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Vortex microscopic structure in BCS to BEC Fermi superfluids
Physica C 460–462 (2007) 275–276 www.elsevier.com/locate/physc
Vortex microscopic structure in BCS to BEC Fermi superfluids Masahiko Machida
a,c,* ,
Tomio Koyama
b,c
, Yoji Ohashi
d
a IMR, Tohoku University, Katahira, Sendai 980-8577, Japan CCSE, Japan Atomic Energy Research Institute, Tokyo 110-0015, Japan c CREST, JST, Kawaguchi 332-0012, Japan Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-0061, Japan b
d
Available online 22 April 2007
Abstract In order to examine how the vortex core changes its structure when approaching BEC from BCS regime in superfluid fermion-atom gasses we make a systematic numerical study for the core structure of a singly quantized vortex, solving numerically the generalized Bogoliubov-de Gennes equation derived from the fermion–boson model. Numerical calculations reveal that the vortex core can be well characterized by two length scales based on the spatial variation of the fermionic gap functions and those scales show distinctive differences between the BCS and BEC regimes. Ó 2007 Elsevier B.V. All rights reserved. PACS: 03.75.Ss; 03.75.Lm; 74.25.Op Keywords: Vortex core; Atomic Fermi gas; Bogoliubov-de Gennes equation
1. Introduction Very recently, the MIT group has succeeded in observing not only a singly quantized vortex but also a triangular vortex lattice in a two-component atomic Fermi gas [1]. The singly quantized vortex in a Fermi superfluid is characterized by several inherent spatial lengths. The most wellknown length among them is the coherence length, n0 DvF0 , where vF and D0 are the Fermi velocity and the
Meissner superfluid gap, respectively. In the singly quantized superconducting vortex we have another well-known characteristic length n1 n11 ¼ DðrÞ 1 lim which is related to the slope of the gap funcr!0 D0 r tion at the center of a vortex (see Fig. 1) [2]. In the low-temperature regime, n1 shows a linear temperature dependence (the Kramer–Pesch (KP) effect [2]), i.e., n1 / T, which is *
Corresponding author. Address: IMR, Tohoku University, Katahira, Sendai 980-8577, Japan. Tel.: +81 3 5246 2517; fax: +81 3 5246 2537. E-mail address: [email protected] (M. Machida). 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.431
sharply contrasted with the behavior of the coherence length n0 being nearly independent of temperature. In this paper, we study how the gradient g1 ¼ limr!0 DðrÞ [2], which r dominates the behavior of the length scale n1, changes at the BCS–BEC crossover and clarify that g1 is distinctly dependent on microscopic parameters such as the Fermi velocity. In addition, we study another characteristic gradient g2 in the fermionic gap function, which is separately defined outside the region in which g1 shows a linear slope (see Fig. 1) function. Our calculations show that the gradients g1 and g2 are well-defined and reveal their characteristic behaviors depending on the interaction strength. 2. Theoretical formalism From the Hamiltonian of the fermion–boson (FB) model [3], we have the generalized Bogoliubov equation as ! unr unr Hr DF þ /B ¼ En ; ð1Þ vnr vnr DF þ /B H r 1 2 g2 r þ 2m 2l /B ðrÞ ¼ DF ðrÞ; ð2Þ 4m U
276
M. Machida et al. / Physica C 460–462 (2007) 275–276
ΔF0
ΔF
ξ 2 (g2 ) ξ1(g1)
φB
1
0.8
2ν
ν=90 ν=80 ν=70 ν=60 ν=50 ν=40 ν=30 ν=20
0.8 0.6
0
0.6
ΔF
ΔF
ξ0
Andreev bound-state
0.4
0.4
0.2
Fig. 1. Schematic view of the vortex core structure in the fermionic superfluid gap function, DF and the molecular condensate function, /B. n0 is the coherence length. gi (i = 1 and 2) is characteristic gradient for DF which is numerically investigated in this paper and ni (i = 1 and 2) is the related length scale.
