Detection of Fermi Pairing via Electromagnetically Induced Transparency
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Detection of Fermi Pairing via Electromagnetically Induced Transparency
Lei Jiang , Han Pu , Weiping Zhang , and Hong Y. Ling Department of Physics and Astronomy, and Rice Quantum Institute, Rice University, Houston, TX 77251, USA State Key Laboratory of Precision Spectroscopy, Department of Physics,East China Normal University, Shanghai 200062, P. R. China and Department of Physics and Astronomy, Rowan University, Glassboro, New Jersey, 08028-1700, USA (Dated: November 16, 2018)An optical spectroscopic method based on the principle of electromagnetically-induced trans-parency (EIT) is proposed as quite a generic probing tool that provides valuable insights into thenature of Fermi paring in ultracold Fermi gases of two hyperfine states. This technique has thecapability of allowing spectroscopic response to be determined in a nearly non-destructive mannerand the whole spectrum may be obtained by scanning the probe laser frequency faster than thelifetime of the sample without re-preparing the atomic sample repeatedly. A quasiparticle picture isconstructed to facilitate a simple physical explanation of the pairing signature in the EIT spectra.
PACS numbers: 03.75.Ss, 05.30.Fk, 32.80.Qk
I. INTRODUCTION
The two-component degenerate Fermi gas, in whichthe interaction between atoms of two different hyperfinestates is made magnetically tunable via Feshbach reso-nance, has been the main source of inspiration for muchrecent excitement at the forefront of ultracold atomicphysics research. In addition to being an ideal systemfor the exploration of the crossover from a Bose-Einstencondensate (BEC) of highly localized pairs to nonlocalBardeen-Cooper-Schrieffer (BCS) pairs, the degenerateFermi gas, when operating in the unitarity regime, con-stitutes a strongly interacting Fermi gas exhibiting a richset of physics, the study of which may shed light on long-standing problems in many different branches of physics,in particular, condensed matter physics.A unique phenomenon of low temperature Femi systemis the formation of correlated Fermi pairs. How to detectpair formation in an indisputable fashion has remained acentral problem in the study of ultracold atomic physics.Unlike the BEC transition of bosons for which the phasetransition is accompanied by an easily detectable dras-tic change in atomic density profile, the onset of pairingin Fermi gases does not result in measurable changes infermion density. Early proposals sought the BCS pair-ing signature from the images of off-resonance scatteringlight [1]. The underlying idea is that to gain pairing in-formation, measurement must go beyond the first-ordercoherence, for example, to the density-density correla-tion. This is also the foundation for other detectingmethods such as spatial noise correlations in the imageof the expanding gas [2], Bragg scattering [3, 4], Ramanspectroscopy [5], Stokes scattering method [6], radio fre-quency (RF) spectroscopy [7, 8], optical detection of ab-sorption [9], and interferometric method [10]. Among allthese methods, RF spectroscopy [7, 8] has been the onlyone implemented in current experiments [11, 12].In this paper, we propose an alternative detectionscheme, whose principle of operation is illustrated inFig. 1(a). In our scheme, a relatively strong coupling and a weak probe laser field between the excited state | e i and, respectively, the ground state | g i and the spinup state |↑i , form a Λ-type energy diagram, which fa-cilitates the use of the principle of electromagneticallyinduced transparency (EIT) to determine the nature ofpairing in the interacting Fermi gas of two hyperfine spinstates: |↑i and |↓i . EIT [13], in which a probe laser fieldexperiences (virtually) no absorption but steep dispersionwhen operating around an atomic transition frequency,has been at the forefront of many exciting developmentsin the field of quantum optics [14]. Such a phenomenon isbased on quantum interference, which is absent in mea-surement schemes such as in Ref. [6], where lasers aretuned far away from single-photon resonance. In the con-text of ultracold atoms, an important example is the ex-perimental demonstration of dramatic reduction of lightspeed in the EIT medium in the form of Bose conden-sate [15]. This experiment has led to a renewed interestin EIT, motivated primarily at the prospect of the newpossibilities that the slow speed and low intensity lightmay add to nonlinear optics [16] and quantum informa-tion processing [17]. More recently, EIT has been usedto spectroscopically probe ultracold Rydberg atoms [18].In this work, we will show how EIT can be exploited toreveal the nature of pairing in Fermi gases.Before we present our detailed calculation, let us firstcompare the proposed EIT method with the RF spec-troscopy method which is widely used in probing Fermigases nowadays. In the latter [7, 8], an atomic sampleis prepared and an RF pulse is applied to the samplewhich couples one of the pairing states to a third atomiclevel | i . This is followed by a destructive measurementof the transferred atom numbers using absorption laserimaging. The RF signal is defined as the average ratechange of the population in state | i during the RF pulse,which can be inferred from the measured loss of atomsin | ↑i . This process is repeated for another RF pulsewith a different frequency. In addition to sparking manytheoretical activities [19, 20, 21, 22, 23], this methodhas recently been expanded into the imbalanced Fermi | d d p W c | W p d p - E k W p u k W p v k d p + E k d + E k ↑ ↓ › › | g › | g › | e › | e › | +1 › | −1 › − a) b) | ↓ › | › | › | › | › | › | › | ↑ › = = | g › = W c W p m I = +1 m I = 1 −
00 +11 − c) W c d - E k E k E k FIG. 1: (Color online) (a) The bare state picture of our model.(b) The dressed state picture of our model equivalent to (a).(c) A possible realization in Li. Here the states labelled by | i i ( i = 1 , , ...,
6) are the 6 ground state hyperfine states. Mostexperiments involving Li are performed with a magnetic fieldstrength tuned near a Feshbach resonance at 834G. Undersuch a magnetic field, the magnetic quantum number for thenuclear spin m I is, to a very good approximation, a goodquantum number. The values of m I are shown in the level di-agrams. Two-photon transition can only occur between stateswith the same m I . Any pair of the lower manifold ( | i , | i ,and | i ) can be chosen to form the pairing states. In the ex-ample shown here, we choose | i = |↑i , | i = |↓i and | i = | g i .The excited state | e i (not shown) can be chosen properly asone of the electronic p state. gas systems [25, 26, 27, 28, 29], where paring can resultin a number of interesting phenomena [30]. A disad-vantage of this method is its inefficiency: The samplemust be prepared repeatedly for each RF pulse. In addi-tion, for the most commonly used fermionic atom species,i.e., Li, the state | i interacts strongly with the pairingstates due to the fact that all three states involved haspairwise Feshbach resonances at relatively close magneticfield strength. This leads to so-called final state effect[24] which greatly complicates the interpretation of theRF spectrum.In the EIT method, by contrast, one can directlymeasure the absorption or transmission spectrum of theprobe light. Applying a frequency scan faster than thelifetime of the atomic sample to the weak probe field, thewhole spectrum can be recorded continuously in a nearlynon-destructive fashion to the atomic sample. Further-more, EIT signal results from quantum interference andis extremely sensitive to the two-photon resonance condi-tion. The width of the EIT transparency window can becontrolled by the coupling laser intensity and be made narrower than E F . As we will show below, this prop-erty can be exploited to detect the onset of pairing asthe pairing interaction shifts and destroys the two-photonresonance condition. In addition, due to different selec-tion rules compared with the RF method, one can picka different final state whose interaction with the pairingstates are negligible [see Fig. 1(c)], hence avoiding thefinal state effects.The paper is organized as follows. In Sec. II, we de-scribed the model under study and define the key quan-tity of the proposal — the absorption coefficient of theprobe light. In Sec. III, we present the expression of theprobe absorption coefficient and construct a quasiparti-cle picture that will become convenient to explain thefeatures of the spectrum. The results are presented inSec. IV, where spectral features at different temperatuesare explained. We also show that how EIT spectrum canbe used to detect the onset of pairing. A breif summaryis presented in Sec. V. Finally, we provide an appendixin Sec. VI where the derivation of the EIT spectrum isprovided. In particular, we include in this derivation thepairing fluctuations in the framework of the pseudogaptheory [19]. II. MODEL
Let us now describe our model in more detail, begin-ning with the definition of ω i and Ω i as the temporal andRabi frequencies of the probe ( i = p ) and coupling ( i = c )laser field of plane waves copropagating with an almostidentical wavevector k L (along z direction). The systemto be considered is a homogeneous one with a total vol-ume V , and can thus be described by operators ˆ a k ,i (ˆ a † k ,i )for annihilating (creating) a fermionic atom in state | i i with momentum ~ k , and kinetic energy ǫ k = ~ k / m ,where m is the atomic mass. Here, ˆ a k ,i are defined inan interaction picture in which ˆ a k ,e = ˆ a ′ k ,e e − iω p t , ˆ a k ,g = ˆ a ′ k ,g e i ( ω c − ω p ) t , and ˆ a k ,σ = ˆ a ′ k ,σ ( σ = ↑ , ↓ ), where ˆ a ′ k ,i are the corresponding Schr¨odinger picture operators.In a probe spectrum, the signal to be measured is theprobe laser field, which is modified by a polarization hav-ing the same mathematical form as the probe field ac-cording to [31] ∂ Ω p ∂z + 1 c ∂ Ω p ∂t = i µ ω p cd e ↑ P p ≡ α Ω p , (1)where P p is the slowly varying amplitude of that polar-ization, d ij is the matrix element of the dipole momentoperator between states | i i and | j i , and µ and c arethe magnetic permeability and the speed of light in vac-uum, respectively. The parameter α in Eq. (1) representsthe complex absorption coefficient of the probe light [31].By performing an ensemble average of atomic dipole mo-ment, we can express α as α = i α Ω p V X k , q D ˆ a † q , ↑ ˆ a k + k L ,e E e i ( k − q ) · r , (2)where α ≡ µ ω p c | d e ↑ | . The real and imaginary partof α correspond to the probe absorption and dispersionspectrum, respectively.To determine the probe spectrum, we startfrom the grand canonical Hamiltonian ˆ H = P k (cid:16) ˆ H k + ˆ H k + ˆ H k (cid:17) , whereˆ H k = ( ǫ ′ k − δ p ) ˆ a † k ,e ˆ a k ,e + ( ǫ ′ k − δ ) ˆ a † k ,g ˆ a k ,g , ˆ H k = −
12 (Ω c ˆ a † k + k L ,e ˆ a k ,g + Ω p ˆ a † k + k L ,e ˆ a k , ↑ ) − h.c , ˆ H k = X σ ǫ ′ k ˆ a † k ,σ ˆ a k ,σ − (∆ˆ a † k , ↑ ˆ a †− k , ↓ + h.c ) , describe the bare atomic energies of states | e i and | g i ,the dipole interaction between atoms and laser fields,and the mean-field Hamiltonian for the spin up anddown subsystem, respectively. Here, ǫ ′ k = ǫ k − µ with µ being the chemical potential, δ p = ~ ( ω p − ω e ↑ ) and δ c = ~ ( ω c − ω eg ) are the single-photon detunings, and δ = δ p − δ c is the two-photon detuning with ω ij be-ing the atomic transition frequency from level | i i to | j i .In arriving at ˆ H k , in order for the main physics to bemost easily identified, we have expressed the collisionsbetween atoms of opposite spins in terms of the gap pa-rameter ∆ = − U V − P k h ˆ a − k , ↓ ˆ a k , ↑ i under the assump-tion of BCS paring, where U characterizes the interactionbetween | ↑i and | ↓i which, in the calculation, will bereplaced in favor of the s -wave scattering length a s viathe regularization procedure: m π ~ a s = 1 U + 1 V X k ǫ k . A more complex model including the pseudo-gap physics [19] will be presented later in the paper. Finally, we notethat the effect of the collisions involving the final state | g i in the RF spectrum has been a topic of much recentdiscussion [21, 22, 23]. In our model, the spectra arenot limited to the RF regime, and this may provide uswith more freedom to choose | g i (and | e i ) that minimizesthe final state effect. In what follows, for the sake ofsimplicity, we ignore the collisions involving states | g i (and | e i ). In practice, the effects of final state interactioncan be minimized by choosing the proper atomic species[32] or hyperfine spin states [33]. In the example shown inFig. 1(c), it is indeed expected that | g i does not interactstrongly with either of the pairing state. III. QUASIPARTICLE PICTURE
The part of the Hamiltonian describing the pairing ofthe fermions can be diagonalized using the standard Bo-goliubov transformation:ˆ a k , ↑ = u k ˆ α k , ↑ + v k ˆ α †− k , ↓ , ˆ a †− k , ↓ = − v k ˆ α k , ↑ + u k ˆ α †− k , ↓ , where u k = p ( E k + ǫ ′ k ) / E k , v k = p ( E k − ǫ ′ k ) / E k ,and E k = p ǫ ′ k + ∆ is the quasiparticle energy disper-sion. Now we introduce two sets of quasiparticle states | ± k i , representing the electron and hole branches, re-spectively. The corresponding field operators are definedas ˆ α k , +1 ≡ ˆ α k , ↑ , ˆ α k , − ≡ ˆ α †− k , ↓ , in terms of which, the grand canonical Hamiltonian canbe written asˆ H = X k h ( ǫ ′ k − δ p ) ˆ a † k ,e ˆ a k ,e + ( ǫ ′ k − δ ) ˆ a † k ,g ˆ a k ,g + E k ˆ α † k , +1 ˆ α k , +1 − E k ˆ α † k , − ˆ α k , − − (cid:18) Ω c a † k + k L ,e ˆ a k ,g + h.c. (cid:19) − (cid:18) Ω p u k a † k + k L ,e ˆ α k , +1 + h.c (cid:19) − (cid:18) Ω p v k a † k + k L ,e ˆ α k , − + h.c (cid:19)(cid:21) . (3)A physical picture emerges from this Hamiltonian verynicely. The state | +1 k i ( |− k i ) has an energy dispersion+ E k ( − E k ) and is coupled to the excited state | e i byan effective Rabi frequency Ω p u k (Ω p v k ), which is nowa function of k . In the quasiparticle picture, our modelbecomes a double Λ system as illustrated in Fig. 1(b).Let +Λ ( − Λ) denote the Λ configuration involving | +1 k i ( |− k i ). The +Λ ( − Λ) system is characterized with asingle-photon detuning of δ p + E k ( δ p − E k ) and a two-photon detuning of δ + E k ( δ − E k ). In thermal equi-librium at temperature T (in the absence of the probe field), we have h ˆ α † k , +1 ˆ α k ′ , +1 i = δ k , k ′ − h ˆ α † k , − ˆ α k ′ , − i = δ k , k ′ f ( E k ) , (4)where f ( ω ) = [exp ( ω/k B T ) + 1] − , (5)is the standard Fermi-Dirac distribution for quasiparti-cles. Thus, as temperature increases from zero, the prob-ability of finding a quasiparticle in state | +1 k i increaseswhile that in state |− k i decreases but the total proba-bility within each momentum group remains unchanged.Similarly, in the quasiparticle picture, the probe spec-trum receives contributions from two transitions α = i α Ω p V X k , q e i ( k − q ) · r × [ u q ρ e, +1 ( k + k L , q ) + v q ρ e, − ( k + k L , q )] , (6)where ρ i, ± ( k , k ′ ) = D ˆ α † k ′ , ± ˆ a k ,i E are the off-diagonaldensity matrix elements in momentum space.The equations for the density matrix elements can beobtained by averaging, with respect to the thermal equi-librium