Spin squeezing in a bimodal condensate: spatial dynamics and particle losses
We propose an analytical method to study the entangled spatial and spin dynamics of interacting bimodal Bose-Einstein condensates. We show that at particular times during the evolution spatial and spin dynamics disentangle and the spin squeezing can be predicted by a simple two-mode model. We calculate the maximum spin squeezing achievable in experimentally relevant situations with Sodium or Rubidium bimodal condensates, including the effect of the dynamics and of one, two and three-body losses.
pacs:PACS-03.75.GgEntanglement and decoherence in Bose-Einstein condensates and PACS-42.50.DvQuantum state engineering and measurements and PACS-03.75.KkDynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow and PACS-03.75.MnMulticomponent condensates; spinor condensates
In atomic systems effective spins are collective variables that can be defined in terms of orthogonal bosonic modes. In this paper the two modes we consider are two different internal states of the atoms in a bimodal Bose-Einstein condensate. States with a large first order coherence between the two modes, that is with a large mean value of the effective spin component in the equatorial plane of the Bloch sphere, can still differ by their spin fluctuations. For an uncorrelated ensemble of atoms, the quantum noise is evenly distributed among the spin components orthogonal to the mean spin. However quantum correlations can redistribute this noise and reduce the variance of one spin quadrature with respect to the uncorrelated case, achieving spin squeezing Ueda (); manip (). Spin-squeezed states are multi-particle entangled states that have practical interest in atom interferometry, and high precision spectroscopy Wineland (). Quantum entanglement to improve the precision of spectroscopic measurements has already been used with trapped ions Liebfried () and it could be used in atomic clocks where the standard quantum limit has already been reached Santarelli ().
A promising all-atomic route to create spin squeezing in bimodal condensates, proposed in Nature (), relies on the Kerr-type non linearity due to elastic interactions between atoms. Quite analogously to what happens to a coherent state in a nonlinear Kerr medium in optics coherent (), an initial “phase state” or coherent spin state, where all the effective spins point at the same direction, dynamically evolves into a correlated spin-squeezed state. A straightforward way to produce the initial phase state in a bimodal condensate is to start with one atomic condensate in a given internal state and perform a -pulse coupling coherently the internal state to a second internal state Cornell (). However, as the strength of the interactions between two atoms , and are in general different, the change in the mean field energy excites the spatial dynamics of the condensate wave functions. In the evolution subsequent to the pulse, the spin dynamics creating squeezing and the spatial dynamics are entangled Nature (); EPJD (); Sorensen (); Dutton () and occur on the same time scale set by an effective interaction parameter . This makes it a priori more difficult to obtain simple analytical results.
In this paper we develop a simple formalism which allows us to calculate analytically or semi analytically the effect of the spatial dynamics on spin squeezing. In Section 2 we present our dynamic model. Using our treatment we show that at particular times in the evolution the spatial dynamics and the spin dynamics disentangle and the dynamical model gives the same results as a simple two-mode model. We also identify configurations of parameters in which the simple two mode-model is a good approximation at all times. Restricting to a two-mode model, in Section 3 we generalize our analytical results of PRLlosses () on optimal spin squeezing in presence of particle losses to the case of overlapping and non-symmetric condensates.
In Sections 4 and 5, we apply our treatment to cases of practical interest. We first consider a bimodal Rb condensate. Rb is one of the most common atoms in BEC experiments and it is a good candidate for atomic clocks using trapped atoms on a chip clockRb (). Restricting to states which are equally affected by a magnetic field to first order, the most common choices are and which can be magnetically trapped, or and that must be trapped optically but for which there exists a low-field Feshbach resonance which can be used to reduce the inter-species scattering length Sengstock (); Widera (). Indeed a particular feature of these Rb states is that the three -wave scattering lengths characterizing interactions between , and atoms are very close to each other. A consequence is that the squeezing dynamics is very slow when the two condensates overlap. The inter-species Feshbach resonance can be used to overcome this problem and speed up the dynamics Widera ().
In schemes involving the of Rubidium, the main limit to the maximum squeezing achievable is set by the large two-body losses rate in these states. As a second case of experimental interest we then consider Na atoms in the states Nature (). Although theses states have opposite shifts in a magnetic field, they present the advantage of negligible two-body losses. Using our analytical optimization procedure, we calculate the maximum squeezing achievable in this system including the effect of spatial dynamics and particle losses.
