations can be used to measure more general correlation functions in many-body systems, even in the absence of true long range order (LRO). Here we demonstrate this idea by developing a detailed theory for the noise correlations in one dimensional systems of bosons and fermions.
The analysis is laid out in the following order. In section II we consider a model system of spinless fermions in one dimension. To elucidate the connection between noise correlation and ordering tendencies we first assume in section II.1 the presence of true long range order. This allows to provide a clear physical picture for the way order in spin, charge (particle-hole) or pairing (particle-particle) channels translate into sharp features (in this case delta function peaks) in the noise correlations. In section II.2 we shall consider the actual system of interacting fermions in one dimension where only power law order parameter correlations exist (quasi-long range order). This system is characterized by a tendency to charge density wave (CDW) ordering for repulsive interactions, and to superconducting order (SC) for attractive interactions. The naive expectation is that the delta function peaks will be replaced by power-law singularities with a power that reflect the algebraic order parameter correlations. Interestingly we find that such a simple relation exists only if there are order parameter correlations that decay with a sufficiently slow power. This is the case beyond a critical interaction strength (repulsive or attractive). At weak interactions, on the other hand, the situation is more subtle and singular signatures of both the dominant and sub-dominant ordering tendencies are seen in the noise correlations (see Fig. 2b,c). We compare these results with the information that can be extracted from measuring the static structure factors in the spin and charge channels.
In section III, we move on to analyze the noise correlations in one dimensional systems of interacting spin- fermions. In the long wavelength effective field theory the interaction is parameterized by two Luttinger parameters and corresponding to the spin and charge sectors, as well as a backscattering parameter . The ordering tendencies of this system are summarized in the phase-diagram in Fig. 3 (See (19)). We find that the noise correlations provide direct information on the real space order parameter correlations only for negative backscattering. In this case a spin-gap opens and the system is characterized by competing CDW and SSC correlations, which are revealed by the noise correlations.
In section IV, we turn to interacting Bose systems. We first study the noise correlation within Bogoliubov theory for a condensate with true long range order (section IV.1). The interesting feature her is a peak in the noise correlations (1) at . This can be simply understood as the signature of pairing correlations in the Bogoliubov wave-function, describing quantum depletion of the condensate. We compute the evolution of this signature with increasing temperature. In addition we point out the existance of sharp dips in the noise correlations along the lines and , these reflect correlations between the condensate and the quantum depletion cloud. In section IV.2 we move on to discuss interacting one dimensional Bose liquids, where the single particle density matrix decays as a power-law with distance. We show how the sharp features in the noise correlation function on the lines , , and broaden to power-law singularities.
Ii Spinless Fermions
A system of interacting spinless fermions is perhaps the simplest Fermi system with a non trivial competition of ordering tendencies. Such systems can be implemented in experiments using fully polarized fermionic atoms that interact via an odd angular momentum channel. The interaction strength can be tuned by using a -wave Feshbach resonance (6). Another possibility is to use a Bose-Fermi mixture, where the phonons of the bosonic superfluid mediate effective interactions between the fermions (18).
In the restricted geometry of 1D the Fermi surface consists only of two points, the left (L) Fermi point at and the right (R) Fermi point at . We introduce the left- and right-moving fields and through:
and are slowly varying fields, because the rapidly oscillating phase factors have been separated out. The natural order parameters that characterize this system are the charge density and superconducting (pairing) operators:
The central problem we address in this section concerns the connection between the correlations of these physical order parameters and the observable noise correlation signal.
ii.1 Long range order
The relation between order parameter correlations and sharp features in the noise correlation function is most apparent when the system supports true long range order. Of course one dimensional Fermi systems generically do not display spontaneous long range order in either the CDW or SC channels. However long range order can be induced by an external potential (e.g. a superlattice potential) or it can form spontaneously in a weakly coupled array of one dimensional systems. Moreover the intuition gained from the exercise will be useful in approaching the more interesting case of power-law order parameter correlations (quasi long range order), which will be considered in the following sections.