0.2
0 0
Let us present numerical solutions for the single-vortex. Fig. 2a and b represent the profiles of the fermionic superfluid gap functions DF along the radial direction, respectively, in the BCS (1/kFa < 0) and the BEC (1/kFa > 0) sides. The profiles show clear difference in these two sides. In the BCS side the gradient g1 is almost independent of m. This tendency is more clearly seen in the enlarged view as shown in the inset of Fig. 2a. On the other hand, in the BEC side, g1 strongly depends on m as seen in Fig. 2b. Here, we note g1 1/kF in the BCS region and g1 m l in the BEC region. The former relation was suggested in the superconducting vortex [5] in the quantum limit. From these relations one understands that the Fermi edge of the fermionic atomic gas is nearly invariant in the BCS regime, whereas it is strongly dependent on m in the BEC regime. In other words, from the observation of g1 one can detect a signal of the crossover from fermionic to bosonic superfluid states. It is also noted that the linear slope of g1 is restricted in the region of 0 < r < 1/kF (see Fig. 2a). This is because g1 is determined only by the Andreev localized-bound states as shown in Ref. [5]. Let us next consider the gradient g2 (see Fig. 1). As seen in Fig. 2a, the gap function DF in r > 1/kF is dependent on m
0
1
2
k Fr
10
20
30
40
k Fr 1
1 where H r 2m r2 l, DF and /B(r) are the mean-field vacuum expectation values for the fermionic gap and the molecular condensate functions, respectively, and l and 2m are the chemical potential and the Feshbach resonance threshold-energy, respectively. These equations are solved self-consistently together with the fermionic gap equation, P DF ðrÞ ¼ U n un ðrÞvn ðrÞ and the chemical potential given by the constraint for the total number R Pof particles, N = NF + 2NB = const., where N F ¼ 2 dr i jvi ðrÞj2 and R 2 B N B ¼ dr j/ gðrÞj [4]. 2
3. Numerical calculation results
0
0.8 1
0.6
ΔF
0.8 0.6
Δ
F
0.4
0.4
ν=20 ν=10 ν=5 ν=1
0.2
0.2
0
0
0
0
10
1
2 3 4 5 k Fr
20
30
40
kFr
Fig. 2. (a) The radial profile of the fermionic superfluid gap DF (b) from m = 20 to 90 (1/kFa < 0) and (b) from m = 1 to m = 20 (1/kFa > 0). The insets in (a) and (b) are the focus views on the central region of the vortex.
even in the BCS side. The value of g2 increases with decreasing m, i.e., increasing the pairing strength in the BCS side (1/kFa < 0). This behavior is sharply contrasted to g1. On the other hand, g2 decreases with decreasing m, i.e., increasing the pairing strength in the BEC side (see Fig. 2b) contrary to the BCS side. Thus, one understands that g2 is correlated with the fermionic superfluid gap DF. Here, we notice that the coherence length n0 has the approximate relation as n0 n1 + n2. In summary, one concludes from the numerical calculations that g1 and g2 behave as g1 1/kF (BCS) and m l (BEC) and g2 DF (BCS and BEC). References [1] M.W. Zwierlein et al., Nature 435 (2005) 1047. [2] For a recent review, see e.g., J.E. Sonier, J. Phys. Condens. Mat. 16 (2004) S4499. [3] Y. Ohashi, A. Griffin, Phys. Rev. Lett. 89 (2002) 130402. [4] M. Machida, T. Koyama, Phys. Rev. Lett. 94 (2005) 140401. [5] N. Hayashi et al., Phys. Rev. Lett. 80 (1998) 2921.
Vortex microscopic structure in BCS to BEC Fermi superfluids Masahiko Machida
a,c,* ,
Tomio Koyama
b,c
, Yoji Ohashi
d
a IMR, Tohoku University, Katahira, Sendai 980-8577, Japan CCSE, Japan Atomic Energy Research Institute, Tokyo 110-0015, Japan c CREST, JST, Kawaguchi 332-0012, Japan Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-0061, Japan b
d
Available online 22 April 2007
Abstract In order to examine how the vortex core changes its structure when approaching BEC from BCS regime in superfluid fermion-atom gasses we make a systematic numerical study for the core structure of a singly quantized vortex, solving numerically the generalized Bogoliubov-de Gennes equation derived from the fermion–boson model. Numerical calculations reveal that the vortex core can be well characterized by two length scales based on the spatial variation of the fermionic gap functions and those scales show distinctive differences between the BCS and BEC regimes. Ó 2007 Elsevier B.V. All rights reserved. PACS: 03.75.Ss; 03.75.Lm; 74.25.Op Keywords: Vortex core; Atomic Fermi gas; Bogoliubov-de Gennes equation
1. Introduction Very recently, the MIT group has succeeded in observing not only a singly quantized vortex but also a triangular vortex lattice in a two-component atomic Fermi gas [1]. The singly quantized vortex in a Fermi superfluid is characterized by several inherent spatial lengths. The most wellknown length among them is the coherence length, n0 DvF0 , where vF and D0 are the Fermi velocity and the
Meissner superfluid gap, respectively. In the singly quantized superconducting vortex we have another well-known characteristic length n1 n11 ¼ DðrÞ 1 lim which is related to the slope of the gap funcr!0 D0 r tion at the center of a vortex (see Fig. 1) [2]. In the low-temperature regime, n1 shows a linear temperature dependence (the Kramer–Pesch (KP) effect [2]), i.e., n1 / T, which is *
Corresponding author. Address: IMR, Tohoku University, Katahira, Sendai 980-8577, Japan. Tel.: +81 3 5246 2517; fax: +81 3 5246 2537. E-mail address: [email protected] (M. Machida). 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.431
sharply contrasted with the behavior of the coherence length n0 being nearly independent of temperature. In this paper, we study how the gradient g1 ¼ limr!0 DðrÞ [2], which r dominates the behavior of the length scale n1, changes at the BCS–BEC crossover and clarify that g1 is distinctly dependent on microscopic parameters such as the Fermi velocity. In addition, we study another characteristic gradient g2 in the fermionic gap function, which is separately defined outside the region in which g1 shows a linear slope (see Fig. 1) function. Our calculations show that the gradients g1 and g2 are well-defined and reveal their characteristic behaviors depending on the interaction strength. 2. Theoretical formalism From the Hamiltonian of the fermion–boson (FB) model [3], we have the generalized Bogoliubov equation as ! unr unr Hr DF þ /B ¼ En ; ð1Þ vnr vnr DF þ /B H r 1 2 g2 r þ 2m 2l /B ðrÞ ¼ DF ðrÞ; ð2Þ 4m U
276
M. Machida et al. / Physica C 460–462 (2007) 275–276
ΔF0
ΔF
ξ 2 (g2 ) ξ1(g1)
φB
1
0.8
2ν
ν=90 ν=80 ν=70 ν=60 ν=50 ν=40 ν=30 ν=20
0.8 0.6
0
0.6
ΔF
ΔF
ξ0
Andreev bound-state
0.4
0.4
0.2
Fig. 1. Schematic view of the vortex core structure in the fermionic superfluid gap function, DF and the molecular condensate function, /B. n0 is the coherence length. gi (i = 1 and 2) is characteristic gradient for DF which is numerically investigated in this paper and ni (i = 1 and 2) is the related length scale.
0.2
0 0
Let us present numerical solutions for the single-vortex. Fig. 2a and b represent the profiles of the fermionic superfluid gap functions DF along the radial direction, respectively, in the BCS (1/kFa < 0) and the BEC (1/kFa > 0) sides. The profiles show clear difference in these two sides. In the BCS side the gradient g1 is almost independent of m. This tendency is more clearly seen in the enlarged view as shown in the inset of Fig. 2a. On the other hand, in the BEC side, g1 strongly depends on m as seen in Fig. 2b. Here, we note g1 1/kF in the BCS region and g1 m l in the BEC region. The former relation was suggested in the superconducting vortex [5] in the quantum limit. From these relations one understands that the Fermi edge of the fermionic atomic gas is nearly invariant in the BCS regime, whereas it is strongly dependent on m in the BEC regime. In other words, from the observation of g1 one can detect a signal of the crossover from fermionic to bosonic superfluid states. It is also noted that the linear slope of g1 is restricted in the region of 0 < r < 1/kF (see Fig. 2a). This is because g1 is determined only by the Andreev localized-bound states as shown in Ref. [5]. Let us next consider the gradient g2 (see Fig. 1). As seen in Fig. 2a, the gap function DF in r > 1/kF is dependent on m
0
1
2
k Fr
10
20
30
40
k Fr 1
1 where H r 2m r2 l, DF and /B(r) are the mean-field vacuum expectation values for the fermionic gap and the molecular condensate functions, respectively, and l and 2m are the chemical potential and the Feshbach resonance threshold-energy, respectively. These equations are solved self-consistently together with the fermionic gap equation, P DF ðrÞ ¼ U n un ðrÞvn ðrÞ and the chemical potential given by the constraint for the total number R Pof particles, N = NF + 2NB = const., where N F ¼ 2 dr i jvi ðrÞj2 and R 2 B N B ¼ dr j/ gðrÞj [4]. 2
3. Numerical calculation results
0
0.8 1
0.6
ΔF
0.8 0.6
Δ
F
0.4
0.4
ν=20 ν=10 ν=5 ν=1
0.2
0.2
0
0
0
0
10
1
2 3 4 5 k Fr
20
30
40
kFr
Fig. 2. (a) The radial profile of the fermionic superfluid gap DF (b) from m = 20 to 90 (1/kFa < 0) and (b) from m = 1 to m = 20 (1/kFa > 0). The insets in (a) and (b) are the focus views on the central region of the vortex.
even in the BCS side. The value of g2 increases with decreasing m, i.e., increasing the pairing strength in the BCS side (1/kFa < 0). This behavior is sharply contrasted to g1. On the other hand, g2 decreases with decreasing m, i.e., increasing the pairing strength in the BEC side (see Fig. 2b) contrary to the BCS side. Thus, one understands that g2 is correlated with the fermionic superfluid gap DF. Here, we notice that the coherence length n0 has the approximate relation as n0 n1 + n2. In summary, one concludes from the numerical calculations that g1 and g2 behave as g1 1/kF (BCS) and m l (BEC) and g2 DF (BCS and BEC). References [1] M.W. Zwierlein et al., Nature 435 (2005) 1047. [2] For a recent review, see e.g., J.E. Sonier, J. Phys. Condens. Mat. 16 (2004) S4499. [3] Y. Ohashi, A. Griffin, Phys. Rev. Lett. 89 (2002) 130402. [4] M. Machida, T. Koyama, Phys. Rev. Lett. 94 (2005) 140401. [5] N. Hayashi et al., Phys. Rev. Lett. 80 (1998) 2921.