In Section 5 we examine a different scenario for Rb condensates in which, instead of changing the scattering length, one would spatially separate the two condensates after the mixing pulse and hold them separately during a well chosen squeezing time. An interesting feature of this scheme is that the squeezing dynamics acts only when the clouds are spatially separated and it freezes out when the two clouds are put back together so that one could prepare a spin squeezed state and then keep it for a certain time PRLlosses (). State-selective potentials for Rb in and clockRb () have recently been implemented on an atom chip, and such scheme could be of experimental interest.
2 Dynamical spin squeezing model
In this section we develop and compare dynamical models for spin squeezing. No losses will be taken into account in this section.
2.1 State evolution
We consider the model Hamiltonian
where is the one-body hamiltonian including kinetic energy and external trapping potential
The interactions constants are related to the corresponding -wave scattering lengths characterizing a cold collision between an atom in state with an atom in state (), and is the mass of one atom.
We assume that we start from a condensate with atoms in the internal state ; the stationary wave function of the condensate is . After a pulse, a phase state is created, which is our initial state:
where , are mixing coefficients with and the operator creates a particle in the internal state with wave function . To describe the entangled evolution of the spin dynamics and the external dynamics of the wave functions, it is convenient to introduce Fock states with well defined number of particles in and , these numbers being preserved during time evolution subsequent to the mixing pulse. Expanded over the Fock states, the initial state (3) reads:
where , and
where and are solutions of the coupled Gross-Pitaevskii equations:
here with the initial conditions
and the time dependent phase factor solves:
With this treatment we fully include the quantum dynamics of the two condensate modes and , as one does for the simple two modes model, but also including the spatial dynamics of the two modes and their dependence on the number of particles. The approximation we make is to neglect all the other modes orthogonal to the condensates which would be populated thermally. An alternative method is to use a number conserving Bogoliubov theory that explicitly includes the operators of the condensates as in Sorensen (). In that case all the modes are present but the modes orthogonal to the condensates are treated in a linearized way. In Sorensen (), the author compares the number conserving Bogoliubov approach to our approach using many Gross-Piaevskii equations, also used in Nature (), and he finds very similar result for the spin squeezing. He also finds that within the Bogoliuobov approximation the thermally excited modes strictly do not affect the squeezing in the scheme we consider here. If the number conserving Bogoliubov has the advantage of being systematic, our approach, supplemented with a further approximation (the modulus-phase approximation introduced in Sect. 2.3) allows us to get some insight and obtain simple analytical results.
2.2 Calculation of spin squeezing
Referring to the Fig.1(Top) we introduce the polar angles and giving the direction of the mean spin; determines the relative mean atom number in the two internal states, , while the azimuthal angle corresponds to the relative phase between the components, .
The minimal variance of the spin in the plane orthogonal to the mean spin , represented in Fig. 1(Bottom), is given by
where we introduced
where is the length of the average spin.
When expressed in the original frame of reference, the minimal variance in the orthogonal plane is:
and where we introduced the correlations
The spin squeezing is then calculated in terms of averages of field operators products, with the state of the system at time , obtained by evolving equation (4) with equation (6). To calculate the averages one needs to compute the action of the field operators on the Fock states (5) notecalc1 (),
The explicit expressions of the averages needed to calculate the spin squeezing parameter are given in Appendix A. These quantum averages correspond to an initial state with a well-defined number of particles . In case of fluctuations in the total number of particles where the density matrix of the system is a statistical mixture of states with a different number of particles, a further averaging of over a probability distribution is needed EPJD (); EPJDloss ().
2.3 Dynamical modulus-phase approach
In principle, equations (7)-(9) can be solved numerically for each Fock state in the sum equation (4), and the squeezing can be computed as explained in the previous section. However, for a large number of atoms and especially in three dimensions and in the absence of particular symmetries (e.g. spherical symmetry) this can be a very heavy numerical task. To overcome this difficulty, in order to develop an analytical approach, we can exploit the fact that for large in the initial state (4) the distributions of the number of atoms and are very peaked around their average values with a typical width of order . Moreover, assuming that possible fluctuations in the total number of particles are described by a distribution having a width much smaller than the average of the total number of particles , we can limit to and close to and . We then split the condensate wave function into modulus and phase
and we assume that the variation of the modulus over the distribution of can be neglected while we approximate the variation of the phase by a linear expansion around EPJD (). The approximate condensate wave functions read
The modulus phase approximation takes into account, in an approximate way, the dependence of the condensate wave functions on the number of particles. It is precisely this effect that is responsible of entanglement between spatial dynamics and spin dynamics.
