Introduction

Due to the experimental realization of strongly interacting atomic Fermi gases, there has been increased theoretical interest in properties of asymmetric two-component superfluid Fermi systems with unusual pairing configurations [1,2,3,4,5,6,7,8,9,10,11]. Of current interest are the properties of spin-polarized condensates having an unequal number of spin-up and spin-down fermions. One of the condensation possibilities is the ``breached-pair" (BP) superfluid state, in which the normal phase coexists with a full pairing of the minority fermions at high spin imbalances. Recently, a gap in a single-particle excitation spectrum of a highly spin-imbalanced sample has been observed experimentally in a $^6$Li atomic condensate [12].

Atomic nuclei can also exhibit interesting pairing properties, including BP superfluidity, although the fraction of the polarization is typically quite small. Examples include:

• Pairing isomers [13,14,15] having an appreciably smaller value of the pairing gap than the ground state. In the roots of pairing isomerism is very different single-particle level density in the vicinity of the Fermi surface in coexisting configurations, usually associated with different nuclear shapes. Such states have been found in, e.g., $A$$\approx$80, 100, and 190 mass regions.
• Gapless superconductivity at high spins of atomic nuclei [16,17]. Here, quasiparticle energies in the rotating frame become negative at certain rotational frequencies; i.e., there is no gap in the quasiparticle spectrum. However, the pairing energy, or the expectation value of the pairing condensate, still remains large.
• Angular-momentum aligned nuclear configurations, such as high-spin isomers, in which pairing correlations may be reduced due to blocking (see Refs. [18,19,20] and references quoted therein).

In this paper we will briefly rederive some of the generic results of the Hartree-Fock-Bogoliubov theory (also called the Bogoliubov de-Gennes theory) for polarized quasiparticle states, using constraining fields to reach the states, as was done in Refs. [7,10] to study the polarized atomic condensates. We shall call this the ``two-Fermi level approach" (2FLA). In particular, odd-mass systems require special attention in the HFB theory. In the usual blocking approach, the selected quasiparticle state becomes occupied and this requires a modification of the HFB density matrix. In 2FLA, the lowest quasiparticle orbit becomes occupied by changing the particle-number parity of the vacuum through the external field without modifying any of the vectors explicitly. We show that for spin systems (e.g., Fermi gases) the 2FLA is equivalent to the standard rotational cranking approximation while this is not the case for atomic nuclei, in which polarization is due to the angular momentum alignment. In both cases, however, 2FLA can be viewed as a ``vacuum selector" by means of the non-collective cranking.

The paper is organized as follows. Section 2 briefly discusses the concept of Bogoliubov quasiparticles. Particular attention is paid to the choice of the HFB vacuum, the way the particle-number parity is encoded in the Bogoliubov matrix transformation, and self-consistent signature symmetry of HFB and its relation to time reversal. The HFB extension to the case of two-component systems (2FLA) is outlined in Sec. 3, and its relation to non-collective cranking and the blocking procedure is discussed. Section 4 shows numerical examples, both from atomic and nuclear physics, based on the HFB approach. Finally, Sec. 5 contains the main conclusions of this work.