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Manipulating the superfluid - Mott insulator transition of a Bose - Einstein condensate in an amplitude - modulated optical lattice

The superfluid - Mott insulator transition in a BEC confined in an amplitude - modulated optical lattice can be manipulated by the modulation strength. Two standing laser waves of main and sideband frequencies of an optical lattice induce a Raman transition; due to resonant Raman driving the critical value of the transition parameter depends on the modulation strength and the detuning from resonance and can be tuned to a given value. It is shown that there is an interval of the initial MI phase, in which a modulation switches the MI phase to the SF phase.

Manipulating a phase transition is a subject of great interest to physicists. A physical system that crosses the boundary between two phases changes its properties in a fundamental way. Second -order phase transitions usually termed classical phase transitions, take place at a finite temperature, and they are accomplished by changing the temperature. A quantum phase transition is one that occurs at absolute zero of temperature. The quantum phase transitions have attracted much interest in recent years. These transitions are accomplished by changing not the temperature, but some parameter in the Hamiltonian of the system. This parameter is the lattice laser intensity for a Bose -Einstein condensate (BEC) in an optical lattice, which controls the superfluid -Mott insulator transition; the charging energy in Josephson -junction arrays. which controls the superconductor -insulator transition; the magnetic field in a quantum -Hall system, which controls the transition between the Hall plateaus; disorder in a conductor near its metal -insulator transition. The competition between two interactions is fundamental to quantum phase transitions and inherently different from classical phase transitions which are driven by the competition between inner energy and entropy. The quantum phase transition are driven by the interplay of the different contributions to the Hamiltonian of the system. For instance, for a BEC in an optical lattice the two terms in the Hamiltonian represent the self -interaction of the atoms in the lattice site and the hopping of the atoms between the sites, for Josephson -junction arrays they represent the charging energy of each grain and the Josephson coupling energy between grains; the competition between them determines the quantum phase transition.
The experimental observation of Bose-Einstein condensation in a dilute gas of ultracold trapped atoms [1][2][3] has generated much interest in manipulating such coherent matter by external electromagnetic fields. Recently atoms have been confined in optical potentials created by standing light waves [ 4,5 ], with the wavelength of the optical potential much smaller than the dimensions of the trap. The unique prospects of this new class of system (the correlated bosons on a lattice [ 6 -9]) became evident. A transition to a Mott insulator becomes possible when a superfluid like a BEC is placed in a periodic potential. This quantum phase transition is characterized by the competition between two interactions: the tendency of the particles to hop into adjacted wells ( kinetic energy ), and the interparticle forces that keep them in separate wells ( potential energy ). As the strength of the interaction term (the repulsive interaction energy U between two atoms ) relative to the tunneling term (the tunneling J between adjacent sites in the Bose -Hubbard Hamiltonian) is changed, the system reaches a quantum critical point in the ratio of U/J, for which the system will undergo a quantum phase transition from the superfluid state ( SF ) to the Mott insulator state (MI). A seminal experiment by Greiner and collaborators [ 5 ] demonstrated a quantum phase transition in a Bose -Einstein condensate from a superfluid state into a Mott insulator state, by varying the lattice laser intensity as proposed theoretically in [ 10 ].
The balance between two interactions, which determines a quantum phase transition can be changed by a time -dependent action, and thereby a quantum phase transition can be manipulated. In this letter, we propose this basic idea which will be used for the SF -MI transition in a BEC and for the external Josephson effect between two BEC's.
We consider a manipulation of the SF -MI transition in a BEC in an amplitudemodulated optical lattice. The such optical lattice can be created by interfering pair of amplitude -modulated beams. In general, there are the two generic types of the manipulation. The effect of the modulation is to modulate the optical lattice potential so that it becomes a time -dependent optical lattice potential which the depth of the potential well is where ε is the modulation strength, ω M is the modulation frequency. If the period of modulation 2π/ω M is larger the nonequilibrium dynamic time τ noneq , then the time behavior of a BEC in an amplitude -modulated optical lattice is quasistatic. The nonequilibrium dynamic time τ noneq for trapped bosonic atoms in an optical lattice potential was considered in Ref. [11] and was shown that the characteristic time of the dynamical restoration of the phase coherence in the domain of a quantum phase transition is the integer of the order of mechanism is determined by change in the population due to a resonant driving. In a BEC in a periodic potential the characteristic energetic scale of the band structure [12,13] is the recoil energy E R . And, usually, E R exceeds J, and the quasistatic behavior is not valid.
However, in this case is more convenient to use a picture that an amplitude -modulated field can be presented as a set of monochromatic fields (main on frequency ω and sideband on frequencies ω ± ω M ). These fields can induce a Raman transition between the two internal states of a BEC |a > and |b >. A resonant field via a two -photon ( Raman ) transition changes the population of the ground state of a BEC. Therefore, the on -site interaction and the tunneling, which depend on the population, are also changed. However, due to this changing population the variations of the on -site interaction and the tunneling are different; therefore, the balance between kinetic and potential energies, which determines a phase transition, is changed. As a result, we have a modification of the quantum phase transition.
