Nonlinear quantum interferometry with Bose condensed atoms
In quantum interferometry, it is vital to control and utilize nonlinear interactions for achieving high-precision measurements. Attribute to their long coherent time and high controllability, ultracold atoms including Bose condensed atoms have been widely used for implementing quantum interferometry. Here, we review the recent progresses in theoretical studies of quantum interferometry with Bose condensed atoms. In particular, we focus on the nonlinear phenomena induced by the atom-atom interaction and how to control and utilize these nonlinear phenomena. Under the mean-field description, due to the atom-atom interaction, matter-wave solitons appear in the interference patterns, and macroscopic quantum self-trapping exists in the Bose-Josephson junctions. Under the many-body description, the atom-atom interaction can generate non-classical entanglement, which may be utilized to achieve high-precision measurements beyond the standard quantum limit.
Keywords: nonlinear quantum interferometry, Bose-Einstein condensate, Bose-Josephson junction
pacs:03.75.Dg, 03.75.Lm, 37.25.+k
- I Introduction
II Macroscopic quantum coherence and atomic matter-wave interferometry
- II.1 Macroscopic quantum coherence and mean-field description
- II.2 Atomic matter-wave interference
- II.3 Nonlinear excitations in atomic matter-wave interference
- II.4 Bose-Josephson junction (BJJ)
- III Many-body quantum interferometry
- IV Conclusions and perspectives
It is well known that an ambitious goal of quantum physics is to control and exploit quantum coherence and entanglement. Linear superposition of multiple quantum states is a perfect description for quantum coherence. Macroscopic quantum coherence (MQC), the quantum coherence among an ensemble of particles, may lead to novel phenomena never existed in single-particle systems and new applications in quantum interferometry. The systems of Bose-Einstein condensates (BECs) (1); (2); (3); (4), such as, superfluids, superconductors and Bose-Einstein condensed atoms, are typical matters of MQC.
Under ultralow temperature close to absolute zero, as the thermal motions are almost frozen, the quantum features become significant and robust. In the last 20 years, there appear dramatic progresses in our ability to control and manipulate atoms. Now, atomic quantum gases and atomic BECs can be easily prepared by the well-developed techniques of trapping and cooling (5); (6), and their degrees of freedom can also be accurately manipulated by applying laser and magnetic fields. In addition to new perspectives for the study of many-body quantum physics (7); (8), these progresses offer new possibilities of atomic interferometers of unprecedented sensitivity (9). In particular, nonlinear dynamics, quantum entanglement and quantum metrology meet together in systems of Bose condensed atoms (10); (11); (12); (13); (14); (15), in which nonlinear dynamics may generate quantum entanglement and quantum entanglement may be used to implement high-precision quantum metrology.
This article is a review on recent progresses in theoretical studies in nonlinear quantum interferometry with Bose-Einstein condensed atoms. This section is an introduction for some fundamental conceptions for quantum interferometry, atomic BECs and nonlinear dynamics in BECs. In the next section, we present a description for MQC of Bose condensed atoms and discuss how to implement quantum interferometry via atomic matter-waves, in particular, the Bose-Josephson junctions. In the third section, we show how to realize many-body quantum interferometry via an ensemble of condensed atoms. The last section summarizes this article and discusses some perspectives in this field.
i.1 Quantum interferometry
In an optical interferometer, two or more light beams are coherently combined for interference and then one can extract their relative phase from their interference patterns. Optical interferometers have been extensively used for precision measurements, surface diagnostics, astrophysics, seismology, etc. There are several configurations for optical interferometers, for an example, an optical Mach-Zehnder interferometer splits a laser beam into two beams propagating along two paths and then recombines the two beams for interference and extracting their relative phase.
Beyond the conventional interferometry via classical waves, the quantum interferometry uses the wave nature of particles to achieve higher precision limit which can not be reached by classical interferometry. For an example, in a quantum Mach-Zehnder interferometer, a quantum state is transferred into a coherent superposition of two states by an unitary operation, then the two states accumulate different phases in a free time evolution, and lastly the two states are coherently recombined for interference and extracting their relative phase. The quantum interferometry has demonstrated by various systems from individual microscopic particles (photons, neutrons, electrons, atoms, molecules, etc) to macroscopic objects of an ensemble of microscopic particles (superfluids, superconductors, atomic BECs, etc).