As explained in Appendix B, all the relevant averages needed to calculate spin squeezing can then be expressed in terms of and of three time and position dependent quantities:
In some cases (see Sect. 2.4) these quantities can be explicitly calculated analytically. To calculate the squeezing in the general case, it is sufficient to evolve a few coupled Gross-Pitaevskii equations (7) for different values of , , to calculate numerically the derivatives of the phases appearing in (24)-(26). Although we do not expect a perfect quantitative agreement with the full numerical model for all values of parameters, we will see that the analytical model catches the main features and allows us to interpret simply the results.
In the particular case of stationary wave functions of the condensates, the parameters , and become space-independent:
To test our modulus-phase dynamical model, in Fig. 2, we consider a situation in which the external dynamics is significantly excited after the pulse which populates the state . Parameters correspond to a bimodal Rb condensate in and with and where a Feshbach resonance is used to reduce by about 10% with respect to its bare value Sengstock (); Widera (). The considered harmonic trap is very steep kHz. In the figure we compare our modulus-phase approach (dashed line) with the full numerical solution (solid line) and with a stationary calculation using (27)-(28) (dash-dotted line) which is equivalent to a two-mode model. The oscillation of the squeezing parameter in the two dynamical calculations (dashed line and solid line) are due to the fact that the sudden change in the mean-field causes oscillations in the wave functions whose amplitude and the frequency are different for each Fock state. From the figure, we find that our modulus-phase approach obtained integrating 5 Gross-Pitaevskii equations (dashed line) reproduces the main characteristics of the full numerical simulation using 3000 Fock states (solid line). The stationary two mode model on the other hand is not a good approximation in this case. Only for some particular times the three curves almost touch. At these times the wave functions of all the Fock states almost overlap and, as we will show in our analytical treatment, spatial dynamics and spin dynamics disentangle.
In Fig.3 we move to a shallow trap and less atoms.
We note that in this case both the modulus-phase curve and the numerical simulation are very close to the stationary two-mode model which is then a good approximation at all times.
2.4 Squeezing in the breathe-together solution
In this section we restrict to a spherically symmetric harmonic potential identical for the two internal sates. For values of the inter particle scattering lengths such that
and for a particular choice of the mixing angle such that the mean field seen by the two condensates with and particles is the same:
the wave functions and solve the same Gross-Pitaevskii equation. In the Thomas-Fermi limit, the wave functions and share the same scaling solution Scaling1 (); Scaling2 () and “breathe-together” EPJD ().
is the chemical potential of the stationary condensate before the pulse, when all the atoms are in state , and is the corresponding Thomas-Fermi radius. The initial conditions for (34) are and .
Note that the scaling solution identical for the two modes and is valid only for , and does not apply to all the wave functions and in the expansion equation (4). Nevertheless, an advantage of choosing the mixing angle in order to satisfy the breathe-together condition equation (31), is that the mean spin has no drift velocity. In Fig.4 we calculate the spin squeezing (Top) and the angle giving the direction of the mean spin projection on the equatorial plane of the Bloch sphere (Bottom), for the same parameters as in Fig.3 except for the mixing angle that we now choose satisfying equation (31) while in Fig.3 we had . Note that practically does not evolve. The maximum amount of squeezing is lower in the breathe-together configuration than in the even-mixing case (see also Dutton ()). However, as we will see in the next section, this conclusion does not hold when particle losses are taken into account.
By linearization of and around the breathe-together solution and using classical hydrodynamics, it is even possible to calculate analytically the parameters and relevant for the squeezing dynamics EPJD (). One obtains:
and where is solution of the differential equations
to be solved together with equation (34), with initial conditions . In practice, when we expand the condensate wave functions around the breathe-together solution equation (32) as in EPJD (), we encounter the hydrodynamics operator Stringari ()
The deviation of the relative phase and the relative density from the breathe-together solution expand over two eigenmodes of : A zero-energy mode which grows linearly in time and gives the dominant features of phase dynamics and squeezing (integral term in the curly brackets in Eq.(36)), and a breathing mode of frequency which is responsible for the oscillations of the squeezing parameter. The fact that in breathe-together conditions and within the modulus-phase approximation is shown in Appendix C.