The modificated by a resonant modulation ratio U r /J r is less than this ratio without a modulation. The phase transition [6,9,14] occurs at U/J = z × 5.8 = const, with z being the number of next neighbours of a lattice site. The system has the renormalized by a modulation parameters U r and J r of a Bose -Hubbard Hamiltonian and reaches a quantum critical point if U r /J r = const with the same constant, then it means that the optical potential critical depth in an amplitude -modulated resonant lattice V cr 0res increases, because the ratio of U/J is the growing function [ 10,14 ] on the optical potential depth. Second is a nonresonant mechanism for which we will use a direct time -dependent description. A nonresonant mechanism of manipulation is in operation for a quasistatic regime ω M < 2πτ −1 noneq , therefore, if V 0 corresponds to a quantum critical point V cr 0 then as the time runs, the system goes from a superfluid to an insulator and back again with frequency ω M . The ratio of U/J is a nonlinear function on the laser intensity with the leading exponential dependence on the strength of the periodic potential V 0 . If we have a time -dependent lattice laser intensity, then, although V 0 (t) = V 0 , due to the nonlinearity a time -averaged ratio of U/J will be different from a ratio of U/J without a modulation. For a small ε we have the time -averaged U/J = U/J + ∆(U/J), where the bar stands for a time average and ∆(U/J) is proportional to the time -averaged ε 2 cos 2 ω M t ( the time -averaged of the modulation harmonic oscillation to the lowest even order power ).
A Bose -Einstein condensate in an amplitude -modulate optical lattice.
In resonant case the system is excited by an off -resonant Raman ( two -photon ) driving with the Rabi frequency Ω R . In an amplitude -modulated optical lattice two standing laser waves of frequencies ω and ω−ω M (ω M ≪ ω) drive an atom in a Raman scheme. The effective two -photon Rabi frequency is given in terms of the single -photon Rabi frequencies Ω and Ω ′ of the fields on the main ω and sideband ω ′ = ω − ω M frequencies as Ω R = Ω·Ω ′ 2∆ . For the amplitude -modulated laser beams with the modulation strength ε we have for standing h , here ∆ is the far -detuning of the laser beams from the atomic resonance (the closest neighboring optical dipole transition), and V 0 (x) = 3 j=1 V jo cos 2 kx j is the optical lattice potential with wave vectors k = 2π λ and λ the wavelength of the laser beams on the main frequency, and we are omitted the terms proportional to the small parameter  [ 15,16 ] BEG resemble the Bloch equations describing a driven two -level system if the external driving field hΩ R is much larger than the difference in the two -body interaction K ( this energy [17] is proportional to U a + U b − 2U ab ). We consider a steady state driving regime, in which the solution ϕ(x) of the Bloch equation of the standard Rabi problem is determined on the parameters of the field ( the Rabi frequency and the detuning from resonance δ = ω M − (ω b − ω a ) ). Therefore, the boson field operator for atom Ψ(x) can be represent by an expansion in the Wannier basis, which is a sum over the lattice i, and multiplied by the function ϕ(x) , which is determined by the Raman field, where the operators a i are the operators on lattice site i. In order to avoid a sufficient population of the excited state b, we are assume Ω R < δ, then In the tight -binding limit the Wannier states are localized and can be approximated [ 10,14 ] by the harmonic oscillator wave function w n (x − x i ) of number n. The size of the ground -state oscillator wave function β in a lattice is much less than the lattice period λ/2, β ≪ λ, where the size β = h 2mν j determined by the atomic mass m and the oscillation frequency in the wells ( see, for example [ 10 ] ) ν j = 4E R V jo /h, then the inequality hΩ R ≫ K reduces to the following εV 0 ≫ K. This inequality can be easily fulfiled, because, usually, K/h ∼ 1 ÷ 0.1s −1 ( for example, K/h ≃ 0.1s −1 for hyperfine states of 87 Rb [18]) and V 0 ≃ 10E R , E R /h ≃ 2 × 10 5 s −1 for a blue detuning [ 19 ] λ = 514nm for sodium. We find for ( εV 0 hδ ) 2 ≪ 1 that the parameters of a Bose -Hubbard Hamiltonian modificated by a resonant modulation are The parameters of a Bose -Hubbard Hamiltonian J and U depend on the optical lattice depth V 0 , and they were calculated analytically in Ref. [14] in the tight -binding limit in terms of the microscopic parameters of the atoms in the optical lattice. We find that for a resonant amplitude -modulated optical lattice the ratio U r /J r is always less than without a modulation, then it means that for a resonant modulation the optical potential critical depth V cr 0 increases. The corresponding tuning ∆V res 0 (ε) of the optical lattice critical depth is determined for As a result, we can manipulate the superfluid -Mott insulator transition of a BEC in an amplitude -modulated optical lattice. The tuning of the optical lattice critical depth is determined by the modulation strength ε. By varying the parameter ε/δ, we can tune the critical value to a given value. Therefore, in the interval of the laser intensity V 0 , for which V cr 0 < V 0 < V cr 0res (ε), we can switch one phase to another by a modulation, and in this interval of the initial MI phase a modulation switches the MI phase to the SF phase.