It has demonstrated that quantum states in particular entangled states offer unprecedented advantages for metrology (16); (17); (18); (19); (20); (21); (22); (23); (24); (25). The quantum states have been extensively used in high-precision measurements, such as, quantum states of photons have been used for high-resolution imaging, quantum states of spins have been used for high-resolution measurements of magnetic field, and quantum states of atoms have been used to build clocks of ultimate accuracy. Moreover, multi-atom systems in particular entangled multi-atom systems, such as ultracold atoms in optical lattices and quantum atomic gases, provide new opportunity for designing next-generation atomic clocks (26).
i.2 Atomic Bose-Einstein condensates and nonlinear dynamics
In an atomic BEC, the matter-waves of individual atoms overlap each other and therefore all atoms lose their distinguishability. The indistinguishability of individual atoms means the appearance of MQC. Under such a low temperature, the atom-atom interaction in a BEC is dominated by the s-wave scattering. Under the mean-field description (1); (2); (3); (4), an atomic BEC obeys a nonlinear Schrödinger equation, the Gross-Pitaevskii equation, in which the nonlinear strength is proportional to the s-wave scattering length. Due to the intrinsic nonlinearity from the atom-atom interaction, the de Broglie matter waves of Bose condensed atoms have led to the development of nonlinear and quantum optics with atoms (27), which are analogues of conventional nonlinear and quantum optics with photons.
It has demonstrated the existence of many nonlinear phenomena such as four-wave mixing, solitons and chaos in atomic BECs. In an optical four-wave mixing, a fourth light wave is produced by mixing three different light waves under the condition of energy and momentum conservation. Similarly, four-wave mixing process may take place in atomic BECs (28), in which three different matter waves of atoms mix to produce a fourth matter wave of atoms. Solitons are stable solutions which maintain their shape unchanged during their propagation. Due to the balance between dispersion and nonlinear atom-atom interaction, matter wave solitons may appear in atomic BECs. The dark (29); (30) and bright solitons (31); (32) have been observed in atomic BECs with repulsive and attractive atom-atom interactions, respectively. Due to the intrinsic nonlinearity from the mean-field interaction, macroscopic quantum chaos has been theoretically predicted in time driven systems (33); (34) and high-dimensional systems of atomic BECs (35); (36). The strength of nonlinearity (i.e. the s-wave scattering length) can be tuned by Feshbach resonances (37) and so that the nonlinear dynamics can be modulated (38); (39).
i.3 Quantum interferometry via Bose condensed atoms
Since the first realization of Bose-Einstein condensation in ultracold atomic gases, the condensed atoms have been extensively used for performing quantum interferometry (9); (40); (41); (42). There are two fundamental schemes for quantum interferometry with condensed atoms: space-domain interferometry and time-domain interferometry. In the space-domain scheme, two or more atomic BECs at different spatial positions are released for expansion and then the matter waves from different condensates overlap each other. In the time-domain scheme, by means of a double separated oscillator technique, the Ramsey oscillations of condensed atoms involving two or more internal hyperfine levels have been demonstrated.
Ii Macroscopic quantum coherence and atomic matter-wave interferometry
ii.1 Macroscopic quantum coherence and mean-field description
It is well known that wave-like properties of single particles such as photons, electrons and atoms have been demonstrated in various experiments. Fantastically, for an atomic BEC, the collection of atoms may also behave as an entire object of wave-like properties (1); (2); (3); (4). These wave-like properties among an ensemble of particles are a signature of macroscopic quantum coherence (MQC), which relates to the superposition of multiple macroscopic quantum states.
with and denoting bosonic fields of atoms. Here, is the atomic mass, stands for the external potential, and describes the interaction between two atoms at positions and .
The time evolution of the field operators is given by the Schrödinger equation,
Solving the above equation for an ensemble of particles involves heavy numerical work. For an atomic BEC dominated by condensed atoms, mean-field (MF) theory allows one to well understand most behaviors related to MQC. However, as will be discussed in Section 3, the theoretical treatment must go beyond MF and take into account quantum fluctuations if non-classical states such as squeezed states and macroscopic coherent superpositions are encountered.