Consider first the case of long range pairing correlations (i.e. SC order). Take () to be near the right (left) Fermi points, and , deviations from those Fermi points. It is now useful to write the noise correlations explicitly in terms of the pairing correlations
Here the operator creates a fermion pair whose constituents are separated by and their center of mass coordinate is . Note that we have dropped terms that undergo rapid spatial oscillations and therefore vanish under integration. The existence of long range pairing correlations implies
at long distances (). Here is the translationally invariant pairing wave function. The long range saturation of the correlation function leads to a singular contribution to (5) at :
We thus conclude that the noise correlations in this case are directly related to the long distance limit of the pairing correlations. This is one way to formally justify the mean-field decoupling of the noise correlation function carried out in Ref. [(13)], which gives:
A very similar analysis follows for the case of long range order in the particle-hole cannel. Take for example CDW order at the wave-vector . In this case we should write the noise correlation function in terms of the density correlations to expose their singular contribution:
Note the minus sign in front of the integral, which resulted from commuting the Fermi operators to obtain an expression written in terms of density wave correlations. Long range CDW correlations imply
which leads to a singular contribution to (9) for
This justifies the corresponding mean field decoupling of the noise correlation function:
Again, as in the SC case, we see that the noise correlations are directly related to the long distance limit of the non decaying order parameter correlation.
We note two obvious distinctions between the cases of order in the particle-particle (pairing) and particle-hole channels. In the former the singular correlations are between particles with opposite momenta and are positive correlations. In the latter case, by contrast, the singular correlations are between particles whose momenta differ by and are negative, that is anti-correlations. To better understand the origin of this effect it is worthwhile to inspect the mean field wave functions, which sustain the respective broken symmetries. For the superconducting order the mean field wave-function is the BCS state representing a pair condensate:
Contrary to a filled Fermi sea the particle number in a specific point is not definite in this wave function. However, if a particle is found at , there is with certainty another one at . This implies positive correlation between and as visualized in Fig. 1 (a).
The CDW state on the other hand may be viewed as a condensate of particle-hole pairs on top of the filled Fermi sea. It is then written in a way that exposes the similarity to the BCS state:
Here denotes a full fermi sea wavefunction. As in the BCS state, the particle number at a specific point is not definite. Now however, if one finds a particle at there will be with certainty a missing particle (hole) in the Fermi sea at . This implies anti-correlation between and as visualized in Fig. 1 (b).
Because of strong quantum fluctuations, an actual one dimensional Fermi system cannot sustain true long range order such as the SC and CDW orders discussed above. Instead, at zero temperature a critical phase with power law correlations, or quasi long range order, is established. If the power-law decay is sufficiently slow then it is reasonable to expect that it would still make a singular contribution to the integrals in (5) or (9). To calculate the resulting singularity and its dependence on the system parameters we use the low energy Luttinger liquid theory. The basic idea is that the asymptotic low energy and long wave-length properties of the interacting one dimensional Fermi system are captured by a universal harmonic theory:
Here is the Luttinger parameter, which determines the power-law decay of long range correlations. is a sound velocity, which we will henceforth set to 1. The bosonization identity, which relates the bosonic fields to the fermion operators is:
with the commutation relation:
is an artificial cutoff of the bosonization procedure which must be sent to zero at the end of the calculation. Implicit in the action (15) is also a physical short distance cutoff , taken in most cases to be of order . Here is the phase field of the Fermi operator. is the smooth () component of the density fluctuation, and by (17), is also canonically conjugate to .
The effective action (15) is a free theory in terms of the bosonic fields. It therefore allows to calculate the needed correlation functions using simple gaussian quadrature. The fermion interactions affect the correlation functions only through the Luttinger parameter . For non interacting fermions , for repulsive interactions and for attractive interactions. In general deviates more from the stronger the interactions. However it is in general not possible to make a precise connection between the microscopic parameters and the Luttinger parameter. Among other things, our analysis points to a way of extracting this parameter from experiments.
For any value of the system shows either CDW or SC quasi-long range order, which means that the correlation functions decay slow enough to give a divergent susceptibility. The calculation of these correlation functions has been given in numerous places (see e.g. (21); (7)). For the long distance behavior of we have , and for we have where the scaling exponents for CDW and SC are
The susceptibilities correspond to the spatial and temporal Fourier transform of the correlation functions, . These will be divergent at large distances exactly if the scaling exponent of the operator is positive. As we can see from (18) and (19), diverges at for , and diverges at for . In this sense of QLRO we say that the system is in the CDW regime for , and in the pairing regime for (21), as depicted in Fig. 2 e). We note that from Eq. 15 and 16 one can read off the duality mapping: , , which leaves the action invariant, and maps the CDW regime onto the SC regime, .
With this formalism we derive the noise correlations , which we show in detail in App. A. We find
The integration in Eq. 20 is over the three spatial variables , and . The exponents and are given by and .
Following the discussion of the MFA, we note that this integral ’contains’ the equal-time correlations of and . This can be seen by setting and , which gives the Fourier transform in of . If we set and , we obtain the Fourier transform of in . To discuss this further we introduce the following variables: , , and . With this, , is of the form
In the limit , we have . This enforces the integrand to be negligible except in regions with . With this, the expression approximately evaluates to
In the dual limit, , we have , which enforces . In this limit the integral approximately evaluates to
which can also be inferred from duality. These contributions are divergent for and , and turn out to be the dominant contribution in theses regimes.
To confirm this expectation, and in order to study the regime , we evaluate this integral numerically for different values of . In Fig. 2 we show for , , and . The example shows indeed a power-law divergence of the particle-hole type, which we find througout the regime. For we find a result similar to the example, an algebraic divergence of the particle-particle type. These two regimes are indicated in Fig. 2 e).
In between these two regimes, for , we find a regime in which both p-h as well as p-p correlations exist, i.e. we find precursors of the near-by competing order. A simple argument for the qualitative shape of the noise correlations function in this regime is the following: If we consider an interacting 1D Fermi gas and treat the interaction perturbatively, the lowest order contribution would consist of states that contain two p-h pairs, as two fermions have been taken from the Fermi sea and put into the unoccupied states above the Fermi sea (see Fig. 1 c)). Such a state exhibits qualitatively the noise correlations that are observed in Fig. 2 b) and c): positive correlations for and , and negative correlations for and . To quantify how this affects the line-shape of the noise correlations we expand to second order in (see App. B for details). The result is
This expression shows a divergence at , and both particle-particle and particle-hole fluctuations. Higher order terms either enhance particle-particle (for ) or particle-hole () fluctuations.
Before concluding the section, we compare the static and dynamic structure factor with the noise correlations discussed in this section. The dynamic structure factor can be measured by a stimulated two-photon process, as has been demonstrated in (9). It is defined as
As discussed in Ref. (7), it is given by
for small . For , it behaves as
where . So at , coming from larger values, two additional peaks appear at , because the exponent in Eq. (29) switches sign. At two additional peaks appear at , and so on. So the measurement of the dynamic structure factor, being the Fourier transform of the density-density correlation function, certainly allows insight into the CDW phase, but contains no information about the SC phase. In contrast, as demonstrated in Fig. 2, the measurement of noise correlations proposed in this paper allows the identification of pairing and CDW ordering in a single approach and on equal footing, because both the CDW and the SC correlations are contained in the expression for the noise correlations.
The static structure factor of a BEC has been measured in [(10)]. It corresponds to the instantaneous density-density correlations:
For , we have
and for :
Here the first set of peaks appears at , the next set of peaks at and so on. This is the power-law divergence that dominates the noise correlations for , as discussed in the previous section. Again there is no signature of SC.
In summary we have derived the noise correlations of a spinless fermionic LL in this section, and compared it to a MFA result. We found different subregimes with qualitatively different behavior in each of the quasi-phases, summarized in Fig. 2, and discussed how phenomena such as QLRO and competing phases are reflected in the noise correlations.
Iii Spin- Fermions
In this section we discuss the noise correlations of an SU(2)-symmetric Fermi mixture (20).