We give an example corresponding to strongly oscillating wave functions in Fig.5 where we compare the spin squeezing from the analytical theory with a numerical simulation. In the analytical formula, the entanglement between spatial degrees of freedom and spin dynamics is apparent as equation (36) is position dependent. The points in which the dynamical curve (dotted line) touches the stationary two mode curve (dash-dotted line) correspond to Im (see the bottom curve) where space and spin dynamics are disentangled. We note however that the validity conditions of classical hydrodynamics are more stringent for a mixture of condensates with rather close scattering lengths than for a single condensate EPJD (). We checked numerically that in order for equation (36) to correctly predict the frequency of the oscillations in the squeezing parameter, we have to enter deeply in the Thomas-Fermi regime.
2.5 “Extracted” spin squeezing
As we pointed out, the definitions equations (10)-(12) explicitly include the spatial overlap between the two modes. Here we give an alternative definition that can be used always, whether or not the modes overlap. To this aim, we introduce the time-dependent operators
where is the solution of Gross-Pitaevskii equation (7) for mode with , particles. We then introduce the spin operators:
In the new definition of spin squeezing calculated by the spin operators defined in equations (44)-(46), which we call the “extracted” spin squeezing, we still take into account entanglement between external motion and spin dynamics, but we give up the information about the overlap between the two modes. In Appendix D, we give the quantum averages useful to calculate the extracted spin squeezing within the modulus-phase approach described in Section 2.3. We will use this extracted spin squeezing in Section 5.
3 Two-mode model with Particle losses
In this section we generalize our results of PRLlosses () to possibly overlapping and non-symmetric condensates. In subsection 3.1 we address the general case, while in subsection 3.2 we restrict to symmetric condensates and perform analytically an optimization of the squeezing with respect to the trap frequency and number of atoms. In the whole section, as in PRLlosses (), we will limit to a two-mode stationary model and we do not address dynamical issues.
3.1 Spin squeezing in presence of losses
We consider a two-component Bose-Einstein condensate initially prepared in a phase state, that is with well defined relative phase between the two components,
When expanded over Fock states, the state (47) shows binomial coefficients which, for large , are peaked around the average number of particles in and , and . In the same spirit as the “modulus-phase” approximation of subsection 2.3, we can use this fact to expand the Hamiltonian of the system to the second order around and
where the chemical potentials and all the derivatives of should be evaluated in and . We can write
The function of the total number of particles, , commutes with the density operator of the system and can be omitted. The second term in equation (49) proportional to describes a rotation of the average spin vector around the axis with velocity . The third term proportional to provides the nonlinearity responsible for spin squeezing. It also provides a second contribution to the drift of the relative phase between the two condensates in the case .
In presence of losses, the evolution is ruled by a master equation for the density operator of the system. In the interaction picture with respect to , with one, two, and three-body losses, we have:
where , , and similarly for ,
is the -body rate constant () and is the condensate wave function for the component with and particles. is the rate constant for a two-body loss event in which two particles coming from different components are lost at once.
In the Monte Carlo wave function approach MCD () we define an effective Hamiltonian and the jump operators ()
We assume that a small fraction of particles will be lost during the evolution so that we can consider , and as constant parameters of the model. The state evolution in a single quantum trajectory is a sequence of random quantum jumps at times and non-unitary Hamiltonian evolutions of duration :
where now or . Application of a jump to the -particle phase state at yields
After a quantum jump, the phase state is changed into a new phase state, with particle less and with the relative phase between the two modes showing a random shift with respect to the phase before the jump. Note that in the symmetrical case and no random phase shift occurs in the case of a jump of . Indeed we will find that at short times in the symmetrical case theses kind of crossed losses are harmless to the the squeezing.
In presence of one-body losses only, also the effective Hamiltonian changes a phase state into another phase state and we can calculate exactly the evolution of the state vector analytically, as we did in PRLlosses () for symmetrical condensates. When two and three-body losses enter into play, we introduce a constant loss rate approximation EPJDloss ()
valid when a small fraction of particles is lost at the time at which the best squeezing is achieved. In this approximation, the mean number of particles at time is
where for example is the fraction of lost particles due to -body losses in the condensate. Let us present the evolution of a single quantum trajectory: Within the constant loss rate approximation, we can move all the jump operators in (58) to the right. We obtain: