Protocol for nonlinear state discrimination in rotating condensate
Michael R. Geller





 
Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA
Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA
Center for Simulational Physics, University of Georgia, Athens, Georgia 30602, USA

 


^*Corresponding author. Email: mgeller@uga.edu
0.5in
Abstract
0.1in
0.9
Nonlinear mean field dynamics enables quantum information processing operations that are impossible in linear one-particle quantum mechanics. In this approach, a register of bosonic qubits (such as neutral atoms or polaritons)  is initialized into a symmetric product state ψ^⊗ n through condensation, then subsequently controlled by varying the qubit-qubit interaction. We propose an experimental implementation of quantum state discrimination, an important  subroutine in quantum computation,
with a toroidal Bose-Einstein condensate. 
The condensed bosons here are atoms, each in 
the same superposition of angular momenta
 0 and ħ, encoding a qubit.
A nice feature of the protocol is that only readout of individual quantized circulation states (not superpositions) is required.

0.5in



Protocol for Nonlinear State Discrimination in Rotating Condensate
    Michael R. Geller^*
    April 24, 2024
==================================================================





§ INTRODUCTION


A variety of atomtronic architectures have been proposed for quantum computing and quantum technology applications <cit.>. 
Two main Bose-Einstein condensate (BEC) types have been considered for realizing qubits: multi-component condensates and multi-mode condensates. Multi-component and spinor condensate approaches <cit.>
encode a single qubit in two (or more) metastable atomic states, such as spin or hyperfine levels, with all atoms in the same translational mode (for example the motional ground state). 
Multi-mode approaches <cit.> encode a single qubit using two (or more) translational modes in a scalar condensate, such as a BEC in a double-well trapping potential. In the limit where there are a large number of condensed bosons in each well, the system becomes equivalent to two (or more) BECs, each with a well defined phase, connected by tunneling barriers that act as Josephson junctions <cit.>.
Arrays of such BECs can be produced in optical lattices and are described by the 
Bose-Hubbard model <cit.>.
Another multi-mode approach, which we adopt here, uses circulating states in a ring geometry 
<cit.>  for the translational modes.

Given the demonstrated high performance and scalability of trapped ion qubits <cit.>, superconducting qubits <cit.>, and of neutral atom arrays <cit.>, what does a BEC qubit offer? 
We argue that it offers 
a platform for an alternative approach to quantum information processing that leverages the special properties of condensates. 
In this approach, a BEC is used to 
prepare a register of qubits in a product state ψ^⊗ n and control its subsequent evolution. From a quantum computing perspective, having multiple identical copies of an unknown input is already a useful resource (whereas classical information is freely cloned).
In standard circuit-model quantum computation, illustrated in Fig. <ref>a, initialized qubits are subsequently entangled using two-qubit gates. 
Here we do the opposite and try to suppress entanglement, Fig. <ref>b. This is achieved by making n large, interactions weak, and by preserving permutation symmetry. In this limit entanglement monogamy <cit.> bounds the pairwise concurrence to zero, and the BEC is exactly described by a nonlinear Schrödinger equation (the Gross–Pitaevskii equation <cit.>), enabling novel dynamics <cit.>. 
The theory is developed in a large n limit with a rigorous bound on the error resulting from the 
mean field approximation. The nonlinear approach trades exponential time complexity for space complexity, requiring n to be large.



 

Does this mean that n has to be exponentially large? Actually the requirements on n are not that bad. This is because the BEC is assumed to be initialized in a product state, and it takes time 
t_ ent for the atomic collisions to produce entanglement. Ideally, the whole experiment is performed in a short-time regime. We measure the accuracy of mean field theory by  ϵ := ρ_ eff(t) - ρ_1(t) _1, and call this the model error. Here ρ_ eff  is the mean field state, ρ_1 is the exact state traced over all atoms but one, t is the gate duration, and ·_1 is the trace norm. In a large family of condensate models  
<cit.>

    ϵ≤ 
    c  e^t/ t_ ent-1 /n ,

where c  and t_ ent are positive constants (model-dependent quantities independent of t and n). Although the error might grow exponentially in time, there is always a short-time window t < t_ ent  where the required number of condensed atoms n ≈  c t / t_ entϵ is sub-exponential in t.

We propose a demonstration of quantum information processing using this nonlinearity. In the remainder of this section we discuss the qubit encoding and state discrimination subroutine. The protocol is explained in Sec. <ref>. 
Conclusions are given in Sec. <ref>, and additional information about the BEC  model and large n limit are provided in the Appendix.