The field operator can be expressed in form of
where is called as the order parameter or ¡°wave-function of the condensate” and describes the fluctuations around the condensate state. Therefore, the condensate density reads as .
Under ultra-low temperatures, if the average inter-particle spacing is sufficiently large, the atom-atom interaction is dominated by the s-wave scattering. That is, the atom-atom interaction is effectively described by a “contact interaction”,
with and being the s-wave scattering length. Physically, and correspond to repulsive and attractive interactions, respectively. Inserting the delta interaction potential into the many-body quantum Hamiltonian (1), one can find
For an atomic gas dominated by condensed atoms, the fluctuation part is negligible.
Under conditions of ultralow temperature and dilute density of atoms, that is, (a) of very small fluctuations, (b) the inter-atom interaction is a contact interaction, and (c) the average inter-particle distance is much larger than the s-wave scattering length, an atomic BEC can be described by a mean-field Hamiltonian,
The time evolution of the condensate wave-function obeys a nonlinear Schrödinger equation, the Gross-Pitaevskii equation (GPE),
Here, the term of describes the nonlinear mean-field interaction between atoms. For a stationary state, , the spatial wave function obeys a time-independent GPE,
with denoting the chemical potential.
ii.2 Atomic matter-wave interference
The first interference experiment of two atomic BECs was implemented in 1997 by Ketterle’s group at MIT (45). In this experiment, two atomic BECs released from a double-well potential freely expand and then form clear interference patterns. Generally speaking, the double-well potential can be created by superposing a potential barrier on a harmonic potential. In MIT’s experiment, the harmonic trap is a magnetic potential and the barrier is provided by a blue-detuned far-off-resonance laser beam. There are several other schemes for forming double-well potentials, such as, radio-frequency-induced adiabatic potentials created by atom chips (46); (47); (48) and superposing a weakly optical lattices on a strongly harmonic trap (49); (50).
There are two different methods for preparing two BECs in a double-well potential. The first method is loading cold atoms into a double-well potential and obtaining two BECs via evaporative cooling. The second method is preparing a BEC in a harmonic trap and then splitting it into two BECs via increasing the potential barrier.
Interference patterns will appear if atomic matter waves from different condensates may reach same spatial positions. A natural way is switching off the external potential. The experimental data shows the interference patterns differ from the ones of two point-like monochromatic sources and two point-like pulsed sources. This indicates that the two BECs released from a double-well potential can not be simply treated as two point-like monochromatic sources or two point-like pulsed sources. In general, for particles from two point-like sources, their fringe period of the interference pattern is the de Broglie wavelength associated with the relative motion of particles, , where is Planck’s constant, is the relative momentum, is the time, is the distance between two sources and is the particle mass. For the case of two BECs released from a double-well potential, the fringe period formulae for two point-like sources is only valid for the central fringe (45).
For simplicity, we consider an elongated system of condensed atoms. That is, the atoms are confined by a strong transverse confinement, . Integrating the transverse coordinates, the elongated system can be described by a one-dimensional (1D) GPE (51); (52):
where , characterizes the nonlinear MF interaction, and is an external potential.
Usually, the condensate wave-function is normalized to the total number of condensed atoms . To compare with the conventional Schrödinger equation of no MF interaction, we normalize for our 1D model to one and its nonlinearity strength is determined by , the s-wave scattering length and the transverse trapping frequency (52). For convenience, we introduce the dimensionless model equation by choosing the natural units of . The external potential is formed by a superposition of a time-independent harmonic potential and a time-dependent Gaussian barrier, see Fig. 1. That is,
where is the trapping frequency, is the barrier width, and the barrier height is a function of time. If , is a double-well potential with two minima at . In the splitting process, the barrier height is adiabatically increased and therefore the single condensate in a harmonic trap splits into two condensates in a deep double-well potential. To observe the interference pattern, one has to switch off the potential barrier at least and let the two condensates overlap each other.
In Fig. 1, we show our numerical simulation for the splitting process. Initially, the condensate stays in the ground state for the harmonic trap. Then the system is transferred into the ground state for a deep double-well potential by slowly increasing the barrier height. If the barrier is sufficiently high, the condensates in two wells almost have no overlap and so that they can be looked as two independent condensates. In our calculation, we choose the interaction strength , the barrier width , the trap frequency , and the barrier height is linearly increased from zero to 25.
The external potential is suddenly switched off when two condensates have been prepared in the double-potential. Thus the two condensates freely expand and interference fringes gradually appear when atoms from different condensates reach the same position. In Fig. 2, we show our numerical simulation for how the interference fringes appear. In our calculation, we choose the final state of the splitting process shown in Fig. 1 as the initial state before switching off the external potential.
ii.3 Nonlinear excitations in atomic matter-wave interference
The MF theory for atomic BECs predicts that solitons may exist in 1D BEC systems. If the s-wave scattering length is negative, the atom-atom interaction is attractive and acts as the role of self-focusing in nonlinear optics, solitons appear as shape-maintaining wavepackets in propagation. If the s-wave scattering length is positive, the atom-atom interaction is repulsive and acts as the role of self-defocusing in nonlinear optics, solitons appear as shape-maintaining notches in propagation.
In matter-wave interference experiments of atomic BECs, due to the intrinsic atom-atom interaction, the interference patterns differ from the ones of linear waves and nonlinear excitations may gradually appear. Theoretically, it has demonstrated that dark solitons can be generated by collision (53); (54) and quantum tunneling (55) in double-well BEC interferometers. Additionally, nonlinear effects in the BEC interference are studied by using exact solutions of the one-dimensional GPE (56). Experimentally, for atomic BECs with repulsive MF interaction, dark solitons are gradually generated by merging and splitting BECs (57); (58); (59); (60).
The soliton generation depends on the initial phase difference and how fast the recombination occurs. In a sufficiently slow recombination, if the initial state is the ground state for the double-well potential of no phase difference, the condensate is adiabatically transferred into its ground state for the final harmonic trap. However, if the initial state is the first-excited state for the double-well potential of phase difference, the condensate will be adiabatically transferred into its first-excited state for the final harmonic trap. The notch of this excited state can be looked as a dark soliton. Usually, more number of solitons will be generated in a faster recombination (55).
Based upon the 1D GPE (7), we explore how dark solitons appear via numerical integration, see Fig. 3. The initial states are set as the condensate ground states for the external potential with parameters , and . Following the experiments on recombination from a double-well potential into a harmonic potential (57); (58), we only switch off the potential barrier and keep the trap frequency unchanged. There are several schemes for switching off the potential barrier. In our simulation, similar to the matter-wave interference in free space, we suddenly switch off the potential barrier, that is, ). Due to the harmonic forces, two condensates collide again and again. The interference patterns strongly depend on the interaction strength. For a linear system () of no inter-particle interaction, the interference patterns periodically oscillate with the period for the harmonic trap. For the nonlinear systems of repulsive interactions (), the periodicity for the interference patterns is destroyed by the nonlinear inter-particle interactions and dark solitons are gradually generated. In general, solitons gradually appear from the overlapped regions and more solitons are generated in more strongly nonlinear systems.
ii.4 Bose-Josephson junction (BJJ)
In 1962, Brian Josephson predicted the existence of supercurrent through an insulator barrier between two superconductors (61); (62). This tunnelling effect of electrons is named as the Josephson effect. The device of two superconductors linked by an insulator barrier is known as a superconductor Josephson junction (SJJ), which has been extensively used to build quantum interference devices for sensitive measurements.
To detect and exploit the MQC among BECs of atoms, similar to the SJJs, it is natural to link different condensates with Josephson couplings. The peculiar tunnelling phenomena in Josephson coupled BECs are called as Bose-Josephson effects, and the physical systems of two BECs linked by Josephson couplings are called as Bose-Josephson junctions (BJJs). There are two different types of BJJs: the external and internal BJJs (44), see their schematic diagrams in Fig. 4. An external BJJ (46); (47); (48); (49); (50) involves two condensates in a double-well potential and the Josephson coupling is provided by the quantum tunnelling through the barrier between two wells, see panel (a) of Fig. 4. An internal BJJ (63); (64); (65) involves a two-component condensate of atoms occupying two hyperfine levels which are coupled by the external fields, see panel (b) of Fig. 4. Besides the Josephson effects in two-mode systems, non-Abelian Josephson effects have been predicted in a spin-2 BEC of five modes (66). Furthermore, the Josephson oscillations have been demonstrated in an array of BECs in optical lattices (67); (68); (69).
Because of the nonlinear MF interaction, several exotic macroscopic quantum phenomena have been demonstrated, such as macroscopic quantum self-trapping (MQST) (70); (71); (72); (73); (74) and macroscopic quantum chaos (33); (34). Moreover, in contrast to the linear Landau-Zener (LZ) tunneling in two-state single-particle quantum systems, loop structures may appear in the nonlinear LZ tunneling in two-mode BEC systems (75); (76); (77). Actually, these loop structures and MQST are two different sides of the bistability induced by the nonlinear MF interaction.
In this subsection, under the MF description, we will focus on discussing the macroscopic quantum phenomena in BJJs. At first, we introduce an unified model for BJJs. Then, we discuss the dynamics in time-independent systems, such as, Rabi oscillation, MQST and static bifurcation. At last, we analyze the dynamics in time-dependent systems, such as, chaos and slow passages across the critical point.
In the MF theory, BJJs can be treated as multiple coupled modes of macroscopic matter waves (55); (78). For an external BJJ with a sufficiently high potential barrier, the condensate wave-function can expanded by two Wannier states for the two wells (71); (79). That is,
where the spatial functions (j = 1 and 2) are the two Wannier states and the time-dependent amplitudes are complex numbers. As the condensate wave-function is normalized to the total number of condensed atoms and each Wannier state is normalized to one, we have .
Under the two-mode approximation (11), integrating all spatial coordinates, the MF Hamiltonian read as,
with the inter-mode tunnelling strength,
the zero-point energy for atoms in the j-th mode,
the intra-mode interaction strength,
the inter-mode interaction strength,
and the inter-mode exchange collision interaction strength,
In the exchange collision denoted by , an atom from the j-th mode collides with an atom from the k-th mode and then these two atoms are transferred into an atom in the l-th mode and an atom in the m-th mode. For a sufficiently deep double-well potential, two Wannier states are well localized at the two well-centers, so that the high-order overlaps between two Wannier states are very small. This indicates that the terms in last four rows of Hamiltonian (12) can be ignored, that is,
For an internal BJJ, in which a two-component BEC involves atoms in two-hyperfine levels coupled by laser or radio-frequency fields, its Hamiltonian in quantum field theory reads as
with the single component Hamiltonian for the first component,
the single component Hamiltonian for the second component,
and the inter-component interaction and the linear coupling between two components,
Here, is the single-atom mass, with denoting the s-wave scattering length between atoms in hyperfine states and , is the external harmonic potential, is the detuning to the resonant transition between the two hyperfine levels and is the Rabi frequency. The symbols and are Bose creation and annihilation operators for atoms in , respectively.
The system can be described the MF theory if the condensed atoms dominate the whole system, i.e., . Under a sufficiently strong confinement, only the ground state for the external potential will be occupied and so that the condensate wave-functions can be expressed as (80); (81)
with describes the spatial distribution and the complex amplitudes satisfy the normalization condition, . Here, is the total number of atoms. Inserting the approximation ansatz (15) into the Hamiltonian (14) and integrating all spatial coordinates, one will find that the complex amplitudes obey the following MF Hamiltonian,
with the Josephson coupling strength,
the single-particle energy for the first-component,
the single-particle energy for the second-component,
the intra-component interaction strength,
and the inter-component interaction strength,
The time evolution of BJJs obeys two coupled differential equations
with given by the Hamiltonian (13) or (16). Obviously, the time evolution conserves the total number of atoms, . This means that commutates with the Hamiltonian. Thus, one can omit the terms and in Hamiltonians (13) and (16) and rewrite them into an unified form,
with , , and for external BJJs or for internal ones. Correspondingly, the time evolution of BJJs obeys the following two coupled discrete GPEs,
Under the transformation, , and , the Hamiltonian (18) becomes as . This means that the dimensionless Hamiltonian is invariant under the transformation. In this article, the coupling strength is assumed to be non-negative.