defined in Eq. (4), the corresponding Heisen-berg’s equations of motion based upon Hamiltonian (3).In the regime where the linear response theory holds, theterms at the second order and higher can be ignored, andthe density matrix elements correct up to the first orderin Ω p are then found to be governed by the followingcoupled equations: i ~ ddt (cid:20) ρ e,η ( k + k L , q ) ρ g,η ( k , q ) (cid:21) = M η (cid:20) ρ e,η ( k + k L , q ) ρ g,η ( k , q ) (cid:21) − Ω p η ( k ) δ k , q , ( η = ± , (7)whereΛ +1 ( k ) = (cid:18) u k f ( E k )0 (cid:19) , Λ − ( k ) = (cid:18) v k f ( − E k )0 (cid:19) , and M η = (cid:20) ǫ ′ k − δ p − ηE k − iγ − Ω c − Ω ∗ c ǫ ′ k − δ − ηE k (cid:21) . Here we have introduced phenomenologically the pa-rameter γ which represents the decay rate of the ex-cited state | e i . Inserting the steady-state solution fromEq. (7) into Eq. (6), we immediately arrive at α ( δ c , δ ) = α +1 ( δ c , δ ) + α − ( δ c , δ ), where α ± ( δ c , δ ) = i α V X k w k ( δ c , δ, ± E k ) f ( ± E k ) (cid:26) u k v k , (8)with w k ( δ c , δ, ω ) = ǫ ′ k − δ − ωλ k ( δ c , δ, ω ) ( ǫ ′ k − δ − ω ) − (cid:12)(cid:12) Ω c (cid:12)(cid:12) , (9)and λ k ( δ c , δ, ω ) = ǫ ′ k + k L − δ c − δ − iγ − ω . IV. RESULTS
Examples of the probe absorption coefficient, Re( α ),are presented in Fig. 2(a) and (b). For the resultsshown in this paper, we choose 1 / ( k F a s ) = − . E F , k F , and T F = E F /k B be Fermi energy, FIG. 2: (Color online) (a) ∆ (black solid curve) and the probeabsorption coefficient real( α ) at δ = 0 (red dotted curve) asfunctions of T , obtained from the mean-field BCS theory. (b)Real( α ) as a function of δ (absorption spectrum) at different T . (c) ∆, ∆ sc and ∆ pg as functions of T obtained from thepseudogap approach. (∆ sc = 0 and ∆ = ∆ pg when T c < T
In summary, we propose to use optical spectroscopy inan EIT setting to probe the fermionic pairing in Fermigases. We have demonstrated that the EIT technique of-fers an extremely efficient probing method and is capableof detecting the onset of pair formation (i.e., determin-ing T ∗ ) due to its spectral sensitivity. With a sufficientlyweak probe field, the whole spectrum may be obtainedwith a nearly non-destructive fashion via a relatively fastscan of probe frequency, without the need of repeatedlyre-preparing the sample. We note that in this work, wehave focused on probing the atomic system using pho-tons. In the future, it will also be interesting to studyhow we can use atomic Fermi gas to manipulate the light.Superfluid fermions can serve a new type of nonlinear me-dia for photons. Finally, we want to remark that, in thiswork, as a proof-of-principle, we have only considered ahomogeneous system. As usual, the trap inhomogeneitycan be easily accounted for within local density approxi-mation. Nevertheless, we note that the capability of de-tecting the onset of pairing remains the same even in thepresence of the trap. Furthermore, as optical fields areused in this scheme, one may focus the probe laser beamsuch that only a small localized portion of the atomiccloud is probed, hence there is no need to average overthe whole cloud. Acknowledgments
We thank Randy Hulet for insightful discussions. Thiswork is supported by the US National Science Foun-dation (H.P., H.Y.L.), the US Army Research Office(H.Y.L.), and the Robert A. Welch Foundation (GrantNo. C-1669), and the W. M. Keck Foundation (L.J.,H.P.), and by the National Natural Science Foundationof China under Grant No. 10588402, the National BasicResearch Program of China (973 Program) under GrantNo. 2006CB921104, the Program of Shanghai SubjectChief Scientist under Grant No. 08XD14017, the Pro-gram for Changjiang Scholars and Innovative ResearchTeam in University, Shanghai Leading Academic Disci-pline Project under Grant No. B480 (W.Z.).
VI. APPENDIX: EIT SPECTRA INCLUDINGPSEUDOGAP
In this appendix, we generalize the result of Eq. (2)for α valid under the mean-field BCS pairing to a morerealistic situation where pair fluctuations are included inthe form of pseudogap. We show two different ways toaccomplish this generalization. The first is an approachused more often by people working in the field of quantumoptics. The second uses the linear response theory [34]more familiar in the field of condensed matter physics. A. A Brief Account of Pseudogap Theory
First, let us highlight the results of pseudogap theory[19] that are relevant to our EIT spectrum calculation.When pairing fluctuations at finite temperature are in-cluded in the framework of the pseudogap model [19], theBCS gap equation and number equation are still valid.However, the gap ∆ is now regarded as the total gapdivided into a BCS gap ∆ sc for condensed (BCS) pairsbelow T c and a pseudogap ∆ pg for preformed (finite mo-mentum) pairs: ∆ = ∆ sc + ∆ pg . The onset of the total gap ∆ occurs at temperature T ∗ ,which is greater than T c . The system with preformedpairs is described by the Green’s function G − ( k , iw n ) = G − ( k , iw n ) − Σ( k , iw n ) , (10)where the non-interacting Green’s function G − ( k , iw n ) = ( iω n − ǫ ′ k ) − , (11)and the self energyΣ( k , iw n ) = Σ sc ( k , iw n ) + Σ pg ( k , iw n )= ∆ sc iw n + ǫ ′ k + ∆ pg iw n + ǫ ′ k + iγ p , (12)with w n being the fermi Matsubara frequency and γ − p the finite lifetime of pseudogap pairs. The spectral func-tion A ( k , ω ) can be obtained from the Green’s functionvia the relation A ( k , ω ) = − G (cid:0) k , ω + i + (cid:1) , which, with the help of Eqs. (10), (11), and (12), is foundto be given by A ( k , ω ) = 2( ω + ǫ ′ k ) γ p ∆ pg [ ω − E k ] ( ω + ǫ ′ k ) + γ p [ ω − E sc k ] , (13)where E sck = q ǫ ′ k + ∆ sc . In the limit of γ p → E sck → E k , we recover from Eq. (13) the spectral functionunder the BCS paring A ( k , w ) = 2 π [ u k δ ( ω − E k ) + v k δ ( ω + E k )] . (14) B. Quantum Optics Approach
In order to develop a formalism which directly incor-porates the spectral function, we rewrite Eq. (8) in terms of the equal time correlation function h q , k ( t ) = D ˆ a † q , ↑ ( t ) ˆ a k + k L ,e ( t ) E as α = i α Ω p V lim t −→∞ X k , q h q , k ( t ) e i ( k − q ) · r , (15)where the limit is introduced to indicate explicitly thatwe are interested in the steady state spectrum. Here,ˆ a † q , ↑ ( t ) and ˆ a k + k L ,e ( t ) obey the Heisenberg equations ofmotion i ~ ddt (cid:18) ˆ a k + k L ,e ˆ a k ,g (cid:19) = ˆ M (cid:18) ˆ a k + k L ,e ˆ a k ,g (cid:19) − Ω p a k , ↑ (cid:18) (cid:19) , (16)with ˆ M = (cid:20) ǫ ′ k + k L − ( δ p + iγ ) − Ω c − Ω ∗ c ǫ ′ k − δ (cid:21) . (17)Note that due to the dissipative nature of our model,strictly speaking, Eqs. (16) should be those of quantumLangevin equations containing the noise operators of thereservoir that gives rise to the decay rate γ . Here, inanticipation that Eqs. (16) will produce the right aver-ages of our interest, we have ignored the noise operators.We solve Eqs. (16) for ˆ a k + k L ,e ( t ) in the limit of t → ∞ when the terms involving the initial operators have alldied away, and then combine it with ˆ a † q , ↑ ( t ) to form h q , k ( t ) = Ω p Z t h e − i ˆ M ( t − t ′ ) i G < ( k , t ′ , t ) δ k , q dt ′ , (18)where [ ... ] denotes the element at the first row and thefirst column of the matrix inside the square bracket, and G < ( k , t ′ , t ) = i D ˆ a † k , ↑ ( t ) ˆ a k , ↑ ( t ′ ) E is one of the Green’sfunctions in real time. By substituting G < ( k , t ′ , t ) inEq. (18) with a Fourier transformation of its counterpartin real frequency, G < ( k , ω ), we are able to carry out thetime integration in Eq. (18) explicitly, leading to h q , k ( t → ∞ ) = δ k , q Ω p Z + ∞−∞ dω π (cid:20) A ( k , ω ) f ( ω )ˆ M − ω (cid:21) , where the use of a well-known relation: G < ( k , ω ) = if ( ω ) A ( k , ω ) [34] has been made. Finally, replacing[1 / ( ˆ M − ω )] with w k ( δ c , δ, ω ), obtained with the helpof Eq. (17), we arrive at α = i α V X k Z + ∞−∞ dω π A ( k , ω ) f ( ω ) w k ( δ c , δ, ω ) (19)where w k ( δ c , δ, ω ) is defined in Eq. (9) of the main text.One can easily check that Eq. (19) reduces to Eq. (8) inthe limit of mean-field BCS pairing when Eq. (14) is usedas the spectral function. C. Condensed Matter Approach