 §.§ Qubit encoding


BEC-based qubits necessarily encode a small 
number of parameters (ψ_0,1∈ℂ) into a large number of degrees of freedom and the map is not unique. However two encodings can often be considered:
Let a_l^† create an atom in BEC component l  ∈{0, 1}  (in a two-component condensate) or in translational mode l ∈{0, 1} (in a two-mode condensate), and let ψ_0,1  be complex coordinates satisfying | ψ_0 |^2 + | ψ_1 |^2  = 1.  One encoding that is interesting from a  quantum foundations perspective is

    CAT_n  :=
    ψ_0 (a_0^†)^n + ψ_1 (a_1^†)^n /√(n!) vac ,  ⟨ CAT_n   |  CAT_n ⟩ = 1,     
     n ≥ 1,

but this is a superposition of two macroscopically distinct BECs (a Schrödinger cat state) which would be highly susceptible to decoherence <cit.>.
Instead we use the encoding 

    F_n  :=
     ( ψ_0   a_0^† + ψ_1   a_1^†)^n /√(n!) vac ,    ⟨ F_n  | F_n ⟩ = 1,     
     n ≥ 1,

which is a condensate of n bosons  ψ_0   a_0^† + ψ_1   a_1^†, each a single atom in a superposition  of components or modes. (While CAT_n and  F_n depend on both n and ψ_0,1, the  latter dependence is suppressed.)
The encoding (<ref>) was originally proposed by
Cirac et al. <cit.> and by
Byrnes et al. <cit.> for two-component condensates; in that case 
F_n is a pseudospin coherent state <cit.>.  But our
 a^†_0
and a^†_1 create atoms in circulating states of orbital angular momentum 0 and 
ħ, respectively, and it is better to regard  F_n as a coherent 
state of atoms in angular momenta superpositions. 
The states (<ref>) 
are mean field states since the atoms are not entangled. 
They  satisfy

    a_l  F_n    = ψ_l √(n) F_n-1    and   
    a_l a_l' F_n = ψ_lψ_l'√(n(n-1)) F_n-2 .

Because each atom in (<ref>)  
carries a copy of the qubit state ψ = ψ_00 + ψ_11, the state
F_n exhibits a bosonic orthogonality catastrophe <cit.> in the large n limit, meaning that close qubit states 
ψ
and
ψ^' encode to orthogonal 
F_n  and F^'_n 
as n →∞ (the semiclassical limit in the spin coherent state picture <cit.>).
Furthermore, due to the polynomial encoding in F_n,
the single-particle superposition principle with respect to ψ_0,1 is violated (see below).



In the atomtronic implementation we assume a toroidal BEC operated in a regime supporting metastable quantized circulation states with l trapped vortices

    Φ_l^n = (a_l^†)^n /√(n) vac ,

where a_l^† creates an atom in the ring with angular momentum l ∈ℤ. 
These states are stabilized by the repulsive atomic interactions <cit.>.
An atom with mass m and l=1 has velocity  ħ/ mR and circles the ring with angular velocity Ω_0 = ħ / mR^2. 
We construct a low-energy effective description for the BEC within the manifold of states (<ref>). This is possible because they are selected out by the path integral in the large n limit, due to their diverging contribution to the action. The action in the subspace spanned by these states is

    S_ eff [ ψ̅_l, ψ_l ]
    = ∫ dt    F_n   i  ∂_t - H_ rotF_n   ,

where H_ rot is the BEC Hamiltonian in the rotating frame. The BEC is rotated with frequency  Ω≈Ω_0/2 to  bring the 0-vortex (no circulation) state Φ_0^n and the 1-vortex state Φ_1^n close in energy. Higher energy l are then neglected, leading to a two-mode model. In the large n limit (see Appendix) the saddle point equations are 

    d/dt[ ψ_0; ψ_1 ] =    - i H_ eff[ ψ_0; ψ_1 ] ,   
     H_ eff  = V_01σ^x + B_z σ^z + g (|ψ_0|^2  -   |ψ_1|^2)   σ^z    .

The first two terms in H_ eff generate rigid x and z rotations of the Bloch sphere. Rotations about x couple l=0 and l=1 angular momenta and are produced by breaking rotational symmetry.  Here V_01 is a matrix element for an applied potential energy barrier. 
The parameter B_z is controlled by the frequency Ω of the BEC rotation discussed above. The nonlinear term describes a z rotation with a rate that increases with increasing Bloch sphere coordinate tr(ρσ^z) = |ψ_0|^2  -   |ψ_1|^2, vanishes on the equator, and reverses direction for tr(ρσ^z) < 0. This  z-axis torsion <cit.>
(1-axis  twisting <cit.>) of the Bloch sphere is the key to fast state discrimination, but is prohibited in ordinary single-particle quantum mechanics. Although the qubit here is informational and not associated with any physical 2-state system, we can define a logical basis {0, 1} and treat it like any other qubit:

    ψ = ψ_0  0 + ψ_1 1
    = [ ψ_0; ψ_1 ] ,   0  :=  [ 1; 0 ] = Φ_0^n ,   1 :=  [ 0; 1 ] = Φ_1^n .

It should be emphasized that (<ref>) is the physical state of the quantum gas,  not  (<ref>). 
However the basis states 0 , 1 
are the quantized circulation states Φ_0,1^n, which is important for the readout step.



 §.§ Single-input state discrimination


As an application, we consider the problem of quantum state discrimination <cit.>, a basic task in quantum information science. In the two-state variant considered here, a quantum state ψ∈{a ,  b} is input to a processor, which knows the values of  a and b ahead of time and tries determine which was provided (with a bounded failure probability).
This is easy if a and b are orthogonal: For a qubit, a single unitary
U_ read = 0a + 1b rotates αa + βb to α0 + β1, which is then measured in the standard basis. The challenging case is when a and b are similar, | ⟨ a | b ⟩ |^2 = 1 - 2^-k  with  k ≫ 1, where n > 2^k identical copies of the input are required <cit.>. 
 In minimum-error discrimination, the subroutine selects a or b, each with some probability of error, and the 
objective is to minimize the average error. 
In unambiguous state discrimination, the subroutine  identifies a or b  perfectly,  but has the possibility of abstaining, returning an 
inconclusive result.
State discrimination can be used to solve 
NP-complete (and harder) problems <cit.>, at the expense of 2^k input copies and exponential runtime.  This cost reflects the limited information gained from measurement.  

Abrams and Lloyd <cit.> showed that certain nonlinearity in the Schrödinger equation would bypass this exponential cost, allowing NP-complete problems to be solved efficiency (in an idealized setting with no errors or decoherence). But the presence of such nonlinearity would constitute a fundamental modification of quantum mechanics that is not supported by experiment <cit.>. 
In a condensate, the nonlinearity is not fundamental, but effective. Although we can realize nonlinear gates, this doesn't constitute a complexity violation, due to the large n requirement of mean field theory. 





§ PROTOCOL



 

The process is illustrated in Fig. <ref>. A single state  ψ∈{a,  b} is input to the discriminator, which ideally returns output 0 if  ψ = a, or returns 1 if  ψ = b. 
The single-input discriminator regarded as a channel must be nonunitary, because the overlap 
⟨ a | b ⟩ is not preserved.
Equivalently, the distance ρ_a - ρ_b _1 between their density matrices in trace norm is not preserved in time (here X  _1 :=   tr√( X^† X)). 
For pure states, ρ_a - ρ_b _1 = 2|sin(θ_ab/2) |, where θ_ab is the angle between their Block vectors.
Linear completely-positive trace preserving (CPTP)  channels satisfy  d/dtρ_a - ρ_b _1 ≤ 0; they are either distance preserving or strictly contractive on the inputs <cit.>.
Because the discriminator orthogonalizes the potential inputs, it is expansive on those inputs:
 d/dtρ_a - ρ_b _1 > 0.
 Thus, the discriminator is described by a nonlinear PTP channel 
<cit.>.

The implementation proposed here does not discriminate an unknown input (produced by a previous computation), but instead uses a black box state preparation step to randomly prepare  a or b, with a small Bloch vector angle θ_ab≥ 0 between them. Then | ⟨ a | b ⟩ |^2 = cos^2(θ_ab / 2) 
≈ 1 - (θ_ab / 2)^2. 
This  can be accomplished by initializing in the Φ^n_0 state and using V_01 and B_z in (<ref>) 
to apply x and z rotations. 
(Ideally, this step is hidden from the remainder of the experiment.)  
The discrimination gate itself follows Refs. <cit.>
and uses the z-axis torsion to increase the angle between a and b. It's clear that  a and b should begin with equal and opposite z components 
z_a = - z_b [here r^μ_a,b =  tr ( ρ_a,bσ^μ),  μ∈{1,2,3}].
Consider a simple option with y_a,b = 0, namely

    a   =   cos( π-θ_ab/4)  0   +      sin(  π-θ_ab/4)  1 , 
    b   =   cos(  π+θ_ab/4)  0  +   sin(  π+θ_ab/4)  1 ,

which has 

    x_a = x_b =  | cos( θ_ab/2)   | ,    
    y_a = y_b = 0,   
    z_a =  sin( θ_ab/2),   
    z_b =  -sin( θ_ab/2).

