Quantum State Reduction by Matter-Phase-Related Measurements in Optical Lattices

A many-body atomic system coupled to quantized light is subject to weak measurement. Instead of coupling light to the on-site density, we consider the quantum backaction due to the measurement of matter-phase-related variables such as global phase coherence. We show how this unconventional approach opens up new opportunities to affect system evolution. We demonstrate how this can lead to a new class of final states different from those possible with dissipative state preparation or conventional projective measurements. These states are characterised by a combination of Hamiltonian and measurement properties thus extending the measurement postulate for the case of strong competition with the system’s own evolution.


S1 Suppressing the effective coupling to atomic density
In the main text we showed thatâ = C(D +B), where g out,in are the atom-light coupling constants for the outgoing and incoming beams, ∆ a is the detuning between the incoming probe beam and the atomic resonance frequency, ∆ p is the detuning between the incoming probe beam and the outgoing cavity beam, a 0 is the amplitude of the coherent probe beam, and κ is the cavity decay rate. However, we are only interested in the case whenâ = CB. Therefore, we need to find the conditions under which this is true. For clarity we will consider a 1D lattice, but the results can be applied and generalised to higher dimensions. Central to engineering theâ operator are the coefficients J m,n given by J m,n = w(r − r m )u * out (r)u in (r)w(r − r n ) dr, where w(r) are the Wannier functions of the lowest band, u in,out (r) are the light mode functions of the incoming and outgoing beams, and r is the position vector. The operatorsB andD depend on the values of J m,m+1 and J m,m respectively and are given byD where b m annihilates an atom at site m, andn m = b † m b m is the number operator at site m. These J m,n coefficients are determined by the convolution of the light mode product, u * out (r)u in (r) with the relevant Wannier function overlap w(r − r m )w(r − r n ). For theB operator we calculate the convolution with the nearest neighbour overlap, W 1 (r) ≡ w(r − a/2)w(r + a/2), where a is the site separation vector, and for theD operator we calculate the convolution with the square of the Wannier function at a single site, W 0 (r) ≡ w 2 (r). Therefore, in order to enhance theB term we need to maximise the overlap between the light modes and the nearest neighbour Wannier overlap, W 1 (r). This can be achieved by concentrating the light between the sites rather than at atom positions.
In order to calculate the J m,n coefficients it is necessary to perform numerical calculations using realistic Wannier functions. However, it is possible to gain some analytic insight into the behaviour of these values by looking at the Fourier transforms of the Wannier function overlaps, F [W 0,1 ](k). This is because the light mode product, u * out (r)u in (r), can be in general decomposed into a sum of oscillating exponentials of the form e ik·r making the integral in Eq. (S2) a sum of Fourier transforms of W 0,1 (r).
We consider a setup shown in Fig. S1 and take both the detected and probe beam to be standing waves, u in,out (r) = cos(k in,out · r + ϕ in,out ), where k is the wavevector of the beam and ϕ is a constant phase shift. This gives the following expressions for theD andB operatorŝ The key result is that theB operator is phase shifted by k ± d/2 with respect to theD operator since it depends on the amplitude of light in between the lattice sites and not at the positions of the atoms, allowing to decouple them at specific angles.
Firstly, we will use this result to show how one can obtain the uniform pattern for whichB =B 1 , wherê i.e. J m,m+1 = J 1 This can be achieved by crossing the light modes such that θ in = −θ out and k in,x = k out,x = π/a and choosing the light mode phases such that ϕ + = π. In order to make theB contribution to light scattering dominant we need to set . This arrangement of light modes maximizes the interference signal,B, by suppressing the density signal,D, via interference compensating for the spreading of the Wannier functions and leads to the parameter value J 1 = F [W 1 ](2π/a)/2. The light mode patterns are illustrated in the main text in Fig. 1(a). Secondly, we show that we can have a spatially varying pattern for whichB =B 2 , wherê We consider an arrangement where the beams are arranged such that k in,x = 0 and k out,x = π/a which gives the following expressions for the density and interference termŝ It is clear that for ϕ out = ±π/2,D = 0, which is intuitive as this places the lattice sites at the nodes of the mode u out (r) and yields the parameter value J 2 = −F [W 1 ](π/a) cos(ϕ in ). This is a diffraction minimum as the light amplitude is zero, B = 0, because contributions from alternating inter-site regions interfere destructively. However, the intensity â †â = |C| 2 B 2 is proportional to the variance ofB and is non-zero. The light mode patterns are illustrated in the main text in Fig. 1(b).