Using the atomic numbers and the phases , the complex amplitudes can be expressed in the form of . Thus, one can introduce two new variables, the fractional population imbalance
and the relative phase
From Eqs. (19) and (20), one can find obeying
where we have assumed the Planck constant . Clearly, the equations of motion for are classical Hamiltonian equations for
which describes a non-rigid pendulum. Here, are a pair of canonical coordinates corresponding to the angular momentum and the angular variable for a classical pendulum. Thus, the unified MF Hamiltonian for BJJs is equivalent to a classical non-rigid pendulum Hamiltonian.
Although Eqs. (19) and (20) are equivalent to Eqs. (23) and (24), numerical divergence appears in Eq. (23) if [i.e. all atoms stay in only one of the two modes]. Therefore, for systems involving , one has to choose Eqs. (19) and (20) for numerical simulation.
Rabi oscillation, macroscopic quantum self-trapping and plasma oscillation
The stationary states of BJJs described by Hamiltonian (18) correspond to the fixed points for the classical non-rigid pendulum described by Hamiltonian (25). Mathematically, the fixed points for are determined by and , and their general solutions can be written in forms of Jacobian and Weierstrassian elliptic functions (71); (72).
Similar to the electronic current in a SJJ, one can define the atomic current in a BJJ as,
For the BJJ described by the Hamiltonian (18), due to the conservation of its total number of atoms in two BECs, there is no phenomenon of a direct current between two BECs. Although its atomic currents are alternating currents, its phase evolution is not as same as a SJJ with AC Josephson effects, in which the relative phase varies linearly with time. Firstly, in addition to the running-phase modes induced the asymmetry corresponding to the bias voltage in a SJJ, as mentioned above, running-phase modes can also appear in a symmetric BJJ with strong nonlinearity. Secondly, a BJJ may support Josephson effects of AC atomic currents and oscillating relative phase. By loading atomic BECs into an asymmetric double-well potential and a washboard potential, the AC and DC Josephson effects in atomic BECs have been experimentally demonstrated (82).
For a given system of parameters , , and , from Eqs. (23) and (24), its stationary states can be obtained by solving
Obviously, from Eq. (28), all stationary states request the relative phase with integers . The states of even integers and odd integers are called as in-phase and -states, respectively. For simplicity, we will focus on discussing the region of .
(1) Rabi oscillation. – For a symmetric system without nonlinearity, and , there are two stationary states and , which have no population imbalance. Its general solutions are periodic oscillations around one of two stationary states with a frequency . Actually, this linear system is similar to the Rabi model for a two-level system coupled by classical lasers (83). Therefore, these periodic oscillations around a stationary state are called as Rabi oscillations. In Fig. 5, we show the Rabi oscillations in a linear Bose-Josephson system described the Hamiltonian (18) of , , and . In which, the black solid dots denote the stationary states.
(2) MQST. – For a symmetric system with non-zero , in addition to the two normal stationary states with no population imbalance, two new stationary states appear if . If , two new stationary states are -phase states, . If , two new stationary states are in-phase states, . The non-zero population imbalance is a direct signature of MQST (70); (71); (72); (73); (74). The critical point, where the two new stationary states appear in addition to the normal stationary state, is as same as the bifurcation point for a Hopf bifurcation (81). Usually, the general solutions are periodic oscillations enclosing at least a stationary state. However, due to the nonlinearity, there may exist a new type of solution, running-phase mode, in which the population oscillates periodically and the relative phase changes monotonously.
In Fig. 6, we show the Rabi oscillations and MQST in a nonlinear Bose-Josephson system described by the Hamiltonian (18) of , , and . The stable and unstable stationary states are denoted by black and red solid dots, respectively. In addition to the normal stationary states of no population imbalance, there appear nontrivial stationary states of non-zero population imbalance, which are self-trapped states. Correspondingly, there are three different types of non-stationary states: (i) Rabi oscillations in the green orbit of and ; (ii) running-phase MQST between the green and red orbits of and , which is labeled as MQST-A ; and (iii) MQST in the red orbits of and , which is labeled as MQST-B. Here, denotes the time averaged value for the variable .
(3) Plasma oscillation. – Around a stationary state, a BJJ may have small harmonic oscillations corresponding to plasma oscillations in a SJJ (71); (72); (84). The solution around a stationary state can be written in form of
where and are small oscillations around . Inserting this solution into Eqs. (23) and (24) and keeping only the linear terms of and , one will obtain the following linearized equations,
As all stationary states satisfy , the terms of disappear in the linearized equations.
For a linear system, due to and , the linearized equations read as
Clearly, these small oscillations are Rabi-like oscillations of an angular frequency , which is just the Rabi frequency.
For a nonlinear system around the normal stationary states of , the linearized equations become as
which describe the small oscillations of an angular frequency . Mathematically, for stationary states satisfying , the angular frequency may take imaginary values. The appearance of imaginary means instability of a stationary state and divergence of all small perturbations to this stationary state. Thus, for an unstable stationary state, there is no small oscillation around it.
For a nonlinear system of and , the in-phase stationary state of and is stable, but the anti-phase stationary state of and becomes unstable if . For a nonlinear system of and , the anti-phase stationary state of and is stable, but the in-phase stationary state of and becomes unstable if . Actually, the transition from stable to unstable of a normal stationary state accompanies with the appearance of two additional stable stationary states of non-zero population imbalance. This confirms that the bifurcation from normal to self-trapped states is a Hopf bifurcation (81).
For a strongly nonlinear system of and , the small oscillations around the stationary states of non-zero population imbalance obey
These small oscillations are sinusoidal oscillations of an angular frequency .
Different from a linear system, the plasma frequency of a nonlinear system not only depends on the Josephson coupling , but also depends on the nonlinearity and the relative phase . In a strongly nonlinear system, there are phase-dependent stationary states of self-trapped non-zero population imbalance and small oscillations around these stationary states. Due to the competition between the nonlinearity and the Josephson coupling, the plasma frequency gradually decreases to zero when the system approaches to the critical point, where self-trapping appears.
Shapiro resonance and chaos
In the presence of a periodic driving field, if the energy difference of a particle (or a composite particle such as a Cooper pair) at two sides of the potential barrier can be compensated by the driving field, the system exhibits resonant tunnelling phenomena named as the Shapiro resonance (72); (84). In a SJJ subjected to a bias voltage, , Shapiro resonances occur when the driving frequency satisfies . Here, are positive integers and is the angular frequency for the AC Josephson oscillations in a SJJ with a DC bias voltage . The resonance condition indicates that the energy difference is compensated by the energy of “photons” from the driving field. This means that the physical pictures for Shapiro resonances and photon-assisted tunnelling (85); (86); (87) have no any difference.
In a BJJ, the asymmetry acts as the role of the bias voltage in a SJJ. Similarly, we assume the asymmetry has a constant component and a periodic component, that is, . Such an asymmetry can be realized by controlling double-well asymmetry (or detuning to resonance) for external (or internal) BJJs. Introducing a transformation
the coupled discrete GPEs (19) and (20) read as
with the new coupling strengthes
and is the complex conjugate for . Using the ordinary Bessel functions , the coupling strength can be expanded as,
with . Thus the Hamiltonian (18) becomes
For a linear system of zero , if coupling term can be regarded as a small perturbation, Shapiro resonances are expected to take place when the energy of an integer number of “photons” matches the static asymmetry, . Near a n-photon resonance, the detuning is very small. Using the rotating-wave approximation, in the vicinity of a n-photon resonance, the fast oscillating terms with in the above Hamiltonian can be ignored. That is, the inter-mode coupling is dominated by the term of . Therefore, at the n-photon resonance, the driven asymmetric system is approximately equivalent to a undriven symmetric system of an effective coupling strength .
For an actual system under a weak driving, a resonance may occur if a rational multiple of the driving frequency matches the intrinsic frequency of the undriven system. Since the intrinsic frequency depends on both the coupling strength and the nonlinearity strength, the actual resonance positions are shifted away from (85). Generally, if resonance condition is not satisfied and the driving is sufficiently weak, the oscillations are quasi-periodic ones including two frequency component: the intrinsic oscillation frequency and the driving frequency.
We now consider systems of strong driving fields which can not treated as perturbations. Under a strong driving, the time evolution of a linear system will be quasi-periodic oscillations, which include the intrinsic oscillation frequency and the driving frequency (88). For a nonlinear system, chaotic oscillations may appear if the driving is sufficiently strong (33); (34); (89). In Fig. 7, we show Poincare sections for a driven nonlinear Bose-Josephson system (18) with , , and . In our simulation, for a given initial condition, we record a data per driving period. In the Poincare sections, a curve or a circle of infinite dots corresponds to a regular motion and a region of infinite dots corresponds to chaotic motions. For a weak driving, the oscillations are regular, see panel (a). When the driving strength increases, chaos appear in the vicinity of the unstable stationary state for the undriven system and the region of chaotic motions increases with the driving strength. However, even in the sea of chaos, it is possible to find some islands of regular motions. In our figure, the chaotic sea is denoted by the region of green dots, regular motions in the small regular islands surrounded by chaotic sea are denoted by curves/circles of red dots, and other regular motions are denoted by curves/circles of blue dots.
It is also possible to find exact Floquet states for systems driven by some periodic modulations (90); (91); (92). These Floquet states have no linear counterparts due to their existence requires a nonzero MF interaction (90). These nonlinear Floquet states may be stable or unstable. Usually, the stale Floquet states stay around the node points of the undriven system and the unstable Floquet states are enclosed by the chaotic motions around the saddle points of the undriven system. There is a triangular structure in the quasienergy band, which corresponds to the onset of a localization phenomenon called as the coherent destruction of tunneling (91).
Universal mechanism for slow passages through a critical point
Mathematically, slow passages through a bifurcation point exhibit memory effects induced by bifurcation delay (93); (94). Physically, slow passages through a critical point of vanished excitation gap connect with the breakdown of adiabaticity and the appearance of non-adiabatic excitations (also called as defect modes) (95). When the system approaches to the critical point, the adiabaticity breaks down and the generated defect modes follows Kibble-Zurek (KZ) mechanism (95); (96); (97); (98). As mentioned in Section 2.4.2, for a BJJ described by the Hamiltonian (18), there is a phase transition from normal to self-trapped states, which corresponds to a Hopf bifurcation (81). Below, we present the slow passage through the critical point between the normal and self-trapped states (99).
For simplicity, we assume the Josephson coupling strength and only discuss the MF ground-states for symmetric BJJs in this section. For a symmetric BJJ of positive , the normal-to-MQST transition occurs in anti-phase states, which are not its MF ground-states. For a symmetric BJJ of negative , the normal-to-MQST transition occurs in in-phase states, which are its MF ground-states. To push the system across the critical point, one may adjust the nonlinearity via Feshbach resonances (100); (101); (37) or vary the Josephson coupling strength . For external BJJs, the Josephson coupling strength can be controlled via tuning the potential barrier (46); (47); (48); (49); (50). For internal BJJs, the Josephson coupling strength can be controlled via tuning the intensity of the coupling lasers (63); (64).
To explore the normal-to-MQST transition in ground-states, one has to select a system of , whose critical point is determined by . For this system across the critical point, its ground state changes from to with and . Clearly, the normal-to-MQST transition is a type of spontaneous symmetry breaking related to internal degrees of freedom (81); (99); (102).
In a slow passage through the critical point, there are two characteristic times: the reaction time and the transition time. The reaction time, , describes how fast the system follows eigenstates of its instantaneous Hamiltonian. The transition time, , depicts how fast the system is driven. The adiabatic condition keeps valid if . Otherwise, if , the adiabatic condition becomes invalid and defect modes appear.
The Bogoliubov theory is a useful method for analyzing the collective excitations over the ground states (1); (2); (3); (4); (103). In a BJJ described by Hamiltonian (18), due to its spatial degrees of freedom have been frozen, the excitations will not change the spatial distribution but only change the population distribution. Therefore, under the MF description, the perturbed ground state read as
with and the ground state chemical potential . Inserting the above perturbed ground state into coupled GPEs (19) and (20), one will obtain the linearized equations for the perturbations,
with Dirac kets