For a nonresonant manipulation we will use a time -dependent potential ( Eq.(l)), and there could be a modulation frequency ω M low enough for quasistatic time behavior.
Therefore, it modulates the on -site two -body interaction U and the hopping matrix element J, so that these quantities become time -dependent U(t), J(t). Note that such potential is created by interfering pair of amplitude -modulated laser beams with the modulation strength ε/2 of the electric field amplitude, and was omitted the quadratic term for the small parameter ε 2 /4. Using the analytical dependencies [ 14,20 ], we have for a timedependent optical lattice depth Calculating the time -average of the product of U(t) by (J(t)) −1 by expanding in a series about the small parameter ε 2 V 0 E R < 1, we find where g = 2hωa s , ω is the confining frequency, a s is the s -wave scattering length between two atoms, ( U J ) nr is the time -average of the ratio U J for a nonresonant modulation, and were omitted the small terms proportional to (ε 2 V 0 E R ) n for n > 1. If ε 2 V 0 E R is not small, then we may expand the expression for the time -dependent J(t) using Bessel functions I n (x) and the functions cos n ω M t. In this case instead the parameter ε 2 4 [ , because, usually, V 0 /E R ∼ 10 and for ε < 0.4 among I n (x < 1.3) the largest is I 2 (x).

An external Josephson effect between two BECs.
Here we consider briefly the manipulation by an amplitude -modulated laser beam an external Josephson effect between two BECs, the full presentation will be given elsewhere.
This system is a prototype system of an array of Josephson junctions. In such systems, a superconducting -insulator phase transition is driven by the competition between the Josephson coupling energy E j , which govern the tunneling through the intrawell barriers, and the on -site interparticle interaction energy E C . A Josephson effect between two spatially separate BECs can be realized only with small barriers. A key element of this effect is the Josephson coupling energy E j . Based on a tunneling Hamiltonian description the dominant Josephson term of first order in the tunneling is proportional to the barrier transmission amplitude. Such approximation by a tunneling or transfer Hamiltonian is valid for high and narrow barriers, for which the barrier transmission amplitude t lr is proportional to where the two BECs ( a and b) confined in a cubic volume L 3 are separated by a square potential barrier of height V B and width d. A Josephson geometry for BECs may be created by splitting a single condensate in a long trap into two separate parts by a narrow light sheet produced by a strongly detuned laser beam. Because the repulsive potential due to the ac Stark shift is proportional to the laser power, thus the height of the barrier is determined by the laser intensity. Therefore, if we create the necessary for a Josephson effect two -well potential by an amplitude -modulated laser beam, then the potential barrier height V B will be an amplitude -modulated time -dependent one ( similarly Eq.(1)). Just as the analytical dependencies of the tunneling J between adjacent sites of a BEC in an amplitude -modulated optical lattice and the transmission amplitude t lr are similar, so the time -average of this barrier transmission amplitude t lr is differ from the value t lr without a modulation. As a result, the Josephson coupling energy E j , which is proportional to the barrier transmission amplitude, is also can be manipulated by the abovementioned manner.
In summary, the superfluid -Mott insulator transition in a BEC in an optical lattice can be manipulated by an amplitude -modulated manner. This manipulating is caused by the change of the balance two interactions, which determines a quantum phase transition, by a time -dependent action. There are two mechanisms of the manipulation. One, resonant mechanism is in operation for the modulation frequency closed to the excited internal state of a BEC, and two standing laser waves of main and sideband frequencies of an amplitude -modulated optical lattice induce a Raman transition. Due to resonant Raman driving the population of the ground state of a BEC changes. Therefore, the on -site interaction and the atomic tunneling are also changed, and their ratio is different from the ratio without a resonant modulation. As a result, the optical lattice critical depth is other than without a modulation, this tuning is determined by the square of the modulation strength. The optical lattice critical depth with a resonant modulation is higher than without a modulation, and, therefore, there is an interval of the initial MI phase, in which a modulation switches the MI phase to the SF phase. Second, nonresonant mechanism is determined by the nonlinear dependence of the ratio of the repulsive interaction to the tunneling on the optical lattice depth. Due to this nonlinearity for the harmonic modulation of the optical lattice depth the time -averaged ratio is different from the ratio without a modulation. The proposed mechanism of manipulating and controlling Bose -Einstein condensates makes it possible to tune the critical value of parameter of the quantum phase transition to a given value.
In conclusion, tuning and conducting, in general, the superfluid -Mott insulator transition in a BEC in an optical lattice by a time -dependent action ( in particular, in an amplitude -modulated optical lattice ) can be a path to the solution of the general problem of controlling a quantum phase transition. I thank A.J.Freeman for encouragement, A. Patashinski for discussions.