Our analysis applies to systems of the form
i.e. the 1D Hubbard model, or a mixture in a 1D continuum with a contact interaction:
which can both be realized in experiment.
iii.1 Long range order
Before we turn to the LL picture, we introduce the types of order that occur in this system, and discuss what kind of signature can be expected if they develop long-range order, with similar arguments as in Sect. II.1.
As for the spinless fermions we split the spinful fermionic operators into left- and right-moving fields:
describes singlet pairing, and therefore contains and operators, whereas describes the and component of triplet pairing, and therefore describes pairing between equal spin states. is the order parameter of the and component of the spin density wave order.
Because these order parameters contain both R and L, as well as and operators, we expect correlations between and , as well as and , so we consider two types of correlation functions; correlations between atoms in the same spin state
and correlations between opposite spins
We again use the same convention that is located near the right Fermi point and located near the left Fermi point. By considering the dominant contraction of these correlation functions, analogous to Eq. (8) and Eq. (12), we can expect the following signatures: For CDW order we expect , for SDW whereas for triplet pairing we expect , and for singlet pairing . Each of these statements can be confirmed and quantified with a mean-field calculation.
We bosonize these fermionic fields according to:
with the same definitions of and that we used for spinless fermions. We introduce spin and charge fields according to:
Written in terms of and the action of any SU(2)-symmetric system separates into a charge and a spin sector:
Each of these sectors is characterized by a velocity and a Luttinger parameter . In addition to the quadratic terms in the action we find a non-linear term, describing backscattering processes in the spin sector, with the prefactor . If this action is derived from a system with a short-ranged interaction such as (33) and (34), these parameters have the following properties: For repulsive interaction, one finds , and positive backscattering , for attractive interaction , , and negative backscattering . As discussed in (22); (21), this sine-Gordon model, can be treated with an RG calculation to identify two limiting cases: The case in which the backscattering term is irrelevant and the system flows towards the non-interacting fixed point (), which happens for repulsive interaction, and the case in which the backscattering term is relevant and a spin gap appears (), which happens for attractive interaction. In the evaluation of the noise correlation functions we will use these limiting values, for the gapless phase, for the spin-gapped regime. One can find the phase diagram of this system in exact analogy to the spinless case,
by studying the correlation functions of the order parameters. The scaling exponents of these operators are given by:
From these expressions one can read off the structure of the phase diagram. In the gapless phase we have , therefore singlet and triplet pairing, as well as SDW and CDW are algebraically degenerate. For we find a TS/SS phase, for we obtain a SDW/CDW phase. For , both and are sent to , whereas and are now given by and . Hence, we can distinguish four regimes: For we have singlet pairing, for we get CDW ordering. In between these two values of the system shows coexisting orders, that is, both the singlet pairing susceptibility and the CDW susceptibility are divergent. For CDW is dominant and SS is subdominant, for it is the other way around.
The noise correlations can be calculated in the same way as described for the spinless case. We obtain analogous expressions to Eq. (20), in which the exponents and are replaced by:
In order to understand in what regimes of the phase diagram we should expect algebraic divergencies, we again consider the equal-time correlation functions of the operators (36)–(39). In momentum space, these correlation functions scale as , where is the corresponding scaling exponent, given in (49)–(52). If the system is in the gapless phase (i.e. ), these correlation functions never exhibit an algebraic divergence, and we should expect to find coexisting fluctuations throughout the phase diagram for , similar to the regime for spinless fermions. If the system is in the spin-gapped phase, we find the following behavior: Both TS and SDW fluctuations are frozen out, i.e. only short-ranged, whereas SS and CDW are increased by , compared to the gapless phase, because . We therefore expect algebraic divergencies for in the channel, and for in the channel.