After switching on g, the two input options
evolve as R_z( ±  gt θ_ab ) and orthogonalize after a time t  ≈π / g  θ_ab.  However this implementation does not have a favorable scaling with θ_ab.  
The optimal protocol for nonlinear discrimination was derived by Childs and Young (CY) in  <cit.>.  Instead of (<ref>), the CY gate 
begins with 

    x_a = x_b 
    = |  cos( θ_ab/2) |,   
    y_a= z_a = sin ( θ_ab/2)/√(2),    
    y_b= z_b = - sin ( θ_ab/2)/√(2) ,

and applies x rotations  to hold y_a,b = z_a,b during the subsequent evolution in order to reach antipodal points on the Bloch sphere. The options orthogonalize in a time t = O( log1/θ_ab ), after which a readout gate
 U_ read (defined with respect to 
 time-evolved a, b)  transforms them to circulation states Φ_0^n or  Φ_1^n, which are then measured   via time-of-flight <cit.>.
 
In an idealized context where (<ref>) is regarded as exact, and where there are no control errors, readout errors, decoherence errors, or noise, the nonlinear discriminator works perfectly every time. We refer to this idealization as a  single-input discriminator to distinguish it from the more familiar minimum error and unambiguous discriminators based on linear CPTP channels
<cit.>.
Of course any actual atomtronic realization is likely to suffer from all such errors, and may fail to give the correct answer or return an answer at all. Although the theoretically achievable performance 
depends sensitively on the system and device details, and is beyond the scope of this work, we note that combining torsion with non-CP dissipation is predicted to implement an autonomous discriminator <cit.>, whose control sequence and operation is (mostly) independent of a and b. In this implementation, the nonlinearity and dissipation 
create two basins of attraction with a shared boundary in the Bloch ball, one with an attracting fixed point near 0 and the other with an attracting fixed point near 1, 
giving the discriminator a degree of intrinsic fault-tolerance.

 

The presence of channel nonlinearity indicates a breakdown of the superposition principle. Figure <ref> illustrates a nice example of this effect:
In Fig. <ref>a, the evolution of a superposition 
ψ_0  0 +  ψ_1  1
is given by a superposition of evolved basis states 
 e^-it0  and  e^it1,  shown as a velocity field. However in Fig. <ref>b, the evolved states e^-it0  and  e^-it1 are now static (phase factors are a global phase), whereas the actual dynamics is not,  except on the equatorial plane.



§ CONCLUSIONS


 
We have discussed an approach to quantum information processing that leverages the special properties of condensates, including their nonlinearity, and proposed an atomtronic implementation of a “nonlinear” qubit. An experimental demonstration of nonlinear state discrimination, while striking, would not by itself constitute a computation, because the qubit isn't coupled to anything. To implement a useful computation, the BEC qubit must be entangled with other qubits (for example trapped ions) in a scalable circuit-model quantum computer, which is not addressed here.

The standard models of quantum computation assume gates and errors based on linear CPTP channels.  Physical hardware, however,  might admit initial correlation and be better described by more general maps <cit.>. 
It is therefore interesting to investigate any additional computational power enabled by quantum channels beyond the linear CPTP paradigm, as we did here. Another example 
was investigated by Chen et al. <cit.>, who experimentally demonstrated unambiguous state discrimination in a linear but non-Hermitian optical system. 
After completing this work, Großardt posted a preprint <cit.> proposing the use of a two-component BEC coupled to a neutral atom computer to simulate a large family of nonlinear Schrödinger equations.
Given their potential for fast quantum state discrimination and simulation of the nonlinear Schrödinger equation, the non-Hermitian and nonlinear approaches to quantum information processing deserve further exploration.



This work was partly supported by the NSF under grant no. DGE-2152159.





§ BEC MODEL


Here we derive the qubit equation of motion (<ref>). We consider a toroidal BEC (thin circular ring with   radius R) with rotating tunneling barriers that act as Josephson junctions <cit.>. Thin means the dynamics is quasi-1d in the azimuthal direction, θ. This requires the energy, temperature, and effective interaction strength to be below an energy scale Δϵ determined by the confining potential. The shape of the potential (without barriers) is mostly arbitrary as long as it is invariant under rotations about the axis threading the ring, which we call the z axis.
The  angular momentum eigenfunctions on the ring are φ_l (θ) =  e^ i l  θ / √(2 π R), with l  ∈ℤ the angular momentum.  In the absence of the tunnel barriers and interaction, these are stationary states.
The condensate consists of n weakly interacting bosonic atoms of mass m, each in their electronic ground state  Ψ_0.  At sufficiently low energy and densitiy, the atomic  collisions are elastic,  and the Hamiltonian is 

    H(t)   =   ∫_ d^3r   {ħ^2 ∇ϕ^†·∇ϕ/2m 
     + U/2 ϕ^†ϕ^†ϕ ϕ  
      + V  ϕ^†ϕ},   [ϕ( r) , ϕ^†( r') ] = δ( r- r').

Here  is the volume of the ring, 
 U  = 4 πħ^2 a_ s /m is a short-range interaction strength (proportional to the s-wave scattering length a_ s), and V( r,t) is a confining potential, including  the rotating barriers. Acting on the vacuum, ϕ^†( r) creates a bosonic atom in state Ψ_0 at point r. We assume a tunable repulsive interaction with a_ s≥ 0. We also assume zero temperature, no dissipation, and no disorder.



Two rotating tunnel barriers  are used to implement an atomtronic quantum interference 
device <cit.>.
When the barriers are turned on, the Hamiltonian  (<ref>) is time dependent. Assuming the barriers are rigidly rotated about the z axis with frequency Ω, 
we have
V( r,t) = e^-i Ω t L_z / ħ  V( r, 0) 
  e^i Ω t L_z / ħ,
where L_z is the angular momentum.

However, we can transform to a noninertial reference frame in which the Hamiltonian, H_ rot, is time independent. 
Decomposing the time-evolution operator in the lab frame as 

    U_ lab = T e^- i/ħ∫_0^t  H dt'  = e^-i Ω t L_z / ħ U_ rot  ,

 we obtain
 
    dU_ rot/dt = - i/ħ  H_ rot   U_ rot ,    H_ rot = e^i Ω t L_z / ħ ( H - 
     Ω L_z ) e^-i Ω t L_z / ħ  = H(0) 
     - Ω  L_z .

 
Next we discuss the two-mode limit:
In the low energy, thin ring limit,  we can expand the field operators and angular momentum as
 
    ϕ( r) = ∑_l    e^ i l  θ/√()  a_l ,   
    L_z = ∑_l ħ l   a^†_l a_l ,   
    [ a_l  , a_l '^† ] = δ_l  l ' ,

which leads to

    H_ rot   =   ħΩ_0/2∑_l   l^2 
    a_l ^† a_l 
    + U/2  ∑_l_1,  l_2, l_3
     a_l_1 + l_3 ^† a_l_2 -l_3^† a_l_2  a_l_1 
     +  ∑_l_1,  l_2  V_l_1   l_2  a_l_1^† a_l_2
    - ħΩ∑_l l  a^†_l a_l ,

where

    V_l_1   l_2   =   ∮ d θ/2 π  V(θ, t  =  0)    e^-i (l_1 - l_2) θ .

Nonzero V_l_1 l_2 induce transitions between angular momentum states. Then we have

    H_ rot   =    ∑_l ħω_l
    a_l ^† a_l 
    + U/2  ∑_l_1,  l_2, l_3
     a_l_1 + l_3 ^† a_l_2 -l_3^† a_l_2  a_l_1 
     +   ∑_l_1,  l_2  V_l_1   l_2  a_l_1^† a_l_2
      -  n ħΩ^2/2  Ω_0 ,

where

    ω_l = (Ω - l Ω_0)^2 /2 Ω_0 ,   ω_0 = Ω^2 /2 Ω_0 ,   ω_1 = (Ω - Ω_0)^2 /2 Ω_0.




As explained above, the BEC is rotated with frequency  Ω≈Ω_0/2 to  bring the l=0 and l=1 states close in energy. 
We restrict (<ref>) to angular momenta l = 0,1 neglecting the others on the basis of their higher energy. 
Then

    ∑_l_1,  l_2, l_3
     a_l_1 + l_3 ^† a_l_2 -l_3^† a_l_2  a_l_1    =   ∑_l ∈ℤ{  
     a_l^† a_-l^† a_0 a_0
     +  a_l+1^† a_-l^† a_0 a_1
     +  a_l^† a_1-l^† a_1 a_0
     +  a_1+l^† a_1-l^† a_1 a_1
     }    
        =    
     a_0^† a_0^† a_0 a_0 
     +  a_1^† a_1^† a_1 a_1 
     + 4  a_0^† a_1^† a_1 a_0 
        =    
    ( a_0^† a_0 )^2 - a_0^† a_0 
    + ( a_1^† a_1 )^2 - a_1^† a_1 
    + 4  a_0^† a_0 a_1^† a_1 .

This leads to a two-mode model

    H_ rot = ∑_l=0,1( ħω_l + V_ll + γ a_l ^† a_l - γ) a_l ^† a_l 
    + γ^'  a_0^† a_0 a_1^† a_1  + (  V_01 a_0^† a_1 + V̅_01  a_1^† a_0)  -  n ħΩ^2/2  Ω_0 ,

where 

    γ = U/2  ,   γ^' =  4 γ = 2 U/ .

In what follows we will treat γ , γ^'≥ 0 as independent parameters, 
allowing 
(<ref>) to apply to other systems as well.
The last term in (<ref>) subtracts the classical kinetic energy of the spinning ring: n ħΩ^2  / 2  Ω_0 = 1/2 I_ ringΩ^2,  I_ ring = n m R^2  .

Finally we discuss the large n limit:  Condensates feature an enhanced two-particle interaction ⟨ a^† a^†  a a ⟩≈ n(n-1) caused by the effectively infinite-ranged interaction between condensed atoms. This makes a naive large n limit unphysical, because the energy per particle diverges <cit.>, and our low-energy assumptions would be violated.
The framework discussed here is instead based on  a modified large n limit where the interaction simultaneously weakens as 1/n (a standard assumption in rigorous studies of mean field theory <cit.>).
This allows for a rigorous study of the large  n limit including bounds on the accuracy of mean field theory. For real V_01 = ∮ d θ/2 π  V(θ)    e^i θ,  and after dropping the classical kinetic energy term (which does not affect the qubit dynamics) we obtain
 
    H_ rot   =    ∑_l=0,1(   ( ħω_l + V_ll ) a_l ^† a_l  
    + γ a_l ^† a_l^† a_l a_l 
    - γ  a_l^† a_l )
    + γ^'  a_0^† a_0 a_1^† a_1  + V_01(  a_0^† a_1 +  a_1^† a_0) .

Evaluating  (<ref>) and assuming n ≫ 1 leads to (setting ħ = 1)

    S_ eff =  n ∫ dt 
    {∑_l=0,1( ψ̅_l   i ∂_t  ψ_l
     -   ( ω_l + V_ll ) | ψ_l|^2
     -    n  γ   |ψ_l |^4 )
     -   n  γ^' | ψ_0  ψ_1 |^2 
     -    V_01  ( ψ̅_0 ψ_1
     + ψ̅_1 ψ_0) } .

Here z̅ denotes complex conjugation.
Due to the O(n^2) interaction energies in (<ref>), we cannot take the n →∞ limit in (<ref>) without violating our low-energy assumptions. Instead we consider a modified limit defined by the simultaneous limits γ, γ^'→ 0, n →∞, and low energy. To evaluate (<ref>) in this limit we assume that the interaction strengths decrease with n as γ = K / n and γ^' = K^' / n, where K and K^' are now fixed coupling constants. 
Then

    S_ eff =   n  ∫ dt 
    {∑_l=0,1( ψ̅_l   i ∂_t  ψ_l
      -   ( ħω_l + V_ll ) | ψ_l |^2
     -    K    |ψ_l |^4 )
     -   K^' | ψ_0  ψ_1 |^2 
     -    V_01  ( ψ̅_0 ψ_1
     + ψ̅_1 ψ_0) } .


To obtain (<ref>) we have used the results summarized below in Table <ref>,  which also gives the corresponding results for encoding (<ref>).
In the large n limit the stationary phase approximation leads to  (<ref>) with

    H_ eff     =    [        ħω_0 +V_00 +  2 K  |ψ_0|^2  + K^'   |ψ_1|^2                                               V_01;                                               V_01 ħω_1  + V_11  +  2 K  |ψ_1|^2    +   K^'   |ψ_0|^2 ]
       =    V_01σ^x + B_z σ^z + g    tr(ρσ^z)  σ^z   +  const. ,

where

    B_z :=  ħω_0 - ħω_1 + V_00 - V_11/2   and  g :=  2K - K^'/2  = - U n/2  .

This concludes the derivation of (<ref>).





References