S2 Finding the measurement projection subspaces
The main text defines the projectors P M = ∑ m∈M P m , where P m are the projectors onto theâ eigenspaces, such that ∑ M P M =1, P M P N = δ M,N P M , [P M ,Ĥ 0 ] = 0, and [P M ,â] = 0. To find P M we need to identify the subspaces M which satisfy the following relation ∑ m∈M P m = ∑ m∈M |h m h m |, where |h m are the eigenstates ofĤ 0 . This can be done iteratively by (i) selecting some P m , (ii) identifying the |h m which overlap with this subspace, (iii) identifying any other P m which also overlap with these |h m from step (ii). We repeat (ii)-(iii) for all the P m found in (iii) until we have identified all the subspaces P m linked in this way and they will form one of our P M projectors. If P M = 1 then there will be other subspaces P m which we have not included so far and thus we repeat this procedure on the unused projectors until we identify all P M . Computationally this can be straightforwardly solved with some basic algorithm that can compute the connected components of a graph. The above procedure, whilst mathematically correct and always guarantees to generate the projectors P M , is very unintuitive and gives poor insight into the nature or physical meaning of P M . In order to get a better understanding of these subspaces we will use another result from the main text. We showed that for an operatorÔ with eigenspace projectors R m for which [Ô,Ĥ 0 ] = 0, and [Ô,â] = 0, then we can write the subspace projectors as P M = ∑ m∈M R m = ∑ m∈M P m .

2/4
We are interested in identifying these subspaces for the operatorB 2 given bŷ We have identified that forB 2 , an operatorÔ that commutes with both the measurement operator and the Hamiltonian is given byÔ = ∑ RBZ g kÔk , whereÔ k =n k +n k−π/a , for any arbitrary constants g k . The subspaces R m of this operator simply consist of momentum space Fock states that have the same number of atoms in each (k, k − π/a) pair of momenta. However, it turns out that the P M consist of multiple such subspaces complicating the picture. Firstly, sinceB 2 contains sin(ka) coefficients atoms in different k modes that have the same sin(ka) value are indistinguishable to the measurement and will lie in the same P m eigenspaces. This will happen for the pairs (k, π/a − k). Therefore, the R m spaces that have the sameÔ k +Ô π/a−k eigenvalues must belong to the same P M .
Secondly, if we re-write these operators in terms of the β k andβ k modes we get and so it's not hard to see thatB 2,k = (β † k β k −β † kβ k ) will have the same eigenvalues for different values ofÔ k = β † k β k +β † kβ k . Specifically, if a given subspace R m corresponds to the eigenvalue O k ofÔ k then the possible values of B 2,k will be Thus, we can see that all R m with even values of O k will share B 2,k eigenvalues and thus they will overlap with the same P m subspaces. The same is true for odd values of O k . However, R m with an even value of O k will never have the same value of B 2,k as a subspace with an odd value of O k . Therefore, a single P M will contain all R m that have the same parities of O k for all k, e.g. if it includes the R m with O k = 6, it will also include the R m for which O k = 0, 2, 4, 6, ..., N, where N is the total number of atoms.
Finally, the k = π/a mode is special, because sin(π) = 0 which means that B 2,k=π/a = 0 always. This in turn implies that all possible values of O π/a are degenerate to the measurement. Therefore, we exclude this mode when matching the parities of the other modes.
To illustrate the above let us consider a specific example. Let us consider two atoms, N = 2, on eight sites M = 8. This configuration has eight momentum modes ka = {− 3π 4 , − π 2 , − π 4 , 0, π 4 , π 2 , 3π 4 , π} and so the RBZ has only four modes RBZ := { π 4 , π 2 , 3π 4 , π}. There are 10 different ways of splitting two atoms into these four modes and thus we have 10 different R m = {O π/4a , O π/2a , O 3π/4a , O π/a } eigenspaces ofÔ and they are shown in Table S1. In the third column we have also listed the eigenvalues of theB 2 eigenstates that lie within the given R m .
We note that ka = π/4 will be degenerate with ka = 3π/4 since sin(ka) is the same for both. Therefore, we already know that we can combine (R 0 , R 2 , R 7 ), (R 1 , R 5 ), and (R 3 , R 8 ), because those combinations have the same O π/4a + O 3π/4a values. This is very clear in the table as these subspaces span exactly the same values of B 2 . Now we have to match the parities. Subspaces that have the same parity combination for the pair (O π/4a + O 3π/4a , O π/2a ) will be degenerate in P M . Note that we excluded O π/a , because as we discussed earlier they are all degenerate due to sin(π) = 0. Therefore, the (even,even) subspace will include (R 0 , R 2 , R 4 , R 7 , R 9 ), the (odd,even) will contain (R 3 , R 8 ), the (even, odd) will contain (R 6 ) only, and the (odd, odd) contains (R 1 , R 5 ). These overlaps should be evident from the table as we can see that these combinations combine all R m that contain any eigenstates ofB 2 with the same eigenvalues. Therefore, we have end up with four distinct P M subspaces P even,even =R 0 + R 2 + R 4 + R 7 + R 9 P odd,even =R 3 + R 8 P even,odd =R 6 P odd,odd =R 1 + R 5 .
At this point it should be clear that these projectors satisfy all our requirement. The conditions ∑ M P M = 1 and P M P N = δ M,N P M should be evident from the form above. The commutator requirements are also easily satisfied since the subspaces R m are of an operator that commutes with both the Hamiltonian and the measurement operator. And finally, one can also verify using the table that all of these projectors are built from complete subspaces ofB 2 (i.e. each subspace P m belongs to only one P M ) and thus P M = ∑ m∈M P m .