A numerical study of the noise correlations confirms these expectations: We indeed find coexisting fluctuations in the gapless phase for any value of . Furthermore, since the expressions (54) and (55) become identical for , we find that in this regime. This is a manifestation of the degeneracy (at the algebraic level) of triplet and singlet pairing for , and of spin density and charge density wave ordering for , as discussed in [(21)]. For the spin-gapped phase () this symmetry is broken and and behave qualitatively different. We find the following behavior: shows an algebraic divergence of the p-h type for , as expected from the equal-time correlation function of the CDW order parameter, an algebraic cusp for , and no ordering for . behaves in a complementary way: an algebraic divergence of the p-p type is found for , an algebraic cusp for and no ordering below that. In particular, for in the vicinity of , we find coexisting orders, as we demonstrate in Fig. 4. This is particularly clear if we consider the noise correlations of the total density , for which the noise correlations are given by . In Fig. 4 g) – i) we clearly see the coexistence of pairing and CDW ordering.
To understand this behavior further in this limit, we use the same argument as for the and regimes for spinless fermions. We re-write and in the same way as (23), where and need to be replaced by and , respectively. In the limit the arguments that lead to the scaling behavior (24) and (25) become exact, and we obtain
These expressions show exactly the structure that was found numerically: algebraic divergencies for () in the () channel, and an algebraic cusp for ().
We turn to address the noise correlations in bosonic systems with either long range or quasi long-range order in the off diagonal density matrix . As in the fermion case we shall start with the case of true long range order, relevant to three dimensional systems at temperature . This analysis is also relevant for lower dimensional systems if they are sufficiently weakly interacting. Then the fact that the off diagonal density matrix decays as a power law is unnoticeable in a condensate of realistic size. We shall discuss in some detail the possibility of observing the pairing correlations associated with quantum depletion and show how such measurements would depend on the temperature. Then we move on to derive the noise correlations in a one dimensional Bose system at zero temperature, taking into account the power-law behavior of the correlations. As in the case of Fermions we use the effective Luttinger liquid theory, which correctly accounts for the singular contributions to the noise correlations due to the long distance power-law behavior of the off diagonal density matrix.
iv.1 Bose-Einstein condensate with true ODLRO
Our starting point for analysis of the noise correlations is the Hamiltonian of a weakly interacting Bose gas with contact interactions
Here is the free particle dispersion with and with the s-wave scattering length. To compute the correlations in the condensed phase we apply the standard Bogoliubov theory (see e.g. (25)). As usual, the operators and are replaced by a number representing the condensate amplitude , while the other modes are treated as fluctuations and expanded to quadratic order. The effective Bogoliubov Hamiltonian is then given by
where is the condensate density. This Hamiltonian is diagonalized by the Bogoliubov transformation with , , and .
The structure of the ground state wave function in the Bogoliubov approximation is given by
Like the state (13), the Bogoliubov wave-function describes perfectly correlated pairs of particles at momenta and , which suggests the appearance of pairing correlations in the noise.
It is straight forward to compute the noise correlations for . Because the Bogoliubov Hamiltonian (59) is quadratic we can use Wick’s theorem to decouple the four point function
where the expectation values correspond to thermal averages, and . A bit more care is needed if either or because the quadratic hamiltonian describes only the fluctuations in the depletion cloud, not in the condensate number. To obtain the fluctuations in the condensate within Bogoliubov theory, we use the conservation of total particle number, which implies that fluctuations in the condensate number are exactly to minus those of the depletion cloud. In other words we substitute for the condensate particle number operator. Then we may use (59) to compute the noise correlation between points and . Putting it all together we get the general expression for the noise correlations:
Here is the quasi-particle number distribution.
At zero temperature each term in (62) has a simple physical interpretation. We already noted that the first term manifests the pairing correlations present in the quantum depletion described by the Bogoliubov wave function (60). The second term is a positive correlation due to boson bunching at a point in -space. The dips at reflect the fact that an extra atom found at in the quantum depletion cloud corresponds to a pair of atoms, now missing from the condensate. Finally the positive peak at appears because extra atoms in the condensate must always come in pairs, annihilated from the depletion cloud. That is, if we find an extra atom in the condensate, then we are sure to find another extra atom in it.
The evolution of the peaks with temperature and their momentum dependencies are controlled by the ratio of two natural length scales of the problem. One is the healing length of the condensate which is determined by interactions . The other is the thermal wavelength . The combination