## Abstract

Quantum optical coherence can be quantified only by accounting for both the particle- and wave-nature of light. For an ideal laser beam^{1,2,3}, the coherence can be thought of roughly as the number of photons emitted consecutively into the beam with the same phase. This number, \({\mathfrak{C}}\), can be much larger than the number of photons in the laser itself, *μ*. The limit for an ideal laser was thought to be of order *μ*^{2} (refs. ^{4,5}). Here, assuming only that a laser produces a beam with properties close to those of an ideal laser beam and that it has no external sources of coherence, we derive an upper bound on \({\mathfrak{C}}\), which is of order *μ*^{4}. Moreover, using the matrix product states method^{6}, we find a model that achieves this scaling and show that it could, in principle, be realized using circuit quantum electrodynamics^{7}. Thus, \({\mathfrak{C}}\) of order *μ*^{2} is only a standard quantum limit; the ultimate quantum limit—or Heisenberg limit—is quadratically better.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

## Code availability

The iMPS codes used in this study are available from the corresponding author upon request.

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## Acknowledgements

We thank I. Cirac, H. Carmichael, M. Mirrahimi, I. McCulloch, M. Hall and A. Tilloy for useful discussions. This work was supported by ARC Discovery Projects DP170101734, DP160102426 and DP190102633 and an Australian Government RTP Scholarship.

## Author information

### Affiliations

### Contributions

H.M.W. conceived, acquired funding for and directed the project. D.W.B. had the key idea for Theorem 1. All authors contributed to the analytics. S.N.S. carried out the numerics. T.J.B. produced the figures. H.M.W. drafted the manuscript and T.J.B. and S.N.S. drafted the Supplementary Information. All authors contributed to revisions.

### Corresponding author

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## Extended data

### Extended Data Fig. 1 Illustration of the method for deriving the upper bound \({\mathfrak{C}}=\Theta ({\mu }^{4})\).

Time increases from the top to the bottom but is branched. Initially, we consider the cavity at time *T* in steady state, and the segment of the beam emitted since time *T* − *τ*. At an immediately following time *T*^{+} > *T*, we suppose an observer, Effie, can perform heterodyne filtering over the beam emitted in the interval [*T* − *τ*, *T*), to obtain a phase estimate *ϕ*_{F} of the cavity at time *T*. The state of the cavity, conditioned on her measurement is \({\rho }_{{\rm{c}}}^{{\phi }_{F}}\). Now, there are two ways a second observer, Rod, could measure the phase of the cavity (green arrows). The first method allowing the cavity to emit the beam up until time *T*^{+} + *τ*, upon which heterodyne retrofiltering could be performed over the interval (*T*^{+}, *T*^{+} + *τ*], yielding an estimate *ϕ*_{R} at time *T*^{+} + *τ*^{+}. The second would consist of a canonical phase measurement performed directly on the cavity at time *T*^{++} immediately following *T*^{+}, with outcome *ϕ*_{D}. Since the result *ϕ*_{R} cannot be better than *ϕ*_{D} as a phase estimate of *ϕ*_{F}, the upper bound on \({\mathfrak{C}}\) follows from known results on optimal covariant phase estimation.

### Extended Data Fig. 2 Conceptual diagram of our laser model.

From the upper figure to the lower, one time step has passed, converting one pair of input qubits (pump and vacuum) into a new pair of output qubits (beam and sink), with position label *q*_{0} + 1. The indefinite length string of pairs of output qubits is described by an iMPS of bond-dimension *D*, equal to the Hilbert space dimension of the laser cavity.

### Extended Data Fig. 3 iMPS optimization results when the limit of a maximum \({\mathfrak{C}}\) is achieved.

We show the profiles of non-zero diagonals of iMPS matrices for *D* = 50 (left) and *D* = 100 (right), when the ansätze \({A}_{mn}^{[2]}=0\) and \({A}_{mn}^{[0]}=\,\text{const.}\,\) (for their allowed non-zero elements) are placed as constraints. The solid lines display an ansatz for the steady-state presented in the main text, \({\rho }_{m}^{{\rm{ansatz}}}\propto {\sin }^{4}(\pi \frac{m+1}{D+1})\).
Source data

### Extended Data Fig. 4 Deviations from the ideal laser model (a coherent state with its phase undergoing pure diffusion) of the first-order Glauber coherence function for our laser model.

For \({\mathcal{N}}=1\), the magnitude of the quantity \(\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(1)}(s,t)\) (with *t* = 0 without loss of generality) is shown over ten coherence times. For increasing *D*, we see this difference is converging toward zero like a power law, which implies *G*^{(1)}-ideality is satisfied for our laser model.
Source data

### Extended Data Fig. 5 Deviations from the ideal laser model (a phase-diffusing coherent state) of the second-order Glauber coherence function for our laser model.

The global maxima of \(\delta {g}_{{\rm{ideal}}}^{(2)}(-\tau ,s^{\prime} ,t^{\prime} ,t)\) versus the coherence (black diamonds) calculated for \(\{s^{\prime} ,t^{\prime} ,t\}\in [-\tau ,\tau ]\). These were calculated employing interior-point optimizations of the iMPS forms for bond dimensions up to 250, where the iMPS forms of Glauber coherence functions can be found in the Methods. Some examples for \(\delta {g}_{{\rm{ideal}}}^{(2)}\), for particular choices of time arguments, are also presented for comparison for bond dimensions up to 1000, where one of these choices (blue circles) is indistinguishable from the numerically found maximum. Error bars are smaller than the symbol size for the black diamonds. Coloured lines are power-law fits to large-*D* points of these examples and the black line is \({{\mathfrak{C}}}^{-1/2}\) for comparison purposes, showing that our definition of *G*^{(2)}-ideality is satisfied.
Source data

## Supplementary information

### Supplementary Information

Supplementary Figs 1 and 2 and discussion.

## Source data

### Source Data Fig. 1

*x*–*y* data for main and inset plots in Fig. 1.

### Source Data Fig. 2

Source data for the results in Fig. 2. Here, the first column always represents the *t* and the second represents \(\delta {G}_{{\rm{model}}-1}^{(2)}\) in the first tab, \(\delta {G}_{{\rm{ideal}}-1}^{(2)}\) in the second tab and \(\delta {G}_{{\rm{model}}-{\rm{ideal}}}^{(2)}\) in the last tab.

### Source Data Extended Data Fig. 3

Multi-column data for the iMPS matrices of the cases *D* = 50, 100 in Extended Data Fig. 3. Here the first column always represents the row index of the non-zero diagonal, the second represents *A*^{[0]}, the third represents *A*^{[1]}, the fourth represents *A*^{[3]}, and the fifth represents *ρ*_{ss}.

### Source Data Extended Data Fig. 4

Source data for the *D* = 50, 100, 200 results in Extended Data Fig. 4. Here the first column always represents the time *s* and the second represents \(\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(1)}(s,0)\).

### Source Data Extended Data Fig. 5

Multi-column data for the results in Extended Fig. 5. Here the first column always represents \({\mathfrak{C}}\), the second represents \(\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(2)}(-\tau ,\tau ,\tau ,-\tau )\) (while the third column is its standard error), the fourth represents \(\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(2)}(-\tau ,-(1-\epsilon )\tau ,(1-\epsilon )\tau ,\tau )\) (while the fifth column is its standard error), and the last represents \(\max [\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(2)}]\).

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Baker, T.J., Saadatmand, S.N., Berry, D.W. *et al.* The Heisenberg limit for laser coherence.
*Nat. Phys.* **17, **179–183 (2021). https://doi.org/10.1038/s41567-020-01049-3

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