Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The Heisenberg limit for laser coherence


Quantum optical coherence can be quantified only by accounting for both the particle- and wave-nature of light. For an ideal laser beam1,2,3, 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 μ2 (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 μ4. Moreover, using the matrix product states method6, we find a model that achieves this scaling and show that it could, in principle, be realized using circuit quantum electrodynamics7. Thus, \({\mathfrak{C}}\) of order μ2 is only a standard quantum limit; the ultimate quantum limit—or Heisenberg limit—is quadratically better.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Coherence and Glauber(1)-ideality in our laser model.
Fig. 2: Glauber(2)-ideality in our laser model.
Fig. 3: Proposal for a circuit QED device to realize a Heisenberg-limited maser.

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.


  1. McClung, F. J. & Hellwarth, R. W. Giant optical pulsations from ruby. J. Appl. Phys. 33, 828–829 (1962).

    Article  ADS  Google Scholar 

  2. Glauber, R. J. The quantum theory of optical coherence. Phys. Rev. 130, 2529 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  3. Carmichael, H. Statistical Methods in Quantum Optics 1 (Springer, 1999).

  4. Schawlow, A. L. & Townes, C. H. Infrared and optical masers. Phys. Rev. 112, 1940–1949 (1958).

    Article  ADS  Google Scholar 

  5. Wiseman, H. M. Light amplification without stimulated emission: beyond the standard quantum limit to the laser linewidth. Phys. Rev. A 60, 4083–4093 (1999).

    Article  ADS  Google Scholar 

  6. Perez-Garcia, D., Verstraete, F., Wolf, M. M. & Cirac, J. I. Matrix product state representations. Quantum Info. Comput. 7, 401–430 (2007).

    MathSciNet  MATH  Google Scholar 

  7. Haroche, S., Brune, M. & Raimond, J. M. From cavity to circuit quantum electrodynamics. Nat. Phys. 16, 243–246 (2020).

    Article  Google Scholar 

  8. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).

    Article  ADS  Google Scholar 

  9. Caves, C. M. Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 1693 (1981).

    Article  ADS  Google Scholar 

  10. Yurke, B., McCall, S. L. & Klauder, J. R. SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033 (1986).

    Article  ADS  Google Scholar 

  11. Berry, D., Hall, M. & Wiseman, H. Stochastic Heisenberg limit: optimal estimation of a fluctuating phase. Phys. Rev. Lett. 111, 113601 (2013).

    Article  ADS  Google Scholar 

  12. Sayrin, C. et al. Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011).

    Article  ADS  Google Scholar 

  13. Chou, K. S. et al. Deterministic teleportation of a quantum gate between two logical qubits. Nature 561, 368–373 (2018).

    Article  ADS  Google Scholar 

  14. Bartlett, S. D. & Wiseman, H. M. Entanglement constrained by superselection rules. Phys. Rev. Lett. 91, 097903 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  15. Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79, 555–609 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  16. McCulloch, I. P. Infinite size density matrix renormalization group, revisited. Preprint at (2008).

  17. Orús, R. A practical introduction to tensor networks: matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  18. Schollwock, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96–192 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  19. Meiser, D., Ye, J., Carlson, D. R. & Holland, M. J. Prospects for a millihertz-linewidth laser. Phys. Rev. Lett. 102, 163601 (2009).

    Article  ADS  Google Scholar 

  20. Ketterle, W. When atoms behave as waves: Bose–Einstein condensation and the atom laser. Rev. Mod. Phys. 74, 1131–1151 (2002).

    Article  ADS  Google Scholar 

  21. Louisell, W. H. Quantum Statistical Properties of Radiation (Wiley, 1973).

  22. Sargent, M., Scully, M. O. & Lamb, W. E. (eds) Laser Physics (Addison-Wesley, 1974).

  23. Vitiello, M. S. et al. Quantum-limited frequency fluctuations in a terahertz laser. Nat. Photon. 6, 525–528 (2012).

    Article  ADS  Google Scholar 

  24. Åberg, J. Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014).

    Article  ADS  Google Scholar 

  25. Wiseman, H. M. & Milburn, G. J. Quantum Measurement and Control (Cambridge Univ. Press, 2010).

  26. Bandilla, A., Paul, H. & Ritze, H. H. Realistic quantum states of light with minimum phase uncertainty. Quantum Opt. J. Eur. Opt. Soc. B 3, 267–282 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  27. Schön, C., Solano, E., Verstraete, F., Cirac, J. I. & Wolf, M. M. Sequential generation of entangled multiqubit states. Phys. Rev. Lett. 95, 110503 (2005).

    Article  ADS  Google Scholar 

  28. Schön, C., Hammerer, K., Wolf, M. M., Cirac, J. I. & Solano, E. Sequential generation of matrix-product states in cavity qed. Phys. Rev. A 75, 032311 (2007).

    Article  ADS  Google Scholar 

  29. Jarzyna, M. & Demkowicz-Dobrzański, R. Matrix product states for quantum metrology. Phys. Rev. Lett. 110, 240405 (2013).

    Article  ADS  Google Scholar 

  30. Manzoni, M. T., Chang, D. E. & Douglas, J. S. Simulating quantum light propagation through atomic ensembles using matrix product states. Nat. Commun. 8, 1743 (2017).

    Article  ADS  Google Scholar 

  31. Streltsov, A., Adesso, G. & Plenio, M. B. Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  32. Vaccaro, J. A., Anselmi, F., Wiseman, H. M. & Jacobs, K. Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. Phys. Rev. A 77, 032114 (2008).

    Article  ADS  Google Scholar 

  33. Wiseman, H. M. How many principles does it take to change a light bulb … into a laser? Phys. Scr. 91, 033001 (2016).

    Article  ADS  Google Scholar 

  34. Hughes, B. L. & Cooper, A. B. Nearly optimal multiuser codes for the binary adder channel. IEEE Trans. Inf. Theory 42, 387–398 (1996).

    Article  Google Scholar 

  35. Gradshteyn, I. S. & Ryzhik, I. M. Table of Integrals, Series and Products 571 (Academic Press, 2014).

  36. Sidje, R. Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24, 130–156 (1998).

    Article  Google Scholar 

  37. D’Alessandro, D. Introduction to Quantum Control and Dynamics (CRC Press, 2007).

Download references


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

Authors and Affiliations



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

Correspondence to Howard M. Wiseman.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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 Tτ. At an immediately following time T+ > T, we suppose an observer, Effie, can perform heterodyne filtering over the beam emitted in the interval [Tτ, T), to obtain a phase estimate ϕF 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 ϕR 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 ϕD. Since the result ϕR cannot be better than ϕD as a phase estimate of ϕF, 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 q0 + 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

xy 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 ρss.

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 \(\max [\delta {g}_{{\rm{model}}-{\rm{ideal}}}^{(2)}]\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baker, T.J., Saadatmand, S.N., Berry, D.W. et al. The Heisenberg limit for laser coherence. Nat. Phys. 17, 179–183 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing